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1.
It is well known that the recurrence relations
are periodic, in the sense that they define periodic sequences for all choices of the initial data, and lead to sequences
with periods 2, 5 and 8, respectively. In this paper we determine all periodic recursions of the form
where are complex numbers, are non-zero and . We find that, apart from the three recursions listed above, only
lead to periodic sequences (with periods 6 and 8). The non-periodicity of (R) when (or and ) depends on the connection between (R) and the recurrence relations
and
We investigate these recursions together with the related
Each of (A), (B), and (C) leads to periodic sequences if k = 1 (with periods 6, 5, and 9, respectively). Also, for k = 2, (B) leads to periodicity with period 8. However, no other cases give rise to periodicity. We also prove that every real
sequence satisfying any of (A), (B), and (C) must be bounded. As a consequence, we find that for an arbitrary k, every rational sequence satisfying any of (A), (B), and (C) must be periodic.
(Received 27 June 2000; in revised form 5 January 2001) 相似文献
2.
Abstract. We prove that the best way to reduce the volume of the n -dimensional unit cube by a linear transformation that maps each of the main vertices to a point within distance ɛ < from is to shorten all edges by a factor (1-ɛ) . In particular, the minimal volume of such an almost cubic parallelepiped is (1-ɛ)
n
. This problem naturally arises in the construction of lattice-based one-way functions with worst-case / average-case connection. 相似文献
3.
Abstract. Weakly left ample semigroups are a class of semigroups that are (2,1) -subalgebras of semigroups of partial transformations, where the unary operation takes a transformation α to the identity map in the domain of α . It is known that there is a class of proper weakly left ample semigroups whose structure is determined by unipotent monoids acting on semilattices or categories. In
this paper we show that for every finite weakly left ample semigroup S , there is a finite proper weakly left ample semigroup and an onto morphism from to S which separates idempotents. In fact, is actually a (2,1) -subalgebra of a symmetric inverse semigroup, that is, it is a left ample semigroup (formerly, left type A). 相似文献
5.
A classical inequality for Euler’s gamma function states that
for all and with . We prove the following extension of this result. Let be the weighted power mean of of order r. The inequality
holds for all and with if and only if
(Received 3 April 2000; in revised form 26 June 2000) 相似文献
6.
Let X, Y be vector spaces. It is shown that if a mapping f ： X → Y satisfies f（（x＋y）/2＋z）＋f（（x-y）/2＋z=f（x）＋2f（z）,（0.1） f（（x＋y）/2＋z）-f（（x-y）/2＋z）f（y）,（0.2） or 2f（（x＋y）/2＋x）=f（x）＋f（y）＋2f（z）（0.3）for all x, y, z ∈ X, then the mapping f ： X →Y is Cauchy additive.
Furthermore, we prove the Cauchy-Rassias stability of the functional equations （0.1）, （0.2） and （0.3） in Banach spaces. The results are applied to investigate isomorphisms between unital Banach algebras. 相似文献
7.
One of the classical problem in computational biology is the character compatibility problem or perfect phylogeny problem. A standard formulation of this problem in terms of two closely related questions is the following. Given a data set consisting
of a finite set X and a set
of partitions induced on X by a set of characters. Is
compatible, that is, does there exist an evolutionary tree that represents (in a well-defined sense) the data? If this is the case,
is this tree unique? A fundamental result in phylogenetics states that the answer to the former of the two questions is yes
precisely if the partition intersection graph
associated to
can be made chordal by obeying a certain rule.
The main insight from this paper is that the relation graph
associated to a set
of partitions may provide a key for deciding whether such a chordalization of
exists. To prove our results, we introduce an extension of the concept of the partition intersection graph associated to
using
.
Received August 27, 2004 相似文献
8.
Let Y=( X,{ R
i
} 0≤i≤D) denote a symmetric association scheme with D≥3, and assume Y is not an ordinary cycle. Suppose Y is bipartite P-polynomial with respect to the given ordering A
0, A
1,…, A
D
of the associate matrices, and Q-polynomial with respect to the ordering E
0, E
1,…, E
D
of the primitive idempotents. Then the eigenvalues and dual eigenvalues satisfy exactly one of (i)–(iv).
(i)
(ii) D is even, and
(iii) θ *
0>θ 0, and
(iv) θ *
0>θ 0, D is odd, and
Received: February 13, 1996 / Revised: October 16, 1996 相似文献
9.
In this article, we study an iterative procedure of the following form , where f is a function and F is a set valued map acting from a Banach space X to a linear normed space Y, for solving generalized equations.
We show that this method is locally Q-linearly convergent to a solution x* of the generalized equation if the set-valued map is Aubin continuous at (0, x*) with a constant M for growth, f: X → Y is a function, whose Fréchet derivative is L-Lipschitz and A ∈ L( X,Y) is such that 2 M∥Δ f( x*) − A∥ < 1. We also study the stability of this method.
The research of this paper is partially supported by a Technical University of Varna internal research grant number 487/2008. 相似文献
10.
For any irrational , let denote the regular continued fraction expansion of x and define f, for all z > 0 by and by J. G ALAMBOS proved that (μ the Gauss measure) In this paper, we first point out that for all , ( has no limit for for almost all , proving more precisely that: For all , one has for almost all 相似文献
11.
Abstract. Given a set A and a function , we study the set of all functions that are continuous for all topologies for which f continuous.
We prove that in a sense to be made precise in the text, for any essentially infinitary function f , any non-constant such g equals f
n
, for some n∈ N. We also prove a similar result for the clone of n -ary functions from . 相似文献
12.
Let and be the von Mangoldt function and M?bius function, respectively, x real and y“small” compared with x. This paper gives, for the first time, a non-trivial estimate of the sum for all whenever . Correspondingly, it is also proved that 相似文献
13.
Let us consider the linear boundary value problem
where
and
is defined by
Classical Lyapunov inequality states that
for any function
where
The constant 4/ L is optimal. Let us note that Lyapunov inequality is given in terms of
the usual norm in the space L1(0, L). In this paper we review some recent results on Lp Lyapunovtype inequalities,
, for ordinary and partial differential equations on a bounded and regular domain in
In the last case, it is showed that the relation between the quantities p and N/2 plays a crucial role, pointing out a deep difference with respect to the ordinary case. In the proof, the best constants
are obtained by using a related variational problem and Lagrange multiplier theorem. Finally, the linear results are combined
with Schauder fixed point theorem in the study of resonant nonlinear problems.
The authors have been supported by the Ministry of Science and Technology of Spain MTM2005- 01331 and by Junta de Andalucia
(FQM116). 相似文献
14.
Let X, Y, and Z be operators on a Hilbert space such that X and Z are positive. It is shown that
Applications of this commutator inequality are given.
相似文献
15.
§ 1 IntroductionIn this note we are concerned with the asymptotically periodic second order equation-u″+α( x) u =β( x) uq +γ( x) up, x∈ R,( 1 )where1 相似文献
17.
Let X be k-regular graph on v vertices and let τ denote the least eigenvalue of its adjacency matrix A( X). If α( X) denotes the maximum size of an independent set in X, we have the following well known bound: . It is less well known that if equality holds here and S is a maximum independent set in X with characteristic vector x, then the vector is an eigenvector for A( X) with eigenvalue τ . In this paper we show how this can be used to characterise the maximal independent sets in certain classes
of graphs. As a corollary we show that a graph defined on the partitions of {1, . . . ,9} with three cells of size three is
a core.
* Researchs upported by NSERC. 相似文献
18.
Abstract. Given a finite set A and a distinguished function , we study the set of all functions that are continuous for all topologies for which f is continuous. The main result is a characterization of the functions f such that this set is trivial, that is, contains only the constant functions and the iterates of f . 相似文献
19.
In a recent paper [7] the author considered the family of parametrized Thue equations
for monic polynomials which satisfy
Under some technical conditions it could be proved that there is a computable constant such that for all integers the only integer solutions of the Diophantine equation satisfy .
In this paper, we give an explicit expression for depending on the polynomials .
(Received 5 September 2000; in revised form 30 December 2000) 相似文献
20.
We study the q-Clifford algebras
, called FRT–Clifford algebras, introduced by Faddeev, Reshetikhin and Takhtajan. It is shown that
acts on the q-exterior algebra
. Moreover, explicit formulas for the embedding of
into
and its relation to the vector and spin representations of
are given and proved. 相似文献
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