This paper develops a significant extension of E. Lutwak’s dual Brunn-Minkowski theory, originally applicable only to star-shaped sets, to the class of bounded Borel sets. The focus is on expressions and inequalities involving chord-power integrals, random simplex integrals, and dual affine quermassintegrals. New inequalities obtained include those of isoperimetric and Brunn-Minkowski type. A n...

In this paper we give definitions of basic concepts such as symmetries, first integrals, Hamilto-nian and recursion operators suitable for ordinary differential equations on associative algebras, and in particular for matrix differential equations. We choose existence of hierarchies of first integrals and/or symmetries as a criterion for integrability and justify it by examples. Using our compo...

The paper studies properties of continuous time processes in pathwise deterministic setting with spectrum degeneracy at a single point where their Fourier transforms vanish. It appears that these processes are predictable is some weak sense, meaning that convolution integrals over future time can be approximated by integrals over past time. In particular, this means that the processes with this...

A computational technique that transform integrals over R , or some of its subsets, into the hypercube [0, 1] can be exploited in order to solve integrals via Monte Carlo integration without the need to simulate from the original distribution; all that is needed is to simulate iid uniform [0, 1] pseudo random variables. In particular the technique arises from the copula representation of multiv...

Soliton equations describe infinite-dimensional hamiltonian systems. They are closely related to infinite-dimensional Lie groups and algebraic curves. These relations account for complete integrability of soliton equations, i.e. the existence of infinitely many integrals of motion in involution. Recently a new insight has been brought into the theory by the observation that these integrals of m...

We present a systematic method for reducing an arbitrary one−loop N(≥ 5)−point massless Feynman integral with generic 4−dimensional momenta to a set comprised of eight fundamental scalar integrals: six box integrals in D = 6, a triangle integral in D = 4, and a general two−point integral in D = 4 + 2ε space time dimensions. All IR divergences present in the original integral are contained in th...

In this note, we show that Radford's formula for the fourth power of the antipode can be proven for any regular multiplier Hopf algebra with integrals (algebraic quantum groups). This of course not only includes the case of a finite-dimensional Hopf algebra but also the case of any Hopf algebra with integrals (co-Frobenius Hopf algebras). The proof follows in a few lines from well-known formula...

One of the main results of this paper is an upper bound for the total number of real isolated zeros of complete Abelian integrals, exponential in the degree of the form (Theorem 1 below). This result improves a previously obtained in [IY1] double exponential estimate for the number of real isolated zeros on a positive distance from the singular locus. In fact, the theorem on zeros of Abelian in...

Several new multiple-integral representations are proved for well-poised hypergeometric series and integrals. The results yield, in particular, transformations of the multiple integrals that cannot be achieved by evident changes of variable. All this generalizes some classical results of Whipple and Bailey in analysis and, on the other hand, certain analytic constructions with known connection ...

S U M M A R Y Fundamental to complex analysis is the Cauchy integral theorem, and the derivation of Cauchytype integrals. For over 40 yr, Cauchy-type integrals have been used to describe analytical continuation, establish the location of singular points, and study non-single-valued solutions of inverse problems in 2-D potential field theory. In this paper, we revive this interesting and fundame...