the bunch kaufman factorization of symmetric matrices signature similar to sign regular matrices abstract the bunch kaufman pivoting strategy is a most commonly used method in practice to factor symmetric indefinite matrices however this method in general may destroy the band structure of banded matrices and its growth factor bound of 2 57 n 1 can almost be attained in this paper we investigate the behavior of the method for factorizing symmetric indefinite matrices that are signature similar to sign regular matrices with signature the growth factor bound depending on flips in the signature is derived for factorizing such banded matrices if the bunch kaufman pivoting strategy is modified as suggested by sorensen and van loan then not only do the same growth factor bounds hold but also the bandwidth is nicely preserved

rings of invariants for modular representations of the klein four group we study the rings of invariants for the indecomposable modular representations of the klein four group for each such representation we compute the noether number and give minimal generating sets for the hilbert ideal and the field of fractions we observe that with the exception of the regular representation the hilbert ideal for each of these representations is a complete intersection

continued fractions and the potential models of confinement reply to a comment the analytic continued fractional formula for energies of the bound states in the harmonium potential v r ar 1 br cr2 as suggested by singh et al 1978 1979 and criticised by flessas 1982 is correct if and only if b 0 this is the special case of the more general result the author extends the analytic construction of the green function and the rigorous specification of its validity to the whole class of potentials with r2v r polynomial in ralpha alpha positive rational number

explicit generators of the invariants of finite groups abstract let r denote a commutative and associative ring with 1 and let a denote a finitely generated commutative r algebra let g denote a finite group of r algebra automorphisms of a in the case that r is a field of characteristic 0 noether constructed a finite set of r algebra generators of the invariants of g this paper proves that the same construction produces a set of generators of the invariants of g when g is invertible in r generators of the invariants of g are also explicitly described in the case that g is solvable and g is invertible in r

gaussian elimination is stable for the inverse of a diagonally dominant matrix let b e m n c be a row diagonally dominant matrix i e i b ii b ij i 1 n where 0 i 1 i 1 n with a max 1 i n i we show that no pivoting is necessary when gaussian elimination is applied to a b 1 moreover the growth factor for a does not exceed 1 the same results are true with row diagonal dominance being replaced by column diagonal dominance

eigenvalue bounds for transformations of solvable potentials we study smooth transformations of potentials for which exact bound state solutions of schrodinger s equation are known eigenvalue approximation formulae are obtained which provide lower or upper energy bounds according to whether the transformation function g is convex or concave detailed results are presented for perturbed coulomb potentials of the form and

computation of invariants for reductive groups abstract we will give an algorithm for computing generators of the invariant ring for a given representation of a linearly reductive group the algorithm basically consists of a single grobner basis computation we will also show a connection between some open conjectures in commutative algebra and finding good degree bounds for generating invariants

castelnuovo mumford regularity by approximation abstract the castelnuovo mumford regularity of a module gives a rough measure of its complexity we bound the regularity of a module given a system of approximating modules whose regularities are known such approximations can arise naturally for modules constructed by inductive combinatorial means we apply these methods to bound the regularity of ideals constructed as combinations of linear ideals and the module of derivations of a hyperplane arrangement as well as to give degree bounds for invariants of finite groups

the orthogonality property of the lommel polynomials and a twofold infinity of relations between rayleigh s σ sums making use of a remarkable theorem which expresses a relationship between a certain type of infinite continued fractions and systems of orthogonal polynomials it is proven that the known infinite continued fraction development of the ratio of bessel functions jv 1 z jv z gives rise to an orthogonality property of the lommel polynomials rm v 1z m n when v is real and positive the corresponding weight function which appears to be non negative in the interval of definition is obtained by the application of two successive integral transforms it consists of an infinite series of dirac functions whose singularities are distributed symmetrically around the origin on the real axis in such a manner that the origin is their limit point on both sides for any positive v the lommel polynomials form a system of so called orthogonal polynomials of a discrete variable the orthogonality property may also be conveniently expressed by means of a stieltjes integral one of its corollaries is a twofold infinity of linear relations between the sums v r defined by v r n 1 1 jv n2r with v 1 r0 r n0 in which jv n represents the nth positive zero of jv z another by product consists of a complement to a theorem of hurwitz concerning the nature and the position of the zeros of the lommel polynomials written as gm v z in the modified notation of the mentioned author from this study also result two interesting approximations of jv 1 applicable for v 1 1

sharpening the generalized noether bound in the invariant theory of finite groups abstract we consider linear representations of a finite group g on a finite dimensional vector space over a field f in which g is invertible by a theorem due to e noether in char 0 and to fleischmann and fogarty in general the ring of invariants is generated by homogeneous elements of degree at most g schmid domokos and hegedus sharpened noether s bound when g is not cyclic and char f 0 we prove that the sharpened bound holds over general fields if g is not cyclic and g is invertible in f then the ring of invariants is generated by elements of degree at most 3 4 g if g is even and at most 5 8 g if g is odd

on the castelnuovo mumford regularity of rings of polynomial invariants we show that when a group acts on a polynomial ring over a eld the ring of invariants has castelnuovo mumford regularity at most zero as a consequence we prove a well known conjecture that the invariants are always generated in degrees at most n jgj 1 where n 1 is the number of polynomial generators andjgj 1 is the order of the group we also prove some other related conjectures in invariant theory

the depth of invariant rings and cohomology abstract let g be a finite group acting linearly on a vector space v over a field k of positive characteristic p and let p g be a sylow p subgroup ellingsrud and skjelbred compositio math 41 1980 233 244 proved the lower bound depth k v g min dim v p 2 dim v for the depth of the invariant ring with equality if g is a cyclic p group let us call the pair g v flat if equality holds in the above in this paper we use cohomological methods to obtain information about the depth of invariant rings and in particular to study the question of flatness for g of order not divisible by p2 it ensues that g v is flat if and only if h1 g k v 0 or dim vp 2 dim v we obtain a formul a for the depth of the invariant ring in the case that g permutes a basis of v and has order not divisible by p2 in this situation g v is usually not flat moreover we introduce the notion of visible flatness of pairs g v and prove that this implies flatness for example the groups sl2 q so3 q su3 q sz q and r q with many interesting representations in defining characteristic are visibly flat in particular if g sl2 q and v is the space of binary forms of degree n or a direct sum of such spaces then g v is flat for all q pr with the exception of a finite number of primes p along the way we obtain results about the buchsbaum property of invariant rings and about the depth of the cohomology modules hi g k v we also determine the support of the positive cohomology h g k v as a module over k v g in the appendix the visibly flat pairs g v of a group with bn pair and an irreducible representation are classified

exact solutions of the schrodinger equation d dx2 x2 λx2 1 gx2 ψ x eψ x the authors prove the existence of a class of exact eigenvalues and eigenfunctions of the schrodinger equation for the potential x2 x2 1 gx2 when certain algebraic relations between and g hold some of the properties of these solutions are discussed it is shown that in a certain sense they may be regarded as sturmians for the schrodinger equation with the potential x2 g g2x2

eigenvalues of the schrodinger equation with coulomb potentials plus linear and harmonic radial terms analytic approximations for the eigenvalues have been obtained using the two point quasifractional approximants method here we extend and generalize previous work where the method was applied to determine eigenenergies of two dimensional coulomb potentials in the presence of a magnetic field of arbitrary strength we have also shown the advantage of our results with respect to other calculations using the shifted 1 n method or pade approximants

invariants of the diagonal cp action on v3 abstract let c p denote the cyclic group of order p where p 3 is prime we denote by v 3 the indecomposable three dimensional representation of c p over a field f of characteristic p we compute a set of generators in fact a sagbi basis for the ring of invariants f v 3 v 3 c p our main result confirms the conjecture of shank r j shank classical covariants and modular invariants in h e a campbell d l wehlau eds invariant theory in all characteristics crm proc lecture notes vol 35 amer math soc 2004 pp 241 249 for this example that all modular rings of invariants of c p are generated by rational invariants norms and transfers

two interacting electrons in a uniform magnetic field and a parabolic potential the general closed form solution we present an analytical analysis of the two dimensional schrodinger equation for two interacting electrons subjected to a homogeneous magnetic field and confined by a two dimensional external parabolic potential we have found the general closed form expression for the eigenstates of the problem and its corresponding eigenenergies for particular values of magnetic field and spatial confinement length the mathematical framework is just based on a rigorous solution of the three term recursion relation among the coefficients that arises from the series solution of biconfluent heun bhe equation connected with the radial part of the schrodinger equation for the internal motion it is also shown that by vanishing of coulomb repulsion strength the obtained explicit analytical solutions of bhe equation reduces to the well known polynomials satisfying the associated laguerre differential equation furthermore in the presence of this interaction the results are compared with those previously obtained in th

the freeness and minimal free resolutions of modules of differential operators of a generic hyperplane arrangement abstract let a be a generic hyperplane arrangement composed of r hyperplanes in an n dimensional vector space and s the polynomial ring in n variables we consider the s submodule d m a of the nth weyl algebra of homogeneous differential operators of order m preserving the defining ideal of a we prove that if n 3 r n m r n 1 then d m a is free holm s conjecture combining this with some results by holm we see that d m a is free unless n 3 r n m r n 1 in the remaining case we construct a minimal free resolution of d m a by generalizing yuzvinsky s construction for m 1 in addition we construct a minimal free resolution of the transpose of the m jet module which generalizes a result by rose and terao for m 1

the asymptotic iteration method for the eigenenergies of the schrodinger equation with the potential v r z r gr λr2 the eigenenergies of the schrodinger equation with coulomb potential plus linear and harmonic radial terms are investigated using the perturbation technique within the framework of the asymptotic iteration method the obtained eigenenergies are compared with those obtained from the shifted 1 n expansion method the moment method and also with the exact values obtained from the super symmetric method it is found that asymptotically this method gives accurate results for arbitrary g z and

on the μx2 λx4 ηx6 interaction a class of exact even and odd parity solutions of the schrodinger equation for the interaction x2 x4 x6 0 is obtained in the form polynomial exponential when and satisfy some specific relations in the general case the eigenvalues of the bounded potential problem are found by the method of series solution for any value of the authors also show that the analytic continued fraction method does not always lead to the corrected results

degree bounds for syzygies of invariants abstract suppose that g is a linearly reductive group good degree bounds for generators of invariant rings were given in proc amer math soc 129 4 2001 955 here we study minimal free resolutions of invariant rings for finite linearly reductive groups g it was recently shown in adv math 156 1 2000 23 electron res announc amer math soc 7 2001 5 adv math 172 2002 151 that rings of invariants are generated in degree at most the group order g in characteristic 0 this degree bound is a classical result by emmy noether see math ann 77 1916 89 given an invariant ring of a finite linearly reductive group g we prove that the ideal of relations of a minimal set of generators is generated in degree at most 2 g

homomorphisms localizations and a new algorithm to construct invariant rings of finite groups abstract let g be a finite group acting on a polynomial ring a over the field k and let a g denote the corresponding ring of invariants let b be the subalgebra of a g generated by all homogeneous elements of degree less than or equal to the group order g then in general b is not equal to a g if the characteristic of k divides g however we prove that the field of fractions quot b coincides with the field of invariants quot a g quot a g we also study various localizations and homomorphisms of modular invariant rings as tools to construct generators for a g we prove that there is always a nonzero transfer c a g of degree g such that the localization a g c can be generated by fractions of homogeneous invariants of degrees less than 2 g 1 if a sym v f g with finite dimensional f g module v then c can be chosen in degree one and 2 g 1 can be replaced by g let n denote the image of the classical noether homomorphism see the definition in the paper we prove that n contains the transfer ideal and thus can be used to calculate generators for a g by standard elimination techniques using grobner bases this provides a new construction algorithm for a g

the module structure of a group action on a polynomial ring a finiteness theorem consider a group acting on a polynomial ring over a finite field we study the polynomial ring as a module for the group and prove a structure theorem with several striking corollaries for example any indecomposable module that appears as a summand must also appear in low degree and so the number of isomorphism types of such summands is finite there are also applications to invariant theory giving a priori bounds on the degrees of the generators

vector invariants for the two dimensional modular representation of a cyclic group of prime order in this paper we study the vector invariants of the 2 dimensional indecomposable representation v2 of the cyclic group cp of order p over a field f of characteristic p f mv2 cp this ring of invariants was first studied by david richman 1990 20 who showed that the ring required a generator of degree m p 1 thus demonstrating that the result of noether in characteristic 0 that the ring of invariants of a finite group is always generated in degrees less than or equal to the order of the group does not extend to the modular case he also conjectured that a certain set of invariants was a generating set with a proof in the case p 2 this conjecture was proved by campbell and hughes 1997 in 3 later shank and wehlau 2002 in 24 determined which elements in richman s generating set were redundant thereby producing a minimal generating set we give a new proof of the result of campbell and hughes shank and wehlau giving a minimal algebra generating set for the ring of invariants f mv2 cp in fact our proof does much more we show that our minimal generating set is also a sagbi basis for f mv2 cp further our results provide a procedure for finding an explicit decomposition of f mv2 into a direct sum of indecomposable cp modules finally noting that our representation of cp on v2 is as the p sylow subgroup of sl2 fp we describe a generating set for the ring of invariants f mv2 sl2 fp and show that p m 2 p 1 is an upper bound for the noether number for m 2

the hill determinant approach to the coulomb plus linear confinement an application of the so called hill determinant method to the bound state eigenvalue problem in the elementary quarkonium potential v r a r br is described proved and illustrated for a few examples an improvement of the method which is based on an extended continued fraction formulation of the eigenvalue condition is also proposed

the eigenvalues of the bounded λx2m oscillators the even and odd parity eigenvalues of the bounded pure x2m oscillators are obtained as the roots of two equations derived explicitly

factorizing complex symmetric matrices with positive definite real and imaginary parts complex symmetric matrices whose real and imaginary parts are positive definite are shown to have a growth factor bounded by 2 for lu factorization this result adds to the classes of matrix for which it is known to be safe not to pivot in lu factorization block mathrm ldl t factorization with the pivoting strategy of bunch and kaufman is also considered and it is shown that for such matrices only 1 times 1 pivots are used and the same growth factor bound of 2 holds but that interchanges that destroy band structure may be made the latter results hold whether the pivoting strategy uses the usual absolute value or the modification employed in linpack and lapack

zero separating invariants for finite groups abstract we fix a field k of characteristic p for a finite group g denote by g and g respectively the minimal number d such that for every finite dimensional representation v of g over k and every v v g 0 or v v 0 respectively there exists a homogeneous invariant f k v g of positive degree at most d such that f v 0 let p be a sylow p subgroup of g which we take to be trivial if the group order is not divisible by p we show that g p if n g p p is cyclic we show g n g p if g is p nilpotent and p is not normal in g we show g g l where l is the smallest prime divisor of g these results extend known results in the non modular case to the modular case

on the hill determinant method explicit examples are given of the false eigenvalues of the hill determinant method the existence of which has been suspected by workers using analytic methods a simple criterion is discovered for distinguishing between false and physical eigenvalues

the rotating harmonic oscillator eigenvalue problem i continued fractions and analytic continuation the continued fraction approach to the solution of the rotating harmonic oscillator eigenvalue problem is examined in detail it is shown how one may obtain eigenvalue information only from an analytic continuation of the continued fraction accomplished with the aid of modified approximants

the noether bound in invariant theory of finite groups abstract let r be a commutative ring v a finitely generated free r module and g glr v a finite group acting naturally on the graded symmetric algebra a sym v let ag denote the minimal number m such that the ring ag of invariants can be generated by finitely many elements of degree at most m furthermore let h g be a normal subgroup such that the index g h is invertible in r in this paper we prove the inequality a g a h g h for h 1 and g invertible in r we obtain noether s bound ag g which so far had been shown for arbitrary groups only under the assumption that the factorial of the group order g is invertible in r

coinvariants and the regular representation of a cyclic p group we consider an indecomposable representation of a cyclic p group z p r over a field of characteristic p we show that the top degree of the corresponding ring of coinvariants is less than frac r 2 3r p r 2 this bound also applies to the degrees of the generators for the invariant ring of the regular representation

comments on a paper concerning the analytic continued fraction technique for bound states the author shows that the treatment of bound and confined states for a class of confinement potentials by application of the analytic continued fraction method as presented in a recent paper dalta et al ibid vol 13 p 3161 1980 is erroneous and may lead to wrong results for the eigenvalues of the relevant schrodinger equation

the hill determinant an application to a class of confinement potentials it is shown that the method of determination of the eigenvalues in terms of infinite continued fractions for a class of confinement potentials is equivalent to the vanishing of the hill determinant the problem of normalisation of the wavefunction is also discussed

decomposing modular coinvariants we consider the ring of coinvariants for a modular representation of a cyclic group of prime order p we show that the classes of the terminal variables in the coinvariants have nilpotency degree p and that the coinvariants are a free module over the subalgebra generated by these classes an incidental result we have is a description of a grobner basis for the hilbert ideal and a decomposition of the corresponding monomial basis for the coinvariants with respect to the monomials in the terminal variables

spectral properties of the biconfluent heun differential equation abstract the spectrum of zeros of the polynomial solutions of the biconfluent heun differential equation is investigated by two different methods first the spectral newton sums i e the sums of the r th powers of the zeros are given in a rigorous and recurrent way second the density of zeros i e the number of zeros per unit interval is calculated in an explicit way within the so called semiclassical or bkw approximation this is done by using a general theorem which applies to a linear second order differential equation under certain conditions applications to the radial schrodinger equations associated to some quantum mechanical systems rotating harmonic oscillator confinement potentials as well as to the bessel differential equation are also shown

a new degree bound for vector invariants of symmetric groups let r be a commutative ring v a finitely generated free rmodule and g glr v a finite group acting naturally on the graded symmetric algebra a s v let g v g denote the minirnal number m such lxhat the ring ag of invariants can be generated by finitely many elements of degree at most m for g en and v n kn the k fold direct sum of the natural perrnutation module one knows that p v n kn en n provided that n is invertible in r this was used by e noether to prove p v g igi if igl g a in this paper we prove p v n 1z en masc n kn n 1 for arbitrary commutative rings r and show equality for n ps a prime power and a z or any ring with n 1r o ou r results imply g v g maz lgl rank v lgi 1 for any ring with igi g r

a sharp bound for the castelnuovo mumford regularity of subspace arrangements abstract we show that the ideal of an arrangement of d linear subspaces of projective space is d regular in the sense of castelnuovo and mumford answering a question of b sturmfels in particular this implies that the ideal of an arrangement of d subspaces is generated in degrees less than or equal to d

on the applicability of the hill determinant and the analytic continued fraction method to anharmonic oscillators the authors finish a line of argument started in an earlier paper and show that the application of the hill determinant and the associated analytic continued fraction method for the calculation of eigenvalues of anharmonic oscillators is of dubious validity and may thus lead to erroneous results in this context some theorems concerning the eigenvalues of the lambda x4 and the doubly anharmonic oscillators are proved

the invariants of modular indecomposable representations of mathbb z _ p 2 we consider the invariant ring for an indecomposable representation of a cyclic group of order p 2 over a field mathbb f of characteristic p we describe a set of mathbb f algebra generators of this ring of invariants and thus derive an upper bound for the largest degree of an element in a minimal generating set for the ring of invariants this bound as a polynomial in p is of degree two

global degree bounds and the transfer principle for invariants abstract the main purpose of this paper is to verify a conjecture of derksen and kemper concerned with the boundedness of rings of invariants we shall make use of some results connected with the transfer principle in invariant theory these results are dealt with in some generality because of their independent interest and potential for further applications

the rotating harmonic oscillator eigenvalue problem ii analytic perturbation theory the theory of self adjoint analytic families is applied to the rotating harmonic oscillator hamiltonian in l2 0 to obtain weak and strong coupling expansions of the eigenvalues various estimates on the radius of convergence of the weak coupling expansion are obtained the strong coupling expansion is shown to be an asymptotic series which with the neglect of exponentially small terms is expressible in terms of a simple formal perturbation of the ordinary harmonic oscillator hamiltonian in l2

a free resolution of the module of logarithmic forms of a generic arrangement 1 1 the setup of this paper let v be an i dimensional vector space over a field k we assume that the characteristic of k is zero for the positive characteristic case see 4 6 let d be a generic arrangement of ii hyperplanes d is a finite family of one codimensional subspaces of v satisfying 1 n d 1 3 2 every 2 hyperplanes of d intersect only at the origin let s denote the symmetric algebra s v of the dual space v of v then 5 can be considered as the k algebra of all polynomial functions on v let 0 6 q 1 let 524 2 denote the s module of all regular q forms on v then each qq is a free s module of rank i for each he d choose cake v such that ker a h let

the noether numbers and the davenport constants of the groups of order less than 32 abstract the computation of the noether numbers of all groups of order less than thirty two is completed it turns out that for these groups in non modular characteristic the noether number is attained on a multiplicity free representation it is strictly monotone on subgroups and factor groups and it does not depend on the characteristic algorithms are developed and used to determine the small and large davenport constants of these groups for each of these groups the noether number is greater than the small davenport constant whereas the first example of a group whose noether number exceeds the large davenport constant is found answering partially a question posed by geroldinger and grynkiewicz

the noether number for the groups with a cyclic subgroup of index two abstract for the finite groups with a cyclic subgroup of index two the exact degree bound for the generators of rings of polynomial invariants is determined

on the schrodinger equations of rotating harmonic three dimensional and doubly anharmonic oscillators and a class of confinement potentials in connection with the biconfluent heun differential equation the schrodinger equations of rotating harmonic three dimensional and doubly anharmonic oscillators and a class of confinement potentials are examined simultaneously with the biconfluent heun equation the eigenvalue problems lead the authors to compute a transcendent constant function of the corresponding physical parameters

on generators of the hilbert ideal for cyclic groups in modular invariant theory abstract let g be a cyclic group of order p 2 and v be a faithful indecomposable representation of g over a field f of characteristic p we show that the hilbert ideal of the invariant ring is generated by polynomials of degree at most g whenever dim v 4 p or dim v p 2 2 p proving a conjecture of derksen and kemper in this particular case

the noether numbers for cyclic groups of prime order abstract the noether number of a representation is the largest degree of an element in a minimal homogeneous generating set for the corresponding ring of invariants we compute the noether number for an arbitrary representation of a cyclic group of prime order and as a consequence prove the 2 p 3 conjecture

quotient singularities and wild p cyclic actions in this paper we develop canonical forms for certain p cyclic automorphisms of formal power series rings over fields of characteristic p our interest in this problem is based on the geometrical question of classifying quotient singularities over fields of non zero characteristic let v be a smooth variety over an algebraically closed field k of arbitrary characteristic given a finite group g of k automorphisms of v we can consider the quotient variety v g whose points correspond to orbits of points in v the variety v g is smooth almost everywhere but may acquire singularities corresponding to points of v with non trivial isotropy one would like to classify all singularities which can arise in this manner in characteristic zero this problem has been extensively studied and a complete classification has been given by brieskorn 4 in the case that v has dimension two the key to the characteristic zero solution is a result of cartan 5 which states that it is possible to choose coordinates for v so that g acts linearly thus the study reduces to a classification of quotients by matrix group actions this reduction has the twofold advantage of greatly restricting the classes of actions to be analyzed as well as of bringing the action into a form for which explicit calculations are particularly straightforward however in non zero characteristics this method breaks down if the characteristic divides the order of g the case of wild actions it is rarely possible to linearize the action of the group consequently very little progress has been made in the study of characteristic p quotients except in certain restricted cases in which g acts linearly see for example 8 the purpose of this work is to construct normal forms for non linear wild group actions and use them to study the resulting quotient singularities we concentrate on the case that g is cyclic of order p

invariants of finite groups over fields of characteristicp abstract let v denote a finite dimensional k vector space and let g denote a finite group of k linear automorphisms of v let v m denote the direct sum of m copies of v and let g act on the symmetric algebra k v m of v m by the diagonal action on v m a result of noether implies that if char k 0 then k v m g can be generated as a k algebra by polynomials whose degrees are g no matter how large m is this paper proves that this result no longer holds when the characteristic of k divides g more precisely it is proved in this case that there is a positive number depending only on g and char k such that every set of k algebra generators of k v m g contains a generator whose degree is m

a free resolution of the module of derivations for generic arrangements let y be an i dimensional vector space over an arbitrary field f and d an arrangement of n hyperplanes h i7 in v we always assume that n i 1 for each i 1 id 1 we fix once and forever aj e v such that ker cl hi the arrangement d is generic if its every i elements intersect only at 0 or equivalently every i functionals ai are linearly independent in v let s be the symmetric algebra of v and der s der s s the s module of derivations of s the following graded s module was introduced by h terao 4 and studied in many papers on arrangements d d d 0 e der s f3 ai e sa i 1 n at the nsf cbms conference on arrangements of hyperplanes in 1988 the author conjectured that for a generic arrangement with ii 1 we have hd d 1 2 soon after that l rose and h terao 2 proved this conjecture moreover they constructed minimal free resolutions of all modules of logarithmic forms with poles along d and used the fact that d is isomorphic to one of them the aim of the present paper is to give another somewhat more straightforward construction of a minimal free resolution of d more precisely we put do d d bedjo o and note that d does mo where t is the euler derivation defined by a ci for every ci e v then we construct a minimal free resolution of do which of course has the same length as a minimal free resolution of d the notation introduced above is used throughout the paper besides we denote by l the intersection lattice of including v as the unique minimal element of l put li x l dimx i and for every xel put cqzx xn hii x hi viewing it as an arrangement in x also for every xel we put l yel ycx l yel xc y and a x

a note on the hilbert ideals of a cyclic group of prime order the hilbert ideal is the ideal generated by positive degree invariant polynomials of a finite group for a cyclic group of prime order p we show that the image of the transfer lie in the ideal generated by invariants of degree at most p 1 consequently we show that the hilbert ideal corresponding to an indecomposable representation is generated by polynomials of degree at most p confirming a conjecture of harm derksen and gregor kemper for this case

the hill determinant method in application to the sextic oscillator limitations and improvement using the sextic anharmonic oscillator a chi 2 b chi 4 c chi 6 as a testing ground it is shown why the hill determinant method has a limited range of applicability a possible improvement is suggested and discussed

on vector invariants over finite fields abstract let f denote a finite field and let s m n f denote a set of generators of the invariants of sl n f acting on mn component vectors this paper proves that if m n 1 then s m n f must contain a generator whose degree is greater than or equal to m n 2 f 1 similar results are obtained for the vector invariants of other groups of matrices with entries in f

constructing modular separating invariants abstract we consider a finite dimensional modular representation v of a cyclic group of prime order p we show that two points in v that are in different orbits can be separated by a homogeneous invariant polynomial that has degree one or p and that involves variables from at most two summands in the dual representation simultaneously we describe an explicit construction for a separating set consisting of polynomials with these properties

on the x2 x4 interaction the even and odd parity eigenvalues for the bounded potential mu x2 lambda x4 with both positive and negative values of mu and lambda 0 are obtained by the method of series solution

symmetric powers and modular invariants of elementary abelian p groups abstract let e be an elementary abelian p group of order q p n let w be a faithful indecomposable representation of e with dimension 2 over a field k of characteristic p and let v s m w with m q we prove that the rings of invariants k v e are generated by elements of degree q and relative transfers this extends recent work of wehlau 18 on modular invariants of cyclic groups of order p if m p we prove that k v e is generated by invariants of degree 2 q 3 extending a result of fleischmann sezer shank and woodcock 6 for cyclic groups of order p our methods are primarily representation theoretic and along the way we prove that for any d q with d m q s d v is projective relative to the set of subgroups of e with order m and that the sequence s d v d 0 is periodic with period q modulo summands which are projective relative to the same set of subgroups these results extend results of almkvist and fossum 1 on cyclic groups of prime order

vector invariants ofu2 fp a proof of a conjecture of richman abstract in this paper we prove a conjecture of d r richman concerning the vector invariants of the group u 2 f p let v be a vector space of dimension 2 with basis x y over the field f p and let f p x y be the symmetric algebra of v over f p if denotes a generator of u 2 f p then we may assume x x and y x y let a n be the symmetric algebra of v n we obtain an automorphism of a n of order p by using the diagonal action of extended to the whole of a n the subalgebra of polynomials left invariant by this action is called the ring of vector invariants of u 2 f p richman conjectured that these rings of invariants have certain sets of generators and gave a proof in the case p 2 we prove his conjecture for all primes