numeric and mesh algorithms for the coxeter spectral study of positive edge bipartite graphs and their isotropy groups we develop algorithmic techniques for the coxeter spectral analysis of the class ubigr n of connected loop free positive edge bipartite graphs d with n 2 vertices i e signed graphs in particular we present numerical and graphical algorithms allowing us a computer search in the study of such graphs d by means of their gram matrix email protected d the complex spectrum specc email protected c of the coxeter matrix cox d email protected email protected email protected d t r and the geometry of weyl orbits in the set mor d d of matrix morsifications email protected m n z of a simply laced dynkin diagram email protected a n d n e 6 e 7 e 8 associated with d and mesh root systems of type email protected our algorithms construct the coxeter gram polynomials cox d t z t and mesh geometries of root orbits of small connected loop free positive edge bipartite graphs d we apply them to the study of the following coxeter spectral analysis problem does email protected email protected hold i e the email protected email protected d arez congruent for any pair of connected positive loop free edge bipartite graphs d d inubigr nsuch thatspecc d specc d the problem if any square integer matrix email protected m n z is z congruent with its transpose a t r is also discussed we present a solution for graphs in ubigr n with email protected 6

on algorithmic coxeter spectral analysis of positive posets abstract following a general framework of coxeter spectral analysis of signed graphs and finite posets i introduced by simson siam j discrete math 27 827 854 2013 we present efficient numerical algorithms for the coxeter spectral study of finite posets i 1 n i that are positive in the sense that the symmetric gram matrix g i 1 2 c i c i t r m n q is positive definite where c i m n z is the incidence matrix of i encoding the relation i in the framework of scientific computing we present a complete coxeter spectral classification of finite positive posets i of size n i 20 it extends one of the main results obtained in gasiorek et al eur j comb 48 127 142 2015 for posets of size n 10 we also show that the connectivity of such posets i is determined by the complex coxeter spectrum specc i c equivalently by the coxeter polynomial cox i t z t od i one of the main results of the paper is a new technique to compute in a polynomial time a z invertible matrix b i m n z such that b i t r c i b i g d i where g d i m n z is a non symmetric gram matrix of a simply laced dynkin diagram d i a n d n e 6 e 7 e 8 associated with a finite positive poset i

periodicity in bilinear lattices and the coxeter formalism abstract we introduce and study in detail so called circulant coxeter periodic elements and circulant families in a bilinear lattice k as well as their dual versions called anti circulant we show that they form a natural environment for a systematic explanation of certain cyclotomic factors of the coxeter polynomial k of k and in consequence of coxeter polynomials of algebras of finite global dimension we discuss the properties of quadratic forms induced by circulant and anti circulant families moreover we interpret the results in the language of representation theory of algebras and point out applications facts concerning tubular families in auslander reiten quivers and quadratic forms of algebras abstract considerations in bilinear lattices are illustrated with a collection of non trivial examples arising from module and derived categories the results show that techniques of linear algebra and number theory provide efficient tools for explaining various representation theoretic facts

a kinetic decomposition for singular limits of non local conservation laws abstract we consider a non local regularization of nonlinear hyperbolic conservation laws in several space variables the regularization is motivated by the theory of phase dynamics and is based on a convolution operator we formulate the initial value problem and begin by deriving a priori estimates which are independent of the regularization parameter following hwang and tzavaras we establish a kinetic decomposition associated with the problem under consideration and we conclude that the sequence of solutions generated by the non local model converges to a weak solution of the corresponding hyperbolic problem depending on the scaling introduced in the non local dispersive term this weak limit is either a classical kruzkov solution satisfying all entropy inequalities or more interestingly a nonclassical entropy solution in the sense defined by lefloch that is a weak solution satisfying a single entropy inequality and containing undercompressive shock waves possibly selected by a kinetic relation finally we illustrate our analytical conclusions with numerical experiments in one spatial variable

inflation algorithm for loop free non negative edge bipartite graphs of corank at least two we continue the study of finite connected loop free edge bipartite graphs with m 3m 3 vertices a class of signed graphs we started in simson 2013 48 and m g siorek et al 2016 19 by means of the non symmetric gram matrix g mm z of its symmetric gram matrix g 12 g g tr mm 12z and the gram quadratic form q zm zq zm z in particular we study connected non negative edge bipartite graphs with n r 3n r 3 vertices of corank r 2r 2 in the sense that the symmetric gram matrix g mn r z g mn r z of is positive semi definite of rank n 1n 1 the edge bipartite graphs of corank r 2r 2 are studied up to the weak gram zz congruence z z where z z means that g btr g bg btr g b for some b mn r z b mn r z such that det b 1det b 1 our main result of the paper asserts that given a connected edge bipartite graph with n r 3n r 3 vertices of corank r 2r 2 there exists a suitably chosen sequence t of the inflation operators of the form tab such that the composite operator t reduces to a connected bigraph d n r such that zd n r and d n r is one of the canonical r vertex extensions a n r n 1n 1 d n r n 4n 4 e 6 r e 7 r and e 8 r with n rn r vertices of the simply laced dynkin diagrams an dn e6 e7 e8an dn e6 e7 e8 with n 1n 1 vertices the algorithm constructs also a matrix b mn r z b mn r z with det b 1det b 1 defining the weak gram zz congruence zd n r that is satisfying the equation gd n r btr g b

elliptic functions and equations of modular curves let p ge 5 be a prime we show that the space of weight one eisenstein series defines an embedding into mathbb p p 3 2 of the modular curve x 1 p for the congruence group gamma 1 p that is scheme theoretically cut out by explicit quadratic equations

on the structure of loop free non negative edge bipartite graphs abstract we continue the study of a class of signed graphs called finite connected loop free edge bipartite graphs bigraphs for short started in simson 2013 33 and continued in g siorek et al 2016 14 and simson zaj c 2017 38 in this paper we present an algorithmic approach to the study of non negative bigraphs with n r 2 vertices of corank r 1 that is bigraphs with the symmetric gram matrix g 1 2 g g t r m n r 1 2 z positive semi definite of rank n 1 where g m n r z is the non symmetric gram matrix of one of the main results of this paper are theorem 2 1 theorem 2 17 asserting that every such a bigraph with n r 2 vertices can be obtained from a corank zero i e positive connected loop free bigraph with n 1 vertices by an r point extension procedure u 1 u r u 1 u r along the r 1 roots u 1 u r z n of the positive definite gram form q z n z v v g v t r it is also shown that the extension procedure yields a combinatorial algorithm allowing us to obtain inductively all connected loop free non negative corank r 1 bigraphs with n r 2 vertices from the corank zero bigraphs with n 1 vertices

coxeter energy of graphs abstract we study the concept of the coxeter energy of graphs and digraphs quivers as an analogue of gutman s adjacency energy which has applications in theoretical chemistry and is a recently widely investigated graph invariant coxeter energy ce g of a di graph g is defined to be the sum of the absolute values of all complex eigenvalues of the coxeter matrix associated with g our main inspiration for the study comes from the coxeter formalism appearing in group theory lie theory representation theory of algebras mathematical physics and other contexts we focus on the coxeter energy of trees and we prove that the path resp the maximal star has the smallest resp the greatest coxeter energy among all trees resp two large subclasses of trees with fixed number of vertices we provide several other related results as the characterization of trees with second smallest and second greatest coxeter energy bounds for coxeter energy and general facts on coxeter spectra of graphs extending known results e g for salem trees and for certain special real coxeter eigenvalues of trees additionally we discuss few other energy like quantities for the coxeter spectra of di graphs including coxeter energy normalized by the trace of the coxeter matrix and the quantities derived from variants of coulson integral formula

the small dispersion limit of the korteweg de vries equation ii in part i the scattering transform method is used to study the weak limit of solutions to the initial value problem for the korteweg devries kdv equation as the dispersion tends to zero in that limit the associated schrodinger operator becomes semiclassical so the exact scattering data is replaced by its corresponding wkb expressions only nonpositive initial data are considered in that case the limiting reflection coefficient vanishes the explicit solution of kay and moses for the reflectionless inverse transform is then analyzed and the weak limit valid for all time is characterized by a quadratic minimum problem with constraints this minimum problem is reduced to a riemann hilbert problem in function theory in parts ii and iii we use function theoretical methods to solve the riemann hilbert problem in terms of solutions to an auxiliary initial value problem the weak limit satisfies the kdv equation with the dispersive term dropped until its derivatives become infinite up to that time the weak limit is a strong l2 limit at later times the weak limit is locally described by whitham s averaged equations or more generally by the equations found by flaschka et al using multiphase averaging for large times behavior of the weak limit is studied directly from the minimum problem

symbolic computation of strong gram congruences for cox regular positive edge bipartite graphs with loops abstract the paper can be viewed as a second part of the author s paper simson 2018 43 our main aim is to construct symbolic algorithms for the coxeter spectral classification of a class of signed graphs called edge bipartite graphs with n 2 vertices we started in simson 2013 37 and bocian et al 2014 6 more precisely we construct algorithms for the classification up to the strong gram z congruence z of all finite connected cox regular edge bipartite graphs with at least one loop bigraphs for short that are positive in the sense that the associated symmetric gram matrix g 1 2 g g t r m n 1 2 z is positive definite where g m n z is the non symmetric gram matrix of defined in section 1 and z means that there is a z invertible matrix b m n z such that g b t r g b we recall from 43 that every such a bigraph with a loop is strongly gram z congruent with a bigraph d that is one of the positive cox regular bigraphs b n n 2 c n n 3 f 4 m 4 g 2 presented in section 1 here by applying the geometry of mesh root system technique we construct symbolic algorithms that compute the correspondence d and the set gl n z d of all z invertible matrices b m n z defining the strong gram z congruence z d for any connected positive bigraph with n 2 vertices and at least one loop

equations of 1 d polarized abelian surfaces in this paper we study the equations of projectively embedded abelian surfaces with a polarization of type 1 d classical results say that given an ample line bundle l on an abelian surface a the line bundle l n is very ample for n 3 and furthermore in case n is even and n 4 the generators of the homogeneous ideal ia of the embedding of a via l n are all quadratic a possible choice for a set of generators of ia are the riemann theta relations on the other hand much less is known about embeddings via line bundles l of type 1 d that is line bundles l which are not powers of another line bundle on a it is well known that if d 5 and a is a general abelian surface then l is very ample while l can never be very ample for d 5 however even if d 5 l may not be very ample for special abelian surfaces we will restrict our attention in what follows only to the general abelian surface and wish to know what form the equations take for such a projectively embedded abelian surface a few special cases are well documented in the literature d 4 in which case the general surface is a singular octic in p3 cf blvs and d 5 in which case the abelian surface is described as the zero set of a section of the horrocks mumford bundle hm whereas its homogeneous ideal is generated by 3 heisenberg invariant quintics and 15 sextics cf ma also recent work by manolache and schreyer ms and by ranestad ra provides a description of the equations and syzygies in the case d 7

p critical integral quadratic forms and positive unit forms an algorithmic approach abstract we introduce the class of p critical integral unit forms q z n z containing the critical forms in the sense of ovsienko 13 several characterisations of p critical forms are given in particular it is proved that q is p critical if and only if there is a uniquely determined extended dynkin diagram a n d n e 6 e 7 e 8 and a special group isomorphism t z n z n such that q t is the quadratic form q z n z n 0 of the diagram a correspondence between positive forms p z n 1 z with a sincere root and p critical forms q z n z is described and efficient linear algebra algorithms for computing p critical unit forms and positive forms are constructed

root systems and inflations of non negative quasi cartan matrices abstract cartan matrices quasi cartan matrices and associated integral quadratic forms and root systems play an important role in such areas like lie theory representation theory and algebraic graph theory we study quasi cartan matrices by means of the inflation algorithm an idea used in ovsienko s classical proof of the classification of positive definite integral quadratic forms and recently applied in several other classification results we prove that it is enough to perform linear number of inflations to reduce positive or non negative principal quasi cartan matrix to its canonical form i e to the cartan matrix of a dynkin or euclidean diagram respectively moreover we show that in the positive case the length of every sequence of inflations has quadratic bound and that the only other natural non negative class having such universal bound is the class of so called pos sincerely principal quasi cartan matrices and in this case the bound is linear we obtain such low bounds by applying among others some new observations on the properties of the reduced root systems in the sense of bourbaki associated with positive quasi cartan matrices

periodic coxeter matrices and their associated quadratic forms we study relations between the periodicity of a coxeter matrix and the positivity or the non negativity of a quadratic form with respect to a cartan matrix of an artinian ring we introduce the notion of the weak periodicity of a coxeter matrix induced by a non negative quadratic form and we characterize the weak periodicity condition and the periodicity condition in terms of eigenvalues of a coxeter matrix

a gram classification of non negative corank two loop free edge bipartite graphs we continue the coxeter spectral study of finite connected loop free edge bipartite graphs with m 2 3 vertices a class of signed graphs started in simson 2013 49 by means of the non symmetric gram matrix g mm 2 z of its symmetric gram matrix g 12 g g tr mm 2 12z the gram quadratic form q zm 2 z and the coxeter spectrum specc c i e the complex spectrum of the coxeter matrix cox g g tr gl m 2 z in the present paper we study non negative edge bipartite graphs of corank two in the sense that the symmetric gram matrix g mm 2 z of is positive semi definite of rank m 1 one of our aims is to get a complete classification of all connected corank two loop free edge bipartite graphs with m 2 3 vertices up to the weak gram z congruence z where z means that g btr g b for some b mm 2 z such that det b 1 by one vertex extensions of the simply laced euclidean diagrams a m m 1 d m m 4 e 6 e 7 e 8 we construct a family of connected loop free corank two diagrams a m 2 d m 2 e 6 2 e 7 2 e 8 2 called simply extended euclidean diagrams such that they classify all connected corank two loop free edge bipartite graphs with m 2 3 vertices up to the weak gram z congruence z a structure of connected corank two loop free edge bipartite graphs is described it is shown that every such contains a connected positive edge bipartite subgraph that is z congruent with a simply laced dynkin diagram dyn called the dynkin type of such that is a two point extension u w of along two roots u w of the positive definite gram form q zm z this yields a combinatorial algorithm u w u w allowing us to construct all connected corank two loop free edge bipartite graphs with m 2 3 vertices and d dyn from the triples u w where is positive of the dynkin type d and u w are roots of the positive definite gram form q zm z

averaging lemmas without time fourier transform and application to discretized kinetic equations we prove classical averaging lemmas in the l2 framework with the help of the fourier transform in variables x and v but not t this method is then used to study discretized problems arising out of the numerical analysis of kinetic equations

canonical matrices for linear matrix problems abstract we consider a large class of matrix problems which includes the problem of classifying arbitrary systems of linear mappings for every matrix problem from this class we construct belitskii s algorithm for reducing a matrix to a canonical form which is the generalization of the jordan normal form and study the set c mn of indecomposable canonical m n matrices considering c mn as a subset in the affine space of m n matrices we prove that either c mn consists of a finite number of points and straight lines for every m n or c mn contains a 2 dimensional plane for a certain m n

toric varieties and modular forms let n r be a lattice and let deg n be a piecewise linear function that is linear on the cones of a complete rational polyhedral fan under certain conditions on deg the data n deg determines a function f that is a holomorphic modular form of weight r for the congruence subgroup 1 l moreover by considering all possible pairs n deg we obtain a natural subring l of modular forms with respect to 1 l we construct an explicit set of generators for l and show that l is stable under the action of the hecke operators finally we relate l to the hirzebruch elliptic genera that are modular with respect to 1 l

nonlinear diffusive dispersive limits for scalar multidimensional conservation laws abstract we consider a class of multidimensional conservation laws with vanishing nonlinear diffusion and dispersion terms under a condition on the relative size of the diffusion and dispersion coefficients we show that the approximate solutions converge in a strong topology to the entropy solution of a scalar conservation law our proof is based on methodology developed in s hwang a e tzavaras kinetic decomposition of approximate solutions to conservation laws applications to relaxation and diffusion dispersion approximations comm partial differential equations 27 2002 1229 1254 which uses the averaging lemma

periods of modular forms and identities between eisenstein series borisov and gunnells observed in 2001 that certain linear relations between products of two holomorphic weight 1 eisenstein series had the same structure as the relations between periods of modular forms a similar phenomenon exists in higher weights we give a conceptual reason for this observation in arbitrary weight this involves an unconventional way of expanding the rankin selberg convolution of a cusp form with an eisenstein series we also prove a partial result towards understanding the action of a hecke operator on a product of two eisenstein series

tubes in derived categories and cyclotomic factors of the coxeter polynomial of an algebra let be a k algebra of finite global dimension we study tubular families in the auslander reiten quiver of the bounded derived category db satisfying certain natural axioms in particular we precisely describe their influence on the cyclotomic factors of the coxeter polynomial of and discuss several numerical limitations for their possible shapes moreover we show that our results provide an alternative relatively simple proof of non trivial classical facts concerning tubular families in module categories and also extend them slightly

kinetic decomposition for kinetic models of bgk type abstract in this paper we consider kinetic models of bgk type which describe the scalar conservation law at the microscopic scale we use new technique developed in comm partial differential equations 27 2002 1229 in order to get the convergence first we obtain the approximate transport equation for the given kinetic models of bgk type then using the averaging lemma we obtain the convergence this paper shows how to relate the given kinetic model with the averaging lemma to get the convergence

systems of subspaces of a unitary space abstract for a finite poset p p 1 p t we study systems u 1 u t u of subspaces of a unitary space u such that u i u j if p i p j two systems u 1 u t u and v 1 v t v are said to be isometric if there exists an isometry u v such that u i v i we classify such systems up to isometry if p is a semichain we prove that the problem of their classification is unitarily wild if p is not a semichain a classification problem is called unitarily wild if it contains the problem of classifying linear operators on a unitary space which is hopeless in a certain sense

kinetic formulation for systems of two conservation laws and elastodynamics for scalar conservation laws the kinetic formulation makes it possible to generate all the entropies from a simple kernel we show how this concept replaces and simplifies greatly the concept of young measures avoiding the difficulties encountered when working in lp the general construction of the two kinetic functions that generate the entropies of 2 2 strictly hyperbolic systems is also developed here we show that it amounts to building a universal entropy i e one that can be truncated by a kinetic value along riemann invariants for elastodynamics this construction can be completed and specialized using the additional galilean invariance this allows a full characterization of convex entropies it yields a kinetic formulation consisting of two semi kinetic equations which as usual are equivalent to the infinite family of all the entropy inequalities

toric modular forms and nonvanishing of l functions in a previous paper 1 we defined the space of toric forms t l and showed that it is a finitely generated subring of the holomorphic modular forms of integral weight on the congruence group 1 l in this article we prove the following theorem modulo eisenstein series the weight two toric forms coincide exactly with the vector space generated by all cusp eigenforms f such that l f 1 6 0 the proof uses work of merel and involves an explicit computation of the intersection pairing on manin symbols

toric modular forms of higher weight in the papers 1 2 we used the geometry of complete polyhedral fans to construct a subring t l of the modular forms on 1 l and showed that for weight two the cuspidal part of t l coincides with the space of cusp forms of analytic rank zero in this paper we show that in weights greater than two the cuspidal part of t l coincides with the space of all cusp forms

diffusion dispersion limits for multidimensional scalar conservation laws with source terms abstract in this paper we consider conservation laws with diffusion and dispersion terms we study the convergence for approximation applied to conservation laws with source terms the proof is based on the hwang and tzavaras s new approach seok hwang athanasios e tzavaras kinetic decomposition of approximate solutions to conservation laws application to relaxation and diffusion dispersion approximations comm partial differential equations 27 5 6 2002 1229 1254 and the kinetic formulation developed by lions perthame and tadmor p l lions b perthame e tadmor a kinetic formulation of multidimensional scalar conservation laws and related equations j amer math soc 7 1 1994 169 191

non classical shocks and kinetic relations scalar conservation laws this paper analyzes the non classical shock waves which arise as limits of certain diffusive dispersive approximations to hyperbolic conservation laws such shocks occur for non convex fluxes and connect regions of different convexity they have negative entropy dissipation for a single convex entropy function but not all convex entropies and do not obey the classical oleinik entropy criterion we derive necessary conditions for the existence of non classical shock waves and construct them as limits of traveling wave solutions for several diffusive dispersive approximations

classical yang baxter equation and the a constraint abstract we show that elliptic solutions of classical yang baxter equation cybe can be obtained from triple massey products on an elliptic curve we introduce the associative version of this equation which has two spectral parameters and construct its elliptic solutions we also study some degenerations of these solutions