global heat kernel estimate for relativistic stable processes in exterior open sets in this paper sharp two sided estimates for the transition densities of relativistic stable processes with mass m 0 1 in c1 1 exterior open sets are established for all time t 0 these transition densities are also the dirichlet heat kernels of m m2 2 with m 0 1 in c1 1 exterior open sets the estimates are uniform in m in the sense that the constants are independent of m 0 1 as a corollary of our main result we establish sharp two sided green function estimates for relativistic stable processes with mass m 0 1 in c1 1 exterior open sets

optimal foldover plans for two level nonregular orthogonal designs this article considers optimal foldover plans for nonregular designs by using the indicator function we define words with fractional lengths the extended word length pattern is then used to select among nonregular foldover designs some general properties of foldover designs are obtained using the indicator function we prove that the full foldover plan that reverses the signs of all factors is optimal for all 12 run and 20 run orthogonal designs the optimal foldover plans for all 16 run regular and nonregular orthogonal designs are constructed and tabulated for practical use optimal foldover plans for higher order orthogonal designs can be constructed in a similar manner

constructing tables of minimum aberration pn m designs fries and hunter 1980 presented a practical algorithm for selecting standard 2 n m fractional factorial designs based on a criterion they called minimum aberration in this article some simple results are presented that enable the fries hunter algorithm to be used for a wider range of n and m and for designs with factors at p levels where p 2 is prime examples of minimum aberration 2 n m designs with resolution r 4 are given for m n m 9 a matrix is given for generating 3 n m designs with m n m 6 which have or nearly have minimum aberration

boundary harnack inequality for markov processes with jumps we prove a boundary harnack inequality for jump type markov processes on metric measure state spaces under comparability estimates of the jump kernel and urysohn type property of the domain of the generator of the process the result holds for positive harmonic functions in arbitrary open sets it applies e g to many subordinate brownian motions levy processes with and without continuous part stable like and censored stable processes jump processes on fractals and rather general schrodinger drift and jump perturbations of such processes

weighted poincare inequality and heat kernel estimates for finite range jump processes it is well known that there is a deep interplay between analysis and probability theory for example for a markovian infinitesimal generator mathcal l the transition density function p t x y of the markov process associated with mathcal l if it exists is the fundamental solution or heat kernel of mathcal l a fundamental problem in analysis and in probability theory is to obtain sharp estimates of p t x y in this paper we consider a class of non local integro differential operators mathcal l on mathbb r d of the form mathcal l u x lim limits varepsilon downarrow 0 int limits y in mathbb r d y x varepsilon u y u x j x y dy where displaystyle j x y frac c x y x y d alpha bf 1 x y leq kappa for some constant kappa 0 and a measurable symmetric function c x y that is bounded between two positive constants associated with such a non local operator mathcal l is an mathbb r d valued symmetric jump process of finite range with jumping kernel j x y we establish sharp two sided heat kernel estimate and derive parabolic harnack principle for them along the way some new heat kernel estimates are obtained for more general finite range jump processes that were studied in barlow et al in trans am math soc 2008 one of our key tools is a new form of weighted poincare inequality of fractional order which corresponds to the one established by jerison in duke math j 53 2 503 523 1986 for differential operators using meyer s construction of adding new jumps we also obtain various a priori estimates such as holder continuity estimates for parabolic functions of jump processes not necessarily of finite range where only a very mild integrability condition is assumed for large jumps to establish these results we employ methods from both probability theory and analysis extensively

gaugeability and conditional gaugeability new kato classes are introduced for general transient borel right processes for which gauge and conditional gauge theorems hold these new classes are the genuine extensions of the green tight measures in the classical brownian motion case however the main focus of this paper is on establishing various equivalent conditions and consequences of gaugeability and conditional gaugeability we show that gaugeability conditional gaugeability and the subcriticality for the associated schrodinger operators are equivalent for transient borel right processes with strong duals analytic characterizations of gaugeability and conditional gaugeability are given for general symmetric markov processes these analytic characterizations are very useful in determining whether a process perturbed by a potential is gaugeable or conditionally gaugeable in concrete cases connections with the positivity of the spectral radii of the associated schrodinger operators are also established

practical stability analysis of finite difference equations by the matrix method it is shown how the various norms of the coefficient matrix of a set of finite difference equations can in many cases be employed in an easy straightforward fashion to find sufficient conditions for stability in situations involving non periodic boundary conditions and variable coefficients with the aid of the easily computed norms particularly the infinite norm one does not have to rely on the necessary condition provided by the spectral radius which has been the main basis of criticism of the matrix method in the past

potential theory of truncated stable processes for any alpha in 0 2 a truncated symmetric stable process is a symmetric levy process in mathbb r d with a levy density given by c x d alpha 1 x 1 for some constant c in this paper we study the potential theory of truncated symmetric stable processes in detail we prove a harnack inequality for nonnegative harmonic functions of these processes we also establish a boundary harnack principle for nonnegative functions which are harmonic with respect to these processes in bounded convex domains we give an example of a non convex domain for which the boundary harnack principle fails

heat kernel upper bounds for jump processes and the first exit time for all t 0 x m and f l2 m the function pt x y can be considered as the transition density of the associated markov processx xt t 0 and the question of estimating of pt x y which is the main subject of this paper is closely related to the properties of x the function pt x y is also referred to as a heat kernel and this terminology goes back to the classical gauss weierstrass heat kernel associated with the heat semigroup e t 0 in r n whose markov process is brownian motion a somewhat more general but still well treated case is when m d is a riemannian metric measure space that is m is a riemannian manifold

screening strategies in the presence of interactions product and process improvement can involve a large number of factors that must be varied simultaneously understanding how factors interact is a key step in identifying those factors that have a substantial impact on the response this article gives the first comprehensive assessment and comparison of screening strategies for interactions using two level supersaturated designs group screening and a variety of data analysis methods including shrinkage regression and bayesian methods we develop novel methodology to allow application of bayesian methods in two stage group screening insights on using the strategies are provided through a variety of simulation scenarios and open issues are discussed supplementary materials are available online

heat kernels and analyticity of non symmetric jump diffusion semigroups let d geqslant 1 and alpha in 0 2 consider the following non local and non symmetric levy type operator on mathbb r d begin aligned fancyscript l kappa alpha f x hbox p v int mathbb r d f x z f x frac kappa x z z d alpha mathord mathrm d z end aligned where 0 kappa 0 leqslant kappa x z leqslant kappa 1 kappa x z kappa x z and kappa x z kappa y z leqslant kappa 2 x y beta for some beta in 0 1 using levi s method we construct the fundamental solution also called heat kernel p kappa alpha t x y of fancyscript l kappa alpha and establish its sharp two sided estimates as well as its fractional derivative and gradient estimates we also show that p kappa alpha t x y is jointly holder continuous in t x the lower bound heat kernel estimate is obtained by using a probabilistic argument the fundamental solution of fancyscript l kappa alpha gives rise a feller process x mathbb p x x in mathbb r d on mathbb r d we determine the levy system of x and show that mathbb p x solves the martingale problem for fancyscript l kappa alpha c 2 b mathbb r d furthermore we show that the c 0 semigroup associated with fancyscript l kappa alpha is analytic in l p mathbb r d for every p in 1 infty a maximum principle for solutions of the parabolic equation partial t u fancyscript l kappa alpha u is also established as an application of the main result of this paper sharp two sided estimates for the transition density of the solution of mathord mathrm d x t a x t mathord mathrm d y t is derived where y is a rotationally symmetric stable process on mathbb r d and a x is a holder continuous d times d matrix valued function on mathbb r d that is uniformly elliptic and bounded

derivative pricing as a transport problem mpdata solutions to black scholes type equations abstract we discuss applications of the multidimensional positive definite advection transport algorithm mpdata to numerically solve partial differential equations arising from stochastic models in quantitative finance in particular we develop a framework for solving black scholes type equations by transforming them into advection diffusion problems the equations are then numerically integrated backward in time using an iterative explicit finite difference approach in which the fickian term is represented as an additional advective term we discuss the correspondence between transport phenomena and financial models uncovering the possibility of expressing the no arbitrage principle as a conservation law we show second order accuracy in time and space of the embraced numerical scheme this is done via a convergence analysis comparing mpdata numerical solutions with classic black scholes analytical formulae for the valuation of european options we demonstrate in addition a way of applying mpdata to solve the free boundary problem leading to a linear complementarity problem for the valuation of american options we finally comment on the potential of mpdata methods with respect to more complex models typically used in quantitative finance

transition probabilities for symmetric jump processes we consider symmetric markov chains on the integer lattice in d dimensions where a 0 2 and the conductance between x and y is comparable to x y d we establish upper and lower bounds for the transition probabilities that are sharp up to constants

blocked nonregular two level factorial designs this article discusses the optimal blocking criteria for nonregular two level designs we extend the optimal blocking criteria of cheng and wu to nonregular designs by adapting the g and g2 minimum aberration criteria discussed by tang and deng to define word length pattern for nonregular designs we extend the notion of word to nonregular designs through a polynomial representation of factorial designs we define treatment resolution and block resolution for evaluating the degrees of aliasing and confounding we propose four new criteria which we use to search for optimal blocking schemes of 12 run 16 run and 20 run two level orthogonal arrays

indicator function and its application in two level factorial designs a two level factorial design can be uniquely represented by a polynomial indicator function therefore properties of factorial designs can be studied through their indicator functions this paper shows that the indicator function is an effective tool in studying two level factorial designs the indicator function is used to generalize the aberration criterion of a regular two level fractional factorial design to all two level factorial designs an important identity of generalized aberration is proved the connection between a uniformity measure and aberration is also extended to all two level factorial designs

multi layer designs for computer experiments space filling designs such as latin hypercube designs lhds are widely used in computer experiments however finding an optimal lhd with good space filling properties is computationally cumbersome on the other hand the well established factorial designs in physical experiments are unsuitable for computer experiments owing to the redundancy of design points when projected onto a subset of factor space in this work we present a new class of space filling designs developed by splitting two level factorial designs into multiple layers the method takes advantage of many available results in factorial design theory and therefore the proposed multi layer designs mlds are easy to generate moreover our numerical study shows that mlds can have better space filling properties than optimal lhds

optimal foldover plans for two level fractional factorial designs a commonly used follow up experiment strategy involves the use of a foldover design by reversing the signs of one or more columns of the initial design defining a foldover plan as the collection of columns whose signs are to be reversed in the foldover design this article answers the following question given a 2k p design with k factors and p generators what is its optimal foldover plan we obtain optimal foldover plans for 16 and 32 runs and tabulate the results for practical use most of these plans differ from traditional foldover plans that involve reversing the signs of one or all columns there are several equivalent ways to generate a particular foldover design we demonstrate that any foldover plan of a 2k p fractional factorial design is equivalent to a core foldover plan consisting only of the p out of k factors furthermore we prove that there are exactly 2k p foldover plans that are equivalent to any core foldover plan of a 2k p design and demonstrate how these foldover plans can be const

comparative study of six explicit and two implicit finite difference schemes for solving one dimensional parabolic partial differential equations eight finite difference schemes used in solving parabolic partial differential equations are compared with respect to accuracy execution time and programming effort the analysis presented is useful in selecting the appropriate numerical scheme depending on the emphasis placed upon accuracy execution time or programming effort

a graph aided method for planning two level experiments when certain interactions are important in planning a fractional factorial experiment prior knowledge may suggest that some interactions are potentially important and should therefore be estimated free of the main effects in this article we propose a graph aided method to solve this problem for two level experiments first we choose the defining relations for a 2 n k design according to a goodness criterion such as the minimum aberration criterion then we construct all of the nonisomorphic graphs that represent the solutions to the problem of simultaneous estimation of main effects and two factor interactions for the given defining relations in each graph a vertex represents a factor and an edge represents the interaction between the two factors for the experiment planner the job is simple draw a graph representing the specified interactions and compare it with the list of graphs obtained previously our approach is a substantial improvement over taguchi s linear graphs

aleksandrov bakelman pucci type estimates for integro differential equations in this work we provide an aleksandrov bakelman pucci type estimate for a certain class of fully nonlinear elliptic integro differential equations the proof of which relies on an appropriate generalization of the convex envelope to a nonlocal fractional order setting and on the use of riesz potentials to interpret second derivatives as fractional order operators this result applies to a family of equations involving some nondegenerate kernels and as a consequence provides some new regularity results for previously untreated equations furthermore this result also gives a new comparison theorem for viscosity solutions of such equations which depends only on the l and ln norms of the right hand side in contrast to previous comparison results which utilize the continuity of the right hand side for their conclusions these results appear to be new even for the linear case of the relevant equations

better foldover fractions for resolution iii 2k p designs this article describes a family of resolution iiidesigns for which the usual advice regarding foldover reversing all factors is ill advised the smallest such designs have 16 runs for designs in this family alternative foldover fractions not only increase the resolution to iv but also separate some of the aliased two factor interactions although resolution iv designs obtained by reversing all factors provide fewer than half the degrees of freedom for estimating two factor interactions the recommended foldovers for designs in this class permit estimation of many more two factor interactions this result implies that designs in this class make attractive initial designs even if they are not minimum aberration

estimates of heat kernels for non local regular dirichlet forms in this paper we present new heat kernel upper bounds for a certain class of non local regular dirichlet forms on metric measure spaces including fractal spaces we use a new purely analytic method where one of the main tools is the parabolic maximum principle we deduce off diagonal upper bound of the heat kernel from the on diagonal one under the volume regularity hypothesis restriction of the jump kernel and the survival hypothesis as an application we obtain two sided estimates of heat kernels for non local regular dirichlet forms with finite effective resistance including settings with the walk dimension greater than 2

small composite designs small second order composite designs were suggested by hartley 1959 westlake 1965 provided even smaller designs for k 5 7 and 9 factors for which intricate construction methods were needed here simple designs formed using plackett and burman 1946 designs are offered for k 5 7 and 9 designs with one run fewer than westlake s for k 5 and 7 and three fewer for k 9 are feasible by deleting repeat points that occur in some of the designs

using the folded over 12 run plackett burman design to consider interactions this article demonstrates that the folded over 12 run plackett burman design is useful for considering up to 12 factors in 24 runs even if one anticipates that some two factor interactions may be significant the properties of this design are investigated and a sequential procedure for analyzing the data from such a design is proposed the performance of the procedure is investigated through the analysis of real and constructed examples and through a small simulation study applications to other folded over plackett burman designs are also briefly discussed

non local dirichlet forms and symmetric jump processes we consider the non local symmetric dirichlet form e f given by with f the closure with respect to e 1 of the set of c 1 functions on r d with compact support where e 1 f f e f f f rd f x 2 dx and where the jump kernel j satisfies for 0 2 x y 1 this assumption allows the corresponding jump process to have jump intensities whose sizes depend on the position of the process and the direction of the jump we prove upper and lower estimates on the heat kernel we construct a strong markov process corresponding to e f we prove a parabolic harnack inequality for non negative functions that solve the heat equation with respect to e finally we construct an example where the corresponding harmonic functions need not be continuous

heat kernels for non symmetric diffusion operators with jumps abstract for d 2 we prove the existence and uniqueness of heat kernels to the following time dependent second order diffusion operator with jumps l t 1 2 i j 1 d a i j t x i j 2 i 1 d b i t x i l t where a a i j is a uniformly bounded elliptic and holder continuous matrix valued function b belongs to some suitable kato s class and l t is a non local stable type operator with bounded kernel moreover we establish sharp two sided estimates gradient estimate and fractional derivative estimate for the heat kernel under some mild conditions

heat kernel estimates for jump processes of mixed types on metric measure spaces in this paper we investigate symmetric jump type processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces the class of metric measure spaces includes the alfors d regular sets which is a class of fractal sets that contains geometrically self similar sets a typical example of our jump type processes is the symmetric jump process with jumping intensity e c 0 x y x y int alpha 1 alpha 2 frac c alpha x y x y d alpha nu d alpha where is a probability measure on alpha 1 alpha 2 subset 0 2 c x y is a jointly measurable function that is symmetric in x y and is bounded between two positive constants and c 0 x y is a jointly measurable function that is symmetric in x y and is bounded between 1 and 2 where either 2 1 0 or 1 2 0 this example contains mixed symmetric stable processes on mathbb r n as well as mixed relativistic symmetric stable processes on mathbb r n we establish parabolic harnack principle and derive sharp two sided heat kernel estimate for such jump type processes

regularity results for stable like operators for 1 2 we consider operators of the form lf x rd f x h f x 1 h 1 f x h a x h h d dh and for 0 1 we consider the same operator but where the f term is omitted we prove under appropriate conditions on a x h that any solution u to lu f will be in c if f c

heat kernel estimates and parabolic harnack inequalities for symmetric dirichlet forms abstract in this paper we consider the following symmetric dirichlet forms on a metric measure space m d e f g e c f g m m f x f y g x g y j d x d y where e c is a strongly local symmetric bilinear form and j d x d y is a symmetric radon measure on m m under general volume doubling condition on m d and some mild assumptions on scaling functions we establish stability results for upper bounds of heat kernel resp two sided heat kernel estimates in terms of the jumping kernels the cut off sobolev inequalities and the faber krahn inequalities resp the poincare inequalities we also obtain characterizations of parabolic harnack inequalities our results apply to symmetric diffusions with jumps even when the underlying spaces have walk dimensions larger than 2

regularity estimates for the solution and the free boundary of the obstacle problem for the fractional laplacian we use a characterization of the fractional laplacian as a dirichlet to neumann operator for an appropriate differential equation to study its obstacle problem we write an equivalent characterization as a thin obstacle problem in this way we are able to apply local type arguments to obtain sharp regularity estimates for the solution and study the regularity of the free boundary

design selection and classification for hadamard matrices using generalized minimum aberration criteria deng and tang 1999 and tang and deng 1999 proposed and justified two criteria of generalized minimum aberration for general two level fractional factorial designs the criteria are defined using a set of values called j characteristics in this article we examine the practical use of the criteria in design selection specifically we consider the problem of classifying and ranking designs that are based on hadamard matrices a theoretical result on j characteristics is developed to facilitate the computation the issue of design selection is further studied by linking generalized aberration with the criteria of efficiency and estimation capacity our studies reveal that generalized aberration performs quite well under these familiar criteria

global heat kernel estimates for symmetric jump processes in this paper we study sharp heat kernel estimates for a large class of symmetric jump type processes in r d for all t 0 a prototype of the processes under consideration are symmetric jump processes on r d with jumping intensity

intrinsic ultracontractivity of feynman kac semigroups for symmetric jump processes consider a symmetric non local dirichlet form d d d given by d f f rd rd f x f y 2j x y dxdy with d d the closure of the set of c1 functions on rd with compact support under the norm d1 f f where d1 f f d f f f2 x dx and j x y is a nonnegative symmetric measurable function on rd rd suppose that there is a hunt process xt t 0 on rd corresponding to d d d and that l d l is its infinitesimal generator we study the intrinsic ultracontractivity for the feynman kac semigroup ttv t 0 generated by lv l v where v 0 is a non negative locally bounded measurable function such that the lebesgue measure of the set x rd v x r is finite for every r 0 by using intrinsic super poincare inequalities and establishing an explicit lower bound estimate for the ground state we present general criteria for the intrinsic ultracontractivity of ttv t 0 in particular if j x y x y d 1 x y 1 e x y 1 x y 1 for some 0 2 and 1 and the potential function v x x for some 0 then ttv t 0 is intrinsically ultracontractive if and only if 1 when 1 we have the following explicit estimates for the ground state 1 c1exp c2 1 x log 1 1 x 1 x c3exp c4 1 x log 1 1 x where ci 0 i 1 2 3 4 are constants we stress that our method efficiently applies to the hunt process xt t 0 with finite range jumps and some irregular potential function v such that lim x v x

the liouville property and a conjecture of de giorgi we consider bounded entire solutions of the nonlinear pde u u u 3 0i n r d and prove that under certain monotonicity conditions these solutions must be constant on hyperplanes the proof uses a liouville theorem for harmonic functions associated with a nonuniformly elliptic divergence form operator c 2000 john wiley sons inc

lower estimates of heat kernels for non local dirichlet forms on metric measure spaces abstract we are concerned with heat kernel estimates for a non local dirichlet form on an ahlfors regular metric measure space we use an analytic approach to obtain the full lower stable like estimate of the heat kernel from the near diagonal lower estimate combining with other known results we obtain certain equivalent conditions for two sided stable like estimates of the heat kernel the results can be simplified in certain cases for example when the dirichlet form admits an effective resistance

sharp heat kernel estimates for relativistic stable processes in open sets in this paper we establish sharp two sided estimates for the transition densities of relativistic stable processes i e for the heat kernels of the operators m m2 2 in c1 1 open sets here m 0 and 0 2 the estimates are uniform in m 0 m for each fixed m 0 letting m 0 we recover the dirichlet heat kernel estimates for 2 2 in c1 1 open sets obtained in 14 sharp two sided estimates are also obtained for green functions of relativistic stable processes in bounded c1 1 open sets

symmetric stable laws and stable like jump diffusions asymptotic expansions are obtained for finite dimensional symmetric stable distributions they are used to prove the existence of continuous transition probability densities of stable and stable like jump diffusions and to construct local multiplicative asymptotics and global two sided estimates for these densities as a consequence the distribution of the first passage times for stable jump diffusions is estimated and the integral test for the limsup behaviour of their sample paths as t to 0 is provided 1991 mathematics subject classification 60e07 60g17 60j35 47d07

a note on foldover of 2k p designs with column permutations foldover is a commonly used follow up strategy in experimental designs all existing foldover designs were constructed by reversing the sign of columns of the initial design we propose a new methodology by allowing the permutation of columns in foldover focusing on resolution iv designs we show that almost all designs are better than existing results with respect to the minimum aberration criterion while augmenting a design by a foldover with column permutations may result in a nonregular combined design the proposed designs all have a resolution of 4 5 or higher for which no two factor interaction is fully aliased with any other two factor interactions

a new proof of moser s parabolic harnack inequality using the old ideas of nash in 1958 nash published his fundamental work on the local holder continuity of solutions of second order parabolic equations with coefficients which need not be smooth 8 the primary purpose of that work was to study the properties of the fundamental solution corresponding to the parabolic operator and from these properties to derive regularity for a general solution though the work is often cited in the literature about weak solutions of elliptic and parabolic equations one feels that nash s ideas were never fully understood and maybe still are not and that because of this the more understandable and seemingly more fruitful ideas of de giorgi 4 and moser 6 7 were subsequently adopted

laplacian perturbed by non local operators suppose that d ge 1 and 0 beta 2 we establish the existence and uniqueness of the fundamental solution q b t x y to the operator mathcal l b delta mathcal s b where begin aligned mathcal s bf x int mathbb r d left f x z f x nabla f x cdot z 1 1 z le 1 right frac b x z z d beta dz end aligned and b x z is a bounded measurable function on mathbb r d times mathbb r d with b x z b x z for x z in mathbb r d we show that if for each x in mathbb r d b x z ge 0 for a e z in mathbb r d then q b t x y is a strictly positive continuous function and it uniquely determines a conservative feller process x b which has strong feller property furthermore sharp two sided estimates on q b t x y are derived

heat kernel estimates for subordinate brownian motions in this article we study transition probabilities of a class of subordinate brownian motions under mild assumptions on the laplace exponent of the corresponding subordinator sharp two sided estimates of the transition probability are established this approach in particular covers subordinators with laplace exponents that vary regularly at infinity with index one e g log 1 1 or log 1 2 0 2 that correspond to subordinate brownian motions with scaling order that is not necessarily stricty between 0 and 2 these estimates are applied to estimate green function potential of subordinate brownian motion we also prove the equivalence of the lower scaling condition of the laplace exponent and the near diagonal upper estimate of the transition estimate

operateurs uniformement sous elliptiques sur les groupes de lie given a connected unimodular lie group g having polynomial volume growth and left invariant vector fields x1 xk which generate the lie algebra of g consider the operator la i j 1kxiaijxj where a g rk rk is a smooth symmetric matrix valued function which satisfies a 2 i j 1k aij x i j 1 2 x g rk for some 0 1 what we show is that the corresponding heat flow semigroup etla admits a kernel which satisfies two sided gaussian estimates in terms of the control distance determined by the xi s moreover the estimates can be made to depend on a only through we also prove harnack inequalities and holder regularity of solutions we end with a discussion of some other sub elliptic situations

differentiability of spectral functions for symmetric alpha stable processes let be a signed radon measure in the kato class and define a schrodinger type operator h 1 2 2 on r d we show that its spectral bound c inf h is differentiable if d 2 and is green tight

projection properties of plackett and burman designs the projection properties of the 2 r q p fractional factorials are well known and have been used effectively in a number of published examples of experimental investigations the plackett and burman designs also have interesting projective properties knowledge of which allows the experimenter to follow up an initial plackett and burman design with runs that increase the initial resolution for the factors that appear to matter and thus permit efficient separation of effects of interest projections of designs into 2 5 dimensions are discussed and the 12 run case is given in detail a numerical example illustrates the practical uses of these projections

construction of two level nonregular designs of strength three with large run sizes abstracttwo level orthogonal arrays of strength 3 permit the study of the main effects and the two factor interactions of the experimental factors these arrays are classified into regular and nonr

dirichlet heat kernel estimates for rotationally symmetric levy processes in this paper we consider a large class of purely discontinuous rotationally symmetric levy processes we establish sharp two sided estimates for the transition densities of such processes killed upon leaving an open set d w hend is a fat open set the sharp two sided estimates are given in terms of surviving probabilities and the global transition density of the levy process when d is a c 1 1 open set and the levy exponent of the process is given by 2 with being a complete bernstein function satisfying a mild growth condition at infinity our two sided estimates are explicit in terms of the distance function to the boundary of d and the levy density of x this gives an affirmative answer to the conjecture posted in chen kim and song global heat kernel estimates for relativistic stable processes in half space like open sets potential anal 36 2012 235 261 our results are the first sharp two sided dirichlet heat kernel estimates for a large class of symmetric levy processes with general levy exponents we also derive an explicit lower bound estimate for symmetric levy processes on r d in terms of their levy exponents

parabolic harnack inequality and heat kernel estimates for random walks with long range jumps we investigate the relationships between the parabolic harnack inequality heat kernel estimates some geometric conditions and some analytic conditions for random walks with long range jumps unlike the case of diffusion processes the parabolic harnack inequality does not in general imply the corresponding heat kernel estimates

connections between two level designs of resolutions iii and v resolution iii designs are resolution iii designs in which no two factor interactions are confounded with one another they arise naturally in the search for small composite designs and were first suggested by hartley 1959 new results given here link resolution iii designs to resolution v designs enabling passage between the two types of designs to be made this means that previously untabulated resolution iii designs can be derived directly from known resolution v designs for example a hartley type composite design can be based on a 27 1 iii design derived from a 26 1 v design the usual composite design is based on a 27 1 v design double the number of factorial runs the resolution iii designs can also be usefully modified for specific industrial applications box and jones 1989 moreover the maximum number of factors it is possible to accommodate in a resolution iii design of a given size can be obtained directly from the known maximum for resolution v designs of the same size

a comparative study of finite difference methods for solving the one dimensional transport equation with an initial boundary value discontinuity six explicit and six implicit finite difference methods are used to solve the transport convective diffusion equation where the intersection of the boundary and initial conditions is discontinuous each of the methods are classified according to their theoretical order of accuracy for continuous initial boundary value data the transport equation with the given conditions has an exact solution so the root mean squared error can be obtained for each method with changes in time and space it is found that the theoretical order of accuracy has little effect on the results and that unless the diffusion number r t x 2 is very small then the second and third order implicit methods are more accurate it is also found that the implicit methods have an advantage in terms of lower cpu times over the explicit methods if a desired level of accuracy is required because they are able to use larger time steps the third order implicit methods are found to reduce to second order as a result of the discontinuity in the initial boundary conditions however this reduction in accuracy can be overcome by the use of a fine grid adjacent to the discontinuity

semifolding 2 k p designs this article addresses the varied possibilities for following a two level fractional factorial with another fractional factorial half the size of the original experiment although follow up fractions of the same size as an original experiment are common practice in many situations a smaller followup experiment will sufiice peter john coined the term semifolding to describe using half of a foldover design existing literature does include brief mention and examples of semifolding but no thorough development of this follow up strategy after a quick examination of the estimation details for semlfoldmg the 24 1 iv design we focus on following 16 run fractions with a semifold design of eight runs two such examples are considered one in which the initial fraction ia resolution iv the other resolution iii a general result is proven for semifolding 2 k p iv designs conducting full foldover designs in two blocks is also recommended

optimal semifoldover plans for two level orthogonal designs foldover is a widely used procedure for selecting follow up experimental runs as foldover designs require twice as many runs as the initial design they can be inefficient if the number of effects to be de aliased is smaller than the size of the original experiment semifolding refers to adding half of a foldover fraction and is a technique that has been investigated in recent literature for regular two level fractional factorial designs as an alternative to foldover in this article semifoldover plans for two level orthogonal factorial designs regular and nonregular are compared and ranked based on the concept of minimal dependent sets optimal plans are developed for selected 12 16 20 and 32 run designs a hidden projection property of optimal semifoldovers of 12 and 20 run orthogonal arrays is uncovered general properties of semifoldover designs are also discussed all initial designs considered in this article are included in the supplementary files available online

random conductance models with stable like jumps heat kernel estimates and harnack inequalities abstract we establish two sided heat kernel estimates for random conductance models with non uniformly elliptic possibly degenerate stable like jumps on graphs these are long range counterparts of the well known two sided gaussian heat kernel estimates by m t barlow for nearest neighbor short range random walks on the supercritical percolation cluster unlike the cases of nearest neighbor conductance models we cannot use parabolic harnack inequalities since even elliptic harnack inequalities do not hold in the present setting as an application we establish the local limit theorem for the models