Related papers: On $q$-nearly bent Boolean functions
In the recent paper [2], it was proved that the closure of the planar diffeomorphisms in the Sobolev norm consists of the functions which are non-crossing (NC), i.e., the functions which can be uniformly approximated by continuous…
We prove a local contraction property for holomorphic functions that are nearly constant, relating weighted Bergman spaces $A^p_\alpha(\B_n)$ and $A^q_\beta(\B_n)$. Our approach converts geometric information on weighted superlevel sets…
We prove two main results on how arbitrary linear threshold functions $f(x) = \sign(w\cdot x - \theta)$ over the $n$-dimensional Boolean hypercube can be approximated by simple threshold functions. Our first result shows that every…
In this paper, we consider the problem of finding perfectly balanced Boolean functions with high non-linearity values. Such functions have extensive applications in domains such as cryptography and error-correcting coding theory. We provide…
Bent functions are Boolean functions in an even number of variables that are indicators of Hadamard difference sets in elementary abelian 2-groups. A bent function in m variables is said to be normal if it is constant on an affine space of…
Inspired by a recent work of Mesnager, we present several new infinite families of quadratic ternary bent, near-bent and 2-plateaued functions from some known quadratic ternary bent functions. Meanwhile, the distribution of the Walsh…
First of all we give some reasons that "natural proofs" built not a barrier to prove P $\not=$ NP using Boolean complexity. Then we investigate the approximation method for its extension to prove super-polynomial lower bounds for the…
We construct infinite classes of almost bent and almost perfect nonlinear polynomials, which are affinely inequivalent to any sum of a power function and an affine function.
We consider negabent Boolean functions that have Trace representation. We completely characterize quadratic negabent monomial functions. We show the relation between negabent functions and bent functions via a quadratic function. Using this…
Roughly speaking, a near-best (abbr. NB) quasi-interpolant (abbr. QI) is an approximation operator of the form $Q_af=\sum_{\alpha\in A} \Lambda_\alpha (f) B_\alpha$ where the $B_\alpha$'s are B-splines and the $\Lambda_\alpha (f)$'s are…
Homogeneous rotation symmetric Boolean functions have been extensively studied in recent years because of their applications in cryptography. Little is known about the basic question of when two such functions are affine equivalent. The…
We prove that, to compute a Boolean function $f$ on $N$ variables with error probability $\epsilon$, any quantum black-box algorithm has to query at least $\frac{1 - 2\sqrt{\epsilon}}{2} \rho_f N = \frac{1 - 2\sqrt{\epsilon}}{2} \bar{S}_f$…
We study the most-informative Boolean function conjecture using a differential equation approach. This leads to a formulation of a functional inequality on finite-dimensional random variables. We also develop a similar inequality in the…
A function $f$ is $d$-resilient if all its Fourier coefficients of degree at most $d$ are zero, i.e., $f$ is uncorrelated with all low-degree parities. We study the notion of $\mathit{approximate}$ $\mathit{resilience}$ of Boolean…
We solve the problem of Fourier transformation for the one-dimensional $q$-deformed Heisenberg algebra. Starting from a matrix representation of this algebra we observe that momentum and position are unbounded operators in the Hilbert…
A motivation comes from {\em M. Ismail and et al.: A generalization of starlike functions, Complex Variables Theory Appl., 14 (1990), 77--84} to study a generalization of close-to-convex functions by means of a $q$-analog of a difference…
We design a nonadaptive algorithm that, given oracle access to a function $f: \{0,1\}^n \to \{0,1\}$ which is $\alpha$-far from monotone, makes poly$(n, 1/\alpha)$ queries and returns an estimate that, with high probability, is an…
We characterize when the level sets of a continuous quasi-monotone functional defined on a suitable convex subset of a normed space can be uniquely represented by a family of bounded continuous functionals. Furthermore, we investigate how…
Much work has been devoted to bent functions in odd characteristic, but there still remains a gap between our knowledge of binary and nonbinary bent functions. In the first part of this paper, we attempt to partially bridge this gap by…
Prompted by an observation about the integral of exponential functions of the form $f(x)=\lambda e^{\alpha x}$, we investigate the possibility to exactly integrate families of functions generated from a given function by scaling or by…