a p q oscillator realization of two parameter quantum algebras it is noted that the study of a quantum algebra sup q 2 with two independent deformation parameters p q leads to a p q oscillator realization for it the analysis is extended to the p q analogues of su 1 1 osp 2 1 and the centreless virasoro algebra the standard single parameter q deformations are obtained in the limit p q

neural networks for optimization problems with inequality constraints the knapsack problem a strategy for finding approximate solutions to discrete optimization problems with inequality constraints using mean field neural networks is presented the constraints x d 0 are encoded by x x terms in the energy function a careful treatment of the mean field approximation for the self coupling parts of the energy is crucial and results in an essentially parameter free algorithm this methodology is extensively tested on the knapsack problem of size up to 103 items the algorithm scales like nm for problems with n items and m constraints comparisons are made with an exact branch and bound algorithm when this is computationally possible n d 30 the quality of the neural network solutions consistently lies above 95 of the optimal ones at a significantly lower cpu expense for the larger problem sizes the algorithm is compared with simulated annealing and a modified linear programming approach for nonhomogeneous problems these produce good solutions whereas for the more difficult homogeneous problems the neural approach is a winner with respect to solution quality and or cpu time consumption the approach is of course also applicable to other problems of similar structure like set covering

a class of quadratic matrix algebras arising from the quantized enveloping algebra uq a2n 1 a natural family of quantized matrix algebras is introduced it includes the two best studied such located inside uq a2n 1 it consists of quadratic algebras with the same hilbert series as polynomials in n2 variables we discuss their general properties and investigate some members of the family in great detail with respect to associated varieties degrees centers and symplectic leaves finally the space of rank r matrices becomes a poisson submanifold and there is an associated tensor category of rank r matrices

on the structure of quantum super groups glq m n abstract we show that a quantum super matrix in standard format is invertible if and only if its block matrices of even entries are invertible we prove the q analog of the well known formula for the berezinian

quantum subgroups of glα β n abstract let c 0 and l n be odd with l 3 we determine all hopf algebra quotients of the quantized coordinate algebra o gl n when 1 is a primitive l th root of unity and satisfy certain mild conditions and we characterize all finite dimensional quotients when 1 is not a root of unity as a byproduct we find a new family of non semisimple and non pointed hopf algebras with non pointed duals which are quotients of o gl n

fixed points and homology of superelliptic jacobians let eta c f n rightarrow mathbb p 1 be a cyclic cover of mathbb p 1 of degree n which is totally and tamely ramified for all the ramification points we determine the group of fixed points of the cyclic covering group mathrm aut eta simeq mathbb z n mathbb z acting on the jacobian j n mathrm jac c f n for each prime ell distinct from the characteristic of the base field the tate module t ell j n is shown to be a free module over the ring mathbb z ell t sum i 0 n 1 t i we also study the subvarieties of j n and calculate the degree of the induced polarization on the new part j n mathrm new of the jacobian

remarks on classical invariant theory a uniform formulation applying to all classical groups simultaneously of the first fundamental theory of classical invariant theory is given in terms of the weyl algebra the formulation also allows skew symmetric as well as symmetric variables examples illustrate the scope of this formulation

endomorphisms of superelliptic jacobians let k be a field of characteristic zero n 5 an integer f x an irreducible polynomial over k of degree n whose galois group contains a doubly transitive simple non abelian group let p be an odd prime mathbb z zeta p the ring of integers in the pth cyclotomic field cf p yp f x the corresponding superelliptic curve and j cf p its jacobian assuming that either n p 1 or p does not divide n n 1 we prove that the ring of all endomorphisms of j cf p coincides with mathbb z zeta p the same is true if n 4 the galois group of f x is the full symmetric group s4 and k contains a primitive pth root of unity

a two parameter quantization of sl 2 1 and its finite dimensional representations the lie superalgebra sl 2 1 is quantized in its non standard simple rod system resulting in a two parameter quantum superalgebra uq1 q2 sl 2 1 when the two parameters coincide uq1 q2 sl 2 1 reduces to a one parameter dependent z2 graded hopf algebra which is algebraically equivalent but coalgebraically inequivalent to the standard uq sl 2 1 the finite dimensional irreducible representations of this two parameter quantum superalgebra are explicitly constructed when both or one of q1 and q2 are considered as indeterminates and cyclic representations are also obtained when both deformation parameters are roots of unity

poisson relations between minors and their consequences a multiplicative poisson bracket on gl n can be restricted onto sl n if the determinant det is central on gl n to decide whether the det is central a simple criterion is derived in the r matrix language with the help of a general formula for poisson brackets between determinants of arbitrary minors

parabosons virasoro type algebras and their deformations two parameter p q deformation of a parabosonic algebra underlying the two particle calogero model is considered the fock space representation bargmann fock representation coherent states spectrum generating algebra and a hidden supersymmetry are discussed the paraboson algebra induces a centreless virasoro type algebra with a doubling of the space of generators a two parameter deformation of this virasoro type algebra is studied

the quantum group suq 2 and a q analogue of the boson operators a new realisation of the quantum group suq 2 is constructed by means of a q analogue to the jordan schwinger mapping determining thereby both the complete representation structure and q analogues to the wigner and racah operators to achieve this realisation a new elementary object is defined a q analogue to the harmonic oscillator the uncertainty relation for position and momentum in a q harmonic oscillator is quite unusual

duality for the matrix quantum group glp q 2 c it is shown that the algebra up q dual to glp q 2 c is isomorphic to u pq 1 2 sl 2 c z as a commutation algebra where z is a subalgebra central in up q the subalgebra z is a hopf subalgebra of up q while the commutation subalgebra u pq 1 2 sl 2 c is not a hopf subalgebra

knapsack packing networks a knapsack packing neural network of 4n units with both low order and conjunctive asymmetric synapses is derived from a non hamiltonian energy function parallel simulations of randomly generated problems of size n in 5 10 20 are used to compare network solutions with those of simple greedy fast parallel enumerative algorithms

differential calculi on the quantum group glp q 2 the author presents a systematical study of left covariant differential calculus on the quantum group glp q 2 a two parameter deformation of the general linear group in particular the author explicitly constructs the most general bicovariant calculus it depends on a new parameter s the corresponding lie algebra of left coinvariant vector fields is a two parameter deformation of the classical lie algebra p and q appear only in the combination r pq in the limit p q to 1 one obtains non standard bicovariant differential calculi on the classical lie group

the multiparametric non standard deformation of an 1 the authors introduce a multiparametric generalization of the non standard r matrix if sl n applying the method of faddeev reshetikhin takhtajan 1990 to the r matrix they construct the quantum group associated to it the new feature of this quantum group is that upon a certain choice of the parameters there can be nilpotent elements in it which is a sign of some hidden superstructure

erratum to remarks on classical invariant theory although this paper circulated as a preprint for 12 years an error in the argument for the capelli identity went unnoticed until professor m wakayama read the paper in order to review it i thank him for his care and for bringing this error to my attention the error occurs on page 566 the result at issue that det ii is central in gyn is true and classical and the general line of argument is sufficient to prove it but the details of the computation are described incorrectly the identity on line 3 page 566 is correct but the description of the calculation drawn from it is not instead of terms of the form

multiparametric quantum deformation of the general linear supergroup in the work l d faddeev and his collaborators and subsequently v g drinfeld m jimbo s l woronowicz a new class of hopf algebras was constructed they can be considered as one parametric deformations of either group ring or the universal enveloping algebra of a simple algebraic group in this paper we define and investigate a multiparametric deformation of the general linear supergroup this is the simplest example of some general constructions described in 5 6

on the representations of glp q 2 glp q 1 mod 1 and non commutative spaces explicit realizations of the quantum groups glp q 2 and glp q 1 mod 1 corresponding to unimodular values of the deformation parameters p and q are given in terms of the canonically conjugate x p operators using the heisenberg weyl commutation relations matrix representations are also discussed some observations are made on similar realizations of the non commutative coordinate spaces on which the quantum groups act as endomorphisms

polyhedral combinatorics and neural networks the often disappointing performance of optimizing neural networks can be partly attributed to the rather ad hoc manner in which problems are mapped onto them for solution in this paper a rigorous mapping is described for quadratic 0 1 programming problems with linear equality and inequality constraints this being the most general class of problem such networks can solve the problem s constraints define a polyhedron p containing all the valid solution points and the mapping guarantees strict confinement of the network s state vector to p however forcing convergence to a 0 1 point within p is shown to be generally intractable rendering the hopfield and similar models inapplicable to the vast majority of problems a modification of the tabu learning technique is presented as a more coherent approach to general problem solving with neural networks when tested on a collection of knapsack problems the modified dynamics produced some very encouraging results

constant solutions of reflection equations and quantum groups to the yang baxter equation an additional relation can be added this is the reflection equation that appears in various places with or without spectral parameter e g in factorizable scattering on a half line integrable lattice models with nonperiodic boundary conditions noncommutative differential geometry on quantum groups etc two forms of spectral parameter independent reflection equations are studied chosen by the requirement that their solutions be comodules with respect to the quantum group coaction leaving invariant the reflection equations for a variety of known solutions of the yang baxter equation the constant solutions of the reflection equations are given various quadratic algebras defined by the reflection equations are also given explicitly

consistent multiparameter quantisation of gl n the author describes a manifold of quantum group structures on the vector space of the universal enveloping algebra of gl n and on its dual the space of polynomials in n2 variables the dimension of the manifold is n2 n 2 2

a two parameter quantization of osp 4 2 a two parameter deformation of the universal algebra of osp 4 2 is carried out yielding a z2 graded hopf algebra with a bijective antipode this hopf algebra depends on the extra parameter in both its algebraic and coalgebraic structures and also admits nontrivial finite dimensional irreps at arbitrary deformation parameters

non isogenous superelliptic jacobians let l be an odd prime let k be a field of characteristic zero with algebraic closure ka let n m 4 be integers that are not divisible by l let f x h x k x be irreducible separable polynomials of degree n and m respectively suppose that the galois group gal f of f acts doubly transitively on the set open image in new window of roots of f and that gal h acts doubly transitively on open image in new window as well let j cf l and j ch l be the jacobians of the superelliptic curves cf l yl f x and ch l yl h x respectively we prove that j cf l and j ch l are not isogenous over ka if the splitting fields of f and h are linearly disjoint over k and k contains a primitive lth root of unity

on wick algebras with braid relations the general theory of deformed commutation relations for creation and annihilation operators of particles equipped with arbitrary braid statistics statistics is studied in terms of wick algebras the construction and examples of such algebras are given

point matching using asymmetric neural networks abstract matching control points is an important step in many pattern recognition applications the matching problem is formulated under translation and rotation as a 0 1 integer programming problem and an artificial neural network is proposed for approximately solving it the solution to the 0 1 integer programming problem is obtained as the high gain limit point of the continuous network the network is capable of handling distortion and noise and can use both interpoint distance information and feature properties associated with the points the results obtained by the network compare favourably with that of the relaxation method of ton and jain ieee trans geosci remote sensing 27 642 651 1989

multiparameter deformations of gl n and jordan schwinger constructions the covariant differential calculus on the quantum hyperplane acted on by glx qij n is interpreted as a deformed heisenberg algebra holomorphically realized on a polynomial basis and the corresponding quantum algebra is constructed through the jordan schwinger construction for n 3 this quantum algebra is shown to give quadratic constraints among the generators however for n 2 this construction is elevated to a hopf algebra by finding simple comultiplication rules this is seen to be equivalent to the woronowicz type two parameter deformation of schirrmacher et al z phys c 49 317 1991 as well as the drinfeld jimbo deformation v g drinfeld in proceedings of the international congress of mathematicians berkeley 1986 academic new york vol 1 pp 798 820 m jimbo lett math phys 10 63 1985 m jimbo lett math phys 11 247 1986

del pezzo surfaces of degree 1 and jacobians we construct absolutely simple jacobians of nonhyperelliptic genus 4 curves using del pezzo surfaces of degree 1

noncommutative geometry for arbitrary braidings the noncommutative geometry corresponding to arbitrary braidings is studied in a purely algebraical way the new and original proposal of the geometry is given in terms of the theory of braided monoidal categories the quantum oscillator is considered as an example of application of our geometry

the noncommutative space of stochastic diffusion systems in the matrix product ground states approach to n species diffusion processes the stationary probability distribution is expressed as a matrix product state with respect to a quadratic algebra determined by the dynamics of the process we show that the quadratic algebra defines a noncommutative space with a glq n quantum group action as its symmetry boundary processes account for the appearance of parameter dependent linear terms in the algebraic relations we argue that for systems with boundary conditions the diffusion algebras are also obtained either by a shift of basis in the n dimensional quantum plane or by an appropriate change of basis in a lower dimensional one which leads to a reduction of the glq n symmetry

a two parametric quantization of sl 2 a two parametric solution of the constant yang baxter equation is used for the construction of structures corresponding to quantized groups the corresponding hopf algebra is generated by five generators

two parameter differential calculus on the h superplane we introduce a noncommutative differential calculus on the two parameter h superplane via a contraction of the p q superplane we manifestly show that the differential calculus is covariant under glh1 h2 1 1 transformations we also give a two parameter deformation of the 1 1 dimensional phase space algebra

hodge classes on certain types of abelian varieties the hodge conjecture for a states that such classes are algebraic in the sense that they are q linear combinations of cohomology classes of algebraic subvarieties of codimension p on a this statement is well known for p 1 lefschetz and it is trivially true for p 0 hence for a given a it holds for allp if the ring formed by the hodge classes under cup product is generated by the hodge classes of codimension 1 we thus have a simple criterion for the hodge conjecture to be true for a which we call the 1 1 criterion responding to a question posed by tate 14 p 107 mumford found that the 1 1 criterion while true for all a such that dim a c 3 is not necessarily satisfied by abelian varieties of dimension 4 although mumford s counterexample involved an abelian variety of cm type 51 weil 15 has stressed that its most important feature is the action of an imaginary quadratic field k on a in such a way that lie a becomes a free k x c module abelian varieties of precisely this type play an important role in deligne s proof 2 that hodge cycles on an arbitrary abelian variety are absolutely hodge there is speculation that the hodge conjecture is false at least in general for such abelian varieties but no method is available at present for settling this question either way indeed one knows no example of an algebraic variety for which the hodge conjecture is false and only scattered examples shioda 12 of

a simple derivation of the quantum clebsch gordan coefficients for su 2 q to su 2 q the quantum deformation of su 2 the van der waerden method for calculating the clebsch gordan cg coefficients is genaralized the polynomial basis for irreducible representations of su 2 q the relevant polynomial invariants and the deduction of the q analog of the cg coefficients are given