stepwise irregular graphs abstract a graph is stepwise irregular si if the degrees of any two of its adjacent vertices differ by exactly one among graphs with non zero edge imbalance si graphs are least irregular some basic properties of si graphs are established

some new spectral bounds for graph irregularity abstract the irregularity of a simple graph g v e is defined as irr g u v e g d g u d g v where dg u denotes the degree of a vertex u v g this graph invariant introduced by albertson in 1997 is a measure of the defect of regularity of a graph recently it also gains interest in chemical graph theory where it is named the third zagreb index in this paper by means of the laplacian eigenvalues and the normalized laplacian eigenvalues of g we establish some new spectral upper bounds for irr g we then compare these new bounds with a known bound by goldberg and it turns out that our bounds are better than the goldberg bound in most cases we also present two spectral lower bounds on irr g

tractability properties of the weighted star discrepancy of the halton sequence abstract we study the weighted star discrepancy of the halton sequence in particular we show that the halton sequence achieves strong polynomial tractability for the weighted star discrepancy for product weights j j 1 under the mildest condition on the weight sequence known so far for explicitly constructive sequences the condition requires sup d 1 max 0 u d j u j j the same result holds for niederreiter sequences and for other types of digital sequences our results are true also for the weighted unanchored discrepancy

shifted lattice rules based on a general weighted discrepancy for integrals over euclidean space we approximate weighted integrals over euclidean space by using shifted rank 1 lattice rules with good bounds on the generalised weighted star discrepancy this version of the discrepancy corresponds to the classic l weighted star discrepancy via a mapping to the unit cube the weights here are general weights rather than the product weights considered in earlier works on integrals over r d known methods based on an averaging argument are used to show the existence of these lattice rules while the component by component technique is used to construct the generating vector of these shifted lattice rules we prove that the bound on the weighted star discrepancy considered here is of order o n 1 d for any d 0 and with the constant involved independent of the dimension this convergence rate is better than the o n 1 2 achieved so far for both monte carlo and quasi monte carlo methods

on some properties of graph irregularity indices with a particular regard to the σ index abstract some well known graph irregularity indices of a connected graph g are investigated our study is focused mainly on the comparison of the bell s degree variance var g and the collatz sinogowitz irregularity index cs g with the recently introduced g irregularity index it is a degree based topological invariant calculated as g f g 2 m 2 g where m2 g is the second zagreb index f g d 3 v and d v is the degree of the vertex v in g by introducing the notion of the complete split like graphs representing a broad subclass of bidegreed connected graphs it is shown that for these graphs the equality g n 2 v a r g holds

on the ito wentzell formula for distribution valued processes and related topics we prove the ito wentzell formula for processes with values in the space of generalized functions by using the stochastic fubini theorem and the ito wentzell formula for real valued processes appropriate versions of which are also proved

branching measure valued processes the objective of this paper is to investigate the structure of a general subcritical branching measure valued processx subject to the usual regularity conditions we prove that if the second moments of the total massxt e are finite thenx is a superprocess and we give an explicit expression of the branching characteristicsq andl in terms of the continuous martingale component of the total massxt e and the levy measure jumps compensator ofx

a note on the irregularity of graphs abstract denote by 1 g the largest eigenvalue of a real 0 1 adjacency matrix of a graph g and by d g the mean degree of g collatz and sinogowitz proposed 1 g d g as a measure of irregularity of g a second such measure is the variance of the vertex degrees of g the most irregular graphs according to these measures are determined for certain classes of graphs and the two measures are shown to be incompatible for some pairs of graphs

infinite family of 2 connected transmission irregular graphs abstract distance between two vertices is the number of edges in the shortest path connecting them in a connected graph g the transmission of a vertex v is the sum of distances from v to all the other vertices of g if transmissions of all vertices are mutually distinct then g is a transmission irregular graph it is known that almost no graphs are transmission irregular infinite families of transmission irregular trees were presented in 4 the following problem was posed in 4 do there exist infinite families of 2 connected transmission irregular graphs in this paper an infinite family of such graphs is constructed

stepwise transmission irregular graphs abstract the distance d u v between vertices u and v of a connected graph g is defined as the number of edges in a shortest path connecting them the transmission of a vertex v of g is the sum of distances from v to all the other vertices of g a graph is stepwise transmission irregular sti if the transmissions of any two of its adjacent vertices differ by exactly one some basic properties of sti graphs are established and infinite families are constructed

one dimensional stochastic partial differential equations and the branching measure diffusion an existence theorem in the spirit of keisler ke is proved for the simple one dimensional diffusion equation driven by white noise modulated by a non linear function of the solution this is used to obtain a density and a stochastic partial differential equation in one dimension for the critical branching diffusion studied by dawson et al

burkholder davis gundy inequalities in umd banach spaces in this paper we prove burkholder davis gundy inequalities for a general martingale m with values in a umd banach space x assuming that m 0 0 we show that the following two sided inequality holds for all 1 le p infty here gamma m t is the l 2 norm of the unique gaussian measure on x having m t x y langle m x rangle langle m y rangle t as its covariance bilinear form this extends to general umd spaces a recent result by veraar and the author where a pointwise version of star was proved for umd banach functions spaces x we show that for continuous martingales star holds for all 0 p infty and that for purely discontinuous martingales the right hand side of star can be expressed more explicitly in terms of the jumps of m for martingales with independent increments star is shown to hold more generally in reflexive banach spaces x with finite cotype in the converse direction we show that the validity of star for arbitrary martingales implies the umd property for x as an application we prove various ito isomorphisms for vector valued stochastic integrals with respect to general martingales which extends earlier results by van neerven veraar and weis for vector valued stochastic integrals with respect to a brownian motion we also provide ito isomorphisms for vector valued stochastic integrals with respect to compensated poisson and general random measures

good lattice rules based on the general weighted star discrepancy we study the problem of constructing rank 1 lattice rules which have good bounds on the weighted star discrepancy here the non negative weights are general weights rather than the product weights considered in most earlier works in order to show the existence of such good lattice rules we use an averaging argument and a similar argument is used later to prove that these lattice rules may be obtained using a component by component cbc construction of the generating vector under appropriate conditions on the weights these lattice rules satisfy strong tractability bounds on the weighted star discrepancy particular classes of weights known as order dependent and finite order weights are then considered and we show that the cost of the construction can be very much reduced for these two classes of weights

holder continuity for spatial and path processes via spectral analysis for d a finite borel measure on r d we consider l 2 d valued stochastic processes y t with te property that y t y t where y t t 0 e t s dm s and m t is a continuous martingale with quadratic variation m t t 0 g s ds we prove timewise holder continuity and maximal inequalities for y and use these results to obtain hilbert space regularity for a class of superrocesses as well as a class of stochastic evolutions of the form dx axdt gdw with w a cylindrical brownian motion maximal inequalities and holder continuity results are also provenfor the path process t y t t

on strong tractability of weighted multivariate integration we prove that for every dimension s and every number n of points there exists a point set pn s whose weighted unanchored l discrepancy is bounded from above by c b n1 2 b independently of s provided that the sequence k has k 1 a k for some even arbitrarily large a here b is a positive number that could be chosen arbitrarily close to zero and c b depends on b but not on s or n this result yields strong tractability of the corresponding integration problems including approximation of weighted integrals d f x x dx over unbounded domains such as d r s it also supplements the results that provide an upper bound of the form c s n when k 1

on graphs whose wiener complexity equals their order and on wiener index of asymmetric graphs abstract if u is a vertex of a graph g then the transmission of u is the sum of distances from u to all the other vertices of g the wiener complexity cw g of g is the number of different complexities of its vertices g is transmission irregular if c w g n g it is proved that almost no graphs are transmission irregular let t n 1 n 2 n 3 be the tree obtained from paths of respective lengths n1 n2 and n3 by identifying an end vertex of each of them it is proved that t 1 n 2 n 3 is transmission irregular if and only if n 3 n 2 1 and n 2 k 2 1 2 k 2 2 2 for some k 3 it is also proved that if t is an asymmetric tree of order n then the wiener index of t is bounded by n 3 13 n 48 6 with equality if and only if t t 1 2 n 4 a parallel result is deduced for asymmetric uni cyclic graphs