the complex geometry of weak piecewise smooth solutions of integrable nonlinear pde s of shallow water and dym type an extension of the algebraic geometric method for nonlinear integrable pde s is shown to lead to new piecewise smooth weak solutions of a class of n component systems of nonlinear evolution equations this class includes among others equations from the dym and shallow water equation hierarchies the main goal of the paper is to give explicit theta functional expressions for piecewise smooth weak solutions of these nonlinear pde s which are associated to nonlinear subvarieties of hyperelliptic jacobians

consistency for bi skew symmetric solutions to systems of generalized sylvester equations over a finite central algebra let be a finite dimensional central algebra with an involutorial antiautomorphism and char 2 n n be the set of all n n matrices over a aij n n is called bisymmetric if aij an i 1 n j 1 aji and biskewsymmetric if aij an i 1 n j 1 aji the following systems of generalized sylvester equations over a1x yb1 c1 i asx ybs cs a1xb1 c1xd1 e1 ii asxbs csxds es are considered necessary and sufficient conditions are given for the existence of constant solutions with bi skew symmetric constrains to i and ii as a particular case auxiliary results dealing with the system of sylvester equations are also presented

computational methods for linear matrix equations given the square matrices a b d e and the matrix c of conforming dimensions we consider the linear matrix equation a mathbf x e d mathbf x b c in the unknown matrix mathbf x our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing

an integrable equation governing short waves in a long wave model the dynamics of a nonlinear and dispersive long surface capillary gravity wave model equation is studied analytically in its short wave limit we exhibit its lax pair and some non trivial conserved quantities through a change of functions an unexpected connection between this classical surface water wave model and the sine gordon or sinh gordon equation is established numerical and analytical studies show that in spite of integrability their solutions can develop singularities and multivaluedness in finite time these features can be traced to the fact that the surface tension term in the energy involves second order derivatives it would be interesting to see in an experiment whether such singularities actually appear for which surface tension would be specifically responsible

dynamics of director fields an asymptotic equation for weakly nonlinear hyperbolic waves governed by variational principles is derived and analyzed the equation is used to study a nonlinear instability in the director field of a nematic liquid crystal it is shown that smooth solutions of the asymptotic equation break down in finite time also constructed are weak solutions of the equation that are continuous despite the fact that their spatial derivative blows up

a piecewise linear upper bound on the network recourse function we consider the optimal value of a pure minimum cost network flow problem as a function of supply demand and arc capacities we present a new piecewise linear upper bound on this function which is called the network recourse function the bound is compared to the standard madansky bound and is shown computationally to be a little weaker but much faster to find the amount of work is linear in the number of stochastic variables not exponential as is the case for the madansky bound therefore the reduction in work increases as the number of stochastic variables increases computational results are presented

roth s removal rule revisited abstract the theory of companion matrices is used to give explicit representations for the matrices needed in roth s removal rule these are then used to give simple proofs for the cyclic decomposition theorem as well as for roth s similarity theorem for matrices over a field

the extension of roth s theorem for matrix equations over a ring abstract this paper extends roth s similarity theorem as follows let r be a ring with identity b ki 0 bi i rm q if either r is a division ring and a rn n is algebraic or r is finitely generated as a module over its center then the matrix equation ki 0 aixbi c over r has a solution if and only if

single commodity network design with random edge capacities this paper examines the single commodity network design problem with stochastic edge capacities we characterize the structures of the optimal designs and compare with the deterministic counterparts we do this partly to understand what constitutes robust network designs but also to construct a heuristic for the stochastic problem leading to optimality gaps of about 10 in our view that is a rather good result for problems that otherwise cannot be solved at all

the matrix equation x axb c and an analogue of roth s theorem abstract we give a necessary and sufficient condition for the consistency of the matrix equation x axb c

global weak solutions for a new periodic integrable equation with peakon solutions abstract we prove the existence of global weak solutions for a new periodic integrable equation with peakon solutions

well posedness global existence and blowup phenomena for a periodic quasi linear hyperbolic equation we establish the local well posedness of a recently derived model for small amplitude shallow water waves for a large class of initial data we prove global existence of the corresponding solution criteria guaranteeing the development of singularities in finite time for strong solutions with smooth initial data are obtained and an existence and uniqueness result for a class of global weak solutions is also given c 1998 john wiley sons inc

simultaneous solutions of matrix equations and simultaneous equivalence of matrices given matrices ai bi and ci i i of corresponding dimensions over a field f we prove that i if aiciobi are simultaneously similar to aioobi then there exists a simultaneous solution x to the matrix sylvester equations aix xbi ci and ii if aiciobi are simultaneously equivalent to aioobi then there exist simultaneous solutions x y to the matrix equations aix ybi ci we also show that analogous results hold for mixed pairs of matrix sylvester equations a1x1 yb1 c1 a2x2 yb2 c2 and for generalized stein equations x ayb c

two sided coupled generalized sylvester matrix equations solving using a simultaneous decomposition for fifteen matrices abstract in this paper we investigate and analyze in detail the structure and properties of a simultaneous decomposition for fifteen matrices a i c p i t i b i c s i q i c i c p i t i 1 d i c s i 1 q i and e i c p i q i i 1 2 3 we show that from this simultaneous decomposition we can derive some necessary and sufficient conditions for the existence of a solution to the system of two sided coupled generalized sylvester matrix equations with four unknowns a i x i b i c i x i 1 d i e i i 1 2 3 apart from proving an expression for the general solutions to this system we derive the range of ranks of these solutions using the ranks of the given matrices a i b i c i d i and e i we provide some numerical examples to illustrate our results moreover we present a similar approach to consider the simultaneous decomposition for 5k matrices and the system of k two sided coupled generalized sylvester matrix equations with k 1 unknowns a i x i b i c i x i 1 d i e i i 1 k k 4 the main results are also valid over the real number field and the real quaternion algebra

existence and singularities of solutions to an integrable equation governing short waves in a long wave model this paper is concerned with the cauchy problem of a new periodic integrable equation governing short waves in a long wave model we first establish the local well posedness for its derivative equation which correspondingly implies the local existence and multivaluedness of solutions to the model then we obtain a blow up scenario for these solutions finally we show that the equation has strong solutions that develop singularities in finite time

on a completely integrable nonlinear hyperbolic variational equation abstract we show that the nonlinear partial differential equation u t uu x xx 1 2 u x 2 x is a completely integrable bi variational bi hamiltonian system the corresponding equation for w u xx belongs to the harry dym hierarchy this equation arises in two different physical contexts in two nonequivalent variational forms it describes the propagation of weakly nonlinear orientation waves in a massive nematic liquid crystal director field and it is the high frequency limit of the camassa holm equation which is an integrable model equation for shallow water waves using the bi hamiltonian structure we derive a recursion operator a lax pair and an infinite family commuting hamiltonian flows together with the associated conservation laws we also give the transformation to action angle coordinates smooth solutions of the partial differential equation break down because their derivative blows up in finite time nevertheless the hamiltonian structure and complete integrability appear to remain valid globally in time even after smooth solutions break down we show this fact explicitly for finite dimensional invariant manifolds consisting of conservative piecewise linear solutions we compute the bi hamiltonian structure on this invariant manifold which is obtained by restricting the bi hamiltonian structure of the partial differential equation

roth s solvability criteria for the matrix equations ax xˆb c and x axˆb c over the skew field of quaternions with an involutive automorphism q qˆ abstract the matrix equation a x x b c has a solution if and only if the matrices a c 0 b and a 0 0 b are similar this criterion was proved over a field by w e roth 1952 and over the skew field of quaternions by huang liping 1996 h k wimmer 1988 proved that the matrix equation x a x b c over a field has a solution if and only if the matrices a c 0 i and i 0 0 b are simultaneously equivalent to a 0 0 i and i 0 0 b we extend these criteria to the matrix equations a x x b c and x a x b c over the skew field of quaternions with a fixed involutive automorphism q q

restarted global fom and gmres algorithms for the stein like matrix equation x m x c abstract in this paper we propose the restarted global full orthogonalization method gl fom and global generalized minimum residual gl gmres method to solve the stein like matrix equation x m x c with m x a x b m x a x b m x a x b or m x a x h b respectively where x is an unknown matrix to be solved first by using a real inner product in complex matrix spaces a generalized global arnoldi process is given then we demonstrate how to employ the restarted gl fom and gl gmres algorithms for solving the stein like matrix equation the proposed algorithms take advantage of the shifted structure of the matrix equation and are implemented by the original coefficient matrices finally some numerical examples are given to illustrate the effectiveness with comparison to some existing methods

on the geometric approach to the motion of inertial mechanical systems according to the principle of least action the spatially periodic motions of one dimensional mechanical systems with no external forces are described in the lagrangian formalism by geodesics on a manifold configuration space the group d of smooth orientation preserving diffeomorphisms of the circle the periodic inviscid burgers equation is the geodesic equation on d with the l 2 right invariant metric however the exponential map for this right invariant metric is not a c 1 local diffeomorphism and the geometric structure is therefore deficient on the other hand the geodesic equation on d for the h 1 rightinvariant metric is also a re expression of a model in mathematical physics we show that in this case the exponential map is a c 1 local diffeomorphism and that if two diffeomorphisms are sufficiently close on d t heycan be joined by a unique length minimizing geodesic a state of the system is transformed to another nearby state by going through a uniquely determined flow that minimizes the energy we also analyse for both metrics the breakdown of the geodesic flow

roth s theorems and decomposition of modules it is shown that there is a connection between roth s theorems on similarity and equivalence of block triangular matrices and decomposition of modules the module property is that if m n mn then n is a summand of m this holds for any commutative ring if m is finitely presented new proofs of roth s theorems are given for commutative rings some results are established in the noncommutative case

global weak solutions and blow up structure for the degasperis procesi equation in this paper we study several qualitative properties of the degasperis procesi equation we first established the precise blow up rate and then determine the blow up set of blow up strong solutions to this equation for a large class of initial data we finally prove the existence and uniqueness of global weak solutions to the equation provided the initial data satisfies appropriate conditions

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out