On Excess-time Correlated Cumulative Processes

In this paper a class of correlated cumulative processes, B _s (t) = ∑^N(t)_i=1 H _s (X _i )X _i , is studied with excess level increments X _i ?s, where {N(t), t ?0} is the counting process generated by the renewal sequence T _n , T _n and X _n are correlated for given n, H _s (t) is the Heaviside function and s?0 is a given constant. Several useful results, for the distributions of B _s (t), and that of the number of excess (non-excess) increments on (0, t) and the corresponding means, are derived. First passage time problems are also discussed and various asymptotic properties of the processes are obtained. Transform results, by applying a flexible form for the joint distribution of correlated pairs (T _n , X _n ) are derived and inverted. The case of non-excess level increments, X _i < s, is also considered. Finally, applications to known stochastic shock and pro-rata warranty models are given.     

Lower bounds for the largest eigenvalue of the gcd matrix on {1, 2,..., n}

Consider the n×n matrix with (i, j)’th entry gcd (i, j). Its largest eigenvalue λ_n and sum of entries s_n satisfy λ_n > s_n/n. Because s_n cannot be expressed algebraically as a function of n, we underestimate it in several ways. In examples, we compare the bounds so obtained with one another and with a bound from S.Hong, R.Loewy (2004). We also conjecture that λ_n > 6π^?2nlogn for all n. If n is large enough, this follows from F.Balatoni (1969).     

Diversity vectors of balls in graphs and estimates of the components of the vectors

The diversity vectors of balls are considered (the ith component of a vector of this kind is equal to the number of different balls of radius i) for the usual connected graphs and the properties of the components of the vectors are studied. The sharp upper and lower estimates are obtained for the number of different balls of a given radius in the n-vertex graphs (trees) and n-vertex trees (graphs with n ? 2d) of diameter d. It is shown that the estimates are precise in every graph regardless of the radius of balls. It is proven a necessary and sufficient condition is given for the existence of an n-vertex graph of diameter d with local (complete) diversity of balls.     

Hausdorff dimension of the maximal run-length in dyadic expansion

For any x ∈ [0, 1), let x = [? ₁, ? ₂, …,] be its dyadic expansion. Call r _n (x):= max{j ? 1: ? _i+1 = … = ? _i+j = 1, 0 ? i ? n ? j} the n-th maximal run-length function of x. P.Erdös and A.Rényi showed that \(\mathop {\lim }\limits_{n \to \infty } \) r _n (x)/log₂ n = 1 almost surely. This paper is concentrated on the points violating the above law. The size of sets of points, whose runlength function assumes on other possible asymptotic behaviors than log₂ n, is quantified by their Hausdorff dimension.     

On necessary and sufficient conditions for the self-normalized central limit theorem

Let X₁, X₂, … be a sequence of independent random variables and S_n = Σ _i=1 ⁿ X_i and V _n ² = Σ _i=1 ⁿ X _i ² . When the elements of the sequence are i.i.d., it is known that the self-normalized sum S_n=V_n converges to a standard normal distribution if and only if max_1?i?n|X_i|/V_n→0 in probability and the mean of X₁ is zero. In this paper, sufficient conditions for the self-normalized central limit theorem are obtained for general independent random variables. It is also shown that if max_1?i?n|X_i|/V_n→0 in probability, then these sufficient conditions are necessary.     

A Polynomial Time Algorithm for the Resource Allocation Problem with a Convex Objective Function

Consider the resource allocation problem:minimize ∑ⁿ_i=1 f_i(x_i) subject to ∑ⁿ_i=1 x_i = N and x_i's being nonnegative integers, where each f_i is a convex function. The well-known algorithm based on the incremental method requires O(N log n + n) time to solve this problem. We propose here a new algorithm based on the Lagrange multiplier method, requiring O[n²(log N)²] time. The latter is faster if N is much larger than n. Such a situation occurs, for example, when the optimal sample size problem related to monitoring the urban air pollution is treated.     

Dn,r is not potentially nilpotent for n ⩾ 4r − 2

An n × n sign pattern A is said to be potentially nilpotent if there exists a nilpotent real matrix B with the same sign pattern as A. Let D_n,r be an n × n sign pattern with 2 ≤ r ≤ n such that the superdiagonal and the (n, n) entries are positive, the (i, 1) (i = 1,..., r) and (i, i ? r + 1) (i = r + 1,..., n) entries are negative, and zeros elsewhere. We prove that for r ≥ 3 and n ≥ 4r ? 2, the sign pattern D_n,r is not potentially nilpotent, and so not spectrally arbitrary.     

On a recursive inverse eigenvalue problem

Let s ₁, ..., s _n be arbitrary complex scalars. It is required to construct an n × n normal matrix A such that s _i is an eigenvalue of the leading principal submatrix A _i , i = 1, 2, ..., n. It is shown that, along with the obvious diagonal solution diag(s ₁, ..., s _n ), this problem always admits a much more interesting nondiagonal solution A. As a rule, this solution is a dense matrix; with the diagonal solution, it shares the property that each submatrix A _i is itself a normal matrix, which implies interesting connections between the spectra of the neighboring submatrices A _i and A _{i + 1}.     

Second-order Asymptotics on Distributions of Maxima of Bivariate Elliptical Arrays

Let {(ξni, ηni), 1 ≤ i ≤ n, n ≥ 1} be a triangular array of independent bivariate elliptical random vectors with the same distribution function as(S_1, ρ_n S_1 +(1-ρ_n~2S_2)~(1/2)), ρn∈(0, 1), where(S1, S2) is a bivariate spherical random vector. For the distribution function of radius (S_1~2+ S_2~2)~(1/2) belonging to the max-domain of attraction of the Weibull distribution, the limiting distribution of maximum of this triangular array is known as the convergence rate of ρn to 1 is given. In this paper,under the refinement of the rate of convergence of ρn to 1 and the second-order regular variation of the distributional tail of radius, precise second-order distributional expansions of the normalized maxima of bivariate elliptical triangular arrays are established.     

Packing circles in the smallest circle: an adaptive hybrid algorithm

The circular packing problem (CPP) consists of packing n circles C _i of known radii r _i , i∈N={1,?…,?n}, into the smallest containing circle ?. The objective is to determine the coordinates (x _i ,?y _i ) of the centre of C _i , i∈N, as well as the radius r and centre (x,?y) of ?. CPP, which is a variant of the two-dimensional open-dimension problem, is NP hard. This paper presents an adaptive algorithm that incorporates nested partitioning within a tabu search and applies some diversification strategies to obtain a (near) global optimum. The tabu search is to identify the n circles’ ordering, whereas the nested partitioning is to determine the n circles’ positions that yield the smallest r. The computational results show the efficiency of the proposed algorithm.     

New algorithm for computing the Hermite interpolation polynomial

Let x ₀, x ₁,? , x _n , be a set of n + 1 distinct real numbers (i.e., x _i ≠ x _j , for i ≠ j) and y _{i, k} , for i = 0,1,? , n, and k = 0 ,1 ,? , n _i , with n _i ≥ 1, be given of real numbers, we know that there exists a unique polynomial p _{N ? 1}(x) of degree N ? 1 where \(N={\sum }_{i=0}^{n}(n_{i}+1)\), such that \(p_{N-1}^{(k)}(x_{i})=y_{i,k}\), for i = 0,1,? , n and k = 0,1,? , n _i . P _N?1(x) is the Hermite interpolation polynomial for the set {(x _i , y _{i, k} ), i = 0,1,? , n, k = 0,1,? , n _i }. The polynomial p _N?1(x) can be computed by using the Lagrange polynomials. This paper presents a new method for computing Hermite interpolation polynomials, for a particular case n _i = 1. We will reformulate the Hermite interpolation polynomial problem and give a new algorithm for giving the solution of this problem, the Matrix Recursive Polynomial Interpolation Algorithm (MRPIA). Some properties of this algorithm will be studied and some examples will also be given.     

Graded Lie algebras with few nontrivial components

We prove that if a (?/n?)-graded Lie algebra L = ? _i=0 ^n?1 L _i has d nontrivial components L _i and the null component L ₀ has finite dimension m, then L has a homogeneous solvable ideal of derived length bounded by a function of d and of codimension bounded by a function of m and d. An analogous result holds also for the (?/n?)-graded Lie rings L = ? _i=0 ^n?1 with few nontrivial components L _i if the null component L ₀ has finite order m. These results generalize Kreknin’s theorem on the solvability of the (?/n?)-graded Lie rings L = ? _i=0 ^n?1 L _i with trivial component L ₀ and Shalev’s theorem on the solvability of such Lie rings with few nontrivial components L _i . The proof is based on the method of generalized centralizers which was created by E. I. Khukhro for Lie rings and nilpotent groups with almost regular automorphisms of prime order [1], as well as on the technique developed in the work of N. Yu. Makarenko and E. I. Khukhro on the almost solvability of Lie algebras with an almost regular automorphism of finite order [2].     

Properties of Axial Diameters of a Simplex

Let S be a nondegenerate simplex in ? ⁿ . It is proved that the minimal possible σ>0, such that a homothetic copy of S of ratio σ contains [0,1] ⁿ , is equal to \(\sum_{i=1}^{n} 1/d_{i}(S)\). Here d _i (S) denotes the length of a longest segment in S parallel to the ith coordinate axis.     

Systems that generate solutions with a small period

Let (j₁,..., j_n) be a permutation of the n-tuple (1, ..., n). A system of differential equations \(\dot x = {f_i}\left( {{x_{{j_i}}}} \right),i = 1, \ldots ,n\) in which each function f_i is continuous on ? is considered. This system is said to have the property of generation of solutions with a small period if, for any number M > 0, there exists a number ω₀ = ω₀(M) > 0 such that if 0 < ω ≤ ω₀ and h_i(t, x₁, ..., x_n) are continuous functions on ? × ?ⁿ ω-periodic in t that satisfy the inequalities |h_i| ≤ M the system \(\dot x = {f_i}\left( {{x_{{j_i}}}} \right),i = 1, \ldots ,n\) has an ω-periodic solution. It is shown that a system has the property of generation of solutions with a small period if and only if f_i(?) = ? for i = 1,..., n. It is also shown that the smallness condition on the period is essential.     

Optimal selection based on relative rank (the “secretary problem”)

n rankable persons appear sequentially in random order. At theith stage we observe the relative ranks of the firsti persons to appear, and must either select theith person, in which case the process stops, or pass on to the next stage. For that stopping rule which minimizes the expectation of the absolute rank of the person selected, it is shown that asn → ∞ this tends to the value

$$\prod\limits_{j = 1}^\infty {(\tfrac{{j + 2}}{j})^{1/j + 1} } \cong 3.8695$$     

Tensor convolutions and Hankel tensors

Let A be an mth order n-dimensional tensor, where m, n are some positive integers and N:= m(n?1). Then A is called a Hankel tensor associated with a vector v ∈ ?^N+1 if A_σ = v _k for each k = 0, 1,...,N whenever σ = (i₁,..., i_m) satisfies i₁ +· · ·+i_m = m+k. We introduce the elementary Hankel tensors which are some special Hankel tensors, and present all the eigenvalues of the elementary Hankel tensors for k = 0, 1, 2. We also show that a convolution can be expressed as the product of some third-order elementary Hankel tensors, and a Hankel tensor can be decomposed as a convolution of two Vandermonde matrices following the definition of the convolution of tensors. Finally, we use the properties of the convolution to characterize Hankel tensors and (0,1) Hankel tensors.     

Sum-free sets in abelian groups

An IP system is a functionn taking finite subsets ofN to a commutative, additive group Ω satisfyingn(α∪β)=n(α)+n(β) whenever α∩β=ø. In an extension of their Szemerédi theorem for finitely many commuting measure preserving transformations, Furstenberg and Katznelson showed that ifS _i ,1≤i≤k, are IP systems into a commutative (possibly infinitely generated) group Ω of measure preserving transformations of a probability space (X, B, μ, andA∈B with μ(A)>0, then for some ø≠α one has μ(? _i=1 ^k S _i({α})A>0). We extend this to so-called FVIP systems, which are polynomial analogs of IP systems, thereby generalizing as well joint work by the author and V. Bergelson concerning special FVIP systems of the formS(α)=T(p(n(α))), wherep:Z ^t →Z ^d is a polynomial vanishing at zero,T is a measure preservingZ ^d action andn is an IP system intoZ ^t . The primary novelty here is potential infinite generation of the underlying group action, however there are new applications inZ ^d as well, for example multiple recurrence along a wide class ofgeneralized polynomials (very roughly, functions built out of regular polynomials by iterated use of the greatest integer function).     

Dynamic bivariate normal copula

Normal copula with a correlation coefficient between-1 and 1 is tail independent and so it severely underestimates extreme probabilities. By letting the correlation coefficient in a normal copula depend on the sample size, H¨usler and Reiss(1989) showed that the tail can become asymptotically dependent. We extend this result by deriving the limit of the normalized maximum of n independent observations, where the i-th observation follows from a normal copula with its correlation coefficient being either a parametric or a nonparametric function of i/n. Furthermore, both parametric and nonparametric inference for this unknown function are studied, which can be employed to test the condition by H¨usler and Reiss(1989). A simulation study and real data analysis are presented too.     

Precise large deviations for sums of random vectors with dependent components of consistently varying tails

Let {X _i = (X _1,i ,...,X _m,i )^?, i ≥ 1} be a sequence of independent and identically distributed nonnegative m-dimensional random vectors. The univariate marginal distributions of these vectors have consistently varying tails and finite means. Here, the components of X ₁ are allowed to be generally dependent. Moreover, let N(·) be a nonnegative integer-valued process, independent of the sequence {X _i , i ≥ 1}. Under several mild assumptions, precise large deviations for S _n = Σ _i=1 ⁿ X _i and S _N(t) = Σ _i=1 ^N(t) X _i are investigated. Meanwhile, some simulation examples are also given to illustrate the results.