A generalization of the binary Preparata code

A classical binary Preparata code P₂(m) is a nonlinear (2^m+1,2^2(2m-1-m),6)-code, where m is odd. It has a linear representation over the ring Z₄ [Hammons et al., The Z₄-linearity of Kerdock, Preparata, Goethals and related codes, IEEE Trans. Inform. Theory 40(2) (1994) 301-319]. Here for any q=2^l>2 and any m such that (m,q-1)=1 a nonlinear code P_q(m) over the field F=GF(q) with parameters (q(Δ+1),q^2(Δ-m),d?3q), where Δ=(q^m-1)/(q-1), is constructed. If d=3q this set of parameters generalizes that of P₂(m). The equality d=3q is established in the following cases: (1) for a series of initial admissible values q and m such that q^m<2¹⁰⁰; (2) for m=3,4 and any admissible q, and (3) for admissible q and m such that there exists a number m₁ with m₁|m and d(P_q(m₁))=3q. We apply the approach of [Nechaev and Kuzmin, Linearly presentable codes, Proceedings of the 1996 IEEE International Symposium Information Theory and Application Victoria, BC, Canada 1996, pp. 31-34] the code P is a Reed-Solomon representation of a linear over the Galois ring R=GR(q²,4) code P dual to a linear code K with parameters near to those of generalized linear Kerdock code over R.     

Existence Criterion for Estimates of Derivatives of Rational Functions

Suppose that K is a compact set in the open complex plane. In this paper, we prove an existence criterion for an estimate of Markov-Bernstein type for derivatives of a rational function R(z) at any fixed point z ₀ ∈ K. We prove that, for a fixed integer s, the estimate of the form |R ^(s) (z ₀)| ≤ C(K, z ₀, s)n‖R‖ _C(K), where R is an arbitrary rational function of degree n without poles on K and C is a bounded function depending on three arguments K, z ₀, and s, holds if and only if the supremum $$\omega (K,z_0 ,s) = \sup \left\{ {\frac{{\operatorname{dist} (z,K)}}{{\left| {z - z_0 } \right|^{s + 1} }}} \right\}$$ over z in the complement of K is finite. Under this assumption, C is less than or equal to const ·s!ω(K, z ₀, s).     

About a construction and some analysis of time inhomogeneous diffusions on monotonely moving domains

We construct and analyze in a very general way time inhomogeneous (possibly also degenerate or reflected) diffusions in monotonely moving domains E⊂R×R^d, i.e. if E_t?{x∈R^d|(t,x)∈E}, t∈R, then either E_s⊂E_t, ∀s?t, or E_s⊃E_t, ∀s?t, s,t∈R. Our major tool is a further developed L²(E,m)-analysis with well chosen reference measure m. Among few examples of completely different kinds, such as e.g. singular diffusions with reflection on moving Lipschitz domains in R^d, non-conservative and exponential time scale diffusions, degenerate time inhomogeneous diffusions, we present an application to what we name skew Bessel process on γ. Here γ is either a monotonic function or a continuous Sobolev function. These diffusions form a natural generalization of the classical Bessel processes and skew Brownian motions, where the local time refers to the constant function γ≡0.     

Multiple zeta functions and asymptotic structure of free Abelian groups of finite rank

We define the multiple zeta function of the free Abelian group Z^d as

ζ_Z^d(s₁,…,s_d)=∑_|Z^d:H|<∞α₁(H)^−s₁?α_d(H)^−s_d,     

Complexity Estimates for the Schmüdgen Positivstellensatz

LetKbe a closed basic set inRⁿgiven by the polynomial inequalities φ₁≥ 0, . . . , φ_m≥ 0 and let Σ be the semiring generated by the φ_kand the squares inR[x₁, . . . ,x_n]. Schmüdgen has shown that ifKis compact then any polynomial function strictly positive onKbelongs to Σ. Easy consequences are (1)f≥ 0 onKif and only iff∈R⁺+ Σ (Positivstellensatz) and (2) iff≥ 0 onKbutf∈ Σ then asdtends to 0⁺, in any representation off + das an element of Σ in terms of the φ_k, the squares and semiring operations, the integerN(d) which is the minimum over all representations of the maximum degree of the summands must become arbitrarily large. A one-dimensional example is analyzed to obtain asymptotic lower and upper bounds of the formcd^−1/2≤N(d) ≤Cd^−1/2log (1/d).     

Multiobjective programming and penalty functions

This paper generalizes the penalty function method of Zang-will for scalar problems to vector problems. The vector penalty function takes the form $$g(x,\lambda ) = f(x) + \lambda ^{ - 1} P(x)e,$$ wheree ?R ^m, with each component equal to unity;f:R ⁿ →R ^m, represents them objective functions {f ⁱ} defined onX \( \subseteq \) R ⁿ; λ ∈R ¹, λ>0;P:R ⁿ →R ¹ X \( \subseteq \) Z \( \subseteq \) R ⁿ,P(x)≦0, ∨x ∈R ⁿ,P(x) = 0 ?x ∈X. The paper studies properties of {E (Z, λ _r )} for a sequence of positive {λ _r } converging to 0 in relationship toE(X), whereE(Z, λ _r ) is the efficient set ofZ with respect tog(·, λ_r) andE(X) is the efficient set ofX with respect tof. It is seen that some of Zangwill's results do not hold for the vector problem. In addition, some new results are given.     

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).     

About the p-adic Yosida equation inside a disk

Let K be a complete ultrametric algebraically closed field and let ?(d(0, R^?)) be the field of meromorphic functions inside the disk d(0,R⁻) = {x ∈ K ∣ ∣x∣ < R}. Let ?_b(d(0, R^?)) be the subfield of bounded meromorphic functions inside d(0,R⁻) and let ?_u(d(0, R^?)) = ?(d(0, R^?)) ? ?_b(d(0, R^?)) be the subset of unbounded meromorphic functions inside d(0,R⁻). Initially, we consider the Yosida Equation: , where m ∈ ?^* and F(X) is a rational function of degree d with coefficients in ?_b(d(0, R^?)). We show that, if d ≥ 2m + 1, this equation has no solution in ?_u(d(0, R^?)).Next, we examine solutions of the above equation when F(X) is apolynomial with constant coefficients and show that it has no unbounded analytic functions in d(0,R⁻). Further, we list the only cases when the equation may eventually admit solutions in ?_u(d(0, R^?)). Particularly, the elliptic equation may not.     

Rings of matrix invariants in positive characteristic

Denote by R_n,m the ring of invariants of m-tuples of n×n matrices (m,n?2) over an infinite base field K under the simultaneous conjugation action of the general linear group. When char(K)=0, Razmyslov (Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974) 723) and Procesi (Adv. Math. 19 (1976) 306) established a connection between the Nagata-Higman theorem and the degree bound for generators of R_n,m. We extend this relationship to the case when the base field has positive characteristic. In particular, we show that if 0<char(K))?n, then R_n,m is not generated by its elements whose degree is smaller than m. A minimal system of generators of R_2,m is determined for the case char(K)=2: it consists of 2^m+m−1 elements, and the maximum of their degrees is m. We deduce a consequence indicating that the theory of vector invariants of the special orthogonal group in characteristic 2 is not analogous to the case char(K)≠2. We prove that the characterization of the R_n,m that are complete intersections, known before when char(K)=0, is valid for any infinite K. We give a Cohen-Macaulay presentation of R_2,4, and analyze the difference between the cases char(K)=2 and char(K)≠2.     

Construction of generators of quasi-interpolation operators of high approximation orders in spaces of polyharmonic splines

The paper presents a simple procedure for the construction of quasi-interpolation operators in spaces of m-harmonic splines in R^d, which reproduce polynomials of high degree. The procedure starts from a generator ?₀, which is easy to derive but with corresponding quasi-interpolation operator reproducing only linear polynomials, and recursively defines generators ?₁,?₂,…,?_m−1 with corresponding quasi-interpolation operators reproducing polynomials of degree up to 3,5,…,2m−1 respectively. The construction of ?_j from ?_j−1 is explicit, simple and independent of m. The special case d=1 and the special cases d=2,m=2,3,4 are discussed in details.     

Cylinders in del Pezzo fibrations

We show that a del Pezzo fibration π: V → W of degree d contains a vertical open cylinder, that is, an open subset whose intersection with the generic fiber of π is isomorphic to Z × A_K¹ for some quasi-projective variety Z defined over the function field K of W, if and only if d ≥ 5 and π: V → W admits a rational section. We also construct twisted cylinders in total spaces of threefold del Pezzo fibrations π: V → P¹ of degree d ≤ 4.     

Reducibility of Punctual Hilbert Schemes of Cone Varieties

In this short note we show that for any pair of positive integers (d, n) with n > 2 and d > 1 or n = 2 and d > 4, there always exist projective varieties X ? ? ^N of dimension n and degree d and an integer s ₀ such that Hilb ^s (X) is reducible for all s ≥ s ₀. X will be a projective cone in ? ^N over an arbitrary projective variety Y ? ? ^N?1. In particular, we show that, opposite to the case of smooth surfaces, there exist projective surfaces with a single isolated singularity which have reducible Hilbert scheme of points.     

Tame linear extension operators for smooth Whitney functions

For a compact set K⊆Rd we present a rather easy construction of a linear extension operator E:E(K)→C^∞(Rd) for the space of Whitney jets E(K) which satisfies linear tame continuity estimates , where ‖⋅s_‖ denotes the s-th Whitney norm. The construction turns out to be possible if and only if the local Markov inequality LMI(s) introduced by Bos and Milman holds for every s>r on K. In particular, E(K) admits a tame linear extension operator if and only if the local Markov inequality LMI(s) holds on K for some s?1.     

Valeurs aux entiers négatifs des séries de Dirichlet associées a un polynôme,I

In this paper, we are studying Dirichlet series Z(P,ξ,s) = Σ_n?$\text{N}$^1rP(n)^?s ξⁿ, where P ∈ $\text{R}$₊ [X₁,…,X_r] and ξⁿ = ξ₁ⁿ¹ … ξ_r^nr, with ξ_i ∈ $\text{C}$, such that |ξ_i| = 1 and ξ_i ≠ 1, 1 ≦ i ≦ r. We show that Z(P, ξ,·) can be continued holomorphically to the whole complex plane, and that the values Z(P, ξ, ?k) for all non negative integers, belong to the field generated over $\text{Q}$ by the ξ_i and the coefficients of P. If, there exists a number field K, containing the ξ_i, 1 ≦ i ≦ r, and the coefficients of P, then we study the denominators of Z(P, ξ, ?k) and we define a $\text{B}$-adic function Z$\text{B}$(P, ξ,·) which is equal, on class of negative integers, to Z(P, ξ, ?k).     

The distribution and average order of the coefficients of Dedekind ζ functions

Let K/$\text{Q}$ be an algebraic number field and ζ_K(s) be the associated Dedekind ζ function. A quantitative estimate is proved which shows that the average order of the coefficients of ζ_k^m(s) (for m ∈ $\text{Z}$⁺) arises from infrequent occurrences of very large values of these coefficients. This leads to new Ω-estimates for the associated error terms, improving results of Szegö and Walfisz.     

Intersection patterns of convex sets

LetK ₁,…K_n be convex sets inR ^d. For 0≦i denote byf _ithe number of subsetsS of {1,2,…,n} of cardinalityi+1 that satisfy ∩{K _i∶i∈S}≠Ø. We prove:Theorem.If f _d+r=0 for somer r>=0, then {fx161-1} This inequality was conjectured by Katchalski and Perles. Equality holds, e.g., ifK ₁=…=K_r=R^d andK _r+1,…,K_n aren?r hyperplanes in general position inR ^d. The proof uses multilinear techniques (exterior algebra). Applications to convexity and to extremal set theory are given.     

Boundedness for pseudodifferential operators on multivariate α-modulation spaces

The α-modulation spaces M ^s,α _p,q (R ^d ), α∈[0,1], form a family of spaces that contain the Besov and modulation spaces as special cases. In this paper we prove that a pseudodifferential operator σ(x,D) with symbol in the Hörmander class S ^b _ρ,0 extends to a bounded operator σ(x,D):M ^s,α _p,q (R ^d )→M ^s-b,α _p,q (R ^d ) provided 0≤α≤ρ≤1, and 1<p,q<∞. The result extends the well-known result that pseudodifferential operators with symbol in the class S ^b _1,0 maps the Besov space B ^s _p,q (R ^d ) into B ^s-b _p,q (R ^d ).     

IBN rings and orderings on grothendieck groups

LetR be a ring with an identity element.R∈IBN means thatR ^m⋟Rⁿ impliesm=n, R∈IBN ₁ means thatR ^m⋟Rⁿ⊕K impliesm≥n, andR∈IBN ₂ means thatR ^m⋟R^m⊕K impliesK=0. In this paper we give some characteristic properties ofIBN ₁ andIBN ₂, with orderings on the Grothendieck groups. In addition, we obtain the following results: (1) IfR∈IBN ₁ and all finitely generated projective leftR-modules are stably free, then the Grothendieck groupK ₀(R) is a totally ordered abelian group. (2) If the pre-ordering of the Grothendieck groupK ₀(R) of a ringR is a partial ordering, thenR∈IBN ₁ orK ₀(R)=0. Supported by National Nature Science Foundation of China.     

Strong cleanness of generalized matrix rings over a local ring

Let K_s(R) be the generalized matrix ring over a ring R with multiplier s. For a general local ring R and a central element s in the Jacobson radical of R, necessary and sufficient conditions are obtained for K_s(R) to be a strongly clean ring. For a commutative local ring R and an arbitrary element s in R, criteria are obtained for a single element of K_s(R) to be strongly clean and, respectively, for the ring K_s(R) to be strongly clean. Specializing to s = 1 yields some known results. New families of strongly clean rings are presented.     

Refining Castelnuovo-Halphen bounds

Fix integers r,d,s,π with r≥4, d?s, r?1≤s≤2r?4, and π≥0. Refining classical results for the genus of a projective curve, we exhibit a sharp upper bound for the arithmetic genus p _a (C) of an integral projective curve C?? ^r of degree d, assuming that C is not contained in any surface of degree <s, and not contained in any surface of degree s with sectional genus >π. Next we discuss other types of bound for p _a (C), involving conditions on the entire Hilbert polynomial of the integral surfaces on which C may lie.