Prime links in some skew-polynomial and skew-laurent rings

Let R be a Noetherian commutative ring and a α₁,…,α_n commuting automorphisms of R. Define T = R[θ₁,…,θ_n;α₁,…,α_n] to be the skew-polynomial ring with θ_ir = α_i(r)θ_i and θ_iθ_j= θ_jθ_i, for all i,j ? (1,…,n) and r ? R, and let S = Rθ₁,θ₁:^-1,…,θ_n:,θ_n;^-1;α₁:,…,α_n] be the corresponding skew-Laurent ring. In this paper we show that S and T satisfy the strong second layer condition and characterize the links between prime ideals in these rings.     

Sharp asymptotics of large deviations in ℝ d

Given a sequence {X₁}i=1,2,3,... of i.i.d. random variables taking values in ? ^d ,d≥2, letS _n =Σ _i=1 ⁿ X _t=1. For Λ a Borel set in ? ^d having smooth boundary, witha=inf_x∈ΛI(x) the minimal value of the large deviation rate functionI(x) over Λ, we find, under suitable hypotheses, asymptotic results asn→∞, of the form $$P(S_n \in n\Gamma ) = n^\gamma e^{ - na} (d_0 + o(1))$$ where the constant γ depends sensitively on the geometry of Λ and the dimensiond, and takes values ?∞<γ≤(d?2/2). For fixeda=inf_x∈ΛI(x), we construct examples having any specific γ in this range.     

Confidence Bands for Isotonic Median Curves Using Sign Tests

Suppose that one observes independent random variables (X₁, Y₁), (X₂, Y₂), …, (X_n, Y_n) in R² with unknown distributions, except that Median(Y_i | X_i = M(x) for some unknown isotonic function M. We describe an explicit algorithm for the computation of confidence bands for the median function M whose running time is of order O(n²). The bands rely on multiscale sign tests and are shown to have desirable asymptotic properties.     

Arithmétique d'une famille de corps cubiques

In this Note, we study the family of polynomials: P(X)=X³?nX²?n, with n=3^sp₁…p_t, where s=0 or 1 and where the p_i, for 1?i?t, are distinct prime numbers and all different from 3, and (4n²+27)/9^s is squarefree. For this family, we determine the arithmetic invariants of the number field $\text{K=}\text{Q}\text{(α),}$ where α is the only real root of the polynomial P(X), and we find the following results: $\text{O}{}_{\mathrm{K}}\text{=}\text{Z}\text{[α]}$ is the ring of integers of K, d_K=?n²(4n²+27) is the discriminant of K; ε=α²+1 is the fundamental unit of O_K and R_K=Log(α²+1) is the regulator of K. To cite this article: O. Lahlou, M. El Hassani Charkani, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

Phase reconstruction from magnitude of band-limited multidimensional signals

In this paper, the problem of phase reconstruction from magnitude of multidimensional band-limited functions is considered. It is shown that any irreducible band-limited function f(z₁…,z_n), z_i ? $\text{C}$, i=1, …, n, is uniquely determined from the magnitude of f(x₁…,x_n): | f(x₁…,x_n)|, x_i ? $\text{R}$, i=1,…, n, except for (1) linear shifts: ^{i(α1z1+…+α_n2n+β}), β, α_i?$\text{R}$, i=1,…, n; and (2) conjugation: $\text{f}{}^1\text{(z}{}_1{}^1\text{,…,z}{}_{\mathrm{n}}{}^1\text{)}$.     

Asymptotic properties of least squares estimators for discrete compound distributions

Let (X, Λ) be a pair of random variables, where Λ is an Ω (a compact subset of the real line) valued random variable with the density functiong(Θ: α) andX is a real-valued random variable whose conditional probability function given Λ=Θ is P {X=x|Θ} withx=x ₀, x₁, …. Based onn independent observations ofX, x ⁽ⁿ⁾, we are to estimate the true (unknown) parameter vectorα=(α ₁, α₂, ...,α_m) of the probability function ofX, P_α(X=∫_ΩP{X=x|Θ}g(Θ:α)dΘ. A least squares estimator of α is any vector \(\hat \alpha \left( {X^{\left( n \right)} } \right)\) which minimizes $$n^{ - 1} \sum\limits_{i = 1}^n {\left( {P_\alpha \left( {x_i } \right) - fn\left( {x_i } \right)} \right)^2 } $$ wherex ⁽ⁿ⁾=(x₁, x₂,…,x _n) is a random sample ofX andf _n(x_i)=[number ofx _i inx ⁽ⁿ⁾]/n. It is shown that the least squares estimators exist as a unique solution of the normal equations for all sufficiently large sample size (n) and the Gauss-Newton iteration method of obtaining the estimator is numerically stable. The least squares estimators converge to the true values at the rate of \(O\left( {\sqrt {2\log \left( {{{\log n} \mathord{\left/ {\vphantom {{\log n} n}} \right. \kern-0em} n}} \right)} } \right)\) with probability one, and has the asymptotically normal distribution.     

Minimum norm problems over transportation polytopes

We consider the problem of updating input-output matrices, i.e., for given (m,n) matrices A ? 0, W ? 0 and vectors u ? $\text{R}$^m, v?$\text{R}$ⁿ, find an (m,n) matrix X ? 0 with prescribed row sums Σⁿ_j=1X_ij = u_i (i = 1,…,m) and prescribed column sums Σ^m_i=1X_ij = v_j (j = 1,…,n) which fits the relations X_ij = A_ij + λ_iW_ij + W_ij + W_ijμ_j for all i,j and some λ?$\text{R}$^m, μ?$\text{R}$ⁿ. Here we consider the question of existence of a solution to this problem, i.e., we shall characterize those matrices A, W and vectors u,v which lead to a solvable problem. Furthermore we outline some computational results using an algorithm of [2].     

Complementing and exactly covering sequences

The following is proved (in a slightly more general setting): Let α₁, …, α_m be positive real, γ₁, …, γ_m real, and suppose that the system [nα_i + γ_i], i = 1, …, m, n = 1, 2, …, contains every positive integer exactly once (= a complementing system). Then $\text{α}{}_{\mathrm{i}}\text{α}{}_{\mathrm{j}}$ is an integer for some i ≠ j in each of the following cases: (i) m = 3 and m = 4; (ii) m = 5 if all α_i but one are integers; (iii) m ? 5, two of the α_i are integers, at least one of them prime; (iv) m ? 5 and α_n ? 2ⁿ for n = 1, 2, …, m ? 4.For proving (iv), a method of reduction is developed which, given a complementing system of m sequences, leads under certain conditions to a derived complementing system of m ? 1 sequences.     

The behaviour of (k,…,kn,n−ik)ci/i! is asymptotically normal

Let N(n,i) = (k,…,kⁿ,n?ik)c^i/i, i = O.…,[n/k]. We prove that the random variable X_n such that $\text{P(X}{}_{\mathrm{<mtext>n</mtext>}}\text{= i) =}\text{N(n, i)}\text{Σ}{}_{\mathrm{<mtext>j</mtext>}}\text{N(n, j)}$ has asymptotically (n → ∞) a normal distribution and we give some combinatorial applications of this result.We also improve a result of Godsil [3] dealing with matchings in graph.     

Strong law for the bootstrap

Let X₁, X₂, X₃, … be i.i.d. r.v. with E|X₁| < ∞, E X₁ = μ. Given a realization X = (X₁,X₂,…) and integers n and m, construct Y_n,i, i = 1, 2, …, m as i.i.d. r.v. with conditional distribution $\text{P}{}^1\text{(Y}{}_{\mathrm{n,i}}\text{= X}{}_{\mathrm{j}}\text{) =}\text{1}\text{n}$ for 1 ? j ? n. ($\text{P}{}^1$ denotes conditional distribution given X). Conditions relating the growth rate of m with n and the moments of X₁ are given to ensure the almost sure convergence of $\text{(}\text{1}\text{m}\text{)Σ}{}^{\mathrm{m}}{}_{\mathrm{i=1}}\text{Y}{}_{\mathrm{n,i}}$ toμ. This equation is of some relevance in the theory of Bootstrap as developed by Efron (1979) and Bickel and Freedman (1981).     

A result on hypothesis testing for a multivariate normal distribution when some observations are missing

Let U_i = (X_i, Y_i), i = 1, 2,…, n, be a random sample from a bivariate normal distribution with mean μ = (μ_x, μ_y) and covariance matrix

. Let X_i, i = n + 1,…, N represent additional independent observations on the X population. Consider the hypothesis testing problem H₀ : μ = 0 vs. H₁ : μ ≠ 0. We prove that Hotelling's T² test, which uses (X_i, Y_i), i = 1, 2,…, n (and discards X_i, i = n + 1,…, N) is an admissible test. In addition, and from a practical point of view, the proof will enable us to identify the region of the parameter space where the T²-test cannot be beaten. A similar result is also proved for the problem of testing μ_x ? μ_y = 0. A Bayes test and other competitors which are similar tests are discussed.     

Convergence of optimal stochastic bin packing

Consider independent identically distributed random variables (X_i) valued in [0,1]. Let B(n) be the optimal (minimum) number of unit size bins needed to pack n items of size X₁, X₂,…,X_n. We prove that there exists a numerical constant C such that for t > 0,

$\text{Pr(∣B(n)?E(B(n))∣>t}\text{n}\text{)≤ C exp(? t).}$

The constant C does not depend on the distribution of X.     

General theorems on rates of convergence in distribution of random variables I. General limit theorems

Let (X_n)_n?$\text{N}$ be a sequence of real, independent, not necessarily identically distributed random variables (r.v.) with distribution functions F_Xn, and S_n = Σ_i=1ⁿX_i. The authors present limit theorems together with convergence rates for the normalized sums ?(n)S_n, where ?: $\text{N}$ → $\text{R}$⁺, ?(n) → 0, n → ∞, towards appropriate limiting r.v. X, the convergence being taken in the weak (star) sense. Thus higher order estimates are given for the expression ∝_$\text{R}$f(x) d[F_?(n)Sn(x) ? F_X(x)] which depend upon the normalizing function ?, decomposability properties of X and smoothness properties of the function f under consideration. The general theorems of this unified approach subsume O- and o-higher order error estimates based upon assumptions on associated moments. These results are also extended to multi-dimensional random vectors.     

On a class of balancing games

A balancing game is a perfect information two-person game. Given a set V?R^d, in the ith round Player I picks a vector vⁱ?V, and then Player II picks ?_i = +1 or ?1. Player II tries to minimize sup {| Σ_i=1ⁿ?_ivⁱ | : n = 0, 1, 2,…}. In this paper we generalize this game and give necessary and sufficient conditions for the existence of a winning strategy for Player I and Player II in the generalized game. Later we give upper and lower bounds to the value of the original game; the bounds in many cases are equal. Further we present simple strategies for both players.     

On the strong limit theorems for double arrays of blockwise M-dependent random variables

For a double array of blockwise M-dependent random variables {X _mn ,m ?? 1, n ?? 1}, strong laws of large numbers are established for double sums ?? _i=1 ^m ?? _j=1 ⁿ X _ij , m ?? 1, n ?? 1. The main results are obtained for (i) random variables {X _mn ,m ?? 1, n ?? 1} being non-identically distributed but satisfy a condition on the summability condition for the moments and (ii) random variables {X _mn ,m ?? 1, n ?? 1} being stochastically dominated. The result in Case (i) generalizes the main result of Móricz et al. [J. Theoret. Probab., 21, 660?C671 (2008)] from dyadic to arbitrary blocks, whereas the result in Case (ii) extends a result of Gut [Ann. Probab., 6, 469?C482 (1978)] to the bockwise M-dependent setting. The sharpness of the results is illustrated by some examples.     

On maximal antichains consisting of sets and their complements

Let 1 ? k₁ ? k₂ ? … ? k_n be integers and let S denote the set of all vectors x = (x₁, x₂, …, x_n) with integral coordinates satisfying 0 ? x_i ? k_i, i = 1, 2, …, n. The complement of x is (k₁ ? x₁, k₂ ? x₂, …, k_n ? x_n) and a subset X of S is an antichain provided that for any two distinct elements x, y of X, the inequalities x_i ? y_i, i = 1, 2, …, n, do not all hold. We determine an LYM inequality and the maximal cardinality of an antichain consisting of vectors and its complements. Also a generalization of the Erdös-Ko-Rado theorem is given.     

Behrend's theorem for sequences containing no k-element arithmetic progression of a certain type

Let k and n be positive integers, and let d(n, k) be the maximum density in {0, 1, 2,…, kⁿ ? 1} of a set containing no arithmetic progression of k terms with first term a = Σa_ikⁱ and common difference d = Σ?_ikⁱ, where 0 ? a_i ? k ? 1, ?_i = 0 or 1, and ?_i = 1 ? a_i = 0. Setting β_k = lim_n→∞d(n, k), we show that lim_k→∞β_k is either 0 or 1.     

Multiplicative infinite loop space theory

Let G be a finitely presented group given by its pre-abelian presentation <X1,…,Xm; Xe11ζ1,…,Xemmζ,ζm+1,…>, where ei≥0 for i = 1,…, m and ζj?G′ for j≥1. Let N be the subgroup of G generated by the normal subgroups [xeii, G] for i = 1,…, m. Then Dn+2(G)≡γn+2(G) (modNG′) for all n≥0, where G” is the second commutator subgroup of G,γn+2(G) is the (n+2)th term of the lower central series of G and Dn+2(G) = G∩(1+△n+2(G)) is the (n+2)th dimension subgroup of G.     

Elasticity of factorizations in R 0 + X R1 + … + X lRl [X]

For an atomic domain R the elasticity ρ(R) is defined by ρ(R) = sup{m/n ¦ u₁ … u _m = v ₁ … v_n where u_i, v_i ∈ R are irreducible}. Let R ₀ ? … ? R _l be an ascending chain of domains which are finitely generated over ? and assume that R _l is integral over R ₀. Let X be an indeterminate. In this paper we characterize all domains D of the form D = R ₀ + XR₁ + … + X^lR_l[X] whose elasticity ρ(D) is finite.