Partially-bent functions

We study a conjecture stated in [6] about the numbers of non-zeros of, respectively, the auto-correlation function and the Walsh transform of the function (–1) ^f(x) , wheref(x) is any boolean function on {0, 1} ⁿ . The result that we obtain leads us to introduce the class of partially-bent functions. We study within these functions the propagation criterion. We characterize those partially-bent functions which are balanced and prove a relation between their number (which is unknown) and the number of non-balanced partially-bent functions on {0, 1} ^n–1. Eventually, we study their correlation immunity.     

Multiplication Operators on the Lipschitz Space of a Tree

In this paper, we study the multiplication operators on the space of complex-valued functions f on the set of vertices of a rooted infinite tree T which are Lipschitz when regarded as maps between metric spaces. The metric structure on T is induced by the distance function that counts the number of edges of the unique path connecting pairs of vertices, while the metric on ℂ is Euclidean. After observing that the space L{\mathcal{L}} of such functions can be endowed with a Banach space structure, we characterize the multiplication operators on L{\mathcal{L}} that are bounded, bounded below, and compact. In addition, we establish estimates on the operator norm and on the essential norm, and determine the spectrum. We then prove that the only isometric multiplication operators on L{\mathcal{L}} are the operators whose symbol is a constant of modulus one. We also study the multiplication operators on a separable subspace of L{\mathcal{L}} we call the little Lipschitz space.     

DIEUDONNÉ—KÖTHE DUALITY FOR VECTOR—VALUED FUNCTION SPACES: LOCALIZATION OF BOUNDED SETS AND BARRELLEDNESS

Abstract

We study Dieudonné-Köthe spaces of Lusin-measurable functions with values in a locally convex space. Let Λ be a solid locally convex lattice of scalar-valued measurable functions defined on a measure space Ω. If E is a locally convex space, define Λ {E} as the space of all Lusinmeasurable functions f: Ω → E such that q(f(·)) is a function in Λ for every continuous seminorm q on E. The space Λ {E} is topologized in a natural way and we study some aspects of the locally convex structure of A {E}; namely, bounded sets, completeness, duality and barrelledness. In particular, we focus on the important case when Λ and E are both either metrizable or (DF)-spaces and derive good permanence results for reflexivity when the density condition holds.     

Arithmetic of semigroups of series in multiplicative systems

We study the arithmetic of a semigroup M_P\mathcal{M}_{\mathcal{P}} of functions with operation of multiplication representable in the form f(x) = ?_{n = 0}^￥ a_nc_n(x)    ( a_n 3 0,?_{n = 0}^￥ a_n = 1 ) f(x) = \sum\nolimits_{n = 0}^\infty {{a_n}{\chi_n}(x)\quad \left( {{a_n} \ge 0,\sum\nolimits_{n = 0}^\infty {{a_n} = 1} } \right)} , where { c_n }_{n = 0}^￥ \left\{ {{\chi_n}} \right\}_{n = 0}^\infty is a system of multiplicative functions that are generalizations of the classical Walsh functions. For the semigroup M_P\mathcal{M}_{\mathcal{P}}, analogs of the well-known Khinchin theorems related to the arithmetic of a semigroup of probability measures in R ⁿ are true. We describe the class I₀(M_P)I_0(\mathcal{M}_{\mathcal{P}}) of functions without indivisible or nondegenerate idempotent divisors and construct a class of indecomposable functions that is dense in M_P\mathcal{M}_{\mathcal{P}} in the topology of uniform convergence.     

Spectral Concentration of Positive Functions on?Compact Groups

The problem of understanding the Fourier-analytic structure of the cone of positive functions on a group has a long history. In this article, we develop the first quantitative spectral concentration results for such functions over arbitrary compact groups. Specifically, we describe a family of finite, positive quadrature rules for the Fourier coefficients of band-limited functions on compact groups. We apply these quadrature rules to establish a spectral concentration result for positive functions: given appropriately nested band limits A ì B ì [^(G)]\mathcal {A}\subset \mathcal {B} \subset\widehat{G}, we prove a lower bound on the fraction of L ²-mass that any B\mathcal {B}-band-limited positive function has in A\mathcal {A}. Our bounds are explicit and depend only on elementary properties of A\mathcal {A} and B\mathcal {B}; they are the first such bounds that apply to arbitrary compact groups. They apply to finite groups as a special case, where the quadrature rule is given by the Fourier transform on the smallest quotient whose dual contains the Fourier support of the function.     

On the structure of Gabor and super Gabor spaces

We study the structure of Gabor and super Gabor spaces inside L²(\mathbbR^2d){L^{2}(\mathbb{R}^{2d})} and specialize the results to the case where the spaces are generated by vectors of Hermite functions. We then construct an isometric isomorphism between such spaces and Fock spaces of polyanalytic functions and use it in order to obtain structure theorems and orthogonal projections for both spaces at once, including explicit formulas for the reproducing kernels. In particular we recover a structure result obtained by N. Vasilevski using complex analysis and special functions. In contrast, our methods use only time-frequency analysis, exploring a link between time-frequency analysis and the theory of polyanalytic functions, provided by the polyanalytic part of the Gabor transform with a Hermite window, the polyanalytic Bargmann transform.     

A New Class of Inner Functions Uniformly Approximable by Interpolating Blaschke Products

We study the class of inner functions Q{\Theta} whose zero set Z(Q){Z(\Theta)} stays hyperbolically close to [`(Z_\mathbbD(Q))]{\overline{Z_\mathbb{D}(\Theta)}} on the corona of H ^∞ and show that these functions are uniformly approximable by interpolating Blaschke products.     

Two results on maximum nonlinear functions

Maximum nonlinear functions are widely used in cryptography because the coordinate functions F _β (x) := tr(β F(x)), , have large distance to linear functions. Moreover, maximum nonlinear functions have good differential properties, i.e. the equations F(x + a) − F(x) = b, , have 0 or 2 solutions. Two classes of maximum nonlinear functions are the Gold power functions , gcd(k, m) = 1, and the Kasami power functions , gcd(k, m) = 1. The main results in this paper are: (1) We characterize the Gold power functions in terms of the distance of their coordinate functions to characteristic functions of subspaces of codimension 2 in . (2) We determine the differential properties of the Kasami power functions if gcd(k,m) ≠ 1.      

The $(1-\mathbb{E})$-transform in combinatorial Hopf algebras

We extend to several combinatorial Hopf algebras the endomorphism of symmetric functions sending the first power-sum to zero and leaving the other ones invariant. As a “transformation of alphabets”, this is the (1-\mathbbE)(1-\mathbb{E})-transform, where \mathbbE\mathbb{E} is the “exponential alphabet,” whose elementary symmetric functions are e_n=\frac1n!e_{n}=\frac{1}{n!}. In the case of noncommutative symmetric functions, we recover Schocker’s idempotents for derangement numbers (Schocker, Discrete Math. 269:239–248, 2003). From these idempotents, we construct subalgebras of the descent algebras analogous to the peak algebras and study their representation theory. The case of WQSym leads to similar subalgebras of the Solomon–Tits algebras. In FQSym, the study of the transformation boils down to a simple solution of the Tsetlin library in the uniform case.     

Further properties of several classes of Boolean functions with optimum algebraic immunity

Based on a method proposed by the first author, several classes of balanced Boolean functions with optimum algebraic immunity are constructed, and they have nonlinearities significantly larger than the previously best known nonlinearity of functions with optimal algebraic immunity. By choosing suitable parameters, the constructed n-variable functions have nonlinearity for even for odd n, where Δ(n) is a function increasing rapidly with n. The algebraic degrees of some constructed functions are also discussed.      

Universally prestarlike functions as convolution multipliers

Universally prestarlike functions (of order α ≤ 1) in the slit domain L:=\mathbbC\[1,￥]{\Lambda:=\mathbb{C}{\setminus}[1,\infty]} have recently been introduced in Ruscheweyh et al. (Israel J Math, to appear). This notation generalizes the corresponding one for functions in the unit disk \mathbbD{\mathbb{D}} (and other circular domains in \mathbbC{\mathbb{C}}). In this paper we study the behaviour of universally prestarlike functions under the Hadamard product. In particular it is shown that these function classes (with α fixed), are closed under convolution, and that their members, as Hadamard multipliers, also preserve the prestarlikeness (of the same order) of functions in arbitrary circular domains containing the origin.     

Local Łojasiewicz exponents,Milnor numbers and mixed multiplicities of ideals

Let be a finite analytic map. We give an expression for the local Łojasiewicz exponent and for the multiplicity of g when the component functions of g satisfy certain condition with respect to a set of n monomial ideals I ₁,..., I _n . We give an effective method to compute Łojasiewicz exponents based on the computation of mixed multiplicities. As a consequence of our study, we give a numerical characterization of a class of functions that includes semi-weighted homogenous functions and Newton non-degenerate functions. Work supported by DGICYT Grant MTM2006-06027.     

Characterizations of Bent and Almost Bent Function on {\mathbb{Z}_p^2}

Bent and almost-bent functions on \mathbbZ_p²{\mathbb{Z}_p^2} are studied in this paper. By calculating certain exponential sum and using a technique due to Hou (Finite Fields Appl 10:566–582, 2004), we obtain a degree bound for quasi-bent functions, and prove that almost-bent functions on \mathbbZ_p²{\mathbb{Z}_p^2} are equivalent to a degenerate quadratic form. From the viewpoint of relative difference sets, we also characterize bent functions on \mathbbZ_p²{\mathbb{Z}_p^2} in two classes of M{\mathcal{M}} ’s and PS{\mathcal{PS}} ’s, and show that the graph set corresponding to a bent function on \mathbbZ_p²{\mathbb{Z}_p^2} can be written as the sum of a graph set of M{\mathcal{M}} ’s type bent function and another group ring element. By using our characterization and some technique of permutation polynomial, we obtain the result: a bent function must be of M{\mathcal{M}} ’s type if its corresponding set contains more than (p − 3)/2 flats. A problem proposed by Ma and Pott (J Algebra 175:505–525, 1995) is therefore partially answered.     

Biharmonic Green Functions on Homogeneous Trees

The study of biharmonic functions under the ordinary (Euclidean) Laplace operator on the open unit disk \mathbbD{\mathbb{D}} in \mathbbC{\mathbb{C}} arises in connection with plate theory, and in particular, with the biharmonic Green functions which measure, subject to various boundary conditions, the deflection at one point due to a load placed at another point. A homogeneous tree T is widely considered as a discrete analogue of the unit disk endowed with the Poincaré metric. The usual Laplace operator on T corresponds to the hyperbolic Laplacian. In this work, we consider a bounded metric on T for which T is relatively compact and use it to define a flat Laplacian which plays the same role as the ordinary Laplace operator on \mathbbD{\mathbb{D}}. We then study the simply-supported and the clamped biharmonic Green functions with respect to both Laplacians.     

Empirical Processes with a Bounded Ψ 1 Diameter

We study the empirical process ${{\rm sup}_{f \in F}|N^{-1}\sum_{i=1}^{N}\,f^{2}(X_i)-\mathbb{E}f^{2}|}We study the empirical process sup_{f ? F}|N^-1?_i=1^N f²(X_i)-\mathbbEf²|{{\rm sup}_{f \in F}|N^{-1}\sum_{i=1}^{N}\,f^{2}(X_i)-\mathbb{E}f^{2}|}, where F is a class of mean-zero functions on a probability space (Ω, μ), and (X_i)_{i = 1}^N{(X_{i})_{i =1}^N} are selected independently according to μ.     

On the ideal of functions with compact support in pointfree function rings

As usual, let RL\mathcal{R}L denote the ring of real-valued continuous functions on a completely regular frame L. We consider the ideals R_s(L)\mathcal{R}_{s}(L) and R_K(L)\mathcal{R}_{K}(L) consisting, respectively, of functions with small cozero elements and functions with compact support. We show that, as in the classical case of C(X), the first ideal is the intersection of all free maximal ideals, and the second is the intersection of pure parts of all free maximal ideals. A corollary of this latter result is that, in fact, R_K(L)\mathcal{R}_{K}(L) is the intersection of all free ideals. Each of these ideals is pure, free, essential or zero iff the other has the same feature. We observe that these ideals are free iff L is a continuous frame, and essential iff L is almost continuous (meaning that below every nonzero element there is a nonzero element the pseudocomplement of which induces a compact closed quotient). We also show that these ideals are zero iff L is nowhere compact (meaning that non-dense elements induce non-compact closed quotients).     

Complete Asymptotic Expansion for Generalized Favard Operators

In the present paper we consider a generalization _boxclose F_{n,\sigma_{n}} of the Favard operators and study the local rate of convergence for smooth functions. As a main result we derive the complete asymptotic expansion for the sequence ( F_n,snf)( x)( F_{n,\sigma _{n}}f)( x) as n tends to infinity. Furthermore, we consider a truncated version of these operators. Finally, all results were proved for simultaneous approximation.     

On the Complexity Functions for T-Ideals of Associative Algebras

Let be the sequence of codimension growth for a variety V of associative algebras. We study the complexity function , which is the exponential generating function for the sequence of codimensions. Earlier, the complexity functions were used to study varieties of Lie algebras. The objective of the note is to start the systematic investigation of complexity functions in the associative case. These functions turn out to be a useful tool to study the growth of varieties over a field of arbitrary characteristic. In the present note, the Schreier formula for the complexity functions of one-sided ideals of a free associative algebra is found. This formula is applied to the study of products of T-ideals. An exact formula is obtained for the complexity function of the variety U c of associative algebras generated by the algebra of upper triangular matrices, and it is proved that the function is a quasi-polynomial. The complexity functions for proper identities are investigated. The results for the complexity functions are applied to study the asymptotics of codimension growth. Analogies between the complexity functions of varieties and the Hilbert--Poincaré series of finitely generated algebras are traced.     

Constructions of p-ary Quadratic Bent Functions

Bent functions have many applications in the fields of coding theory, communications and cryptography. This paper studies the constructions of bent functions having the form for odd n and for even n, over the finite field of odd characteristic p, where . Based on the irreducibility of some polynomials on , we focus on characterizing the bent functions for n=p ^v q ^r and n=2p ^v q ^r , where is an odd prime and p a primitive root modulo q ². Moreover, the enumerations of those functions are also considered. Partially supported by the NSF of China under Grants No. 60603012 and No. 60573053.