Generalized Maiorana–McFarland class and normality of p-ary bent functions

A class of bent functions which contains bent functions with various properties like regular, weakly regular and not weakly regular bent functions in even and in odd dimension, is analyzed. It is shown that this class includes the Maiorana–McFarland class as a special case. Known classes and examples of bent functions in odd characteristic are examined for their relation to this class. In the second part, normality for bent functions in odd characteristic is analyzed. It turns out that differently to Boolean bent functions, many – also quadratic – bent functions in odd characteristic and even dimension are not normal. It is shown that regular Coulter–Matthews bent functions are normal.     

The Hamming Distance Spectrum Between Self-Dual Maiorana–McFarland Bent Functions

A bent function is self-dual if it is equal to its dual function. We study the metric properties of the self-dual bent functions constructed on using available constructions. We find the full Hamming distance spectrum between self-dual Maiorana–McFarland bent functions. Basing on this, we find the minimal Hamming distance between the functions under study.     

Sobolev and Hardy–Littlewood–Sobolev inequalities

This paper is devoted to improvements of Sobolev and Onofri inequalities. The additional terms involve the dual counterparts, i.e. Hardy–Littlewood–Sobolev type inequalities. The Onofri inequality is achieved as a limit case of Sobolev type inequalities. Then we focus our attention on the constants in our improved Sobolev inequalities, that can be estimated by completion of the square methods. Our estimates rely on nonlinear flows and spectral problems based on a linearization around optimal Aubin–Talenti functions.     

CM–fields and skew–symmetric matrices

Cohen and Odoni prove that every CM–field can be generated by an eigenvalue of some skew–symmetric matrix with rational coefficients. It is natural to ask for the minimal dimension of such a matrix. They show that every CM–field of degree 2n is generated by an eigenvalue of a skew–symmetric matrix over Q of dimension at most 4n+2. The aim of the present paper is to improve this bound.     

On theorems of Delsarte–McEliece and Chevalley–Warning–Ax–Katz

We present a theorem that generalizes the result of Delsarte and McEliece on the p-divisibilities of weights in abelian codes. Our result generalizes the Delsarte–McEliece theorem in the same sense that the theorem of N. M. Katz generalizes the theorem of Ax on the p-divisibilities of cardinalities of affine algebraic sets over finite fields. As the Delsarte–McEliece theorem implies the theorem of Ax, so our generalization implies that of N. M. Katz. The generalized theorem gives the p-divisibility of the t-wise Hamming weights of t-tuples of codewords (c ⁽¹⁾, . . . ,c ^(t)) as these words range over a product of abelian codes, where the t-wise Hamming weight is defined as the number of positions i in which the codewords do not simultaneously vanish, i.e., for which ${(c^{(1)}_i,\ldots,c^{(t)}_i)\not=(0,\ldots,0)}$ . We also present a version of the theorem that, for any list of t symbols s ₁, . . . ,s _t , gives p-adic estimates of the number of positions i such that ${(c^{(1)}_i,\ldots,c^{(t)}_i)=(s_1,\ldots,s_t)}$ as these words range over a product of abelian codes.     

Hardy–Littlewood–Sobolev and Stein–Weiss inequalities and integral systems on the Heisenberg group

In this paper, we study two types of weighted Hardy–Littlewood–Sobolev (HLS) inequalities, also known as Stein–Weiss inequalities, on the Heisenberg group. More precisely, we prove the |u|$\left|u\right|$ weighted HLS inequality in Theorem 1.1 and the |z|$\left|z\right|$ weighted HLS inequality in Theorem 1.5 (where we have denoted u=(z,t)$u=(z,t)$ as points on the Heisenberg group). Then we provide regularity estimates of positive solutions to integral systems which are Euler–Lagrange equations of the possible extremals to the Stein–Weiss inequalities. Asymptotic behavior is also established for integral systems associated to the |u|$\left|u\right|$ weighted HLS inequalities around the origin. By these a priori estimates, we describe asymptotically the possible optimizers for sharp versions of these inequalities.     

Zero–nonzero and real–nonreal sign determination

We consider first the zero–nonzero determination problem, which consists in determining the list of zero–nonzero conditions realized by a finite list of polynomials on a finite set Z⊂C^k$Z\subset {\mathrm{C}}^k$ with C an algebraic closed field. We describe an algorithm to solve the zero–nonzero determination problem and we perform its bit complexity analysis. This algorithm, which is in many ways an adaptation of the methods used to solve the more classical sign determination problem, presents also new ideas which can be used to improve sign determination. Then, we consider the real–nonreal sign determination problem, which deals with both the sign determination and the zero–nonzero determination problem. We describe an algorithm to solve the real–nonreal sign determination problem, we perform its bit complexity analysis and we discuss this problem in a parametric context.     

Matrix-exponential groups and Kolmogorov–Fokker–Planck equations

Aim of this paper is to provide new examples of H?rmander operators L{\mathcal{L}} to which a Lie group structure can be attached making L{\mathcal{L}} left invariant. Our class of examples contains several subclasses of operators appearing in literature and arising both in theoretical and in applied fields: evolution Kolmogorov operators, degenerate Ornstein–Uhlenbeck operators, Mumford and Fokker–Planck operators, Ornstein–Uhlenbeck operators with time-dependent periodic coefficients. Our examples basically come from exponential of matrices, as well as from linear constant-coefficient ODE’s, in \mathbbR{\mathbb{R}} or in \mathbbC{\mathbb{C}} . Furthermore, we describe how these groups can be combined together to obtain new structures and new operators, also having an interest in the applied field.     

Ringel duality and Auslander–Dlab–Ringel algebras

We introduce a new class of quasi-hereditary algebras, containing in particular the Auslander–Dlab–Ringel (ADR) algebras. We show that this new class of algebras is preserved under Ringel duality, which determines in particular explicitly the Ringel dual of any ADR algebra. As a special case of our theory, it follows that, under very restrictive conditions, an ADR algebra is Ringel dual to another one. The latter provides an alternative proof for a recent result of Conde and Erdmann, and places it in a more general setting.     

Generalized Gauss–Radau and Gauss–Lobatto Formulae

Computational methods are developed for generating Gauss-type quadrature formulae having nodes of arbitrary multiplicity at one or both end points of the interval of integration. Positivity properties of the boundary weights are investigated numerically, and related conjectures are formulated. Applications are made to moment-preserving spline approximation. AMS subject classification (2000) 65D32, 41A15.     

Spectral Triple and Sinai–Ruelle–Bowen Measures

In this article, we recover the Sinai–Ruelle–Bowen measure associated to a real-valued Hölder continuous function defined on the Julia set of a hyperbolic quadratic polynomial, as a noncommutative measure by constructing an appropriate spectral triple.     

Affine Moser–Trudinger and Morrey–Sobolev inequalities

An affine Moser–Trudinger inequality, which is stronger than the Euclidean Moser–Trudinger inequality, is established. In this new affine analytic inequality an affine energy of the gradient replaces the standard L ⁿ energy of gradient. The geometric inequality at the core of the affine Moser–Trudinger inequality is a recently established affine isoperimetric inequality for convex bodies. Critical use is made of the solution to a normalized version of the L ⁿ Minkowski Problem. An affine Morrey–Sobolev inequality is also established, where the standard L ^p energy, with p > n, is replaced by the affine energy.     

Mathias–Prikry and Laver–Prikry type forcing

We study the Mathias–Prikry and Laver–Prikry forcings associated with filters on ω. We give a combinatorial characterization of Martin?s number for these forcing notions and present a general scheme for analyzing preservation properties for them. In particular, we give a combinatorial characterization of those filters for which the Mathias–Prikry forcing does not add a dominating real.     

Rota–Baxter coalgebras and Rota–Baxter bialgebras

We give a construction of Rota–Baxter coalgebras from Hopf module coalgebras and also derive the structures of the pre-Lie coalgebras via Rota–Baxter coalgebras of different weight. Finally, the notion of Rota–Baxter bialgebra is introduced and some examples are provided.     

Mean and variance optimization of non–linear systems and worst–case analysis

In this paper, we consider expected value, variance and worst–case optimization of nonlinear models. We present algorithms for computing optimal expected value, and variance policies, based on iterative Taylor expansions. We establish convergence and consider the relative merits of policies based on expected value optimization and worst–case robustness. The latter is a minimax strategy and ensures optimal cover in view of the worst–case scenario(s) while the former is optimal expected performance in a stochastic setting. Both approaches are used with a small macroeconomic model to illustrate relative performance, robustness and trade-offs between the alternative policies.     

On Fano and weak Fano Bott–Samelson–Demazure–Hansen varieties

Let G be a simple algebraic group over the field of complex numbers. Fix a maximal torus T and a Borel subgroup B of G containing T. Let w be an element of the Weyl group W of G, and let $Z(\stackrel{?}{w})$ be the Bott–Samelson–Demazure–Hansen (BSDH) variety corresponding to a reduced expression $\stackrel{?}{w}$ of w with respect to the data $(G,B,T)$.In this article we give complete characterization of the expressions $\stackrel{?}{w}$ such that the corresponding BSDH variety $Z(\stackrel{?}{w})$ is Fano or weak Fano. As a consequence we prove vanishing theorems of the cohomology of tangent bundle of certain BSDH varieties and hence we get some local rigidity results.     

Renormalizability and Unitarity of the Englert–Broute–Higgs–Kibble Model

We show that the Englert–Broute–Higgs–Kibble model is renormalizable and unitary.     

Converse Sturm–Hurwitz–Kellogg theorem and related results

We prove that if Vⁿ is a Chebyshev system on the circle and f is a continuous real-valued function with at least n + 1 sign changes then there exists an orientation preserving diffeomorphism of S¹ that takes f to a function L²-orthogonal to V. We also prove that if f is a function on the real projective line with at least four sign changes then there exists an orientation preserving diffeomorphism of that takes f to the Schwarzian derivative of a function on . We show that the space of piecewise constant functions on an interval with values ± 1 and at most n + 1 intervals of constant sign is homeomorphic to n-dimensional sphere. To V. I. Arnold for his 70th birthday