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.     

Pincement sur le spectre et le volume en courbure de Ricci positive

We shall show that a complete Riemannian manifold of dimension n with Ric?n−1 and its n-st eigenvalue close to n is both Gromov-Hausdorff close and diffeomorphic to the standard sphere. This extends, in an optimal way, a result of P. Petersen [Invent. Math. 138 (1999) 1] (as a by-product, we fill a gap stated in the erratum [Invent. Math. 155 (2004) 223]). We shall also show that a manifold with Ric?n−1 and volume close to is both Gromov-Hausdorff close and diffeomorphic to a space form Sn/π₁(M). This extends results of T. Colding [Invent. Math. 124 (1996) 175] and T. Yamaguchi [Math. Ann. 284 (1989) 423].     

Lines imply spaces in density Ramsey theory

Some results of geometric Ramsey theory assert that if F is a finite field (respectively, set) and n is sufficiently large, then in any coloring of the points of Fⁿ there is a monochromatic k-dimensional affine (respectively, combinatorial) subspace (see [9]). We prove that the density version of this result for lines (i.e., k = 1) implies the density version for arbitrary k. By using results in [3, 6] we obtain various consequences: a “group-theoretic” version of Roth's Theorem, a proof of the density assertion for arbitrary k in the finite field case when ∥F∥ = 3, and a proof of the density assertion for arbitrary k in the combinatorial case when ∥F∥ = 2.     

A “planar” representation for generalized transition kernels

In [9], Mauldin, Preiss and von Weizsäcker have given a theorem representing transition kernels (atomless and between standard Borel spaces) by a planar model. Here, motivated by measure-theoretic as well as probabilistic considerations, we generalize by allowing the parametrizing spaceX to be arbitrary, with an arbitrary σ-field of “Borel” subsets, and allowing the corresponding measures to have atoms. (We also, for convenience rather than generality, allow arbitrary finite measures rather than probability ones.) The transition kernel is replaced by a substantially equivalent one fromX toX ×I that is “sectioned”, hence completely orthogonal. This is shown to be isomorphic to a model in which the image space consists of 3 specifically defined subsets ofX × ?: an ordinate set (in which vertical sections have Lebesgue measure), an “atomic” set contained inX × (??), and a “singular” set with null sections. The method incidentally produces and exploits a “reverse” transition kernel fromX toX ×I. Some further extensions are briefly discussed; in particular, allowing “uniformly σ-finite” measures (in the “standard” case) leads to a generalization that includes the planar representation theorem of Rokhlin [10] and the author [5]; cf. also [7, 2].     

Distribution of points for convergent interpolatory integration rules on (−∞, ∞)

Letw be a “nice” positive weight function on (?∞, ∞), such asw(x)=exp(??x?^α) α>1. Suppose that, forn≥1, $$I_n [f]: = \sum\limits_{j = 1}^n {w_{jn} } f(x_{jn} )$$ is aninterpolatory integration rule for the weightw: that is for polynomialsP of degree ≤n-1, $$I_n [P]: = \int\limits_{ - \infty }^\infty {P(x)w(x)dx.} $$ Moreover, suppose that the sequence of rules {I _n} _n=1 ^t8 isconvergent: $$\mathop {\lim }\limits_{n \to \infty } I_n [f] = \int\limits_{ - \infty }^\infty {f(x)w(x)dx} $$ for all continuousf:R→R satisfying suitable integrability conditions. What then can we say about thedistribution of the points {x _jn} _j=1 ⁿ ,n≥1? Roughly speaking, the conclusion of this paper is thathalf the points are distributed like zeros of orthogonal polynomials forw, and half may bearbitrarily distributed. Thus half the points haveNevai-Ullmann distribution of order α, and the rest are arbitrarily distributed. We also describe the possible distributions of the integration points, when the ruleI _n has precision other thann-1.     

Polynomials orthogonal on the semicircle,II

Generalizing previous work [2], we study complex polynomials {π _k },π _k (z)=z ^k +?, orthogonal with respect to a complex-valued inner product (f,g)=∫ ₀ ^π f(e ^iθ)g(e ^iθ)w(e ^iθ)dθ. Under suitable assumptions on the “weight function”w, we show that these polynomials exist whenever Re ∫ ₀ ^π w(e ^iθ)dθ≠0, and we express them in terms of the real polynomials orthogonal with respect to the weight functionw(x). We also obtain the basic three-term recurrence relation. A detailed study is made of the polynomials {π _k } in the case of the Jacobi weight functionw(z)=(1?z)^α(1+z)^β, α>?1, and its special case \(\alpha = \beta = \lambda - \tfrac{1}{2}\) (Gegenbauer weight). We show, in particular, that for Gegenbauer weights the zeros ofπ _n are all simple and, ifn≥2, contained in the interior of the upper unit half disc. We strongly suspect that the same holds true for arbitrary Jacobi weights. Finally, for the Gegenbauer weight, we obtain a linear second-order differential equation forπ _n (z). It has regular singular points atz=1, ?1, ∞ (like Gegenbauer's equation) and an additional regular singular point on the negative imaginary axis, which depends onn.     

Permanent formulae from the Veronesean

The two formulae for the permanent of a d × d matrix given by Ryser (1963) and Glynn (2010) fit into a similar pattern that allows generalization because both are related to polarization identities for symmetric tensors, and to the classical theorem of P. Serret in algebraic geometry. The difference between any two formulae of this type corresponds to a set of dependent points on the “Veronese variety” (or “Veronesean”) v _d ([d ? 1]), where v _d ([n]) is the image of the Veronese map v _d acting on [n], the n-dimensional projective space over a suitable field. To understand this we construct dependent sets on the Veronesean and show how to construct small independent sets of size nd + 2 on v _d ([n]). For d = 2 such sets of 2n + 2 points in [n] have been called “associated” and we observe that they correspond to self-dual codes of length 2n + 2.     

Estimates of the solutions of difference-differential equations of neutral type

In this paper, we study scalar difference-differential equations of neutral type of general form $$\sum\limits_{j = 0}^m {\int_0^h {u^{(j)} (t - \theta )d\sigma _j (\theta ) = 0,t > h,} } $$ where the σ_j(θ) are functions of bounded variation. For the solutions of this equation, we obtain the following estimate: $$\left\| {u(t)} \right\|W_2^m (T,T + h) \leqslant CT^{q - 1} e^{\kappa T} \left\| {u(t)} \right\|W_2^m (0,h),$$ where C is a constant independent of u ₀(t) and the values of q and ? are determined by the properties of the characteristic determinant of this equation. Earlier, this estimate was proved for equations of less general form. For example, for piecewise constant functions σ _j(θ) or for the case in which the function σ _m(θ) has jumps at both points θ = 0 and θ = h. In the present paper, this estimate is obtained under the only condition that σ _m(θ) experiences a jump at the point θ = 0; this condition is necessary for the correct solvability of the initial-value problem.     

The characterization of theta-distinguished representations of GL(<Emphasis Type="Italic">n</Emphasis>)

Let θ and θ’ be a pair of exceptional representations in the sense of Kazhdan and Patterson [KP84], of a metaplectic double cover of GL _n . The tensor θ ? θ’ is a (very large) representation of GL _n . We characterize its irreducible generic quotients. In the square-integrable case, these are precisely the representations whose symmetric square L-function has a pole at s = 0. Our proof of this case involves a new globalization result. In the general case these are the representations induced from distinguished data or pairs of representations and their contragredients. The combinatorial analysis is based on a complete determination of the twisted Jacquet modules of θ. As a corollary, θ is shown to admit a new “metaplectic Shalika model”.     

A Bernstein function related to Ramanujan’s approximations of exp(n)

Ramanujan’s sequence θ(n),n=0,1,2,…?, is defined by $\frac{e^{n}}{2}=\sum_{j=0}^{n-1}\frac{n^{j}}{j!}+\frac{n^{n}}{n!} \theta(n)$ . It is possible to define, in a simple manner, the function θ(x) for all nonnegative real numbers x. We show that the function $\lambda(x):=x (\theta(x)-\frac{1}{3} )$ is a Bernstein function on [0,∞), that is, λ(x) is nonnegative with completely monotonic derivative on [0,∞). This implies some earlier results concerning complete monotonicity of the function θ(x) on [0,∞).     

Stability conditions for linear retarded functional differential equations

The purpose of this note is to study the exponential stability for the linear retarded functional differential equation $\text{x}\text{?}\text{(t) = ∫}{}_{\mathrm{?1}}{}^0\text{[dη(θ)] x(t ? r(θ))}$, where the delay function r(θ) ? 0 is continuous and η(θ) is of bounded variation on the interval [?1, 0]. It is shown that the spectral limit function for the equation above has a continuous dependence on the pair (η, r). The set of all functions of bounded variation η for which the equation above is exponentially stable for every delay function r, the so-called region of stability globally in the delays, is a cone. Therefore for a fixed r, the set of all η which make our equation exponentially stable, that is, the region of stability for the delay function r, contains a cone. A discussion of the characterization of these regions of stability, as well as of the largest cone contained in each region of stability for a fixed delay function r, is given. Some remarks are made with respect to a similar question for the equation $\text{x}\text{?}\text{(t) = Ax(t) + ∫}{}_{\mathrm{?\; 1}}{}^0\text{[dμ(θ)] x(t?r(θ))}$, where A is a real n by n matrix, μ(θ) is bounded variation on [?1, 0] and r(θ) as before. Several examples illustrate the results obtained.     

On a class of degenerate extremal graph problems

Given a class ? of (so called “forbidden”) graphs, ex (n, ?) denotes the maximum number of edges a graphG ⁿ of ordern can have without containing subgraphs from ?. If ? contains bipartite graphs, then ex (n, ?)=O(n ^2?c ) for somec>0, and the above problem is calleddegenerate. One important degenerate extremal problem is the case whenC _2k , a cycle of 2k vertices, is forbidden. According to a theorem of P. Erd?s, generalized by A. J. Bondy and M. Simonovits [3₂, ex (n, {C _2k })=O(n ^1+1/k ). In this paper we shall generalize this result and investigate some related questions.     

Rectilinear line segment intersection,layered segment trees,and dynamization

The aim of the present paper is to provide an efficient solution to the following problem: “Given a family of n rectilinear line segments in two-space report all intersections in the family with a query consisting of an arbitrary rectilinear line segment.” We provide an algorithm which takes O(nlog₂n) preprocessing time, o(nlog₂n) space and O(log₂n + k) query time, where k is the number of reported intersections. This solution serves to introduce a powerful new data structure, the layered segment tree, which is of independent interest. Second it yields, by way of recent dynamization techniques, a solution to the on-line version of the above problem, that is the operations INSERT and DELETE and QUERY with a line segment are allowed. Third it also yields a new nonscanning solution to the batched version of the above problem. Finally we apply these techniques to the problem obtained by replacing “line segment” by “rectangle” in the above problem, giving an efficient solution in this case also.     

Upper functions for positive random functionals. I. General setting and Gaussian random functions

In this paper we are interested in finding upper functions for a collection of real-valued random variables {Ψ(χ _θ ), θ ∈ Θ}. Here {χ _θ , θ ∈ Θ} is a family of continuous random mappings, Ψ is a given sub-additive positive functional and Θ is a totally bounded subset of a metric space. We seek a nonrandom function U: Θ → ?₊ such that sup _θ∈Θ{Ψ(χ _θ ) ? U(θ)}₊ is “small” with prescribed probability. We apply the results obtained in the general setting to the variety of problems related to Gaussian random functions and empirical processes.     

Explicit representations of the integral containing the error term in the divisor problem

We shall investigate several properties of the integral $$ \int_1^\infty {t^{ - \theta } \Delta _k \left( t \right) log^j t dt} $$ with a natural number k, a non-negative integer j and a complex variable θ, where Δ _k (x) is the error term in the divisor problem of Dirichlet and Piltz. The main purpose of this paper is to apply the “elementary methods” and the “elementary formulas” to derive convergence properties and explicit representations of this integral with respect to θ for k = 2.     

On topologies generated by Moisil resemblance relations

In [8] and [9] Moisil has introduced the resemblance relations. Following [9] we associate to every resemblance relation an extensive operator which commutes with arbitrary unions of sets. We are leading to consider spaces endowed with such closure operators; we shall call these spaces total ?ech spaces (TC-spaces).TC-spaces are in one-to-one, onto correspondence with reflexive relations. TC-spaces generated by transitive relations are in one-to-one, onto correspondence with the total topological spaces of W. Hartnett (which are called total Kuratowski spaces, TK-spaces).We study the category of TC-spaces and its full subcategory determined by TK-spaces. Both categories are Cartesian closed, but they are not elementary toposes.     

Structure symplectique généralisée sur le fibre des connexions

We prove that on the bundle of connections of an arbitrary principal bundle π: P → M there exists a canonical differential 2--form taking values in the adjoint bundle ;ing: ad_g:: ad P → M which defines a generalized symplectic structure and which verifies a property of “universal curvature”. The results of the present Note generalize those of [3] to an arbitrary Lie group.     

Reflection classes whose cubes cover the alternating group

A reflection class (REC) over a finite set A is a conjugacy class of a reflection (permutation of order ? 2) of A. It was known that for no REC X, X² = Alt(n) holds, and that for some RECs X, X⁴ = Alt(n) holds (n ? 5). Let i > 0, and let c(θ) denote the number of cycles of θ?S(n). Let X_i = {ψ ∈ S(n): ψ² = 1, ψ has exactly i fixed points}. We prove that θ?X_i³ if and only if: (1) i ≡ n (mod 2); (2) The parity of X_i equals the parity of θ; and (3) $\text{i ?}\text{1}\text{3}\text{(n + 2 c(θ))}$. As a consequence, {X: X is a REC, X³ = Alt(n)} and {X: X is a REC, X³ = S(n) ? Alt(n)} are determined.