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1.
A set {a1,…,am} of m distinct positive integers is called a Diophantine m-tuple if aiaj+1 is a perfect square for all i, j with 1?i<j?m. It is conjectured that if {a,b,c,d} is a Diophantine quadruple with a<b<c<d, then d=d+, where d+=a+b+c+2abc+2rst and , , . In this paper, we show that if {a,b,c,d,e} is a Diophantine quintuple with a<b<c<d<e, then d=d+.  相似文献   

2.
Let h, k be fixed positive integers, and let A be any set of positive integers. Let hA ≔ {a 1 + a 2 + ... + a r : a i A, rh} denote the set of all integers representable as a sum of no more than h elements of A, and let n(h, A) denote the largest integer n such that {1, 2,...,n} ⊆ hA. Let n(h, k) := : n(h, A), where the maximum is taken over all sets A with k elements. We determine n(h, A) when the elements of A are in geometric progression. In particular, this results in the evaluation of n(h, 2) and yields surprisingly sharp lower bounds for n(h, k), particularly for k = 3.  相似文献   

3.
4.
Using a slight modification of an argument of Croot, Ruzsa and Schoen we establish a quantitative result on the existence of a dilated copy of any given configuration of integer points in sparse difference sets. More precisely, given any configuration {v1,…,v?} of vectors in Zd, we show that if Ad[1,N] with |A|/Nd?CN−1/?, then there necessarily exists r≠0 such that {rv1,…,rv?}⊆AA.  相似文献   

5.
Let {a1,a2,a3,…} be an unbounded sequence of positive integers with an+1/an approaching α as n→∞, and let β>max(α,2). We show that for all sufficiently large x?0, if A⊂[0,x] is a set of nonnegative integers containing 0 and satisfying
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6.
Min Tang   《Discrete Mathematics》2009,309(21):6288-6293
Let A={a1,a2,…}(a1<a2<) be an infinite sequence of nonnegative integers, let k≥2 be a fixed integer and denote by rk(A,n) the number of solutions of ai1+ai2++aikn. Montgomery and Vaughan proved that r2(A,n)=cn+o(n1/4) cannot hold for any constant c>0. In this paper, we extend this result to k>2.  相似文献   

7.
Let G be a finite abelian group. Write and denote by rk(2G) the rank of the group 2G.Extending a result of Meshulam, we prove the following. Suppose that AG is free of “true” arithmetic progressions; that is, a1+a3=2a2 with a1,a2,a3A implies that a1=a3. Then |A|<2|G|/rk(2G). When G is of odd order this reduces to the original result of Meshulam.As a corollary, we generalize a result of Alon and show that if an integer k?2 and a real ε>0 are fixed, |2G| is large enough, and a subset AG satisfies |A|?(1/k+ε)|G|, then there exists A0A such that 1?|A0|?k and the elements of A0 add up to zero. When G is of odd order or cyclic this reduces to the original result of Alon.  相似文献   

8.
The well-known “splitting necklace theorem” of Alon [N. Alon, Splitting necklaces, Adv. Math. 63 (1987) 247-253] says that each necklace with kai beads of color i=1,…,n, can be fairly divided between k thieves by at most n(k−1) cuts. Alon deduced this result from the fact that such a division is possible also in the case of a continuous necklace [0,1] where beads of given color are interpreted as measurable sets Ai⊂[0,1] (or more generally as continuous measures μi). We demonstrate that Alon's result is a special case of a multidimensional consensus division theorem about n continuous probability measures μ1,…,μn on a d-cube d[0,1]. The dissection is performed by m1+?+md=n(k−1) hyperplanes parallel to the sides of d[0,1] dividing the cube into m1⋅?⋅md elementary cuboids (parallelepipeds) where the integers mi are prescribed in advance.  相似文献   

9.
Given a graph G, for an integer c∈{2,…,|V(G)|}, define λc(G)=min{|X|:XE(G),ω(GX)≥c}. For a graph G and for an integer c=1,2,…,|V(G)|−1, define,
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10.
Let denote the rational normal curve of order d. Its homogeneous defining ideal admits an SL2-stable filtration J2J4⊆…⊆IC by sub-ideals such that the saturation of each J2q equals IC. Hence, one can associate to d a sequence of integers (α1,α2,…) which encodes the degrees in which the successive inclusions in this filtration become trivial. In this paper we establish several lower and upper bounds on the αq, using inter alia the methods of classical invariant theory.  相似文献   

11.
Let G be a finite abelian group of order n and let AZ be non-empty. Generalizing a well-known constant, we define the Davenport constant of G with weight A, denoted by DA(G), to be the least natural number k such that for any sequence (x1,…,xk) with xiG, there exists a non-empty subsequence (xj1,…,xjl) and a1,…,alA such that . Similarly, for any such set A, EA(G) is defined to be the least tN such that for all sequences (x1,…,xt) with xiG, there exist indices j1,…,jnN,1?j1<?<jn?t, and ?1,…,?nA with . In the present paper, we establish a relation between the constants DA(G) and EA(G) under certain conditions. Our definitions are compatible with the previous generalizations for the particular group G=Z/nZ and the relation we establish had been conjectured in that particular case.  相似文献   

12.
In this paper, we reconsider the iterative method Xk=Xk−1+βY(IAXk−1), k=1,2,…,βC?{0} for computing the generalized inverse over Banach spaces or the generalized Drazin inverse ad of a Banach algebra element a, reveal the intrinsic relationship between the convergence of such iterations and the existence of or ad, and present the error bounds of the iterative methods for approximating or ad. Moreover, we deduce some necessary and sufficient conditions for iterative convergence to or ad.  相似文献   

13.
The spectra of some trees and bounds for the largest eigenvalue of any tree   总被引:2,自引:0,他引:2  
Let T be an unweighted tree of k levels such that in each level the vertices have equal degree. Let nkj+1 and dkj+1 be the number of vertices and the degree of them in the level j. We find the eigenvalues of the adjacency matrix and Laplacian matrix of T for the case of two vertices in level 1 (nk = 2), including results concerning to their multiplicity. They are the eigenvalues of leading principal submatrices of nonnegative symmetric tridiagonal matrices of order k × k. The codiagonal entries for these matrices are , 2 ? j ? k, while the diagonal entries are 0, …, 0, ±1, in the case of the adjacency matrix, and d1d2, …, dk−1dk ± 1, in the case of the Laplacian matrix. Finally, we use these results to find improved upper bounds for the largest eigenvalue of the adjacency matrix and of the Laplacian matrix of any given tree.  相似文献   

14.
An interesting and recently much studied generalization of the classical Schur class is the class of contractive operator-valued multipliers S(λ) for the reproducing kernel Hilbert space H(kd) on the unit ball BdCd, where kd is the positive kernel kd(λ,ζ)=1/(1−〈λ,ζ〉) on Bd. The reproducing kernel space H(KS) associated with the positive kernel KS(λ,ζ)=(IS(λ)S(ζ))⋅kd(λ,ζ) is a natural multivariable generalization of the classical de Branges-Rovnyak canonical model space. A special feature appearing in the multivariable case is that the space H(KS) in general may not be invariant under the adjoints of the multiplication operators on H(kd). We show that invariance of H(KS) under for each j=1,…,d is equivalent to the existence of a realization for S(λ) of the form S(λ)=D+C−1(Iλ1A1−?−λdAd)(λ1B1+?+λdBd) such that connecting operator has adjoint U which is isometric on a certain natural subspace (U is “weakly coisometric”) and has the additional property that the state operators A1,…,Ad pairwise commute; in this case one can take the state space to be the functional-model space H(KS) and the state operators A1,…,Ad to be given by (a de Branges-Rovnyak functional-model realization). We show that this special situation always occurs for the case of inner functions S (where the associated multiplication operator MS is a partial isometry), and that inner multipliers are characterized by the existence of such a realization such that the state operators A1,…,Ad satisfy an additional stability property.  相似文献   

15.
For r = (r1,…, rd) ∈ ?d the mapping τr:?d →?d given byτr(a1,…,ad) = (a2, …, ad,−⌊r1a1+…+ rdad⌋)where ⌊·⌋ denotes the floor function, is called a shift radix system if for each a ∈ ?d there exists an integer k > 0 with τrk(a) = 0. As shown in Part I of this series of papers, shift radix systems are intimately related to certain well-known notions of number systems like β-expansibns and canonical number systems. After characterization results on shift radix systems in Part II of this series of papers and the thorough investigation of the relations between shift radix systems and canonical number systems in Part III, the present part is devoted to further structural relationships between shift radix systems and β-expansions. In particular we establish the distribution of Pisot polynomials with and without the finiteness property (F).  相似文献   

16.
17.
This paper designs a set of graph operations, and proves that for 2k/d<3, starting from Kk/d, by repeatedly applying these operations, one can construct all graphs G with c(G)k/d. Together with the result proved in [20], where a set of graph operations were designed to construct graphs G with c(G)k/d for k/d3, we have a complete analogue of Hajós' Theorem for the circular chromatic number. This research was partially supported by the National Science Council under grant NSC 89-2115-M-110-003  相似文献   

18.
We demonstrate that for any prescribed set of finitely many disjoint closed subdomains D1,…,Dm of a given spatial domain Ω in RN, if d1,d2,a1,a2,c,d,e are positive continuous functions on Ω and b(x) is identically zero on D?D1∪?∪Dm and positive in the rest of Ω, then for suitable choices of the parameters λ, μ and all small ε>0, the competition model
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19.
20.
Andrea Vietri 《Order》2005,22(3):201-221
A class of ranked posets {(D h k , ≪)} has been recently defined in order to analyse, from a combinatorial viewpoint, particular systems of real homogeneous inequalities between monomials. In the present paper we focus on the posets D 2 k , which are related to systems of the form {x a x b * abcd x c x d : 0 ≤ a, b, c, dk, * abcd ∈ {<, >}, 0 < x 0 < x 1 < ...< x k}. As a consequence of the general theory, the logical dependency among inequalities is adequately captured by the so-defined posets . These structures, whose elements are all the D 2 k 's incomparable pairs, are thoroughly surveyed in the following pages. In particular, their order ideals – crucially significant in connection with logical consequence – are characterised in a rather simple way. In the second part of the paper, a class of antichains is shown to enjoy some arithmetical properties which make it an efficient tool for detecting incompatible systems, as well as for posing some compatibility questions in a purely combinatorial fashion.  相似文献   

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