Convexoid and generalized derivations

Let E,F be two Banach spaces and let S be a symmetric norm ideal of L(E,F). For AL(F) and BL(E) the generalized derivation δ_S,A,B is the operator on S that sends X to AX−XB. A bounded linear operator is said to be convexoid if its (algebraic) numerical range coincides with the convex hull of its spectrum. We show that δ_S,A,B is convexoid if and only if A and B are convexoid.     

Flawless 0-sequences and Hilbert functions of Cohen-Macaulay integral domains

Let A = A₀A₁ be a commutative graded ring such that (i) A₀ = k a field, (ii) A = k[A₁] and (iii) dim_k A₁ < ∞. It is well known that the formal power series ∑^∞_{n = 0} (dim_kA_n)λ ⁿ is of the form (h₀ + h₁λ + + h_sλ^s)/(1 − λ)^dimA with each h_iε . We are interested in the sequence (h₀, h₁,…,h_s), called the h-vector of A, when A is a Cohen–Macaulay integral domain. In this paper, after summarizing fundamental results (Section 1), we study h-vectors of certain Gorenstein domains (Section 2) and find some examples of h-vectors arising from integrally closed level domains (Sections 3 and 4).     

On generalized Boolean functions II

This cycle of papers is based on the concept of generalized Bolean functions introduced by the author in the first article of the series. Every generalized Boolean function f:Bⁿ→B can be written in a manner similar to the canonical disjunctive form using some function defined on A×B, where A is a finite subset of B containing 0 and 1. The set of those functions f is denoted by GBF_n[A]. In this paper the following questions are presented: (1) What is the relationship between GBF_n[A₁] and GBF_n[A₂] when A₁A₂. (2) What can be said about GBF_n[A₁∩A₂] and GBF_n[A₁A₂] in comparison with GBF_n[A₁]∩GBF_n[A₂] and GBF_n[A₁]GBF_n[A₂], respectively.     

Efficient robust estimation of parameter in the random censorship model

For an open set Θ of ^k, let \s{P_θ: θ Θ\s} be a parametric family of probabilities modeling the distribution of i.i.d. random variables X₁,…, X_n. Suppose X_i's are subject to right censoring and one is only able to observe the pairs (min(X_i, Y_i), [X_i Y_i]), i = 1,…, n, where [A] denotes the indicator function of the event A, Y₁,…, Y_n are independent of X₁,…, X_n and i.i.d. with unknown distribution Q₀. This paper investigates estimation of the value θ that gives a fitted member of the parametric family when the distributions of X₁ and Y₁ are subject to contamination. The constructed estimators are adaptive under the semi-parametric model and robust against small contaminations: they achieve a lower bound for the local asymptotic minimax risk over Hellinger neighborhoods, in the Hájel—Le Cam sense. The work relies on Beran (1981). The construction employs some results on product-limit estimators.     

Persistent convergence on randomly deleted sets

At time t_k, a unit with magnitude X_k and lifetime L_k enters a system. Let λ be a real valued function on the finite real sequences. One such sequence, B*_t, consists of the X_k's for which t_k t < t_k + L_k. When λ(X₁,…, X_n) converges (in some sense) to φ, we find conditions under which λ(B*_t) converges or fails to converge to φ in the same sense.     

Summation theorems for multidimensional basic hypergeometric series by determinant evaluations

We derive summation formulas for a specific kind of multidimensional basic hypergeometric series associated to root systems of classical type. We proceed by combining the classical (one-dimensional) summation formulas with certain determinant evaluations. Our theorems include A_r extensions of Ramanujan's bilateral ₁ψ₁ sum, C_r extensions of Bailey's very-well-poised ₆ψ₆ summation, and a C_r extension of Jackson's very-well-poised ₈φ₇ summation formula. We also derive multidimensional extensions, associated to the classical root systems of type A_r, B_r, C_r, and D_r, respectively, of Chu's bilateral transformation formula for basic hypergeometric series of Gasper–Karlsson–Minton type. Limiting cases of our various series identities include multidimensional generalizations of many of the most important summation theorems of the classical theory of basic hypergeometric series.     

A note on geometric ergodicity of autoregressive conditional heteroscedasticity (ARCH) model

For the pth-order linear ARCH model,

, where ₀ > 0, _i 0, I = 1, 2, …, p, {_t} is an i.i.d. normal white noise with E_t = 0, E_t² = 1, and _t is independent of {X_s, s < t}, Engle (1982) obtained the necessary and sufficient condition for the second-order stationarity, that is, ₁ + ₂ + ··· + _p < 1. In this note, we assume that _t has the probability density function p(t) which is positive and lower-semicontinuous over the real line, but not necessarily Gaussian, then the geometric ergodicity of the ARCH(p) process is proved under E_t² = 1. When _t has only the first-order absolute moment, a sufficient condition for the geometric ergodicity is also given.     

Descents and one-dimensional characters for classical Weyl groups

This paper examine all sums of the form

where W is a classical Weyl group, X is a one-dimensional character of W, and d(π) is the descent statistic. This completes a picture which is known when W is the symmetric group S_n (the Weyl group A_n−1). Surprisingly, the answers turn out to be simpler and generalize further for the other classical Weyl groups B_n(C_n) and D_n. The B_n, case uses sign-reversing involutions, while the D_n case follows from a result of independent interest relating statistics for all three groups.     

On the Glivenko–Cantelli theorem for generalized empirical processes based on strong mixing sequences

Given \s{X_i, i 1\s} as non-stationary strong mixing (n.s.s.m.) sequence of random variables (r.v.'s) let, for 1 i n and some γ ε [0, 1],

F₁(x)=γP(X_i<x)+(1-γ)P(X_ix)

and

I_i(x)=γI(X_i<x)+(1-γ)I(X_ix)

. For any real sequence \s{C_i\s} satisfying certain conditions, let

.

In this paper an exponential type of bound for P(D_n ), for any >0, and a rate for the almost sure convergence of D_n are obtained under strong mixing. These results generalize those of Singh (1975) for the independent and non-identically distributed sequence of r.v.'s to the case of strong mixing.     

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.     

The kernels of irreducible representations of the general linear group

A derivation for the kernel of the irreducible representation T_(λ) of the general linear group GL_n(C) is given. This is then applied to the problem of determining necessary and sufficient conditions under which T_(λ)(A) = T_(λ)(B), where A and B are linear transformations, not necessarily invertible. Finally, conditions are obtained under which normality of T_(λ)(A) implies normality of A.     

A matrix problem associated with computing Cohen-Macaulay type

Suppose A∈M_n×m(F), B∈M_n×t(F) for some field F. Define Г(AB) to be the set of n×n diagonal matrices D such that the column space of DA is contained in the column space of B. In this paper we determine dim Г(AB). For matrices AB of the same rank we provide an algorithm for computing dim Г(AB).     

More limit theory for the sample correlation function of moving averages

Let X_t = Σ^∞_j=-∞ c_jZ_{t - j} be a moving average process where {Z_t} is iid with common distribution in the domain of attraction of a stable law with index , 0 < < 2. If 0 < < 2, E|Z₁| < ∞ and the distribution of |Z₁|and |Z₁Z₂| are tail equivalent then the sample correlation function of {X₁} suitably normalized converges in distribution to the ratio of two dependent stable random variables with indices and /2. This is in sharp contrast to the case E|Z₁| ^{= ∞} where the limit distribution is that of the ratio of two independent stable variables. Proofs rely heavily on point process techniques. We also consider the case when the sample correlations are asymptotically normal and extend slightly the classical result.     

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.     

Sperner''s theorem with constraints

A set X of subsets of an n-element set S is called an anti-chain if no two elements of X are related by set-wise inclusion. Sperner showed [8] that max |X|=(ⁿ_[n/2]), where |X| denotes the number of elements in X and the maximum is taken over all anti-chains of subsets of S.

Let non-negative integers i_o<n and m_io≠0, m_io+1,…m_n be given. In this paper we give an algorithm for calculating max |X| where the maximum is taken only over anti-chains containing exactly m_i i-element subsets of S for i_o i n.     

Flat portions on the boundary of the numerical ranges of certain Toeplitz matrices

We give criterions for a flat portion to exist on the boundary of the numerical range of a matrix. A special type of Teoplitz matrices with flat portions on the boundary of its numerical range are constructed. We show that there exist 2 × 2 nilpotent matrices A₁,A₂, an n  × n nilpotent Toeplitz matrix N_n, and an n  × n cyclic permutation matrix S_n(s) such that the numbers of flat portions on the boundaries of W(A₁⊕N_n) and W(A₂⊕S_n(s)) are, respectively, 2(n - 2) and 2n.     

Stable limits of empirical processes of moving averages with infinite variance

The paper obtains a functional limit theorem for the empirical process of a stationary moving average process X_t with i.i.d. innovations belonging to the domain of attraction of a symmetric -stable law, 1<<2, with weights b_j decaying as j^−β, 1<β<2/. We show that the empirical process (normalized by N^1/β) weakly converges, as the sample size N increases, to the process c_x⁺L⁺+c_x⁻L⁻, where L⁺,L⁻ are independent totally skewed β-stable random variables, and c_x⁺,c_x⁻ are some deterministic functions. We also show that, for any bounded function H, the weak limit of suitably normalized partial sums of H(X_s) is an β-stable Lévy process with independent increments. This limiting behavior is quite different from the behavior of the corresponding empirical processes in the parameter regions 1/<β<1 and 2/<β studied in Koul and Surgailis (Stochastic Process. Appl. 91 (2001) 309) and Hsing (Ann. Probab. 27 (1999) 1579), respectively.     

Compatibilité entre structures d''intervalles et relations d''orde

Compatibility between interval structures and partial orderings.

If H=(X,E) is a hypergraph, n the cardinality of X,I_n the ordered set {1..n} and < an order relation on X, we call F(X,<) the set of the one-to-one functions from X to I_n which are compatible with <. If AI_n we denote by (A) the length of the smallest interval of I_n which contains A.

We first deal with the following problem: Find ƒF(X,<) which minimise . The a_e, eR are positive coefficients.

This problem can be understood as a scheduling problem and is checked to be NP-complete. We learn how to recognize in polynomial time those hypergraphs H=(X,E) which induce an optimal value of z min equal to .

Next we work on a dual question which arises about interval graphs, when some partial orderings on the vertex set of these graphs intend to represent inclusion, overlapping or anteriority relations between closed intervals of the real line.     

A formula for the number of Latin squares

We establish an explicit formula for the number of Latin squares of order n:

, where B_n is the set of n×n(0,1) matrices, σ₀(A is the number of zero elements of the matrix A and per A is the permanent of the matrix A.