symmetries and algebras of the integrable dispersive long wave equations in 2 1 dimensional spaces similarly to the kadomtsev petviashvili kp equation a set of generalized symmetries with arbitrary functions of t is given by a simple constructable formula for the integrable dispersive long wave equations in 2 1 space dimensions these symmetries constitute an infinite dimensional lie algebra which is a generalization to the known winfinity algebra

algebraic k theory of group rings and the cyclotomic trace map we prove that the farrell jones assembly map for connective alge braic k theory is rationally injective under mild homological niteness condi tions on the group and assuming that a weak version of the leopoldt schneider conjecture holds for cyclotomic elds this generalizes a result of bokstedt hsiang and madsen and leads to a concrete description of a large direct summand of kn z g z q in terms of group homology in many cases the number theoretic conjectures are true so we obtain rational injectivity results about assembly maps in particular for whitehead groups under niteness assumptions on the group only the proof uses the cyclotomic trace map to topological cyclic homology bokstedt hsiang madsen s functor c and new general injectivity results about the assembly maps for topological hochschild homology and c

eta invariants with spectral boundary conditions we study the asymptotics of the heat trace where p is an operator of dirac type where f is an auxiliary smooth smearing function which is used to localize the problem and where we impose spectral boundary conditions using functorial techniques and special case calculations the boundary part of the leading coefficients in the asymptotic expansion is found

heat equation asymptotics of nonminimal operators on differential forms let d d d e be a natural operator on differential forms for example dv j k kvj c1 j kvk c2rjkvk has this form it will be shown that when the lower order term e is absent the integrated heat kernel coefficients an d can be simply expressed in terms of the familiar coefficients an j for the laplacian operators on forms the first two coefficients when a zeroth order term is present are evaluated

ends of maps ii versions of the finiteness obstruction and simple homotopy theory within e overx are developed this provides a setting for obstructions to the map analogs of the end ands cobordism theorems for manifolds these are applied to study equivariant mapping cylinder neighborhoods in topological group actions triangulations of locally triangulable spaces and block bundle structures on approximate fibrations

the existence and uniqueness of solutions of yang mills equations with bag boundary conditions the cauchy problem for the yang mills equations in the coulomb gauge is studied on a compact connected and simply connected riemannian manifold with boundary an existence and uniqueness theorem for the evolution equations is proven for fields with cauchy data in an appropriate sobolev space the proof is based the hodge decomposition of the yang mills fields and the theory of non linear semigroups

on a spin conformal invariant on manifolds with boundary let m be an n dimensional connected compact manifold with non empty boundary equipped with a riemannian metric g a spin structure and a chirality operator we define and study some properties of a spin conformal invariant given by lambda rm min m partial m underset overline g in g rm inf lambda 1 pm overline g rm vol m overline g frac 1 n where lambda 1 pm overline g is the smallest eigenvalue of the dirac operator under the chiral bag boundary condition mathbb b pm overline g more precisely we show that if n 2 then lambda rm min m partial m leq lambda rm min mathbb s n partial mathbb s n

travelling wave solutions for a generalized double dispersion equation abstract we analyze the generalization of a double dispersion equation by means of the theory of symmetry reductions of partial differential equations we consider travelling wave reductions depending on the form of an arbitrary function we present some explicit solutions soliton solutions kinks and anti kinks

heat kernel coefficients of the laplace operator on the d dimensional ball we present a very quick and powerful method for the calculation of heat kernel coefficients it makes use of rather common ideas as integral representations of the spectral sum mellin transforms non trivial commutation of series and integrals and skillful analytic continuation of zeta functions on the complex plane we apply our method to the case of the heat kernel expansion of the laplace operator on a d dimensional ball with either dirichlet neumann or in general robin boundary conditions the final formulas are quite simple using this case as an example we illustrate in detail our scheme which serves for the calculation of an in principle arbitrary number of heat kernel coefficients in any situation when the basis functions are known we provide a complete list of new results for the coefficients b3 b10 corresponding to the d dimensional ball with all the mentioned boundary conditions and d 3 4 5

on higher spin partition functions we observe that the partition function of the set of all free massless higher spins s 0 1 2 3 in flat space is equal to one the ghost determinants cancel against the physical ones or equivalently the regularized total number of degrees of freedom vanishes this reflects large underlying gauge symmetry and suggests analogy with supersymmetric or topological theory the z 1 property extends also to the ads background i e the 1 loop vacuum partition function of vasiliev theory is equal to 1 assuming a particular regularization of the sum over spins this was noticed earlier as a consistency requirement for the vectorial ads cft duality we find that z 1 is true also in the conformal higher spin theory with higher derivative kinetic terms expanded near flat or conformally flat s4 background we also consider the partition function of free conformal theory of symmetric traceless rank s tensor field which has 2 derivative kinetic term but only scalar gauge invariance in flat 4d space this non unitary theory has weyl invariant action in curved background and it corresponds to partially massless field in ads5 we discuss in detail the special case of s 2 or conformal graviton compute the corresponding conformal anomaly coefficients and compare them with previously found expressions for generic representations of conformal group in 4 dimensions

novel interacting phenomena in 2 1 dimensional akns system abstract by means of an new auxiliary equation method based on mapping method and the second order linear ordinary differential equation some types of variable separation solutions with two arbitrary lower dimensional functions of the 2 1 dimensional ablowitz kaup newell segur system are derived based on the derived variable separation excitation some new special types of non localized solutions such as doubly periodic wave and line periodic wave are constructed by choosing appropriate functions in addition we also investigate the annihilation phenomenon and non elastic interaction graphically

nonclassical analysis and painleve property for the kupershmidt equations by means of the nonclassical lie approach and the direct method all the similarity reductions of the kupershmidt equations are found two reduction equations which can also be obtained by the classical lie approach are the painleve type ii and iv respectively the other three reduction equations which cannot be obtained by the classical lie approach are only linear or riccati type

invariants of the heat equation for non minimal operators a special class of non minimal operators which are relevant for quantum field theory is introduced the general form of the heat kernel coefficients of these operators on manifolds without boundary is described new results are presented for the traces of the first two heat kernel coefficients for vector yang mills and perturbative gravity it is argued that non minimal operators can be used to define gauge fixing independent actions and solve the conformal mode problem in quantum gravity

residues of the eta function for an operator of dirac type abstract we compute the asymptotics of tr l 2 pe tp 2 where p is a first order operator of dirac type this is equivalent to evaluating the residues of the eta function

antisymmetric tensor fields on spheres functional determinants and non local counterterms the hodge de rham laplacian on spheres acting on antisymmetric tensor fields is considered explicit expressions for the spectrum are derived in a quite direct way confirming previous results associated functional determinants and the heat kernel expansion are evaluated using this method new non local counterterms in the quantum effective action are obtained which can be expressed in terms of betti numbers

new variable separation approach application to nonlinear diffusion equations the concept of the derivative dependent functional separable solution ddfss as a generalization to the functional separable solution is proposed as an application it is used to discuss the generalized nonlinear diffusion equations based on the generalized conditional symmetry approach as a consequence a complete list of canonical forms for such equations which admit the ddfss is obtained and some exact solutions to the resulting equations are described

smeared heat kernel coefficients on the ball and generalized cone we consider smeared zeta functions and heat kernel coefficients on the bounded generalized cone in arbitrary dimensions the specific case of a ball is analyzed in detail and used to restrict the form of the heat kernel coefficients an on smooth manifolds with boundary supplemented by conformal transformation techniques it is used to provide an effective scheme for the calculation of the an as an application the complete a5 2 coefficient is given

deformations of the riccati equation by using miura type transformations using some different miura type transformations a c integrable ordinary differential equation the riccati equation is deformed to some different s integrable models such as the 1 1 dimensional and 2 1 dimensional sinh gordon equations and mikhailov dodd bullough equations

darboux transformations and the symmetric fourth painleve equation this paper is concerned with the group symmetries of the fourth painlev equation piv a second order nonlinear ordinary differential equation it is well known that the parameter space of piv admits the action of the extended affine weyl group as shown by noumi and yamada the action of as b cklund transformations of piv provides a derivation of its symmetric form sp4 the dynamical system sp4 is also equivalent to the isomonodromic deformation of an associated three by three matrix linear system lax pair the action of the generators of on this lax pair is derived using the darboux transformation for an associated third order operator

heat kernel expansion for fermionic billiards in an external magnetic field using seeley s heat kernel expansion the authors compute the asymptotic density of states of the dirac operator coupled to a magnetic field on a two dimensional manifold with boundary fermionic billiard local boundary conditions compatible with vector current conservation depend on a free parameter alpha it is shown that the perimeter correction identically vanishes for alpha 0 in that case the next order constant term is found to be proportional to the euler characteristic of the manifold these results are independent of the external magnetic field and of the shape of the billiard provided the boundary is sufficiently smooth for the flat circular billiard the constant term is found to be 1 2 in agreement with a numerical result of berry and mondragon

on diagonal singularities of the green functions for schrodinger operators we investigate the behavior of the green functions of schrodinger operators near the diagonal the only nontrivial cases where the on diagonal singularities are nonzero and do not depend on the spectral parameter are two and three dimensions in the case of two dimensions we show that the singularity is independent of both the scalar and the gauge potentials in dimension three we obtain conditions for preserving the singularity under perturbations by nonregular potentials some examples illustrating dependence of the singularity on general scalar and gauge potentials are presented

lie symmetry analysis and exact explicit solutions for general burgers equation in this paper the lie symmetry analysis is performed for the general burgers equation the exact solutions and similarity reductions generated from the symmetry transformations are provided furthermore the all exact explicit solutions and similarity reductions based on the lie group method are obtained some new method and techniques are employed simultaneously such exact explicit solutions and similarity reductions are important in both applications and the theory of nonlinear science

spectral boundary conditions for generalizations of laplace and dirac operators spectral boundary conditions for laplace type operators on a compact manifold x with boundary are partly dirichlet partly oblique neumann conditions where the partitioning is provided by a pseudodifferential projection they have an interest in string and brane theory relying on pseudodifferential methods we give sufficient conditions for the existence of the associated resolvent and heat operator and show asymptotic expansions of their traces in powers and power log terms allowing a smearing function the leading log coefficient is identified as a non commutative residue which vanishes when 1 the study has new consequences for well posed spectral boundary problems for first order dirac like elliptic operators generalizing the atiyah patodi singer problem it is found e g that the zeta function is always regular at zero in the selfadjoint case there is a stability of the zeta function value and the eta function regularity at zero under perturbations of the boundary projection of order dim x

similarity reductions of partial differential equations the author determine the relation between two methods of reduction the similarity reduction method using nonclassical symmetry groups and the direct approach used by clarkson and kruskal 1989 they prove that the solutions which are obtained by similarity in correspondence to nonclassical groups constitute a larger family than the one obtained by the method of clarkson and kruskal the two procedures are equivalent only if the generators zeta x t u tau x t u eta x t u of the nonclassical groups are such that zeta tau is independent of u to explain these results he proves the existence of families of solutions of the burgers equation which are found by means of nonclassical symmetry reduction and which cannot be determined via the general reduction form of clarkson and kruskal

heat kernel coefficients for chiral bag boundary conditions we study the asymptotic expansion of the smeared l 2 trace of f e tp 2 where p is an operator of dirac type f is an auxiliary smooth smear ing function which is used to localize the problem and chiral bag boundary conditions are imposed special case calculations functorial methods and the theory of and invariants are used to obtain the boundary part of the heat kernel coefficientsa1 and a2

lie group classifications and exact solutions for two variable coefficient equations in this paper the lie symmetry analysis and group classifications are performed for two variable coefficient equations the hanging chain equation and the bond pricing equation the symmetries for the two equations are obtained the exact explicit solutions generated from the similarity reductions are presented moreover the exact analytic solutions are considered by the power series method

concrete construction and properties of the difference equation derived from the cellular automaton using the filtration technique following the proposal of a filtration technique by nobe satsuma and tokihiro we concretely construct partial difference equations which preserve any time evolution patterns of cellular automaton ca stably by the filtration technique we illustrate how to develop a method of filtration for applying to the typical two spatial dimensional ca rule the game of life and verify that the filtration method provides the stable difference equation associated with the ca compared with the inverse ultradiscretization besides in order to discuss whether the filtration technique can lead one to partial differential equations from ca rules we show a derivation of the burgers equation from rule 184 ca via the discrete burgers equation constructed by the filtration method as an example

spectral action for torsion with and without boundaries we derive a commutative spectral triple and study the spectral action for a rather general geometric setting which includes the skew symmetric torsion and the chiral bag conditions on the boundary the spectral action splits into bulk and boundary parts in the bulk we clarify certain issues of the previous calculations show that many terms in fact cancel out and demonstrate that this cancellation is a result of the chiral symmetry of spectral action on the boundary we calculate several leading terms in the expansion of spectral action in four dimensions for vanishing chiral parameter of the boundary conditions and show that 0 is a critical point of the action in any dimension and at all orders of the expansion

coefficients for the farrell jones conjecture abstract we introduce the farrell jones conjecture with coefficients in an additive category with g action this is a variant of the farrell jones conjecture about the algebraic k or l theory of a group ring rg it allows to treat twisted group rings and crossed product rings the conjecture with coefficients is stronger than the original conjecture but it has better inheritance properties since known proofs using controlled algebra carry over to the set up with coefficients we obtain new results about the original farrell jones conjecture the conjecture with coefficients implies the fibered version of the farrell jones conjecture

neutrino billiards time reversal symmetry breaking without magnetic fields a dirac hamiltonian describing massless spin half particles neutrinos moving in the plane r x y under the action of a 4 scalar not electric potential v r is in position representation h i h c v r z where x y and z are the pauli matrices acts on two component column spinor wavefunctions r 1 2 and has eigen values ck n does not possess time reversal symmetry t if v r describes a hard wall bounding a finite domain d billiards this is equivalent to a novel boundary condition for 2 1 t breaking is interpreted semiclassically as a difference of between the phases accumulated by waves travelling in opposite senses round closed geo desics in d with odd numbers of reflections the semiclassical large k asymptotics of the eigenvalue counting function spectral staircase n k are shown to have the weyl leading term ak 2 4 where a is the area of d but zero perimeter correction the dirac equation is transformed to an integral equation round the boundary of d and forms the basis of a numerical method for computing the k n when d is the unit disc geodesics are integrable and the eigenvalues which satisfy j l k n j l 1 k n are locally poisson distributed when d is an africa shape cubic conformal map of the unit disc the eigenvalues are locally distributed according to the statistics of the gaussian unitary ensemble of random matrix theory as predicted on the basis of t breaking and lack of geometric symmetry

new similarity reductions and painleve analysis for the symmetric regularised long wave and modified benjamin bona mahoney equations the author discusses similarity reductions and painleve analysis for the symmetric regularised long wave and modified benjamin bona mahoney equations 1972 both of which arise in several physical applications including shallow water waves both equations are thought to non integrable i e not solvable by inverse scattering since numerical studies show that the interaction of solitary waves in inelastic in particular we determine some new similarity reductions of the symmetric regularised long wave equation these new similarity reductions are not obtainable using the classical lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method which involves no group theoretical techniques it is shown that every similarity reduction of both the symmetric regularised long wave and modified benjamin bona mahoney equations obtained using the classical lie group method reduces the associated partial differential equation to an ordinary differential equation of painleve type whereas the new similarity solution of the symmetric regularised long wave equation reduces it to an ordinary differential equation which is not of painleve type it is also shown that neither the symmetric regularised long wave equation nor the modified benjamin bona mahoney equation possesses the painleve property for partial differential equations as defined by weiss et al 1983

gauge theories on manifolds with boundary the boundary value problem for laplace type operators acting on smooth sections of a vector bundle over a compact riemannian manifold with generalized local boundary conditions including both normal and tangential derivatives is studied the condition of strong ellipticity of this boundary value problem is formulated the resolvent kernel and the heat kernel in the leading approximation are explicitly constructed as a result the previous work in the literature on heat kernel asymptotics is shown to be a particular case of a more general structure for a bosonic gauge theory on a compact riemannian manifold with smooth boundary the problem of obtaining a gauge field operator of laplace type is studied jointly with local and gauge invariant boundary conditions which should lead to a strongly elliptic boundary value problem the scheme is extended to fermionic gauge theories by means of local and gauge invariant projectors after deriving a general condition for the validity of strong ellipticity for gauge theories it is proved that for euclidean yang mills theory and rarita schwinger fields all the above conditions can be satisfied for euclidean quantum gravity however this property no longer holds i e the corresponding boundary value problem is not strongly elliptic some non standard local formulae for the leading asymptotics of the heat kernel diagonal are also obtained it is shown that due to the lack of strong ellipticity the heat kernel diagonal is non integrable near the boundary

the painleve property a modified boussinesq equation and a modified kadomtsev petviashvili equation abstract it is shown that the modified boussinesq equation and the modified kadomtsev petviashvili equation both of which are known to be completely integrable possess the painlev e property for differential equations as defined by weiss tabor and carnevale

lie symmetry analysis and exact solutions for the short pulse equation abstract in this paper the lie symmetry analysis and the generalized symmetry method are performed for a short pulse equation spe the symmetries for this equation are given for the traveling wave solutions the exact parametric representations are investigated to guarantee the existence of the above solutions all parameter conditions are determined furthermore the exact analytic solutions are obtained by using the power series method

finite decomposition complexity and the integral novikov conjecture for higher algebraic k theory decomposition complexity for metric spaces was recently introduced by guentner tessera and yu as a natural generalization of asymptotic dimension we prove a vanishing result for the continuously controlled algebraic k theory of bounded geometry metric spaces with finite decomposition complexity this leads to a proof of the integral k theoretic novikov conjecture regarding split injectivity of the k theoretic assembly map for groups with finite decomposition complexity and finite cw models for their classifying spaces by work of guentner tessera and yu this includes all geometrically finite linear groups

new similarity solutions for the modified boussinesq equation in this paper the author presents some new similarity solutions of the modified boussinesq equation which is a completely integrable soliton equation these new similarity solutions include reductions to the second and fourth painleve equations which are not obtainable using the standard lie group method for finding group invariant solutions of partial differential equations they are determined using a new and direct method which involves no group theoretical techniques

comparison of three semi analytical methods for solving 1 1 dimensional dispersive long wave equations in this work we consider how adomian s decomposition method adm the homotopy analysis method ham and the homotopy perturbation method hpm can be used to investigate wave solutions of 1 1 dimensional dispersive long wave equations it is also worth noting that the advantage of the approximation of the series methodologies is a fast convergence of the solutions

the farrell jones conjecture for mapping class groups we prove the farrell jones conjecture for mapping class groups the proof uses the masur minsky theory of the large scale geometry of mapping class groups and the geometry of the thick part of teichmuller space the proof is presented in an axiomatic setup extending the projection axioms in bestvina et al publ math inst hautes etudes sci 122 1 64 2015 more specifically we prove that the action of mathrm mod sigma on the thurston compactification of teichmuller space is finitely mathcal f amenable for the family mathcal f consisting of virtual point stabilizers

zeta function for the laplace operator acting on forms in a ball with gauge boundary conditions the laplace operator acting on antisymmetric tensor fields in a d dimensional euclidean ball is studied gauge invariant local boundary conditions absolute and relative ones in the language of gilkey are considered the eigenfuctions of the operator are found explicitly for all values of d using in a row a number of basic techniques as mellin transforms deformation and shifting of the complex integration contour and pole compensation the zeta function of the operator is obtained from its expression in particular 0 and 0 are evaluated exactly a table is given in the paper for d 3 4 8 the functional determinants and casimir energies are obtained for d 3 4 6

scattering on compact manifolds with infinitely thin horns the quantum mechanical scattering on a compact manifold with semi axes attached to the manifold hedgehog shaped manifold is considered the complete description of the spectral structure of schrodinger operators on such a manifold is done the proof of existence and uniqueness of scattering states is presented an explicit form for the scattering matrix is obtained and unitarity of this matrix is proven it is shown that the positive part of the spectrum of the schrodinger operator on the initial compact manifold as well as the spectrum of a point perturbation of such an operator may be recovered from the scattering amplitude for one attached half line moreover the positive part of the spectrum of the initial schrodinger operator is fully determined by the conductance properties of an electronic device consisting of the initial manifold and two wires attached to it

new symmetry reductions and exact solutions of the davey stewartson system i reductions to ordinary differential equations in this article symmetry reductions and exact solutions are presented for the 2 1 dimensional davey stewartson ds system which has the completely integrable dsi and dsii systems as special cases these symmetry reductions are obtained using the direct method originally developed by clarkson and kruskal to study symmetry reductions of the boussinesq equation which involves no group theoretic techniques the ds system is reduced directly to ordinary differential equations with no intermediate step using these reductions exact solutions of the ds system including some expressible in terms of the second and fourth painleve equations and elementary functions are obtained

symmetries of differential equations from sophus lie to computer algebra the topic of this article is the symmetry analysis of differential equations and the applications of computer algebra to the extensive analytical calculations which are usually involved in it the whole area naturally decomposes into two parts depending on whether ordinary or partial differential equations are considered we show how a symmetry may be applied to lower the order of an ordinary differential equation or to obtain similarity solutions of partial differential equations the computer algebra packages sods and spde respectively which have been developed to perform almost all algebraic manipulations necessary to determine the symmetry group of a given differential equation are presented furthermore it is argued that the application of computer algebra systems has qualitatively changed this area of applied mathematics

heat kernels and functional determinants on the generalized cone we consider zeta functions and heat kernel expansions on the bounded generalized cone in arbitrary dimensions using an improved calculational technique the specific case of a global monopole is analysed in detail and some restrictions thereby placed on thea 5 2 coefficient the computation of functional determinants is also addressed general formulas are given and known results are incidentally and rapidly reproduced

mixed boundary conditions in quantum field theory when quantum field theory is formulated in terms of transition amplitudes between initial and final configurations it is important to impose mathematically consistent boundary conditions on the fields the choice of boundary conditions is also important when dealing with elliptic complexes and index theorems on bounded manifolds although dirichlet and neumann boundary conditions are adequate for many purposes they are not always appropriate however a number of the remaining cases can be handled by imposing dirichlet conditions on certain field components and neumann conditions on the others quantum gravity and supergravity spinor field theory and various elliptic complexes all lend themselves to mixed boundary conditions of this sort the mathematical significance of mixed boundary conditions is investigated and a simple rule is obtained for determining which field components should obey which boundary conditions it will be shown that gauge transformations on a bounded manifold give rise to intrinsic gauge transformations within the boundary this result is used to define gauge invariant and brs invariant quantum amplitudes for boundary configurations the most general form for mixed boundary conditions on the dirac operator and the de rham complex will also be derived

a notion of geometric complexity and its application to topological rigidity we introduce a geometric invariant called finite decomposition complexity fdc to study topological rigidity of manifolds we prove for instance that if the fundamental group of a compact aspherical manifold m has fdc and if n is homotopy equivalent to m then m n is homeomorphic to n n for n large enough this statement is known as the stable borel conjecture on the other hand we show that the class of fdc groups includes all countable subgroups of gl n k for any field k

new classes of symmetries for partial differential equations new classes of symmetries for partial differential equations are introduced by writing a given partial differential equation s in a conserved form a related system t with potentials as additional dependent variables is obtained the lie group of point transformations admitted by t induces a symmetry group of s new symmetries may be obtained for s that are neither point nor lie backlund symmetries they are determined by a completely algorithmic procedure significant new symmetries are found for the wave equation with a variable wave speed and the nonlinear diffusion equation

localized excitations in 2 1 dimensional systems by means of a special variable separation approach a common formula with some arbitrary functions has been obtained for some suitable physical quantities of various 2 1 dimensional models such as the davey stewartson ds model the nizhnik novikov veselov nnv system asymmetric nnv equation asymmetric ds equation dispersive long wave equation broer kaup kupershmidt system long wave short wave interaction model maccari system and a general n m component ablowitz kaup newell segur akns system selecting the arbitrary functions appropriately one may obtain abundant stable localized interesting excitations such as the multidromions lumps ring soliton solutions breathers instantons etc it is shown that some types of lower dimensional chaotic patterns such as the chaotic chaotic patterns periodic chaotic patterns chaotic line soliton patterns chaotic dromion patterns fractal lump patterns and fractal dromion patterns may be found in higher dimensional soliton systems the interactions between the traveling ring type soliton solutions are completely elastic the traveling ring solitons pass through each other and preserve their shapes velocities and phases some types of localized weak solutions peakons are also discussed especially the interactions between two peakons are not completely elastic after the interactions the traveling peakons also pass through each other and preserve their velocities and phases however they completely exchange their shapes

the coupled modified korteweg de vries equations similarity reduction lie backlund symmetries and integrability the lie point symmetries of n coupled modified korteweg de vries equations n cmkdv is derived and shown that the similarity reduction associated with symmetries passes the painleve property indicating its integrability the nonclassical symmetry analysis of bluman and cole and the direct method of clarkson and kruskal to n cmkdv equations show that there exists no new similarity reductions a sequence of lie backlund symmetries for n cmkdv equations is derived explicitly establishing its complete integrability the question of constructing the recursion operator of n cmkdv a characteristic of integrable systems is also discussed

derivative dependent functional separable solutions for the kdv type equations abstract a complete list of canonical forms for a special kdv type of equations which admit derivative dependent functional separable solutions is obtained by using the generalized conditional symmetry approach some exact solutions to the resulting equations are explicitly obtained

determinant structure of the rational solutions for the painleve iv equation rational solutions for the painleve iv equation are investigated by hirota bilinear formalism it is shown that the solutions in one hierarchy are expressed by 3 reduced schur functions and those in another two hierarchies by casorati determinants of the hermite polynomials or by a special case of the schur polynomials

spectral asymmetry on the ball and asymptotics of the asymmetry kernel let be the dirac operator on a d 2d dimensional ball with radius r we calculate the spectral asymmetry for d 2 and d 4 when local chiral bag boundary conditions are imposed with these boundary conditions we also analyse the small t asymptotics of the heat trace where p is an operator of dirac type and f is an auxiliary smooth smearing function

nonclassical symmetry reductions for the kadomtsev petviashvili equation abstract in this paper we discuss dimensional reductions of the physically and mathematically significant kadomtsev petviashvili equation in particular we exhibit some new reductions of the kadomtsev petviashvili equation which are obtained by a generalization of the direct method for determining similarity reductions recently developed by clarkson and kruskal j math phys 30 1989 2201 new solutions of the kadomtsev petviashvili equation are obtained in terms of weierstrass elliptic functions solutions of the lame equations and the first second and fourth painleve transcendents a group theoretical explanation of the new solutions is provided in terms of conditional symmetries leaving only specific subsets of solutions of the kadomtsev petviashvili equation invariant

a group theoretic analysis of the generalised gardner equation with arbitrary order nonlinear terms abstract in this paper we analyse a gardner equation with dual power law nonlinearity of any order which has been widely investigated due to its applications in quantum field theory solid state plasma and fluid physics a group theoretic approach is used to perform a comprehensive and detailed analysis of the equation we derive the symmetry generators of the equation in terms of its arbitrary parameters and used them to obtain symmetry reductions and exact solutions furthermore the conservation laws of the equation are derived via the noether approach after increasing the order and by the use of the multiplier method the importance of these conservation laws in finding exact solutions is proved via double reduction theory most of these solutions are new and contain many known solutions as special cases these solutions include important soliton solutions and nontrivial solutions in terms of special functions which are meromorphic in the entire complex plane since the gardner equation can model a variety of wave phenomena in plasma solid state and fluid physics these solutions possess significant features in the non linear mechanics aspects of the work they can also be used as a basis for solving other related model problems and assessing numerical and approximate analytical methods for nonlinear equations describing solitons in wave mechanics

painleve analysis and new analytic solutions for variable coefficient kadomtsev petviashvili equation with symbolic computation abstract the variable coefficient kadomtsev petviashvili kp equation is hereby under investigation painleve analysis is given out and an auto backlund transformation is presented via the truncated painleve expansion based on the auto backlund transformation new analytic solutions are given including the soliton like and periodic solutions it is also reduced to a 1 1 dimensional partial differential equation via classical lie group method and the painleve i equation by ck direct method

infinitesimal weak symmetries of nonlinear differential equations in two independent variables non classical infinitesimal weak symmetries of pde introduced by olver and rosenau 1987 are analysed in the case of pde in two independent variables it is demonstrated that obtaining such symmetries is equivalent to obtaining the two dimensional modules of non classical partial symmetries the boussinesq and the nonlinear heat equations are treated from the point of view of non classical symmetries

new solutions for solving boussinesq equation via potential symmetries method this work deals with the boussinesq equation that describes the propagation of the solitary waves with small amplitude on the surface of shallow water firstly the equation is written in a conserved form a potential function is then assumed reducing it to a system of partial differential equations the lie group method has been applied for determining symmetry reductions of the system of partial differential equations the solution of the problem by means of lie group method reduces the number of independent variables in the given partial differential equation by one leading to nonlinear ordinary differential equations the resulting non linear ordinary differential equations are then solved numerically using matlap package

linear groups of finite cohomological dimension our main result provides necessary and sufficient conditions for a finitelygenerated subgroup of gln c n 0 to have finite virtual cohomological dimension a group has finite virtual cohomological dimension vcd if it has a subgroup of finite index which has finite cohomological dimension this dimension is in fact the same for all torsion free subgroups of finite index it is of course necessary for a group r with vcd t q is independent of the choice of solvable series thus for a polycyclic group r h t is the number of infinite factors in a normal series with cyclic quotients we announce our main result

finite temperature properties of the dirac operator under local boundary conditions we study the finite temperature free energy and fermion number for dirac fields in a one dimensional spatial segment under two different members of the family of local boundary conditions defining a self adjoint euclidean dirac operator in two dimensions for one of such boundary conditions compatible with the presence of a spectral asymmetry we discuss in detail the contribution of this part of the spectrum to the zeta regularized determinant of the dirac operator and thus to the finite temperature properties of the theory

the fourth painleve equation and associated special polynomials in this article rational solutions and associated polynomials for the fourth painleve equation are studied these rational solutions of the fourth painleve equation are expressible as the logarithmic derivative of special polynomials the okamoto polynomials the structure of the roots of these okamoto polynomials is studied and it is shown that these have a highly regular structure the properties of the okamoto polynomials are compared and contrasted with those of classical orthogonal polynomials further representations are given of the associated rational solutions in the form of determinants through schur functions

the third painleve equation and associated special polynomials in this paper we are concerned with rational solutions algebraic solutions and associated special polynomials with these solutions for the third painleve equation piii these rational and algebraic solutions of piii are expressible in terms of special polynomials defined by second order bilinear differential difference equations which are equivalent to toda equations the structure of the roots of these special polynomials is studied and it is shown that these have an intriguing highly symmetric and regular structure using the hamiltonian theory for piii it is shown that these special polynomials satisfy pure difference equations fourth order bilinear differential equations as well as differential difference equations further representations of the associated rational solutions in the form of determinants through schur functions are given

green functions of higher order differential operators the green functions of the partial differential operators of even order acting on smooth sections of a vector bundle over a riemannian manifold are investigated via the heat kernel methods we study the resolvent of a special class of higher order operators formed by the products of second order operators of laplace type defined with the help of a unique riemannian metric but with different bundle connections and potential terms the asymptotic expansion of the green functions near the diagonal is studied in detail in any dimension as a by product a simple criterion for the validity of the huygens principle is obtained it is shown that all the singularities as well as the non analytic regular parts of the green functions of such high order operators are expressed in terms of the usual heat kernel coefficients ak for a special laplace type second order operator

symmetry reductions of a generalized cylindrical nonlinear schrodinger equation in this work symmetry reductions for a generalized cylindrical nonlinear schrodinger equation are presented these are obtained using an extension of the direct method originally developed by clarkson and kruskal 1989 which involves no group theoretic techniques

strong ellipticity and spectral properties of chiral bag boundary conditions we prove strong ellipticity of chiral bag boundary conditions on even dimensional manifolds from a knowledge of the heat kernel in an infinite cylinder some basic properties of the zeta function are analysed on cylindrical product manifolds of arbitrary even dimension

group invariant solutions of differential equations we introduce the concept of a weak symmetry group of a system of partial differential equations that generalizes the nonclassical method introduced by bluman and cole for finding group invariant