Homogeneous bent functions

We establish a new connection between invariant theory and the theory of bent functions. This enables us to construct Boolean functions with a prescribed symmetry group action. Besides the quadratic bent functions the only other previously known homogeneous bent functions are the six variable degree three functions constructed in l16r. We show that these bent functions arise as invariants under an action of the symmetric group on four letters and determine the stabilizer which turns out to be a matrix group of order 10752. We apply the machinery of invariant theory in order to construct homogeneous bent functions of degree three in 8, 10, and 12 variables. This approach gives a great computational advantage over the unstructured search problem and yields Boolean functions which have a concise description in terms of certain designs and graphs. We consider the question of linear equivalence of the constructed bent functions and study the properties of the associated elementary abelian difference sets.

Recently in Çeşmelioğlu, Meidl (Adv. Math. Commun.,18, 2024), the study of EA-equivalence and CCZ-equivalence for functions from {mathbb {V}}_n^{(p)} to the cyclic group {mathbb {Z}}_{p^k} has been initiated, where {mathbb {V}}_n^{(p)} denotes an n-dimensional vector space over {mathbb {F}}_p. Amongst others it has been shown that there exist functions from {mathbb {V}}_n^{(2)} to {mathbb {Z}}_4 which are CCZ-equivalent but not EA-equivalent. We extend these results to larger classes of functions from {mathbb {V}}_n^{(p)} to {mathbb {Z}}_{p^k}. We then discuss constructions of generalized bent functions from {mathbb {V}}_n^{(p)} to {mathbb {Z}}_{p^k}, p odd or p=2 and n is even, which correspond to large affine spaces of bent functions. In particular we employ versions of the direct sum, the semi-direct sum and of a recent secondary bent function construction in Wang et. al., (IEEE Trans. Inform. Theory69, 2023), to generate large affine spaces of bent functions. Finally we present a solution for constructing generalized bent functions from {mathbb {V}}_n^{(2)} to {mathbb {Z}}_{2^k}, n odd, from arbitrary generalized bent functions from {mathbb {V}}_{n-1}^{(2)} to {mathbb {Z}}_{2^{k-1}}.

This paper aims at the identification of classes of bent functions in order to allow their construction without searching or sieving. In order to reach this aim, we studied first the relationship between bent functions and complexity classes defined by the Specific Normal Forms of all Boolean functions. As result of this exploration we found classes of bent functions which are embedded in different complexity classes defined by the Specific Normal Form. In the second step to reach our global aim, we utilized the found classes of bent functions in order to express bent functions in terms of derivative operations of the Boolean Differential Calculus. In detail, we studied bent functions of two and four variables. This exploration leads finally to Boolean differential equations that will allow the direct calculation of all bent functions of two and four variables. A given generalization allows to calculate subsets of bent functions for each even number of Boolean variables.

Rotation Symmetric Boolean Functions –; Count and Cryptographic Properties

On the degree of homogeneous bent functions

This paper is devoted to the study of slowly varying and almost periodic at infinity distributions from harmonic spaces; a number of spaces of homogeneous functions are considered. The notion of a harmonic space of distributions is introduced; this space is constructed by a homogeneous functional spaces. Properties of harmonic spaces of distributions endowed with the structure of Banach modules are studied. We prove that such spaces are isometrically isomorphic to the corresponding homogeneous functional spaces. Based on the definitions of slowly varying and almost periodic at infinity functions from a homogeneous space, we introduce the notions of slowly varying and almost periodic at infinity distributions from a harmonic space. Using methods of abstract harmonic analysis, we construct Fourier series of almost periodic distributions at infinity and examine their properties. In this paper, we essentially used results of the theory of isometric representations and the theory of Banach modules.

By the relationship between the first linear spectra of a function at partial points and the Hamming weights of the sub-functions, and by the Hamming weight of homogenous Boolean function, it is proved that there exist no homogeneous bent functions of degreem inn=2m variables form>3.

The aim of this paper is to explain the classification of the unitary SL_2(R) representations done by Gelfand by using the induced representation technique. We induce the SL_2(R) representation from the subgroup N. We get a representation constructed on a space of homogeneous functions in two variables. Then, we move to induce the SL_2(R) representation in stages. Consequently, the representation of SL_2(R) acts on a space of functions of one variable.

The homogeneous polynomial solutions for the Grushin operator

We consider the functional linear regression model where the explanatory variable is a random surface and the response is a real random variable, in various situations where both the explanatory variable and the noise can be unbounded and dependent. Bivariate splines over triangulations represent the random surfaces. We use this representation to construct least squares estimators of the regression function with a penalisation term. Under the assumptions that the regressors in the sample span a large enough space of functions, bivariate splines approximation properties yield the consistency of the estimators. Simulations demonstrate the quality of the asymptotic properties on a realistic domain. We also carry out an application to ozone concentration forecasting over the USA that illustrates the predictive skills of the method.

The problem of identifying the satisfiability threshold of random 3-SAT formulas has received a lot of attention during the last decades and has inspired the study of other threshold phenomena in random combinatorial structures. The classical assumption in this line of research is that, for a given set of n Boolean variables, each clause is drawn uniformly at random among all sets of three literals from these variables, independently from other clauses. Here, we keep the uniform distribution of each clause, but deviate significantly from the independence assumption and consider richer families of probability distributions. For integer parameters n, m, and k, we denote by ℱ_k(n,m) the family of probability distributions that produce formulas with m clauses, each selected uniformly at random from all sets of three literals from the n variables, so that the clauses are k-wise independent. Our aim is to make general statements about the satisfiability or unsatisfiability of formulas produced by distributions in ℱ_k(n,m) for different values of the parameters n, m, and k. Our technical results are as follows: First, all probability distributions in ℱ₂(n,m) with m ∈ Ω(n³) return unsatisfiable formulas with high probability. This result is tight. We show that there exists a probability distribution 𝒟 ∈ ℱ₃(n,m) with m ∈ O(n³) so that a random formula drawn from 𝒟 is almost always satisfiable. In contrast, for m ∈ Ω(n²), any probability distribution 𝒟 ∈ ℱ₄(n,m) returns an unsatisfiable formula with high probability. This is our most surprising and technically involved result. Finally, for any integer k ≥ 2, any probability distribution 𝒟 ∈ ℱ_k(n,m) with m ∈ O(n^{1-1/k}) returns a satisfiable formula with high probability.

System architecture design using multi-criteria optimization is demonstrated using a case study of an aero engine health management (EHM) system. A design process for optimal deployment of EHM system functional operations over physical architecture component locations, e.g., on-engine, on-aircraft and on-ground, is described. The EHM system architecture design needs to be optimized with respect to many qualitative criteria in terms of operational attributes within the constraints of resource limitations. In this paper the system architecture design problem is formulated as a multi-criteria optimization problem. Considering the large discrete search space of decision variables and many-objective functions and constraints, an evolutionary multi-objective genetic algorithm along with a progressive preference articulation technique, is used for solving the optimization problem. The optimization algorithm found a family of Pareto solutions which provided valuable insight into design trade-offs. Using the progressive preference articulation technique, the optimization search can be focused for the industrial decision maker on to a region of interest in the objective space. Performance of the proposed method is evaluated using various test metrics. Using this approach it was possible to identify the most significant design constraints (“hot spots”) and the opportunities afforded by either the relaxation or the tightening of these constraints, along with their performance implications.

Today, the contraction of green zones is a key issue for land use planning with relation to climate change effects on urban areas. Furthermore, the loss of evapotranspiring surfaces and vegetated soils is one of the main consequences of urban sprawl processes. The authors present the case of Catania metropolitan area, the 2nd most populated urban region in Sicily. In particular, three municipalities are analyzed, as they present the most relevant urban sprawl processes. Inside this complex “urban jam,” there are still large non urbanized spaces. These patches (cultivated and abandoned agriculture land and lava fields from Mt. Etna), deeply fragmented, are often left for future development. These areas are particularly important in the examined context, considering the lack of green spaces for ecological functions and leisure. This chapter focuses on land cover analysis based on land use maps and oriented to assess evapotranspiration degree of the different land uses. Land use categories have been geographically sampled, and eight land cover types have been extracted with GIS by photo interpretation of high resolution orthophotos. Other sets of geodatabases have been used, including vectorial/raster cartographies and field surveys. Results from this evapotranspiration assessment can be useful for addressing land use planning of non urbanized areas within a sprawled metropolitan area, identifying new forms of agriculture, leisure, and environment protection.

The reliability of spin polarization method results for atomic spin densities, obtained with several widely used Gaussian basis sets, is examined by comparison with the results of full configuration interaction (FCI) calculations. The spin densities obtained with these basis sets using the spin polarization model and some other methods disagree with the FCI treatment. Since the FCI wave function is exact for a given basis, it is not clear that the spin polarization model will be generally reliable. A large active space multiconfigurational (CASSCF) calculation is shown to be inadequate as an alternative to FCI treatment. The importance of accounting at least to some extent for excitations to all orbitals in the complete space of basis functions is illustrated by very slow convergence of CASSCF results with increasing size of active space. The FCI results reported here can be used as benchmarks to test various approaches to spin density calculation.

In 2010, Shpilka and Volkovich established a prominent result on the equivalence of polynomial factorization and identity testing. It follows from their result that a multilinear polynomial over the finite field of order 2 can be factored in time cubic in the size of the polynomial given as a string. Later, we have rediscovered this result and provided a simple factorization algorithm based on computations over derivatives of multilinear polynomials. The algorithm has been applied to solve problems of compact representation of various combinatorial structures, including Boolean functions and relational data tables. In this paper, we describe an improvement of this factorization algorithm and report on preliminary experimental analysis.