quantile based optimization of noisy computer experiments with tunable precision this article addresses the issue of kriging based optimization of stochastic simulators many of these simulators depend on factors that tune the level of precision of the response the gain in accuracy being at a price of computational time the contribution of this work is two fold first we propose a quantile based criterion for the sequential design of experiments in the fashion of the classical expected improvement criterion which allows an elegant treatment of heterogeneous response precisions second we present a procedure for the allocation of the computational time given to each measurement allowing a better distribution of the computational effort and increased efficiency finally the optimization method is applied to an original application in nuclear criticality safety this article has supplementary material available online the proposed criterion is available in the r package diceoptim

rigidity of proper holomorphic maps between bounded symmetric domains our first main result gives assumptions guaranteeing that proper holomorphic maps between cartan type i bounded symmetric domains have simple block matrix shape answering positively a question of mok the proof is based on the second main result establishing similar phenomenon for local cr maps between arbitrary boundary components of two bounded symmetric domains of the above type since boundary components other than shilov boundaries are levi degenerate our analysis is based on their 2 nondegeneracy combining levi forms with higher order tensors

a reflection principle for proper holomorphic mappings of strongly pseudoconvex domains and applications this result is proved in 5 the techniques of 12 and 5 involve detailed computat ion of differential invariants on the boundary another paper philosophically related to the above but involving more elementary techniques is 1 in which proper holomorphic maps of b n to b are shown to be automorphisms more recently this result has been proved on strongly pseudoconvex domains in 2 and in 4 in the paper 3 which is the starting point for the present work completely elementary means are used to obtain a reflection principle for suitable maps of b to b k this in turn leads to a proof of the result of 12 assuming only c 2 smoothness to the boundary further detailed information is obtained about the maps even when k n 1 the elementary techniques of 3 may be adapted to use on strictly

a reflection principle with applications to proper holomorphic mappings j l holomorphic mappings from b into b k we then apply this principle to the problem of characterizing the proper holomorphic mappings from b into b k two such mappings f and g are said to be equivalent if there exist holomorphic automorphisms b and p of b and bk respectively such that f pogo49 there are at present three significant papers on the subject the first by alexander 1 proves that i f f is a proper holomorphic mapping of b into b then f is a holomorphic automorphism of b webster i 5 proved that for n 3 i f f is a proper holomorphic mapping of b into b 1 which is c3 then f maps b into a hyperplane in r hence using alexander s result and properties of the holomorphic automorphisms f is equivalent to the mapping zl z2 z zi z2 zn o alexander s example zl z2 z2 zlz2 z 2 shows that webster s result does not hold for n 2 in a recent paper faran 2 has characterized all proper holomorphic mappings f from b 2 to b 3 that satisfy the additional smoothness condition fec3 2 he proves that such mappings are equivalent to one of the four mappings z w z w 0 z w z zw w2 z w r z 2 l zw w2

proper holomorphic mappings from the two ball to the three ball we prove that a proper mapping of the two ball in cn into the three ball which is c2 on the closed two ball is equivalent to one of four normalized polynomial mappings this improves the known result of faran the proof is basic using taylor expansions

discovering variable fractional orders of advection dispersion equations from field data using multi fidelity bayesian optimization abstract the fractional advection dispersion equation fade can describe accurately the solute transport in groundwater but its fractional order has to be determined a priori here we employ multi fidelity bayesian optimization to obtain the fractional order under various conditions and we obtain more accurate results compared to previously published data moreover the present method is very efficient as we use different levels of resolution to construct a stochastic surrogate model and quantify its uncertainty we consider two different problem set ups in the first set up we obtain variable fractional orders of one dimensional fade considering both synthetic and field data in the second set up we identify constant fractional orders of two dimensional fade using synthetic data we employ multi resolution simulations using two level and three level gaussian process regression models to construct the surrogates

rigidity of cr maps between shilov boundaries of bounded symmetric domains our goal is to establish what seems to be the first rigidity result for cr embeddings between shilov boundaries of bounded symmetric domains of higher rank the result states that any such cr embedding is the standard linear embedding up to cr automorphisms our basic assumption extends precisely the well known optimal bound for the rank one case there are no other restrictions on the ranks in particular the difficult case when the target rank is larger than the source rank is also allowed

on the c version of the reflection principle for mappings between cr manifolds we prove a smooth version of the classical schwarz reflection principle for cr mappings between an abstract cr manifold m and a generic cr manifold embedded in euclidean complex space as a consequence of our results we settle a conjecture of x huang formulated in 1994

the equivalence problem and rigidity for hypersurfaces embedded into hyperquadrics we consider the class of levi nondegenerate hypersurfaces m in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i that admit a local cr transversal embedding near a point p e m into a standard nondegenerate hyperquadric in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i with codimension k n n small compared to the cr dimension n of m we show that for hypersurfaces in this class there is a normal form which is closely related to the embedding such that any local equivalence between two hypersurfaces in normal form must be an automorphism of the associated tangent hyperquadric we also show that if the signature of m and that of the standard hyperquadric in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i are the same then the embedding is rigid in the sense that any other embedding must be the original embedding composed with an automorphism of the quadric

multiscaling fractional advection dispersion equations and their solutions 1 the multiscaling fractional advection dispersion equation ade is a multidimensional model of solute transport that encompasses linear advection fickian dispersion and super fickian dispersion the super fickian term in these equations has a fractional derivative of matrix order that describes unique plume scaling rates in different directions the directions need not be orthogonal so the model can be applied to irregular noncontinuum fracture networks the statistical model underlying multiscaling fractional dispersion is a continuous time random walk ctrw in which particles have arbitrary jump length distributions and finite mean waiting time distributions the meaning of the parameters in a compound poisson process a subset of ctrws is used to develop a physical interpretation of the equation variables the green s function solutions are the densities of operator stable probability distributions the limit distributions of normalized sums of independent and identically distributed random vectors these densities can be skewed heavy tailed and scale nonlinearly resembling solute plumes in granular aquifers they can also have fingers in any direction resembling transport along discrete pathways such as fractures

holomorphic mappings between hyperquadrics with small signature difference in this paper we study holomorphic mappings sending a hyperquadric of signature ell in bbb c n into a hyperquadric of signature ell in bbb c n we show theorem 1 1 that if the signature difference ell ell is not too large then the mapping can be normalized by automorphisms of the target hyperquadric to a particularly simple form and in particular the image of the mapping is contained in a complex plane of a dimension that depends only on ell and ell and not on the target dimension n we also prove a hopf lemma type result theorem 1 3 for such mappings

multidimensional advection and fractional dispersion extension of the fractional diffusion equation to two or three dimensions is not as simple as extension of the second order equation this is revealed by the solutions of the equations unlike the gaussian the most general stable vector cannot be generated with an atomistic measure on the coordinate axes a random combination of maximally skewed stable variables on the unit sphere generates a stable vector that is a general model of a diffusing particle subsets are symmetric stable vectors that have previously appeared in the literature and the well known multidimensional brownian motion a multidimensional fractional differential operator is defined in the process

stochastic collocation algorithms using l_1 minimization for bayesian solution of inverse problems the bayesian methodology has proven to be a convenient framework to solve inverse problems from available data with limited information the main computational challenges in the bayesian solution of inverse problems arise from the need for repeated evaluations of the forward model as required by markov chain monte carlo mcmc methods for posterior sampling in this paper we present an efficient technique for constructing stochastic surrogate models to accelerate the bayesian inference approach for statistical inverse problems the stochastic collocation algorithms using l 1 minimization l1 scm based on generalized polynomial chaos are used to construct a polynomial approximation of the forward solution over the support of the prior distribution this approximation then defines a surrogate posterior probability density that is inexpensive to evaluate a rigorous error analysis shows that the convergence rate of the posterior density is at least twice as fast as the convergence rate of the gpc expa

on the regularity of cr mappings between cr manifolds of hypersurface type we prove smooth and analytic versions of the classical schwarz reflection principle for transversal cr mappings between two levi nondegenerate cr manifolds of hypersurface type

machine learning of linear differential equations using gaussian processes this work leverages recent advances in probabilistic machine learning to discover governing equations expressed by parametric linear operators such equations involve but are not limited to ordinary and partial differential integro differential and fractional order operators here gaussian process priors are modified according to the particular form of such operators and are employed to infer parameters of the linear equations from scarce and possibly noisy observations such observations may come from experiments or black box computer simulations as demonstrated in several synthetic examples and a realistic application in functional genomics employ probabilistic machine learning to discover governing equations expressed by parametric linear operators proper placement of gaussian process priors allows one to efficiently infer model parameters via maximum likelihood estimation a general treatment of inverse problems governed by linear operators leading to model discovery from just a handful of noisy measurements

on twisted lifting if a is a generator of the galois group of a finite cyclic extension e f of local or global fields and e is a character of ce ex or ex ax whose restriction to cf has order n then the irreducible admissible or automorphic representations 1 cf denotes the multiplicative group fx if f is local and the idele class group fx a x if f is global put g gl and fix a character e of ce whose restriction to cf is of order n it determines a character e of g e or g e g ae by e g e det g the restriction of e to cf determines by class field theory a cyclic extension fe of f of degree n fix a generator a of the galois group gal e f denote by r g the set of equivalence classes of irreducible admissible or automorphic representations of g e or g ae then a acts on r g by v g vr a g g in g e or g a and we denote by re g the set of vr in r g with v n e theorem fftere ex s i a natural bijection between are g and the set of gal f f orbits of characters on nl fcl where l fee in the case e f this is due to kazhdan k in the local case and if it is cuspidal with two supercuspidal components kazhdan s proof is remarkably elegant com plete local and global results are derived from the study of the unramified case only our proof follows kazhdan s and uses kottwitz s base change lemma ko and arthur s explicit trace formula a we also show lemma 15 1 that the above 77 are induced ip r from a representation t t t e t e v e where e e 1 and t is a super cuspidal representation of gl w e n mm f is a parabolic subgroup with levi component gl m m then if l e n fe is an extension of f of degree we must have that divides m corollary 16 2 and m l is prime to e proposition 17 1 for example in the special case where e fe each it of the above type is induced from a character of a borel subgroup hence it is not in the discrete series we proved this result in f when e fe is a quadratic extension of f using the trace formula twisted as here in fact this motivated us to look for a twisted analogue of kazhdan s lifting further description of our vr is given in 16 and 17

rigidity of proper holomorphic mappings between nonequidimensional bounded symmetric domains we prove that any proper holomorphic mapping from d p p 1 i to d p p i p geq 3 is necessarily a totally geodesic isometric embedding with respect to their bergman metrics and therefore is the standard linear embedding up to their automorphisms the proof of the main result in this paper will be achieved by a differential geometric study of a special class of complex geodesic curves on the bounded symmetric domains with respect to their bergman metrics

predicting the output from a complex computer code when fast approximations are available s we consider prediction and uncertainty analysis for complex computer codes which can be run at different levels of sophistication in particular we wish to improve efficiency by combining expensive runs of the most complex versions of the code with relatively cheap runs from one or more simpler approximations a bayesian approach is described in which prior beliefs about the codes are represented in terms of gaussian processes an example is presented using two versions of an oil reservoir simulator 1 c complex mathematical models implemented in large computer codes have been used to study real systems in many areas of scientific research sacks et al 1989 usually because physical experimentation is too costly and sometimes impossible as in the case of large environmental systems a computer experiment involves running the code with various input values for the purpose of learning something about the real system often a simulator can be run at different levels of complexity with versions ranging from the most sophisticated high level code to the most basic for example in 4 we consider two codes which simulate oil pressure at a well of a hydrocarbon reservoir both codes use finite element analysis in which the rocks comprising the reservoir are represented by small interacting grid blocks the flow of oil within the reservoir can be simulated by considering the interaction between the blocks the two codes differ in the resolution of the grid so that we have a very accurate slow version using many small blocks and a crude approximation using large blocks which runs much faster alternatively a mathematical model could be expanded to include more of the scientific laws underlying the physical processes simple fast versions of the code may well include the most important features and are useful for preliminary investigations in real time applications the number of runs from a high level simulator may be limited by expense then there is a need to trade off the complexity of the expensive code with the availability of the simpler approximations the purpose of the current paper is to explore ways in which runs from several levels of a code can be used to make inference about the output from the most complex code we may also have uncertainty about values for the input parameters which apply in any given application uncertainty analysis of computer codes describes how this uncertainty on the inputs affects our uncertainty about the output

application of a fractional advection dispersion equation abstract a transport equation that uses fractional order dispersion derivatives hasfundamental solutions that are le vy s a stable densities these densities represent plumesthat spread proportional to time 1 a have heavy tails and incorporate any degree ofskewness the equation is parsimonious since the dispersion parameter is not a functionof time or distance the scaling behavior of plumes that undergo le vy motion isaccounted for by the fractional derivative a laboratory tracer test is described by adispersion term of order 1 55 while the cape cod bromide plume is modeled by anequation of order 1 65 to 1 8 1 introduction anomalous or non fickian dispersion has been an activearea of research in the physics community since the introduc tion of continuous time random walks ctrw by montrolland weiss 1965 these random walks extended the predictivecapability of models built on the stochastic process of brown ian motion which is the basis for the classical advection dispersion equation ade the ctrw assign a joint space time distribution called the transition density to individualparticle motions when the tails are heavy enough i e powerlaw non fickian dispersion results for all time scales andspace scales

on the third gap for proper holomorphic maps between balls let f be a proper rational map from the complex ball mathbb b n into mathbb b n with n 7 and 3n 1 le n le 4n 7 then f is equivalent to a map g 0 dots 0 where g is a proper holomorphic map from mathbb b n into mathbb b 3n

unitary quasi lifting applications let tj 3 be the quasi split unitary group in three variables defined using a quadratic extension e f of number fields complete local and global results are obtained for the o endo unstable lifting from u 2 to gl 3 e this is used to establish quasi endo lifting for automorphic forms from u 2 to u 3 by means of base change from uf3 to gl 3 e base change quasi lifting is also proven continuing the work of i the exposition is elementary and uses only a simple form of an identity of trace formulas and base change transfer of orbital integrals of spherical functions

random walk approximation of fractional order multiscaling anomalous diffusion random walks are developed to approximate the solutions of multiscaling fractional order anomalous diffusion equations the essential elements of the diffusion are described by the matrix order scaling indexes and the mixing measure which describes the diffusion coefficient in every direction two forms of the governing equation also called the multiscaling fractional diffusion equation based on fractional flux and fractional divergence are considered where the diffusion coefficient and the drift vary in space the particle tracking algorithm is also extended to approximate anomalous diffusion with a streamline dependent mixing measure using a streamline projection technique in this and other general cases the random walk method is the only known way to solve the nonhomogeneous equations five numerical examples demonstrate the flexibility simplicity and efficiency of the random walk method

rational proper holomorphic maps from b n into b 2 n in this paper we will investigate rational proper holomorphic maps from b n into b2 n when n 4

extending proper holomorphic mappings of positive codimension in this paper we obtain results on holomorphic continuation of proper holomorphic mappings between pseudoconvex domains with real analytic boundaries in complex spaces of different dimensions equivalently we obtain results concerning the analyticity of cauchy riemann mappings between real analytic pseudoconvex hypersurfaces in complex spaces of different dimensions to begin with we recall the corresponding results for mappings of equidimensional domains let d and d be bounded pseudoconvex domains with smooth boundaries in c if the boundaries of d and d are strictly pseudoconvex or more generally of finite type in the sense of d angelo 15 then every proper holomorphic map of d onto d extends smoothly t o according to the results of bell and catlin 7 8 and diederich and forn ess 19 for mappings between strictly pseudoconvex domains this was proved by fefferman 26 and nirenberg webster and yang 37 if the boundaries of the pseudoconvex domains d d c c are real analytic then every proper holomorphic mapping of d onto d extends holomorphically to a neighborhood of 3 according to baouendi and rothschild 2 and diederich and forn ess 21 this reflection principle was first discovered by lewy 34 and pin6uk 38 for mappings between strictly pseudoconvex domains in the case of biholomorphic mappings between weakly pseudoconvex domains with realanalytic boundaries the result follows from the work of baouendi jacobowitz and treves 5 results in this direction were obtained in recent years by several authors see the papers 4 6 18 22 33 47 49 in this paper we are treating the case when the domains d and d have different dimensions to be specific we assume that d e c and d c c n are bounded pseudoconvex domains with real analytic boundaries and n n l in this situation a proper holomorphic map f d d need not be regular at the boundary for instance there exist proper holomorphic maps of balls of different dimensions

super rigidity for cr embeddings of real hypersurfaces into hyperquadrics we consider holomorphic mappings sending a given levi nondegenerate pseudoconcave hypersurface m in into a nondegenerate hyperquadric of the same signature in and show that if m is sufficiently close to a hyperquadric in a certain sense then any two such mappings differ only by an automorphism of the hyperquadric

uniqueness of certain polynomials constant on a line abstract we study a question with connections to linear algebra real algebraic geometry combinatorics and complex analysis let p x y be a polynomial of degree d with n positive coefficients and no negative coefficients such that p 1 when x y 1 a sharp estimate d 2 n 3 is known in this paper we study the p for which equality holds we prove some new results about the form of these sharp polynomials using these new results and using two independent computational methods we give a complete classification of these polynomials up to d 17 the question is motivated by the problem of classification of cr maps between spheres in different dimensions

backward fractional advection dispersion model for contaminant source prediction the forward fractional advection dispersion equation fade provides a useful model for non fickian transport in heterogeneous porous media the space fade captures the long leading tail skewness and fast spreading typically seen in concentration profiles from field data this paper develops the corresponding backward fade model to identify source location and release time the backward method is developed from the theory of inverse problems and then explained from a stochastic point of view the resultant backward fade differs significantly from the traditional backward advection dispersion equation ade because the fractional derivative is not self adjoint and the probability density function for backward locations is highly skewed finally the method is validated using tracer data from a well known field experiment where the peak of the backward fade curve predicts source release time while the median or a range of percentiles can be used to determine the most likely source location for the observed plume the backward ade cannot reliably identify the source in this application since the forward ade does not provide an adequate fit to the concentration data