On beta-function of tube of light cone

We construct $B$-function of the Hermitian symmetric space \OO(n,2)/\OO(n)\times \OO(2) or equivalently of the tube $(Re z_0)^2> (Re z_1)^2+...+ (Re z_n)^2$ in $C^{n+1}Comment: 7 page

Quantum Algorithm for Dynamic Programming Approach for DAGs. Applications for Zhegalkin Polynomial Evaluation and Some Problems on DAGs

In this paper, we present a quantum algorithm for dynamic programming approach for problems on directed acyclic graphs (DAGs). The running time of the algorithm is $O(\sqrt{\hat{n}m}\log \hat{n})$, and the running time of the best known deterministic algorithm is $O(n+m)$, where $n$ is the number of vertices, $\hat{n}$ is the number of vertices with at least one outgoing edge; $m$ is the number of edges. We show that we can solve problems that use OR, AND, NAND, MAX and MIN functions as the main transition steps. The approach is useful for a couple of problems. One of them is computing a Boolean formula that is represented by Zhegalkin polynomial, a Boolean circuit with shared input and non-constant depth evaluating. Another two are the single source longest paths search for weighted DAGs and the diameter search problem for unweighted DAGs.Comment: UCNC2019 Conference pape

The Łojasiewicz Exponent at Infinity of Non-negative and Non-degenerate Polynomials

Let f be a real polynomial, non-negative at infinity with non-compact zero-set. Suppose that f is non-degenerate in the Kushnirenko sense at infinity. In this paper we give a formula for the Łojasiewicz exponent at infinity of f and a formula for the exponent of growth of f in terms of its Newton polyhedron

The circle transform on trees

We consider the overdetermined problem of integral geometry on trees given by the transform that integrates functions on a tree over circles, and exhibit difference equations that describe the range. We then show how this problem modifies if we restrict the transform to some natural subcomplex of the complex of circles, proving inversion formulas and characterizing ranges. (C) 2003 Elsevier B.V. All rights reserved