dynamical systems defining jacobi s ϑ constants we propose a system of equations that defines weierstrass jacobi s eta and theta constant series in a differentially closed way this system is shown to have a direct relationship to a little known dynamical system obtained by jacobi the classically known differential equations by darboux halphen chazy and ramanujan are the differential consequences or reductions of these systems the proposed system is shown to admit the lagrangian hamiltonian and nambu formulations we explicitly construct a pencil of nonlinear poisson brackets and complete set of involutive conserved quantities as byproducts of the theory we exemplify conserved quantities for the ramamani dynamical system and quadratic system of halphen brioschi

on the shape of a tridiagonal pair abstract let k denote a field and let v denote a vector space over k with finite positive dimension we consider a pair of linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable ii there exists an ordering v i i 0 d of the eigenspaces of a such that a v i v i 1 v i v i 1 for 0 i d where v 1 0 and v d 1 0 iii there exists an ordering v i i 0 of the eigenspaces of a such that av i v i 1 v i v i 1 for 0 i where v 1 0 and v 1 0 iv there is no subspace w of v such that aw w a w w w 0 w v we call such a pair a tridiagonal pair on v it is known that d and for 0 i d the dimensions of v i v i v d i v d i coincide we denote this common dimension by i in this paper we prove that i 0 d i for 0 i d it is already known that 0 1 if k is algebraically closed

the structure of some linear transformations abstract let f denote an algebraically closed field and let v denote a finite dimensional vector space over f recently ito and terwilliger considered a system of linear transformations a a a a on v which generalizes the notions of a tridiagonal pair and a q inverting pair in their paper they mentioned some open problems about this system in this paper we solve problem 1 2 with the following results let v i i 0 d denote the common eigenspaces of a a and let v i i 0 d denote the common eigenspaces of a a we show that each of a a a a is determined up to affine transformation by the sequences v i i 0 d v i i 0 d we also show that the following are equivalent i there exists a nonzero bilinear form on v such that a u v u a v and a u v u a v for all u v v ii there exist scalars in f with nonzero such that a a i and a a i and iii both a a and a a are tridiagonal pairs

hessenberg pairs of linear transformations abstract let k denote a field and v denote a nonzero finite dimensional vector space over k we consider an ordered pair of linear transformations a v v and a v v that satisfy i iii below 1 i each of a a is diagonalizable on v 2 ii there exists an ordering v i i 0 d of the eigenspaces of a such that a v i v 0 v 1 v i 1 0 i d where v 1 0 v d 1 0 3 iii there exists an ordering v i i 0 of the eigenspaces of a such that av i v 0 v 1 v i 1 0 i where v 1 0 v 1 0 we call such a pair a hessenberg pair on v in this paper we obtain some characterizations of hessenberg pairs we also explain how hessenberg pairs are related to tridiagonal pairs

the classification of leonard triples of racah type let k denote an algebraically closed field of characteristic zero let v denote a vector space over k with finite positive dimension by a leonard triple on v we mean an ordered triple of linear transformations a a ae in end v such that for each b a a ae there exists a basis for v with respect to which the matrix representing b is diagonal and the matrices representing the other two linear transformations are irreducible tridiagonal in this paper we define a family of leonard triples said to have racah type and classify them up to isomorphism moreover we show that each of them satisfies the z3 symmetric askey wilson relations as an application we construct all leonard triples that have racah type from the universal enveloping algebra u sl2

from a discrete to a continuum model of cell dynamics in one dimension multiscale modeling is emerging as one of the key challenges in mathematical biology however the recent rapid increase in the number of modeling methodologies being used to describe cell populations has raised a number of interesting questions for example at the cellular scale how can the appropriate discrete cell level model be identified in a given context additionally how can the many phenomenological assumptions used in the derivation of models at the continuum scale be related to individual cell behavior in order to begin to address such questions we consider a discrete one dimensional cell based model in which cells are assumed to interact via linear springs from the discrete equations of motion the continuous rouse p e rouse j chem phys 21 1272 1953 model is obtained this formalism readily allows the definition of a cell number density for which a nonlinear fast diffusion equation is derived excellent agreement is demonstrated between the continuum and discrete models subsequently via the incorporation of cell division we demonstrate that the derived nonlinear diffusion model is robust to the inclusion of more realistic biological detail in the limit of stiff springs where cells can be considered to be incompressible we show that cell velocity can be directly related to cell production this assumption is frequently made in the literature but our derivation places limits on its validity finally the model is compared with a model of a similar form recently derived for a different discrete cell based model and it is shown how the different diffusion coefficients can be understood in terms of the underlying assumptions about cell behavior in the respective discrete models

self similar solution in the leith model of turbulence anomalous power law and asymptotic analysis we consider a leith model of turbulence leith c 1967 phys fluids 10 1409 in which the energy spectrum obeys a nonlinear diffusion equation we analytically prove the existence of a self similar solution with a power law asymptotic on the low wavenumber end and a sharp boundary on the high wavenumber end which propagates to infinite wavenumbers in a finite time t we prove that this solution has a power law asymptotic with an anomalous exponent x which is less than the kolmogorov value x 5 3 this is a result that was previously discovered by numerical simulations in connaughton and nazarenko 2004 phys rev lett 92 044501 we also prove the convergence to this self similar solution of the spectrum evolving from an arbitrary finitely supported initial data as t t

interfaces with a corner point in one dimensional porous medium flow we prove that for a large class of initial distributions the solutions of the initial value problem for the one dimensional porous medium equation have interfaces which start to move abruptly after a positive waiting time this happens for example if the initial pressure is o x 2 as x 0 we also give sufficient conditions for the smoothness of the interface which improve previous results

potential symmetries of a porous medium equation potential symmetries which are not local symmetries are carried out for the porous medium equation where when it can be written in a conserved form these symmetries are realized as local symmetries of a related auxiliary system and lead to the construction of corresponding invariant solutions as well as to the linearization of the equation by non invertible mappings

the classification of leonard triples that have bannai ito type and odd diameter abstract let k denote an algebraically closed field of characteristic zero let v denote a vector space over k with finite positive dimension by a leonard triple on v we mean an ordered triple of linear transformations a a a e in end v such that for each b a a a e there exists a basis for v with respect to which the matrix representing b is diagonal and the matrices representing the other two linear transformations are irreducible tridiagonal the diameter of the leonard triple a a a e is defined to be one less than the dimension of v in this paper we define a family of leonard triples said to have bannai ito type we classify up to isomorphism the leonard triples that have bannai ito type and odd diameter we show that each of these leonard triples satisfies the z 3 symmetric askey wilson relations

large time solution of an inhomogeneous nonlinear diffusion equation the paper considers the large time solution of the nonlinear diffusion equation x u t x c x u u x 0 for initial data with compact support the asymptotic behaviour of x and c x as x defines a parameter space and the nature of the solution at large time depends crucially on the location therein the paper examines the nature and uniformity of the asymptotic solutions for x in a restricted region of the entire parameter space and in certain cases gives the leading error terms a new integral invariant of the solution is also given

bilinear operators and the power series for the weierstrass sigma function we use the bilinear operator formalism to derive a new representation of the power series for the weierstrass function

regularity of solutions and interfaces of the porous medium equation via local estimates on demontre la regularite c pour des interfaces mobiles de solutions locales de l equation de milieu poreux

multidimensional schrodinger equations with abelian potentials we consider two and three dimensional complex schrodinger equations with abelian potentials and a fixed energy level the potential wave function and the spectral bloch variety are calculated in terms of the kleinian hyperelliptic functions associated with a genus two hyperelliptic curve in the special case in two dimensions when the curve covers two elliptic curves exactly solvable schrodinger equations are constructed in terms of the elliptic functions of these curves the solutions obtained are illustrated by a number of plots

an implicit level set method for modeling hydraulically driven fractures we describe a novel implicit level set algorithm to locate the free boundary for a propagating hydraulic fracture a number of characteristics of the governing equations for hydraulic fractures and their coupling present considerable challenges for numerical modeling namely the degenerate lubrication equation the hypersingular elastic integral equation the indeterminate form of the velocity of the unknown fracture front which precludes the implementation of established front evolution strategies that require an explicit velocity field and the computationally prohibitive cost of resolving all the length scales an implicit algorithm is also necessary for the efficient solution of the stiff evolution equations that involve fully populated matrices associated with the coupled non local elasticity and degenerate lubrication equations the implicit level set algorithm that we propose exploits the local tip asymptotic behavior applicable at the computational length scale in order to locate the free boundary local inversion of this tip asymptotic relation yields the boundary values for the eikonal equation whose solution gives the fracture front location as well as the front velocity field the efficacy of the algorithm is tested by comparing the level set solution to analytic solutions for hydraulic fractures propagating in a number of distinct regimes the level set algorithm is shown to resolve the free boundary problem with first order accuracy further it captures the field variables such as the fracture width with the first order accuracy that is consistent with the piecewise constant discretization that is used

the shape of some linear transformations abstract recently ito and terwilliger proposed a problem about some linear transformations a a a a on a finite dimensional vector space v over a field f which can be seen as a generalization of the notions of tridiagonal pair and q inverting pair see problem 1 2 in this paper we will introduce the notion of the shape of the linear transformations a a a a our main results are the properties of the shape i the shape is symmetric and unimodal ii let denote an anti isomorphism from end v to end v with v a vector space over f with the same dimension as v assume that the linear transformations a a a a on v satisfy all conditions of problem 1 2 then the linear transformations a a a a also satisfy all conditions of problem 1 2 moreover a a a a have the same shape as a a a a

moving finite element methods for evolutionary problems ii applications abstract in this the second of two papers on the subject we present applications of the moving finite element method to a number of test problems key features are linear elements a direct approach to parallelism and node overtaking avoiding penalty functions rapid inversion of the mass matrix by preconditioned conjugate gradients and explicit euler time stepping the resulting codes are fast and efficient and are able to follow fronts and similar features with great accuracy the paper includes a substantial section on changes of dependent variable and front tracking techniques for non linear diffusion problems test problems include non linear hyperbolic conservation laws and non linear parabolic equations in one and two dimensions

abelian functions associated with a cyclic tetragonal curve of genus six we develop the theory of abelian functions defined using a tetragonal curve of genus six discussing in detail the cyclic curve y 4 x 5 4x 4 3x 3 2x 2 1x 0 we construct abelian functions using the multivariate function associated with the curve generalizing the theory of the weierstrass function we demonstrate that such functions can give a solution to the kp equation outlining how a general class of solutions could be generated using a wider class of curves we also present the associated partial differential equations satisfied by the functions the solution of the jacobi inversion problem a power series expansion for u and a new addition formula

waiting time solutions of a nonlinear diffusion equation similarity solutions of the equation partial c partial t partial partial x c n partial c partial x n 0 are studied for both t 0 and t 0 and there is a stationary free boundary at x 0 and for t 0 have support in x at beta where beta 1 a 0 i e there is a moving free boundary x at beta by prescribing appropriate initial data c c 0 x x 0 at t t 0 these solutions demonstrate the existence of a waiting time t 0 before the free boundary moves a comparison theorem then enables a number of results to be obtained and conjectures to be made about the existence and length of waiting times for general initial data a three parameter family of solutions which exhibit blow up at t 0 is also constructed applications of these results to models for the spreading of a viscous blob on a hor

some properties of sharp hessenberg pairs abstract let f denote a field and let v denote a vector space over f with finite positive dimension we consider an ordered pair of f linear transformations a v v and a v v that satisfy the following conditions i each of a a is diagonalizable on v ii there exists an ordering v i i 0 d of the eigenspaces of a such that a v i v 0 v 1 v i 1 for 0 i d where v 1 0 and v d 1 0 iii there exists an ordering v i i 0 of the eigenspaces of a such that av i v 0 v 1 v i 1 for 0 i where v 1 0 and v 1 0 we call such a pair a hessenberg pair on v it is known that if the hessenberg pair a a on v is irreducible then d and for 0 i d the dimensions of v i and v d i coincide we say a hessenberg pair a a on v is sharp whenever it is irreducible and dim v 0 dim v d 1 in this paper we give the definitions of a hessenberg system and a sharp hessenberg system we discuss the connection between a hessenberg pair and a hessenberg system we also define a finite sequence of scalars called the parameter array for a sharp hessenberg system which consists of the eigenvalue sequence the dual eigenvalue sequence and the split sequence we calculate the split sequence of a sharp hessenberg system we show that a sharp hessenberg pair is a tridiagonal pair if and only if there exists a nonzero nondegenerate bilinear form on v that satisfies au v u av and a u v u a v for all u v v

hyperelliptic solutions of kdv and kp equations re evaluation of baker s study on hyperelliptic sigma functions explicit function forms of hyperelliptic solutions of korteweg de vries kdv and kadomtsev petviashvili kp equations are constructed for a given curve y2 f x whose genus is three this paper is based upon the fact that about one hundred years ago baker h f 1903 acta math 27 135 56 baker essentially derived kdv hierarchy and kp equations by using a bilinear differential operator d identities of pfaffians symmetric functions the hyperelliptic function and functions log d d 2 2 the connection between his theory and the modern soliton theory is also discussed

leonard triples extended from a given totally almost bipartite leonard pair of bannai ito type let kk denote a field of characteristic zero and let d denote an integer at least 3 let a 1 d 02d 10102d202d 13 d 3d 10d 20dd 1 and a diag 1 d 2d 1 7 5 3 1 a diag 1 d 2d 1 7 5 3 1 be two matrices in matd 1 k matd 1 k then a a a a is a totally almost bipartite leonard pair on kd 1kd 1 of bannai ito type in this paper we determine all the matrices ae matd 1 k ae matd 1 k such that a a aea a ae form a leonard triple on kd 1kd 1

thin hessenberg pairs abstract a square matrix is called hessenberg whenever each entry below the subdiagonal is zero and each entry on the subdiagonal is nonzero let v denote a nonzero finite dimensional vector space over a field k we consider an ordered pair of linear transformations a v v and a v v which satisfy both i and ii below i there exists a basis for v with respect to which the matrix representing a is hessenberg and the matrix representing a is diagonal ii there exists a basis for v with respect to which the matrix representing a is diagonal and the matrix representing a is hessenberg we call such a pair a thin hessenberg pair or th pair this is a special case of a hessenberg pair which was introduced by the author in an earlier paper we investigate several bases for v with respect to which the matrices representing a and a are attractive we display these matrices along with the transition matrices relating the bases we introduce an oriented version of a a called a th system we classify the th systems up to isomorphism

new classes of similarity solutions of the inhomogeneous nonlinear diffusion equations the lie similarity method has been used to extend the similarity solutions of the one dimensional inhomogeneous nonlinear diffusion equations we determine the lie point symmetry vector fields and calculate the similarity ansatz then we discuss the resulting nonlinear ordinary differential equations exact solutions are found and their relations to some real physical situations are discussed

nonclassical symmetries of a porous medium equation with absorption in this paper new symmetry reductions and exact solutions are presented for the porous medium equation with absorption those spatial forms for which the equation can be reduced to an ordinary differential equation are studied the symmetry reductions and exact solutions presented are derived by using the nonclassical method developed by bluman and cole and are unobtainable by lie classical method

hyperelliptic solutions of modified korteweg de vries equation of genus g essentials of the miura transformation explicit hyperelliptic solutions of the modified korteweg de vries equations without any ambiguous parameters were constructed in terms of only the hyperelliptic al functions over the non degenerate hyperelliptic curve y2 f x of arbitrary genus g in the derivation any functions or baker akhiezer functions were not essentially used then the miura transformation naturally appears as the connection between the hyperelliptic functions and hyperelliptic al functions

invariant solutions for the nonlinear diffusion model of turbulence we study leith s model of turbulence represented by a nonlinear degenerate q1 diffusion equation leith 1967 phys fluids 10 1409 16 connaughton and nazarenko 2004 phys rev lett 92 044501 506 the model is constructed such that in the case of vanishing viscosity there are two steady state solutions the kolmogorov spectrum that corresponds to the cascade state and a thermodynamic equilibrium distribution using group analysis we have obtained integral equations which describe all essentially different invariant q2 solutions of the leith equation with or without viscosity the integral equations defining these solutions reveal new possibilities for analytical and numerical studies in these equations the presence of arbitrary constants allows one to solve them for different boundary conditions we have proved existence and uniqueness for such boundary value problems