On the number of two-dimensional threshold functions

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 functions only asymptotic bounds are known. We give an exact formula for the number of two-dimensional threshold functions and derive more accurate asymptotics.Comment: 17 pages, 2 figure

On lattice point counting in Δ\Delta-modular polyhedra

Let a polyhedron $P$ be defined by one of the following ways: (i) $P = \{x \in R^n \colon A x \leq b\}$, where $A \in Z^{(n+k) \times n}$, $b \in Z^{(n+k)}$ and $rank\, A = n$; (ii) $P = \{x \in R_+^n \colon A x = b\}$, where $A \in Z^{k \times n}$, $b \in Z^{k}$ and $rank\, A = k$. And let all rank order minors of $A$ be bounded by $\Delta$ in absolute values. We show that the short rational generating function for the power series $$\sum\limits_{m \in P \cap Z^n} x^m$$ can be computed with the arithmetic complexity $O\left(T_{SNF}(d) \cdot d^{k} \cdot d^{\log_2 \Delta}\right),$ where $k$ and $\Delta$ are fixed, $d = \dim P$, and $T_{SNF}(m)$ is the complexity to compute the Smith Normal Form for $m \times m$ integer matrix. In particular, $d = n$ for the case (i) and $d = n-k$ for the case (ii). The simplest examples of polyhedra that meet conditions (i) or (ii) are the simplicies, the subset sum polytope and the knapsack or multidimensional knapsack polytopes. We apply these results to parametric polytopes, and show that the step polynomial representation of the function $c_P(y) = |P_{y} \cap Z^n|$, where $P_{y}$ is parametric polytope, can be computed by a polynomial time even in varying dimension if $P_{y}$ has a close structure to the cases (i) or (ii). As another consequence, we show that the coefficients $e_i(P,m)$ of the Ehrhart quasi-polynomial $$\left| mP \cap Z^n\right| = \sum\limits_{j = 0}^n e_i(P,m)m^j$$ can be computed by a polynomial time algorithm for fixed $k$ and $\Delta$

The determination optimum sampling interval of thermo-couple's signal in condition monitoring system of the blast furnace

It is noted, that the sampling interval of the temperature sensors is determined by a process of "accumulation-output melting products. Because of the irregularity of this process, it is advisable to uneven sampling of the signal. An algorithm for non-uniform sampling based on the estimation of the instantaneous errors of the prediction signal of the temperature sensor is offered. Carried out processing real data of blast furnace No. 3 plant of JiNan Iron & Steel Group Co.Ltd (China) has confirmed the effectiveness of this algorithm.Отмечено, что интервал дискретизации температурных датчиков определяется, в первую очередь, процессом "накопление-выпуск" продуктов плавки. В силу нерегулярности этого процесса целесообразно проводить неравномерную дискретизацию сигнала. Предложен алгоритм неравномерной дискретизации, основанный на оценке мгновенной ошибки интерполяции сигнала термодатчика. Проведена обработка реальных данных доменной печи №3 комбината JiNan Iron & Steel Group Co.Ltd (Китай), подтвердившая эффективность данного алгоритма

Lower Bounds for the Complexity of Learning Half-Spaces with Membership Queries

Abstract. Exact learning of half-spaces over finite subsets of IR n from membership queries is considered. We describe the minimum set of labelled examples separating the target concept from all the other ones of the concept class under consideration. For a domain consisting of all integer points of some polytope we give non-trivial lower bounds on the complexity of exact identification of half-spaces. These bounds are near to known upper bounds.