a pieri type formula for isotropic flag manifolds we give the formula for multiplying a schubert class on an odd orthogonal or symplectic flag manifold by a special schubert class pulled back from the grassmannian of maximal isotropic subspaces this is also the formula for multiplying a type b respectively type c schubert polynomial by the schur p polynomial p m respectively the schur q polynomial q m geometric constructions and intermediate results allow us to ultimately deduce this formula from formulas for the classical flag manifold these intermediate results are concerned with the bruhat order of the infinite coxeter group b identities of the structure constants for the schubert basis of cohomology and intersections of schubert varieties we show that most of these identities follow from the pieri type formula and our analysis leads to a new partial order on the coxeter group b and formulas for many of these structure constants

on varieties of groups satisfying an engel type identity abstract let m n be positive integers v a multilinear commutator word and w v m denote by v g and w g the verbal subgroups of a group g corresponding to v and w respectively we prove that the class of all groups g in which the w values are n engel and w g is locally nilpotent is a variety theorem a further we show that in the case where m is a prime power the class of all groups g in which the w values are n engel and v g has an ascending normal series whose quotients are either locally soluble or locally finite is a variety theorem b we examine the question whether the latter result remains valid with m allowed to be an arbitrary positive integer in this direction we show that if m n are positive integers u a multilinear commutator word and v the product of 896 u words then the class of all groups g in which the v m values are n engel and the verbal subgroup u g has an ascending normal series whose quotients are either locally soluble or locally finite is a variety theorem c

an approximate solution based on jacobi polynomials for time fractional convectiondiffusion equation operational matrix of shifted jacobi polynomials is considered solution of time fractional order convectiondiffusion problem is numerically estimated main problem is converted to a homogeneous problem by interpolation and afterward an integro differential equation is yielded a system of nonlinear algebraic equations is achieved by approximating the known and unknown functions with the help of shifted jacobi functions problem can be used extensively in science and engineering as in oil reservoir simulations in this article we present a numerical method to numerically solve a time fractional convectiondiffusion equation our method is based on the operational matrices of shifted jacobi polynomials at first problem is converted to a homogeneous problem by interpolation and afterward an integro differential equation is yielded then we approximate the known and unknown functions with the help of shifted jacobi functions a system of nonlinear algebraic equations is obtained finally the unknown coefficients are determined by mathematicatm we implemented the proposed method for several examples that they indicate the high accuracy method it should be noted that this method is generalizable to some appropriate problems

on the convergence of a high accuracy conservative scheme for the zakharov equations abstract in this paper a high accuracy conservative difference scheme is presented to solve the initial boundary value problem of the zakharov equations which preserves the original conservative properties the proposed scheme is based on finite difference method the scheme is second order accuracy in time and fourth order accuracy in space a detailed numerical analysis of the scheme is presented including a convergence analysis result numerical examples are given to confirm the proposed scheme is efficient reliable and of high accuracy

stability and convergence of radial basis function finite difference method for the numerical solution of the reaction diffusion equations stability convergence and application of radial basis function finite difference rbf fd scheme is studied for solving the reaction diffusion equations rdes we show that the explicit rbf fd method is stable and stability condition depends on the shape parameter of related radial basis function the generalized multiquadric gmq is applied as radial basis function and weight coefficients are explicitly presented for equispaced node distribution also two methods are presented to compute the optimal shape parameter the combination of these methods with the gmq fd method will produce two efficient algorithms for numerical solution of rdes the variable gmq fd vgmq fd and the constant gmq fd cgmq fd we test the scheme on traveling wave and compare its accuracy with the conventional finite difference method fdm

a note on the gaussons of some new logarithmic evolution equations abstract in this paper we study the gaussian solitary waves for some nonlinear evolution equations with logarithmic nonlinearities these studied logarithmic evolution equations are the generalized logarithmic bbm equations the logarithmic 2 1 dimensional kp like equations the logarithmic 3 1 dimensional kp like equations the generalized logarithmic 2 1 dimensional klein gordon equations and the generalized logarithmic 3 1 dimensional klein gordon equations we not only prove that they possess gaussons solitary wave solutions of gaussian shape but also derive the relationships among the parameters

a generalized fitzhugh nagumo equation abstract in this paper we investigate mappings of the classical fitzhugh nagumo equation to a generalized fitzhugh nagumo equation these mappings are invertible and transform the solutions of the classical fitzhugh nagumo equation into solutions of the generalized fitzhugh nagumo equation considered here these mappings are found by considering the lie point symmetries admitted by the classical fitzhugh nagumo equation and the generalized fitzhugh nagumo equation considered here a particular example of a generalized fitzhugh nagumo equation that satisfies the boundary conditions of the classical fitzhugh nagumo equation is considered numerical solutions of the generalized fitzhugh nagumo equation that do not satisfy the boundary conditions of the classical fitzhugh nagumo equation are obtained by implementing the method of lines

on the geometry of algebraic groups and homogeneous spaces given a connected algebraic group g over an algebraically closed field and a g homogeneous space x we describe the chow ring of g and the rational chow ring of x with special attention to the picard group also we investigate the albanese and the anti affine fibrations of g and x

nonemptiness of symmetric degeneracy loci let v be a rank n vector bundle on a d dimensional complex projective scheme x assume that v is equipped with a quadratic form with values in a line bundle l and that s 2 v l is ample suppose that the maximum rank of the quadratic form at any point of x is r 0 the main result of this paper is that if d n r the locus of points where the rank of the quadratic form is at most r 1 is nonempty we give some applications to subschemes of matrices and to degeneracy loci associated to embeddings in projective space the paper concludes with an appendix on gysin maps the main result of the appendix which may be of independent interest identifies a gysin map with the natural map from ordinary to relative cohomology

variations on the baer suzuki theorem the baer suzuki theorem says that if p is a prime x is a p element in a finite group g and langle x x g rangle is a p group for all g in g then the normal closure of x in g is a p group we consider the case where x g is replaced by y g for some other p element y while the analog of baer suzuki is not true we show that some variation is we also answer a closely related question of pavel shumyatsky on commutators of conjugacy classes of p elements

chow groups of quotient varieties let g be a reductive algebraic group operating on a scheme x with quotient y x g let ch x be the chow group of x ch y the chow group of y as defined in f in many concrete cases e g binary quantics or certain subspaces of moduli spaces of curves x has a simple structure and we have a lot of information on ch x while ch y is more interesting and much harder to calculate so it is natural to ask what relation is there between ch x and ch y in this article we give a partial answer to this question with the following restrictions we assume that the stabilizer of each point of x is finite and we only consider chow groups with rational coefficients furthermore we assume that x is an algebraic scheme over an algebraically closed field of characteristic 0 set a ch q where q is the field of rational numbers if g is a finite group then a y a x see f example 1 7 61 thus it is only necessary to consider the case in which g is connected theorem 1 says that in this case there is a natural surjective homomorphism a y a x this means that a y is at least as large as a x to be able to bound the kernel of the homomorphism a y a x we make two further assumptions that x is normal and that g cl n or sl n the case g sl n includes the case g pgl n which the one usually found in applications the case of a principal gl n bundle is treated in sga6 exposi ix proposition 4 11 considering the k group of coherent sheaves instead of the chow group what is proved there amounts in the present contest to the following let c c be the chern classes of the rank n vector bundle

character tables of parabolic subgroups of steinberg s triality groups we compute the conjugacy classes of elements and the character tables of the parabolic subgroups of steinberg s triality groups d43 q where q is a power of an odd prime our main tools are clifford theory applied to the levi decomposition of the parabolic subgroups and the decomposition of restrictions of the unipotent characters of d43 q for many of the calculations we use the chevie package

the chow ring of the classifying space bso 2n c we compute the chow ring of the classifying space bso 2n c in the sense of totaro using the fibration gl 2n so 2n bso 2n bgl 2n and a computation of the chow ring of gl 2n so 2n in a previous paper we find this chow ring is generated by chern classes and a characteristic class defined by edidin and graham which maps to 2n 1 times the euler class under the usual class map from the chow ring to ordinary cohomology moreover we show this class represents 1 2n 1 n 1 times the nth chern class of the representation of so 2n whose highest weight vector is twice that of the half spin representation

modified method of simplest equation and its applications to the bogoyavlenskii equation in this paper we construct the exact traveling wave solutions of the bogoyavlenskii equation using modified method of simplest equation the simplest equation herewith is a special riccati equation in the meantime the proposed method in this work is proved to be a powerful mathematical tool for obtaining exact solutions of other nonlinear partial differential equations in mathematical physics and other research fields

new exact solutions to the fitzhugh nagumo equation abstract by the first integral method a series of new exact solutions of the fitzhugh nagumo equation have been obtained it is shown that this method is one of the most effective approaches to obtain the exact solutions of the nonlinear evolution equations especially for nonintegrable models

analytic and approximate solutions for nagumo telegraph reaction diffusion equation it is known now that telegraph equation is more suitable than ordinary diffusion equation in modelling reaction diffusion in several branches of sciences we generalized nagumo equation to the telegraph reaction diffusion equation and present its analytic approximate solution a by using picard iteration method also using the linear piecewise approximation traveling wave solution are presented the results corresponding to the standard nagumo reaction diffusion equation are regained

characteristic classes and quadric bundles in this paper we construct stiefel whitney and euler classes in chow cohomology for algebraic vector bundles with nondegenerate quadratic form these classes are not in the algebra generated by the chern classes of such bundles and are new characteristic classes in algebraic geometry on complex varieties they correspond to classes with the same name pulled back from the cohomology of the classifying space bso n c the classes we construct are the only new characteristic classes in algebraic geometry coming from the classical groups t2 eg we begin by using the geometry of quadric bundles to study chern classes of maximal isotropic subbundles if v x is a vector bundle with quadratic form and if e and f are maximal isotropic subbundles of v then we prove theorem 1 that ci e and ci f are equal mod 2 moreover if the rank of v is 2n then cn e cn f proving a conjecture of fulton we define stiefel whitney and euler classes as chow cohomology classes which pull back to chern classes of maximal isotropic subbundles of the pullback bundle using the above theorem we show theorem 2 that these classes exist and are unique even though v need not have a maximal isotropic subbundle these constructions also make it possible to give schubert presentations

the tanh coth method for solitons and kink solutions for nonlinear parabolic equations abstract the tanh coth method is used to derive solitons and kink solutions for some of the well known nonlinear parabolic partial differential equations the equations include the fisher equation newell whithead equation allen cahn equation fitzhugh nagumo equation fisher s equation and the burgers fisher equation the new tanh coth approach provides abundant solitons and kink solutions in addition to the existing ones the power of this manageable method is confirmed

modified simple equation method for nonlinear evolution equations abstract this paper reflects the implementation of a reliable technique which is called modified simple equation method msem for solving evolution equations the proposed algorithm has been successfully tested on two very important evolution equations namely fitzhugh nagumo equation and sharma tasso olver equation numerical results are very encouraging

products of conjugacy classes in finite and algebraic simple groups abstract we prove the arad herzog conjecture for various families of finite simple groups if a and b are nontrivial conjugacy classes then a b is not a conjugacy class we also prove that if g is a finite simple group of lie type and a and b are nontrivial conjugacy classes either both semisimple or both unipotent then a b is not a conjugacy class we also prove a strong version of the arad herzog conjecture for simple algebraic groups and in particular show that almost always the product of two conjugacy classes in a simple algebraic group consists of infinitely many conjugacy classes as a consequence we obtain a complete classification of pairs of centralizers in a simple algebraic group which have dense product a special case of this has been used by prasad to prove a uniqueness result for tits systems in quasi reductive groups our final result is a generalization of the baer suzuki theorem for p elements with p 5

approximate conditional symmetries and approximate solutions of the perturbed fitzhugh nagumo equation we consider a perturbation of the fitzhugh nagumo equation the perturbation is proportional to the electric potential across the cell membrane the purpose of this investigation is to determine the effects of a change in electric potential across the cell membrane exact solutions of the perturbed equation are easily obtained from the well known solutions of the unperturbed fitzhugh nagumo equation the method of approximate conditional symmetries is used to obtain first order approximate solutions of the perturbed fitzhugh nagumo equation the approximate solutions are compared with the exact solutions of the perturbed equation the exact solutions of the perturbed equation do not indicate a change in the wave front connecting one constant state to another there is only a proportional increase or decrease in the constant nonzero state the approximate solutions do show a change in the shape of the wave front connecting two constant states as well as a proportional increase or decrease in the constant nonzero state

k theoretic chern class formulas for vexillary degeneracy loci abstract using raising operators and geometric arguments we establish formulas for the k theory classes of degeneracy loci in classical types we also find new determinantal and pfaffian expressions for classical cases considered by giambelli the loci where a generic matrix drops rank and where a generic symmetric or skew symmetric matrix drops rank in an appendix we construct a k theoretic euler class for even rank vector bundles with quadratic form refining the chow theoretic class introduced by edidin and graham we also establish a relation between top chern classes of maximal isotropic subbundles which is used in proving the type d degeneracy locus formulas

products and commutators of classes in algebraic groups we classify pairs of conjugacy classes in almost simple algebraic groups whose product consists of finitely many classes this leads to several interesting families of examples which are related to a generalization of the baer suzuki theorem for finite groups we also answer a question of pavel shumyatsky on commutators of pairs of conjugacy classes in simple algebraic groups it turns out that the resulting examples are exactly those for which the product also consists of only finitely many classes

steenrod operations on the chow ring of a classifying space we use the steenrod algebra to study the chow ring ch bg of the classifying space of an algebraic group g we describe a localization property which relates a given g to its elementary abelian subgroups and we study a number of particular cases namely symmetric groups and chevalley groups it turns out that the chow rings of these groups are completely determined by the abelian subgroups and their fusion

a jacobi gauss lobatto collocation method for solving generalized fitzhugh nagumo equation with time dependent coefficients in this paper we propose a new jacobi gauss lobatto collocation method for solving the generalized fitzhugh nagumo equation the jacobi gauss lobatto points are used as collocation nodes for spatial derivatives the proposed method has the advantage of obtaining the solution in terms of the jacobi parameters a and b in addition the problem is reduced to a system of ordinary differential equations in time this system can be solved by any standard numerical techniques numerical solutions obtained by this method when compared with the exact solution reveal that the obtained solution produces high accurate results numerical results show that the proposed method is of high accuracy and is efficient for solving the generalized fitzhugh nagumo equation also the results demonstrate that the proposed method is powerful algorithm for solving the nonlinear partial differential equations

zero dimensional donaldson thomas invariants of calabi yau 4 folds abstract we study hilbert schemes of points on a smooth projective calabi yau 4 fold x we define dt 4 invariants by integrating the euler class of a tautological vector bundle l n against the virtual class we conjecture a formula for their generating series which we prove in certain cases when l corresponds to a smooth divisor on x a parallel equivariant conjecture for toric calabi yau 4 folds is proposed this conjecture is proved for smooth toric divisors and verified for more general toric divisors in many examples combining the equivariant conjecture with a vertex calculation we find explicit positive rational weights which can be assigned to solid partitions the weighted generating function of solid partitions is given by exp m q 1 where m q denotes the macmahon function

an efficient jacobi pseudospectral approximation for nonlinear complex generalized zakharov system in this paper we derive an efficient spectral collocation algorithm to solve numerically the nonlinear complex generalized zakharov system gzs subject to initial boundary conditions the jacobi pseudospectral approximation is investigated for spatial approximation of the gzs it possesses the spectral accuracy in space the jacobi gauss lobatto quadrature rule is established to treat the boundary conditions and then the problem with its boundary conditions is reduced to a system of ordinary differential equations in time variable this scheme has the advantage of allowing us to obtain the spectral solution in terms of the jacobi parameters and which therefore means that the algorithm holds a number of collocation methods as special cases finally two illustrative examples are implemented to assess the efficiency and high accuracy of the jacobi pseudo spectral scheme

a generalisation of a theorem of wielandt abstract in 1974 helmut wielandt proved that in a finite group g a subgroup a is subnormal if and only if it is subnormal in a g for all g g in this paper we prove that the subnormality of an odd order nilpotent subgroup a of g is already guaranteed by a seemingly weaker condition a is subnormal in g if for every conjugacy class c of g there exists c c for which a is subnormal in a c we also prove the following property of finite non abelian simple groups if a is a subgroup of odd prime order p in a finite almost simple group g then there exists a cyclic p subgroup of f g which does not normalise any non trivial p subgroup of g that is generated by conjugates of a