bluuebunny/crossref_metadata_2025_split huggingface dataset:
Almost diagonal systems in asymptotic integration
Authors: Gingold H. 
Published: January 2009

Consider the ordinary linear matrix differential systemψ(x) is a scalar mapping, X and A(x) are n by n matrices. Both belong to C1([a,∞)) for some integer l. The stability and asymptotic behaviour of its solutions have been subject to much investigation. See Bellman [2], Levinson [24], Hartman and Wintner [20], Devinatz [9], Fedoryuk [11], Harris and Lutz [16,17,18] and Cassell [30]. The special interest in eigenvalue problems and in the deficiency index problem stimulated a continued interest in asymptotic integration. See e.g. Naimark [36], Eastham and Grundniewicz [10] and [8,9]. Harris and Lutz [16,17,18] succeeded in explaining how to derive many known theorems in asymptotic integration by repeatedly using certain “(1 + Q)” linear transformations.
Initial boundary value problems for coupled nerve fibres
Authors: Grindrod P. 
Published: January 2009

In this paper we analyse the electrical behaviour within systems of long and short coupled nerve axons by using a geometric approach to obtain a priori bounds on solutions. In [4[ we developed a general model for a bundle of n-uniform unmylinated nerve fibres. If FitzHugh-Nagumo dynamics, [3[ are used to describe the ionic membrane currents, then the model takes the formHere W=(w1,…wn)T denotes the membrane action potentials for each fibre in the bundle and Z=(Z1,…Zn)T represents the recovery variables for each fibre, which control the return to the resting equilibrium after any transmission of signals.
Theorems connecting different classes of Self-Reciprocal Functions
Authors: Mehrotra Brij Mohan 
Published: January 2009

1. The question, “How, from a given function which is self-reciprocal for a transform of a particular order, can we construct other functions which are self-reciprocal for transforms of different orders?” was first raised by Hardy and Titchmarsh who gave some rules for constructing such functions. Following their method, I have shown, in a recent paper, that there are certain general theorems of the following type:—If f (x) is its own Jμtransform, g (x) is its own Jv transform. In this note I add a few more such theorems, the interest lying mainly in the results themselves and not in a rigorous proof thereof; and hence only the formal procedure is given here.
On the application of the operational calculus to the expansion of a function in a series of Legendre's functions of the second kind
Authors: Banerjee D. P. 
Published: January 2009

In the present note we shall obtain the expansion in a series of Legendre functions of the second kind of an integral function φ (ω) represented by Laplace's integralwhere f (x) is an analytic function of x, regular in the circlewhere an are constants and qn (ω) = in+1Qn (iω).
Some Remarks on de Longchamps Chain
Authors: Brown L. M. 
Published: January 2009

The chain of circle theorems now associated with the name of de Longchamps may be stated as follows.(i) Any two lines L (A), L (B) meet in a point (AB).(ii) Any three lines L(A), L (B), L(C) determine three points (AB) (BC) (CA) which lie on a circle S (ABC) with centre (ABC),(iii) Any four lines L(A), L(B), L(C), L(D) determine four circles S(ABC) S(ABD) S(ACD) S(BCD) which meet in a point P (ABCD) and four centres (ABC) (ABD) (ACD) (BCD) which lie on a circle S(ABCD) with centre (ABCD).(iv) Any five lines L(A), L(B), L(C), L(D), L(E) determine five points like P (ABCD), which lie on a circle SM (ABCDE) (Miquel's Theorem), five circles like S (ABCD), which meet in a point P (ABCDE), and their five centres like (ABCD), which lie on a circle S (ABCDE) with centre (ABCDE).(v) Any six lines L (A), … . , L (F) determine six circles like SM (ABCDE) which meet in a point PM (ABCDEF), six circles like S (ABCDE), which meet in a point P (ABCDEF) and six centres like (ABCDE) which lie on a circle S (ABCDEF) with centre (ABCDEF).
An infinite integral formula
Authors: Lambe C. G. 
Published: January 2009

§ 1. The object of this note is to discuss the formulathe integral being supposed convergent for certain ranges of values of x and z. The contour is such that the poles of Γ(– s)lie to its right and the other poles of the integrand to its left. It will be seen that all the Pincherle-Mellin-Barnes integrals are particular cases of this formula.
An extension of de Longchamps' chain of theorems
Authors: Richmond H. W. 
Published: January 2009

1. Points, n in number, A, B, C, D, E, … ., are taken at random in a plane, and through each is drawn a line in a random direction. The only condition imposed is that no two of these lines may be parallel.(i). Two points A, B, define a circle S (AB) which passes through A, B and the intersection of the random lines through A and B. Its centre is denoted by (AB). Each pair of the points gives such a circle and centre.
A self-reciprocal function
Authors: Mohan Brij 
Published: January 2009

1. The object of this note is to discover a new function which is its own reciprocal in the Hankel Transform of order zero.I will make use of the following theorem of Hardy and Titchmarsh1:–A necessary and sufficient condition that a Junction f (x) should be its own Jv transform is that it should be of the formwhere 0 < c < 1, and
Compléments à un théorème de T. S. Broderick concernant les événements dépendants
Authors: Fréchet Par Maurice 
Published: January 2009

Introduction. Nous nous proposons ici de pousser un peu plus loin les conséquences que Broderick a tirées d'un théorème dont il nous a fait connaître l'existence au dernier Colloque de St Andrews.L'un des buts de Broderick était de prouver que certaines formules de King restent valables quand les événements considéréd ne sont plus indépendants, démonstration dont il n'avait pas trouvé trace dans la littérature, sauf relativement à des cas particuliers.
On Lambe's infinite integral formula
Authors: Erdélyi A. 
Published: January 2009

In a recent paper (these Proceedings (2), 6 (1939), 75–8), C. G. Lambe established, and gave some applications of, the formulain which Ds is the symbol for the derivative of fractional order s. Lambe's proof of (1) is not quite rigorous and it does not bring out the conditions which have to be imposed upon f(x) in order to make (1) true. Furthermore this proof does not give any evidence as to the definition of fractional derivative which is to be used in connection with (1).
A property of Cesàro-Perron integrals
Authors: Bosanquet L. S. 
Published: January 2009

It is well known that if f(t) is (a) integrable in the Lebesgue sense, or more generally (b) integrable in the Perron sense, over every interval (α, β) interior to (a, b), and ifexists, then f(t) is integrable in the Perron sense over (a, b) to the value (1·1).
Extension of a Formula by Cayley to Symmetric Determinants
Authors: Gårding Lars 
Published: January 2009

It has been proved by CAYLEY that if x11, x12, x21 … are independent variables, x = det (xik), ξ = det (ξik), (i, k = 1, … n) where ξik =∂/∂xik then by formal derivation ξxα = α(α + 1)…(α + n − 1)xα−1. This is a special case of the formulawhere m=1,…,n and with i = i1,..im; k= k1,…km and xi,… is the algebracial complement of i = i1,..im; k = k1,…km, in .
A Theorem on General Regular Transformations of Series
Authors: Evans Arwel 
Published: January 2009

Many papers have been written on transformations of sequences which can be written aswhere is a function of a finite or infinite number of variables for fixed m. If the number of variables is finite it becomes large with m.Almost all these papers are on linear transformations of the sr, e.g. Cesàro and Abel means, but there are obvious transformations of non-linear form which are regular, and it is a theorem on these which is considered here.
A Generalised Hypergeometric Function
Authors: Fabian William 
Published: January 2009

The hypergeometric function1F(a, b; c; z) is analytic in the domain |arg(−z)| < π, and, when |z| < 1, may be represented by the seriesWhen |z| = 1 in the domain |arg(−z)| <π, this series converges2 to F(a; b; c; z) if R(a+b−c) < 0 (integral values of a, b and c are excluded in the present paper).
A Note on Convergence Factors
Authors: Walsh C. E. 
Published: January 2009

Suppose that un ≥ un+1 > 0, for n = 1, 2, … and that e1, e2, … are factors which make convergent.Let where all the a's are positive, diverges, and Define En by the equationFor the case where an = 1 it has been shewn by Fuchs1 and Karamata1 that, under various conditions, En = 0(1) The object of this note is to extend some of their results.
An Example of a Plane Shock of Variable Strength
Authors: Smith P. 
Published: January 2009

AbstractA particular solution of the equations of one-dimensional anisentropic flow of a polytropic gas is linked by a shock to gas at rest in which the density is non-uniform. The approach is inverse in that the density distribution is derived from the position of the shock and the prescribed flow behind it. The velocity and strength of the shock each vary with time. The result is an example of the propagation of a shock through an inhomogeneous gas.
The Difference between Consecutive Prime Numbers V
Authors: Rankin R. A. 
Published: January 2009

Let pn denote the nth prime and let ε be any positive number. In 1938 (3) Ishowed that, for an infinity of values of n,where, for k≧1, logk+1x = log (logk x) and log1x = log x. In a recent paper (4) Schönhage has shown that the constant ⅓ may be replaced by the larger number ½eγ, where γ is Euler's constant; this is achieved by means of a more efficient selection of the prime moduli used. Schönhage uses an estimate of mine for the number B1 of positive integers n≦u that consist entirely of prime factors p≦y, whereHerer x is large and α and δ are positive constants to be chosen suitably.
Heat Conduction in Hollow Cylinders with Radiation
Authors: Marchi E. , Zgrablich G. 
Published: January 2009

AbstractA new finite integral transformation (an extension of those given by Sneddon (1)), whose kernel is given by cylindrical functions, is used to solve the problem of finding the temperature at any point of a hollow cylinder of any height, with boundary conditions of radiation type on the outside and inside surfaces, with independent radiation constants. It is to be noticed that all possible problems on boundary conditions in hollow cylinders can be solved by particularising the method described here.
Some axially symmetric potential problems
Authors: Leppington F. G. , Levine H. 
Published: January 2009

AbstractSome axially symmetric boundary value problems of potential theory are formulated as integral equations of the first kind. In each case the kernel admits an expansion, for small values of a parameter of the problem, that leads to an approximate integral equation whose solution provides a direct asymptotic estimate for the physical quantity of primary interest. A manipulation of the original and modified integral equations provides an efficient formula for calculating higher order terms in the asymptotic expansion.
Nuclearity and Banach spaces
Authors: Valdivia Manuel 
Published: January 2009

SummaryLet E be a nuclear space provided with a topology different from the weak topology. Let {Ai: i ∈ I} be a fundamental system of equicontinuous subsets of the topological dual E' of E. If {Fi: i ∈ I} is a family of infinite dimensional Banach spaces with separable predual, there is a fundamental system {Bi: i ∈ I} of weakly closed absolutely convex equicontinuous subsets of E'such that is norm-isomorphic to Fi, for each i ∈ I. Other results related with the one above are also given.
Finitep-soluble groups with irreducible modular representations of given degrees
Authors: Humphreys J. F. 
Published: January 2009

LetGbe a finite group,pbe a prime andKbe a field of characteristicp. Letbe a decomposition of the group ring ofGoverKas a sum of indecomposable two-sided ideals. An irreducibleK(G)-module is said to be in the blockBiif it occurs as a composition factor ofBi. The block containing the trivialK(G)-module is called the principal block ofG.
Geometric inequalities involving the stretch of a set
Authors: Lillington J. N. 
Published: January 2009

A number of papers have been written concerning the properties of triangles which circumscribe convex sets, see for example (1), (2), (3).In this note we shall characterise two set functionals, one of which has already been introduced in (6). This will enable us to produce some new results in the spirit of those obtained in the papers mentioned above. These two functionals are interesting in that although they are defined in a rather abstruse manner, they turn out to have an intuitive geometric meaning, namely that they are the minimal widths of certain circumscribing triangles.
On the order of magnitude of Jacobsthal's function
Authors: Vaughan R. C. 
Published: January 2009

Let n be an integer with n > 1. Jacobsthal (3) defines g(n) to be the least integer so that amongst any g(n) consecutive integers a + 1, a + 2, … a + g(n) there is at least one coprime with n. In other words, ifthenIt is probably true thatwhere ω(n) denotes the number of different prime divisors of n, and Erdos (1) has pointed out that by the small sieve it is possible to show that there is a constant C such that
On the oscillation of solutions of certain linear differential equations in the complex domain
Authors: Bank Steven B. , Langley J. K. 
Published: January 2009

Our starting point is the differential equationwhere A(z) is a transcendental entire function of finite order, and we are concerned specifically with the frequency of zeros of a non-trivial solution f(z) of (1.1). Of course it is well known that such a solution f(z) is an entire function of infinite order, and using standard notation from [7],for all , b∈C\{0}, at least outside a set of r of finite measure.
Note on an “Alternant” with Factorial Elements
Authors: Zia-uddin  
Published: January 2009

The alternant with powers of variables for its elementshas often been investigated and its properties are well known. In the present note an “alternant” with factorial elements,will be discussed. Here a, b, c, …, z, α, β, γ, …, ω are integers such thatandwith the convention that a(0) = 1, O(0) = 1, and that, where negative indices occur (in the reductions of §2), the factorial is to be regarded as zero.
On Vanishing Coaxial Minors
Authors: Motzkin Th 
Published: January 2009

In this note we wish to study some properties of a square matrix of order n and of rank r,whose coaxial minors |M| of certain orders, or some of them, are known to be zero. Our main results will lie in two directions, according as the vanishing minors are, within certain limits, of an arbitrary order (Part II) or of two consecutive orders only (Part III).
A question concerning Aronhold's Theorems on Bitangents
Authors: Grace J. H. 
Published: January 2009

The question to be discussed is this:—Can Aronhold's theorems on the bitangents of a quartic curve of genus three be extended to the tritangent planes of a space seztic of genus four?The answer is “No”: decisiveness arises from the fact that the gist of Aronhold's results can be stated in a very simple form, namely:—(a) Given seven lines in a plane it is possible to derive from them uniquely and symmetrically a quartic curve that has them for bitangents.
Remark on a theorem of D. H. Fremlin concerning K-analytic Hausdorff spaces
Authors: Devlin Keith J. 
Published: January 2009

At the 29th Meeting of the British Mathematical Colloquium held at Edinburgh in March 1977, D. H. Fremlin announced the following result.Let Φ denote the proposition that there exists a K-analytic Hausdorff space, having metrizable compacta, which is not Souslin. (We refer the reader to Fremlin's paper (1) for the relevant definitions)Theorem 1.1. (Fremlin) (i) Assume . (ii) Assume Martin's Axiom together with . Then Φ is false.
Decomposition of a Von Neumann Algebra Relative to a*-Automorphism
Authors: Thaheem A. B. 
Published: January 2009

Let X be any real or complex Banach space. If T is a bounded linear operator on X then denote by N(T) the null space of T and by R(T) the range space of T.Now if X is finite dimensional and N(T) = N(T2) then also R(T) = R(T2). Therefore X admits a direct sum decomposition.Indeed it is easy to see that N(T) = N(T2) implies that and, using dimension theory of finite dimensional spaces, that N(T) and R(T) span the whole space (see, for example, (2, pp. 271–73))
Inequalities Involving the Inverses of Positive Definite Matrices
Authors: Merris Russell 
Published: January 2009

Let G be a permutation group of degree m. Let x be an irreducible complex character of G. If A = (aij) is an m-square matrix, the generalised matrix function of A based on G and x is defined byFor example if G = Sm, the full symmetric group, and x is the alternating character, then d = determinant. If G = Sm and x is identically 1, then d = permanent.
Tensor products of separable reduced primary modules
Authors: Webb Cary 
Published: January 2009

R will denote a Dedekind domain and Pone of its prime ideals. A P-primary module will be an R-module all of whose non-zero elements have annihilators that are powers of the prime P. In all that follows E is such a module.The height of 0 ≠ x ∈ E will be max{n : x ∈ PnE}. It is denoted by h(x). If this maximum does not exist we will say h(x)∞.Clearly the condition is equivalent to E having no non-zero elements of infinite height. Adopting the terminology of (2, Ch. XI) where such modules over the ring of integers are studied, we will call these modules separable and reduced.
An Analogue of the Radon-Nikodym Property for Non-Locally Convex Quasi-Banach Spaces
Authors: Kalton N. J. 
Published: January 2009

In recent years there has been considerable interest in Banach spaces with the Radon-Nikodym Property; see (1) for a summary of the main known results on this class of spaces.We may define this property as follows: a Banach space X has the Radon-Nikodym Property if whenever T ∈ ℒ (L1, X)(where L1 = L1(0, 1)) then T is differentiable i.e.where g:(0, 1)→X is an essentially bounded strongly measurable function. In this paper we examine analogues of the Radon-Nikodym Property for quasi-Banach spaces. If 0>p > 1, there are several possible ways of defining “differentiable” operators on Lp, but they inevitably lead to the conclusion that the only differentiable operator is zero.
The partially ordered family of all ℛ-classes of S(x)
Authors: Magill K. D. 
Published: January 2009

Let S(X) denote the semigroup of all continuous selfmaps of the topological space X. Let ℒ(S(X)) and ℛ(S(X)) denote the partially ordered families of all ℒ-classes and ℛ-classes, respectively, of S(X) where the partial orders are the usual ones [3, p. 29]. In [6] we made the followingConjecture. The following statements are equivalent about any two compact 0-dimensional metric spaces X and Y:(1) ℒ(S(X)) and ℒ(S(Y)) are order isomorphic.(2) ℛ(S(X)) and (S{Y)) are order isomorphic.(3) The semigroupsS(X) and S(Y) are isomorphic.(4) The spaces X and Y are homeomorphic.
A singular perturbation problem for the Lavrent'ev-Bitsadze equation
Authors: Weinacht R. J. 
Published: January 2009

In this note we consider a singular perturbation problem for the equationwhere K(y) = sgn y and. Ε is a small (positive) parameter. This equation for ε≠O is elliptic for y<0 and hyperbolic for y>0. Many of the results carry over to more difficult and interesting problems for equations of mixed type. The particularly simple model treated here permits the elimination of some complications in the analysis involving singular integral equations while preserving the main qualitative features of more general cases. For a special Tricomi-like problem for (1.1) we construct asymptotic expansions in ε, including boundary layer corrections, of the solution. A proof of uniform asymptotic validity of the lowest order terms is given.
Unitary bordism of circle actions
Authors: Kosniowski Czes , Yahia Mahgoub 
Published: January 2009

The purpose of this paper is to describe , the bordism module of unitary T-manifolds, where T denotes the circle group S1. We give both an algebraic and a geometric description. The algebraic result iswhere I = (i(1), i(2),…i(2n)) runs through all finite ordered 2n-tuples (n≧0) of non-negative integers which satisfy the conditions (a) i(l) + i(2n)≠0 and (b) if i(2n)≠0 then i(2n)=≠. The isomorphism is also described geometrically and this leads to geometric generators of .
Nonlinear boundary value problems for elliptic systems
Authors: Cañada A. 
Published: January 2009

The purpose of this paper is to discuss non-linear boundary value problems for elliptic systems of the typewhere Ak is a second order uniformly elliptic operator and is such that the problemhas a one-dimensional space of solutions that is generated by a non-negative function. The boundary ∂G is supposed to be smooth and the functions gk, 1≦k≦m are defined on Ḡ×Rm and are continuously differentiate (usually, Bk represents Dirichlet or Neumann conditions and is the first eigenvalue associated with Ak and such boundary conditions).
Some examples of minimally degenerate Morse functions
Authors: Kirwan Frances 
Published: January 2009

Let X be a compact Riemannian manifold. If f:X→ℝ is a nondegenerate Morse function in the sense of Bott [2] then one has Morse inequalities which can be expressed in the formwhere Pt(X) is the Poincaré polynomial Σtidim Hi(X;ℚ of X ann {Cβ|β ∈B} are the connected components of the set of critical points for f For any polynomial Q(t)∈ℤ[t] we write Q(t)≧0 if all the coefficients of Q are nonnegative.
On the Linear Differential Equation of the Second Order
Authors: Brodetsky S. 
Published: January 2009

The linear differential equation of the second orderis not in general integrable by any method at present available. At the same time, several equations of this type have been integrated, either in terms of finite functions or by means of expansions in series. Some properties of the integrals of the general equation have also been obtained. It is the object of this paper to develop some general properties of these integrals, which throw some light on the nature of the solutions, even if not obtainable in explicit terms.
The Solution of Difference Equations by Continued Fractions
Authors: Strang J. A. 
Published: January 2009

The theorems which furnish in C.F. form the roots of a quadratic equation, and the similar process which leads to particular integrals of an ordinary differential equation of the second order, may be applied to certain types of difference equation. The types which suggest themselves for examination arethe bilinear equation, anda special form of the linear equation; the coefficients are functions of r, and s is constant.
A Geometrical Proof of a Theorem of Hurwitz
Authors: Ford Lester R. 
Published: January 2009

In the study of rational approximations to irrational numbers the following problem presents itself: Let ω be a real irrational number, and let us consider the rational fractions satisfying the inequalityhow small can the positive quantity k be chosen with the certainty that there will always be an infinite number of fractions satisfying the inequality whatever the value (irrational) of ω?
Geometrical Illustrations of Cyclant Substitutions
Authors: Taylor D. G. 
Published: January 2009

§1. The following three illustrations of cyclant substitutions are supplementary to those given in § 5 of a previous paper.In the cyclant substitution in three variables with equimodular multipliersin which the coefficients are real and satisfy the relationsthe following expressions, as may at once be verified, are absolute invariants:
Two Mechanical Integrators or Plammeters
Authors: Fraser A.Y. 
Published: January 2009

For the measurement of a plane area, bounded by an irregular curve, various methods are adopted. Besides the well-known methods of approximation in use among land measurers, the following may be mentioned—The area is divided by parallel ordinates. These are measured, and the results are then treated in several ways to produce more or less accurate results. See Williamson's Integral Calculus, third edition, p. 211.