Linear Algebra

We develop highly efficient parallel pricing methods on Graphics Processing Units (GPUs) for multi-asset American options via a Partial Differential Equation (PDE) approach. The linear complementarity problem arising due to the free boundary is handled by a penalty method. Finite difference methods on uniform grids are considered for the space discretization of the PDE, while classical finite differences, such as Crank-Nicolson, are used for the time discretization. The discrete nonlinear penalized equations at each timestep are solved using a penalty iteration. A GPU-based parallel Alternating Direction Implicit Approximate Factorization technique is employed for the solution of the linear algebraic system arising from each penalty iteration. We demonstrate the efficiency and accuracy of the parallel numerical methods by pricing American options written on three assets.

The coupled Navier-Stokes and Cahn-Hilliard equations, plus the constitutive equation for non-Newtonian fluids, are solved using second-order implicit time stepping. Within each time step, Newton iteration is used to handle the nonlinearity, and the linear algebraic system is solved by preconditioned Krylov methods. The phase-field model, with a physically diffuse interface, affords the method several advantages in computing interfacial dynamics. One is the ease in simulating topological changes such as interfacial rupture and coalescence. Another is the capability of computing contact line motion without invoking ad hoc slip conditions. As validation of the 3D numerical scheme, we have computed drop deformation in an elongational flow, relaxation of a deformed drop to the spherical shape, and drop spreading on a partially wetting substrate. The results are compared with numerical and experimental results in the literature as well as our own axisymmetric computations where appropriate. Excellent agreement is achieved provided that the 3D interface is adequately resolved by using a sufficiently thin diffuse interface and refined grid. Since our model involves several coupled partial differential equations and we use a fully implicit scheme, the matrix inversion requires a large memory. This puts a limit on the scale of problems that can be simulated in 3D, especially for viscoelastic fluids.

This is a study about the didactical organization of a research based group of activities designed using APOS theory to help university students make constructions needed to understand and graph two-variable functions, but found to be lacking in previous studies. The model of the “moments of study” of the Anthropological Theory of Didactics is applied to analyze the activities in terms of their institutional viability.

A semianalytical model based on the method of eigenfunction expansions and domain decomposition is developed for Stokes shear flow over a grating composed of a periodic array of parallel slats, with finite slippage on solid surfaces and infinite slippage on the bottom of troughs mimicking a no-shear liquid-gas interface penetrating into the space between slats. The model gives the macroscopic slip lengths for flow parallel or normal to the slats in terms of the microscopic slip length of the liquid-solid interface, area fraction of the no-shear liquid-gas interface, and depth of the liquid-gas interface in the grooves. When the no-shear interface lies flat on the top of the slats, the macroscopic slip lengths are the maximum and can be estimated with reasonably good accuracy by simple formulas. However, the slip lengths, particularly the transverse one, are very sensitive to penetration of the no-shear interface into the grooves. They can be reduced by a large factor when the interface just slightly gets into the grooves. On comparing with some molecular-dynamics simulation measures, it is pointed out that the applied pressure, which has to be less than the capillary pressure in the superhydrophobic state, can be correlated with the penetration depth of the no-shear interface.

By studying the personal work of the students, we introduced the notion of Personal Practice Paths, idealized from the conception of personal praxeologies and praxeological recombination outlined by the praxeological dynamics inherent to the task resolution process. The study was done with students of a mathematics teaching degree course from a Brazilian university. As a methodological approach, we adopted a link-up between didactic engineering and ATD concepts, which allowed us to think about the emergence of a new dialectic that underlies the firstfruits of the Personal Study and Research Paths.

This paper describes the design and implementation of high performance numerical software in Java. Our primary goals are to characterize the performance of object‐oriented numerical software written in Java and to investigate whether Java is a suitable language for such endeavors. We have implemented JLAPACK, a subset of the LAPACK library in Java. LAPACK is a high‐performance Fortran 77 library used to solve common linear algebra problems. JLAPACK is an object‐oriented library, using encapsulation, inheritance, and exception handling. It performs within a factor of four of the optimized Fortran version for certain platforms and test cases. When used with the native BLAS library, JLAPACK performs comparably with the Fortran version using the native BLAS library. We conclude that high‐performance numerical software could be written in Java if a handful of concerns about language features and compilation strategies are adequately addressed.

We present a closed-form, computable expression for the expected number of times any transition event occurs during the transient phase of a reducible Markov chain. Examples of events include time to absorption, number of visits to a state, traversals of a particular transition, loops from a state to itself, and arrivals to a state from a particular subset of states. We give an analogous expression for time-average events, which describe the steady-state behavior of reducible chains as well as the long-term behavior of irreducible chains.

Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact

spherical particles are presented . It is shown that the agreement between exact and approximate c*lculattons for prolat. spheroidi is good . Additionally the validity and utility of the Mi. solution modification ret ly proposed by Chyle Grams , and Pinnick are examined , crime polydisperes mon ~~h.rica aeroSg ~.s Axa -.eaeetflSd. .

BLAS 3 Matrix/matrix Dongarra, Du Croz, Hammarling and Hanson, 1990 LAPACK on cache based machines • The BLAS supports a separation of concerns: -HW/SW optimization experts tuned the BLAS for specific platforms. -Linear algebra experts built software on top of the BLAS ... high performance "for free". • It is difficult to over-state the impact of the BLAS … they revolutionized the practice of computational linear algebra.

Starting from a few elements, binomial matrices and vectors, without resorting to the traditional expansion in power series, polynomials and Bernoulli numbers are defined, proving various theorems on them. Along the way, the classic problem of the sum of powers of successive integers, which is generalized to sums of powers having any arithmetic progression as a basis, is also solved, in various ways,. The itinerary develops through numerous propositions rigorously linked on the classic model of Euclid's Elements. Numerous examples and hypertext references have the didactic purpose of making reading easier.

Since its founding, NASA has been dedicated to the advancement of aeronautics and space science. The NASA scientific and technical information (STI) program plays a key part in helping NASA maintain this important role. The NASA STI Program operates under the auspices of the Agency Chief Information Officer. It collects, organizes, provides for archiving, and disseminates NASA's STI. The NASA STI Program provides access to the NASA Aeronautics and Space Database and its public interface, the NASA Technical Report Server, thus providing one of the largest collection of aeronautical and space science STI in the world. Results are published in both non-NASA channels and by NASA in the NASA STI Report Series, which includes the following report types: • TECHNICAL PUBLICATION. Reports of completed research or a major significant phase of research that present the results of NASA programs and include extensive data or theoretical analysis. Includes compilations of significant scientific and technical data and information deemed to be of continuing reference value. NASA counterpart of peer-reviewed formal professional papers, but having less stringent limitations on manuscript length and extent of graphic presentations. • TECHNICAL MEMORANDUM. Scientific and technical findings that are preliminary or of specialized interest, e.g., quick release reports, working papers, and bibliographies that contain minimal annotation. Does not contain extensive analysis. • CONTRACTOR REPORT. Scientific and technical findings by NASA-sponsored contractors and grantees. • CONFERENCE PUBLICATION. Collected papers from scientific and technical conferences, symposia, seminars, or other meetings sponsored or co-sponsored by NASA. • SPECIAL PUBLICATION. Scientific, technical, or historical information from NASA programs, projects, and missions, often concerned with subjects having substantial public interest.

Fig. 1: (a) We attempt to mimic the "Jack Nicholson" expression of partially closed eyes with an arched eyebrow. First the eyelids are partially closed. (b) The model has three controls over eyebrow shape. The desired arched eyebrow is easily obtained, but the eyelid is changed as a side effect. (c) The model is capable of approximating the desired expression however, by readjusting the eyelid control (or, by using our technique).

Fig. 1: (a) We attempt to mimic the "Jack Nicholson" expression of partially closed eyes with an arched eyebrow. First the eyelids are partially closed. (b) The model has three controls over eyebrow shape. The desired arched eyebrow is easily obtained, but the eyelid is changed as a side effect. (c) The model is capable of approximating the desired expression however, by readjusting the eyelid control (or, by using our technique).

A library of algorithms developed as algorithmic cyberFilms is presented. Algorithmic cyberFilms are a new type of software components for presentation, specification/programming and automatic code generation of computational algorithms. The algorithmic cyberFilm format is implemented as a set of multimedia frames (and scenes), and each component is represented by frames of algorithmic skeletons representing dynamical features of an algorithm, by frames of integrated view providing static features of the algorithm in a compact format, and by corresponding template codes supporting the program generation. We developed a library which is a collection of basic and advanced algorithms taught at universities, including computation on grids, trees and graphs. In this paper, we present basic constructs of visual languages which are used for representing cyberFilms as well as for demonstrating the library components. We also provide a general overview of the library and its features. In addition, we discuss results of experiments which were conducted to verify the usability of the library components and their usefulness in education.

Large scale applications running on HPC systems often require a substantial amount of memory and can have a large computational overhead. Lossy data compression techniques can reduce the size of the data and associated communication cost, but the effect of the loss of accuracy on the numerical algorithm can be hard to predict. In this paper we examine the FGMRES algorithm, which requires the storage of a basis for the Krylov subspace and for the search space spanned by the solutions of the preconditioning systems. We show that the vectors spanning this search space can be compressed by looking at the combination of FGMRES and compression in the context of inexact Krylov subspace methods. This allows us to derive a bound on the normwise relative compression error in each iteration. We use this bound to formulate a number of different practical compression strategies, and validate and compare them through numerical experiments.

We introduce the concept of strongly r-matrix induced (SRMI) Poisson structure, report on the relation of this property with the stabilizer dimension of the considered quadratic Poisson tensor, and classify the Poisson structures of the Dufour-Haraki classification (DHC) according to their membership of the family of SRMI tensors. One of the main results of our work is a generic cohomological procedure for SRMI Poisson structures in arbitrary dimension. This approach allows decomposing Poisson cohomology into, basically, a Koszul cohomology and a relative cohomology. Moreover, we investigate this associated Koszul cohomology, highlight its tight connections with Spectral Theory, and reduce the computation of this main building block of Poisson cohomology to a problem of linear algebra. We apply these upshots to two structures of the DHC and provide an exhaustive description of their cohomology. We thus complete our list of data obtained in previous works, see and [AP07], and gain fairly good insight into the structure of Poisson cohomology.

We introduce a novel hybrid quantum–analog algorithm to perform a graph clustering that exploits connections between the evolution of dynamical systems on graphs and the underlying graph spectra. This approach constitutes a new class of algorithms that combine emerging quantum and analog platforms to accelerate computations. Our hybrid algorithm is equivalent to spectral clustering and significantly reduces the computational complexity from 𝒪(<italic>N</italic>³) to 𝒪(<italic>N</italic>), where <italic>N</italic> is the number of nodes in the graph. We achieve this speedup by circumventing the need for explicit eigendecomposition of the normalized graph Laplacian matrix, which dominates the classical complexity, and instead leveraging quantum evolution of the Schrödinger equation followed by efficient analog computation for the dynamic mode decomposition (DMD) step. Specifically, while classical spectral clustering requires 𝒪(<italic>N</italic>³) operations to perform eigendecomposition, our method exploits the natural quantum evolution of states according to the graph Laplacian Hamiltonian in linear time, combined with the linear scaling for DMD that leverages efficient matrix–vector multiplications on analog hardware. We prove and demonstrate that this hybrid approach can extract the eigenvalues and scaled eigenvectors of the normalized graph Laplacian by evolving Schrödinger dynamics on quantum computers followed by DMD computations on analog devices, providing a significant computational advantage for large-scale graph clustering problems. Our demonstrations can be reproduced using our code that has been released at on github.

We study the semirelativistic Hamiltonian operator composed of the relativistic kinetic energy and a static harmonic-oscillator potential in three spatial dimensions and construct, for bound states with vanishing orbital angular momentum, its eigenfunctions in "compact form," i.e., as power series, with expansion coefficients determined by an explicitly given recurrence relation. The corresponding eigenvalues are fixed by the requirement of normalizability of the solutions.

The Standard Template Adaptive Parallel Library (STAPL) is a superset of C++'s Standard Template Library (STL) which allows highproductivity parallel programming in both distributed and shared memory environments. This framework provides parallel equivalents of STL containers and algorithms enabling ease of development for parallel systems. In this paper, we will discuss our methodology for implementing a fast and efficient matrix multiplication algorithm in STAPL. Our implementation employs external linear algebra libraries, specifically the Basic Linear Algebra Subprograms (BLAS) library which includes highly optimized sequential matrix operations. The paper will describe the benefits of creating a parallel matrix multiplication algorithm whose library calls are specialized based on both the matrix storage and traversal. This specialization technique ensures that the most appropriate implementation in terms of data access and structure will be used, resulting in increased efficiency compared to a non-specialized approach.

In general, the derivative of an eigenvector of a vibrating symmetric system is the solution of a singular problem. Further complications are encountered in dealing with asymmetric damped systems for which the left and right eigenvectors and their derivatives become distinct and complex. Several approaches have been proposed to overcome this singularity such as Nelson's method and the modal method. In the present work, a new approach is presented for calculating simultaneously the derivatives of the eigenvalues and their associated derivatives of the left and right eigenvectors for asymmetric damped systems. With the proposed method, the exact eigenderivatives can be obtained by solving a first-order linear algebraic system of equations. The method is applied on a 104 DOF ventilator-rotor system, which is used as an example of an asymmetric damped system with distinct eigenvalues. The diameter of the shaft has been chosen as the design parameter. The comparison of the computational time shows that the proposed method is more efficient than both Nelson's approach and the modal method.

The application of new methodologies in higher education brings with it questions such as: "Can we do it?", "Can we teach?", "Can we motivate students?", "Can we evaluate?". During the last 8 years, in the Linear Algebra course of the Mechanical and Electromechanical Engineering of the Polytechnic of Coimbra-Institute of Engineering, it has been trying to answer these questions. The teaching methodology presented consists in the approach of Linear Algebra contents, using digital Image Processing applications. This new environment is intended to encourage students to learn the concepts and their manipulation in a productive and dynamic way. From the analysis of the studies and the student's opinions (students' adherence and motivation, questionnaires), it has been found that the use of this new teaching methodology helps the 'learning to learn' as long as it is well defined, with clear objectives for those who implements (teacher) and who interacts (students) with it.

This paper simultaneously makes new contributions to political science, probability and statistics, and linear algebra. We describe a new method, "Sinkhorn ratings," for pairwise-comparison based ranking of chessplayers, web pages, or football teams; it also may be used to rank the candidates in an election in which each vote is a partial ordering of the candidates. We also describe the first "strongly polynomial time" algorithm for finding the Perron-Frobenius eigenvector of a matrix with non-negative entries, and the second strongly polynomial time algorithm (and the first practical one) for Sinkhorn balancing a matrix with non-negative entries. The former also may be regarded as the first strongly polynomial algorithm for finding the stationary distribution of an N -state Markov chain with known transition matrix. Along the way we also present a new formulation of the Perron-Frobenius eigenvector and Markov stationary distribution problems as concave-∪ minimization problems, and present a powerful new technique for proving monotonicity statements e.g. about Markov chains.

En este articulo se presentan los resultados del diagnostico y analisis de algunas causas que obstaculizan el aprendizaje del Algebra Lineal en los estudiantes de ingenieria de la Universidad Tecnologica de Pereira. Igualmente pretende identificar la importancia que se le da a la asignatura dentro de los programas de ingenieria, mediante encuestas que se realizaron a los profesores que orientan diferentes asignaturas de matematicas como a los directores de los diferentes programas. Ademas se presentan conclusiones y recomendaciones acerca del estudio.

The Newton algorithm based on the "continuation" method may be written as being governed by the equation ( ) where F i (x j ) = 0, i, j = 1, …n are nonlinear algebraic equations (NAEs) to be solved, and B ij = ∂F i /∂x j is the corresponding Jacobian matrix. It is known that the Newton's algorithm is quadratically convergent; however, it has some drawbacks, such as being sensitive to the initial guess of solution, and being expensive in the computation of the inverse of B ij at each iterative step. How to preserve the convergence speed, and to remove the drawbacks is a very important issue in the solutions of NAEs. In this paper we discretize the above equation being written as ( ) F x = by a backward difference scheme in a new time scale of s = 1 -e -t , and an ODEs system is derived by introducing a fictitious time-like variable. The new algorithm is obtained by applying a numerical integration scheme to the resultant ODEs. The new algorithm does not need the inverse of B ij , and is thus resulting in a significant reduction in computational time than the Newton's algorithm. A similar technique is also used to modify the homotopy method. Numerical examples given confirm that the modified Newton method is highly efficient, insensitive to the initial condition, to find the solutions with a very small the residual error.

A method to compute the eigenvalues of the spheroidal wave equations is proposed, as an application of a theorem on eigenvalues of certain classes of infinite matrices. The computation of its inverse problem (namely, solving another parameter ¥ §¦ for given eigenvalue ¨) is likewise given. As a result, precise and explicit error estimates are obtained for the approximated eigenvalues.

2007 Juan Gabriel Molina / Asuman Oktaç CONCEPCIONES DE LA TRANSFORMACIÓN LINEAL EN CONTEXTO GEOMÉTRICO Revista Latinoamericana de Investigacion en Matematica Educativa, julio, año/vol. 10, número 002 Comité Latinoamericano de Matemática ...

Few courses undertaken by students in their first two years of engineering education are as abstract as linear algebra. The content is not only critical for future success, but considerably difficult at the same time, because most students are unfamiliar with abstract mathematical concepts at that point in their lives. Many courses around the nation augment their traditional instruction by exposing their students to computer algebra systems (CAS) such as MatLab, Mathematica, MathCad, Maple, etc. While these tools are great for analysis, they are not designed to be primary instructional tools. These tools favor those students that have a stronger theoretical understanding of the concepts, but do little to help them develop a better understanding of these. The author presents the development of an online multimedia collaboration tool that actively teaches the students basic linear algebra concepts. The system utilizes non-linear broadcast quality video and interactive 3D graphics in a multi-user environment. A pseudo intelligent mechanism analyzes the learner's choices and modifies instructional delivery based on a predetermined instructional-objectives map. The system in not meant to replace classroom instruction or utilization of computational CAS systems. Its primary purpose is to provide supplemental tutoring aid while fostering visualization skills in the learner.

Baksalary et al. (Linear Algebra Appl., doi:10.1016/j.laa.2004.02.025, 2004) investigated the invertibility of a linear combination of idempotent matrices. This result was improved by Koliha et al. (Linear Algebra Appl., doi:10.1016/j.laa.2006.01.011, 2006) by showing that the rank of a linear combination of two idempotents is constant. In this paper, we consider similar problems for k-potent matrices. We study the rank and the nullity of a linear combination of two commuting k-potent matrices. Furthermore, the problem of the nonsingularity of linear combinations of two or three k-potent matrices is considered under some conditions. In these situations, we derive explicit formulae of their inverses.

Let A be an m × n matrix in which the entries of each row are all distinct. Drisko [4] showed that, if m ≥ 2n -1, then A has a transversal : a set of n distinct entries with no two in the same row or column. We generalize this to matrices with entries in a matroid. For such a matrix A, we show that if each row of A forms an independent set, then we can require the transversal to be independent as well. We determine the complexity of an algorithm based on the proof of this result. Lastly, we observe that m ≥ 2n -1 appears to force the existence of not merely one but many transversals. We discuss a number of conjectures related to this observation (some of which involve matroids and some of which do not).

Matrix computations are both fundamental and ubiquitous in computational science and its vast application areas. Along with the development of more advanced computer systems with complex memory hierarchies, there is a continuing demand for new algorithms and library software that efficiently utilize and adapt to new architecture features. This article reviews and details some of the recent advances made by applying the paradigm of recursion to dense matrix computations on today's memory-tiered computer systems. Recursion allows for efficient utilization of a memory hierarchy and generalizes existing fixed blocking by introducing automatic variable blocking that has the potential of matching every level of a deep memory hierarchy. Novel recursive blocked algorithms offer new ways to compute factorizations such as Cholesky and QR and to solve matrix equations. In fact, the whole gamut of existing dense linear algebra factorization is beginning to be reexamined in view of the recursive paradigm. Use of recursion has led to using new hybrid data structures and optimized superscalar kernels. The results we survey include new algorithms and library software implementations for level 3 kernels, matrix factorizations, and the solution of general systems of linear equations and several common matrix equations. The software implementations we survey are robust and show impressive performance on today's high performance computing systems.

Consideration is given to the numerical computation of geodesic on surfaces. The method of finite-difference is used for governing non-linear system of differential equations. Then the resulting non-linear algebraic equations are solved by both iterative and NewtonÕs method. It is shown that iterative method (IM) gives better result than New-tonÕs method. Finally, we demonstrated our finding on several surfaces.

The theme of this article is how to apply quantum algorithms for symbolic computations of polynomials, e.g., the calculation of Gröbner basis, which is an important tool in algebraic geometry and provides us with symbolicnumeric methods to solve polynomial equations. However, the computation of Gröbner basis by classical computers is demanding, since the computational *

A rigorous solution is presented for the problem of diffraction of plane harmonic sound waves by a cavity formed by a terminated rigid cylindrical waveguide of finite length whose interior surface is lined by an acoustically absorbent material. The solution is obtained by a modification of the Wiener-Hopf technique and involve an infinite series of unknowns, which are determined from an infinite system of linear algebraic equations. Numerical solution of this system is obtained for various values of the parameters of the problem and their effects on the diffraction phenomenon are shown graphically.

Symmetries in modern physics are a fundamental theme under study that allows to appreciate the Particle Physics. In addition, gauge theories are tools that allow to do a description more complete. With this paper, we analyze a gauge symmetry that appears in complex holomorphic systems. A complex system can be reduced to different real systems, using different gauge conditions and several real systems are connected by gauge transformations in the complex space. We prove that the space of solutions of one system is related to another one using a gauge transformation. Gauge transformations are in some cases canonical transformations. However, in other cases are more general transformations that change the symplectic structure, but there is still a map between systems. We establish a construction extend of gauge transformations and show how to extend the analysis to the quantum case using path integrals by means of the Batalin-Fradkin-Vilkovisky theorem and with in the canonical formalism, where we show explicitly that solutions of the Schrödinger equation are gauge related.

In this paper we give an explicit solution to the rank constrained matrix approximation in Frobenius norm, which is a generalization of the classical approximation of an m × n matrix A by a matrix of rank k at most. 2000 Mathematics Subject Classification: 15A18. Keywords and phrases: SVD decomposition, generalized rank-constrained matrix approximations, generalized inverse.

En este articulo se presentan los resultados del diagnostico y analisis de algunas causas que obstaculizan el aprendizaje del Algebra Lineal en los estudiantes de ingenieria de la Universidad Tecnologica de Pereira. Igualmente pretende identificar la importancia que se le da a la asignatura dentro de los programas de ingenieria, mediante encuestas que se realizaron a los profesores que orientan diferentes asignaturas de matematicas como a los directores de los diferentes programas. Ademas se presentan conclusiones y recomendaciones acerca del estudio.

Modul ini membahas konsep dasar Sistem Persamaan Linear (SPL) beserta metode penyelesaiannya, baik secara manual maupun menggunakan perangkat lunak Maple 2021. Pembahasan mencakup berbagai teknik penyelesaian SPL, seperti metode substitusi, eliminasi, serta metode berbasis matriks seperti Eliminasi Gauss, Gauss-Jordan, dan Aturan Cramer. Selain itu, modul ini memperkenalkan metode numerik seperti Jacobi dan Gauss-Seidel untuk penyelesaian sistem dengan banyak variabel. Dengan menggunakan Maple 2021, modul ini memberikan panduan dalam menyusun, menyelesaikan, dan memvisualisasikan sistem persamaan linear secara komputasional. Penyajian dilakukan melalui contoh soal yang disertai langkah penyelesaian manual serta implementasi dalam Maple, termasuk visualisasi grafik dalam bentuk 2D dan 3D. Melalui modul ini, diharapkan pembaca dapat memahami konsep SPL, menguasai teknik penyelesaiannya, serta mampu membandingkan hasil perhitungan manual dengan solusi yang diperoleh menggunakan perangkat lunak.

Local models are schemes defined in linear algebra terms that describe the étale local structure of integral models for Shimura varieties and other moduli spaces. We point out that the flatness conjecture of Rapoport-Zink on local models fails in the presence of ramification and that in that case one has to modify their definition. We study modifications of the local models for G = \textrm R E / F \textrm G L ( n ) G=\text \textrm {R}_{E/F}\text \textrm {GL}(n) , with E / F E/F a totally ramified extension, and for a maximal parahoric level subgroup. The special fibers of these models are subschemes of the affine Grassmannian. We give applications to the structure of Schubert varieties in the affine Grassmannian and to the calculation of sheaves of nearby cycles and describe a relation with geometric convolution. In the general EL case, we replace the conjecture of Rapoport-Zink with a conjecture about the modified local models.

In this paper, we present a simple combinatorial proof of a Weyl type formula for hook Schur polynomials, which has been obtained by using a Kostant type cohomology formula for gl m|n . In general, we can obtain in a combinatorial way a Weyl type formula for various highest weight representations of a Lie superalgebra, which together with a general linear algebra forms a Howe dual pair.