geometry of free loci and factorization of noncommutative polynomials abstract the free singularity locus of a noncommutative polynomial f is defined to be the sequence of hypersurfaces z n f x m n k g det f x 0 the main theorem of this article shows that f is irreducible if and only if z n f is eventually irreducible a key step in the proof is an irreducibility result for linear pencils arising from this is a free singularity locus nullstellensatz for noncommutative polynomials apart from consequences to factorization in a free algebra the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry

a conjectured property of hermitian pencils abstract if a b is any pair of hermitian matrices the power of dividing det i xa yb will be given by the number of basic singular summands in the pair contrary to conjecture this power can be greater than one even when the pair is unitarily irreducible

when is the wigner function of multidimensional systems nonnegative it is shown that for systems with an arbitrary number of degrees of freedom a necessary and sufficient condition for the wigner function to be nonnegative is that the corresponding state wavefunction is the exponential of a quadratic form this result generalizes the one obtained by hudson rep math phys 6 249 1974 for one dimensional systems

numerical methods for simulating ginzburg landau vortices numerical solution to the ginzburg landau gl equation becomes infeasible as the gl parameter kappa and the number of gl vortices increase to a physically interesting regime it is in this regime that we focus our attention to design a simulated annealing approach for minimizing the associated renormalized energy to study the gl vortex configurations this approach applied to a commonly used model produces vortex configurations which are of interest in recent studies

quantum harmonic analysis on phase space relative to an irreducible representation of the canonical commutation relations convolutions between quantum mechanical operators and between functions and operators are defined for which the usual weyl transform acts as a fourier transform basic properties of these operations are developed in close analogy to harmonic analysis on r2n using the quantum version of wiener s approximation theorem a natural one to one correspondence between the closed phase space translation invariant subspaces of classical and quantum observables is established

vortices in complex scalar fields abstract consider a spatially extended field which evolves in time according to a pde the solutions contain particle like defects whose motions are parts of the full dynamics we inquire into the formulation of an asymptotic particle field defect dynamics which derives from the full pde we carry out this program for two complex scalar field equations in two space dimensions the nonlinear schrodinger equation and the nonlinear heat equation the topological defects are zeros of the complex scalar field with nonzero integer winding numbers called vortices vortices evolving under the nonlinear schrodinger equation behave like point vortices in ideal fluid pairs of vortices evolving under the nonlinear heat equation with like opposite winding numbers undergo a repulsive attractive interaction

simultaneous similarity of matrices abstract in this paper we solve completely and explicitly the long standing problem of classifying pairs of nxn complex matrices a b under a simultaneous similarity roughly speaking the classification decomposes to a finite number of steps in each step we consider an open algebraic set then we construct a finite number of rational functions in the entries of a and b whose values are constant on all pairs similar to a b the values of the functions phi sub i a b i equals 1 s determine a finite number of similarity classes

analysis and approximation of the ginzburg landau model of superconductivity the authors consider the ginzburg landau model for superconductivity first some well known features of superconducting materials are reviewed and then various results concerning the model the resultant differential equations and their solution on bounded domains are derived then finite element approximations of the solutions of the ginzburg landau equations are considered and error estimates of optimal order are derived

a counterexample to kippenhahn s conjecture on hermitian pencils abstract a counterexample is constructed to a conjecture of kippenhahn math nachr 6 193 288 1951 52 a pair of hermitian 8 8 matrices h k is found such that 1 h k generate m 8 c and 2 the minimal polynomial of the pencil xh yk has degree 4 recent work of h shapiro e g linear algebra appl 43 201 221 1982 has established the conjecture for n n matrices n 5

fast parallel recursive aggregation methods for simulation of dynamical systems a novel recursive aggregation algorithm for numerical simulations of dynamic systems is proposed and analyzed the algorithm exploits a special structure of the linear equation problem resulting from the discretization of the dynamic system and an aggregation disaggregation procedure the algorithm has a time complexity of i q 2m q 3 logn for solving linear systems with q states for n discrete time instants using o q sup 3 n processors where i q is the parallel time complexity for inverting a q spl times q matrix m q is the parallel time complexity for matrix multiplication of two q spl times q matrices the competing parallel cyclic reduction method for the same problem has a time complexity of i q 3m q 4 logn thus the proposed algorithm has a definite speed advantage over the cyclic reduction method an approximation technique for the unknown boundary conditions in boundary value problems is also proposed the algorithm was implemented to simulate some dynamical stable and unstable systems and the numerical results show that the accumulation of roundoff errors is insignificant as compared to the discretization errors

continuity statements and counterintuitive examples in connection with weyl quantization we use the properties of an integral transform relating a classical function f with the matrix elements between coherent states of its quantal counterpart q f to derive continuity properties of the weyl transform from classes of distributions to classes of quadratic forms we also give examples of pathological behavior of the weyl transform with respect to other topologies e g bounded functions leading to unbounded operators

wigner quantisation of arrival time and oscillator phase by means of the wigner distribution function an expression is derived for the quasiprobability that a free particle in one dimension arrives at the origin during a given time interval and also for the quasiprobability that the phase of a one dimensional harmonic oscillator lies in a given segment of the circle it is shown that the best bounds on these quasiprobabilities p are 0 1559 or p or 1 0077 independent of the size of the interval it is shown that on disjoint unions of infinitely many intervals the quasiprobabilities may add up to any positive or negative real value it is briefly indicated how negative probabilities can be avoided altogether in an alternative approach to the quantisation of arrival time and oscillator phase

pencils of complex and real symmetric and skew matrices abstract this expository paper establishes the canonical forms under congruence for pairs of complex or real symmetric or skew matrices the treatment is in the spirit of the well known book of gantmacher on matrix theory and may be regarded as a supplement to gantmacher s chapters on pencils of matrices

energy momentum tensor in phase space a connection between schrodinger energy momentum tensor and terletsky distribution function schrodinger energy momentum tensor for a matter field interacting with an external electromagnetic field is represented in phase space by means of terletsky distribution function the explicit form of this distribution function in the case of a hydrogen type atom is found the structure of the schrodinger energy momentum tensor in phase space is compared with the structure of a real fluid and its local pressure and local energy are presented in the fundamental state of a hydrogen type atom the negative values of the terletsky distribution function are not rejected in view of their participation in the phase space energy momentum tensor the negative part of the quantum distribution function could be interpreted as phase space density for anti constituents in a hypothetical sub quantum level of the matter

joint numerical range and its generating hypersurface in this paper we study the joint numerical range of m tuples of hermitian matrices via their generating hypersurfaces an example is presented which shows the invalidity of an analogous kippenhahn theorem for the joint numerical range of three hermitian matrices

quantization and noiseless measurements in accordance with the fact that quantum measurements are described in terms of positive operator measures poms we consider certain aspects of a quantization scheme in which a classical variable is associated with a unique positive operator measure pom ef which is not necessarily projection valued the motivation for such a scheme comes from the well known fact that due to the noise in a quantum measurement the resulting outcome distribution is given by a pom and cannot in general be described in terms of a traditional observable a selfadjoint operator accordingly we note that the noiseless measurements are those which are determined by a selfadjoint operator the pom ef in our quantization is defined through its moment operators which are required to be of the form with being a fixed map from classical variables to hilbert space operators in particular we consider the quantization of classical questions that is functions taking only values 0 and 1 we compare two concrete realizations of the map in view of their ability to produce noiseless measurements one being the weyl map and the other defined by using phase space probability distributions

aggregation disaggregation methods for computing the stationary distribution of a markov chain we implement and analyse aggregation disaggregation procedures constructed to accelerate the convergence of successive approximation methods suitable for computing the stationary distribution of a finite markov chain we define six of these methods and analyse them in detail in particular we show that some existing procedures lie in the aggregation disaggregation framework we set and hence can be considered as special cases also for all described methods we identify cases where they are promising numerical examples for the applications of some of the methods for nearly completely decomposable stochastic matrices are given as well

aggregation disaggregation for eigenvalue problems convergence of the power method for the algebraic eigenvalue problem is accelerated by an aggregation disaggregation process a lower dimensional aggregated eigenvalue problem is created from time to time during the course of the power iteration and the solution of this problem is used to produce acceleration analysis and numerical experiments show the effectiveness of the method the more so the more poorly conditioned is the original eigenvalue problem

local convergence of the exact and inexact iterative aggregation method for linear systems and markov operators the iterative aggregation method for the solution of linear systems is extended in several directions to operators on banach spaces to the method with inexact correction i e to methods where the inner linear system is in turn solved iteratively and to the problem of finding stationary distributions of markov operators local convergence is shown in all cases convergence results apply to the particular case of stochastic matrices moreover an argument is given which suggests why the iterative aggregation method works so well for nearly uncoupled markov chains as well as for markov chains with other zero nonzero structures

a linearized crank nicolson galerkin method for the ginzburg landau model the time dependent ginzburg landau model is used extensively in studying the nonequilibrium state of superconductivity the computer simulation of this model requires high performance computing power and reliable and efficient numerical methods to solve the ginzburg landau equtions in this paper a linearized crank nicolson galerkin method is proposed for solving these nonlinear and coupled partial differential equations the method uses the galerkin finite element approximation in spatial discretization and the crank nicolson implicit scheme in time discretization together with certain techniques which linearize and decouple the ginzburg landau equations while retaining the stabillity and accuracy of the crank nicolson scheme the proposed approach results in symmetric and positive definite matrix problems thus substantially improving the computational efficiency furthermore and even more important the proposed approach is suitable for large scale parallel computation numerical results from simulating the vortex dynamics of superconductivity by using the linearized crank nicolson galerkin method are presented

equality of higher numerical ranges of matrices and a conjecture of kippenhahn on hermitian pencils let mn be the algebra of all n n complex matrices for 1 k n the kth numerical range of a mn is defined by wk a 1 k jk 1xj axj x1 xk is an orthonormal set in n it is known that tr a n wn a wn 1 a w1 a we study the condition on a under which wm a wk a for some given 1 m k n it turns out that this study is closely related to a conjecture of kippenhahn on hermitian pencils a new class of counterexamples to the conjecture is constructed based on the theory of the numerical range

mixed states with positive wigner functions the wigner distribution function of a pure quantum state is everywhere positive if and only if the state is coherent according to a result of hudson the characterization of mixed states with a positive wigner function is a special case of the problem of determining functions satisfying a twisted positive definiteness condition for a prescribed set of twisting parameters i e functions with given wigner spectrum in the sense of narcowich if a state is a convex combination of coherent states it has the property that the wigner spectrum contains the unit interval which in turn implies that the wigner function is positive it is shown by explicit examples that the converses of both implications are false the examples are taken from a low dimensional section of the state space in which all wigner spectra can be computed in this set counterexamples to a conjecture by narcowich concerning the wigner spectrum of products are also found as well as a state whose wigner spectrum is a convergent sequence of discrete points

the wigner distribution of n arbitrary observables we study a generalization of the wigner function to arbitrary tuples of hermitian operators we show that for any collection of hermitian operators a1 an and any quantum state there is a unique joint distribution on rn with the property that the marginals of all linear combinations of the ak coincide with their quantum counterparts in other words we consider the inverse radon transform of the exact quantum probability distributions of all linear combinations we call it the wigner distribution because for position and momentum this property defines the standard wigner function we discuss the application to finite dimensional systems establish many basic properties and illustrate these by examples the properties include the support the location of singularities positivity the behavior under symmetry groups and informational completeness

conjugate dynamical systems classical analogue of the quantum energy translation an aspect of quantum mechanics that has not been fully understood is the energy shift generated by the time operator in this study we introduce the use of the eigensurfaces of dynamical variables and commutators in classical mechanics to study the classical analogue of the quantum translation of energy we determine that there is a conjugate dynamical system that is conjugate to hamilton s equations of motion and then we generate the analogue of the time operator and use it in the translation of points along the energy direction i e the classical analogue of the pauli theorem the theory is illustrated with a nonlinear oscillator model

a survey of canonical forms and invariants for unitary similarity abstract matrices a and b are said to be unitarily similar if u au b for some unitary matrix u this expository paper surveys results on canonical forms and invariants for unitary similarity the first half gives a detailed description of methods developed by several authors brenner littlewood mitchell mcrae radjavi serge chuk and benedetti and cragnolini using inductively defined reduction procedures to transform matrices to canonical form the matrix is partitioned and successive unitary similarities applied to reduce the submatrices to some nice form at each stage one refines the partition and restricts the set of permissible unitary similarities to those that preserve the already reduced blocks the process ends in a finite number of steps producing both the canonical form and the subgroup of the unitary group that preserves that form depending on the initial step various canonical forms may be defined the method can also be used to define canonical forms relative to certain subgroups of the unitary group and canonical forms for finite sets of matrices under simultaneous unitary similarity the remainder of the paper surveys results on unitary invariants and other topics related to unitary similarity such as the specht pearcy trace invariants the numerical range and unitary reducibility

hermitian pencils with a cubic minimal polynomial let a be an n n complex matrix and write a h ik where h and k are hermitian matrices we show that if the minimal polynomial of the pencil xh yk has degree 3 then there is a unitary matrix u such that u 1au is block diagonal with blocks of size 3 3 or smaller this is a special case of a conjecture made by kippenhahn in 1951

on a conjecture of kippenhahn about the characteristic polynomial of a pencil generated by two hermitian matrices i abstract let a be an n n matrix write a h ik where i 2 1 and h and k are hermitian let f x y z det zi xh yk we first show that a pair of matrices over an algebraically closed field which satisfy quadratic polynomials can be put into block upper triangular form with diagonal blocks of size 1 1 or 2 2 via a simultaneous similarity this is used to prove that if f x y z g x y z n 2 where g has degree 2 then for some unitary matrix u the matrix u au is the direct sum of n 2 copies of a 2 2 matrix a 1 where a 1 is determined up to unitary similarity by the polynomial g x y z we use the connection between f x y z and the numerical range of a to investigate the case where f x y z has the form z ax y r g x y z s where g x y z is irreducible of degree 2

eigenvalue bounds and inequalities using vector aggregation of matrices abstract for matrices partitioned into block form the operation of block vector aggregation which associates with a given matrix a matrix of smaller size is introduced properties of aggregated matrices are analyzed in particular it is shown that in the hermitian case the eigenvalues of a block vector aggregated matrix interlace those of the original matrix by using vector aggregation new eigenvalue bounds and inequalities for normal and hermitian matrices are derived and put in context with the existing ones in particular inequalities interrelating eigenvalues of a block partitioned hermitian matrix with those of its diagonal blocks are obtained also it is shown that the spectral constants characterizing the block partitioning of a hermitian matrix are bounded below by the corresponding constants related to associated vector aggregated matrices