A sufficient and necessary condition for an OWA bag mapping having the self-identity

A mapping ƒ : _n=1^∞Iⁿ → I is called a bag mapping having the self-identity if for every (x₁,…,x_n) ε _i=1^∞Iⁿ we have (1) ƒ(x₁,…,x_n) = ƒ(x_i1,…,x_in) for any arrangement (i₁,…,i_n) of {1,…,n}; monotonic; (3) ƒ(x₁,…,x_n, ƒ(x₁,…,x_n)) = ƒ(x₁,…,x_n). Let {ω_i,n : I = 1,…,n;n = 1,2,…} be a family of non-negative real numbers satisfying Σ_i=1ⁿω_i,n = 1 for every n. Then one calls the mapping ƒ : _i=1^∞Iⁿ → I defined as follows an OWA bag mapping: for every (x₁,…,x_n) ε _i=1^∞Iⁿ, ƒ(x₁,…,x_n) = Σ_i=1ⁿω_i,ny_i, where y_i is the it largest element in the set {x₁,…,x_n}. In this paper, we give a sufficient and necessary condition for an OWA bag mapping having the self-identity.     

A note on linearization of actions of finitely semisimple Hopf algebras on local algebras

Let H be a Hopf algebra over a field k and let H A → A, h a → h.a, be an action of H on a commutative local Noetherian kalgebra (A, m). We say that this action is linearizable if there exists a minimal system x₁, …, x_n of generators of the maximal ideal m such that h.x_i ε kx₁ + …+ kx_n for all h ε H and i = 1, …, n. In the paper we prove that the actions from a certain class are linearizable (see Theorem 4), and we indicate some consequences of this fact.     

An efficient algorithm for the parametric resource allocation problem

The parametric resource allocation problem asks to minimize the sum of separable single-variable convex functions containing a parameter λ, Σ_{i = 1}ⁿ (ƒ_i(x_i + λg_i(x_i)), under simple constraints Σ_{i = 1}ⁿ x_i = M, l_i≤x_i≤u_i and x_i: nonnegative integers for i = 1, 2, …, n, where M is a given positive integer, and l_i and u_i are given lower and upper bounds on x_i. This paper presents an efficient algorithm for computing the sequence of all optimal solutions when λ is continuously changed from 0 to ∞. The required time is O(G√Mlog² n + n log n + n log(M/n)), where G = Σ_{i = 1}ⁿ u_i − Σ_{i = 1n} l_i and an evaluation of ƒ_i(·) or g_i(·) is assumed to be done in constant time.     

The space localization of unbounded boundary perturbations in nonlinear heat conduction with transfer

We consider the nonlinear parabolic equation u_t = (k(u)u_x)_x + b(u)_x, where u = u(x, t, x ε R¹, t > 0; k(u) ≥ 0, b(u) ≥ 0 are continuous functions as u ≥ 0, b (0) = 0; k, b > 0 as u > 0. At t = 0 nonnegative, continuous and bounded initial value is prescribed. The boundary condition u(0, t) = Ψ(t) is supposed to be unbounded as t → +∞. In this paper, sufficient conditions for space localization of unbounded boundary perturbations are found. For instance, we show that nonlinear equation u_t = (uⁿu_x)_x + (u^β)_x, n ≥ 0, β >; n + 1, exhibits the phenomenon of “inner boundedness,” for arbitrary unbounded boundary perturbations.     

Cycle decompositions of crowns

For an integer n3, the crown S_n⁰ is defined to be the graph with vertex set {x₀,x₁,…,x_n−1,y₀,y₁,…,y_n−1} and edge set {x_iy_j: 0i,jn−1, i≠j}. In this paper we give some sufficient conditions for the edge decomposition of the crown into isomorphic cycles.     

Full quadrature sums for pth powers of polynomials with Freud weights

In this paper, we provide a solution of the quadrature sum problem of R. Askey for a class of Freud weights. Let r> 0, b (− ∞, 2]. We establish a full quadrature sum estimate

1 p < ∞, for every polynomial P of degree at most n + rn^1/3, where W² is a Freud weight such as exp(−¦x¦), > 1, λ_jn are the Christoffel numbers, x_jn are the zeros of the orthonormal polynomials for the weight W², and C is independent of n and P. We also prove a generalisation, and that such an estimate is not possible for polynomials P of degree M = m(n) if m(n) = n + ξ_nn^1/3, where ξ_n → ∞ as n → ∞. Previous estimates could sum only over those x_jn with ¦x_jn¦ σx_1n, some fixed 0 < σ < 1.     

A parallel computing scheme for minimizing a class of large scale functions

This paper gives a parallel computing scheme for minimizing a twice continuously differentiable function with the form

where x = (x^T₁,…,x^T_m)^T and x_i R^n_i, ∑^m_{i = 1}n_i = n, and n a very big number. It is proved that we may use m parallel processors and an iterative procedure to find a minimizer of ƒ(x). The convergence and convergence rate are given under some conditions. The conditions for finding a global minimizer of ƒ(x by using this scheme are given, too. A similar scheme can also be used parallelly to solve a large scale system of nonlinear equations in the similar way. A more general case is also investigated.     

On the fourth power moment of Fourier coefficients of cusp form

Let a(n)be the Fourier coefficients of a holomorphic cusp form of weightκ=2n≥12 for the full modular group and A(x)=∑_(n≤x)a(n).In this paper,we establish an asymptotic formula of the fourth power moment of A(x)and prove that ∫T1A~4(x)dx=3/(64κπ~4)s_4;2()T~(2κ)+O(T~(2κ-δ_4+ε))with δ_4=1/8,which improves the previous result.     

The growth of laguerre matrix polynomials on bounded intervals

Let A be a matrix in ^r×r such that Re(z) > −1/2 for all the eigenvalues of A and let {π_n^(A,1/2) (x)} be the normalized sequence of Laguerre matrix polynomials associated with A. In this paper, it is proved that π_n^(A,1/2) (x) = O(n^(A)/2ln^r−1(n)) and π_n+1^(A,1/2) (x) − π_n^(A,1/2) (x) = O(n^((A)−1)/2ln^r−1(n)) uniformly on bounded intervals, where (A) = max{Re(z); z eigenvalue of A}.     

Equality of higher numerical ranges of matrices and a conjecture of Kippenhahn on Hermitian pencils

Let M_n be the algebra of all n × n complex matrices. For 1 k n, the kth numerical range of A M_n is defined by W_k(A) = (1/k)∑_j^k=1x_j^*Ax_j : x₁, …, x_k is an orthonormal set in ⁿ]. It is known that tr A/n = W_n(A) W_n−1(A) W₁(A). We study the condition on A under which W_m(A) = W_k(A) for some given 1 m < k n. It turns out that this study is closely related to a conjecture of Kippenhahn on Hermitian pencils. A new class of counterexamples to the conjecture is constructed, based on the theory of the numerical range.     

A Helly theorem for geodesic convexity in strongly dismantlable graphs

A (finite or infinite) graph G is strongly dismantlable if its vertices can be linearly ordered x₀,…, x so that, for each ordinal β < , there exists a strictly increasing finite sequence (i_j)_{0 j n} of ordinals such that i₀ = β, i_n = and x_ij+1 is adjacent with x_ij and with all neighbors of x_ij in the subgraph ofG induced by {x_y: β γ }. We show that the Helly number for the geodesic convexity of such a graph equals its clique number. This generalizes a result of Bandelt and Mulder (1990) for dismantlable graphs. We also get an analogous equality dealing with infinite families of convex sets.     

First passage time for some stationary processes

For a 1-dependent stationary sequence {X_n} we first show that if u satisfies p₁=p₁(u)=P(X₁>u)0.025 and n>3 is such that 88np₁³1, then

P{max(X₁,…,X_n)u}=ν·μⁿ+O{p₁³(88n(1+124np₁³)+561)}, n>3,

where

ν=1−p₂+2p₃−3p₄+p₁²+6p₂²−6p₁p₂,μ=(1+p₁−p₂+p₃−p₄+2p₁²+3p₂²−5p₁p₂)⁻¹

with

p_k=p_k(u)=P{min(X₁,…,X_k)>u}, k1

and

|O(x)||x|.

From this result we deduce, for a stationary T-dependent process with a.s. continuous path {Y_s}, a similar, in terms of P{max_0skTY_s<u}, k=1,2 formula for P{max_0stY_su}, t>3T and apply this formula to the process Y_s=W(s+1)−W(s), s0, where {W(s)} is the Wiener process. We then obtain numerical estimations of the above probabilities.     

On application of the Lanczos method to solution of some partial differential equations

Let A be a square symmetric n × n matrix, φ be a vector from ⁿ, and f be a function defined on the spectral interval of A. The problem of computation of the vector u = f(A)φ arises very often in mathematical physics.

We propose the following method to compute u. First, perform m steps of the Lanczos method with A and φ. Define the spectral Lanczos decomposition method (SLDM) solution as u_m = φ Qf(H)e₁, where Q is the n × m matrix of the m Lanczos vectors and H is the m × m tridiagonal symmetric matrix of the Lanczos method. We obtain estimates for u − u_m that are stable in the presence of computer round-off errors when using the simple Lanczos method.

We concentrate on computation of exp(− tA)φ, when A is nonnegative definite. Error estimates for this special case show superconvergence of the SLDM solution. Sample computational results are given for the two-dimensional equation of heat conduction. These results show that computational costs are reduced by a factor between 3 and 90 compared to the most efficient explicit time-stepping schemes. Finally, we consider application of SLDM to hyperbolic and elliptic equations.     

Intersections of nest algebras in finite dimensions

If are maximal nests on a finite-dimensional Hilbert space H, the dimension of the intersection of the corresponding nest algebras is at least dim H. On the other hand, there are three maximal nests whose nest algebras intersect in the scalar operators. The dimension of the intersection of two nest algebras (corresponding to maximal nests) can be of any integer value from n to n(n+1)/2, where n=dim H. For any two maximal nests there exists a basis {f₁,f₂,…,f_n} of H and a permutation π such that and where M_i=  span{f₁,f₂,…,f_i} and N_i= span{f_π(1),f_π(2),…,f_π(i)}. The intersection of the corresponding nest algebras has minimum dimension, namely dim H, precisely when π(j)=n−j+1,1jn. Those algebras which are upper-triangular matrix incidence algebras, relative to some basis, can be characterised as intersections of certain nest algebras.     

A characterization on n-critical economical generalized tic-tac-toe games

There is a so called generalized tic-tac-toe game playing on a finite set X with winning sets A₁, A₂,…, A_m. Two players, F and S, take in turn a previous untaken vertex of X, with F going first. The one who takes all the vertices of some winning set first wins the game. Erd s and Selfridge proved that if |A₁|=|A₂|==|A_m|=n and m<2ⁿ⁻¹, then the game is a draw. This result is best possible in the sense that once m=2ⁿ⁻¹, then there is a family A₁, A₂,…, A_m so that F can win. In this paper we characterize all those sets A₁,…, A_2n−1 so that F can win in exactly n moves. We also get similar result in the biased games.     

Resolutions of Stanley-Reisner rings and Alexander duality

Associated to any simplicial complex Δ on n vertices is a square-free monomial ideal I_Δ in the polynomial ring A = k[x₁, …, x_n], and its quotient k[Δ] = A/I_Δ known as the Stanley-Reisner ring. This note considers a simplicial complex Δ^* which is in a sense a canonical Alexander dual to Δ, previously considered in [1, 5]. Using Alexander duality and a result of Hochster computing the Betti numbers dim_k Tor_i^A (k[Δ],k), it is shown (Proposition 1) that these Betti numbers are computable from the homology of links of faces in Δ^*. As corollaries, we prove that I_Δ has a linear resolution as A-module if and only if Δ^* is Cohen-Macaulay over k, and show how to compute the Betti numbers dim_k Tor_i^A (k[Δ],k) in some cases where Δ^* is wellbehaved (shellable, Cohen-Macaulay, or Buchsbaum). Some other applications of the notion of shellability are also discussed.     

Boundedness and asymptotic behavior of the solutions of a fuzzy difference equation

In this paper we study the existence, the uniqueness, the boundedness and the asymptotic behavior of the positive solutions of the fuzzy difference equation x_n+1=∑_i=0^kA_i/x_n−i^p_i, where k{1,2,…,}, A_i, i{0,1,…,k}, are positive fuzzy numbers, p_i, i{0,1,…,k}, are positive constants and x_i, i{−k,−k+1,…,0}, are positive fuzzy numbers.     

An extended result of Kleitman and Saks concerning binary trees

In a recent paper, D.J. Kleitman and M.E. Saks gave a proof of Huang's conjecture on alphabetic binary trees.

Given a set E = {e_i}, I = 0, 1, 2, …, m and assigned positive weights to its elements and supposing the elements are indexed such that w(e₀) ≤ w(e₁) ≤ … ≤w (e_m), where w(e_i) is the weight of e_i, we call the following sequence E^* a ‘saw-tooth’ sequence

E^*=(e₀,e_m,e₁,…,e_j,e_m−j,…).

Huang's conjecture is: E^* is the most expensive sequence for alphabetic binary trees. This paper shows that this property is true for the L-restricted alphabetic binary trees, where L is the maximum length of the leaves and log₂(m + 1) ≤L≤m.     

The connectivity of a graph on uniform points on [0,1]

A random graph G_n(x) is constructed on independent random points U₁,…,U_n distributed uniformly on [0,1]^d, d1, in which two distinct such points are joined by an edge if the l_∞-distance between them is at most some prescribed value 0<x<1. The connectivity distance c_n, the smallest x for which G_n(x) is connected, is shown to satisfy

(1)

For d2, the random graph G_n(x) behaves like a d-dimensional version of the random graphs of Erdös and Rényi, despite the fact that its edges are not independent: c_n/d_n→1, a.s., as n→∞, where d_n is the largest nearest-neighbor link, the smallest x for which G_n(x) has no isolated vertices.     

Pancyclic orderings of in-tournaments

An in-tournament is an oriented graph such that the negative neighborhood of every vertex induces a tournament. In this paper, pancyclic orderings of a strong in-tournament D are investigated. This is a labeling, say x₁,x₂,…,x_n, of the vertex set of D such that D[{x₁,x₂,…,x_t}] is Hamiltonian for t=3,4,…,n. Moreover, we consider the related problem on weak pancyclic orderings, where the same holds for t4 and x₁ belongs to an arbitrary oriented 3-cycle. We present sharp lower bounds for the minimum indegree ensuring the existence of a pancyclic or a weak pancyclic ordering in strong in-tournaments.