conformal mapping based coordinate generation method for flows in periodic configurations abstract a conformal mapping based coordinate generation method for periodic configurations has been developed a channel of an arbitrary shape in the physical plane representing internal configurations is mapped into a straight channel in the computational plane and a wall of an arbitrary shape in the physical plane representing external configurations is mapped into a straight wall in the computational plane the parameters of the transformations have to be determined through a method of successive approximations a simple iteration scheme that converges quite rapidly even with a poor initial guess is presented solutions for different configurations are displayed to illustrate the capabilities of the method

maximal abelian subalgebras of e p q algebras abstract maximal abclian subalgebras masas of one of the classical real inhomogencous lie algebras are constructed namely those of the pseudoeuclidean lie algebra e p q use is made of the semidireet sum structure of e p q with the translations t p q as an abclian ideal we first construct splitting masas that arc themselves direct sums of abelian subalgebras of o p q and of subalgebras of t p q the splitting subalgebras are used to construct the complementary nonsplitting ones here the results are less complete than in the splitting case we present general decomposition theorems and construct indecomposable masas for all algebras e p q p q 0 the case of q 0 and 1 were treated earlier in a physical context the case q 2 is analyzed here in detail as an illustration of the general results

commuting pairs and triples of matrices and related varieties abstract in this note we show that the set of all commuting d tuples of commuting n n matrices that are contained in an n dimensional commutative algebra is a closed set and therefore gerstenhaber s theorem on commuting pairs of matrices is a consequence of the irreduciblity of the variety of commuting pairs we show that the variety of commuting triples of 4 4 matrices is irreducible we also study the variety of n dimensional commutative subalgebras of m n f and show that it is irreducible of dimension n 2 n for n 4 but reducible of dimension greater than n 2 n for n 7

modules and matrices abstract noncommutative ring theory was described in terms of matrices in its earliest days we give some examples showing how matrix theory can now in turn profit from the theory of rings and modules

implicit spectrally accurate method for moving boundary problems using immersed boundary conditions concept a fully implicit spectral algorithm for the analysis of moving boundary problem is described the algorithm is based on the concept of immersed boundary conditions ibc i e the computational domain is fixed while the time dependent physical domain is submerged inside the computational domain and is described in the context of the diffusion type problems the physical conditions along the edges of the physical domain are treated as internal constraints the method eliminates the need for adaptive grid generation that follows evolution of the physical domain and provides sharp resolution of the location of the boundary various tests confirm the spectral accuracy in space and the first and second order accuracy in time the computational cost advantage of the ibc method as compared with the more traditional algorithm based on the mapping concept is demonstrated

three dimensional front tracking we describe a three dimensional front tracking algorithm discuss its numerical implementation and present studies to validate the correctness of this approach based on the results of the two dimensional computations we expect three dimensional front tracking to significantly improve computational efficiencies for problems dominated by discontinuities in some cases for which the interface computations display considerable numerical sensitivity we expect a greatly enhanced capability

a resolution of paz s conjecture in the presence of a nonderogatory matrix abstract let m n f be the algebra of n n matrices over the field f and let s be a generating set of m n f as an f algebra the length of a finite generating set s of m n f is the smallest number k such that words of length not greater than k generate m n f as a vector space it is a long standing conjecture of paz that the length of any generating set of m n f cannot exceed 2 n 2 we prove this conjecture under the assumption that the generating set s contains a nonderogatory matrix in addition we find linear bounds for the length of generating sets that include a matrix with some conditions on its jordan canonical form finally we investigate cases when the length 2 n 2 is achieved

contractions of lie algebras and separation of variables the n dimensional sphere inonu wigner contractions from the rotation group o n 1 to the euclidean group e n are used to relate the separation of variables in laplace beltrami operators on n dimensional spheres and euclidean spaces respectively in this article we consider all subgroup type coordinates corresponding to different chains of subgroups of o n 1 and e n respectively in particular the contractions relate the graphical formalism of trees on spheres to the clusters on euclidean spaces introduced in this article the contractions are considered analytically on several levels the vector fields realizing the lie algebras the complete sets of commuting operators characterizing separable coordinate systems the coordinate systems themselves and the separated eigenfunctions

fujita type exponent for fully nonlinear parabolic equations and existence results abstract in this paper we prove that a class of parabolic equations involving a second order fully nonlinear uniformly elliptic operator has a fujita type exponent these exponents are related with an eigenvalue problem in all r n and play the same role in blow up theorems as the classical p 1 2 n introduced by fujita for the laplacian we also obtain some associated existence results

singular sets for curvature equation of order k abstract we consider the removability of singular sets for the curvature equations of the form h k u which is determined by the kth elementary symmetric function in an n dimensional domain we prove that for 1 k n 1 and a compact set k whose n k dimensional hausdorff measure is zero any generalized solution to the curvature equation on k is always extendable to a generalized solution on the whole domain

coherent states for the hydrogen atom discrete and continuous spectra we construct the systems of generalized coherent states for the discrete and continuous spectra of the hydrogen atom these systems are expressed in elementary functions and are invariant under the so 3 2 discrete spectrum and so 4 1 continuous spectrum subgroups of the dynamical symmetry group so 4 2 of the hydrogen atom both systems of coherent states are particular cases of the kernel of integral operator which intertwines irreducible representations of the so 4 2 group

approximating commuting operators the problem of approximating m tuples of commuting n n complex matrices by commuting m tuples of generic matrices is studied we narrow the gap for commuting triples by showing that they can be perturbed if n 29

on approximately simultaneously diagonalizable matrices abstract a collection a1 a2 ak of n n matrices over the complex numbers c has the asd property if the matrices can be perturbed by an arbitrarily small amount so that they become simultaneously diagonalizable such a collection must perforce be commuting we show by a direct matrix proof that the asd property holds for three commuting matrices when one of them is 2 regular dimension of eigenspaces is at most 2 corollaries include results of gerstenhaber and neubauer sethuraman on bounds for the dimension of the algebra generated by a1 a2 ak even when the asd property fails our techniques can produce a good bound on the dimension of this subalgebra for example we establish dim c a 1 a k 5 n 4 for commuting matrices a1 ak when one of them is 2 regular this bound is sharp one offshoot of our work is the introduction of a new canonical form the h form for matrices over an algebraically closed field the h form of a matrix is a sparse jordan like upper triangular matrix which allows us to assume that any commuting matrices are also upper triangular the jordan form itself does not accommodate this

maximal abelian subgroups of the isometry and conformal groups of euclidean and minkowski spaces the maximal abelian subalgebras masas of the euclidean and pseudo euclidean lie algebras are classified into conjugacy classes under the action of the corresponding lie groups and and also under the conformal groups and respectively the results are presented in terms of decomposition theorems for orthogonally indecomposable masas exist only for p 1 and p 2 for on the other hand orthogonally indecomposable masas exist for all values of p the results are used to construct new coordinate systems in which wave equations and hamilton jacobi equations allow the separation of variables

on the inequality f x d 2u ge f u g u du q we consider fully nonlinear degenerate elliptic equations with zero and first order terms we provide a priori upper bounds and characterize the existence of entire subsolutions under growth conditions on the lower order coefficients which extend the classical keller osserman condition for semilinear equations

similarity of matrices over local rings abstract wasow investigated the problem of when for a pair of matrices of analytic functions pointwise similarity implies analytic similarity by using the topology of spec r this is studied for an arbitrary commutative ring results of wasow ohm schneider and friedland are generalized in this setting a canonical form is determined for matrices satisfying the wasow condition

extinction of superdiffusions and semilinear partial differential equations abstract a superdiffusion is a measure valued branching process associated with a nonlinear operator lu u where l is a second order elliptic differential operator and is a function from r d r to r in the case l 1 0 the so called subcritical case the expectation of the total mass does not increase and the mass vanishes at a finite time with probability 1 most results on connections between the superdiffusion and differential equations involving lu u were obtained for the subcritical case in the present paper we assume only that l 1 is a bounded function in this more general setting the probability of extinction can be smaller than 1 and we show that his happens if and only if there exists a strictly positive solution of the equation lu u 0 in r d we also establish a relation between the probability of extinction in a domain d and strictly positive solutions of equation lu u 0 in d that are equal to 0 on the boundary of d we call such solutions special for the equation c u u where is the beltrami laplace operator on a complete riemannian manifold e strictly positive solutions in e were studied in connection with a geometrical problem which two functions represent scalar curvatures of two riemannian metrics related by a conformal mapping see 1 and references there

over determined formulation of the immersed boundary conditions method abstract the over determined formulation of the immersed boundary conditions ibc method is proposed the method relies on the fourier expansions in the periodic direction and chebyshev expansions in the transverse direction the boundaries of the physical domain are immersed inside a regular computational domain and the boundary conditions enter the algorithm in the form of constraints construction of these constraints provides degrees of freedom in excess of that required to formulate a closed system of algebraic equations use of the additional degrees of freedom that leads to an over determined system is explored in order to improve the accuracy of the ibc method and to expand its applicability to more extreme geometries the over constraint formulation has been tested on three model problems that lead to the laplace biharmonic and navier stokes equations and thus cover the most commonly encountered types of operators in all cases tested the over determined formulation was found to improve the performance of the ibc method

lengths of quasi commutative pairs of matrices abstract in this paper we discuss some partial solutions of the length conjecture which describes the length of a generating system for matrix algebras we consider mainly the sets of two matrices which are quasi commuting it is shown that in this case the length function is linearly bounded we also analyze which particular natural numbers can be realized as the lengths of certain special generating sets and prove that for commuting or product nilpotent pairs all possible numbers are realizable however there are non realizable values between lower and upper bounds for the other quasi commuting pairs in conclusion we also present several related open problems

some aspects of olga taussky s work in algebra abstract olga taussky s pivotal role as a torch bearer for matrix theory is well understood and well appreciated as is her role in number theory and in particular there her key contribution to the development of matrix theoretic techniques in algebraic number theory her contributions in these subjects have been highlighted in numerous tributes to her in my own case i can strongly affirm the positive impact that her work and her personal advice and interest has had in inspiring and encouraging my own research however in this note i wish to draw attention to her contributions to other parts of algebra and the influence that the results themselves and the questions she posed have had in these areas a number of those results arose in connection with problems in linear algebra or number theory but are of interest in more general terms i would also like to put the results in the context of current research of necessity i have had to make a short selection from her numerous papers and my choice inevitably is a subjective one

a direct spectral method for determination of flows over corrugated boundaries a completely spectral algorithm for analysis of flows over corrugated boundaries is proposed the algorithm treats the flow problem as an internal rather then a boundary value problem where the flow conditions are specified along a line in the interior of the computational domain the method eliminates the need for a coordinate generation and or premapping required for regularization of the computational domain in standard implementations of spectral discretizations various tests confirm the spectral accuracy of the algorithm

numerical simulation of unsteady nonisothermal capillary interfaces a family of algorithms for simulation of unsteady nonisothermal capillary interfaces has been developed the algorithms are based on a coordinate transformation method the time dependent unknown physical domain is mapped onto a rectangular computational domain with the explicit form of the mapping function not being known four types of temporal discretization are used leading to the first order accurate one step implicit method second order accurate crank nicolson and trapezoidal methods and second order accurate two step implicit method in all cases second order finite difference approximations were used for spatial discretizations various tests demonstrated that the algorithms deliver theoretically predicted accuracy even for very large interfacial distortions the crank nicolson and trapezoidal methods have been found to be conditionally stable and thus are not recommended

the minimal dimension of maximal commutative subalgebras of full matrix algebras abstract let f be a field and a a maximal commutative subalgebra of the full matrix algebra mn f it is shown that dim a 2n 2 3 1 it is also shown that if the radical of a has cube zero then dim a 3n 2 3 4 and that this result is best possible for infinitely many natural numbers n

on varieties of commuting nilpotent matrices abstract let n d n be the variety of all d tuples of commuting nilpotent n n matrices it is well known that n d n is irreducible if d 2 if n 3 or if d 3 and n 4 on the other hand n 3 n is known to be reducible for n 13 we study in this paper the reducibility of n d n for various values of d and n in particular we prove that n d n is reducible for all d n 4 in the case d 3 we show that it is irreducible for n 6

separation of variables and subgroup bases on n dimensional hyperboloids a graphical formalism is introduced for describing subgroup type coordinates on n dimensional lorentzian hyperboloids imbedded into n 1 dimensional minkowski spaces the o n 1 group element is parametrized according to different subgroup chains involving lorentz rotation and euclidean subgroups the coordinates are then induced by the corresponding group action eigenfunctions of the laplace beltrami operator are obtained as products of jacobi functions associated legendre functions and modified bessel functions

roth s theorems for sets of matrices abstract it is shown that roth s theorems on the equivalence and similarity of block diagonal matrices hold for finite sets of matrices over a commutative ring

a variety of commuting triples abstract the problem of approximating triples of commuting n n matrices by generic matrices is equivalent to the problem of whether the variety c 3 n of commuting triples is irreducible the answer to the problem is known to be positive for dimensions no greater than 5 and negative for dimensions no smaller than 30 using simultaneous commutative perturbations of pairs of matrices in the centralizer of the third matrix we are able to show that the answer is positive in dimension 6 as well all the proofs are given over an arbitrary algebraically closed field of characteristic zero

superposition principles for matrix riccati equations a superposition rule is obtained for the matrix riccati equation mre w a wb cw wdw where w t a t b t c t and d t are real n n matrices expressing the general solution in terms of five known solutions for all n 2 the symplectic mre w wt a at d dt c bt is treated separately and a superposition rule is derived involving only four known solutions for the unitary and gl n r subcases with d a and c bt or d a and c bt respectively superposition rules are obtained involving only two solutions the derivation of these results is based upon an interpretation of the mre in terms of the action of the groups sl 2n r sp 2n r u n and gl n r on the grassman manifold gn r2n

representations under ring extensions latimer macduffee and taussky correspondences c g latimer and c c macduffee defined in 8 a bijective correspondence between similarity classes of nonsingular rational integral n x n matrices having a common separable characteristic polynomialf x and the classes of faithful ideals in the ring z x f x z x z the ring of rational integers several authors have used this simple duality with the ideas indigenous to one theory to obtain valuable consequences in the other see 14 and 15 f or a survey thereby motivating further analysis of the correspondence as in 1 2 4 17 and 181 nevertheless the theory has generally been catagorized as a subset of module theory of orders in semisimple algebras following zassenhaus s proof of jordan s theorem on the finiteness of such similarity classes see 7 14 and 21 i the purpose of this note is to replace the semisimple assumption present in these previous studies with the weaker condition that the modules considered become trivial under a suitable extension of the base ring the ease in which matrix and ideal theory are transferable under the latimer macduffee duality prompted our adherence to their approach when possible in this development over arbitrary commutative rings we review prior matrix ideal correspondences in section 2 preliminary to the general framework in section 3 there an n x ii matrix class is replaced by the isomorphism class of a rank n projective r module m which is also a faithful module over a rank n projective r algebra a the ideal theory is obtained by extending the base ring r of scalars to a commutative ring s for which m becomes trivial s ll mn s or a as s or a modules and m is then identified with a fractional a ideal in s or a in the classical case s is the field of quotients of a domain r theorem 1 reduces to the latimer macduffee duality in the latter case theorem 2 shows that the theory is equally adaptable to any commutative rank iz projective r algebra a which is isomorphic to its dual over a suitable extension of r as an application we show that a nonderogatory matrix is similar to the companion matrix of its

an immersed boundary finite volume method for simulations of flow in complex geometries abstract a new immersed boundary method for simulating flows over or inside complex geometries is developed by introducing a mass source sink as well as a momentum forcing the present method is based on a finite volume approach on a staggered mesh together with a fractional step method both momentum forcing and mass source are applied on the body surface or inside the body to satisfy the no slip boundary condition on the immersed boundary and also to satisfy the continuity for the cell containing the immersed boundary in the immersed boundary method the choice of an accurate interpolation scheme satisfying the no slip condition on the immersed boundary is important because the grid lines generally do not coincide with the immersed boundary therefore a stable second order interpolation scheme for evaluating the momentum forcing on the body surface or inside the body is proposed three different flow problems decaying vortices and flows over a cylinder and a sphere are simulated using the immersed boundary method proposed in this study and the results agree very well with previous numerical and experimental results verifying the accuracy of the present method

holder gradient estimates for fully nonlinear elliptic equations in this paper we prove interior and global holder estimates for lipschitz viscosity solutions of second order nonlinear uniformly elliptic equations the smoothness hypotheses on the operators are more general than previously considered for classical solutions so that our estimates are also new in this case and readily extend to embrace obstacle problems in particular isaac s equations of stochastic differential game theory constitute a special case of our results and moreover our techniques in combination with recent existence theorems of ishii lead to existence theorems for continuously differentiable viscosity solutions of the uniformly elliptic isaac s equation

user s guide to viscosity solutions of second order partial differential equations the notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems existence theorems and theorems about continuous dependence may now be proved by very efficient and striking arguments the range of important applications of these results is enormous this article is a self contained exposition of the basic theory of viscosity solutions

complex orthogonal and symplectic matrices depending on parameters versal deformations of elements of complex orthogonal and symplectic lie algebras are studied for a general element m of o n c or sp 2n c a normal form ma is found which unlike the jordan normal form mj depends holomorphically on m and on the similarity transformation ma gmg 1 from the corresponding group orthogonal and symplectic cases are treated simultaneously in order to underline their close relation bundles of matrices of low codimension are listed and bifurcation diagrams of two parameter families of orthogonal matrices are shown finally versal deformations of all elements of o 6 c are explicitly shown

maximal abelian subalgebras of pseudounitary lie algebras abstract the task of classifying and constructing all maximal abelian subalgebras of su p q p q 1 is reduced to that of classifying orthogonally indecomposable masas these are either maximal abelian nilpotent subalgebras represented by nilpotent matrices in any finite dimensional representation or for p q they can be nonorthogonally decomposable and their study can be reduced to a construction of manss of sl p c two types of manss of su p q are shown to exist one rowed and non one rowed numerous classification theorems are proven and applied to obtain all masas of su p 1 su p 2 and su p q with p q 6 physical applications are discussed

normal forms of elements of classical real and complex lie and jordan algebras elements of the classical complex and real lie and jordan algebras with involutions are classified into conjugacy classes under the action of the corresponding classical lie group normal forms of representatives of each conjugacy class are chosen so as to resemble the jordan normal forms of n n complex matrices for completeness similar results are given for gl n c gl n r and gl n h

symmetry reduction for the kadomtsev petviashvili equation using a loop algebra the kadomtsev petviashvili kp equation ut 3uux 2 1 4 uxxx x 3 uyy 4 0 allows an infinite dimensional lie group of symmetries i e a group transforming solutions amongst each other the lie algebra of this symmetry group depends on three arbitrary functions of time t and is shown to be related to a subalgebra of the loop algebra a 1 4 low dimensional subalgebras of the symmetry algebra are identified specifically all those of dimension n 3 and also a physically important six dimensional lie algebra containing translations dilations galilei transformations and quasirotations new solutions of the kp equation are obtained by symmetry reduction using the one dimensional subalgebras of the symmetry algebra these solutions contain up to three arbitrary functions of t

matrix equivalence and isomorphism of modules abstract it is well known that if a and b are n m matrices over a ring r then coker a coker b does not imply a and b are equivalent an elementary proof is given that the implication does hold if 1 is in the stable range of r furthermore for certain r including commutative rings if a is block diagonal and b is block upper triangular with the same diagonal blocks as a then coker a coker b implies a and b are equivalent under a special equivalence this extends results of roth and gustafson as a corollary a theorem on decomposition of modules is obtained

completely integrable relativistic hamiltonian systems and separation of variables in hermitian hyperbolic spaces the hamilton jacobi and laplace beltrami equations on the hermitian hyperbolic space hh 2 are shown to allow the separation of variables in precisely 12 classes of coordinate systems the isometry group of this two complex dimensional riemannian space su 2 1 has four mutually nonconjugate maximal abelian subgroups these subgroups are used to construct the separable coordinates explicitly all of these subgroups are two dimensional and this leads to the fact that in each separable coordinate system two of the four variables are ignorable ones the symmetry reduction of the free hh 2 hamiltonian by a maximal abelian subgroup of su 2 1 reduces this hamiltonian to one defined on an o 2 1 hyperboloid and involving a nontrivial singular potential separation of variables on hh 2 and more generally on hh n thus provides a new method of generating nontrivial completely integrable relativistic hamiltonian systems

upper bounds for the length of non associative algebras abstract we introduce the notion of length for non associative finite dimensional unitary algebras and obtain a sharp upper bound for the lengths of algebras belonging to this class we also put forward a new method of characteristic sequences based on linear algebra technique which provides an efficient tool for computing the length function in non associative case then we apply the introduced method to obtain an upper bound for the length of an arbitrary locally complex algebra in the last case the length is bounded in terms of the fibonacci sequence we present concrete examples demonstrating the sharpness of our bounds

graphs with relations coverings and group graded algebras the paper studies the interrelationship between coverings of finite directed graphs and gradings of the path algebras associated to the directed graphs to include gradings of all basic finite dimensional algebras over an algebraically closed field a theory of coverings of graphs with relations is introduced the object of this paper is to relate group gradings on algebras to coverings of a graph which is associated to the algebra the linking of the theories allows one to relate purely algebraic questions to questions in algebraic topology group theory or combinatorics in the representation theory of artin algebras the association to each algebra of a finite directed graph called the quiver of the algebra has been a useful tool the reason that the quiver of such an algebra is of interest is that there is a natural definition of representations of the quiver so that the category of representations of the quiver satisfying certain relations is equivalent to the category of finitely generated modules over the algebra 1 gives a slight extension of these concepts to finitely generated algebras the main emphasis of the paper is to show that the theory of coverings of graphs with relations introduced by c riedtmann 9 and expanded by p gabriel 2 and the theory of group graded algebras are essentially the same although the original connection between coverings and gradings was inspired by the similarity of results for z graded artin algebras 3 4 and p gabriel s announced results 2 the context of this paper is more general and deals with all finitely generated algebras over a field we associate to each such algebra a finite directed graph which we still call a quiver of the algebra we show that for each regular covering t of a quiver t0 of an algebra a with certain prescribed restrictions we get a g grading of the algebra a where g is the automorphism group of the covering y over ro conversely given a certain type of g grading of a where g is a group we construct a regular covering y of the quiver t0 of the algebra such that g is isomorphic to the automorphism group of t over ro furthermore if y is a regular covering of t0 with automorphism group g we show that the category of representations of y satisfying a certain set of relations is equivalent to the category of finite dimensional graded g modules we

on the structure of commutative matrices ii a finite set a of n n nilpotent commutative matrices that have one dimensional joint kernel is considered the theorem due to suprunenko and tyshkevich that the algebra a generated by a and the identity matrix has dimension equal to n is proved a canonical basis for a is given and related structure constants are discussed

gridless spectral algorithm for stokes flow with moving boundaries a gridless spectrally accurate algorithm for the stokes flow with moving boundaries is presented the algorithm uses fixed computational domain with boundaries of the flow domain moving inside the computational domain the spatial discretization is based on the fourier expansions in the streamwise direction and the chebyshev expansions in the transverse direction temporal discretization uses one and two steps implicit formulations the boundary conditions on the moving boundaries are imposed using the immersed boundary conditions concept numerical tests confirm the spectral accuracy in space and theoretically predicted accuracy in time different variants of the solution procedure are presented and their relative advantages are discussed

the maximum principle for viscosity solutions of fully nonlinear second order partial differential equations we prove that viscosity solutions in w1 of the second order fully nonlinear equation f d2u du u 0 are unique when i f is degenerate elliptic and decreasing in u or ii f is uniformly elliptic and nonincreasing in u we do not assume that f is convex the method of proof involves constructing nonlinear approximation operators which map viscosity subsolutions and supersolutions onto viscosity subsolutions and supersolutions respectively this method is completely different from that used in lions 8 9 for second order problems with f convex in d 2 u and from that used by crandall lions 3 and crandall evans lions 2 for fully nonlinear first order problems

maximal abelian subalgebras of pseudoorthogonal lie algebras abstract the classification of maximal abelian subalgebras masas of o p q is reduced to the construction of orthogonally indecomposable oid masas five types of oid masas exist and their classification amounts to an analysis of maximal abelian nilpotent subalgebras manss of o p q su p q sl n r or sl n c theorems are provided making possible the classification of all manss of o p q and low dimensional examples are analyzed in detail

spectrally accurate algorithm for the analysis of flows in two dimensional vibrating channels a spectral algorithm based on the immersed boundary conditions ibc concept has been developed for the analysis of flows in channels bounded by vibrating walls the vibrations take the form of travelling waves of arbitrary profile the algorithm uses a fixed computational domain with the flow domain immersed in its interior boundary conditions enter the algorithm in the form of constraints the spatial discretization uses a fourier expansion in the stream wise direction and a chebyshev expansion in the wall normal direction use of the galileo transformation converts the unsteady problem into a steady one an efficient solver which takes advantage of the structure of the coefficient matrix has been used it is demonstrated that the method can be extended to more extreme geometries using the overdetermined formulation various tests confirm the spectral accuracy of the algorithm

fundamental solutions of homogeneous fully nonlinear elliptic equations we prove the existence of two fundamental solutions and of the pde input amssym f d 2 phi 0 quad rm in bbb r n setminus 0 for any positively homogeneous uniformly elliptic operator f corresponding to f are two unique scaling exponents 1 that describe the homogeneity of and we give a sharp characterization of the isolated singularities and the behavior at infinity of a solution of the equation f d2u 0 which is bounded on one side a liouville type result demonstrates that the two fundamental solutions are the unique nontrivial solutions of f d2u 0 in input amssym bbb r n setminus 0 that are bounded on one side in both a neighborhood of the origin as well as at infinity finally we show that the sign of each scaling exponent is related to the recurrence or transience of a stochastic process for a two player differential game 2010 wiley periodicals inc

commuting pairs in the centralizers of 2 regular matrices inmn k kan algebraically closed field we call a matrixl regular if each eigenspace is at mostl dimensional we prove that the variety of commuting pairs in the centralizer of a 2 regular matrix is the direct product of various affine spaces and various determinantal varieties zl mobtained from matrices over truncated polynomial rings we prove that these varieties zl mare irreducible and apply this to the case of thek algebra generated by three commuting matrices we show that if one of the three matrices is 2 regular then the algebra has dimension at mostn we also show that such an algebra is always contained in a commutative subalgebra ofmn k of dimension exactlyn

maximal abelian subalgebras of real and complex symplectic lie algebras we provide guidelines for classifying maximal abelian subalgebras masa s of the symplectic lie algebras sp 2n r and sp 2n c into conjugacy classes under the lie groups sp 2n r and sp 2n c respectively the task of classifying all masa s is reduced to the classification of orthogonally indecomposable oid masa s two types of orthogonally indecomposable masa s of sp 2n c exist 1 indecomposable maximal abelian nilpotent subalgebras mans s 2 decomposable masa s their classification reduces to a classification of mans s of sl n c four types of orthogonally indecomposable masa s of sp 2n r exist 1 absolutely indecomposable masa s mans s 2 relatively indecomposable masa s their classification reduces to a classification of mans s of su p q for p q n 3 decomposable absolutely oid masa s involving mans s of sl n r 4 decomposable relatively oid masa s involving mans s of sl n 2 c for n even low dimensional cases of sp 2n f n 1 2 3 f r or c are treated exhau

a generalization of the motzkin taussky theorem abstract in this paper we generalize the motzkin taussky theorem to matrices with polynomial entries

removable singularities for fully nonlinear elliptic equations we obtain a removability result for the fully nonlinear uniformly elliptic equations f d 2 u f u 0 the main theorem states that every solution to the equation in a punctured ball without any restrictions on the behaviour near the centre of the ball is extendable to the solution in the entire ball provided the function f satisfies certain sharp conditions depending on f previously such results were known for linear and quasilinear operators f in comparison with the semi or quasilinear theory the techniques for the fully nonlinear equations are new and based on the use of the viscosity notion of generalised solution rather than the distributional or the weak solutions

the effect of assuming the identity as a generator on the length of the matrix algebra abstract let m n f be the algebra of n n matrices and let s be a generating set of m n f as an f algebra the length of a finite generating set s of m n f is the smallest number k such that words of length not greater than k generate m n f as a vector space traditionally the identity matrix is assumed to be automatically included in all generating sets s and counted as a word of length 0 in this paper we discuss how the problem changes if this assumption is removed

on the dimension of the algebra generated by two positive semi commuting matrices abstract gerstenhaber s theorem states that the dimension of the unital algebra generated by two commuting n n matrices is at most n we study the analog of this question for positive matrices with a positive commutator we show that the dimension of the unital algebra generated by the matrices is at most n n 1 2 and that this bound can be attained we also consider the corresponding question if one of the matrices is a permutation or a companion matrix or both of them are idempotents in these cases the upper bound for the dimension can be reduced significantly in particular the unital algebra generated by two semi commuting positive idempotent matrices is at most 9 dimensional this upper bound can be attained

length realizability for pairs of quasi commuting matrices abstract for the pairs of quasi commuting matrices we completely characterize natural numbers that can be realized as the lengths of these pairs of generators

solvable lie algebras with abelian nilradicals a procedure is presented for classifying solvable lie algebras with abelian nilradicals theorems on the structure of such algebras are proven and their centres are constructed many examples are analysed

an upper bound for the length of a finite dimensional algebra abstract letfbe a field and letabe a finite dimensionalf algebra writed dimf a and letebe the largest degree of the minimal polynomial for anya a define the function f d e e 2d e 1 1 4 e 2 2 we prove that ifsis any finite generating set foraas anf algebra the words insof length less thanf d e spanaas anf vector space in the special case ofn by nmatrices this bound becomes f n 2 n n 2n 2 n 1 1 4 n 2 2 o n 3 2 this is a substantial improvement over previous bounds which have all beeno n2 we also prove that for particular setssof matrices the bound can be sharpened to one that is linear inn as an application of these results we reprove a theorem of small stafford and warfield about semiprime affinef algebras

a probabilistic approach to one class of nonlinear differential equations we establish connections between positive solutions of one class of nonlinear partial differential equations and hitting probabilities and additive functionals of superdiffusion processes as an application we improve results on superprocesses by using the recent progress in the theory of removable singularities for differential equations

maximal abelian subalgebras of complex orthogonal lie algebras abstract the study of maximal abelian subalgebras masas of o n c is reduced to the study of orthogonally indecomposable masas their classification is in turn reduced to that of maximal abelian nilpotent subalgebras manss of sl n 2 c for n even and of o n c itself general results and guidelines are given making it possible to construct all manss and hence all masas for any given n representatives of all o n c conjugacy classes of masas of o n c are constructed for 2 n 7 as an application of the general techniques

roth s theorems and decomposition of modules it is shown that there is a connection between roth s theorems on similarity and equivalence of block triangular matrices and decomposition of modules the module property is that if m n mn then n is a summand of m this holds for any commutative ring if m is finitely presented new proofs of roth s theorems are given for commutative rings some results are established in the noncommutative case

isolated singularities for some types of curvature equations abstract we consider the removability of isolated singularities for the curvature equations of the form h k u 0 which is determined by the k th elementary symmetric function in an n dimensional domain we prove that for 1 k n 1 isolated singularities of any viscosity solutions to the curvature equations are always removable provided the solution can be extended continuously at the singularities we also consider the class of generalized solutions and prove the removability of isolated singularities

trace on the boundary for solutions of nonlinear differential equations let l be a second order elliptic differential operator in rd with no zero order terms and let e be a bounded domain in rd with smooth boundary e we say that a function h is l harmonic if lh 0 in e every positive l harmonic function has a unique representation h x j x y i dy e where k is the poisson kernel for l and z is a finite measure on e we call z the trace of h on me our objective is to investigate positive solutions of a nonlinear equation lu u in e for 1 2 we associate with every solution u a pair if i where f is a closed subset of e and zis a radon measure on 0 e f we call f zi the trace of u on me f is empty if and only if u is dominated by an l harmonic function we call such solutions moderate a moderate solution is determined uniquely by its trace in general many solutions can have the same trace we establish necessary and sufficient conditions for a pair f z to be a trace and we give a probabilistic formula for the maximal solution with a given trace

commutative matrix subalgebras and length function it is proved that if the length of a commutative matrix subalgebra is maximal then this subalgebra is maximal under inclusion the examples are given showing that the converse does not hold to establish this result we prove several fundamental properties of the length function

new classes of examples satisfying the three matrix analog of gerstenhaber s theorem abstract in 1961 gerstenhaber proved the following theorem if k is a field and x and y are commuting d d matrices with entries in k then the unital k algebra generated by these matrices has dimension at most d the analog of this statement for four or more commuting matrices is false the three matrix version remains open we use commutative algebraic techniques to prove that the three matrix analog of gerstenhaber s theorem is true for some new classes of examples in particular we translate this three commuting matrix statement into an equivalent statement about certain maps between modules and prove that this commutative algebraic reformulation is true in special cases we end with ideas for an inductive approach intended to handle the three matrix analog of gerstenhaber s theorem more generally

schwarz christoffel methods for conformal mapping of regions with a periodic boundary abstract numerical conformal mapping methods for regions with a periodic boundary have been developed these methods are based on the generalized schwarz christoffel equation and can deal with boundary curves of arbitrary forms i e made up of one or more rectifiable jordan curves high order quadrature rules have been implemented in order to increase accuracy of the mapping this is of particular relevance to highly accurate grid generation techniques required by for example implementation of high order compact finite difference discretization schemes

schwarz christoffel mappings a general approach abstract a general method is developed for constructing equations of the schwarz christoffel type these equations define conformal transformations between regions of the complex plane each of whose boundaries may consist of straight or continuously curved line segments and corners the more familiar types of schwarz christoffel equations are rederived and some new types are obtained

spectrally accurate algorithm for moving boundary problems for the navier stokes equations a spectral algorithm based on the immersed boundary conditions ibc concept is developed for simulations of viscous flows with moving boundaries the algorithm uses a fixed computational domain with flow domain immersed inside the computational domain boundary conditions along the edges of the time dependent flow domain enter the algorithm in the form of internal constraints spectral spatial discretization uses fourier expansions in the stream wise direction and chebyshev expansions in the normal to the wall direction up to fourth order implicit temporal discretization methods have been implemented it has been demonstrated that the algorithm delivers the theoretically predicted accuracy in both time and space performances of various linear solvers employed in the solution process have been evaluated and a new class of solver that takes advantage of the structure of the coefficient matrix has been proposed the new solver results in a significant acceleration of computations as well as in a substantial reduction in memory requirements

subgroups of lie groups and separation of variables separable systems of coordinates for the helmholtz equation d e in pseudo riemannian spaces of dimension d have previously been characterized algebraically in terms of sets of commuting second order symmetry operators for the operator d they have also been characterized geometrically by the form that the metric ds2 gik x dxidxk can take we complement these characterizations by a group theoretical one in which the second order operators are related to continuous and discrete subgroups of g the symmetry group of d for d 3 we study all separable coordinates that can be characterized in terms of the lie algebra l of g and show that they are of eight types seven of which are related to the subgroup structure of g our method clearly generalizes to the case d 3 although each separable system corresponds to a pair of commuting symmetry operators there do exist pairs of commuting symmetries s1 s2 that are not associated with separable coordinates for subgroup related operators we show in detail just

the immersed boundary method a projection approach a new formulation of the immersed boundary method with a structure algebraically identical to the traditional fractional step method is presented for incompressible flow over bodies with prescribed surface motion like previous methods a boundary force is applied at the immersed surface to satisfy the no slip constraint this extra constraint can be added to the incompressible navier stokes equations by introducing regularization and interpolation operators the current method gives prominence to the role of the boundary force acting as a lagrange multiplier to satisfy the no slip condition this role is analogous to the effect of pressure on the momentum equation to satisfy the divergence free constraint the current immersed boundary method removes slip and non divergence free components of the velocity field through a projection the boundary force is determined implicitly without any constitutive relations allowing the present formulation to use larger cfl numbers compared to some past methods symmetry and positive definiteness of the system are preserved such that the conjugate gradient method can be used to solve for the flow field examples show that the current formulation achieves second order temporal accuracy and better than first order spatial accuracy in l 2 norms for one and two dimensional test problems results from two dimensional simulations of flows over stationary and moving cylinders are in good agreement with those from previous experimental and numerical studies

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

two generated commutative matrix subalgebras abstract let f be a field and let mn f be the algebra of n n matrices over f let a b mn f with ab ba and let a be the algebra generated by a b over f a theorem of gerstenhaber ann math 73 324 348 1991 states that the dimension of a is at most n gerstenhaber s proof uses the methods of algebraic geometry in this paper we obtain a purely matrix theoretic proof of the result we also examine the case when equality occurs the case where f is algebraically closed and a is indecomposable under similarity holds the key to the general situation and in the indecomposable case we obtain a cayley hamilton like theorem expressing bk as a polynomial in i b bk 1 with coefficients in f a where k denotes the number of blocks in the jordan form of a if all jordan blocks of a have the same size we obtain a nonderogatory like condition on b which is equivalent to dim f a n we also show that in this case dim f a n is equivalent to the maximality of a as a commutative subalgebra of mn f

some thoughts on gerstenhaber s theorem abstract we weigh the current evidence for and against an extension of gerstenhaber s 1961 theorem to three commuting matrices over a field

solvable lie algebras with heisenberg ideals all finite dimensional indecomposable solvable lie algebras l n f having the heisenberg algebra h n as the nilradical are constructed the number of non nilpotent elements f that can be added to h n satisfies f or n 1 the casimir and generalized casimir operators of the algebras l n f are obtained