Bent and generalized bent Boolean functions

In this paper, we investigate the properties of generalized bent functions defined on ${\mathbb{Z}_2^n}$ with values in ${\mathbb{Z}_q}$ , where q ≥ 2 is any positive integer. We characterize the class of generalized bent functions symmetric with respect to two variables, provide analogues of Maiorana–McFarland type bent functions and Dillon’s functions in the generalized set up. A class of bent functions called generalized spreads is introduced and we show that it contains all Dillon type generalized bent functions and Maiorana–McFarland type generalized bent functions. Thus, unification of two different types of generalized bent functions is achieved. The crosscorrelation spectrum of generalized Dillon type bent functions is also characterized. We further characterize generalized bent Boolean functions defined on ${\mathbb{Z}_2^n}$ with values in ${\mathbb{Z}_4}$ and ${\mathbb{Z}_8}$ . Moreover, we propose several constructions of such generalized bent functions for both n even and n odd.     

A new construction of bent functions based on {\mathbb{Z}} -bent functions

Dobbertin has embedded the problem of construction of bent functions in a recursive framework by using a generalization of bent functions called ${\mathbb{Z}}$ -bent functions. Following his ideas, we generalize the construction of partial spreads bent functions to partial spreads ${\mathbb{Z}}$ -bent functions of arbitrary level. Furthermore, we show how these partial spreads ${\mathbb{Z}}$ -bent functions give rise to a new construction of (classical) bent functions. Further, we construct a bent function on 8 variables which is inequivalent to all Maiorana–McFarland as well as PS _ap type bents. It is also shown that all bent functions on 6 variables, up to equivalence, can be obtained by our construction.     

Bent functions embedded into the recursive framework of {\mathbb{Z}} -bent functions

Suppose that n is even. Let ${\mathbb{F}_2}$ denote the two-element field and ${\mathbb{Z}}$ the set of integers. Bent functions can be defined as ± 1-valued functions on ${\mathbb{F}_2^n}$ with ± 1-valued Fourier transform. More generally we call a mapping f on ${\mathbb{F}_2^n}$ a ${\mathbb{Z}}$ -bent function if both f and its Fourier transform ${\widehat{f}}$ are integer-valued. ${\mathbb{Z}}$ -bent functions f are separated into different levels, depending on the size of the maximal absolute value attained by f and ${\widehat{f}}$ . It is shown how ${\mathbb{Z}}$ -bent functions of lower level can be built up recursively by gluing together ${\mathbb{Z}}$ -bent functions of higher level. This recursion comes down at level zero, containing the usual bent functions. In the present paper we start to study bent functions in the framework of ${\mathbb{Z}}$ -bent functions and give some guidelines for further research.     

Bent functions on partial spreads

For an arbitrary prime \(p\) we use partial spreads of \(\mathbb{F }_p^{2m}\) to construct two classes of bent functions from \(\mathbb{F }_p^{2m}\) to \(\mathbb{F }_p\) . Our constructions generalize the classes \(PS^{(-)}\) and \(PS^{(+)}\) of binary bent functions which are due to Dillon.     

Generating Functions for Spherical Harmonics and Spherical Monogenics

In this paper, we study generating functions for the standard orthogonal bases of spherical harmonics and spherical monogenics in \({\mathbb{R}^{m}}\) . Here spherical monogenics are polynomial solutions of the Dirac equation in \({\mathbb{R}^{m}}\) . In particular, we obtain the recurrence formula which expresses the generating function in dimension m in terms of that in dimension m–1. Hence we can find closed formulæ of generating functions in \({\mathbb{R}^{m}}\) by induction on the dimension m.     

A new class of bent and hyper-bent Boolean functions in polynomial forms

Bent functions are maximally nonlinear Boolean functions and exist only for functions with even number of inputs. This paper is a contribution to the construction of bent functions over ${\mathbb{F}_{2^{n}}}$ (n = 2m) having the form ${f(x) = tr_{o(s_1)} (a x^ {s_1}) + tr_{o(s_2)} (b x^{s_2})}$ where o(s _i ) denotes the cardinality of the cyclotomic class of 2 modulo 2 ⁿ ? 1 which contains s _i and whose coefficients a and b are, respectively in ${F_{2^{o(s_1)}}}$ and ${F_{2^{o(s_2)}}}$ . Many constructions of monomial bent functions are presented in the literature but very few are known even in the binomial case. We prove that the exponents s ₁ = 2 ^m ? 1 and ${s_2={\frac {2^n-1}3}}$ , where ${a\in\mathbb{F}_{2^{n}}}$ (a ?? 0) and ${b\in\mathbb{F}_{4}}$ provide a construction of bent functions over ${\mathbb{F}_{2^{n}}}$ with optimum algebraic degree. For m odd, we give an explicit characterization of the bentness of these functions, in terms of the Kloosterman sums. We generalize the result for functions whose exponent s ₁ is of the form r(2 ^m ? 1) where r is co-prime with 2 ^m  + 1. The corresponding bent functions are also hyper-bent. For m even, we give a necessary condition of bentness in terms of these Kloosterman sums.     

Gowers $$U_3$$ norm of some classes of bent Boolean functions

The Gowers \(U_3\) norm of a Boolean function is a measure of its resistance to quadratic approximations. It is known that smaller the Gowers \(U_3\) norm for a Boolean function larger is its resistance to quadratic approximations. Here, we compute Gowers \(U_3\) norms for some classes of Maiorana–McFarland bent functions. In particular, we explicitly determine the value of the Gowers \(U_3\) norm of Maiorana–McFarland bent functions obtained by using APN permutations. We prove that this value is always smaller than the Gowers \(U_3\) norms of Maiorana–McFarland bent functions obtained by using differentially \(\delta \)-uniform permutations, for all \(\delta \ge 4\). We also compute the Gowers \(U_3\) norms for a class of cubic monomial functions, not necessarily bent, and show that for \(n=6\), these norm values are less than that of Maiorana–McFarland bent functions. Further, we computationally show that there exist 6-variable functions in this class which are not bent but achieve the maximum second-order nonlinearity for 6 variables.     

A Family of Functions with Two Different Spectra of Singularities

Our goal is to study the multifractal properties of functions of a given family which have few non vanishing wavelet coefficients. We compute at each point the pointwise Hölder exponent of these functions and also their local \(L^p\) regularity, computing the so-called \(p\) -exponent. We prove that in the general case the Hölder and \(p\) -exponent are different at each point. We also compute the dimension of the sets where the functions have a given pointwise regularity and prove that these functions are multifractal both from the point of view of Hölder and \(L^p\) local regularity with different spectra of singularities. Furthermore, we check that multifractal formalism type formulas hold for functions in that family.     

Inductive topological Hausdorff dimensions and fibers of generic continuous functions

In an earlier paper Buczolich, Elekes and the author introduced a new concept of dimension for metric spaces, the so called topological Hausdorff dimension. They proved that it is precisely the right notion to describe the Hausdorff dimension of the level sets of the generic real-valued continuous function (in the sense of Baire category) defined on a compact metric space $K$ . The goal of this paper is to determine the Hausdorff dimension of the fibers of the generic continuous function from $K$ to $\mathbb {R}^n$ . In order to do so, we define the $n$ th inductive topological Hausdorff dimension, $\dim _{t^nH} K$ . Let $\dim _H K,\,\dim _t K$ and $C_n(K)$ denote the Hausdorff and topological dimension of $K$ and the Banach space of the continuous functions from $K$ to $\mathbb {R}^n$ . We show that $\sup _{y\in \mathbb {R}^n} \dim _{H}f^{-1}(y) = \dim _{t^nH} K -n$ for the generic $f \in C_n(K)$ , provided that $\dim _t K\ge n$ , otherwise every fiber is finite. In order to prove the above theorem we give some equivalent definitions for the inductive topological Hausdorff dimensions, which can be interesting in their own right. Here we use techniques coming from the theory of topological dimension. We show that the supremum is actually attained on the left hand side of the above equation. We characterize those compact metric spaces $K$ for which $\dim _{H} f^{-1}(y)=\dim _{t^nH}K-n$ for the generic $f\in C_n(K)$ and the generic $y\in f(K)$ . We also generalize a result of Kirchheim by showing that if $K$ is self-similar and $\dim _t K\ge n$ then $\dim _{H} f^{-1}(y)=\dim _{t^nH}K-n$ for the generic $f\in C_n(K)$ for every $y\in {{\mathrm{int}}}f(K)$ .     

Fourier analysis of subgroup conjugacy invariant functions on finite groups

Given a finite group $G$ and a subgroup $H\le G$ , we develop a Fourier analysis for $H$ -conjugacy invariant functions on $G$ , without the assumption that $H$ is a multiplicity-free subgroup of $G$ . We also study the Fourier transform for functions in the center of the algebra of $H$ -conjugacy invariant functions on $G$ . We show that a recent calculation of Cesi is indeed a Fourier transform of a function in the center of the algebra of functions on the symmetric group that are conjugacy invariant with respect to a Young subgroup.     

CCZ-equivalence of bent vectorial functions and related constructions

We observe that the CCZ-equivalence of bent vectorial functions over ${{\bf F}_2^n}$ (n even) reduces to their EA-equivalence. Then we show that in spite of this fact, CCZ-equivalence can be used for constructing bent functions which are new up to EA-equivalence and therefore to CCZ-equivalence: applying CCZ-equivalence to a non-bent vectorial function F which has some bent components, we get a function F?? which also has some bent components and whose bent components are CCZ-inequivalent to the components of the original function F. Using this approach we construct classes of nonquadratic bent Boolean and bent vectorial functions.     

A weak second term identity of the regularized Siegel-Weil formula for unitary groups

Following W. T. Gan and S. Takeda, we obtain a weak second term identity of the regularized Siegel-Weil formula for the unitary dual pair $(U(n,n),U(V))$ , where $V$ is a split hermitian space of dimension $2r$ with $r+1\le n \le 2r-1$ . As an application, we obtain a Rallis inner product formula for theta lifts from $U(W)$ to $U(V)$ for a skew-hermitian space $W$ of dimension $n$ .     

f-Vectors Implying Vertex Decomposability

We prove that if a pure simplicial complex $\Delta $ of dimension $d$ with $n$ facets has the least possible number of $(d-1)$ -dimensional faces among all complexes with $n$ faces of dimension $d$ , then it is vertex decomposable. This answers a question of J. Herzog and T. Hibi. In fact, we prove a generalization of their theorem using combinatorial methods.     

Young walls and graded dimension formulas for finite quiver Hecke algebras of type A^{(2)}_{2\ell } and D^{(2)}_{\ell +1}

We study graded dimension formulas for finite quiver Hecke algebras \(R^{\Lambda _0}(\beta )\) of type \(A^{(2)}_{2\ell }\) and \(D^{(2)}_{\ell +1}\) using combinatorics of Young walls. We introduce the notion of standard tableaux for proper Young walls and show that the standard tableaux form a graded poset with lattice structure. We next investigate Laurent polynomials associated with proper Young walls and their standard tableaux arising from the Fock space representations consisting of proper Young walls. Then, we prove the graded dimension formulas described in terms of the Laurent polynomials. When evaluating at \(q=1\) , the graded dimension formulas recover the dimension formulas for \(R^{\Lambda _0}(\beta )\) described in terms of standard tableaux of strict partitions.     

A backward \lambda -lemma for the forward heat flow

The inclination or \(\lambda \) -lemma is a fundamental tool in finite dimensional hyperbolic dynamics. In contrast to finite dimension, we consider the forward semi-flow on the loop space of a closed Riemannian manifold \(M\) provided by the heat flow. The main result is a backward \(\lambda \) -lemma for the heat flow near a hyperbolic fixed point \(x\) . There are the following novelties. Firstly, infinite versus finite dimension. Secondly, semi-flow versus flow. Thirdly, suitable adaption provides a new proof in the finite dimensional case. Fourthly and a priori most surprisingly, our \(\lambda \) -lemma moves the given disk transversal to the unstable manifold backward in time, although there is no backward flow. As a first application we propose a new method to calculate the Conley homotopy index of \(x\) .     

Hausdorff measures of subgroups of \mathbb{R }/\mathbb{Z } and \mathbb{R }

For a sequence $\underline{u}=(u_n)_{n\in \mathbb{N }}$ of integers, let $t_{\underline{u}}(\mathbb{T })$ be the group of all topologically $\underline{u}$ -torsion elements of the circle group $\mathbb{T }:=\mathbb{R }/\mathbb{Z }$ . We show that for any $s\in ]0,1[$ and $m\in \{0,+\infty \}$ there exists $\underline{u}$ such that $t_{\underline{u}}(\mathbb{T })$ has Hausdorff dimension $s$ and $s$ -dimensional Hausdorff measure equal to $m$ (no other values for $m$ are possible). More generally, for dimension functions $f,g$ with $f(t)\prec g(t), f(t)\prec \!\!\!\prec t$ and $g(t)\prec \!\!\!\prec t$ we find $\underline{u}$ such that $t_{\underline{u}}(\mathbb{T })$ has at the same time infinite $f$ -measure and null $g$ -measure.     

Adjunction and singular loci of hyperplane sections,II

Let \((X,L)\) be a smooth polarized variety of dimension \(n\) . Let \(A\in |L|\) be an irreducible hypersurface and let \(\Sigma \) be the singular locus of \(A\) . We assume that \(\Sigma \) is a smooth subvariety of dimension \(k\ge 2\) , and odd codimension \(\ge 3\) . Motivated from the results of Beltrametti et al. (J. Math. Soc. Jpn. 2014), we study the nefness and bigness of the adjoint bundle \(K_{\Sigma }+ (k-2)L_{\Sigma }\) in this framework. Several explicit examples show that the results are effective.     

1/2-Heavy sequences driven by rotation

We investigate the set of \(x \in S^1\) such that for every positive integer \(N\) , the first \(N\) points in the orbit of \(x\) under rotation by irrational \(\theta \) contain at least as many values in the interval \([0,1/2]\) as in the complement. By using a renormalization procedure, we show both that the Hausdorff dimension of this set is the same constant (strictly between zero and one) for almost-every \(\theta \) , and that for every \(d \in [0,1]\) there is a dense set of \(\theta \) for which the Hausdorff dimension of this set is \(d\) .     

Automorphisms of Some Topological Regular Parallelisms of {{\rm PG}(3, \mathbb{R})}

In a precedent article we constructed various topological regular parallelisms of the real projective 3-space \({{\rm PG}(3, \mathbb{R})}\) via hyperflock determining line sets of \({{\rm PG}(5, \mathbb{R})}\) (see Betten and Riesinger in Mh Math 161:43–58, 2010). In the present paper we discuss for some of these parallelisms their automorphism groups consisting of all automorphic collineations and all automorphic dualities, especially we compute their group dimension. Thus we are able to present: (1) topological regular 5-dimensional parallelisms of \({{\rm PG}(3, \mathbb{R})}\) of group dimension 0, (2) topological regular 4-dimensional parallelisms of \({{\rm PG}(3, \mathbb{R})}\) of group dimension 0 or 1, (3) topological regular 3-dimensional parallelisms of \({{\rm PG}(3, \mathbb{R})}\) of group dimension 1.     

On zeros of Kloosterman sums

A Kloosterman zero is a non-zero element of ${{\mathbb F}_q}$ for which the Kloosterman sum on ${{\mathbb F}_q}$ attains the value 0. Kloosterman zeros can be used to construct monomial hyperbent (bent) functions in even (odd) characteristic, respectively. We give an elementary proof of the fact that for characteristic 2 and 3, no Kloosterman zero in ${{\mathbb F}_q}$ belongs to a proper subfield of ${{\mathbb F}_q}$ with one exception that occurs at q = 16. It was recently proved that no Kloosterman zero exists in a field of characteristic greater than 3. We also characterize those binary Kloosterman sums that are divisible by 16 as well as those ternary Kloosterman sums that are divisible by 9. Hence we provide necessary conditions that Kloosterman zeros must satisfy.