In this paper, we solve the following dual integral equations

where δ is a real positive constant and f(x) is a continuous and integrable function of x in [0, a]. The dual integral equations (1) and (2) arise in a crack problem of elasticity.

In the Mathematische Annalen, 11 (1877), 440–444, E. Toeplitz has proved the following theorem: If three quadrics are polar quadrics of three points with respect to a cubic surface, then their (2, 2, 2) invariant vanishes; the invariant being of the second degree in the coefficients of each of the three quadrics.

Let , (n = 0, 1, ….) be the Laguerre, Hn(x) the Hermite polynomial. Let , be the space of all functions f(x) the pth powers of which are integrable over (a, b), with the norm

Let R be an associative, commutative ring with identity, and let A be a (unitary) R-module. It is well known that if A is a Noetherian R-module then every submodule of A has a primary decomposition in A. The object of the present paper is to dualise this result; that is, to show that if A is an Artinian R-module then every submodule of A can be expressed as the sum of a finite number of coprimary submodules of A.

denote the n-th Cesàro mean of order k for the series aΣan, that is,

where

and let

The present paper describes briefly a notation for representing continued fractions in many dimensions, which has the advantage providing a direct method of attack and of rendering intuitive, results which are usually proved by induction. The notation is the outcome of a generalisation which I previously made [1] in connection with the solution of certain difference equations. Only formal theorems are considered here. For a discussion of convergence reference may be made to the works [2, 3, 4, 5] cited at the end. The paper by Paley and XJrsell is particularly important since these authors discuss very fully the non-cyclic simple continued fraction

Let be a polynomial with distinct real zeros whose separation is defined by δ(f) = min i≠j(ai-aj ). We establish upper estimates for δ(f′-kf) in terms of n, k, and δ(f). The results give sufficient conditions for the inverse operator (D – kl)−1 to preserve the reality of the zeros of a polynomial.

Croft, Falconer and Guy asked: what is the smallest integer $n$ such that an $n$-reptile in the plane has a hole? Motivated by this question, we describe a geometric method of constructing reptiles in $\mathbb{R}^d$, especially reptiles with holes. In particular, we construct, for each even integer $n\ge4$, an $n$-reptile in $\mathbb{R}^2$ with holes. We also answer some questions concerning the topological properties of a reptile whose interior consists of infinitely many components.

Let n be a positive integer. Then we know that, if m>– 1,

Consider the integral

which is equal to

A problem posed in the early eighteenth century asks for right-angled triangles, each of whose sides exceeds double the area by a perfect square. We summarize known results and find such triangles with the smallest possible standard generators.

In this paper the study of radicals of finite near-rings is initiated. The main result (Theorem 4.3) gives a description of hereditary radicals having hereditary semisimple classes too. Also it is shown that there exist non-hereditary radicals having hereditary semisimple classes.

Throughout this paper, we work in the category of (p-localized) spaces having the homotopy type of connected CW-complexes of finite type with base point. We consider a principal bundle

where Gn = SU(n), U(n) or Sp(n) and d = 1, 1 or 2 respectively. In this case, the bundle is obtained as an induced bundle by a mapping f of base space S2dn−1 from the classical group extension as follows:

In this paper, we consider a reaction infiltration problem consisting of a parabolic equation for the concentration, an elliptic equation for the pressure, and an ordinary differential equation for the porosity. Existence and uniqueness of a global classical solution is proved for bounded domains Ω⊂RN, under suitable boundary conditions.

In (1) I obtained † an asymptotic formula for the number of zeros of an arbitrary canonical product II(z) of integral order but not of mean type, all of whose zeros lie on a single radius, from a knowledge of the asymptotic behaviour of (i)log | П(z)| as | z | = r→ ∞ along another radius l, with certain side conditions. After proving the analogous theorem in which log | П(z)| in (i) is replaced by , I show in this note that, at a cost of replacing l by two radii l1 and l2, both of these theorems may be generalised to include a class of canonical products of integral order whose zeros lie along a whole line. In one of the resulting theorems ‡ (Theorem II) I find the asymptotic number of zeros on each half of the line of zeros; another theorem (Theorem III) includes a previous result of mine.§

A necessary and sufficient condition for confluence of two poles of a class of meromorphic solutions of the KdV equation is introduced and proved.

Let G be one of the following compact simply connected Lie groups: SU(3), Sp(2), G2. In the first two cases there is a well known stable decomposition of G as Q ∨ Sd where d = dim G and Q is a certain subspace of G. For SU(3), Q is the stunted complex quasiprojective space Σ(ℂP2/ℂP1) which fits into a cofibration sequence S3→Q→S5 with stable attaching map η:S5 → S4 For Sp(2), Q is the quaternionic quasi-projective space ℍℚ1 and fits into a cofibration sequence S3→Q→S7 with stable attaching map 2ν:S7→S4 (Here η and ν are generators of respectively.)

A graph G consists, for the purposes of this paper, of two disjoint sets V(G), E(G), whose elements are called vertices and edges respectively of G, together with a relationship whereby with each edge is associated an unordered pair of distinct vertices (called its end-vertices) which the edge is said to join, and whereby no two vertices are joined by more than one edge. An edge γ and vertex ξ are incident if ξ is an end-vertex of γ. A monomorphism [isomorphism] of a graph G into [onto] a graph H is a one-to-one function φ from V(G)∪E(G) into [onto] V(H)∪E(H) such that φ(V(G))⊂V(H), φ(E(G))⊂E(H) and an edge and vertex of G are incident in G if and only if their images under φ are incident in H. G and H are isomorphic (in symbols, G ≅ H) if there exists an isomorphism of G onto H. A subgraph of G is a graph H such that V(H) ⊂ V(G), E(H)⊂E(G) and an edge and vertex of H are incident in H if and only if they are incident in G; if V(H) = V(G), H is a spanning subgraph. A collection of graphs are edge-disjoint if no two of them have an edge in common. A decomposition of G is a set of edge-disjoint subgraphs of G which between them include all the edges and vertices of G. Ln is a graph whose vertices are the lattice points of n-dimensional Euclidean space, two vertices A and B being joined by an edge if and only if AB is of unit length (and therefore necessarily parallel to one of the co-ordinate axes). An endless Hamiltonian line of a graph G is a spanning subgraph of G which is isomorphic to L1. The object of this paper is to prove that Ln is decomposable into n endless Hamiltonian lines, a result previously established (1) for the case where n is a power of 2.

A bosonic Laplacian, which is a generalization of Laplacian, is constructed as a second-order conformally invariant differential operator acting on functions taking values in irreducible representations of the special orthogonal group, hence of the spin group. In this paper, we firstly introduce some properties for homogeneous polynomial null solutions to bosonic Laplacians, which give us some important results, such as an orthogonal decomposition of the space of polynomials in terms of homogeneous polynomial null solutions to bosonic Laplacians, etc. This work helps us to introduce Bergman spaces related to bosonic Laplacians, named as bosonic Bergman spaces, in higher spin spaces. Reproducing kernels for bosonic Bergman spaces in the unit ball and a description of bosonic Bergman projection are given as well. At the end, we investigate bosonic Hardy spaces, which are considered as generalizations of harmonic Hardy spaces. Analogs of some well-known results for harmonic Hardy spaces are provided here. For instance, connections to certain complex Borel measure spaces, growth estimates for functions in the bosonic Hardy spaces, etc.