algebras with non periodic bounded modules abstract we study weakly symmetric special biserial algebra of infinite representation type we show that usually some socle deformation of such an algebra has non periodic bounded modules the exceptions are precisely the algebras whose brauer graph is a tree with no multiple edges if the algebra has a non periodic bounded module then its hochschild cohomology cannot satisfy the finite generation property fg introduced in 9

pseudo hermiticity versus pt symmetry iii equivalence of pseudo hermiticity and the presence of antilinear symmetries we show that a diagonalizable non hermitian hamiltonian h is pseudo hermitian if and only if it has an antilinear symmetry i e a symmetry generated by an invertible antilinear operator this implies that the eigenvalues of h are real or come in complex conjugate pairs if and only if h possesses such a symmetry in particular the reality of the spectrum of h implies the presence of an antilinear symmetry we further show that the spectrum of h is real if and only if there is a positive definite inner product on the hilbert space with respect to which h is hermitian or alternatively there is a pseudo canonical transformation of the hilbert space that maps h into a hermitian operator

periodic modules over gorenstein local rings abstract it is proved that the minimal free resolution of a module m over a gorenstein local ring r is eventually periodic if and only if the class of m is torsion in a certain z t 1 module associated to r this module denoted j r is the free z t 1 module on the isomorphism classes of finitely generated r modules modulo relations reminiscent of those defining the grothendieck group of r the main result is a structure theorem for j r when r is a complete gorenstein local ring the link between periodicity and torsion stated above is a corollary

algebraic and scattering aspects of a symmetric solvable potential we study a particular solvable potential and analyse the effect of symmetry on its bound state as well as scattering solutions we determine the transmission and reflection coefficients for the symmetric case and also formulate the problem in terms of an su 1 1 potential group which allows unified treatment of the discrete and the continuous spectra in a natural way we find that bound and scattering states of the symmetric problem supply a basis for the unitary irreducible representations of the su 1 1 potential group and this gives a straightforward group theoretical interpretation of the fact that the complex invariant potential has a real energy spectrum

a method of moments approach for the efficient and accurate modeling of moderately thick cylindrical wire antennas this paper introduces a moment method formulation which is capable of accurately modeling moderately thick cylindrical wire antennas new algorithms are presented for the efficient computation of the cylindrical wire kernel and related impedance matrix integrals these algorithms make use of exact series representations as well as efficient numerical procedures and lead to a significant reduction in overall computation time for thicker wires another major advantage of this moment method technique is that it is no longer restricted by the segment length to radius ratio limitations inherent in past formulations thereby making it possible to achieve solution convergence for a much wider class of wire antenna structures several examples illustrating the superior convergence properties of this new moment method formulation are presented and discussed

on the oscillations appearing in numerical solutions of solvable and nonsolvable integral equations for thin wire antennas differences between certain solvable and nonsolvable ill posed integral equations with the same nonsingular kernel are discussed the main results come from constructing a solvable equation in the context of straight thin wire antennas the kernel of this equation is the usual approximate also called reduced kernel while its exact solution is the familiar sinusoidal current numerical solutions to this solvable equation are compared to corresponding numerical solutions of the usual halle n and pocklington equations with the approximate kernel it is known from previous publications that these last two equations are nonsolvable and that their numerical solutions present severe oscillations when the number of basis functions is sufficiently large it is found that the difficulties encountered in the former solvable equation are much less important compared to those of the nonsolvable ones the same conclusion is brought out from other integral equations arising in different contexts thin wire circular loop antenna method of auxiliary sources and straight wire antenna of infinite length we discuss the consistency of our results with picard s theorem the results in this paper supplement previous publications regarding the difficulties of numerically solving thin wire integral equations with the approximate kernel

an exact formulation for the vector potential of a cylindrical antenna with uniformly distributed current and arbitrary radius an exact formulation is given for the vector potential and corresponding electric fields associated with a uniform current cylindrical dipole antenna of arbitrary radius the exact expressions converge rapidly in the induction and near field regions of the antenna they are completely general and independent of the usual restrictions involving the wavelength field point distance dipole radius and length the properties of these expressions allow them to be readily used for the efficient as well as accurate computational modeling of cylindrical wire antennas an exact expression for the cylindrical wire kernel which is analytically as well as numerically convenient to work with is presented it is demonstrated that the integral of the kernel can be expressed in terms of a series which involves a generalized exponential integral and higher order associated integrals several methods for evaluating the generalized exponential integral are discussed including some useful asymptotic expansions a numerically stable forward recurrence relation for the higher order associated integrals is also presented

auslander reiten components containing modules with bounded betti numbers let r be a connected selfinjective artin algebra and m an indecomposable nonprojective r module with bounded betti numbers lying in a regular component of the auslander reiten quiver of r we prove that the auslander reiten sequence ending at m has at most two indecomposable summands in the middle term furthermore we show that the component of the auslander reiten quiver containing m is either a stable tube or of type za we use these results to study modules with eventually constant betti numbers and modules with eventually periodic betti numbers

a class of exactly solvable potentials related to the jacobi polynomials the author investigates a family of solvable potentials related to the jacobi polynomials this one dimensional potential family depends on three parameters and is restricted to the domain xo so it can be interpreted as the radial part of a central potential in three dimensions with l 0 closed expressions are obtained for the bound state energy spectrum and the wavefunctions the supersymmetric partner of this potential is also determined and it is found not to belong to the same potential family it is shown that this potential family is a special subclass of the general six parameter natanzon potential class and similarities with another subclass the ginocchio potentials are pointed out some aspects of supersymmetric quantum mechanics and shape invariance are also discussed in connection with the potential family under study

on the application of numerical methods to hallen s equation the so called hallen integral equation for the current on a finite linear antenna center driven by a delta function generator takes two forms depending on the choice of kernel the two kernels are usually referred to as the exact and the approximate or reduced kernel with the approximate kernel the integral equation has no solution nevertheless the same numerical method is often applied to both forms of the integral equation in this paper the behavior of the numerical solutions thus obtained is investigated and the similarities and differences between the two numerical solutions are discussed the numerical method is galerkin s method with pulse functions we first apply this method to the two corresponding forms of the integral equation for the current on a linear antenna of infinite length in this case the method yields an infinite toeplitz system of algebraic equations in which the width of the pulse basis functions enters as a parameter the infinite system is solved exactly for nonzero pulse width the exact solution is then developed asymptotically for the case where the pulse width is small when the asymptotic expressions for the case of the infinite antenna are used as a guide for the behavior of the solutions of the finite antenna the latter problem is greatly facilitated for the approximate kernel the main results of this paper carry over to a certain numerical method applied to the corresponding equation of the pocklington type

modules with reducible complexity abstract for a commutative noetherian local ring we define and study the class of modules having reducible complexity a class containing all modules of finite complete intersection dimension various properties of this class of modules are given together with results on the vanishing of homology and cohomology

two singular potentials the space splitting effect the quantum mechanical problem of a particle moving in one dimension 1d under the influence of a singular potential is reexamined an attractive delta function potential of infinite strength and the 1d coulomb potential is considered in particular in both cases the singularity acts as an impenetrable barrier thus dividing space into two independent regions analysis of the scattering states gives results in agreement with this conclusion the implications that the bound states of the 1d coulomb potential cannot be classified according to parity even though the potential is symmetric is also discussed herein

susyqm and other symmetries in quantum mechanics the relation of supersymmetric quantum mechanics susyqm is discussed with two other symmetry based approaches symmetry and a differential realization of the su 1 1 and su 2 algebra it is demonstrated that symmetry imposes conditions on the even and odd parts of the real and imaginary components of the superpotential w x and these are expressed in terms of a system of first order linear differential equations which is homogeneous when the factorization energy is real and inhomogeneous when it is complex the formal solution of this system is presented for various special cases as well as for the general case it is shown that a trivial solution of this system corresponds to the unbroken symmetry for the scarf ii potential the formalism of susyqm is also linked with that of the potential algebra approach and it is demonstrated that the j and j ladder operators of some su 1 1 or su 2 algebras act on series of degenerate levels of different potentials essentially in the same way as the a and a shift operators of susyqm examples are presented for su 1 1 and su 2 potential algebras as well as for spectrum generating algebras of the same type possible generalizations of this construction are also pointed out

concrete construction and properties of the difference equation derived from the cellular automaton using the filtration technique following the proposal of a filtration technique by nobe satsuma and tokihiro we concretely construct partial difference equations which preserve any time evolution patterns of cellular automaton ca stably by the filtration technique we illustrate how to develop a method of filtration for applying to the typical two spatial dimensional ca rule the game of life and verify that the filtration method provides the stable difference equation associated with the ca compared with the inverse ultradiscretization besides in order to discuss whether the filtration technique can lead one to partial differential equations from ca rules we show a derivation of the burgers equation from rule 184 ca via the discrete burgers equation constructed by the filtration method as an example

huneke ulrich almost complete intersections of cohen macaulay type two abstract let r m k be a commutative noetherian local ring n 2 an integer x a 2n 1 2n 1 alternating matrix with entries from m y a 1 2n 1 matrix with entries from m i the ideal i i1 yx and a the quotient ring r i assume that the grade of i is at least 2n in this case i is a perfect ideal of grade equal to 2n and i is minimally generated by 2n 1 elements we prove that the minimal resolution of a by free r modules is a dg algebra furthermore we identify the algebra torr a k and prove that if r is regular and char k 0 or n 2 2 char k then the poincare series pma z i 0 dimk torai m k zi is a rational function for every finitely generated a module m as a consequence we deduce that if the projective dimension of m is infinite then eventually the betti numbers of m form an increasing sequence with strong exponential growth

an integrated approach to ladder and shift operators for the morse oscillator radial coulomb and radial oscillator potentials the morse oscillator radial coulomb and radial harmonic oscillator problems can be solved exactly using a variety of algebraic methods these problems correspond to different realizations of the so 2 1 algebra and a comparison of the generators of the algebra may be used to identify mappings between each pair of systems the resultant transition operators act as ladder or energy changing operators in the cases of the coulomb and harmonic oscillator potentials whereas they act as shift operators acting at constant energy in the case of the morse potential this is a consequence of the so 2 1 dynamical symmetry whereby the morse hamiltonian is expressible solely in terms of the casimir operator of the algebra an alternative algebraic approach the use of the method of supersymmetric quantum mechanics or factorization produces in each case a set of shift operators relations between the various ladder and shift operators may be identified by means of the appropriate mappings and these results can be generalized so as to relate the one dimensional morse oscillator to the radial coulomb and radial harmonic oscillator potentials involving an arbitrary number of angular dimensions

unified treatment of the coulomb and harmonic oscillator potentials in d dimensions quantum mechanical models and practical calculations often rely on some exactly solvable models like the coulomb and the harmonic oscillator potentials the d dimensional generalized coulomb potential contains these potentials as limiting cases thus it establishes a continuous link between the coulomb and harmonic oscillator potentials in various dimensions we present results which are necessary for the utilization of this potential as a model and practical reference problem for quantum mechanical calculations we define a hilbert space basis the generalized coulomb sturmian basis and calculate the green s operator on this basis and also present an su 1 1 algebra associated with it we formulate the problem for the one dimensional case too and point out that the complications arising due to the singularity of the one dimensional coulomb problem can be avoided with the use of the generalized coulomb potential

solution of hallen s integral equations by means of nystrom method to solve hallen s integral equations accurately nystrom method is applied in this paper to obtain an exact and finite solution both the exact kernel and a finite gap width must be taken into consideration in the case of a gap feed a modification function that converts the approximate kernel to the exact kernel is derived as the modification function is singular a one point correction method is proposed as a singularity subtraction method using a step function source we can use various quadrature rules to solve the problem the results show small difference between them the validity of this theory is confirmed by comparing the measured results and the calculated results the effect of the feed system is shown to be strong

fundamentals of thin wire integral equations with the finite gap generator part ii we discuss certain fundamental properties of hallen s and pocklington s equations with the finite gap generator fgg which has long been used as a feed model for cylindrical antennas for the case of the exact kernel we focus on whether important quantities are finite or infinite and compare to corresponding quantities obtained using the delta function generator dfg for the approximate kernel we focus on the unsolvability of the two equations and the relation of unsolvability to the important phenomenon of unphysical oscillations we also extend to the fgg the so called effective current method this is an easy to apply remedy for the oscillatory solutions that allows one to obtain reasonable answers for important quantities such as current distributions and the input impedance it is shown here that these answers are close to those obtained with the exact kernel we show that in a certain sense the method is more advantageous to apply in the present fgg case than in the dfg case our discussions especially in part ii of this paper bring out similarities to 1 the phenomenon of superdirectivity and 2 recently discussed numerical phenomena occurring within the method of auxiliary sources

regularity theory and superalgebraic solvers for wire antenna problems we consider the problem of evaluating the current distribution j z that is induced on a straight wire antenna by a time harmonic incident electromagnetic field the scope of this paper is twofold one of its main contributions is a regularity proof for a straight wire occupying the interval 1 1 in particular for a smooth time harmonic incident field this theorem implies that j z i z sqrt 1 z 2 where i z is an infinitely differentiable function the previous state of the art in this regard placed i in the sobolev space w 1 p p 1 the second focus of this work is on numerics we present three superalgebraically convergent algorithms for the solution of wire problems two based on hallen s integral equation and one based on the pocklington integrodifferential equation both our proof and our algorithms are based on two main elements 1 a new decomposition of the kernel of the form g z f 1 z ln z f 2 z where f 1 z and f 2 z are analytic functions on the real line and 2 removal of the end point square root singularities by means of a coordinate transformation the hallen and pocklington based algorithms we propose converge superalgebraically faster than mathcal o n m and mathcal o m m for any positive integer m where n and m are the numbers of unknowns and the number of integration points required for construction of the discretized operator respectively in previous studies at most the leading order contribution to the logarithmic singular term was extracted from the kernel and treated analytically the higher order singular derivatives were left untreated and the resulting integration methods for the kernel exhibit mathcal o m 3 convergence at best a rather comprehensive set of tests we consider shows that in many cases to achieve a given accuracy the numbers n of unknowns required by our codes are up to a factor of five times smaller than those required by the best solvers previously available the required number m of integration points in turn can be several orders of magnitude smaller than those required in previous methods in particular four digit solutions were found in computational times of the order of four seconds and in most cases of the order of a fraction of a second on a contemporary personal computer much higher accuracies result in very small additional computing times

construction of modules with finite homological dimensions a new homological dimension called g dimension is defined for every finitely generated module m over a local noetherian ring r it is modeled on the ci dimension of avramov gasharov and peeva and has parallel properties in particular a ring r is gorenstein if and only if every finitely generated r module has finite g dimension the g dimension lies between the ci dimension and the g dimension of auslander and bridger this relation belongs to a longer sequence of inequalities where a strict inequality in any place implies equalities to its right and left over general local rings we construct classes of modules that show that a strict inequality can occur at almost every place in the sequence

nonvanishing cohomology and classes of gorenstein rings abstract we give counterexamples to the following conjecture of auslander given a finitely generated module m over an artin algebra there exists a positive integer n m such that for all finitely generated modules n if ext i m n 0 for all i 0 then ext i m n 0 for all i n m some of our examples moreover yield homologically defined classes of commutative local rings strictly between the class of local complete intersections and the class of local gorenstein rings

systematic search for symmetric potentials with real energy spectra changes of coordinates represent one of the most effective ways of deriving solvable potentials from ordinary differential equations for separate special functions here we relax the standard hermiticity requirement and find an innovative construction which leads to unusual complex potentials their energy spectrum is shown to stay real after a weakening of the hermiticity of the schr dinger equation to its mere invariance under the combined parity and time reversal symmetry this ultimately results in richer bound state spectra some of our new exactly solvable potentials generalize the current textbook models details are given for constructions based on the hypergeometric and confluent hypergeometric special functions

solvable potentials associated with su 1 1 algebras a systematic study we consider a specific differential realization of the su 1 1 algebra and use it to explore such algebraic structures associated with shape invariant potentials our approach combines elements of various methods of solving the schrodinger equation such as supersymmetric quantum mechanics or the factorization method algebraic techniques and special function theory in fact it amounts to reformulating transformations mapping the schrodinger equation into the differential equation of orthogonal polynomials in group theoretical terms our systematic study recovers a number of earlier results in a natural unified way and also leads to new findings the procedure presented here implicitly contains a similar treatment of the compact su 2 algebra as well possible generalizations of this approach involving different realizations of the su 1 1 algebra other algebraic structures and larger classes of potentials are also outlined

pseudo hermiticity versus pt symmetry ii a complete characterization of non hermitian hamiltonians with a real spectrum we give a necessary and sufficient condition for the reality of the spectrum of a non hermitian hamiltonian admitting a complete set of biorthonormal eigenvectors

boundedness versus periodicity over commutative local rings over commutative graded local artinian rings examples are constructed of periodic modules of arbitrary minimal period and modules with bounded betti numbers which are not eventually periodic they provide counterexamples to a conjecture of d eisenbud that every module with bounded betti numbers over a commutative local ring is eventually periodic of period 2 it is proved however that the conjecture holds over rings of small length

numerical solution of the helmholtz equation in 2d and 3d using a high order nystrom discretization we show how to solve time harmonic scattering problems by means of a high order nystrom discretization of the boundary integral equations of wave scattering in 2d and 3d the novel aspect of our new method is its use of local corrections to the discretized kernel in the vicinity of the kernel singularity enhanced by local corrections the new algorithm has the simplicity and speed advantages of the traditional nystrom method but also enjoys the advantages of high order convergence for controlling solution error we explain the practical details of implementing a scattering code based on a high order nystrom discretization and demonstrate by numerical example that a scattering code based on this algorithm can achieve high order convergence to the correct answer we also demonstrate its performance advantages over a high order galerkin code

the exact kernel for cylindrical antenna in the numerical calculation for a cylindrical antenna the thin wire kernel is widely used as a convenient approximation to the exact kernel of the electric field equation however this thin wire approximation is valid only if the radius of wire is much less than its length and the wavelength furthermore arbitrarily using this approximation may not obtain the convergent outcomes the expression of the exact kernel for the cylindrical antenna which would be useful in the numerical work is presented

brauer thrall for totally reflexive modules let r be a commutative noetherian local ring that is not gorenstein it is known that the category of totally reflexive modules over r is representation infinite provided that it contains a non free module the main goal of this paper is to understand how complex the category of totally reflexive modules can be in this situation local rings r m with m3 0 are commonly regarded as the structurally simplest rings to admit diverse categorical and homological characteristics for such rings we obtain conclusive results about the category of totally reflexive modules modeled on the brauer thrall conjectures starting from a non free cyclic totally reflexive module we construct a family of indecomposable totally reflexive r modules that contains for every n n a module that is minimally generated by n elements moreover if the residue field r m is algebraically closed then we construct for every n n an infinite family of indecomposable and pairwise non isomorphic totally reflexive r modules each of which is minimally generated by n elements the modules in both families have periodic minimal free resolutions of period at most 2

a search for shape invariant solvable potentials the author investigates a simple method of constructing potentials which is related to the work of bhattacharjie and sudarshan 1962 and for which the schrodinger equation can be solved in terms of known special functions it turns out that this method can be related to supersymmetric quantum mechanics and this relationship can help to decide which special functions satisfying linear homogeneous second order differential equations can be solutions of the schrodinger equation with potentials of the form v x w2 x w x the author illustrates this procedure with the example of orthogonal polynomials and obtains explicit expressions of wavefunctions of a wide class of shape invariant potentials

pt symmetric potentials and the so 2 2 algebra starting from a differential realization of the generators of the so 2 2 algebra we connect the eigenvalue equation of the casimir invariant either with the hypergeometric equation or the schr odinger equation in the latter case we consider problems for whichso 2 2 appears as a potentia la lgebra connecting states with the same energy in different potentials we analyse the role of the two so 2 1 subalgebras and point out their importance for pt symmetric problems where the doubling of bound states is known to occur we present two mechanisms for this and illustrate them with the example of the scarf and the p oschl teller ii potentials we also analyse scattering states transmission