Cyclic block designs with block size 3 from Skolem-type sequences

A Skolem-type sequence is a sequence (s ₁, . . . , s _t ) of positive integers \({i\in D}\) such that for each \({i\in D}\) there is exactly one \({j\in \{1, \ldots , t - i\}}\) such that s _j  = s _j+i  = i. Positions in the sequence not occupied by integers \({i\in D}\) contain null elements. In 1939, Peltesohn solved the existence problem for cyclic Steiner triple systems for v ≡ 1, 3(mod 6), v ≠ 9. Using the same technique in 1981, Colbourn and Colbourn extended the solution to all admissible λ > 1. It is known that Skolem-type sequences may be used to construct cyclic Steiner triple systems as well as cyclic triple systems with λ = 2. The main result of this paper is an extension of former results to cyclic triple systems with λ > 2. In addition we introduce a new kind of Skolem-type sequence.     

On infinite-rank singular perturbations of the Schrödinger operator

Schrödinger operators with infinite-rank singular potentials V=Σ _i,j=1 ^∞ b _ij〈φ_j,·〉φ_i are studied under the condition that the singular elements ψ _j are ξ _j(t)-invariant with respect to scaling transformationsin ?³.     

On the value distribution of positive definite quadratic forms

Denote by 0 = λ ₀ < λ ₁ ≤ λ ₂ ≤ . . . the infinite sequence given by the values of a positive definite irrational quadratic form in k variables at integer points. For l ≥ 2 and an (l ?1)-dimensional interval I = I ₂×. . .×I _l we consider the l-level correlation function \({K^{(l)}_I(R)}\) which counts the number of tuples (i ₁, . . . , i _l ) such that \({\lambda_{i_1},\ldots,\lambda_{i_l}\leq R^2}\) and \({\lambda_{i_{j}}-\lambda_{i_{1}}\in I_j}\) for 2 ≤ j ≤ l. We study the asymptotic behavior of \({K^{(l)}_I(R)}\) as R tends to infinity. If k ≥ 4 we prove \({K^{(l)}_I(R)\sim c_l(Q)\,{\rm vol}(I)R^{lk-2(l-1)}}\) for arbitrary l, where c _l (Q) is an explicitly determined constant. This remains true for k = 3 under the restriction l ≤ 3.     

The one-dimensional wave equation with general boundary conditions

We show that a realization of the Laplace operator Au := u′′ with general nonlocal Robin boundary conditions α _j u′(j) + β _j u(j) + γ _1–j u(1 ? j) = 0, (j = 0, 1) generates a cosine family on L ^p (0, 1) for every \({p\,{\in}\,[1,\infty)}\). Here α _j , β _j and γ _j are complex numbers satisfying α ₀, α ₁ ≠ 0. We also obtain an explicit representation of local solutions to the associated wave equation by using the classical d’Alembert’s formula.     

A Heuristic Decomposition Algorithm for Scheduling Problems on Mixed Graphs

We consider a scheduling problem where a set of n jobs has to be processed on a set of m machines and arbitrary precedence constraints between operations are given. Moreover, for any two operations i and j values a _ij >0 and a _ji >0 may be given where a _ij is the minimal difference between the starting times of operations i and j when operation i is processed first. Often, the objective is to minimize the makespan but we consider also arbitrary regular criteria. Even the special cases of the classical job shop problem J//C_max belong to the set of NP-hard problems. Therefore, approximation or heuristic algorithms are necessary to handle large-dimension problems. Based on the mixed graph model we give a heuristic decomposition algorithm for such a problem, i.e. the initial problem is partitioned into subproblems that can be solved exactly or approximately with a small error bound. These subproblems are obtained by a relaxation of a subset of the set of undirected edges of the mixed graph. The subproblems are successively solved and a proportion of the results obtained for one subproblem is kept for further subproblem definitions. Numerical results of the algorithm presented here are given.     

Designs associated with maximum independent sets of a graph

A (v, β _o , μ)-design over regular graph G = (V, E) of degree d is an ordered pair D = (V, B), where |V| = v and B is the set of maximum independent sets of G called blocks such that if i, j ∈ V, i ≠ j and if i and j are not adjacent in G then there are exactly μ blocks containing i and j. In this paper, we study (v, β _o , μ)-designs over the graphs K _n × K _n , T(n)-triangular graphs, L ₂(n)-square lattice graphs, Petersen graph, Shrikhande graph, Clebsch graph and the Schläfli graph and non-existence of (v, β _o , μ)-designs over the three Chang graphs T ₁(8), T ₂(8) and T ₃(8).     

A construction for modified generalized Hadamard matrices using QGH matrices

Let G be a group of order mu and U a normal subgroup of G of order u. Let G/U = {U ₁,U ₂, . . . ,U _m } be the set of cosets of U in G. We say a matrix H = [h _ij ] of order k with entries from G is a quasi-generalized Hadamard matrix with respect to the cosets G/U if \({\sum_{1\le t \le k} h_{it}h_{jt}^{-1} = \lambda_{ij1}U_1+\cdots+\lambda_{ijm}U_m (\exists\lambda_{ij1},\ldots, \exists \lambda_{ijm} \in \mathbb{Z})}\) for any i ≠ j. On the other hand, in our previous article we defined a modified generalized Hadamard matrix GH(s, u, λ) over a group G, from which a TD _λ (uλ, u) admitting G as a semiregular automorphism group is obtained. In this article, we present a method for combining quasi-generalized Hadamard matrices and semiregular relative difference sets to produce modified generalized Hadamard matrices.     

Classification and nondegeneracy of <Emphasis Type="Italic">SU</Emphasis>(<Emphasis Type="Italic">n</Emphasis>+1) Toda system with singular sources

We consider the following Toda system

Open image in new window

where γ _i >?1, δ ₀ is Dirac measure at 0, and the coefficients a _ij form the standard tri-diagonal Cartan matrix. In this paper, (i) we completely classify the solutions and obtain the quantization result:

$\sum_{j=1}^n a_{ij}\int_{\mathbb{R}^2}e^{u_j} dx = 4\pi(2+\gamma _i+\gamma_{n+1-i}), \quad\forall\;1\leq i \leq n.$

This generalizes the classification result by Jost and Wang for γ _i =0, \(\forall\;1\leq i\leq n\). (ii) We prove that if γ _i +γ _i+1+?+γ _j ?? for all 1≤i≤j≤n, then any solution u _i is radially symmetric w.r.t. 0. (iii) We prove that the linearized equation at any solution is non-degenerate. These are fundamental results in order to understand the bubbling behavior of the Toda system.

     

Derivations cocentralizing multilinear polynomials on left ideals

Let R be a prime ring with extended centroid C, λ a nonzero left ideal of R and f (X ₁, . . . , X _t ) a nonzero multilinear polynomial over C. Suppose that d and δ are derivations of R such that

$d(f(x_{1},\ldots,x_{t}))f(x_{1},\ldots,x_{t})-f(x_{1},\ldots,x_{t})\delta(f(x_{1},\ldots,x_{t}))\in C$

for all \({x_1,\ldots,x_t\in\lambda}\). Then either d = 0 and λ δ(λ) = 0 or λ C = RCe for some idempotent e in the socle of RC and one of the following holds:

1. (1)
   f (X₁, . . . , X _t ) is central-valued on eRCe;
    
2. (2)
   λ(d + δ)(λ) = 0 and f (X₁, . . . , X _t )² is central-valued on eRCe;
    
3. (3)
   char R = 2 and eRCe satisfies st ₄(X ₁, X ₂, X ₃, X ₄), the standard polynomial identity of degree 4.
    

     

A concentration function estimate and intersective sets from matrices

We give several sufficient conditions on an infinite integer matrix (d _ij ) for the set R = {Σ _{ij∈α, i>j} d _ij : α ? ?, |α| < ∞} to be a density intersective set, including the cases d _nj = j ⁿ (1 + O(1/n ^1+ε )) and \(0 < d_{nj} = o(\sqrt {n/\log n} )\). For the latter, a concentration function estimate that is of independent interest is applied to sums of sequences of 2-valued random variables whose means may grow as \(\sqrt {n/\log n} \).     

Bounded indecomposable semigroups of non-negative matrices

A semigroup \({\mathfrak{S}}\) of non-negative n × n matrices is indecomposable if for every pair i, j ≤ n there exists \({S\in\mathfrak{S}}\) such that (S) _ij ≠ 0. We show that if there is a pair k, l such that \({\{(S)_{kl} : S\in\mathfrak{S}\}}\) is bounded then, after a simultaneous diagonal similarity, all the entries are in [0, 1]. We also provide quantitative versions of this result, as well as extensions to infinite-dimensional cases.     

Entropy numbers and interpolation

This paper settles a long-standing question by showing that in certain circumstances the entropy numbers of a map do not behave well under real interpolation. To do this, lemmas of combinatorial type are established and used to obtain lower bounds for the entropy numbers of a particular diagonal map acting between Lorentz sequence spaces. These lower bounds contradict the estimates from above that would be obtained if the behaviour of entropy numbers under real interpolation was as good as conjectured. The paper also provides sharp two-sided estimates of the entropy number e _n (T) of diagonal operators \({T:l_{p}\rightarrow l_{q}, T(\left( a_{k}\right)_{k=1}^{\infty}) = (( \lambda_{k}a_{k}) _{k=1}^{\infty}) ,}\) where 0 < p < q ≤ ∞ and \({\{\lambda _{i}\}_{i=1}^{\infty}}\) is a non-increasing sequence of non-negative numbers with λ _i  = λ _n for all i ≤ n.     

A Probabilistic Characterization of the Dominance Order on Partitions

We present a probabilistic characterization of the dominance order on partitions. Let ν be a partition and Y _ν its Ferrers diagram, i.e. a stack of rows of cells with row i containing ν _i cells. Let the cells of Y _ν be filled with independent and identically distributed draws from the random variable X = B i n(r, p) with r ≥ 1 and p ∈ (0, 1). Given j, t ≥ 0, let P(ν, j, t) be the probability that the sum of all the entries in Y _ν is j while the sum of the entries in each row of Y _ν is no more than t. It is shown that if ν and μ are two partitions of n, ν dominates μ if and only if P(ν, j, t) ≤ P(μ, j, t) for all j, t ≥ 0. It is shown that the same result holds if X is any log-concave integer valued random variable with {i : P(X = i) > 0} = {0, 1,…,r} for some r ≥ 1.     

Kostka functions associated to complex reflection groups and a conjecture of Finkelberg-Ionov

Kostka functions K_(λ,μ)~±(t), indexed by r-partitions λ and μ of n, are a generalization of Kostka polynomials K_(λ,μ)(t) indexed by partitions λ,μ of n. It is known that Kostka polynomials have an interpretation in terms of Lusztig's partition function. Finkelberg and Ionov(2016) defined alternate functions K_(λ,μ)(t) by using an analogue of Lusztig's partition function, and showed that K_(λ,μ)(t) ∈Z≥0[t] for generic μ by making use of a coherent realization. They conjectured that K_(λ,μ)(t) coincide with K_(λ,μ)~-(t). In this paper, we show that their conjecture holds. We also discuss the multi-variable version, namely, r-variable Kostka functions K_(λ,μ)~±(t_1,…,t_r).     

Bose–Einstein distribution as a problem of analytic number theory: The case of less than two degrees of freedom

The problem of finding the number and the most likely shape of solutions of the system \(\sum\nolimits_{j = 0}^\infty {{\lambda _j}{n_j}} \leqslant M,\;\sum\nolimits_{j = 1}^\infty {{n_j}} = N\), where λ_j,M,N > 0 and N is an integer, as M,N →∞, can naturally be interpreted as a problem of analytic number theory. We solve this problem for the case in which the counting function of λ_j is of the order of λ^d/2, where d, the number of degrees of freedom, is less than two.     

Extrapolation in Grand Lebesgue Spaces with <Emphasis Type="Italic">A</Emphasis>∞ Weights

Results on extrapolation withA∞ weights in grand Lebesgue spaces are obtained. Generally, these spaces are defined with respect to the productmeasure μ₁ ×· · ·×μ_n onX₁ ×· · ·×X_n, where (X_i, d_i, μ_i), i = 1,..., n, are spaces of homogeneous type. As applications of the obtained results, new one-weight estimates with A_∞ weights for operators of harmonic analysis are derived.     

Acceleration of convergence of general linear sequences by the Shanks transformation

The Shanks transformation is a powerful nonlinear extrapolation method that is used to accelerate the convergence of slowly converging, and even diverging, sequences {A _n }. It generates a two-dimensional array of approximations \({A^{(j)}_n}\) to the limit or anti-limit of {A _n } defined as solutions of the linear systems

$A_l=A^{(j)}_n +\sum^{n}_{k=1}\bar{\beta}_k(\Delta A_{l+k-1}),\ \ j\leq l\leq j+n,$

where \({\bar{\beta}_{k}}\) are additional unknowns. In this work, we study the convergence and stability properties of \({A^{(j)}_n}\) , as j → ∞ with n fixed, derived from general linear sequences {A _n }, where \({{A_n \sim A+\sum^{m}_{k=1}\zeta_k^n\sum^\infty_{i=0} \beta_{ki}n^{\gamma_k-i}}}\) as n → ∞, where ζ _k  ≠ 1 are distinct and |ζ ₁| = ... = |ζ _m | = θ, and γ _k  ≠ 0, 1, 2, . . .. Here A is the limit or the anti-limit of {A _n }. Such sequences arise, for example, as partial sums of Fourier series of functions that have finite jump discontinuities and/or algebraic branch singularities. We show that definitive results are obtained with those values of n for which the integer programming problems

$\begin{array}{ll}{\quad\quad\quad\quad\max\limits_{s_1,\ldots,s_m}\sum\limits_{k=1}^{m}\left[(\Re\gamma_k)s_k-s_k(s_k-1)\right],}\\ {{\rm subject\,to}\,\, s_1\geq0,\ldots,s_m\geq0\quad{\rm and}\quad \sum\limits_{k=1}^{m} s_k = n,}\end{array}$

have unique (integer) solutions for s ₁, . . . , s _m . A special case of our convergence result concerns the situation in which \({{\Re\gamma_1=\cdots=\Re\gamma_m=\alpha}}\) and n = mν with ν = 1, 2, . . . , for which the integer programming problems above have unique solutions, and it reads \({A^{(j)}_n-A=O(\theta^j\,j^{\alpha-2\nu})}\) as j → ∞. When compared with A _j ? A = O(θ ^j j ^α ) as j → ∞, this result shows that the Shanks transformation is a true convergence acceleration method for the sequences considered. In addition, we show that it is stable for the case being studied, and we also quantify its stability properties. The results of this work are the first ones pertaining to the Shanks transformation on general linear sequences with m > 1.

     

Critical exponent for a quasilinear parabolic equation with inhomogeneous density in a cone

In this paper, we study the initial-boundary value problem of porous medium equation ρ(x)u _t  = Δu ^m  + V(x)h(t)u ^p in a cone D = (0, ∞) × Ω, where \({V(x)\,{\sim}\, |x|^\sigma, h(t)\,{\sim}\, t^s}\). Let ω ₁ denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on Ω and let l denote the positive root of l ² + (n ? 2)l = ω ₁. We prove that if \({m < p \leq 1+(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}\), then the problem has no global nonnegative solutions for any nonnegative u ₀ unless u ₀ = 0; if \({p >1 +(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}\), then the problem has global solutions for some u ₀ ≥ 0.     

Block partitions of sequences

Given a sequence A = (a ₁, …, a _n ) of real numbers, a block B of A is either a set B = {a _i , a _i+1, …, a _j } where i ≤ j or the empty set. The size b of a block B is the sum of its elements. We show that when each a _i ∈ [0, 1] and k is a positive integer, there is a partition of A into k blocks B ₁, …, B _k with |b _i ?b _j | ≤ 1 for every i, j. We extend this result in several directions.