comments on an fpt algorithm with a modularized structure for computing 2 d cyclic convolutions by t k truong et al the modularized fpt algorithm proposed by truong et al in the above named work see ibid vol 36 p 1540 1542 sept 1988 is extended to the finite fields for computing 2 d cyclic convolutions in this method only fpts and number theoretic transforms ntts of the same length are required

common eigenvectors of two matrices abstract a computable criterion is given for two square matrices to possess a common eigenvector as well as a criterion for one matrix to have an eigenvector lying in a given subspace some applications are discussed

eine arithmetische eigenschaft automorpher formen zu gewissen nicht arithmetischen gruppen let sl2 open image in new window denote a fuchsian triangle group anda an elliptic fixed point of for any weightk and any multiplier systemv whose values are roots of unity there is a basis of automorphic forms with expansions f left z right left z bar a right k sum limits n 0 infty r n b n left frac z a z a right n

werte hypergeometrischer funktionen a partial result on a problem of siegel is given hypergeometric functions with rational parameters have transcendental values in almost all algebraic points up to some natural exceptions these exceptions are the well known algebraic functions and an unexpected second class of examples related to certain shimura curves

fuchsian triangle groups and grothendieck dessins variations on a theme of belyi according to a theorem of belyi a smooth projective algebraic curve is defined over a number field if and only if there exists a non constant element of its function field ramified only over 0 1 and the existence of such a belyi function is equivalent to that of a representation of the curve as a possibly compactified quotient space of the poincare upper half plane by a subgroup of finite index in a fuchsian triangle group on the other hand fuchsian triangle groups arise in many contexts such as in the theory of hypergeometric functions and certain triangular billiard problems which would appear at first sight to have no relation to the galois problems that motivated the above discovery of belyi in this note we review several results related to belyi s theorem and we develop certain aspects giving examples for preliminary accounts see the preprint wo1 the conference proceedings article bauitz and the comptes rendus note cowo2

wilson s map operations on regular dessins and cyclotomic fields of definition dessins d enfants can be seen as bipartite graphs embedded in compact orientable surfaces 5 according to grothendieck and others a dessin uniquely determines a complex structure on the surface even an algebraic structure as a projective algebraic curve defined over a number field combinatorial properties of the dessin should therefore determine the equations and also structural properties of the curve such as the field of moduli or the field of definition however apart from a few series of examples very few general results concerning such correspondences 10 are known as a step in this direction we present a graph theoretic characterisation of certain quasiplatonic curves defined over cyclotomic fields based on wilson s operations on maps these leave invariant the graph but change the cyclic ordering of edges around the vertices therefore they change the embeddings and hence the dessins and hence the conformal and algebraic structure of the underlying curves under suitable assumptions satisfied by many series of regular 15 dessins these changes coincide with the effect of galois conjugation this coincidence allows one to draw conclusions about galois orbits and fields of definition of dessins the possibilities afforded by these techniques and their limitations are illustrated by a new look at some known examples and a study of dessins based on the regular embeddings of complete graphs 1 the mathematical framework 20 for the last twenty years embeddings of graphs into surfaces have become important for the theory of algebraic curves as an essential tool in the theory of dessins d enfants in this first section we briefly recall the necessary definitions and give an outline of the problems and results presented in this paper one may introduce dessins d enfants as hypermaps on compact oriented 2 manifolds these 25 objects first studied by cori 5 have several equivalent algebraic or topological definitions topologically they are a generalisation of maps whose underlying graphs are replaced with hypergraphs so that the hyperedges are now allowed to be incident with any finite number of hypervertices and hyperfaces one way to visualise this concept is to represent a hypermap d by its walsh map w d a connected bipartite graph b embedded in a compact oriented 30 surface dividing it into simply connected cells 26 in the language of hypermaps the white and black vertices represent the hypervertices and hyperedges of d the edges represent incidences between them and the cells represent the hyperfaces algebraic hypermaps are an alternative and equivalent way to describe dessins these are triples g x y consisting of a permutation group g sn the monodromy or hypercarto 35 graphic group of the dessin acting transitively on the n edges of the walsh map w d a nd generated by two elements x and y the generators x and y describe the anticlockwise cyclic permutations of the edges around their incident white and black vertices we call d a dessin

algebraic curves uniformized by congruence subgroups of triangle groups we construct certain subgroups of hyperbolic triangle groups which we call congruence subgroups these groups include the classical congruence subgroups of sl2 z hecke triangle groups and 19 families of arithmetic triangle groups associated to shimura curves we determine the eld of moduli of the curves associated to these groups and thereby realize the groups psl2 fq and pgl2 fq regularly as galois groups

long convolutions using number theoretic and polynominal transforms a method is proposed for partitioning a long convolution in several short convolutions to be computed by the fermat number transform the method is based on the use of polynomial transforms the computational cost is much lower than that required by the algorithms already available in the technical literature

nonlinear dynamics and symbolic dynamics of neural networks a piecewise linear equation is proposed as a method of analysis of mathematical models of neural networks a symbolic representation of the dynamics in this equation is given as a directed graph on an n dimensional hypercube this provides a formal link with discrete neural networks such as the original hopfield models analytic criteria are given to establish steady states and limit cycle oscillations independent of network dimension model networks that display multiple stable limit cycles and chaotic dynamics are discussed the results show that such equations are a useful and efficient method of investigating the behavior of neural networks

improved long convolutions using generalized number theoretic and polynomial transforms generalized number theoretic and polynomial transforms can be used in the computation of a long convolution this will reduce the computation complexity by a factor of two it only requires fewer than 2 real multiplications 45 real additions and 10 data shifts per point for a typical 2048 array

on matrices with common invariant cones with applications in neural and gene networks motivated by a differential continuous time switching model for gene and neural networks we investigate matrix theoretic problems regarding the relative location and topology of the dominant eigenvectors of words constructed multiplicatively from two matrices a and b these problems are naturally associated with the existence of common invariant subspaces and common invariant proper cones of a and b the commuting case and the two dimensional case are rich and considered analytically we also analyze and recast the problem of the existence of a common invariant polyhedral cone in a multilinear framework as well as present necessary conditions for the existence of low dimensional common invariant cones

arithmetic 1 e curves and belyĭ maps using the theory of bely maps we calculate the algebraic curves associated to the fuchsian groups of signature that are commensurable with a triangle group along with the picard fuchs differential equations on these curves which are related to pullbacks of hypergeometric differential equations we focus particularly on the groups that are arithmetic

galois action on families of generalised fermat curves abstract it is well known that the complete bipartite graphs k n n occur as dessins d enfants on the fermat curves of exponent n however there are many more curves having k n n as the underlying graph of their dessins even if we require the strongest regularity condition that the graphs define regular maps on the underlying riemann surfaces for odd prime powers n these maps have recently been classified g a jones r nedela m skoviera regular embeddings of k n n where n is an odd prime power european j combin in press they fall into certain families characterised by their automorphism groups in the present paper we show that these families form galois orbits we determine the minimal field of definition of the corresponding curves and in easier cases also their defining equations

beauville surfaces and finite groups abstract extending results of bauer catanese and grunewald and of fuertes and gonzalez diez we show that beauville surfaces of unmixed type can be obtained from the groups l 2 q and sl 2 q for all prime powers q 5 and the suzuki groups sz 2 e and the ree groups r 3 e for all odd e 3 we also show that l 2 q and sl 2 q admit strongly real beauville structures yielding real beauville surfaces for all q 5

alternating quotients of fuchsian groups abstract it is shown that any finitely generated non elementary fuchsian group has among its homomorphic images all but finitely many of the alternating groups a n this settles in the affirmative a long standing conjecture of graham higman

generation of finite quasisimple groups with an application to groups acting on beauville surfaces we verify a conjecture of bauer catanese and grunewald by proving a stronger result concerning quasisimple groups more specifically we prove that every finite quasisimple group apart from a5 and its cover sl 2 5 is a beauville group

chaos in high dimensional neural and gene networks abstract neural and gene networks are often modeled by differential equations if the continuous threshold functions in the differential equations are replaced by step functions the equations become piecewise linear pl equations the flow through the state space is represented schematically by paths and directed graphs on am n dimensional hypercube closed pathways called cycles may reflect periodic orbits with associated fixed points in a chosen poincare section a return map in the poincare section can be constructed by the composition of fractional linear maps the stable and unstable manifolds of the fixed points can be determined analytically these methods allow us to analyze the dynamics in higher dimensional networks as exemplified by a four dimensional network that displays chaotic behavior the three dimensional poincare map is projected to a two dimensional plane this much simpler piecewise linear two dimensional map conserves the important qualitative features of the flow

analysis of continuous time switching networks abstract models of a number of biological systems including gene regulation and neural networks can be formulated as switching networks in which the interactions between the variables depend strongly on thresholds an idealized class of such networks in which the switching takes the form of heaviside step functions but variables still change continuously in time has been proposed as a useful simplification to gain analytic insight these networks called here glass networks after their originator are simple enough mathematically to allow significant analysis without restricting the range of dynamics found in analogous smooth systems a number of results have been obtained before particularly regarding existence and stability of periodic orbits in such networks but important cases were not considered here we present a coherent method of analysis that summarizes previous work and fills in some of the gaps as well as including some new results furthermore we apply this analysis to a number of examples including surprising long and complex limit cycles involving sequences of hundreds of threshold transitions finally we show how the above methods can be extended to investigate aperiodic behaviour in specific networks though a complete analysis will have to await new results in matrix theory and symbolic dynamics

belyi s theorem for complex surfaces belyi s theorem states that a compact riemann surface c can be defined over a number field if and only if there is on it a meromorphic function f with three critical values such functions resp riemann surfaces are called belyi functions resp belyi surfaces alternatively belyi surfaces can be characterized as those which contain a proper zariski open subset uniformised by a torsion free subgroup of the classical modular group sl2 z in this article we establish a result analogous to belyi s theorem in complex dimension two it turns out that the role of belyi functions is now played by composed lefschetz pencils with three critical values while the analogous to torsion free subgroups of the modular group will be certain extensions of them acting on a bergmann domain of 2 these groups were first introduced by bers and griffiths

pseudointegrable systems in classical and quantum mechanics abstract we study particles moving in planar polygonal enclosures with rational angles and show by several methods that trajectories in the classical phase space explore two dimensional invariant surfaces which are generically not tori as in integrable systems but instead have the topology of multiply handled spheres the quantum mechanics of one such pseudointegrable system is studied in detail by computing energy levels using an exact formalism this system consists of motion on a unit coordinate torus containing a square reflecting obstacle with side l we find that neighbouring levels avoid degeneracies as l varies and that the probability distribution for the spacing s of adjacent levels vanishes linearly as s 0 level repulsion the weyl area rule plus edge and corner corrections gives a very accurate approximation for the mean level density oscillatory corrections to the mean level density are given as a sum over closed classical paths for pseudointegrable systems these closed paths form families covering part of the phase space invariant surfaces