The numerical methods in the current known literature require the stochastic differential equations SDEs driven by Poisson randommeasure satisfying the global Lipschitz condition and the linear growth condition. In this paper, Euler’s method is introduced for SDEs driven by Poisson random measure with non-Lipschitz coefficients which cover more classes of such equations than before. Themain aim...

We prove that on a punctured oriented surface with Euler characteristic χ < 0, the maximal cardinality of a set of essential simple arcs that are pairwise nonhomotopic and intersecting at most once is 2|χ|(|χ|+1). This gives a cubic estimate in |χ| for a set of curves pairwise intersecting at most once on a closed surface. We also give polynomial estimates in |χ| for sets of arcs and curves pai...

Abstract. The sum formula is a basic identity of multiple zeta values that expresses a Riemann zeta value as a homogeneous sum of multiple zeta values of a given dimension. This formula was already known to Euler in the dimension two case, conjectured in the early 1990s for higher dimensions and then proved by Granville and Zagier independently. Recently a weighted form of Euler’s formula was o...

This is the fourth part in the series of articles [4], [5], [6] (see also [3]) where the theory of valuations on manifolds is developed. In this part it is shown that the filtration on valuations is compatible with the product. Then it is proved that the Euler-Verdier involution on smooth valuations is an automorphism of the algebra of valuations. Then an integration functional on valuations wi...

We provide an algorithm for computing the Euler characteristic of the curves Sp in P 3 ' C, consisting of all polynomials with a periodic critical point of period p in the space of critically-marked, complex, cubic polynomials. The curves were introduced in [Mi, BKM], and the algorithm applies the main results of [DP]. The output is shown for periods p ≤ 26.

The main purpose of this paper is to investigate the convergence of the Euler method to stochastic differential equations with piecewise continuous arguments SEPCAs . The classical Khasminskiitype theorem gives a powerful tool to examine the global existence of solutions for stochastic differential equations SDEs without the linear growth condition by the use of the Lyapunov functions. However,...

We investigate two arithmetic functions naturally occurring in the study of the Euler and Carmichael quotients. The functions are related to the frequency of vanishing of the Euler and Carmichael quotients. We obtain several results concerning the relations between these functions as well as their typical and extreme values.

This talk and papers concern sums of the type (a; b; c) := 1 X x=1 x 1 X y=1 y 1 X z=1 1 xaybzc ; and their n-variable analogues and polylogarithmic extensions. For which integers a; b; c are these sums reducible: expressible in terms of the simpler series (a; b) := 1 X x=1 x 1 X y=1 1 xayb and (a) := 1 X x=1 1 xa ? Sums of this type are (triple, double or single) Euler sums or Multi-dimensiona...

We present a probabilistic approach to studying the descent statistic based upon a two-variable probability density. This density is log concave and, in fact, satisfies a higher order concavity condition. From these properties we derive quadratic inequalities for the descent statistic. Using Fourier series, we give exact expressions for the Euler numbers and the alternating r-signed permutation...

This is the fourth part in the series of articles [4], [5], [6] (see also [3]) where the theory of valuations on manifolds is developed. In this part it is shown that the filtration on valuations introduced in [5] is compatible with the product. Then it is proved that the Euler-Verdier involution on smooth valuations introduced in [5] is an automorphism of the algebra of valuations. Then an int...