On the range of heterogeneous samples

Let R_n be the range of a random sample X₁,…,X_n of exponential random variables with hazard rate λ. Let S_n be the range of another collection Y₁,…,Y_n of mutually independent exponential random variables with hazard rates λ₁,…,λ_n whose average is λ. Finally, let r and s denote the reversed hazard rates of R_n and S_n, respectively. It is shown here that the mapping t?s(t)/r(t) is increasing on (0,∞) and that as a result, R_n=X_(n)−X₍₁₎ is smaller than S_n=Y_(n)−Y₍₁₎ in the likelihood ratio ordering as well as in the dispersive ordering. As a further consequence of this fact, X_(n) is seen to be more stochastically increasing in X₍₁₎ than Y_(n) is in Y₍₁₎. In other words, the pair (X₍₁₎,X_(n)) is more dependent than the pair (Y₍₁₎,Y_(n)) in the monotone regression dependence ordering. The latter finding extends readily to the more general context where X₁,…,X_n form a random sample from a continuous distribution while Y₁,…,Y_n are mutually independent lifetimes with proportional hazard rates.     

On the Bateman-Horn Conjecture

Let r be a positive integer and f₁,…,f_r be distinct polynomials in Z[X]. If f₁(n),…,f_r(n) are all prime for infinitely many n, then it is necessary that the polynomials f_i are irreducible in Z[X], have positive leading coefficients, and no prime p divides all values of the product f₁(n)···f_r(n), as n runs over Z. Assuming these necessary conditions, Bateman and Horn (Math. Comput.16 (1962), 363-367) proposed a conjectural asymptotic estimate on the number of positive integers n?x such that f₁(n),…,f_r(n) are all primes. In the present paper, we apply the Hardy-Littlewood circle method to study the Bateman-Horn conjecture when r?2. We consider the Bateman-Horn conjecture for the polynomials in any partition {f₁,…,f_s}, {f_s+1,…,f_r} with a linear change of variables. Our main result is as follows: If the Bateman-Horn conjecture on such a partition and change of variables holds true with some conjectural error terms, then the Bateman-Horn conjecture for f₁,…,f_r is equivalent to a plausible error term conjecture for the minor arcs in the circle method.     

On the dependence between the extreme order statistics in the proportional hazards model

Let X₁,…,X_n be a random sample from an absolutely continuous distribution with non-negative support, and let Y₁,…,Y_n be mutually independent lifetimes with proportional hazard rates. Let also X₍₁₎<?<X_(n) and Y₍₁₎<?<Y_(n) be their associated order statistics. It is shown that the pair (X₍₁₎,X_(n)) is then more dependent than the pair (Y₍₁₎,Y_(n)), in the sense of the right-tail increasing ordering of Avérous and Dortet-Bernadet [LTD and RTI dependence orderings, Canad. J. Statist. 28 (2000) 151-157]. Elementary consequences of this fact are highlighted.     

On characterization of multivariate stable distributions via random linear statistics

LetX ₁,X ₂, ...,X _n be independent and identically distributed random vectors inR ^d , and letY=(Y ₁,Y ₂, ...,Y _n )′ be a random coefficient vector inR ⁿ , independent ofX _j ^/′ . We characterize the multivariate stable distributions by considering the independence of the random linear statistic $$U = Y_1 X_1 + Y_2 X_2 + \cdot \cdot \cdot + Y_n X_n $$ and the random coefficient vectorY.     

Construction of nonnegative symmetric matrices with given spectrum

Let σ = (λ₁, … , λ_n) be the spectrum of a nonnegative symmetric matrix A with the Perron eigenvalue λ₁, a diagonal entry c and let τ = (μ₁, … , μ_m) be the spectrum of a nonnegative symmetric matrix B with the Perron eigenvalue μ₁. We show how to construct a nonnegative symmetric matrix C with the spectrum

(λ₁+max{0,μ₁-c},λ₂,…,λ_n,μ₂,…,μ_m).     

On questions related to saturated chains of primes in polynomial rings

We propose to give positive answers to the open questions: is R(X,Y) strong S when R(X) is strong S? is R stably strong S (resp., universally catenary) when R[X] is strong S (resp., catenary)? in case R is obtained by a (T,I,D) construction. The importance of these results is due to the fact that this type of ring is the principal source of counterexamples. Moreover, we give an answer to the open questions: is R〈X₁,…,X_n〉 residually Jaffard (resp., totally Jaffard) when R(X₁,…,X_n) is ? We construct a three-dimensional local ring R such that R(X₁,…,X_n) is totally Jaffard (and hence, residually Jaffard) whereas R〈X₁,…,X_n〉 is not residually Jaffard (and hence, not totally Jaffard).     

More on Stochastic Comparisons and Dependence among Concomitants of Order Statistics

For a sample of iid observations {(X_i, Y_i)} from an absolutely continuous distribution, the multivariate dependence of concomitants Y_[]=(Y_[1], Y_[2], …, Y_[n]) and the stochastic order of subsets of Y_[] are studied. If (X, Y) is totally positive dependent of order 2, Y_[] is multivariate totally positive dependent of order 2. If the conditional hazard rate function of Y given X, h_YX(yx), is decreasing in x for every y, Y_[] is multivariate right corner set increasing. And if Y is stochastically increasing in X, the concomitants are increasing in multivariate stochastic order.     

Reduced Weyl asymptotics for pseudodifferential operators on bounded domains II. The compact group case

Let G⊂O(n) be a compact group of isometries acting on n-dimensional Euclidean space Rn, and X a bounded domain in Rn which is transformed into itself under the action of G. Consider a symmetric, classical pseudodifferential operator A₀ in L²(Rn) that commutes with the regular representation of G, and assume that it is elliptic on X. We show that the spectrum of the Friedrichs extension A of the operator is discrete, and using the method of the stationary phase, we derive asymptotics for the number Nχ(λ) of eigenvalues of A equal or less than λ and with eigenfunctions in the χ-isotypic component of L²(X) as λ→∞, giving also an estimate for the remainder term for singular group actions. Since the considered critical set is a singular variety, we recur to partial desingularization in order to apply the stationary phase theorem.     

Asymptotic enumeration of dense 0-1 matrices with specified line sums

Let s=(s₁,s₂,…,sm) and t=(t₁,t₂,…,tn) be vectors of non-negative integers with . Let B(s,t) be the number of m×n matrices over {0,1} with jth row sum equal to sj for 1?j?m and kth column sum equal to tk for 1?k?n. Equivalently, B(s,t) is the number of bipartite graphs with m vertices in one part with degrees given by s, and n vertices in the other part with degrees given by t. Most research on the asymptotics of B(s,t) has focused on the sparse case, where the best result is that of Greenhill, McKay and Wang (2006). In the case of dense matrices, the only precise result is for the case of equal row sums and equal column sums (Canfield and McKay, 2005). This paper extends the analytic methods used by the latter paper to the case where the row and column sums can vary within certain limits. Interestingly, the result can be expressed by the same formula which holds in the sparse case.     

Spectra of principal submatrices of nonnegative matrices

Let A be an n×n nonnegative matrix with the spectrum (λ₁,λ₂,…,λ_n) and let A₁ be an m×m principal submatrix of A with the spectrum (μ₁,μ₂,…,μ_m). In this paper we present some cases where the realizability of (μ₁,μ₂,…,μ_m,ν₁,ν₂,…,ν_s) implies the realizability of (λ₁,λ₂,…,λ_n,ν₁,ν₂,…,ν_s) and consider the question whether this holds in general. In particular, we show that the list

(λ₁,λ₂,…,λ_n,-μ₁,-μ₂,…,-μ_m)     

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.     

Strong linear preservers of rank reverse permutability on triangular matrices

Let F be a field with ∣F∣ > 2 and T_n(F) be the set of all n × n upper triangular matrices, where n ? 2. Let k ? 2 be a given integer. A k-tuple of matrices A₁, …, A_k ∈ T_n(F) is called rank reverse permutable if rank(A₁ A₂ ? A_k) = rank(A_k A_k−1 ? A₁). We characterize the linear maps on T_n(F) that strongly preserve the set of rank reverse permutable matrix k-tuples.     

Generalized radix representations and dynamical systems. IV

For r = (r₁,…, r_d) ∈ ?^d the mapping τ_r:?^d →?^d given byτ_r(a₁,…,a_d) = (a₂, …, a_d,−⌊r₁a₁+…+ r_da_d⌋)where ⌊·⌋ denotes the floor function, is called a shift radix system if for each a ∈ ?^d there exists an integer k > 0 with τ_r^k(a) = 0. As shown in Part I of this series of papers, shift radix systems are intimately related to certain well-known notions of number systems like β-expansibns and canonical number systems. After characterization results on shift radix systems in Part II of this series of papers and the thorough investigation of the relations between shift radix systems and canonical number systems in Part III, the present part is devoted to further structural relationships between shift radix systems and β-expansions. In particular we establish the distribution of Pisot polynomials with and without the finiteness property (F).     

Schubert classes in the equivariant cohomology of the Lagrangian Grassmannian

Let LGn denote the Lagrangian Grassmannian parametrizing maximal isotropic (Lagrangian) subspaces of a fixed symplectic vector space of dimension 2n. For each strict partition λ=(λ₁,…,λk) with λ₁?n there is a Schubert variety X(λ). Let T denote a maximal torus of the symplectic group acting on LGn. Consider the T-equivariant cohomology of LGn and the T-equivariant fundamental class σ(λ) of X(λ). The main result of the present paper is an explicit formula for the restriction of the class σ(λ) to any torus fixed point. The formula is written in terms of factorial analogue of the Schur Q-function, introduced by Ivanov. As a corollary to the restriction formula, we obtain an equivariant version of the Giambelli-type formula for LGn. As another consequence of the main result, we obtained a presentation of the ring .     

Some lower bounds for the spectral radius of matrices using traces

Let A be an n×n matrix with eigenvalues λ₁,λ₂,…,λ_n, and let m be an integer satisfying rank(A)?m?n. If A is real, the best possible lower bound for its spectral radius in terms of m, trA and trA² is obtained. If A is any complex matrix, two lower bounds for are compared, and furthermore a new lower bound for the spectral radius is given only in terms of trA,trA²,‖A‖,‖A^∗A-AA^∗‖,n and m.     

Linear equations with unknowns from a multiplicative group whose solutions lie in a small number of subspaces

Let K be a field of characteristic 0 and let (K^*)ⁿ denote the n-fold Cartesian product of K^*, endowed with coordinatewise multiplication. Let Γ be a subgroup of (K^*)ⁿ of finite rank. We consider equations (*) a₁x₁ + … + a_nx_n = 1 in x = (x₁x_n)Γ, where a = (a₁,a_n)(K^*)ⁿ. Two tuples a, b(K^*)ⁿ are called Γ-equivalent if there is a uΓ such that b = u · a. Gy?ry and the author [Compositio Math. 66 (1988) 329-354] showed that for all but finitely many Γ-equivalence classes of tuples a(K^*)ⁿ, the set of solutions of (*) is contained in the union of not more than 2⁽ⁿ⁺¹! proper linear subspaces of Kⁿ. Later, this was improved by the author [J. reine angew. Math. 432 (1992) 177-217] to (n!)²ⁿ⁺². In the present paper we will show that for all but finitely many Γ-equivalence classes of tuples of coefficients, the set of non-degenerate solutions of (*) (i.e., with non-vanishing subsums) is contained in the union of not more than 2ⁿ proper linear subspaces of Kⁿ. Further we give an example showing that 2ⁿ cannot be replaced by a quantity smaller than n.     

Regularity in the obstacle problem for parabolic non-divergence operators of Hörmander type

In this paper we continue the study initiated in [15] concerning the obstacle problem for a class of parabolic non-divergence operators structured on a set of vector fields X={_X1,…,_Xq}$X=\{X_1,\dots ,X_q\}$ in Rⁿ${\mathbf{R}}^n$ with C^∞$C^{\infty }$-coefficients satisfying Hörmander?s finite rank condition, i.e., the rank of Lie[_X1,…,_Xq]$\mathrm{Lie}[X_1,\dots ,X_q]$ equals n   at every point in Rⁿ${\mathbf{R}}^n$. In [15] we proved, under appropriate assumptions on the operator and the obstacle, the existence and uniqueness of strong solutions to a general obstacle problem. The main result of this paper is that we establish further regularity, in the interior as well as at the initial state, of strong solutions. Compared to [15] we in this paper assume, in addition, that there exists a homogeneous Lie group G=(Rⁿ,°,_δλ)$\mathbf{G}=({\mathbf{R}}^n,°,{\delta }_{\lambda })$ such that _X1,…,_Xq$X_1,\dots ,X_q$ are left translation invariant on G and such that _X1,…,_Xq$X_1,\dots ,X_q$ are _δλ${\delta }_{\lambda }$-homogeneous of degree one.     

Fuzzy set approach to the utility,preference relations,and aggregation operators

Score x = (x₁, … , x_n) describing an alternative α is modelled by means of a continuous quasi-convex fuzzy quantity μ_α = μ_x, thus allowing to compare alternatives (scores) by means of fuzzy ordering (comparison) methods. Applying some defuzzification method leads to the introduction of operators acting on scores. A special stress is put on the Mean of Maxima defuzzification method allowing to introduce several averaging aggregation operators. Moreover, our approach allows to introduce weights into above mentioned aggregation, even in the non-anonymous (non-symmetric) case. Finally, Ordered Weighted Aggregation Operators (OWAO) are introduced, generalizing the standard OWA operators.