Related papers: A characterization of 2-threshold functions via pa…

A $k$-threshold function on a rectangular grid of size $m \times n$ is the conjunction of $k$ threshold functions on the same domain. In this paper, we focus on the case $k=2$ and show that the number of two-dimensional 2-threshold…

A two-dimensional threshold function of k-valued logic can be viewed as coloring of the points of a k x k square lattice into two colors such that there exists a straight line separating points of different colors. For the number of such…

Let $m,n\ge 2$, $m\le n$. It is well-known that the number of (two-dimensional) threshold functions on an $m\times n$ rectangular grid is {eqnarray*} t(m,n)=\frac{6}{\pi^2}(mn)^2+O(m^2n\log{n})+O(mn^2\log{\log{n}})=…

It is known that a minimal teaching set of any threshold function on the twodimensional rectangular grid consists of 3 or 4 points. We derive exact formulae for the numbers of functions corresponding to these values and further refine them…

The class of threshold functions is known to be characterizable by functional equations or, equivalently, by pairs of relations, which are called relational constraints. It was shown by Hellerstein that this class cannot be characterized by…

In this paper, we consider situations in which a given logical function is realized by a multithreshold threshold function. In such situations, constant functions can be easily obtained from multithreshold threshold functions, and…

Two point correlation functions of the off-critical primary fields \phi_{1, 1+s} are considered in the perturbed minimal models M_{2, 2N+3} + \phi_{1,3}. They are given as infinite series of form factor contributions. The form factors of…

Let $f$ be a $\{0,1\}$-valued function over an integer $d$-dimensional cube $\{0,1,\dots,n-1\}^d$, for $n \geq 2$ and $d \geq 1$. The function $f$ is called threshold if there exists a hyperplane which separates $0$-valued points from…

We give an infinite class of counterexamples to the Gotsman-Linial conjecture when d = 2. On the other hand, we establish an asymptotic form of the conjecture for quadratic threshold functions whose non-zero quadratic terms define a graph…

Threshold graphs are recursive deterministic network models that have been proposed for describing certain economic and social interactions. One drawback of this graph family is that it has limited generative attachment rules. To mitigate…

We say that a formal power series $\sum a_n z^n$ with rational coefficients is a 2-function if the numerator of the fraction $a_{n/p}-p^2 a_n$ is divisible by $p^2$ for every prime number $p$. One can prove that 2-functions with rational…

A semiorder is a model of preference relations where each element $x$ is associated with a utility value $\alpha(x)$, and there is a threshold $t$ such that $y$ is preferred to $x$ iff $\alpha(y) > \alpha(x)+t$. These are motivated by the…

I develop a function that, for any integer $n \geq 2$, takes a value of 1 if $n$ is prime, 0 if $n$ is composite. I also discuss two applications: First, the characteristic function provides a new expression for the prime counting function.…

For fixed natural numbers $r$ and $s$, where $2\leq s \leq r$, we consider a representation of numbers from the interval $[0;\frac{r}{s-1}]$ obtained by encoding numbers by means of the alphabet $A=\{0,1,...,r\}$ via the expansion…

Threshold graphs are a prevalent and widely studied class of simple graphs. They have several equivalent definitions which makes them a go-to class for finding examples and counter examples when testing and learning. This versatility has…

In this note functions that transform open segments of a linear space into open segments of another linear space are studied and characterized. Assuming that the range is non-collinear, it is proved that such a map can always be expressed…

A pairing function for the non-negative integers is said to be binary perfect if the binary representation of the output is of length 2k or less whenever each input has length k or less. Pairing functions with square shells, such as the…

We present a unified framework to study threshold functions for the existence of solutions to linear systems of equations in random sets which includes arithmetic progressions, sum-free sets, $B_{h}[g]$-sets and Hilbert cubes. In…

The general properties of two-dimensional generalized Bessel functions are discussed. Various asymptotic approximations are derived and applied to analyze the basic structure of the two-dimensional Bessel functions as well as their nodal…

A function between two metric spaces is said to be totally bounded regular if it preserves totally bounded sets. These functions need not be continuous in general. Hence the purpose of this article is to study such functions vis-\'a-vis…