bluuebunny/crossref_metadata_2025_split huggingface dataset:
Regions of the n-sphere and related integrals
Authors: Currie I.D. 
Published: January 2009

In this note the volumes of certain regions in the n-sphere will be found in two ways: (a) by using a symmetry argument, (b) by expressing the volumes as repeated integrals over the (n-l)-cube. By considering the 4 and 5 spheres and equating the integrals obtained by method (b) to the solution obtained by method (a) we evaluate integrals of the formfor certain values of a, b and c; it does not appear easy (if indeed it is possible) to evaluate these integrals by direct methods.
On reflection principles for parabolic equations in one space variable
Authors: Colton David 
Published: January 2009

In this note we shall consider the problem of uniquely continuing solutions of the parabolic equationacross an analytic arc σ: x=s1(t) satisfies the boundary dataWe assume that u(x,t) is a classical solution of (1) in the domain D ={(x,t): s1(t)< x < s 2(t), 0 < t < t0}, continuously differentiate in D ∪ σ and define the “reflection” of D across σ by
T-Ideals and C-Ideals
Authors: Klein Aaron 
Published: January 2009

Given a ring R we consider the category Ŕ of R-rings (rings A with given ring-homomorphisms R → A), and R-homomorphisms (ring-homomorphisms that form commutative triangles with the given maps from R). All rings are associative and have 1, all homomorphisms send 1 to 1.We define a c-R-ring as an object A in Ŕ with a family of maps {ρx ∈ homŔ(A,A)|x∈A}. Equivalently, a c-R-ring is an R-ring A with a binary operation a · b(= aρb) on A satisfying
Exceptional Sets in Uniform Distribution
Authors: Baker R. C. 
Published: January 2009

Let B be a measurable set of real numbers in (0,1) of Lebesgue measure |B| and letx1, …,xnbe real. Thendenotes the number ofj(1 ≦j≦n) for which the fractional part {xj}∈B. Thediscrepancyofx1, …,xniswhere the supremum is taken over all intervalsIin [0,1].
Matrices with prescribed characteristic polynomial and principal blocks
Authors: De Oliveira G. N. 
Published: January 2009

Let A be a matrix over a field Φ partitioned as followswhere A11 is n×n and A22 is m×m. The objective of the present paper is to give further results on the problems mentioned in Section 1 of (3). Concretely we shall consider the following question: “we prescribe the characteristic polynomial f(λ) = λn+m−c1λn+m−1 + … of A and the principal blocks A11, A22. Find a necessary and sufficient condition for the existence of A satisfying these prescribed conditions”.
Solutions of first level of meromorphic differential equations
Authors: Balser W. 
Published: January 2009

Let a meromorphic differential equationbe given, where r is an integer, and the series converges for |z| sufficiently large. Then it is well known that (0.1) is formally satisfied by an expressionwhere F( z) is a formal power series in z–1 times an integer power of z, and F( z) has an inverse of the same kind, L is a constant matrix, andis a diagonal matrix of polynomials qj( z) in a root of z, 1≦ j≦ n. If, for example, all the polynomials in Q( z) are equal, then F( z) can be seen to be a convergent series (see Section 1), whereas if not, then generally the coefficients in F( z) grow so rapidly that F( z) diverges for every (finite) z.
Linear diophantine equations with cyclic coefficient matrices and its applications to Riemann surfaces
Authors: Takigawa Nobumasa 
Published: January 2009

Let co, c1, …, cn-1 be the nonzero complex numbers and let C = (cu+1,v+1) = (cn+u-v), O≦u,v≦n — 1, be a cyclic matrix, where n + u — v is taken modulo n. In this paper we shall give the solution of the linear equationswhere Lu (0≦u≦n —1) is a fixed complex number. In Theorem 1 weshall give a necessary and sufficient condition for (1) to have an integral solution.
Fourier multipliers on spaces of distributions
Authors: Lamb W. 
Published: January 2009

In [8], Rooney defines a class of complex-valued functions ζ each of which is analytic in a vertical strip α(ζ)< Res < β(ζ) in the complex s-plane and satisfies certain growth conditions as |Im s| →∞ along fixed lines Re s = c lying within this strip. These conditions mean that the functionsfulfil the requirements of the one-dimensional Mihlin-Hörmander theorem (see [6, p. 417]) and so can be regarded as Fourier multipliers for the Banach spaces . Consequently, each function gives rise to a family of bounded operators W[ζ,σ] σ ∈(α(ζ),β(ζ)), on , 1<p<∞.
A note on Whittaker's cardinal series in harmonic analysis
Authors: Dodson M. M. , Silva A. M. , Soucek V. 
Published: January 2009

The sampling theorem, often referred to as the Shannon or Whittaker-Kotel'nikov- Shannon sampling theorem, is of considerable importance in many fields, including communication engineering, electronics, control theory and data processing, and has appeared frequently in various forms in engineering literature (a comprehensive account of its numerous extensions and applications is given in [3]). The result states that a band-limited signal, i.e. a real function f of the formwhere w>0, is under reasonable conditions on the even function F, determined by its values on the sampling set (l/2w)ℤ and can be reconstructed from the samples f(k/2w), k∈ℤ, by the series
On the Growth of Functions of Mean Type
Authors: Fuchs W. H. J. 
Published: January 2009

1. V. Bernstein, N. Levinson, R. P. Boas, Jr., and others have investigated under what conditions on the sequence for all functions f(z) regular and of suitably restricted growth in the halfplane x≥0. (For references see [1].)In this note the restrictions on f(z) are that f(z) is regular, and not identically zero in continuous in and that for some positive B
On Equivalent Truth-Tables of Many-Valued Logics
Authors: Kalicki J. 
Published: January 2009

Many-valued or non-Aristotelian calculi of propositions (logics) were originally introduced by generalisation of the truth-table method. It was known by the end of the nineteenth century that ordinary “binary” formulae of the calculus of propositions, such ascould be verified directly by means of the truth-table:although the terminology and symbolism used were different.
On an Asymptotic Integral
Authors: Hsu L. C. 
Published: January 2009

1. This note gives an asymptotic evaluation of an integral of the formas n tends to infinity, where is a sequence of real-valued functions. The theorem to be established is a natural extension of B. Levi's generalised Laplace-Darboux theorem (1, 341-51); it gives a rule for evaluating a wider class of asymptotic integrals.
N-ary Transformations of Sequences
Authors: Boyd A. V. 
Published: January 2009

Let T(α1, α2, …, αn−1) be the n-ary transformation which takes the sequence {si}, (i = 0, 1, …), into the sequence {s'i} wherewith sm = 0 when m is a negative integer, and where α1, …, αn an are real numbers with sum unity. In a previous note (1) conditions were found on α and β for the ternary transformation T(α, β) to be equivalent to convergence. A method is given here for treating the similar problem for the general n-ary transformation.
On the generalized heat-equation
Authors: Nasim C. , Aggarwala B. D. 
Published: January 2009

The general heat equation is defined aswhere v is a fixed positive number and α is a fixed number. If v = α = 0, then (1.1) reduces to the ordinary heat equationwhere u(x,t) is regarded as the temperature at a point x at time t, in an infinite insulated rod extended along the x-axis in the xt-plane. If we set , then (1.1) becomes
Non-orthogonalisable vector fields on spheres
Authors: Raussen Martin 
Published: January 2009

A (k – l)-field on Sn-1 may be given as a section ϕ of the fibre bundlewith fibre Vn-1, k-1 or, equivalently, as a semi-orthogonal map, i.e., a mapwhich is isometric in the second variable and such that for the basis vector e1∈Rk and every x∈Rn
An Integral Formula For Qn (cos θ)
Authors: Copson E. T. 
Published: January 2009

The functionis, as is well-known, a general solution of Laplace's equation of degree −1 in (x, y, z). In 1926* I proved that the particular solution r−1Q0 (z/r) cannot be represented in this form whereas the solution r−1Q0 (y/r) can. In the present note I find a very simple expression for the latter solution in the form (1.1), and I deduce from it an apparently new integral formula for Qn (cos θ).
A note on inverse central factorial series
Authors: Aitken A. C. 
Published: January 2009

The purpose of this brief note is to draw attention to a type of inverse factorial series which, so far as the writer can judge, has not been intensively studied. The central difference formulae of interpolation and the corresponding infinite series in central factorial polynomials have in the past received much attention, and so also has the ordinary inverse factorial serieswhich from the time of Stirling (Methodus Differentialis, 1730) has been known to be of value in transforming slowly convergent series into series of more rapid convergence.
On γ-transformations of series
Authors: Nigam Tapeshwari Prasad 
Published: January 2009

The theorems given here deal with the efficiency, mutual consistency, and regularity of γ-transformations.§1. Theorem 1. The necessary and sufficient conditions that γ-matrices are efficient for all divergent series whose partial sums are bounded, arewhere Δgk(a)≡gk(a)−gk+1(a).
Note on certain related conditions in the theory of Cesaro summability
Authors: Hyslop J. M. 
Published: January 2009

In a recent paper I discussed, for a given series Σan the relation between the conditionswhere 0 < ρ < 1 and p is a positive integer. It was there proved (Theorem 14) that (2) implies (1) but that the converse is not necessarily true. My object in this note is to render more precise the connection between (1) and (2) by showing that (2) is equivalent to (1′), where (1′) is (1) with the additional restriction that Σn−1wn is convergent. The proof of this equivalence relation is based on a technique which was introduced and developed by Andersen and is similar to the proofs in my former paper. The note concludes with a brief discussion of the corresponding results for the case ρ = 1.
Logarithmetics of Finite Quasigroups (II)
Authors: Popova Helen 
Published: January 2009

In the first paper of this series (L.Q.I)1 we have shown that the logarithmetic LQ of a finite quasigroup Q is a quasigroup with respect to addition and that it is a subdirect union of the logarithmetics of the elements of Q.In this second part we shall discuss further the structure of LQ in its additive aspect, and obtain results concerning the order N of LQ. For plain quasigroups (§3) the structure of LQ(-\-) is studied in more detail and it is shown that N is a power of n, the order of Q.
An Application of Hankel Transforms in Axially Symmetric Potential Flow
Authors: Mackie A. G. 
Published: January 2009

In his book on Hydrodynamics, Lamb obtained a solution for the potential flow of an incompressible fluid through a circular hole in a plane wall. More recently Sneddon (Fourier Transforms, New York, 1951) obtained Lamb's solution by an elegant application of Hankel transforms.Since the streamlines in this solution are symmetric about the wall, it is not of particular physical interest. In this note, Sneddon's method is used to give a solution in which the fluid is infinite in extent on one side of the aperture but issues as a jet of finite diameter on the other side.
On a Certain Series of Abel
Authors: Clunie J. 
Published: January 2009

If we express ezλ, qua function of λ, in terms of λeλ by Langrange's theorem we find, for |λ| sufficiently small,If we now regard λ as a parameter and z as the independent variable then (1.1) can be writtenSupposing, therefore, that a function is of the form then
Nörlund Methods of Summability Associated with Polynomials
Authors: Borwein D. 
Published: January 2009

Let s, sn(n = 0, 1, …) be arbitrary complex numbers, and letbe a polynomial, with complex coefficients, which satisfies the normalizing conditionAssociated with such a polynomial is a Nörlund method of summability Np: the sequence {sn} is said to be Np-convergent to s, and we write sn →s (Np), if
A Note on the Kawada-into Theorem
Authors: Pym John S. 
Published: January 2009

If µ is a bounded regular Borel measure on a locally compact group G, and L1(G) denotes the class of complex-valued functions which are integrable with respect to the left Haar measure m of G, then, for each f∈L1(G),defines almost everywhere (a.e.) with respect to m a function μ*f which is again in L1(G). The measure μ will be called isotone on G mapping f→μ*f is isotone, i.e. f≧0 a.e. (m) if and only if μ*f≧0 a.e. (m).
Approximate Solutions of Boundary Value Problems in Potential Theory with Applications to Electrostatics and Elastostatics
Authors: Williams W. E. 
Published: January 2009

SummaryA simple approximate formula is obtained for the capacity of the condenser formed by a “small” conductor placed inside a much larger one. The formula involves a constant whose choice is, to a certain extent, arbitrary and it is shown that, for problems involving spheroids inside cylinders and between parallel plates, the constant may be found in a simple fashion so as to give very accurate results. A similar formula is obtained for the loss in potential energy due to a crack or cavity in a circular beam or a thick plate. For the particular cases of the boundary value problems considered which have been treated by other means very close numerical agreement is obtained between those results and ones deduced in the present paper.
Some inequalities for a hypergeometric integral
Authors: Flett T. M. 
Published: January 2009

In various problems in harmonic analysis (4, 5, 6), I have required an estimate for the integral for the hypergeometric functionin the case where a>c−b>0, b>0, and 0 ≦ x < 1 (the integral is then unbounded as x→l−). Although there are innumerable identities for hypergeometric functions, few inequalities for these functions seem to be known, and in estimating the integral (1) I employed ad hoc arguments that made no use of the theory of hypergeometric functions. The estimates obtained were adequate for my purposes, but were far from sharp, and the object of this note is to show that, by assembling a few known facts concerning hypergeometric functions, we can obtain sharp inequalities for the integral (1). We also give (in §4) several related inequalities that can be obtained by transforming the integral.
Divisibility of ordered groups
Authors: Wright I. W. 
Published: January 2009

In this paper it is shown that divisibility of a complete lattice ordered (abelian) group is closely related to the existence of a sufficient number of small elements in the positive cone.We shall denote the set of all real numbers by R which symbol will be reserved for this purpose. All terms used are as defined in Birkhoff(1). For the reader's convenience we now define the two terms most used in the sequel.
General hereditary for radical theory
Authors: Rossa R. F. , Tangeman R. L. 
Published: January 2009

LetWbe a class of not necessarily associative rings which is universal in the sense that it is closed under homomorphic images and is hereditary to subrings. All rings considered will be assumed to belong toW. The notationI⊲Rwill meanIis an ideal ofR. A relation σ onWwill be called anH-relation if σ satisfies the properties:(1)IσRimpliesIis a subring ofR.(2) IfIσRand ø is a homomorphism ofR, thenIØ σRø.(3) IfIσRandJis an ideal of R, thenI∩JσJ.
Locally nilpotent skew linear groups II
Authors: Wehrfritz B. A. F. 
Published: January 2009

Our paper [6] studied in some depth certain locally nilpotent skew linear groups, but our conclusions there left some obvious gaps. By means of a trick, which now seems obvious, but then did not, we are able to tidy up the situation very satisfactorily. This present paper should be viewed as a follow up to [6]. In particular we do not repeat the motivation, basic definitions and references to related work given here.The following was conjectured in [6], where substantial steps were taken towards its solution.
Bessel-integral functions
Authors: Humbert Pierre 
Published: January 2009

SummaryIn a very remarkable work on the operational Calculus, Dr Balth. van der Pol has introduced a new function, playing with respect to Bessel function of order zero the same part as the cosine- or sine-integral with respect to the ordinary cosine or sine. He showed that this function—which he called Bessel-integral junction—can be used to express the differential coefficient of any Bessel function with respect to its index. But he did not investigate the further properties of his new function. I propose to give here some of them, which appear to be interesting, and to introduce and study the functions connected, in the same way, with Bessel functions of any order.
Evaluation of Permanents
Authors: Minc Henryk 
Published: January 2009

The permanent of an m x n matrix A = (aij),m≤n, is defined bywhere the summation is over all one-to-one functions σ from {1, … ,m} to { 1, …,n}. In other words, the permanent of A is the sum of all thediagonal productsofA, that is, all the products ofmentries ofAno two of which lie in the same row or in the same column. Thus the permanent ofAmay be evaluated by first multiplying all the row sums ofAand then subtracting from the product all terms that contain as factors two or more entries from the same column ofA. This is the idea behind the formulas of Binet and of Ryser.
Lattices whose ideal lattice is Stone
Authors: Beazer R. 
Published: January 2009

An elementary fact about ideal lattices of bounded distributive lattices is that they belong to the equational class ℬω of all distributive p-algebras (distributive lattices with pseudocomplementation). The lattice of equational subclasses of ℬω is known to be a chainof type (ω+l, where ℬ0 is the class of Boolean algebras and ℬ1 is the class of Stone algebras. G. Grätzer in his book [7] asks after a characterisation of those bounded distributive lattices whose ideal lattice belongs to ℬ (n≧1). The answer to the problem for the case n = 0 is well known: the ideal lattice of a bounded lattice L is Boolean if and only if L is a finite Boolean algebra. D. Thomas [10] recently solved the problem for the case n = 1 utilising the order-topological duality theory for bounded distributive lattices and in [5] W. Bowen obtained another proof of Thomas's result via a construction of the dual space of the ideal lattice of a bounded distributive lattice from its dual space. In this paper we give a short, purely algebraic proof of Thomas's result and deduce from it necessary and sufficient conditions for the ideal lattice of a bounded distributive lattice to be a relative Stone algebra.
Divisible S-systems and R-modules
Authors: Gould Victoria 
Published: January 2009

Throughout this paper S will denote a given monoid and R a given ring with unity. A set A is a right S-system if there is a map φ:A × S→A satisfyingandfor any element a of A and any elements s, t of S. For φ(a, s) we write as and we refer to right S-systems simply as S-systems. One has the obvious definitions of an S-subsystem, an S-homomorphism and a congruence on an S-system. The reader ispresumed to be familiar with the basic definitions concerning right R-modules over R. As with S-systems we will refer to right R-modules just as R-modules.
Explicit expressions for Sturm-Liouville operator problems
Authors: Jodar Lucas 
Published: January 2009

Throughout this paper H will denote a complex separable Hilbert space and L(H) denotes the algebra of all bounded linear operators on H. If T lies in L(H), its spectrum σ(T) is the set of all complex numbers z such zI–T is not invertible in L(H) and its compression spectrum σcomp(T) is the set of all complex numbers z such that the range (zI-T)(H) is not dense in H ([3, p. 240]). This paper is concerned with the Sturm–Liouville operator problemwhere λ is a complex parameter and X(t), Q, Ei, Fi for i = l,2, and t∈[0,a], are bounded operators in L(H). For the scalar case, the classical Sturm-Liouville theory yields a complete solution of the problem, see [4], and [7]. For the finite-dimensional case, second order operator differential equations are important in the theory of damped oscillatory systems and vibrational systems ([2, 6]). Infinite-dimensional differential equations occur frequently in the theory of stochastic processes, the degradation of polymers, infinite ladder network theory in engineering [1, 17], denumerable Markov chains, and moment problems [10, 20]. Sturm-Liouville operator problems have been studied by several authors and with several techniques ([12, 13, 14, 15, 16]).
Real Linear Substitutions with Equimodular Multipliers, and their expression in terms of their Invariants
Authors: Taylor D. G. 
Published: January 2009

§1. Letdenote a linear substitution of non-vanishing determinant; and let the roots k∈ of its characteristic equationbe for the present assumed distinct. Then with each root k∈ is associated an invariant point or poleP∈, and a linear invariant, or invariant (n−2)-plane ξ∈. If the n points P∈ do not lie on an (n−2)-plane, the determinant of their coordinates,
Note on the Peano-Baker Method of solving Linear Differential Equations
Authors: Milne Arch 
Published: January 2009

In Mathematische Annalen, Vol. 32 (1888) Peano discusses the solution of a system of homogeneous linear differential equationswhere rij denotes a real function of the variable t, and shows how, by a series of repeated substitutions, this system of equations may be replaced by the equivalent equationwhere X denotes the complex [x1, x2, … xn] and R the matrixof which equation the solution X can be represented as a sum of integrals.
On a Group of Parabolas associated with the Triangle
Authors: Philp George 
Published: January 2009

§1. The general theorem underlying the subject of this paper is as follows:—If through c1, c2, c3, … points on AB, a side of ΔABC, rays be drawn from two vertices O1, O2, the former meeting AC in b1, b2b3, …, and the latter meeting BC in a1, a2, a3, …, then the lines a1b2, a2b2, … envelope a conic touching the sides AC and BC. This follows since the ranges a1a2a3 …, b1b2b3 …, are homographic.
Transformations of Axes for Whittaker's Solution of Laplace's Equation
Authors: Jeffery G. B. 
Published: January 2009

Whittaker has shown that a general solution of Laplace's equation, ∇2V = 0, may be expressed in the formSince the harmonic property of a function is in no way dependent upon any particular set of axes it follows that the same solution must be capable of being expressed in the formwhere X, Y, Z are any second set of rectangular coordinates.
The triangle and its escribed parabolas
Authors: Pressland A. J. 
Published: January 2009

The problem “to inflect a straight line between two sides of a triangle so that the intercepted portion is equal to the segments cut off” has been discussed in the third volume of the Proceedings.If we discuss the same analytically; taking CB and CA as axes of x and y (Fig. 1) and calling each segment k, the equation of the line considered is
The Plane Triangle ABC: Intimoscribed Circles, etc.
Authors: Anderson R.E. 
Published: January 2009

§ I. On an infinite series of Triad Circles derived from the inscribed circle. Determination of a direct relation between r and the three radii of the nth triad.Each of the first triad touches two sides of ABC and the inscribed circle : generally, each circle of the mth triad touches two sides of ABC and also touches one of the circles of the (m – 1)th triad.
Note on a property of a quadrilateral
Authors: Steggall J.E.A. 
Published: January 2009

The property is that if any quadrilateral, ABCD, skew or otherwise, have its sides AB, DC divided in E, F so thatAE : EB = DF : FC = AD : BOthen the direction of the line EF shall bisect the angle between the directions of BC, AD.This extension of Euclid's VI. 3 follows immediately from the proposition that if AE : EB = DF : FC, then all such lines as EF are parallel to one plane, namely, the plane parallel to BC, AD; and that they each cut similar lines drawn with reference to BC, AD.