minimum proper loss estimators for parametric models in this paper we propose a methodology for systematically deriving estimators minimizing proper loss functions defined on parametric statistical models by restricting ourselves to losses taking a functional form which allows for straightforward proofs of key properties proving propriety is tantamount to deriving a pairwise divergence quantity between probability measures admitting a natural interpretation of minimum proper loss estimators as divergence minimizers we show that proper losses of varying complexity can be readily constructed given propriety of losses taking a rudimentary form and that for many important models verifying desired properties in the rudimentary case is immediate as a special case we derive computationally tractable estimators requiring only the first and second moments verify strict propriety properties and empirically confirm their utility though parameter estimation tasks using both controlled simulations and real world meteorological network data sets comparisons against numerous standard estimators show that the proposed estimators are at least competitive with and often markedly superior to all standard references uniformly across tasks data sets and model classes suggesting a strong alternative to standard benchmarks in a wide variety of estimation problems

optimized double sweep schwarz method by complete radiation boundary conditions we present an optimized double sweep nonoverlapping schwarz method for solving the helmholtz equation in semi infinite waveguides the domain is decomposed into nonoverlapped layered subdomains along the axis of the waveguide and local wave propagation problems equipped with complete radiation conditions for high order absorbing boundary conditions are solved forward and backward sequentially for communication between subdomains neumann data of local solutions in one domain are transferred to the neighboring subdomain in the forward direction and dirichlet data are exploited in the backward direction the complete radiation boundary conditions enable us to not only minimize reflection coefficients for most important modes in an optimal way but also find neumann data without introducing errors that would be produced if finite difference formulas were used for computing neumann data the convergence of the double sweep schwarz method is proved and numerical experiments using it as a preconditioner are presented to confirm the convergence theory

the physical mirror equivalence for the local mathbb p 2 in this paper we consider the total space of the canonical bundle of mathbb p 2 and we use a proposal by hosono together with results of seidel and auroux katzarkov orlov to deduce the physical mirror equivalence between d b mathbb p 2 k mathbb p 2 and the derived fukaya category of its mirror which assigns the expected central charge to bps states by construction our equivalence is compatible with the mirror map relating the complex and the kahler moduli spaces and with the computation of gromov witten invariants

superpotential algebras and manifolds we study a special class of calabi yau algebras in the sense of ginzburg those arising as the fundamental group algebras of acyclic manifolds motivated partly by the usefulness of superpotential descriptions in motivic donaldson thomas theory we investigate the question of whether these algebras admit superpotential presentations we establish that the fundamental group algebras of a wide class of acyclic manifolds including all hyperbolic manifolds do not admit such descriptions disproving a conjecture of ginzburg regarding them we also describe a class of manifolds that do admit such descriptions and discuss a little their motivic donaldson thomas theory finally some links with topological field theory are described

complete radiation boundary conditions for the helmholtz equation i waveguides we consider the use of complete radiation boundary conditions for the solution of the helmholtz equation in waveguides a general analysis of well posedness convergence and finite element approximation is given in addition methods for the optimization of the boundary condition parameters are considered the theoretical results are illustrated by some simple numerical experiments

the finite element method with nonuniform mesh sizes for unbounded domains the finite element method with nonuniform mesh sizes is employed to approximately solve elliptic boundary value problems in unbounded domains consider the following model problem du f in w sup c u g on partialw partialu partialr 1 r u o 1 r as r vertical barxvertical bar infinity where w sup c is the complement in r sup 3 three dimensional eucidean space of a bounded set w with smooth boundary partialw f and g are smooth functions and f has bounded support this problem is approximately solved by introducing an artifical bound g sub r near infinity e g a sphere of sufficiently large radius r the intersection of this sphere with w sup c is denoted by w sub r sup c and the given problem is replaced by du sub r f in w sup c sub r u sub r g on partialw partialu sub r partialr 1 r u sub r 0 on g sub r this problem is then solved approximately by the finite element method resulting in an approximate solution u sub r sup h for each h 0 in order to obtain a reasonably small error for u u sub r sup h u u sub r u sub r u sub r sup h it is necessary to make r large this necessitates the solution of amore large number of linear equations so that this method is often not very good when a uniform mesh size h is employed it is shown that a nonuniform mesh may be introduced in such a way that optimal error estimates hold and the number of equations is bounded by ch sup 3 with c independent of h and r less

contractions and deformations suppose that f is a projective birational morphism with at most one dimensional fibres between d dimensional varieties x and y satisfying rf ox oy consider the locus l in y over which f is not an isomorphism taking the scheme theoretic fibre c over any closed point of l we construct algebras afib and acon which prorepresent the functors of commutative deformations of c and noncommutative deformations of the reduced fibre respectively our main theorem is that the algebras acon recover l and in general the commutative deformations of neither c nor the reduced fibre can do this as the d 3 special case this proves the following contraction theorem in a neighbourhood of the point the morphism f contracts a curve without contracting a divisor if and only if the functor of noncommutative deformations of the reduced fibre is representable

motivic donaldson thomas invariants of the conifold and the refined topological vertex abstract we compute the motivic donaldson thomas theory of the resolved conifold in all chambers of the space of stability conditions of the corresponding quiver the answer is a product formula whose terms depend on the position of the stability vector generalizing known results for the corresponding numerical invariants our formulae imply in particular a motivic form of the dt pt correspondence for the resolved conifold the answer for the motivic pt series is in full agreement with the prediction of the refined topological vertex formalism

efficient microcomputer based finite difference resistivity modeling via polozhii decomposition finite difference modeling for direct current resistivity problems can be cast in a form allowing rapid and efficient solution on a microcomputer almost a million nodes can be accommodated on a microcomputer with only 256 kilobytes of memory the polozhii decomposition procedure polozhii 1965 allows a three dimensional 3 d problem with no more than a one dimensional 1 d property variation to be transformed into a series of decoupled and independent one dimensional problems the decomposition utilizes matrix transforms analogous to analytic spatial fourier and hankel transforms these matrix transforms along with recursive formulas for the solution of the 1 d problems yield a process which can be thought of as a continuation operator when compared with current popular methods of potential field continuation using the fourier transform the result is that only a single plane of node potentials needs to be determined fully by setting up more complicated models as separate regions with differing 1

boundary conditions for the numerical solution of elliptic equations in exterior regions elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the weill posedness of the problem examples of practical applications include the helmholtz equation and laplace s equation computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition at infinity by a boundary condition on a finite artificial surface direct imposition of the condition at infinity along the finite boundary results in large errors a sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain estimates of the error due to the finite boundary are obtained for several cases computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions the examples are based on a finite element formulation but finite difference methods can also be used

an investigation of finite element modeling for electrical and electromagnetic data in three dimensions the finite element method can be used to solve the differential equations which describe electrical and electromagnetic em field behavior the equations are respectively poisson s equation and the vector damped wave equation the finite element equations are derived in both cases using the minimum theorem while both tetrahedral and hexahedral elements may be used for the modeling of the resistivity problem only hexahedral elements give satisfactory results for the em problem a disadvantage of the relatively simple mesh design used in the approach described here is the presence of long thin elements such elements have very poor interpolating properties and they adversely affect the rate of convergence of the overrelaxation technique used in solving the resulting system of linear equations for the modeling of resistivity data over an earth with one plane of symmetry the system of equations typically has about 9000 unknowns about 50 000 unknowns are needed to give a satisfactory solution to an

bayesian estimation and experimental design in linear regression models estimation and design as a bayesian decision problem choice of a prior distribution conjugate prior distributions bayes estimation of the regression parameter optimality and robustness of the bayes estimator bayesian interpretation of estimators using non bayesian prior knowledge bayes estimation in case of prior ignorance further problems the design problem for the linear bayes estimator characterization of optimal designs construction of optimal continuous designs construction of exact optimal designs

a finite element method for laplace and helmholtz type boundary value problems with singularities laplace and helmholtz type boundary value problems with singularities are considered a sequence of approximations to the exact boundary conditions at an artificial boundary is given then the original problem is reduced to a boundary value problem in a domain away from the singularities furthermore finite element approximations are applied to this problem and error estimates are obtained finally some numerical examples show the effectiveness of this method

noncommutative resolutions of ade fibered calabi yau threefolds in this paper we construct noncommutative resolutions of a certain class of calabi yau threefolds studied by cachazo et al geometric transitions and n 1 quiver theories http arxiv org abs hep th 0108120v2 2001 the threefolds under consideration are fibered over a complex plane with the fibers being deformed kleinian singularities the construction is in terms of a noncommutative algebra introduced by ginzburg calabi yau algebras http arxiv org abs math 0612139v2 2006 which we call the n 1 ade quiver algebra

finite element methods applied to nearly one way wave propagation abstract a numerical method is developed for approximately solving stationary wave propagation problems for which most of the energy propagates in an angle band of less than 90 about a fixed direction this property often occurs in realistic physical models the numerical method is based on factoring out the principal propagation direction in the solution and then applying the finite element method the method is applied to wave propagation in a duct or waveguide and to the exterior scattering problem it is shown both theoretically and by means of numerical examples that this method requires considerably less grid points and computational cost than standard discretization methods

a score regression approach to assess calibration of continuous probabilistic predictions calibration the statistical consistency of forecast distributions and the observations is a central requirement for probabilistic predictions calibration of continuous forecasts is typically assessed using the probability integral transform histogram in this article we propose significance tests based on scoring rules to assess calibration of continuous predictive distributions for an ideal normal forecast we derive the first two moments of two commonly used scoring rules the logarithmic and the continuous ranked probability score this naturally leads to the construction of two unconditional tests for normal predictions more generally we propose a novel score regression approach where the individual scores are regressed on suitable functions of the predictive variance this conditional approach is applicable even for certain nonnormal predictions based on the dawid sebastiani score two case studies illustrate that the score regression approach has typically more power in detecting miscalibrated forecasts than the other approaches considered including a recently proposed technique based on conditional exceedance probability curves

predictive model assessment for count data we discuss tools for the evaluation of probabilistic forecasts and the critique of statistical models for count data our proposals include a nonrandomized version of the probability integral transform marginal calibration diagrams and proper scoring rules such as the predictive deviance in case studies we critique count regression models for patent data and assess the predictive performance of bayesian age period cohort models for larynx cancer counts in germany the toolbox applies in bayesian or classical and parametric or nonparametric settings and to any type of ordered discrete outcomes

making and evaluating point forecasts typically point forecasting methods are compared and assessed by means of an error measure or scoring function with the absolute error and the squared error being key examples the individual scores are averaged over forecast cases to result in a summary measure of the predictive performance such as the mean absolute error or the mean squared error i demonstrate that this common practice can lead to grossly misguided inferences unless the scoring function and the forecasting task are carefully matched effective point forecasting requires that the scoring function be specified ex ante or that the forecaster receives a directive in the form of a statistical functional such as the mean or a quantile of the predictive distribution if the scoring function is specified ex ante the forecaster can issue the optimal point forecast namely the bayes rule if the forecaster receives a directive in the form of a functional it is critical that the scoring function be consistent for it in the sense that t

numerical solution of the three dimensional ginzburg landau models using artificial boundary for three dimensional simulations of the vortex phenomena in superconductors based on the ginzburg landau gl theory the physical variables must be solved in the whole space in general exact boundary conditions on an artificial boundary are discussed to reformulate the problem in a finite domain approximations of these boundary conditions with high accuracy are given and their convergence properties are examined error estimates are derived for the finite element approximations with the approximate boundary conditions

mixed finite element method and high order local artificial boundary conditions for exterior problems of elliptic equation the mixed finite element method is used to solve the exterior elliptic problem with high order local artificial boundary conditions new unknowns are introduced to reduce the order of the derivatives to two this leads to an equivalent mixed variational problem such that the normal finite element can be used and special finite elements are no longer needed on the adjacent layer of the artificial boundary error estimates are obtained for some local artificial boundary conditions with prescribed order numerical examples are presented and the results demonstrate the effectiveness of this method

the simplified hybrid combined methods for laplace s equation with singularities abstract in li and liang 1983 the simplified hybrid combined method is presented for combining the ritz galerkin method and the finite element method in this paper we will apply this method to solve singularity problems of laplace s equation error bounds and stability analyses will be provided while taking into account the integration approximation along the coupling boundary a significant coupling relation between the ritz galerkin and the finite element method has been found for the laplace equation with singularities an optimal rate of convergence has also been achieved numerical experiments have been carried out for solving the benchmark problem motz s problem to verify the theoretical results

solution of helmholtz equation by differential quadrature method the polynomial based differential quadrature pdq and the fourier expansion based differential quadrature fdq methods are applied in this work to solve the two dimensional helmholtz equation it is demonstrated through some sample examples that accurate numerical solution can be obtained by using only 2 to 3 mesh points per wavelength it was found that the fdq approach is more suitable for solving the helmholtz equation

the mckay correspondence as an equivalence of derived categories the classical mckay correspondence relates representations of a finite subgroup g sl 2 c to the cohomology of the well known minimal resolution of the kleinian singularity c2 g gonzalez sprinberg and verdier 10 interpreted the mckay correspondence as an isomorphism on k theory observing that the representation ring of g is equal to the g equivariant k theory of c2 more precisely they identify a basis of the k theory of the resolution consisting of the classes of certain tautological sheaves associated to the irreducible representations of g

the use of summary representation for electromagnetic modeling the method of summary representation developed by g n polozhii is a quasi analytical method for solving self adjoint finite difference boundary value problems expressed on regular meshes in principle the method should allow considerable savings in computing time as well as improved accuracy when compared to commonly used finite difference schemes we have used summary representation as the basis for a new hybrid scheme to solve the two dimensional helmholtz equation for electromagnetic modeling the theory behind this hybrid scheme is presented preliminary results for the two dimensional problem show that substantial computing time and storage savings can be made

finite element analysis of a scattering problem a finite element method for the solution of a scattering problem for the reduced wave equation is formulated and analyzed the method involves a reformulation of the problem on a bounded domain with a nonlocal boundary condition the space of trial functions includes piecewise polynomial functions and functions arising from spherical harmonics

numerical simulation for the problem of infinite elastic foundation in this paper we consider the numerical simulation for the problem of infinite elastic foundation a half circle artificial boundary is introduced and discrete artificial boundary condition on it is designed by using the direct method of lines then the original problem is reduced to a boundary value problem on a bounded computational domain furthermore finite element approximation of this boundary value problem defined in the bounded domain is considered the numerical results show that the artificial boundary condition given in this paper is very effective and more accurate than the neumann boundary condition which is often used in engineering literatures

parameter neutral optimum design for non linear models some bayesian approaches to d optimum design of experiments are considered from the viewpoint of invariance under reparameterization of the underlying statistical model an invariant criterion is proposed which does not require the detailed specification of a prior and which is shown to be equivalent to g optimality under a jeffreys prior the methods are applied and discussed in the contexts of exponential decay and quantal response models

a finite element method for solving helmholtz type equations in waveguides and other unbounded domains a finite element method is described for solving helmholtz type boundary value problems in unbounded regions including those with infinite boundaries typical examples include the propagation of acoustic or electromagnetic waves in waveguides the radiation condition at infinity is based on separation of variables and differs from the classical sommerfeld radiation condition it is shown that the problem may be replaced by a boundary value problem on a fixed bounded domain the behavior of the solution near infinity is incorporated in a nonlocal boundary condition this problem is given a weak or variational formulation and the finite element method is then applied it is proved that optimal error estimates hold

error estimates for the finite element approximation of linear elastic equations in an unbounded domain in this paper we present error estimates for the finite element approximation of linear elastic equations in an unbounded domain the finite element approximation is formulated on a bounded computational domain using a nonlocal approximate artificial boundary condition or a local one in fact there are a family of nonlocal approximate boundary conditions with increasing accuracy and computational cost and a family of local ones for a given artificial boundary our error estimates show how the errors of the finite element approximations depend on the mesh size the terms used in the approximate artificial boundary condition and the location of the artificial boundary a numerical example for navier equations outside a circle in the plane is presented numerical results demonstrate the performance of our error estimates

a hybrid three dimensional electromagnetic modeling scheme we present an efficient numerical method for computing electromagnetic em scattering of arbitrary three dimensional 3 d local inhomogeneities buried in a uniform or two layered earth in this scheme the inhomogeneity is enclosed by a volume whose conductivity is discretized by a finite element mesh and whose boundary is only a slight distance away from the inhomogeneity the scheme uses two sets of independent equations the first is a set of finite element equations derived from a variational integral and the second is a mathematical expression for the fields at the boundany in terms of electric fields inside the boundary the green s function is used to derive the second set of equations an iterative algorithm has been developed to solve these two sets of equations the solutions are the electric fields at nodes inside the finite element mesh the scattered fields anywhere may then be obtained by performing volume integrations over the inhomogeneous region the scheme is used for modeling 3 d inho

computation of superpotentials for d branes we present a general method for the computation of tree level superpotentials for the world volume theory of b type d branes this includes quiver gauge theories in the case that the d brane is marginally stable the technique involves analyzing the a infinity structure inherent in the derived category of coherent sheaves this effectively gives a practical method of computing correlation functions in holomorphic chern simons theory as an example we give a more rigorous proof of previous results concerning 3 branes on certain singularities including conifolds we also provide a new example

a predictive approach to the bayesian design problem with application to normal regression models a predictive decision theoretic approach is developed for the bayesian design problem the loss functions used are fair bayes or proper scoring rules and are quadratic measures of distance between probability measures optimal bayesian designs are those which minimise the preposterior risk for the decision problem such designs typically depend on both the prior distribution and the loss function the results are applied to certain normal regression models where explicit optimal designs are constructed

game theory maximum entropy minimum discrepancy and robust bayesian decision theory we describe and develop a close relationship between two problems that have customarily been regarded as distinct that of maximizing entropy and that of minimizing worst case expected loss using a formulation grounded in the equilibrium theory of zero sum games between decision maker and nature these two problems are shown to be dual to each other the solution to each providing that to the other although topsoe described this connection for the shannon entropy over 20 years ago it does not appear to be widely known even in that important special case we here generalize this theory to apply to arbitrary decision problems and loss functions we indicate how an appropriate generalized definition of entropy can be associated with such a problem and we show that subject to certain regularity conditions the above mentioned duality continues to apply in this extended context this simultaneously provides a possible rationale for maximizing entropy and a tool for finding robust bayes acts we also describe the essential identity between the problem of maximizing entropy and that of minimizing a related discrepancy or divergence between distributions this leads to an extension to arbitrary discrepancies of a well known minimax theorem for the case of kullback leibler divergence the redundancy capacity theorem of information theory for the important case of families of distributions having certain mean values specified we develop simple sufficient conditions and methods for identifying the desired solutions we use this theory to introduce a new concept of generalized exponential family linked to the specific decision problem under consideration and we demonstrate that this shares many of the properties of standard exponential families finally we show that the existence of an equilibrium in our game can be rephrased in terms of a pythagorean property of the related divergence thus generalizing previously announced results for kullback leibler and bregman divergences

the numerical solution of the helmholtz equation for wave propagation problems in underwater acoustics abstract the helmholtz equation k2n2 u 0 with a variable index of refraction n and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems a numerical algorithm has been developed and a computer code implemented that can effectively solve this equation in the intermediate frequency range the equation is discretized by using the finite element method thus allowing for the modeling of complicated geometries including interfaces and complicated boundary conditions a global radiation boundary condition is imposed at the far field boundary that is exact for an arbitrary number of propagating modes the resulting large nonselfadjoint system of linear equations with indefinite symmetric part is solved by using the preconditioned conjugate gradient method applied to the normal equations a new preconditioner based on the multigrid method is developed this preconditioner is vectorizable and is extremely effective over a wide range of frequencies provided the number of grid levels is reduced for large frequencies a heuristic argument is given that indicates the superior convergence properties of this preconditioner the relevant limit to analyze convergence is for k increasing and a fixed prescribed accuracy level the efficiency and robustness of the numerical algorithm are confirmed for large acoustic models including interfaces with strong velocity contrasts

modeling of borehole resistivity measurements using infinite electrical grids an infinite electrical resistive grid generated by a finite difference approximation to poisson s equation in cylindrical coordinates with no azimuthal variation has been used to model borehole resistivity measurements the medium surrounding the borehole is infinite in extent and its resistivity varies only in the radial direction as a result the branch resistance values of the infinite grid also vary only in the radial direction a continued fraction of laurent operators determines the node voltages along the borehole s surface use of an operational calculus based upon the finite fourier transformation converts the continued fraction into a readily computed form this yields new formulas for apparent resistivity given various resistivity measuring arrays and a resistivity profile in the radial direction in contrast to classical methods the need to match boundary conditions at the interface of a cylindrically layered earth is eliminated so that the procedure is just as easily applied to a continuo

the artificial boundary conditions for computing the flow around a submerged body in this paper we consider the numerical simulations of the steady linearized potential flow around a body which moves in a liquid of finite constant depth at constant speed and distance below a free surface two vertical segments are introduced as the artificial boundary to limit the unbounded physical domain to a bounded computational domain a series of approximate artificial boundary conditions at the given artificial boundary are designed by imposing the continuity of the solution and its normal derivative then the original problem is reduced to a problem defined in a bounded computational domain which is solved numerically by the finite element method the numerical examples show that our artificial boundary conditions are very effective

higher order elicitability and osband s principle a statistical functional such as the mean or the median is called elicitable if there is a scoring function or loss function such that the correct forecast of the functional is the unique minimizer of the expected score such scoring functions are called strictly consistent for the functional the elicitability of a functional opens the possibility to compare competing forecasts and to rank them in terms of their realized scores in this paper we explore the notion of elicitability for multi dimensional functionals and give both necessary and sufficient conditions for strictly consistent scoring functions we cover the case of functionals with elicitable components but we also show that one dimensional functionals that are not elicitable can be a component of a higher order elicitable functional in the case of the variance this is a known result however an important result of this paper is that spectral risk measures with a spectral measure with finite support are jointly elicitable if one adds the correct quantiles a direct consequence of applied interest is that the pair value at risk expected shortfall is jointly elicitable under mild conditions that are usually fulfilled in risk management applications

the artificial boundary conditions for incompressible materials on an unbounded domain in this paper we consider the numerical simulations of the incompressible materials on an unbounded domain in frak r 2 a series of artificial boundary conditions at a circular artificial boundary for solving incompressible materials on an unbounded domain is given then the original problem is reduced to a problem on a bounded domain which be solved numerically by a mixed finite element method the numerical example shows that our artificial boundary conditions are very effective

skew group algebras of calabi yau algebras abstract the calabi yau property of skew group algebras is discussed it is shown that the skew group algebra a g of a koszul calabi yau algebra a with a finite subgroup g of automorphisms of a is calabi yau if and only if g is a finite subgroup of the special linear group sl a which is defined by means of the homological determinant using the a algebra structure on the yoneda algebra some results in bocklandt et al 2010 bsw are generalized say every connected graded p koszul calabi yau algebra is derived from a superpotential the superpotential for the skew group algebra a g is also constructed

spectral methods for exterior elliptic problems this paper deals with spectral approximations for exterior elliptic problems in two dimensions as in the conventional finite difference or finite element methods it is found that the accuracy of the numerical solutions is limited by the order of the numerical farfield conditions we introduce a spectral boundary treatment at infinity which is compatible with the infinite order interior spectral scheme computational results are presented to demonstrate the spectral accuracy attainable although we deal with a simple laplace problem throughout the paper our analysis covers more complex and general cases

the finite element method with non uniform mesh sizes applied to the exterior helmholtz problem the finite element method with non uniform mesh sizes is employed to approximately solve helmholtz type equations in unbounded domains the given problem on an unbounded domain is replaced by an approximate problem on a bounded domain with the radiation condition replaced by an approximate radiation boundary condition on the artificial boundary this approximate problem is then solved using the finite element method with the mesh graded systematically in such a way that the element mesh sizes are increased as the distance from the origin increases this results in a great reduction in the number of equations to be solved it is proved that optimal error estimates hold inl 2 h 1 andl provided that certain relationships hold between the frequency mesh size and outer radius

a domain decomposition method for the helmholtz equation and related optimal control problems we present an iterative domain decomposition method to solve the helmholtz equation and related optimal control problems the proof of convergence of this method relies on energy techniques this method leads to efficient algorithms for the numerical resolution of harmonic wave propagation problems in homogeneous and heterogeneous media

finite difference analysis of borehole flows involving domain contractions around three dimensional anomalies a finite difference analysis in cylindrical coordinates of borehole flows governed by poisson s equation yields an infinite cylindrical electrical grid as a model the customary technique of truncating the medium in order to obtain a finite grid can be avoided at least to some extent by using the recently developed theory of infinite electrical networks when the medium around the borehole is either uniform or cylindrically layered except for a three dimensional anomalous region near the borehole infinite electrical network theory allows one to represent the medium outside any cylinder s containing the anomaly and centered on the borehole by an infinite continued fraction of circulant laurent operators the continued fraction in turn can be represented by scalar conductances connected to the grid s nodes on s by choosing s just large enough to contain the anomaly one can replace the large system of equations resulting from the usual medium truncations far removed from the anomaly by a much smaller systems of equations for the grid within and on s an operational calculus for the circulant laurent operators and the fast fourier transform expedites the computations and moreover allows a rapidly computed determination of the solution outside s

the intrinsic normal cone let x be an algebraic stack in the sense of deligne mumford we construct a purely 0 dimensional algebraic stack over x in the sense of artin the intrinsic normal cone frak c x the notion of perfect obstruction theory for x is introduced and it is shown how to construct given a perfect obstruction theory for x a pure dimensional virtual fundamental class in the chow group of x we then prove some properties of such classes both in the absolute and in the relative context via a deformation theory interpretation of obstruction theories we prove that several kinds of moduli spaces carry a natural obstruction theory and sometimes a perfect one

t structures on some local calabi yau varieties abstract let z be a smooth fano variety satisfying the condition that the rank of the grothendieck group of z is one more than the dimension of z let z denote the total space of the canonical line bundle of z considered as a non compact calabi yau variety we use the theory of exceptional collections to describe t structures on the derived category of coherent sheaves on z the combinatorics of these t structures is determined by a natural action of an affine braid group closely related to the well known action of the artin braid group on the set of exceptional collections on z

donaldson thomas type invariants via microlocal geometry we prove that donaldson thomas type invariants are equal to weighted euler characteristics of their moduli spaces in particular such invariants depend only on the scheme structure of the moduli space not the symmetric obstruction theory used to define them we also introduce new invariants generalizing donaldson thomas type invariants to moduli problems with open moduli space these are useful for computing donaldson thomas type invariants over stratifications

quivers from matrix factorizations we discuss how matrix factorizations offer a practical method of computing the quiver and associated superpotential for a hypersurface singularity this method also yields explicit geometrical interpretations of d branes i e quiver representations on a resolution given in terms of grassmannians as an example we analyze some non toric singularities which are resolved by a single mathbb p 1 but have length greater than one these examples have a much richer structure than conifolds a picture is proposed that relates matrix factorizations in landau ginzburg theories to the way that matrix factorizations are used in this paper to perform noncommutative resolutions

coherent dispersion criteria for optimal experimental design we characterize those coherent design criteria which depend only on the dispersion matrix assumed proper and nonsingular of the state of nature which may be a parameter vector or a set of future observables and describe the associated decision problems connections are established with the classical approach to optimal design theory for the normal linear model based on concave functions of the information matrix implications of the theory for more general models are also considered

strictly proper scoring rules prediction and estimation scoring rules assess the quality of probabilistic forecasts by assigning a numerical score based on the predictive distribution and on the event or value that materializes a scoring rule is proper if the forecaster maximizes the expected score for an observation drawn from the distributionf if he or she issues the probabilistic forecast f rather than g f it is strictly proper if the maximum is unique in prediction problems proper scoring rules encourage the forecaster to make careful assessments and to be honest in estimation problems strictly proper scoring rules provide attractive loss and utility functions that can be tailored to the problem at hand this article reviews and develops the theory of proper scoring rules on general probability spaces and proposes and discusses examples thereof proper scoring rules derive from convex functions and relate to information measures entropy functions and bregman divergences in the case of categorical variables we prove a rigorous version of the

domain contractions around three dimensional anomalies in spherical finite difference computations of poisson s equation it is computationally advantageous to solve a finite difference approximation to poisson s equation nabla cdot sigma nabla p h 0 in an infinite domain wherein sigma varies with all three spherical coordinates inside a sphere s j but only with the radial and latitudinal coordinates outside s j and h vanishes outside s j by representing the medium outside s j by a driving point conductance matrix this is equivalent to replacing the spherical grid outside s j by a set of terminating conductances connected to the nodes on s j thus the large system matrix generated by the customary truncation of the infinite medium far beyond s j is replaced by the much smaller matrix corresponding to the nodes on and within s j this technique is especially advantageous when the problem is to be solved many times for various hypothesized variations of sigma within s j because the terminating conductances depend only upon the values of sigma outside s j an operationa

projective toric varieties as fine moduli spaces of quiver representations this paper proves that every projective toric variety is the fine moduli space for stable representations of an appropriate bound quiver to accomplish this we study the quiver q with relations r corresponding to the finite dimensional algebra mathop rm end nolimits textstyle bigoplus nolimits i 0 r l i where cal l scr o x l 1 ldots l r is a list of line bundles on a projective toric variety x the quiver q defines a smooth projective toric variety called the multilinear series cal l and a map x longrightarrow cal l we provide necessary and sufficient conditions for the induced map to be a closed embedding as a consequence we obtain a new geometric quotient construction of projective toric varieties under slightly stronger hypotheses on cal l the closed embedding identifies x with the fine moduli space of stable representations for the bound quiver q r

the a deformation theory of a point and the derived categories of local calabi yaus abstract let a be an augmented algebra over a semi simple algebra s we show that the ext algebra of s as an a module enriched with its natural a infinity structure can be used to reconstruct the completion of a at the augmentation ideal we use this technical result to justify a calculation in the physics literature describing algebras that are derived equivalent to certain non compact calabi yau three folds since the calculation produces superpotentials for these algebras we also include some discussion of superpotential algebras and their invariants

application of a complete radiation boundary condition for the helmholtz equation in locally perturbed waveguides abstract this paper deals with an application of a complete radiation boundary condition crbc for the helmholtz equation in locally perturbed waveguides the crbc one of efficient high order absorbing boundary conditions has been analyzed in straight waveguides in hagstrom and kim 2019 in this paper we apply crbc to the helmholtz equation posed in locally perturbed waveguides and establish the well posedness of the problem and convergence of crbc approximate solutions the new crbc proposed in this paper improves the one studied in hagstrom and kim 2019 in two aspects the first one is that the new crbc involves more damping parameters with the same computational cost as that of crbc in hagstrom and kim 2019 which results in 50 smaller reflection errors the second one is that the new crbc takes a neumann terminal condition of three term recurrence relations of auxiliary variables instead of a dirichlet terminal condition used in hagstrom and kim 2019 so that it can treat cutoff modes effectively finally we present numerical experiments illustrating the convergence theory

noncommutative deformations and flops we prove that the functor of noncommutative deformations of every flipping or flopping irreducible rational curve in a 33 fold is representable and hence we associate to every such curve a noncommutative deformation algebra aconacon this new invariant extends and unifies known invariants for flopping curves in 33 folds such as the width of reid and the bidegree of the normal bundle it also applies in the settings of flips and singular schemes we show that the noncommutative deformation algebra aconacon is finite dimensional and give a new way of obtaining the commutative deformations of the curve allowing us to make explicit calculations of these deformations for certain 3 1 3 1 curves we then show how our new invariant aconacon also controls the homological algebra of flops for any flopping curve in a projective 33 fold with only gorenstein terminal singularities we construct an autoequivalence of the derived category of the 33 fold by twisting around a universal family over the noncommutative deformation algebra aconacon and prove that this autoequivalence is an inverse of bridgeland s flop flop functor this demonstrates that it is strictly necessary to consider noncommutative deformations of curves in order to understand the derived autoequivalences of a 33 fold and thus the bridgeland stability manifold

non reflecting boundary conditions for waveguides new non reflecting boundary conditions are introduced for the solution of the helmholtz equation in a waveguide these boundary conditions are perfectly transparent for all propagating modes they do not require the determination of these propagating modes but only their propagation constants a quasi local form of these boundary conditions is well suited as terminating boundary condition beyond finite element meshes related convergence properties to the exact solution and optimal error estimates are established

the weighted finite element method for elasticity equations on unbounded domains abstract by introducing the weighted riesz galerkin method with a special weight function oh jou 23 made it possible to apply mam the method of auxiliary mapping introduced by babuska oh to general elliptic boundary value problems on unbounded domains in this paper this method is further generalized to yield highly accurate approximate solutions for linear elasticity problems on unbounded domains with bounded or unbounded boundaries it is shown that in the framework of the p version of the fem this method does efficiently handle elasticity problems on unbounded domains and yields accurate solutions at low cost

on bayesian d optimum design criteria and the equivalence theorem in non linear models the concavity of some bayesian d optimality criteria is investigated and is found in some cases to depend on the prior distribution in the case of a non concave criterion the standard equivalence theorem may fail but a local version continues to apply