condensed cramer rule for some restricted quaternion linear equations abstract in this paper we derive an alternative more condensed cramer rule for the unique solution of some restricted left and right systems of quaternion linear equations the findings of this paper extend some known results in the literature

on g inverses and the nonsingularity of a bordered matrixabc0 abstract we consider the bordered matrix m a b c 0 with the rank additivity condition and g c 1 c 2 c 3 c 4 which is a g inverse of m we give the necessary and sufficient conditions for c 1 to be a g inverse outer inverse reflexive g inverse and moore penrose inverse of a as a special case we give necessary and sufficient conditions for m to be nonsingular and explicit expressions of c i in m 1 moreover we give a method for computing any outer inverse of a matrix

subproper and regular splittings for restricted rectangular linear system iterative schemes for approximating a solution to restricted rectangular but consistent linear system of equations ax b email protected t are considered the methods are based upon so called subproper splitting a m n which is a generalization of the concept of subproper splitting introduced by neumann and studied further by berman and neumann and others we give a necessary and sufficient condition on the splitting such that the iterative sequence converges to a solution of ax b in the case of email protected at for every x 0 where email protected c m x n and t is a subspace of c n monotonicity and the concept of regular subproper splitting are used to determine a necessary and a sufficient condition for the convergence of the iterative scheme finally we present two numerical examples to verify our conclusions

determinantal representation of the generalized inverses over the quaternion skew field with applications we first present some determinantal representations of one 1 5 inverse of a quaternion matrix within the framework of a theory of the row and column determinants as applications we show some new explicit expressions of generalized inverses a r t 1 s 1 2 a l t 2 s 2 2 and a t 1 t 2 s 1 s 2 2 over the quaternion skew field finally we give the representations of the unique solution to some restricted left and right systems of quaternionic linear equations the findings of this paper extend some known results in the literature

projection methods solving rectangular systems of linear equations abstract this survey paper contains a general characterization of stationary and instationary linear iterative one step methods solving over or underdetermined linear algebraic systems techniques for convergence acceleration and some generalizations of the kaczmarz art and the de la garza methods among others the question is answered under what conditions an iteration process yields a generalized matrix inverse of the type a a 1 2 a 1 2 3 a 1 2 4 a respectively and under what conditions the resulting limit vector is a minimum norm solution a least squares solution or a solution of the initial linear system

a cramer rule for minimum norm t least squares s solution of inconsistent linear equations abstract in this paper the minimum norm t least squares s solution of inconsistent linear equations ax y a c m n r y r a is given a determinantal form and reduces to common cramer s rule if a is nonsingular

multivalued elliptic equation with exponential critical growth in r2 abstract in this work we study the existence of nontrivial solutions for the following class of multivalued elliptic problems p u v x u h x t f x u in r 2 where 0 v is a continuous function verifying some conditions h h 1 r 2 and t f x u is the generalized gradient of f x t with respect to t and f x t 0 t f x s d s assuming that f is a discontinuous function with exponential critical growth we have applied variational methods for locally lipschitz functional to get two solutions for p when is small enough

a cramer rule for least norm solutions of consistent linear equations abstract the least euclidean norm solution of a consistent linear equation ax b is given a determinantal form which reduces to cramer s rule if a is nonsingular

convergence and quotient convergence of iterative methods for solving singular linear equations with index one singular systems with index one arise in many applications such as markov chain modelling in this paper we use the group inverse to characterize the convergence and quotient convergence properties of stationary iterative schemes for solving consistent singular linear systems when the index of the coefficient matrix equals one we give necessary and sufficient conditions for the convergence of stationary iterative methods for such problems next we show that for the stationary iterative method the convergence and the quotient convergence are equivalent

a cramer rule for the least norm least squared error solution of inconsistent linear equations abstract a recent cramer rule for obtaining the least norm solution of a consistent system of linear equations ax b is shown to also provide the least squared error solution in the inconsistent case i e the solution a b where a is the moore penrose inverse of a

geometric quantization and constrained systems the problem of obtaining the quantum theory of systems with first class constraints is discussed in the context of geometric quantization the precise structure needed on the constraint surface of the full phase space to obtain a polarization on the reduced phase space is displayed in a form that is particularly convenient for applications for unconstrained systems any polarization on the phase space leads to a mathematically consistent quantum description although not all of these descriptions may be viable from a physical standpoint it is pointed out that the situation is worse in the presence of constraints a general polarization on the full phase space need not lead to even a mathematically consistent quantum theory examples are given to illustrate the general constructions as well as the subtle difficulties

splitting based on the outer inverse of matrices in this paper we introduce a conception on the splitting of a matrix a which is based on the outer inverse of the matrix a it will be evidence that splittings early are the special cases several necessary and sufficient conditions to endure the convergence of the fixed pointed system generated by the splitting are also induced

group averaging in the p q oscillator representation of sl 2 r we investigate refined algebraic quantization with group averaging in a finite dimensional constrained hamiltonian system that provides a simplified model of general relativity the classical theory has gauge group sl 2 r and a distinguished o p q observable algebra the gauge group of the quantum theory is the double cover of sl 2 r and its representation on the auxiliary hilbert space is isomorphic to the p q oscillator representation when p 2 q 2 and p q 0 mod 2 we obtain a physical hilbert space with a nontrivial representation of the o p q quantum observable algebra for p q 1 the system provides the first example known to us where group averaging converges to an indefinite sesquilinear form

hilbert space structure in classical mechanics i in this paper we study the hilbert space structure underlying the koopman von neumann kvn operatorial formulation of classical mechanics kvn limited themselves to study the hilbert space of zero forms that are the square integrable functions on phase space they proved that in this hilbert space the evolution is unitary for every system in this paper we extend the kvn hilbert space to higher forms which are basically functions of the phase space points and the differentials on phase space we prove that if we equip this space with a positive definite scalar product the evolution can turn out to be nonunitary for some systems vice versa if we insist in having a unitary evolution for every system then the scalar product cannot be positive definite identifying the one forms with the jacobi fields we provide a physical explanation of these phenomena we also prove that the unitary nonunitary character of the evolution is invariant under canonical transformations

index splitting for the drazin inverse and the singular linear system abstract an index splitting of a singular matrix a for the drazin inverse and its relative iterations for the minimal p norm solution of the singular linear system ax b b r a k k ind a are presented some convergence criteria are also given and a method is developed for constructing the index splitting of matrix a

free boundary problems and the set valued mappings we have pointed out that many free boundary problems may be reduced to find the fixed points of certain set valued mappings and have solved the free boundary problem arising in the plasma physics 7 and the free boundary problem of the water cone in an oil reservoir with bottom water 111 as examples it is the purpose of this paper to develop the idea more extensively in this paper some typical free boundary problems such as the obstacle problem the seepage surface problem the stefan problem the elenbaas equation etc are all picked up into our scheme although most of them have been attacked by other methods there are still some new things in our approach especially the multiplicity of the solutions of the elenbaas equation is interesting we prove that there are l 1 and two distinct positive solutions of the equation

successive matrix squaring algorithm for computing the drazin inverse1this project is supported by the national nature science foundation of china doctoral point foundation of china and youth science foundation of universities in shanghai of china 1 this paper derives a successive matrix squaring sms algorithm to approximate the drazin inverse which can be expressed in the form of successive squaring of a composite matrix t given an n by n matrix a the study shows that the drazin inverse of a can be computed in parallel time ranging from o logn to o log 2n provided that there are enough processors to support matrix multiplication in time o logn the sms algorithm is generalized to higher order schemes where the composite matrix is repeatedly raised to an integer power l 2 this form of expression leads to a simplified notation compared to that of earlier methods we argue that there is no obvious advantage in choosing l other than 2 our derived error bound for the approximation of a d is new

modular invariance on the torus and abelian chern simons theory the implementation of modular invariance on the torus as a phase space at the quantum level is discussed in a group theoretical framework unlike the classical case at the quantum level some restrictions on the parameters of the theory should be imposed to ensure modular invariance two cases must be considered depending on the cohomology class of the symplectic form on the torus if it is of integer cohomology class n then full modular invariance is achieved at the quantum level only for those wave functions on the torus which are periodic if n is even or antiperiodic if n is odd if the symplectic form is of rational cohomology class n r a similar result holds the wave functions must be either periodic or antiperiodic on a torus r times larger in both directions depending on the parity of nr application of these results to the abelian chern simons theory is discussed

an improvement on the perturbation of the group inverse and oblique projection the perturbations of the group inverse a and oblique projection aa of a square matrix a have been previously studied under certain assumptions on the matrix a and a perturbation matrix e upper bounds for b bb b a a and bb aa aa where b a e are given in the literature recently upper bounds for the general case have been published by y wei appl math comp 98 1999 29 however the special cases in the literature and the continuity of the group inverse do not follow from the general upper bounds in this paper we derive new general upper bounds which not only cover all the special cases but also are sharper than wei s results such that the continuity of the group inverse directly follows a numerical example is given to illustrate the sharpness of the new general upper bounds

refined algebraic quantization in the oscillator representation of sl 2 r we investigate refined algebraic quantization raq with group averaging in a constrained hamiltonian system with unreduced phase space t r4 and gauge group sl 2 r the reduced phase space m is connected and contains four mutually disconnected regular sectors with topology r s1 but these sectors are connected to each other through an exceptional set where m is not a manifold and where m has non hausdorff topology the raq physical hilbert space hphys decomposes as hphys hi where the four subspaces hi naturally correspond to the four regular sectors of m the raq observable algebra aobs represented on hphys contains natural subalgebras represented on each hi the group averaging takes place in the oscillator representation of sl 2 r on l2 r2 2 and ensuring convergence requires a subtle choice for the test state space the classical analog of this choice is to excise from m the exceptional set while nevertheless retaining information about the connections between the regular sectors a quantum

schwinger terms from geometric quantization of field theories geometric quantization is applied to infinite countable dimensional linear kahler manifolds to obtain a closed expression for the anomalous commutator of arbitrary polynomial observables examples for the physical relevance of the result are given including the polarization dependence of schwinger terms in bilinear constraint algebras the central terms of virasoro and kac moody algebras and the determination of the critical dimension of the bosonic string

a cramer rule for solution of the general restricted matrix equation a cramer rule for finding the unique w weighted drazin inverse solution of a class of restricted matrix equationswawxwbw d r x r aw k 1 n x n wb k 2 is presented and the results in linear algebra appl 116 1989 27 appl math comput 125 2002 303 can be deduced from this paper where email protected c m x n email protected c n x m ind aw k 1 email protected c p x q email protected c q x p ind wb email protected 2 and email protected c n x p

factorization of complex canonical transformations we investigate the lie series representation of the canonical transformations in a finite dimensional complex phase space it is shown that any transformation of this type can be factorized into a product of three factors associated with a pure imaginary generating function a holomorphic function and an element of the cyclic group c4 the imaginary function can be considered as an observable in the sense of classical mechanics some hints are given which suggest that the holomorphic function can be connected with the notion of the state of a physical system moreover a special kind of mappings is studied which provides a link between entropy action and state functions the occurrence of these important physical quantities shows that the mathematical structure goes beyond a formal analogy to quantum physics at least in the finite dimensional case

iterative methods for solving restricted linear equations in this paper the iterative methods for computing some solution of the general restricted linear equations ax b x et are established and the iterative methods for finding the solution of several special linear equations considered are constructed

group quantization of parametrized systems ii pasting hilbert spaces the method of group quantization described in the preceding paper j math phys 36 4612 1995 is extended so that it becomes applicable to some parametrized systems that do not admit a global transversal surface a simple completely solvable toy model is studied that admits a pair of maximal transversal surfaces intersecting all orbits the corresponding two quantum mechanics are constructed the similarity of the canonical group actions in the classical phase spaces on the one hand and in the quantum hilbert spaces on the other hand suggests how the two hilbert spaces are to be pasted together the resulting quantum theory is checked to be equivalent to that constructed directly by means of dirac s operator constraint method the complete system of partial hamiltonians for any of the two transversal surfaces is chosen and the quantum schrodinger or heisenberg pictures of time evolution are constructed

a new look at the schouten nijenhuis frolicher nijenhuis and nijenhuis richardson brackets in this paper we re express the schouten nijenhuis the frolicher nijenhuis and the nijenhuis richardson brackets on a symplectic space using the extended poisson brackets structure present in the path integral formulation of classical mechanics

phase space representation of quantum state vectors phase space representation of quantum state vectors is obtained within the framework of the relative state formulation for this purpose the hilbert space of a quantum system is enlarged by introducing an auxiliary quantum system relative position state and relative momentum state are defined in the extended hilbert space of the composite quantum system and expressions of basic operators such as canonical position and momentum operators acting on these states are obtained phase space functions which represent a state vector of the relevant quantum system are obtained in terms of the relative position states and the relative momentum states the absolute square of the phase space function represents the probability distribution of the phase space variables time evolution of a quantum system is investigated in terms of the phase space functions the relations to the phase space representations formulated by the other methods are obtained

on the convergence of splittings for semidefinite linear systems abstract recently lee et al young ju lee jinbiao wu jinchao xu ludmil zikatanov on the convergence of iterative methods for semidefinite linear systems siam j matrix anal appl 28 2006 634 641 introduce new criteria for the semi convergence of general iterative methods for semidefinite linear systems based on matrix splitting the new conditions generalize the classical notion of p regularity introduced by keller h b keller on the solution of singular and semidefinite linear systems by iterations siam j numer anal 2 1965 281 290 in view of their results we consider here stipulations on a splitting a m n which lead to fixed point systems such that the iterative scheme converges to a weighted moore penrose solution to the system ax b our results extend the result of lee et al to a more general case and we also show that it requires less restrictions on the splittings than keller s p regularity condition to ensure the convergence of iterative scheme

cramer rule for the unique solution of restricted matrix equations over the quaternion skew field in this paper we establish the determinantal representations of the generalized inverses a r t 1 s 1 2 a l t 2 s 2 2 and a t 1 t 2 s 1 s 2 2 over the quaternion skew field by the theory of the column and row determinants in addition we derive some generalized cramer rules for the unique solution of some restricted quaternion matrix equations the findings of this paper extend some known results in the literature

on a class of semilinear schrodinger equations involving critical growth and discontinuous nonlinearities abstract in this paper we deal with a class of schrodinger equation in r n involving critical sobolev exponent and jump discontinuities the basic tool employed here is an approximation technique with periodic functions and variational arguments based on a linking theorem for locally lipschitz functionals

a new superconformal mechanics in this paper we propose a new supersymmetric extension of conformal mechanics the grassmannian variables that we introduce are the basis of the forms and of the vector fields built over the symplectic space of the original system our supersymmetric hamiltonian itself turns out to have a clear geometrical meaning being the lie derivative of the hamiltonian flow of conformal mechanics using superfields we derive a constraint which gives the exact solution of the supersymmetric system in a way analogous to the constraint in configuration space which solved the original nonsupersymmetric model besides the supersymmetric extension of the original hamiltonian we also provide the extension of the other conformal generators present in the original system these extensions also have a supersymmetric character being the square of some grassmannian charge we build the whole superalgebra of these charges and analyze their closure the representation of the even part of this superalgebra on the odd part turns out to be integer and not spinorial in character

quantization of diffeomorphism invariant theories of connections with local degrees of freedom quantization of diffeomorphism invariant theories of connections is studied and the quantum diffeomorphism constraint is solved the space of solutions is equipped with an inner product that is shown to satisfy the physical reality conditions this provides in particular a quantization of the husain kucha model the main results also pave the way to quantization of other diffeomorphism invariant theories such as general relativity in the riemannian case i e signature the approach appears to contain all the necessary ingredients already in the lorentzian case it will have to be combined in an appropriate fashion with a coherent state transform to incorporate complex connections

variational methods for non differentiable functionals and their applications to partial differential equations the interest in extending the theory of nonlinear partial differential equations pde to pde with discontinuous nonlinearities dnde has been increasing daily the reasons are twofold 1 many free boundary problems may be reduced to boundary value problems of dnde cf chang 16 71 chang and jiang 9 and fleishman and mahar 121 2 sometimes in dealing with a pde problem it is beneficial to put the original pde into a large category for instance dnde cf chang 7 simply speaking a dnde is as follows

computing generalized inverses of matrices by iterative methods based on splittings of matrices in this paper we study the iterative methods for computing the generalized inverses of the form a 2 t s we prove the result that if a has the splitting a m n and if the generalized inverses a 2 t s and m 2 t s exist then the iterationx j 1 m 2 t snx j m 2 t sconverges to a 2 t s for any x 0 if and only ift t s s email protected 2 t sn 1 or equivalently m 2 t email protected 2 t sn 1 the splitting a m n is called t s splitting of a if m satisfies email protected s c m where t and s are subspaces of c n and c m respectively with dimt dims we also present a specific method to construct convergent t s splitting of a apply these two results to the most of commonly used generalized inverses such as a a d a d w a h k we can get criteria of convergence for the corresponding iterations many old and new splittings and specific choice of the related convergent splittings

a characterization and determinantal formula for the generalized inverse at s 2 and its applications in this paper we present a characterization and determinantal formula for the generalized inverse a t s 2 as applications on the one hand we get the corresponding results for many common important generalized inverses such as a a m n a d a and so on on the other hand we apply these results to solve restricted linear equations ax b email protected t

a combinatorial approach to diffeomorphism invariant quantum gauge theories quantum gauge theory in the connection representation uses functions of holonomies as configuration observables physical observables gauge and diffeomorphism invariant are represented in the hilbert space of physical states physical states are gauge and diffeomorphism invariant distributions on the space of functions of the holonomies of the edges of a certain family of graphs then a family of graphs embedded in the space manifold satisfying certain properties induces a representation of the algebra of physical observables we construct a quantum model from the set of piecewise linear graphs on a piecewise linear manifold and another manifestly combinatorial model from graphs defined on a sequence of increasingly refined simplicial complexes even though the two models are different at the kinematical level they provide unitarily equivalent representations of the algebra of physical observables in separable hilbert spaces of physical states their s knot basis is countable hence the combinator

on drazin inverse of singular toeplitz matrix in this paper we show that the drazin inverse of singular toeplitz matrix can be represented as a sum of products of lower and upper triangular toeplitz matrices and give the explicit expression which generalizes the well known result by gohberg and semencul regarding the inverse of toeplitz matrix

a characterization and representation of the generalized inverse a 2 t s and its applications abstract this paper presents an explicit expression for the generalized inverse a 2 t s based on this we established the characterization the representation theorem and the limiting process for a 2 t s as an application we estimate the error bound of the iterative method for approximating a 2 t s

on the solution of singular linear systems of algebraic equations by semiiterative methods for a square matrixt n n where i t is possibly singular we investigate the solution of the linear fixed point problemx tx c by applying semiiterative methods sim s to the basic iterationx0 n xk tck 1 c k 1 such problems arise if one splits the coefficient matrix of a linear systemax b of algebraic equations according toa m n m nonsingular which leads tox m 1nx m 1b tx c even ifx tx c is consistent there are cases where the basic iteration fails to converge namely ift possesses eigenvalues 1 with 1 or if 1 is an eigenvalue oft with nonlinear elementary divisors in these cases and also ifx tx c is incompatible we derive necessary and sufficient conditions implying that a sim tends to a vector hat x which can be described in terms of the drazin inverse of i t we further give conditions under which hat x is a solution or a least squares solution of i t x c

index splitting for the drazin inverse of linear operator in banach space in this paper two kinds of index splitting methods for the drazin inverse of linear operator in banach space are presented also iterative methods for computing the solution of singular operator equation tx b email protected r t k k ind t based on corresponding splitting methods are established

solving a kind of restricted matrix equations and cramer rule the problem of finding determinantal formula for solutions of some restricted linear systems ax b has been discussed in linear multilinear algebra 15 1984 319 linear algebra appl 116 1989 27 linear multilinear algebra 34 1993 177 appl math comput 125 2002 303 this paper deals with some more extensive cases of this kind of problem and establishes determinantal formulas for solutions of the restricted matrix equationsax d r x r a k 1 xb d n x n b k 2 axb d r x r a k 1 n x n b k 2 where a c n x n with ind a k 1 b c m x m with ind b k 2 and d c n x m the results in linear multilinear algebra 15 1984 319 linear algebra appl 74 1986 213 linear multilinear algebra 34 1993 177 appl math comput 125 2002 303 are partially the special cases in our paper the classic cramer rule is also a special case of our results

an iterative method for solving general restricted linear equations the newton iterative method for computing outer inverses with prescribed range and null space is used in the non stationary richardson iterative method to develop an iterative method for solving general restricted linear equations starting with any suitably chosen initial iterate our method generates a sequence of iterates converging to the solution the necessary and sufficient conditions for the convergence along with the error bounds are established the applications of the iterative method for solving some special linear equations are also discussed a number of numerical examples are worked out they include singular square rectangular randomly generated rank deficient matrices full rank matrices and a set of singular matrices given in matrix computation toolbox mctoolbox with the condition numbers ranging from order 1016 to 1050 the mean cpu time mct and the error bounds are the performance measures used our results when compared with the results obtained by chen 1997 leads to substantial improvement in terms of both computational speed and accuracy

semiconvergence criteria of iterations and extrapolated iterations and constructive methods of semiconvergent iteration matrices in this paper we simultaneously study iterations and extrapolated iterations for the consistent rectangular or sigular linear systems 1 1 ax b in the case that a is rectangular let a m n be a subproper splitting of a and let the generalized inverse m t s 1 2 exist and in the case that a is singular let a m n is a usual splitting of a consider the iteration 2 3 x j 1 gx j c where g m t s 1 2 nandc m t s 1 2 b if a is rectangular and g m 1n and c m 1b if a is singular and the extrapolated iteration 2 19 x j 1 g wx email protected where email protected r w 0 1 and 2 20 g w email protected email protected we establish the semiconcovergence criteria of the iterations 2 3 and 2 19 and present the efficient and convenient construction methods of the semiconvergent iteration matrices g and g w and the choices of the extrapolated parameter w

quantization scheme based on the extension of phase space a new quantization scheme is proposed the similarity of geometric and covariant quantization of a certain class of constrained dynamical systems is shown a method to calculate anomalies is developed

mechanical similarity as a generalization of scale symmetry in this paper we study the symmetry known landau and lifshits 1976 course of theoretical physics vol 1 mechanics oxford pergamon as mechanical similarity lms and present for any monomial potential we analyse it in the framework of the koopman von neumann formulation of classical mechanics and prove that in this framework the lms can be given a canonical implementation we also show that the lms is a generalization of the scale symmetry which is present only for the inverse square and a few other potentials finally we study the main obstructions which one encounters in implementing the lms at the quantum mechanical level

on the characteristic polynomials and eigenvalues of some specially bordered matrices abstract using a determinant identity and leverrier s algorithm for m d system we obtain the explicit expressions of the characteristic polynomial and the eigenvalues of some specially structured bordered matrices m a u v d

pcr algorithm for parallel computing minimum norm t least squares s solution of inconsistent linear equations this paper presents a new highly parallel algorithm for computing the minimum norm t least squares s solution of inconsistent linear equations ax b email protected r m x n r email protected r a by this algorithm the solution x a s t b is obtained in t 1 m 1 log 2m n 6 log 2 n r 1 log 2m log 2n r 1 log 2n steps with p mn processors when m 2 n 1 and with p 2n n 1 processors otherwise

a condensed cramer s rule for the minimum norm least squares solution of linear equations abstract in this paper we will derive the condensed cramer s rule of werner for minimal norm least squares solution of linear equations ax b from the cramer s rule of ben israel and verghese in addition a new condensed cramer s rule will be obtained in this paper

cramer s rule for a system of quaternion matrix equations with applications abstract in this paper we investigate cramer s rule for the general solution to the system of quaternion matrix equations a 1 x b 1 c 1 a 2 x b 2 c 2 and cramer s rule for the general solution to the generalized sylvester quaternion matrix equation a x b c y d e respectively as applications we derive the determinantal expressions for the hermitian solutions to some quaternion matrix equations the findings of this paper extend some known results in the literature

quantization of non standard hamiltonian systems the quantization of classical theories that admit more than one hamiltonian description is considered this is done from a geometrical viewpoint both at the quantization level geometric quantization and at the level of the dynamics of the quantum theory a spin system is taken as an example in which all the steps can be completed it is shown that the geometry of the quantum theory imposes restrictions on the physically allowed non standard quantum theories

the dual variational principle and elliptic problems with discontinuous nonlinearities on etudie des problemes aux valeurs limites elliptiques du type u f u p x x u 0 x ou r n est un domaine borne et f peut avoir des discontinuites on cherche des solutions en utilisant le principe d action duale de clarke

an algebraic extension of dirac quantization examples an extension of the dirac procedure for the quantization of constrained systems is necessary to address certain issues that are left open in dirac s original proposal these issues play an important role especially in the context of nonlinear diffeomorphism invariant theories such as general relativity recently an extension of the required type was proposed using algebraic quantization methods in this paper the key conceptual and technical aspects of the algebraic program are illustrated through a number of finite dimensional examples the choice of examples and some of the analysis is motivated by certain peculiar problems endemic to quantum gravity however prior knowledge of general relativity is not assumed in the main discussion indeed the methods introduced and conclusions arrived at are applicable to any system with first class constraints in particular they resolve certain technical issues which are present also in the reduced phase space approach to quantization of these systems

a cramer rule for finding the solution of a class of singular equations a cramer rule for finding the unique solution x r ak of singular equations ax b ind a k b r ak is given and reduces to the common cramer rule if a is a nonsingular

a variational approach to discontinuous problems with critical sobolev exponents abstract we employ variational techniques to study the existence and multiplicity of positive solutions of semilinear equations of the form u h x h u a uq u2 1 in rn where a 0 are parameters h x is both nonnegative and integrable on rn h is the heaviside function 2 is the critical sobolev exponent and 0 q

a study of semiiterative methods for nonsymmetric systems of linear equations given a nonsingular linear systema x b a splittinga m n leads to the one step iteration 1 x m t x m 1 c witht m 1n andc m 1 b we investigate semiiterative methods sim s with respect to 1 under the assumption that the eigenvalues oft are contained in some compact set of with 1 there exist sim s which are optimal with respect to but except for some special sets such optimal methods are not explicitly known in general using results about maximal convergence of polynomials and uniformly distributed nodes from approximation and function theory we describe here sim s which are asymptotically optimal with respect to it is shown that euler methods extensively studied by niethammer varga nv are special sim s various algorithms for sim s are also derived here a 1 1 correspondence between euler methods and sim s generated by generalized faber polynomials is further established here this correspondence gives that asymptotically optimal euler methods are quite near the optimal sim s

constrained quantization on symplectic manifolds and quantum distribution functions a quantization scheme based on the extension of phase space with application of constrained quantization technique is considered the obtained method is similar to the geometric quantization for constrained systems the problem of scalar product on the reduced hilbert space is investigated and a possible solution to this problem is given generalization of the gupta bleuler like conditions is done by minimization of quadratic fluctuations of quantum constraints the scheme for the construction of generalized coherent states is considered and relation with berezin quantization is found the quantum distribution functions are introduced and their physical interpretation is discussed

on the perturbation and subproper splittings for the generalized inverse at s 2 of rectangular matrix a abstract in this paper the perturbation and subproper splittings for the generalized inverse at s 2 the unique matrix x such that xax x r x t and n x s are considered we present lower and upper bounds for the perturbation of at s 2 convergence of subproper splittings for computing the special solution at s 2 b of restricted rectangular linear system ax b x t are studied for the solution at s 2 b we develop a characterization therefore we give a unified treatment of the related problems considered in literature by ben israel berman hanke neumann plemmons etc

on the perturbation of the group inverse and oblique projection for given matrices a and e which are n matrices and matrices and e we discuss upper bounds for b bb b a a and bb aa aa also a lower bound for b is derived

weighted moore penrose drazin and group inverses of the kronecker product a b and some applications abstract the representations for the weighted moore penrose inverse the drazin inverse and the group inverse of the kronecker product a b of the two matrices a and b are given and by using these results the determinant forms for projectors aa mn a mn a a d a and a g a are deduced

explicit expressions of the generalized inverses and condensed cramer rules in this paper we obtain an explicit representation of the 2 inverse a 2 t s of a matrix a c m n with the prescribed range t and null space s as special cases new expressions for the moore penrose inverse a and drazin inverse a d are derived through explicit expressions we re derive the condensed cramer rules of werner for minimal norm least squares solution of linear equations ax b and propose two new condensed cramer rules for the unique solution of a class of singular system ax b x r a k b r a k k ind a finally condensed determinantal expressions for a a d a a a a a ndaa d are also presented

preconditionings and splittings for rectangular systems recently eiermann marek and niethammer have shown how to applysemiiterative methods to a fixed point systemx tx c which isinconsistent or in whichthe powers of the fixed point operator t have no limit to obtain iterative methods which converge to some approximate solution to the fixed point system in view of their results we consider here stipulations on apreconditioning qax qb of the systemax b and separately on asplitting a m n which lead to fixed point systems such that with the aid of a semiiterative method the iterative scheme converges to a weighted moore penrose solution to the systemax b we show in several ways that to obtain a meaningful limit point from a semiiterative method requires less restrictions on the splittings or the reconditionings than those which have been required in the classical picard iterative method see e g the works of berman and plemmons berman and neumann and tanabe we pay special attention to the case when the weighted moore penrose solution which is sought is the minimal norm least squares solution toax b

group quantization of parametrized systems i time levels a method of quantizing parametrized systems is developed that is based on a kind of gauge invariant quantities the so called perennials a perennial must also be an integral of motion the problem of time in its particular form frozen time formalism global problem of time multiple choice problem is met as well as a related difficulty characteristic for this type of theory the paucity of perennials the present paper is an attempt to find some remedy in the ideas on forms of relativistic dynamics by dirac some aspects of dirac s theory are generalized to all finite dimensional first class parametrized systems the generalization is based on replacing the poincare group and the algebra of its generators as used by dirac by a canonical group of symmetries and by an algebra of elementary perennials a number of insights are gained the following are the main results first conditions are revealed under which the time evolution of the ordinary quantum mechanics or a generalization of it