Characterization of the absence of a spectral gapfor a class of periodic Jacobi operators

Let q ∈ {2, 3} and let 0 = s₀ < s₁ < … < s_q = T be integers. For m, n ∈ Z, we put ¯m,n = {j ∈ Z| m? j ? n}. We set l_j = s_j − s_j−1 for j ∈ 1, q. Given (p₁,, p_q) ∈ R^q, let b: Z → R be a periodic function of period T such that b(·) = p_j on s_j−1 + 1, s_j for each j ∈ 1, q. We study the spectral gaps of the Jacobi operator (Ju)(n) = u(n + 1) + u(n − 1) + b(n)u(n) acting on l²(Z). By [λ_2j , λ_2j−1] we denote the jth band of the spectrum of J counted from above for j ∈ 1, T. Suppose that p_m ≠ p_n for m ≠ n. We prove that the statements (i) and (ii) below are equivalent for λ ∈ R and i ∈ 1, T − 1.     

Limit sets of automatic sequences

We show that for every k-automatic sequence there exists a natural number p>0 such that the sequences of the form (k^pn+j)_n?0 with j=0,…,p−1 are scaling sequences for f. Moreover, we demonstrate that every limit set is the union of certain basic limit sets.     

Mean matrices and infinite divisibility

We consider matrices M with entries m_ij = m(λ_i, λ_j) where λ₁, … ,λ_n are positive numbers and m is a binary mean dominated by the geometric mean, and matrices W with entries w_ij = 1/m (λ_i, λ_j) where m is a binary mean that dominates the geometric mean. We show that these matrices are infinitely divisible for several much-studied classes of means.     

Dedekind symbols with plus reciprocity laws

Dedekind symbols are generalizations of the classical Dedekind sums (symbols), and the symbols are determined uniquely by their reciprocity laws, up to an additive constant. For Dedekind symbols D and F, we can consider two kinds of reciprocity laws: D(p,q)−D(q,−p)=R(p,q) and F(p,q)+F(q,−p)=T(p,q). The first type, which we call minus reciprocity laws, have been studied extensively. On the contrary, the second type, which we call plus reciprocity laws, have not yet been investigated. In this note we study fundamental properties of Dedekind symbols with plus reciprocity law F(p,q)+F(q,−p)=T(p,q). We will see that there is a fundamental difference between Dedekind symbols with minus and plus reciprocity laws.     

Non-orientable fundamental surfaces in lens spaces

We give a concrete example of an infinite sequence of (pn,qn)-lens spaces L(pn,qn) with natural triangulations T(pn,qn) with pn tetrahedra such that L(pn,qn) contains a certain non-orientable closed surface which is fundamental with respect to T(pn,qn) and of minimal crosscap number among all closed non-orientable surfaces in L(pn,qn) and has n−2 parallel sheets of normal disks of a quadrilateral type disjoint from the pair of core circles of L(pn,qn). Actually, we can set p₀=0, q₀=1, pk+1=3pk+2qk and qk+1=pk+qk.     

Likelihood ratio orderings of spacings of heterogeneous exponential random variables

Let X₁,X₂,…,X_n be independent exponential random variables such that X_i has failure rate λ for i=1,…,p and X_j has failure rate λ^* for j=p+1,…,n, where p≥1 and q=n-p≥1. Denote by D_i:n(p,q)=X_i:n-X_i-1:n the ith spacing of the order statistics , where X_0:n≡0. It is shown that D_i:n(p,q)?_lrD_i+1:n(p,q) for i=1,…,n-1, and that if λ?λ^* then , and for i=1,…,n, where ?_lr denotes the likelihood ratio order. The main results are used to establish the dispersive orderings between spacings.     

A bijective enumeration of labeled trees with given indegree sequence

For a labeled tree on the vertex set {1,2,…,n}, the local direction of each edge (ij) is from i to j if i<j. For a rooted tree, there is also a natural global direction of edges towards the root. The number of edges pointing to a vertex is called its indegree. Thus the local (resp. global) indegree sequence λ=e₁¹e₂²… of a tree on the vertex set {1,2,…,n} is a partition of n−1. We construct a bijection from (unrooted) trees to rooted trees such that the local indegree sequence of a (unrooted) tree equals the global indegree sequence of the corresponding rooted tree. Combining with a Prüfer-like code for rooted labeled trees, we obtain a bijective proof of a recent conjecture by Cotterill and also solve two open problems proposed by Du and Yin. We also prove a q-multisum binomial coefficient identity which confirms another conjecture of Cotterill in a very special case.     

Extended skolem sequences

A k-extended Skolem sequence of order n is an integer sequence (s₁, s₂,…, s_2n+1) in which s_k = 0 and for each j ? {1,…,n}, there exists a unique i ? {1,…, 2n} such that s_i = s_i+j = j. We show that such a sequence exists if and only if either 1) k is odd and n ≡ 0 or 1 (mod 4) or (2) k is even and n ≡ 2 or 3 (mod 4). The same conditions are also shown to be necessary and sufficient for the existence of excess Skolem sequences. Finally, we use extended Skolem sequences to construct maximal cyclic partial triple systems. © 1995 John Wiley & Sons, Inc.     

Valuations, equimultiplicity and normal flatness

Let R be a regular noetherian local ring of dimension n≥2 and (R_i)≡R=R₀⊂R₁⊂R₂⊂?⊂R_i⊂? be a sequence of successive quadratic transforms along a regular prime ideal p of R (i.e if p_i is the strict transform of p in R_i, then p_i≠R_i, i≥0). We say that p is maximal for (R_i) if for every non-negative integer j≥0 and for every prime ideal q_j of R_j such that (R_i) is a quadratic sequence along q_j with p_j⊂q_j, we have p_j=q_j. We show that p is maximal for (R_i) if and only if V=∪_i≥0R_i/p_i is a valuation ring of dimension one. In this case, the equimultiple locus at p is the set of elements of the maximal ideal of R for which the multiplicity is stable along the sequence (R_i), provided that the series of real numbers given by the multiplicity sequence associated with V diverges. Furthermore, if we consider an ideal J of R, we also show that is normally flat along at the closed point if and only if the Hironaka’s character ν^∗(J,R) is stable along the sequence (R_i). This generalizes well known results for the case where p has height one (see [B.M. Bennett, On the characteristic functions of a local ring, Ann. of Math. Second Series 91 (1) (1970) 25-87]).     

Critical potentials of the eigenvalues and eigenvalue gaps of Schrödinger operators

Let M be a compact Riemannian manifold with or without boundary, and let −Δ be its Laplace-Beltrami operator. For any bounded scalar potential q, we denote by λi(q) the ith eigenvalue of the Schrödinger type operator −Δ+q acting on functions with Dirichlet or Neumann boundary conditions in case ∂M≠∅. We investigate critical potentials of the eigenvalues λi and the eigenvalue gaps Gij=λj−λi considered as functionals on the set of bounded potentials having a given mean value on M. We give necessary and sufficient conditions for a potential q to be critical or to be a local minimizer or a local maximizer of these functionals. For instance, we prove that a potential q∈L^∞(M) is critical for the functional λ₂ if and only if q is smooth, λ₂(q)=λ₃(q) and there exist second eigenfunctions f₁,…,fk of −Δ+q such that . In particular, λ₂ (as well as any λi) admits no critical potentials under Dirichlet boundary conditions. Moreover, the functional λ₂ never admits locally minimizing potentials.     

On the shape of decomposable trees

A n-vertex graph is said to be decomposable if, for any partition (λ₁,…,λ_p) of the integer n, there exists a sequence (V₁,…,V_p) of connected vertex-disjoint subgraphs with |V_i|=λ_i. The aim of the paper is to study the homeomorphism classes of decomposable trees. More precisely, we show that homeomorphism classes containing decomposable trees with an arbitrarily large minimal distance between all pairs of distinct vertices of degree different from 2, is exactly the set of combs.     

p-adic valuations and k-regular sequences

A sequence is said to be k-automatic if the nth term of this sequence is generated by a finite state machine with n in base k as input. Regular sequences were first defined by Allouche and Shallit as a generalization of automatic sequences. Given a prime p and a polynomial f(x)∈Q_p[x], we consider the sequence , where v_p is the p-adic valuation. We show that this sequence is p-regular if and only if f(x) factors into a product of polynomials, one of which has no roots in Z_p, the other which factors into linear polynomials over Q. This answers a question of Allouche and Shallit.     

Mappings between Banach spaces that send mixed summable sequences into absolutely summable sequences

In this work we study mappings f from an open subset A of a Banach space E into another Banach space F such that, once a∈A is fixed, for mixed (s;q)-summable sequences of elements of a fixed neighborhood of 0 in E, the sequence is absolutely p-summable in F. In this case we say that f is (p;m(s;q))-summing at a. Since for s=q the mixed (s;q)-summable sequences are the weakly absolutely q-summable sequences, the (p;m(q;q))-summing mappings at a are absolutely (p;q)-summing mappings at a. The nonlinear absolutely summing mappings were studied by Matos (see [Math. Nachr. 258 (2003) 71-89]) in a recent paper, where one can also find the historical background for the theory of these mappings. When s=+∞, the mixed (∞,q)-summable sequences are the absolutely q-summable sequences. Hence the (p;m(∞;q))-summing mappings at a are the regularly (p;q)-summing mappings at a. These mappings were also studied in [Math. Nachr. 258 (2003) 71-89] and they were important to give a nice characterization of the absolutely (p;q)-summing mappings at a. We show that for q<s<+∞ the space of the (p;m(s;q))-summing mappings at a are different from the spaces of the absolutely (p;q)-summing mappings at a and different from the spaces of regularly (p;q)-summing mappings at a. We prove a version of the Dvoretzky-Rogers theorem for n-homogeneous polynomials that are (p;m(s;q))-summing at each point of E. We also show that the sequence of the spaces of such n-homogeneous polynomials, n∈N, gives a holomorphy type in the sense of Nachbin. For linear mappings we prove a theorem that gives another characterization of (s;q)-mixing operators in terms of quotients of certain operators ideals.     

Two-tuple balance of non-binary sequences with ideal two-level autocorrelation

Let p be a prime, q=p^m and F_q be the finite field with q elements. In this paper, we will consider q-ary sequences of period qⁿ-1 for q>2 and study their various balance properties: symbol-balance, difference-balance, and two-tuple-balance properties. The array structure of the sequences is introduced, and various implications between these balance properties and the array structure are proved. Specifically, we prove that if a q-ary sequence of period qⁿ-1 is difference-balanced and has the “cyclic” array structure then it is two-tuple-balanced. We conjecture that a difference-balanced q-ary sequence of period qⁿ-1 must have the cyclic array structure. The conjecture is confirmed with respect to all of the known q-ary sequences which are difference-balanced, in particular, which have the ideal two-level autocorrelation function when q=p.     

Compositions of Polynomials with Coefficients in a Given Field

Let F ⊂ K be fields of characteristic 0, and let K[x] denote the ring of polynomials with coefficients in K. Let p(x) = ∑_k = 0ⁿa_kx^k ∈ K[x], a_n ≠ 0. For p ∈ K[x]\F[x], define D_F(p), the F deficit of p, to equal n − max{0 ≤ k ≤ n : a_k∉F}. For p ∈ F[x], define D_F(p) = n. Let p(x) = ∑_k = 0ⁿa_kx^k and let q(x) = ∑_j = 0^mb_jx^j, with a_n ≠ 0, b_m ≠ 0, a_n, b_m ∈ F, b_j∉F for some j ≥ 1. Suppose that p ∈ K[x], q ∈ K[x]\F[x], p, not constant. Our main result is that p ° q ∉ F[x] and D_F(p ° q) = D_F(q). With only the assumption that a_nb_m ∈ F, we prove the inequality D_F(p ° q) ≥ D_F(q). This inequality also holds if F and K are only rings. Similar results are proven for fields of finite characteristic with the additional assumption that the characteristic of the field does not divide the degree of p. Finally we extend our results to polynomials in two variables and compositions of the form p(q(x, y)), where p is a polynomial in one variable.     

Economical tight examples for the biased Erd?s-Selfridge theorem

A positional game is essentially a generalization of tic-tac-toe played on a hypergraph (V,F). A pivotal result in the study of positional games is the Erd?s-Selfridge theorem, which gives simple criteria for the existence of a Breaker's winning strategy on a hypergraph F. It has been shown that the Erd?s-Selfridge theorem can be tight and that numerous extremal systems exist for that theorem. We focus on a generalization of the Erd?s-Selfridge theorem proven by Beck for biased (p:q) games, which we call the (p:q)-Erd?s-Selfridge theorem. We show that for pn-uniform hypergraphs there is a unique extremal system for the (p:q)-Erd?s-Selfridge theorem (q?2) when Maker must win in exactly n turns (i.e., as quickly as possible).     

Necessary and sufficient conditions for the boundedness of B-Riesz potential in the B-Morrey spaces

We consider the generalized shift operator, associated with the Laplace-Bessel differential operator . The maximal operator Mγ (B-maximal operator) and the Riesz potential (B-Riesz potential), associated with the generalized shift operator are investigated. At first, we prove that the B-maximal operator Mγ is bounded from the B-Morrey space Lp,λ,γ to Lp,λ,γ for all 1<p<∞ and 0?λ<n+|γ|. We prove that the B-Riesz potential , 0<α<n+|γ| is bounded from the B-Morrey space Lp,λ,γ to Lq,λ,γ if and only if α/(n+|γ|−λ)=1/p−1/q, 1<p<(n+|γ|−λ)/α. Also we prove that the B-Riesz potential is bounded from the B-Morrey space L1,λ,γ to the weak B-Morrey space WLq,λ,γ if and only if α/(n+|γ|−λ)=1−1/q.     

Constructing polynomials of minimal growth

In [2] a cyclic diagonal operator on the space of functions analytic on the unit disk with eigenvalues (λ _n ) is shown to admit spectral synthesis if and only if for each j there is a sequence of polynomials (p _n ) such that lim _n→∞ p _n (λ _k ) = δ _j,k and lim sup _n→∞ sup _k>j |p _n (λ _k )|^1/k ≤ 1. The author also shows, through contradiction, that certain classes of cyclic diagonal operators are synthetic. It is the intent of this paper to use the aforementioned equivalence to constructively produce examples of synthetic diagonal operators. In particular, this paper gives two different constructions for sequences of polynomials that satisfy the required properties for certain sequences to be the eigenvalues of a synthetic operator. Along the way we compare this to other results in the literature connecting polynomial behavior ([4] and [9]) and analytic continuation of Dirichlet series ([1]) to the spectral synthesis of diagonal operators.     

Pattern Frequency Sequences and Internal Zeros

Let q be a pattern and let S_n, q(c) be the number of n-permutations having exactly c copies of q. We investigate when the sequence (S_n, q(c))_c ≥ 0 has internal zeros. If q is a monotone pattern it turns out that, except for q = 12 or 21, the nontrivial sequences (those where n is at least the length of q) always have internal zeros. For the pattern q = 1(l + 1)l…2 there are infinitely many sequences which contain internal zeros and when l = 2 there are also infinitely many which do not. In the latter case, the only possible places for internal zeros are the next-to-last or the second-to-last positions. Note that by symmetry this completely determines the existence of internal zeros for all patterns of length at most 3.