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Showing papers in "Aequationes Mathematicae in 1978"




Journal ArticleDOI
TL;DR: In this paper, the dynamic programming concept without optimization is known as invariant imbedding, which is very similar to (1) except for the absence of the maximization operation.
Abstract: where p and q represent the state and decision vectors, respectively, T represents the transformation of the process, and f(p) represents the optimal return function with initial state p. This functional equation can be studied in several ways, either with respect to the type of processes giving rise to (1), or with respect to the precise form of (1), or with respect to the computational aspects of (1). In this survey article, this function will be treated according to the different types of processes. In addition to the optimization problems in dynamic programming as shown in (1), the dynamic programming concept can also be used to solve various types of boundary value problems arising in engineering and physical sciences. The dynamic programming concept without optimization is known as invariant imbedding. The resulting functional equation of invariant imbedding is very similar to (1) except for the absence of the maximization operation. Dynamic programming involves a completely different approach to formulating the problem: Instead of only considering a single problem with a fixed duration, the dynamic programming approach is to colasider a family of problems, with duration of the process ranging from zero to the duration of the original problem. In order to consider these different duration processes, the corresponding initial conditions for these processes must also be calculated and interpolated

163 citations


Journal ArticleDOI
TL;DR: A simple proof by functional equations is given for Ramanujan's 1 ψ 1 sum as discussed by the authors, which is a useful extension of Jacobi's triple product formula, and has recently become important in the treatment of certain orthogonal polynomials defined by basic hypergeometric series.
Abstract: A simple proof by functional equations is given for Ramanujan’s1 ψ 1 sum. Ramanujan’s sum is a useful extension of Jacobi's triple product formula, and has recently become important in the treatment of certain orthogonal polynomials defined by basic hypergeometric series.

61 citations


Journal ArticleDOI
Marek Kuczma1

55 citations


Journal ArticleDOI
TL;DR: In this paper, the authors define the singular graph of an M-matrix in standard lower block triangular form, with diagonal blocks irreducible, as the set of indices α such that the diagonal block Aαα is singular.
Abstract: LetA be anM-matrix in standard lower block triangular form, with diagonal blocksAii irreducible. LetS be the set of indices α such that the diagonal blockAαα is singular. We define the singular graph ofA to be the setS with partial order defined by α > β if there exists a chain of non-zero blocksAαi, Aij, ⋯, Alβ.

53 citations



Journal ArticleDOI
TL;DR: Menger et al. as discussed by the authors folgen hier der Auffassung von A. N. Serstnev, wie sie in den Arbeiten yon B. Schweizer [8], [9] wiedergegeben ist.
Abstract: Probabilistisch metrische Riiume als Erweiterung von metrischen Riiumen wurden erstmals von K. Menger [4] und A. Wald [14] eingeffihrt. Wir folgen hier der Auffassung von A. N. Serstnev, wie sie in den Arbeiten yon B. Schweizer [8], [9] wiedergegeben ist. Zuniichst statten wir die Menge A + aller nicht negativen Verteilungsfunktionen-d.h, aller monotonwachsenden, linksseitig stetigen Funktionen F: R ~ [0, 1] mit F ( 0 ) = 0 mit einer Halbordnung ~<

31 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that the probability that P n > P n + 1 is positive (almost certainly this probability is 1/2) for every E > 0.
Abstract: is less than Ex . Our proof entails Brun's method . A corollary is that the probability that P(n) > P(n + 1) is positive (almost certainly this probability is 1/2) . Our methods allow us to say something about integers n which have the same sum of their prime factors as n + 1 . We prove the number of such n :5 x is O(x/(log x)' -') for every E > 0. We know a proof in the case E = 0 as well, but it is more complicated and not presented . In addition we present a brief discussion on the largest prime factors of 3 or more consecutive numbers .

31 citations




Journal ArticleDOI
TL;DR: In this article, the authors considered the special case when ¢(z) is a non-linear polynomial, i.e., the equations f(p(z ) ) = q( f( z ) )+ h(z), where p (z ) = a p z P + a p l z P l +... + a o is a polynomial of degree p (p > l ).
Abstract: The purpose of this paper is to consider meromorphic solutions of the functional equations f(q~(z)) = q( f (z ) ) + h(z ) (1) f (~ ( z ) ) = g(z )q( f ( z ) ) , (2) where f ( f~ constant), g (g~0) , h are meromorphic functions (in the whole open plane), ~(z) is a non-constant entire function and q(z) is a rational [unction of order k (k >>-1). Most of our results will concern the special case, when ¢(z) is a non-linear polynomial, i.e. the equations f (p(z ) ) = q( f ( z ) )+ h(z) (3) f (p(z ) ) = g(z)q( f (z) ) (4) where p ( z ) = a p z P + a p _ l z P l + . . . + a o is a polynomial of degree p ( p > l ) . In what follows we shall assume, unless otherwise stated that f, g, h, q~, q, p denote such functions. Certain special cases of these equations have been considered before. Thus, the equation

Journal ArticleDOI
TL;DR: In this paper, a solution of a (1) ~b(ct, 1, y) est une solution de l '6quation (2), de la forme (3), pour laquelle l 'ensemble E des fibres transitives Fk qui n'ont qu'un point, n'est pas vide, (2) g(a) et Fk ont les m6mes sens que dans a), (3) hk, pour chaque k de K, est un homomorphisme de (C~{0
Abstract: of a (1) ~b(ct, 1, y) est une solution de l '6quation (2), de la forme (3), pour laquelle l 'ensemble E des fibres transitives Fk qui n'ont qu'un point, n'est pas vide, (2) g(a) et Fk ont les m6mes sens que dans a), (3) hk, pour chaque k de K, est un homomorphisme de (C~{0},-) dans (C/Ck, +), off + d6signe ici l 'addition dans C/Ck induite par l 'addition dans C et hk(x) est une fonction de l 'ensemble C'-{0} ~ l 'ensemble C telle que

Journal ArticleDOI
TL;DR: In this article, Mendelsohn et al. showed that the existence of semi-symmetric idempotent quasigroups satisfying Steiner triple systems is known.
Abstract: A quasigroupQ is a set together with a binary operation which satisfies the condition that any two elements of the equationxy =z uniquely determines the third. A quasigroup is in indempotent when any elementx satisfies the indentityxx =x. Several types of Tactical Systems are defined as arrangement of points into “blocks” in such a way as to balance the incidence of (ordered or unordered) pairs of points, and shown to be coexistent with idempotent quasigroups satisfying certain identifies. In particular the correspondences given are: 1. totally symmetric idempotent quasigroups and Steiner triple systems, 2. semi-symmetric idempotent quasigroups and directed triple systems, 3. idempotent quasigroups satisfying Schroder's Second Law, namely (xy)(yx)=x, and triple tourna-ments, and 4. idempotent quasigroups satisfying Stein's Third Law, namely (xy)(yx)=y, and directed tournaments. These correspondences are used to obtain corollaries on the existence of such quasig-roups from constructions of the Tactical Systems. In particular this provides a counterexample to an ”almost conjecture“ of Norton and Stein (1956) concerning the existence of those quasigroups in 3 and 4 above. Indeed no idempotent qnasigroups satisfying Stein's Third Law and with order divisible by four were known to N. S. Mendelsohn when he wrote a paper on such quasigroups for the Third Waterloo Conference on Combinatorics (May, 1968). Finally, a construction for triple tournaments is interpreted as a Generalized Semi-Direct Product of idempotent quasigroups.

Journal ArticleDOI
TL;DR: In this article, it was shown that if A = (%) is an n-by-n entry-wise nonnegative matrix whose Perron-Frobenius eigenvalue is 1, then o(A) c Ln (1) where o-(A) denotes the spectrum or set of all eigenvalues of A. Since the unit circle U is not contained in L, this yields an improvement on the usual inclusion region, namely: ~ r (A)c U, for the eigen values of A (which may be obtained via Gersgorin
Abstract: of the complex plane by Lk. This is precisely the region in which complex numbers u + iv satisfy u + Ivl tan (w/k)-< 1. It has been shown [1, 5, 7] that if A = (%) is an n-by-n entry-wise nonnegative matrix whose Perron-Frobenius eigenvalue is 1, then o-(A) c Ln (1) where o-(A) denotes the spectrum or set of all eigenvalues of A. Since the unit circle U is not contained in L,, this yields an improvement on the usual inclusion region, ~ r (A)c U, for the eigenvalues of A (which may be obtained via Gersgorin, norms or Perron-Frobenius), namely: ~r(A) c L~ 71 U. (2)




Journal ArticleDOI
TL;DR: In this article, a necessary and sufficient condition on n in order that there exist a group G such that G = (a) is cyclic of order n, a is a commutator, but not every generator of G is a product of fewer than m commutators, where m is any preassigned integer.
Abstract: I t is well known that if G is a group, not every element of the commutator subgroup G ' need be a commutator . Examples to this effect are given in [2], [3], and [9]. I to [7] has shown that every e lement in the alternating group A , (n-> 5) is a commuta tor in A~. Similar results have been obtained for certain classes of linear groups ([6], [11], [12]). However , it is an open conjecture whether every e lement of a perfect group (or even simple group) is a commutator . Burnside [1] and Gallagher [4] have obtained necessary and sufficient conditions involving the character table of a finite group G in order that an e lement of G be a product of rn commutators. MacDonald [8] has t reated the case where G' is cyclic, and has proved under this assumption that if G is nilpotent or G ' is infinite, then some generator of G ' is a commutator . (This implies that all generators are commutators by a result of H o n d a [5].) On the other hand, MacDonald has constructed a finite group G with G ' cyclic such that no generator of G ' is a product of fewer than m commutators , where m is any preassigned integer. In this paper, we determine a necessary and sufficient condition on n in order that there exist a group G such that G ' = (a) is cyclic of order n, a is a commutator , but not every e lement of G ' is a commutator . The condition is:


Journal ArticleDOI
TL;DR: In this article, it was shown that for π = 2 cos (2π/n), where π is a positive integer > 1, the two equations can be reduced to a single equation.
Abstract: The equations of the title appear in the author's paper “Chromatic Sums for Rooted Planar Triangulations, V: Special Equations.” (Canadian Journal of Mathematics, 26 (1974), 893–907). They appear in that paper as Equations (24) and (25). They are simultaneous equations for two unknown functionsl andy2 of two variablesy1 andz. A parameterμ is involved. The main result is that forμ = 2 cos (2π/n), wheren is a positive integer >1, the two equations can be reduced to a single equation (numbered (49)). Solutions of this are known forn <7. From such solutions we can expect to get information about the averaged chromatic polynomials of planar triangulations with a given number of triangles.

Journal ArticleDOI
TL;DR: In this article, a study of approximation of classes of functions and asymptotic simultaneous approximation of functions by the Mn-operators of Meyer-Konig and Zeller is presented.
Abstract: This note is a study of approximation of classes of functions and asymptotic simultaneous approximation of functions by theMn-operators of Meyer-Konig and Zeller which are defined by $$(M_n f)(x) = (1 - x)^{n + 1} \sum\limits_{k = 0}^\infty {f\left( {\frac{k}{{n + k}}} \right)} \left( \begin{array}{l} n + k \\ k \\ \end{array} \right)x^k , n = 1,2,....$$ Among other results it is proved that for 0<α≤1 $$\mathop {\lim }\limits_{n \to \infty } n^{\alpha /2} \mathop {\sup }\limits_{f \in Lip_1 \alpha } \left| {(M_n f)(x) - f(x)} \right| = \frac{{\Gamma \left( {\frac{{\alpha + 1}}{2}} \right)}}{{\pi ^{1/2} }}\left\{ {2x(1 - x)^2 } \right\}^{\alpha /2} $$ and if for a functionf, the derivativeDm+2f exist at a pointx∈(0, 1), then $$\mathop {\lim }\limits_{n \to \infty } 2n[D^m (M_n f) - D^m f] = \Omega f,$$ where Ω is the linear differential operator given by $$\Omega = x(1 - x)^2 D^{m + 2} + m(3x - 1)(x - 1)D^{m + 1} + m(m - 1)(3x - 2)D^m + m(m - 1)(m - 2)D^{m - 1} .$$

Journal ArticleDOI
TL;DR: In this article, it was shown that if the affine hull of a convex set is affine, then the dimension of the set can be characterized by the affinities of the convex hull.
Abstract: Denoting by dimA the dimension of the affine hull of the setA, we prove that if {Ki:i ∈ T} and {Kij :i ∈ T} are two finite families of convex sets inRn and if dim ∩ {Ki:i ∈ S} = dim ∩ {Kij :i ∈ S}for eachS ⊂ T such that|S| ≤ n + 1 then dim ∩{Ki :i ∈ T} = dim ∩ {Ki′ : {i ∈ T}}.

Journal ArticleDOI
Abstract: A translation plane of dimension d over its kernel GF(q)= K can be represented in terms of a vector space of dimension 2d over K. The lines through the origin (zero vector) form a spread a class of mutually disjoint (except for the zero vector) d-dimensional subspaces which cover the vector space. The linear translation complement is the group of linear translations which acts as a collineation group of the plane. Not many different kinds of examples are known for the case where d and q are both odd. Except for Hering's plane of order 27, all of the examples known to the author have solvable linear translation complements and are either reducible or have a pair of lines through the origin in an orbit of length two. The first case includes the semi-field planes and some of the generalized Andr6 planes. The other generalized Andr6 planes of odd dimension fall in the second category. There must be some more examples, or restrictions on the nature of the linear translation complement, or both. The author has developed some restrictions when the group is non-solvable [7] or contains affine homologies [6]. In this paper, we examine the situation where the dimension over the kernel is an odd prime and we have an irreducible solvable subgroup of the linear translation complement. Many finite incidence structures are closely related to vector spaces over finite fields and the automorphisms (collineations) of these structures can be represented by linear groups i.e. groups of non-singular linear transformations. Frequently one knows that certain collineation groups must be subgroups of GL(n, q) for some specified n .and q. It would be helpful if one knew all of the subgroups of GL(n, q). This is essentially known for complex linear groups of



Journal ArticleDOI
TL;DR: In this article, a functional equation is written based on the functional properties of S petertodd k(n) and several methods of solution are presented, which lead to several recurrence relations for the functions and a simple one-step differential-recurrence relation from which the polynomials can easily be computed successively.
Abstract: This paper presents a direct and simple approach to obtaining the formulas forS k(n)= 1 k + 2 k + ... +n k wheren andk are nonnegative integers. A functional equation is written based on the functional properties ofS k (n) and several methods of solution are presented. These lead to several recurrence relations for the functions and a simple one-step differential-recurrence relation from which the polynomials can easily be computed successively. Arbitrary constants which arise are (almost) the Bernoulli numbers when evaluated and identities for these modified Bernoulli numbers are obtained. The functional equation for the formulas leads to another functional equation for the generating function for these formulas and this is used to obtain the generating functions for theS k 's and for the modified Bernoulli numbers. This leads to an explicit representation, not a recurrence relation, for the modified Bernoulli numbers which then yields an explicit formula for eachS k not depending on the earlier ones. This functional equation approach has been and can be applied to more general types of arithmetic sequences and many other types of combinatorial functions, sequences, and problems.

Journal ArticleDOI
TL;DR: This article used Appell polynomials to solve certain types of linear ODEs with variable coefficients, and showed that they can be solved by Appell Polynomial Methods.
Abstract: We show how to solve certain types of linear ordinary differential equations with variable coefficients by using Appell polynomials.

Journal ArticleDOI
TL;DR: A simple arithmetical proof and a generalization of Bender's generalized q-binomial Vandermonde convolution are given in this paper, where the authors also present a simple proof and generalization.
Abstract: A simple arithmetical proof and a generalization of Bender's generalizedq-binomial Vandermonde convolution are given.