representations of the quantum group suq 2 and the little q jacobi polynomials abstract in this paper we study the finite dimensional unitary representations of the quantum group suq 2 then we obtain the peter weyl theorem for suq 2 and the matrix elements of these unitary representations are explicitly expressed in terms of the little q jacobi polynomials which are known as q analogues of orthogonal polynomials using these expressions the orthogonality relations of these polynomials are obtained in terms of the haar measure on the quantum group suq 2

new constructive methods in classical ideal theory abstract the recent development of computer algebra allows us to take up problems of classical ideal theory for which constructive solutions by nearly unfeasible algorithms exist like the problem of deciding whether a polynomial belongs to a given ideal g hermann math ann 95 1926 736 788 even macaulay s methods by using h bases are not satisfactory at least when considering modules the concept of grobner bases introduced in computer algebra by buchberger allows us to find feasible algorithms for many problems of classical ideal theory by transforming them into some low dimensional systems of linear equations our aim is to demonstrate the effectivity of this new concept considering only polynomial rings p k x 1 x n k a field we generalize in part i the notion of grobner bases to modules present some equivalent properties characterizing these bases and show their good computational qualities in part ii we construct starting with grobner bases of ideals explicit free solutions where all the bases of submodules under consideration are also grobner bases and present a reduction strategy for obtaining shorter resolutions where all the bases have also favourable properties t bases part iii deals with applications we propose a method for constructing minimal resolutions starting with the resolutions of part ii and present a convenient way for computing the hilbert function

localized patterns for the quintic complex swift hohenberg equation abstract we show using numerical simulations that a variety of localized patterns arise in a model equation the quintic swift hohenberg equation with complex coefficients we demonstrate that various sizes of localized standing wave patterns are possible when the imaginary part of the complex coefficient is small localized traveling waves as well as localized standing waves with a fixed size are observed when the imaginary part is rather large we also present stable localized patterns in two spatial dimensions and study their interaction

dynamical grobner bases over dedekind rings abstract in this paper we extend the notion of dynamical grobner bases introduced by the second author to dedekind rings with zero divisors as an application we dynamically solve the ideal membership problem and compute a generating set for the syzygy module over multivariate polynomial rings with coefficients in dedekind rings we also give a partial positive answer to a conjecture about grobner rings

on the realizability of biorthogonal m dimensional two band filter banks we show an algebraic approach for the design of ladder structures for causal biorthogonal filter banks the key ingredient of the approach is known in literature as euclid s algorithm using this algorithm we derive some strong result on the design freedom for ladder structures in particular we show that the dimensionality of the problem plays an important role we end by with some conjectures concerning the extensions to multichannel and noncausal filter banks

coarsening in potential and nonpotential models of oblique stripe patterns we study the coarsening of two dimensional oblique stripe patterns by numerically solving potential and nonpotential anisotropic swift hohenberg equations close to onset all models exhibit isotropic coarsening with a single characteristic length scale growing in time as t 1 2 further from onset the characteristic lengths along the preferred directions x and y grow with different exponents close to 1 3 and 1 2 respectively in this regime one dimensional dynamical scaling relations hold we draw an analogy between this problem and model a in a stationary modulated external field for deep quenches nonpotential effects produce a complicated dislocation dynamics that can lead to either arrested or faster than power law growth depending on the model considered in the arrested case small isolated domains shrink down to a finite size and fail to disappear a comparison with available experimental results for electroconvection in nematics is presented

a zero dimensional valuation ring is 1 grobner abstract in this study we consider the following conjecture and open problem 1 a valuation ring r is 1 grobner if and only if r is an archimedean ring 2 is a 1 grobner valuation ring coherent first we prove that a zero dimensional valuation ring is 1 grobner furthermore we give a positive answer to 1 and a negative answer to 2 finally we present an example of a valuation ring that is 1 grobner but not coherent or noetherian

computing syzygies over v x1 xk v a valuation domain abstract we give an algorithm for computing the v saturation of any finitely generated submodule of v x 1 x k m k n m n where v is a valuation domain our algorithm is based on a notion of echelon form which ensures its correctness the proposed algorithm terminates when two hilbert series on the quotient field and the residue field of v coincide as application our algorithm computes syzygies over v x 1 x k

operations on orbits of unimodular vectors abstract taking the m power of an entry is a well defined operation on the unimodular vectors in an modulo addition operations if n is at least 3 for an arbitrary commutative ring a and any integer m

stability of rolls and hexagonal patterns in non potential systems abstract the stability of rolls hexagons and non equilateral hexagonal patterns is considered in the framework of a system of amplitude equations containing non potential resonant quadratic terms the stability analysis of roll patterns reveals a new kind of instability which can generate steady non equilateral hexagons noise sensitive transitions between roll patterns or periodic oscillations the busse balloons for equilateral and non equilateral hexagons are calculated

stable localized solutions of arbitrary length for the quintic swift hohenberg equation abstract we show that localized solutions of arbitrary length are stable over a finite parameter interval of subcritical values for the quintic swift hohenberg equation with a destabilizing cubic term this equation is thought to model a weakly hysteretic transition to stationary patterns we argue that the stabilization of the localized states of arbitrary length can be traced back to the interaction between long wavelength modulations and spatial variations on the length scale of one unit cell these results are critically compared with other known mechanisms to stabilize localized states in various situations we also discuss for which experimental systems the states predicted here could be detected including e g the stationary onset of binary fluid convection

non intersecting paths random tilings and random matrices we investigate certain measures induced by families of non intersecting paths in domino tilings of the aztec diamond rhombus tilings of an abc hexagon a dimer model on a cylindrical brick lattice and a growth model the measures obtained e g the krawtchouk and hahn ensembles have the same structure as the eigenvalue measures in random matrix theory like gue which can in fact can be obtained from non intersecting brownian motions the derivations of the measures are based on the karlin mcgregor or lindstrom gessel viennot method we use the measures to show some asymptotic results for the models

a method to discern complexity in two dimensional patterns generated by coupled map lattices complex patterns generated by the time evolution of a one dimensional digitalized coupled map lattice are quantitatively analyzed a method for discerning complexity among different patterns is implemented the quantitative results indicate two zones in parameter space where the dynamics show the most complex patterns these zones are located on the two edges of an absorbent region where the system displays spatio temporal intermittency

noether normalization theorem and dynamical grobner bases over bezout domains of krull dimension 1 abstract we propose a new version of noether normalization theorem over bezout domains of krull dimension one the ring z of integers as main example we also show that one can avoid branching when computing dynamical grobner bases over a bezout domain of krull dimension one

secondary turing type instabilities due to strong spatial resonance we investigate the dynamics of pattern forming systems in large domains near a codimension two point corresponding to a strong spatial resonance where competing instabilities with wavenumbers in th

exact soliton solutions of the one dimensional complex swift hohenberg equation abstract using painleve analysis the hirota multi linear method and a direct ansatz technique we study analytic solutions of the 1 1 dimensional complex cubic and quintic swift hohenberg equations we consider both standard and generalized versions of these equations we have found that a number of exact solutions exist to each of these equations provided that the coefficients are constrained by certain relations the set of solutions include particular types of solitary wave solutions hole dark soliton solutions and periodic solutions in terms of elliptic jacobi functions and the weierstrass function although these solutions represent only a small subset of the large variety of possible solutions admitted by the complex cubic and quintic swift hohenberg equations those presented here are the first examples of exact analytic solutions found thus far

correlation kernels for discrete symplectic and orthogonal ensembles in 49 h widom derived formulae expressing correlation functions of orthogonal and symplectic ensembles of random matrices in terms of orthogonal polynomials we obtain similar results for discrete ensembles with rational discrete logarithmic derivative and compute explicitly correlation kernels associated to the classical meixner and charlier orthogonal polynomials

criticality in cellular automata abstract using recent results obtained for the transition to turbulence via spatiotemporal intermittency in extended dynamical systems critical cellular automata rules are built thanks to a systematic procedure the continuous phase transition observed in a coupled map lattice is translated into a sequence of cellular automata rules for which the critical properties of the original system are reproduced as precisely as desired it is shown that criticality as understood from the point of view of statistical mechanics is intimately related to the various characteristics of wolfram s class iv rules this suggests in turn the picture of a critical surface in the space of rules and provides the basis for a discussion of the problem of classification schemes we review recent cellular automata studies where questions linked to criticality arise and argue that they are unified in light of our results leaving the relationship between computational and statistical characterizations of critical rules as the central problem for future studies on this subject

a constructive comparison of the rings r x and r x and application to the lequain simis induction theorem we constructively prove that for any ring r with krull dimension d the ring r x locally behaves like the ring r x or a localization of a polynomial ring of type s 1r x with s a multiplicative subset of r such that the krull dimension of s 1r is d 1 as an application we give a simple and constructive proof of the lequain simis induction theorem which is an important variation of the quillen induction theorem

a bilateral extension of the q selberg integral a multi dimensional bilateral q series extending the q selberg integral is studied using concepts of truncation regularization and connection formulae following aomoto s method which involves regarding theq series as a solution of aq difference equation fixed by its asymptotic behavior an infinite product evaluation is obtained the q difference equation is derived applying the shifted symmetric polynomials introduced by knop and sahi as a special case of the infinite product formula askey evans s q selberg integral evaluation and its generalization by tarasov varchenko and stokman is reclaimed and an explanation in the context of aomoto s setting is thus provided 2010 mathematical subject classification primary 33d15 33d67 secondary 39a13

oscillon type structures and their interaction in a swift hohenberg model abstract motivated by the observation of localized circular excitations oscillons in vertically vibrated granular layers p umbanhowar f melo h swinney nature 382 1996 793 we numerically investigate an extension of a swift hohenberg model that exhibits a subcritical transition to square patterns for sufficiently strong damping of the basic state stable oscillon structures are found the localization mechanism is quite general and is due to non adiabatic effects bound structures of oscillons of equal and opposite polarity are found with bound states of like polarity being less robust much of the phenomena are consistent with the experimental observations and suggest that oscillons are not specific to patterns in granular media or to parametrically driven systems experimental tests are suggested that would determine whether this minimal framework is sufficient to describe the phenomena

the grobner ring conjecture in one variable we prove that a valuation domain v has krull dimension 1 if and only if for every finitely generated ideal i of v x the ideal generated by the leading terms of elements of i is also finitely generated this proves the grobner ring conjecture in one variable the proof we give is both simple and constructive the same result is valid for semihereditary rings

an algorithmic proof of suslin s stability theorem for polynomial rings abstract let k be a field then gaussian elimination over k and the euclidean division algorithm for the univariate polynomial ring k x allow us to write any matrix in sln k or sln k x n 2 as a product of elementary matrices suslin s stability theorem states that the same is true for sln k xl xm with n 3 and m 1 in this paper we present an algorithmic proof of suslin s stability theorem thus providing a method for finding an explicit factorization of a given polynomial matrix into elementary matrices grobner basis techniques may be used in the implementation of the algorithm

algorithms for the quillen suslin theorem abstract we give an algorithm for computing a free basis of a projective c x1 xn module which is presented as the image kernel or cokernel of a polynomial matrix our method can be implemented using grobner bases and it provides an elementary constructive proof of the quillen suslin theorem

localized patterns and hole solutions in one dimensional extended systems abstract the existence stability properties dynamical evolution and bifurcation diagram of localized patterns and hole solutions in one dimensional extended systems are studied from the point of view of front interactions an adequate envelope equation is derived from a prototype model that exhibits these particle like solutions this equation allows us to obtain an analytical expression for the front interaction which is in good agreement with numerical simulations

simultaneous influence of additive and multiplicative noise on stationary dissipative solitons we investigate the simultaneous influence of spatially homogeneous multiplicative noise as well as of spatially correlated additive noise on the formation of localized patterns in the framework of the cubic quintic complex ginzburg landau equation depending on the ratio between the strength of additive and multiplicative noise we find a number of distinctly different types of behavior including explosions collapse filling in and spatio temporal disorder as well as intermittent behavior of all types listed techniques used to analyze the results include snapshots x t plots and plots of the spatially and temporally averaged amplitude as a function of the strength of multiplicative noise while keeping the strength of additive noise fixed typically 50 realizations are used for averaging to obtain the corresponding data points in these diagrams for the widths of these distribution as a function of additive noise we obtain a linear decrease in the limit of fairly large but fixed values of the multiplicative noise to summarize our findings concisely we show three dimensional plots of the mean pattern amplitude and the generalized susceptibility as a function of the strengths of additive and multiplicative noise we critically compare the results of our investigations with those obtained in the two limiting cases of purely additive and of purely multiplicative noise

interpretations of some parameter dependent generalizations of classical matrix ensembles two types of parameter dependent generalizations of classical matrix ensembles are defined by their probability density functions pdfs as the parameter is varied one interpolates between the eigenvalue pdf for the superposition of two classical ensembles with orthogonal symmetry and the eigenvalue pdf for a single classical ensemble with unitary symmetry while the other interpolates between a classical ensemble with orthogonal symmetry and a classical ensemble with symplectic symmetry we give interpretations of these pdfs in terms of probabilities associated to the continuous robinson schensted knuth correspondence between matrices with entries chosen from certain exponential distributions and non intersecting lattice paths and in the course of this probability measures on partitions and pairs of partitions are identified the latter are generalized by using macdonald polynomial theory and a particular continuum limit the jacobi limit of the resulting measures is shown to give pdfs related to those appearing in the work of anderson on the selberg integral and also in some classical work of dixon by interpreting anderson s and dixon s work as giving the pdf for the zeros of a certain rational function it is then possible to identify random matrices whose eigenvalue pdfs realize the original parameter dependent pdfs this line of theory allows sampling of the original parameter dependent pdfs their dixon anderson type generalizations and associated marginal distributions from the zeros of certain polynomials defined in terms of random three term recurrences

instabilities and spatio temporal chaos in hexagon patterns with rotation abstract the dynamics of hexagon patterns in rotating systems are investigated within the framework of modified swift hohenberg equations that can be considered as simple models for rotating convection with broken up down symmetry e g non boussinesq rayleigh benard or marangoni convection in the weakly nonlinear regime a linear stability analysis of the hexagons reveals long and short wave instabilities which can be steady or oscillatory the oscillatory short wave instabilities can lead to stable hexagon patterns that are periodically modulated in space and time or to a state of spatio temporal chaos with a fourier spectrum that precesses on average in time the chaotic state can exhibit bistability with the steady hexagon pattern there exist regimes in which the steady hexagon patterns are unstable at all wave numbers

a generic mechanism for spatiotemporal intermittency a simple mechanism for spatiotemporal intermittency is described it occurs in systems that exhibit bistability between a homogeneous stationary state and an oscillatory one close to parameter values where the oscillation disappears through an andronov homoclinic bifurcation c 1998 elsevier science b v all rights reserved

an algorithm for unimodular completion over noetherian rings abstract we give an algorithm for the well known result asserting that if r is a polynomial ring in a finite number of variables over a noetherian ring a of krull dimension d then for n max 3 d 2 sl n r acts transitively on um n r for technical reasons we demand that the noetherian ring a has a theory of grobner bases and contains an infinite set e y 1 y 2 such that y i y j a for each i j the most important guiding examples are affine rings k x 1 x m i and localizations of polynomial rings s 1 k x 1 x m with k an infinite field moreover we give an algorithmic proof of suslin s stability theorem over these rings for the purpose to prepare the ground for this algorithmic generalizations of the quillen suslin theorem corresponding to the particular case a is a field we will give in the first section a constructive proof of an important lemma of suslin which is the only nonconstructive step in suslin s second elementary solution of serre s conjecture this lemma says that for a commutative ring a if v 1 x v n x a x where v 1 is monic and n 3 then there exist 1 l e n 1 a x such that res v 1 e 1 1 v 2 v n t res v 1 e 1 l v 2 v n t a thanks to this constructive proof suslin s second proof of serre s conjecture becomes fully constructive

on the relation between orthogonal symplectic and unitary matrix ensembles for the unitary ensembles of n n hermitian matrices associated with a weight function w there is a kernel expressible in terms of the polynomials orthogonal with respect to the weight function which plays an important role for the orthogonal and symplectic ensembles of hermitian matrices there are 2 2 matrix kernels usually constructed using skew orthogonal polynomials which play an analogous role these matrix kernels are determined by their upper left hand entries we derive formulas expressing these entries in terms of the scalar kernel for the corresponding unitary ensembles we also show that whenever w w is a rational function the entries are equal to the scalar kernel plus some extra terms whose number equals the order of w w general formulas are obtained for these extra terms we do not use skew orthogonal polynomials in the derivations

a note on the stable decomposition of skew symmetric matrices computationally stable decompositions for skew symmetric matrices which take advantage of the skew symmetry in order to halve the work and storage are presented for solving linear systems of equations

analytical approach to localized structures in a simple reaction diffusion system we study from an analytical point of view a simple reaction diffusion model which admits stable oscillating localized structures as a consequence of the coexistence between a stable limit cycle and a stable fixed point using a generalized matching approach we are able to find approximate analytical expressions for localized oscillating structures in this reaction diffusion model capturing all the essential ingredients of these breathing particlelike solutions

intermittency in the transition to turbulence it is commonly known that the intermittent transition from laminar to turbulent flow in pipes occurs because at intermediate values of a prescribed pressure drop a purely laminar flow offers too little resistance but a fully turbulent one offers too much we propose a phenomenological model of the flow which is able to explain this in a quantitative way through a hysteretic transition between laminar and turbulent states characterized by a disturbance amplitude variable that satisfies a natural type of evolution equation the form of this equation is motivated by physical observations and derived by an averaging procedure and we show that it naturally predicts disturbances having the characteristics of slugs and puffs the model predicts oscillations similar to those which occur in intermittency in pipe flow but it also predicts that stationary biphasic states can occur in sufficiently short pipes

influence of dirichlet boundary conditions on dissipative solitons in the cubic quintic complex ginzburg landau equation we investigate the influence of dirichlet boundary conditions on various types of localized solutions of the cubic quintic complex ginzburg landau equation as it arises as an envelope equation near the weakly inverted onset of traveling waves we find that various types of nonmoving pulses and holes can accommodate dirichlet boundary conditions by having for holes two halves of a ensuremath pi hole at each end of the box moving pulses of fixed shape as they arise for periodic boundary conditions are replaced by a nonmoving asymmetric pulse which has half a ensuremath pi hole at the end of the box in the original moving direction to guarantee that dirichlet boundary conditions are met moving breathing pulses as they arise for periodic boundary conditions propagate toward one end of the container and stop moving while the breathing persists indefinitely finally breathing and moving holes are replaced by two nonbreathing half ensuremath pi holes at each end of the container and one hump in the bulk

defect mediated snaking a new growth mechanism for localized structures stationary spatially localized patterns in parametrically driven systems are studied focusing on the 2 1 and 1 1 resonance tongues as described by the forced complex ginzburg landau equation homoclinic snaking is identified in both cases and the nature of the growth of the localized structures along the snaking branches is described the structures grow from a central defect that inserts new rolls on either side while pushing existing rolls outwards this growth mechanism differs fundamentally from that found in other systems exhibiting homoclinic snaking in which new rolls are added at the fronts that connect the structure to the background homogeneous state

asymptotics of skew orthogonal polynomials exact integral expressions of the skew orthogonal polynomials involved in orthogonal 1 and symplectic 4 random matrix ensembles are obtained the even rank skew orthogonal polynomials are average characteristic polynomials of random matrices from there asymptotics of the skew orthogonal polynomials are derived

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

localized states in the generalized swift hohenberg equation the swift hohenberg equation with quadratic and cubic nonlinearities exhibits a remarkable wealth of stable spatially localized states the presence of these states is related to a phenomenon called homoclinic snaking numerical computations are used to illustrate the changes in the localized solution as it grows in spatial extent and to determine the stability properties of the resulting states the evolution of the localized states once they lose stability is illustrated using direct simulations in time i introduction ever since the observation that the subcritical complex ginzburg landau equation exhibits stable spatially localized states 1 there has been considerable interest in the properties of these states the presence of these states has important consequences for other systems described by partial differential equations on the line since the ginzburg landau equation describes the behavior of such systems near bifurcation from the trivial state of the system specifically the complex ginzburg landau equation describes the evolution of a long wavelength oscillatory instability as well as oscillatory instabilities at finite wavelength in systems with broken reflection symmetry in contrast near a steady state bifurcation with finite wavelength the evolution of the instability is described by the real ginzburg landau equation and this equation possesses only unstable spatially localized states it is of interest therefore to examine what happens to these unstable states at larger amplitude where the real ginzburg landau equation no longer provides an adequate description of the system in this paper we show that the localized states can become stable at such amplitudes and indeed that there is a large multiplicity of coexisting stable localized states under very general conditions we are able to relate the existence of these states to a phenomenon sometimes called homoclinic snaking that is well known from the theory of reversible systems with 1 1 resonance and use this theory to construct a large number of such states the stability properties of these states are also determined and the evolution of nonstationary localized states is studied by numerical integration in time it is an interesting fact that closely related phenomena have already been described in several areas involving pattern formation the theory was originally developed in the context of water waves where localized states have been studied by moving into a reference frame of the waves and converting the problem into an ordinary differential equation ode the resulting localized states are called solitary waves and in some cases turn out to be solitons kirchgassner 2 has pioneered a successful approach to this type of problem that led to a number of advances in this area specifically the ode is viewed as a dynamical system in space and localized states are sought as homoclinic orbits connecting the trivial state to itself whether such orbits are possible depends in part on the stability properties of the trivial state eigenvalues with positive real part indicate that a nontrivial state can grow from x while eigenvalues with negative real part indicate that such a state may return under appropriate conditions back to the trivial state as x the spectrum of the linearization about the trivial state is influenced by spatial symmetries of the system in many cases and in particular in the case considered here the ode is reversible as a result the bifurcations that are encountered as a parameter is increased are nongeneric in the present case the spatial dynamics of the system near the trivial state turn out to be described by the reversible 1 1 resonance the unfolding of this resonance has been worked out in detail by iooss and peroueme 3 and can be used to understand the appearance of a variety of homoclinic orbits in this system and hence of localized states with different spatial structure

global feedback control for pattern forming systems global feedback control of pattern formation in a wide class of systems described by the swift hohenberg sh equation is investigated theoretically by means of stability analysis and numerical simulations two cases are considered i feedback control of the competition between hexagon and roll patterns described by a supercritical sh equation and ii the use of feedback control to suppress the blowup in a system described by a subcritical sh equation in case i it is shown that feedback control can change the hexagon and roll stability regions in the parameter space as well as cause a transition from up to down hexagons and stabilize a skewed mixed mode hexagonal pattern in case ii it is demonstrated that feedback control can suppress blowup and lead to the formation of spatially localized patterns in the weakly nonlinear regime the effects of a delayed feedback are also investigated for both cases and it is shown that delay can induce temporal oscillations as well as blowup

ideal membership in polynomial rings over the integers we will reproduce a proof using hermann s classical method in section 3 below note that the computable character of this bound reduces the question of whether fo g fi fn for given fj g f x to solving an enormous system of linear equations over f hence in this way one obtains a naive algorithm for solving the ideal membership problem for f x provided f is given in some explicitly computable manner later buchberger in his ph d thesis 1965 introduced the important concept of a gr bner basis and gave an algorithm for deciding ideal membership for f x which is widely used today see e g 6 the doubly exponential nature of above is essentially unavoidable as a family of examples due to mayr and meyer 27 shows in fact they prove that ideal membership for q x is exponential space hard the amount of space needed by any algorithm to decide ideal membership for q x or z x grows exponentially in the size of the input if we restrict to o n ofa special form often dramatic

classical skew orthogonal polynomials and random matrices skew orthogonal polynomials arise in the calculation of the n point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry in particular the distribution functions are completely determined by a certain sum involving the skew orthogonal polynomials in the case that the eigenvalue probability density function involves a classical weight function explicit formulas for the skew orthogonal polynomials are given in terms of related orthogonal polynomials and the structure is used to give a closed form expression for the sum this theory treates all classical cases on an equal footing giving formulas applicable at once to the hermite laguerre and jacobi cases

path integral representation for a stochastic sandpile we introduce an operator description for a stochastic sandpile model with a conserved particle density and develop a path integral representation for its evolution the resulting exact expression for the effective action highlights certain interesting features of the model for example it is nominally massless and the dynamics is via cooperative diffusion using the path integral formalism we construct a diagrammatic perturbation theory yielding a series expansion for the activity density in powers of time

an efficient estimate based on fft in topological verification method in this paper localized patterns of the quintic swift hohenberg equation are studied a numerical verification method with the conley index theory developed in zgliczynski and mischaikow rigorous numerics for partial differential equations the kuramoto sivashinsky equation found comput math 1 2001 255 288 is used in order to prove these patterns a new technique to efficiently obtain estimates for nonlinear terms is presented the key idea is based on the pseudo spectral method it is shown that this technique is inevitable for the verification of the localized patterns

corrigendum to dynamical grobner bases j algebra 301 2 2006 447 458 and to dynamical grobner bases over dedekind rings j algebra 324 1 2010 12 24 abstract in this corrigendum i correct a mild error in a technical lemma of yengui 2006 5 this error affected some results and examples in hadj kacem and yengui 2010 3 and yengui 2006 5

classical discrete symplectic ensembles on the linear and exponential lattice skew orthogonal polynomials and correlation functions the eigenvalue probability density function for symplectic invariant random matrix ensembles can be generalized to discrete settings involving either a linear or an exponential lattice the corresponding correlation functions can be expressed in terms of certain discrete and q skew orthogonal polynomials respectively we give a theory of both of these classes of polynomials and the correlation kernels determining the correlation functions in the cases in which the weights for the corresponding discrete unitary ensembles are classical crucial for this are certain difference operators which relate the relevant symmetric inner products to the skew symmetric ones and have a tridiagonal action on the corresponding discrete or q orthogonal polynomials

correlations for superpositions and decimations of laguerre and jacobi orthogonal matrix ensembles with a parameter a superposition of a matrix ensemble refers to the ensemble constructed from two independent copies of the original while a decimation refers to the formation of a new ensemble by observing only every second eigenvalue in the cases of the classical matrix ensembles with orthogonal symmetry it is known that forming superpositions and decimations gives rise to classical matrix ensembles with unitary and symplectic symmetry the basic identities expressing these facts can be extended to include a parameter which in turn provides us with probability density functions which we take as the definition of special parameter dependent matrix ensembles the parameter dependent ensembles relating to superpositions interpolate between superimposed orthogonal ensembles and a unitary ensemble while the parameter dependent ensembles relating to decimations interpolate between an orthogonal ensemble with an even number of eigenvalues and a symplectic ensemble of half the number of eigenvalues by the construction of new families of biorthogonal and skew orthogonal polynomials we are able to compute the corresponding correlation functions both in the finite system and in various scaled limits specializing back to the cases of orthogonal and symplectic symmetry we find that our results imply different functional forms to those known previously

partial skew orthogonal polynomials and related integrable lattices with pfaffian tau functions skew orthogonal polynomials sops arise in the study of the n point distribution function for orthogonal and symplectic random matrix ensembles motivated by the average of characteristic polynomials of the bures random matrix ensemble studied in forrester and kieburg commun math phys 342 1 151 187 2016 we propose the concept of partial skew orthogonal polynomials psops as a modification of the sops and then the psops with a variety of special skew symmetric kernels and weight functions are addressed by considering appropriate deformations of the weight functions we derive nine integrable lattices in different dimensions as a consequence the tau functions for these systems are shown to be expressed in terms of pfaffians and the wave vectors psops in fact the tau functions also admit the multiple integral representations among these integrable lattices some of them are known while the others are novel to the best of our knowledge in particular one integrable lattice is related to the partition function of the bures ensemble besides we derive a discrete integrable lattice which can be used to compute certain vector pade approximants this yields the first example regarding the connection between integrable lattices and generalised inverse vector valued pade approximants about which hietarinta joshi and nijhoff pointed out that this field remains largely to be explored in the recent monograph hietarinta et al in discrete systems and integrability vol 54 cambridge university press cambridge 2016 section 4 4

stably free modules over of rank dim are free we prove that for any finite dimensional ring r and n dim r 2 the group e n r x acts transitively on um n r x in particular we obtain that for any finite dimensional ring r all finitely generated stably free modules over r x of rank dim r are free this result was only known for noetherian rings the proof we give is short simple and constructive

discrete orthogonal polynomial ensembles and the plancherel measure we consider discrete orthogonal polynomial ensembles which are discrete analogues of the orthogonal polynomial ensembles in random matrix theory these ensembles occur in certain problems in combinatorial probability and can be thought of as probability measures on partitions the meixner ensemble is related to a two dimensional directed growth model and the charlier ensemble is related to the lengths of weakly increasing subsequences in random words the krawtchouk ensemble occurs in connection with zig zag paths in random domino tilings of the aztec diamond and also in a certain simplifled directed flrst passage percolation model we use the charlier ensemble to investigate the asymptotics of weakly increasing subsequences in random words and to prove a conjecture of tracy and widom as a limit of the meixner ensemble or the charlier ensemble we obtain the plancherel measure on partitions and using this we prove a conjecture of baik deift and johansson that under the plancherel measure the distribution of the lengths of the flrst k rows in the partition appropriately scaled converges to the asymptotic joint distribution for the k largest eigenvalues of a random matrix from the gaussian unitary ensemble in this problem a certain discrete kernel which we call the discrete bessel kernel plays an important role

directed percolation with an absorbing boundary abstract we consider directed percolation with an absorbing boundary in 1 1 and 2 1 dimensions the distribution of cluster lifetimes and sizes depend on the boundary the new scaling exponents can be related to the exponents characterizing standard directed percolation 1 1 dimension in addition we investigate the backbone cluster and red bonds and calculate the distribution of living sites along the absorbing boundary

snakes and ladders in an inhomogeneous neural field model continuous neural field models with inhomogeneous synaptic connectivities are known to support traveling fronts as well as stable bumps of localized activity we analyze stationary localized structures in a neural field model with periodic modulation of the synaptic connectivity kernel and find that they are arranged in a snakes and ladders bifurcation structure in the case of heaviside firing rates we construct analytically symmetric and asymmetric states and hence derive closed form expressions for the corresponding bifurcation diagrams we show that the ideas proposed by beck and co workers to analyze snaking solutions to the swift hohenberg equation remain valid for the neural field model even though the corresponding spatial dynamical formulation is non autonomous we investigate how the modulation amplitude affects the bifurcation structure and compare numerical calculations for steep sigmoidal firing rates with analytic predictions valid in the heaviside limit

the grobner ring conjecture in the lexicographic order case we prove that a valuation domain mathbf v has krull dimension le 1 if and only if for any n fixing the lexicographic order as monomial order in mathbf v x 1 ldots x n for every finitely generated ideal i of mathbf v x 1 ldots x n the ideal generated by the leading terms of the elements of i is also finitely generated this proves the grobner ring conjecture in the lexicographic order case the proof we give is both simple and constructive the same result is valid for prufer domains as a scoop contrary to the common idea that grobner bases can be computed exclusively on noetherian ground we prove that computing grobner bases over mathbf r x 1 ldots x n where mathbf r is a prufer domain has nothing to do with noetherianity it is only related to the fact that the krull dimension of mathbf r is le 1

syzygies of canonical curves and special linear series 0 introduction 105 1 line bundles on scrolls 109 2 scrolls and pencils 113 3 syzygies of a subvariety of a scroll 115 4 canonical curves 118 5 on the equation of rational surfaces 121 6 d gonal curves for d 5 126 7 proof of the main result 131 references 136

nonintersecting brownian motions on the unit circle we consider an ensemble of n nonintersecting brownian particles on the unit circle with diffusion parameter n 1 2 which are conditioned to begin at the same point and to return to that point after time t but otherwise not to intersect there is a critical value of t which separates the subcritical case in which it is vanishingly unlikely that the particles wrap around the circle and the supercritical case in which particles may wrap around the circle in this paper we show that in the subcritical and critical cases the probability that the total winding number is zero is almost surely 1 as n and in the supercritical case that the distribution of the total winding number converges to the discrete normal distribution we also give a streamlined approach to identifying the pearcey and tacnode processes in scaling limits the formula of the tacnode correlation kernel is new and involves a solution to a lax system for the painleve ii equation of size 2 2 the proofs are based on the determinantal structure of the ensemble asymptotic results for the related system of discrete gaussian orthogonal polynomials and a formulation of the correlation kernel in terms of a double contour integral

simplifying the complexity of pipe flow transitional pipe flow is modeled as a one dimensional excitable and bistable medium models are presented in two variables turbulence intensity and mean shear that evolve according to established properties of transitional turbulence a continuous model captures the essence of the puff slug transition as a change from excitability to bistability a discrete model which additionally incorporates turbulence locally as a chaotic repeller reproduces almost all large scale features of transitional pipe flow in particular it captures metastable localized puffs puff splitting slugs localized edge states a continuous transition to sustained turbulence via spatiotemporal intermittency directed percolation and a subsequent increase in turbulence fraction toward uniform featureless turbulence

on the leading terms ideals of polynomial ideals over a valuation ring abstract we construct an example of a finitely generated ideal i of v x where v is a one dimensional valuation ring whose leading terms ideal is not finitely generated this gives a negative answer to the open question of whether if v is a valuation ring with krull dimension 1 then for any finitely generated ideal i of v x the leading terms ideal of i is also finitely generated the valuation rings satisfying this latter property will be called 1 grobner and are studied in this paper

the importance of the selberg integral it has been remarked that a fair measure of the impact of atle selberg s work is the number of mathematical terms that bear his name one of these is the selberg integral an n dimensional generalization of the euler beta integral we trace its sudden rise to prominence initiated by a question to selberg from enrico bombieri more than thirty years after its initial publication in quick succession the selberg integral was used to prove an outstanding conjecture in random matrix theory and cases of the macdonald conjectures it further initiated the study of q analogues which in turn enriched the macdonald conjectures we review these developments and proceed to exhibit the sustained prominence of the selberg integral as evidenced by its central role in random matrix theory calogero sutherland quantum many body systems knizhnik zamolodchikov equations and multivariable orthogonal polynomial theory

chaotic behavior of an extended system abstract some general aspects of spatiotemporal chaos are considered with emphasis on long range correlations and response an attempt is made to characterize the statistical steady state of a chaotic system by analogy with the description of a system near thermal equilibrium the one dimensional kuramoto sivashinsky equation is considered as an example and our present understanding of its statistical behavior is reviewed

asymptotics of cellular buckling close to the maxwell load we study the deformation of an elastic strut on a nonlinear winkler foundation subjected to an axial compressive load p using multi scale analysis and numerical methods we describe the localized cellular post buckled state of the system when p is removed from the critical load p 2 the solutions and their modulation frequencies differ significantly from those predicted by weakly nonlinear analysis very close to p 2 in particular when p approaches the maxwell load pm the localized solutions approach a large amplitude heteroclinic connection between an unbuckled solution and a periodic solution an asymptotic description of pm in terms of the system parameters is given the agreement between the numerical calculations and the asymptotic approximations is striking

stable localized solutions in nonlinear optics with large dissipation abstract we study the spatio temporal behavior of the electric field inside a laser with saturable absorber in the good cavity limit we find that the underlying nonlinear partial differential equation admits stable spatially localized solutions for both the one and the two dimensional case we critically examine similarities and differences with the localized solutions found recently near a weakly inverted bifurcation to traveling waves

localized periodic patterns for the non symmetric generalized swift hohenberg equation abstract a new asymptotic multiple scale expansion is used to derive envelope equations for localized spatially periodic patterns in the context of the generalized swift hohenberg equation an analysis of this envelope equation results in parametric conditions for localized patterns furthermore it yields corrections for wave number selection which are an order of magnitude larger for asymmetric nonlinearities than for the symmetric case the analytical results are compared with numerical computations which demonstrate that the condition for localized patterns coincides with vanishing hamiltonian and lagrangian for periodic solutions one striking feature of the choice of scaling parameters is that the derived condition for localized patterns agrees with the numerical results for a significant range of parameters which are an o 1 distance from the bifurcation thus providing a novel approach for studying these localized patterns

front motion metastability and subcritical bifurcations in hydrodynamics metastability occurs whenever under given external conditions as defined by the reynolds number in hydrodynamics more than one linearly stable solution may exist in variational problems the preferred state is usually claimed to be the one with the lowest energy but in hydrodynamics and except for very special conditions no energy functional exists from an analogy between amplitude and reaction diffusion equations it is argued that the border separating in space two possible solutions of the flow equations as turbulent and laminar in pipe flows or boundary layers moves with a constant mean velocity depending on the control parameter the sign of this velocity allows to decide which state is stable and which is metastable possible physical implications of these ideas for various problems of hydrodynamics are discussed

making the use of maximal ideals constructive the purpose of this paper is to decipher constructively a lemma of suslin which played a central role in his second solution of serre s problem on projective modules over polynomial rings this lemma says that for a commutative ring a if a x where v 1 is monic and n 3 then there exist c 1 c e n 1 a x such that denoting by w i the first coordinate of c it v 2 v n we have a by the constructive proof we give suslin s proof of serre s problem becomes fully constructive moreover the new method with which we treat this academic example may be a model for miming constructively abstract proofs in which one works modulo a generic maximal ideal in order to prove that an ideal contains 1

the hermite ring conjecture in dimension one abstract we prove constructively that for any ring r of krull dimension 1 and n 3 the group e n r x acts transitively on um n r x in particular we obtain that for any ring r with krull dimension 1 all finitely generated stably free modules over r x are free this settles the long standing hermite ring conjecture for rings of krull dimension 1

continuation of localised coherent structures in nonlocal neural field equations we study localised activity patterns in neural field equations posed on the euclidean plane such models are commonly used to describe the coarse grained activity of large ensembles of cortical neurons in a spatially continuous way we employ matrix free newton krylov solvers and perform numerical continuation of localised patterns directly on the integral form of the equation this opens up the possibility to study systems whose synaptic kernel does not lead to an equivalent pde formulation we present a numerical bifurcation study of localised states and show that the proposed models support patterns of activity with varying spatial extent through the mechanism of homoclinic snaking the regular organisation of these patterns is due to spatial interactions at a specific scale associated with the separation of excitation peaks in the chosen connectivity function the results presented form a basis for the general study of localised cortical activity with inputs and more specifically for investigating the localised spread of orientation selective activity that has been observed in the primary visual cortex with local visual input

faceting and coarsening dynamics in the complex swift hohenberg equation the complex swift hohenberg equation models pattern formation arising from an oscillatory instability with a finite wave number at onset and finds applications in lasers optical parametric oscillators and photorefractive oscillators we show that with real coefficients this equation exhibits two classes of localized states localized in amplitude only or localized in both amplitude and phase the latter are associated with phase winding states in which the real and imaginary parts of the order parameter oscillate periodically but with a constant phase difference between them the localized states take the form of defects connecting phase winding states with equal and opposite phase lag and can be stable over a wide range of parameters the formation of these defects leads to faceting of states with initially spatially uniform phase depending on parameters these facets may either coarsen indefinitely as described by a cahn hilliard equation or the coarsening ceases leading to a frozen faceted structure

on the transition to turbulence in pipe flow abstract in an experiment involving flow in a pipe with circular cross section we calculated the velocity auto correlation spectrum the correlation dimension d2 and an estimation of the kolmogorov entropy k2 from the streamwise velocity fluctuations we find that at the beginning of the transition the intermittently appearing turbulent puffs have a periodic structure afterwards the flow becomes fractal in phase space through a period doubling cascade as the reynolds number re increases the dimension of this fractal rises hierarchically for completely turbulent flow the highest correlation dimension in our experiments is about 11

exponential asymptotics of localised patterns and snaking bifurcation diagrams abstract localised patterns emerging from a subcritical modulation instability are analysed by carrying the multiple scales analysis beyond all orders the model studied is the swift hohenberg equation of nonlinear optics which is equivalent to the classical swift hohenberg equation with a quadratic and a cubic nonlinearity applying the asymptotic technique away from the maxwell point first it is shown how exponentially small terms determine the phase of the fast spatial oscillation with respect to their slow sech type amplitude in the vicinity of the maxwell point the beyond all orders calculation yields the pinning range of parameters where stable stationary fronts connect the homogeneous and periodic states the full bifurcation diagram for localised patterns is then computed analytically including snake and ladder bifurcation curves this last step requires the matching of the periodic oscillation in the middle of a localised pattern both with an up and a down front to this end a third super slow spatial scale needs to be introduced in which fronts appear as boundary layers in addition the location of the maxwell point and the oscillation wave number of localised patterns are required to fourth order accuracy in the oscillation amplitude