dispersive blowup of solutions of generalized korteweg de vries equations abstract the strong effect of dispersion on short wavelength disturbances featured by the korteweg de vries equation and some of its generalizations is exploited to provide solutions of these equations that correspond to infinitely smooth initial data which exhibit a specified loss of spatial smoothness at particular times the points in space time at which smoothness is lost may even comprise an arbitrary discrete subset of the upper half plane x t x r t 0 our results are related to recent work on smoothing of solutions of such equations some of which are sharpened here and they show that in certain aspects these earlier results are not far from being optimal the theory makes use of new results concerning well posedness of such equations in weighted sobolov spaces and some detailed analysis of the linear korteweg de vries equation

the spectra of uniform hypertrees abstract in this paper we study the spectra of uniform hypertrees by using the generalized weighted incident matrix we show that is a nonzero eigenvalue of the hypertree h corresponding to an eigenvector with all elements nonzero if and only if is a root of the polynomial h i 0 m 1 i m i x m i r where m i is the number of matchings of order i in h

the maximum spectral radii of uniform supertrees with given degree sequences abstract this paper generalizes the breadth first search ordering on trees to k uniform supertrees by using 2 switch operation and edge moving operation on supertrees we determine the unique k uniform supertree with the maximum spectral radius among all k uniform supertrees with a given degree sequence moreover we generalize the majorization theorem on trees to k uniform supertrees

on a class of solutions of kdv abstract practically every book on the inverse scattering transform method for solving the cauchy problem for kdv and other integrable systems refers to this method as nonlinear fourier transform if this is indeed so the method should lead to a nonlinear analogue of the fourier expansion formula u t x u k e i kx k t dk in this paper a special class of solutions of kdv whose role is similar to that of e i kx k t is discussed the theory of these solutions referred to here as harmonic breathers is developed and it is shown that these solutions may be used to construct more general solutions of kdv similarly to how the functions e i kx t are used to perform the same task in the theory of fourier transform a nonlinear superposition formula for general solutions of kdv similar to the fourier expansion formula is conjectured

an oriented hypergraphic approach to algebraic graph theory abstract an oriented hypergraph is a hypergraph where each vertex edge incidence is given a label of 1 or 1 we define the adjacency incidence and laplacian matrices of an oriented hypergraph and study each of them we extend several matrix results known for graphs and signed graphs to oriented hypergraphs new matrix results that are not direct generalizations are also presented finally we study a new family of matrices that contains walk information

on the principal eigenvectors of uniform hypergraphs abstract let a h be the adjacency tensor of r uniform hypergraph h if h is connected the unique positive eigenvector x x 1 x 2 x n t with x r 1 corresponding to spectral radius h is called the principal eigenvector of h the maximum and minimum entries of x are denoted by x max and x min respectively in this paper we investigate the bounds of x max and x min in the principal eigenvector of h meanwhile we also obtain some bounds of the ratio x i x j for i j n as well as the principal ratio h x max x min of h as an application of these results we finally give an estimate of the gap of spectral radii between h and its proper sub hypergraph h

a characterization of oriented hypergraphic balance via signed weak walks abstract an oriented hypergraph is a hypergraph where each vertex edge incidence is given a label of 1 or 1 and each adjacency is signed the negative of the product of the incidences an oriented hypergraph is balanced if the product of the adjacencies in each circle is positive we provide a combinatorial interpretation for entries of k th power of the oriented hypergraphic laplacian via the number of signed weak walks of length k using closed weak walks we prove a new characterization of balance for oriented hypergraphs and matrices that generalizes harary s theorem for signed graphs

joins of hypergraphs and their spectra abstract here we represent a general hypergraph by a matrix and study its spectrum we extend the definition of equitable partition and joining operation for hypergraphs and use those to compute eigenvalues of different hypergraphs we derive the characteristics polynomial of a complete m uniform m partite hypergraph k n 1 n 2 n m m studying edge corona of hypergraphs we find the complete spectrum of s loose cycles c l s n m for m 2 s 1 and the characteristics polynomial of s loose paths p l s n m some of the eigenvalues of p l s n m are also derived moreover using vertex corona we show how to generate infinitely many pairs of non isomorphic co spectral hypergraphs

a connection between hadamard matrices oriented hypergraphs and signed graphs abstract matrices associated to oriented hypergraphs produce a connection between signed graphs and hadamard matrices the existence of a family of signed graphs that are switching equivalent to k n and whose adjacency matrices sum to the zero matrix is shown to be equivalent to the existence of a hadamard matrix this equivalent problem is used to make explicit signed graph constructions which specialize to known hadamard constructions

balance 0 1 matrices bicoloring and total dual integrality a 0 1 matrixa is balanced if in every submatrix with two nonzero entries per row and column the sum of the entries is a multiple of four this definition was introduced by truemper 1978 and generalizes the notion of a balanced 0 1 matrix introduced by berge 1970 in this paper we extend a bicoloring theorem of berge 1970 and total dual integrality results of fulkerson hoffman and oppenheim 1974 to balanced 0 1 matrices

spectra of uniform hypergraphs abstract we present a spectral theory of uniform hypergraphs that closely parallels spectral graph theory a number of recent developments building upon classical work has led to a rich understanding of symmetric hyperdeterminants of hypermatrices a k a multidimensional arrays symmetric hyperdeterminants share many properties with determinants but the context of multilinear algebra is substantially more complicated than the linear algebra required to address spectral graph theory i e ordinary matrices nonetheless it is possible to define eigenvalues of a hypermatrix via its characteristic polynomial as well as variationally we apply this notion to the adjacency hypermatrix of a uniform hypergraph and prove a number of natural analogs of basic results in spectral graph theory open problems abound and we present a number of directions for further study

a general product of tensors with applications abstract we study a general product of two n dimensional tensors a and b with orders m 2 and k 1 this product satisfies the associative law and is a generalization of the usual matrix product using this product many concepts and known results of tensors can be simply expressed and or proved and a number of applications of it will be given using the associative law of this tensor product and some properties on the resultant of a system of homogeneous equations on n variables we define the similarity and congruence of tensors which are also the generalizations of the corresponding relations for matrices and prove that similar tensors have the same characteristic polynomials thus the same spectra we study two special kinds of similarity permutational similarity and diagonal similarity and their applications in the study of the spectra of hypergraphs and nonnegative irreducible tensors we also define the direct product of tensors in matrix case it is also called the kronecker product and give its applications in the study of the spectra of two kinds of the products of hypergraphs we also give applications of this general product in the study of nonnegative tensors including a characterization of primitive tensors the upper bounds of primitive degrees and the cyclic indices of some nonnegative irreducible tensors

some properties of the spectral radius for general hypergraphs abstract in this paper the adjacency tensor of a general hypergraph is investigated we study the perron frobenius theorem for the general hypergraphs and obtain some relevant results based on it in particular the techniques of weighted incidence matrix and moving edge are extended to general hypergraphs for determining the structure with the maximum spectral radius a nearly m uniform supertree is both connected and acyclic in which each edge contains either m 1 or m vertices to begin with the structures obtaining the maximum spectral radius in two classes of nearly uniform supertrees are determined where one class is with given number of edges and the other is with given number of vertices

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

modulation of multiphase waves in the presence of resonance the phenomenon of spatio temporal phase modulation made possible by resonance is investigated in detail through the analysis of an example problem a simple family of exact solutions to the ablowitz ladik equations is found to be modulationally stable in some regimes this family of solutions is determined by fixing antiperiod 2 boundary conditions which determines two wavenumbers within the family of solutions the frequencies do not depend on amplitude this feature ensures that the antiperiod 2 boundary conditions will be enforced under modulation the family of solutions is described by four parameters two being actions that foliate the phase space and two being macroscopically observable functions of the phase constants the modulation of the actions is described by a closed hyperbolic system of first order equations which is consistent with the full set of four genus 1 modulation equations the modulation of the phase information easily observed due to the presence of two resonances is described by two more equations that are driven by the actions the results are confirmed by numerical experiments

weak asymptotics method and the interaction of infinitely narrow δ solitons abstract we present a new method for studying the interaction of solitons for nonintegrable kdv type equations with small dispersion and test this method for the kdv equation

a characterization of oriented hypergraphic laplacian and adjacency matrix coefficients abstract an oriented hypergraph is an oriented incidence structure that generalizes and unifies graph and hypergraph theoretic results by examining its locally signed graphic substructure in this paper we obtain a combinatorial characterization of the coefficients of the characteristic polynomials of oriented hypergraphic laplacian and adjacency matrices via a signed hypergraphic generalization of basic figures of graphs additionally we provide bounds on the determinant and permanent of the laplacian matrix characterize the oriented hypergraphs in which the upper bound is sharp and demonstrate that the lower bound is never achieved

generic behavior of the density of states in random matrix theory and equilibrium problems in the presence of real analytic external fields the equilibrium measure in the presence of an external field plays a role in a number of areas in analysis for example in random matrix theory the limiting mean density of eigenvalues is precisely the density of the equilibrium measure typical behavior for the equilibrium measure is 1 it is positive on the interior of a finite number of intervals 2 it vanishes like a square root at endpoints and 3 outside the support there is strict inequality in the euler lagrange variational conditions if these conditions hold then the limiting local eigenvalue statistics is loosely described by a bulk in which there is universal behavior involving the sine kernel and edge effects in which there is a universal behavior involving the airy kernel through techniques from potential theory and integrable systems we show that this regular behavior is generic for equilibrium measures associated with real analytic external fields in particular we show that for any one parameter family of external fields v c the equilibrium measure exhibits this regular behavior except for an at most countable number of values of c we discuss applications of our results to random matrices orthogonal polynomials and integrable systems 2000 john wiley sons inc

the first two largest spectral radii of uniform supertrees with given diameter abstract a supertree is a connected and acyclic hypergraph let s m d k be the set of k uniform supertrees with m edges and diameter d in this paper we determine the supertrees with the first two largest spectral radii among all supertrees in s m d k for 3 d m 1

on the smoothing properties of solutions to the modified korteweg de vries equation abstract we study the smoothness of solutions of the initial value problem for the generalized korteweg de vries equation u t u k u x u xxx 0 k z k 2 the solution u can be written in its integral form u x t u 1 t u 2 t where u1 t and u2 t are the linear and the integral parts of u respectively it is established that for data in hs with s 1 u2 possesses better regularity in the persistence properties and the smoothing effects than the solution u our main tools are recent results obtained by c kenig g ponce and l vega well posedness and scattering results for the generalized korteweg de vries equation via the contraction principle comm pure appl math 46 1993 527 620 related to the well posedness of solutions of the generalized korteweg de vries equation

spectral radius of uniform hypergraphs abstract we prove a result concerning the behavior of the spectral radius of a hypergraph under relocations of edges we determine the unique hypergraphs with maximum spectral radius among connected k uniform hypergraphs with fixed number of pendant edges the unique k uniform hypertrees with respectively maximum second maximum and third maximum spectral radius the unique k uniform unicyclic hypergraphs k uniform linear unicyclic hypergraphs respectively with respectively maximum and second maximum spectral radius we also determine the unique hypergraphs with maximum spectral radius among k uniform unicyclic hypergraphs with given girth

spectral properties of complex unit gain graphs abstract a complex unit gain graph is a graph where each orientation of an edge is given a complex unit which is the inverse of the complex unit assigned to the opposite orientation we extend some fundamental concepts from spectral graph theory to complex unit gain graphs we define the adjacency incidence and laplacian matrices and study each of them the main results of the paper are eigenvalue bounds for the adjacency and laplacian matrices

weak asymptotic methods for scalar equations and systems abstract in this paper we show how one can construct families of continuous functions which satisfy asymptotically scalar equations with discontinuous nonlinearity and systems having irregular solutions this construction produces weak asymptotic methods which are issued from maslov asymptotic analysis we obtain a sequence of functions which tend to satisfy the equation s in the weak sense in the space variable and in the strong sense in the time variable to this end we reduce the partial differential equations to a family of ordinary differential equations in a classical banach space for scalar equations we prove that the initial value problem is well posed in the l 1 sense for the approximate solutions we construct then we prove that this method gives back the widely accepted solutions when they are known for systems we obtain existence in the general case and uniqueness in the analytic case using an abstract cauchy kovalevska theorem

algebraic varieties in the birkhoff strata of the grassmannian gr 2 harrison cohomology and integrable systems the local properties of the families of algebraic subsets wg in the birkhoff strata 2g of gr 2 containing the hyperelliptic curves of genus g are studied it is shown that the tangent spaces tg for wg are isomorphic to the linear spaces of 2 coboundaries particular subsets in wg are described by the integrable dispersionless coupled kdv systems of hydrodynamical type defining a special class of 2 cocycles and 2 coboundaries in tg it is demonstrated that the blows ups of such 2 cocycles and 2 coboundaries and gradient catastrophes for associated integrable systems are interrelated

some properties of the laplace and normalized laplace spectra of uniform hypergraphs abstract in 8 hu and qi studied the normalized laplace tensors and normalized laplace spectra of k uniform hypergraphs they also mentioned the question about whether or not 2 is also an h eigenvalue of the normalized laplace tensor of a k uniform hypergraph when 2 is an eigenvalue of the normalized laplace tensor in this case k is necessarily even in this paper we use an expression for the normalized laplace tensor in terms of the tensor product together with the diagonal similarity of tensors the perron frobenius theorem for nonnegative tensors and nonnegative weakly irreducible tensors and the concept and properties of odd colorable hypergraphs introduced in 13 to give a complete answer to this question we show that i when k 2 mod 4 then the answer to this question is affirmative ii when k 0 mod 4 then the answer to this question is negative and in this case we give an infinite family of counterexamples we also study the signless normalized laplace spectra and the signless normalized laplace h spectra of hypergraphs we give structural characterizations of the hypergraphs having the same normalized laplace spectrum and signless normalized laplace spectrum or having the same normalized laplace h spectrum and signless normalized laplace h spectrum or both finally we determine the first two k uniform supertrees of order n with the largest laplace spectral radii and also determine the unique k uniform hypertree of order n with the smallest laplace spectral radii in the case when k is even

analysis of axisymmetric free vibration of isotropic shallow spherical sheels with a circular hole abstract a general solution for the third order differential equations of a thin isotropic shallow spherical shell with a circular hole subject to axisymmetric free vibration is presented in this paper the frequency equation in terms of bessel functions and modified bessel functions is solved for the fundamental vibrational frequency and mode shape a finite element model is presented for comparison with the analytical solution the relationship between the vibrational frequency and mode shape and the size of the cutout is discussed

binary modulated oscillations in a semi discrete version of burgers equation abstract in this paper we study both analytically and numerically the binary modulated oscillations which arise in an integrable semi discrete approximation to inviscid burgers explicit solutions of the binary modulation equations derived in goodman and lax 1988 are found for several choices of initial data observations based on numerical experiments are presented as to which initial data lead to solutions containing purely binary oscillations comparisons are made to the integrable theory of the kac van moerbeke and toda lattices non integrable schemes for which the binary oscillations also remain hyperbolic are presented and explicit solutions of their binary modulation equations are found

from boltzmann to euler hilbert s 6th problem revisited abstract this article addresses the hydrodynamic limit of the boltzmann equation namely the compressible euler equations of gas dynamics an exact summation of the chapman enskog expansion originally given by gorban and karlin is the key to the analysis an appraisal of the role of viscosity and capillarity in the limiting process is then given where the analogy is drawn to the limit of the korteweg de vries burgers equations as a small parameter tends to zero

kinetic decomposition for singularly perturbed higher order partial differential equations abstract in this paper we consider singularly perturbed higher order partial differential equations we establish the condition under which the approximate solutions converge in a strong topology to the entropy solution of a scalar conservation laws using methodology developed in hwang and tzavaras comm partial differential equations 27 2002 1229 first we obtain the approximate transport equation for the given dispersive equations then using the averaging lemma we obtain the convergence

bifurcations of new eigenvalues for the benjamin ono equation a criterion for the emergence of new eigenvalues is found for the linear scattering problem associated with the benjamin ono bo equation this bifurcation occurs due to perturbations of nongeneric potentials which include the soliton solutions of the bo equation the asymptotic approximation of an exponentially small new eigenvalue is derived the method is based on the expansion of a localized function through a complete set of unperturbed eigenfunctions explicit expressions are obtained for the soliton potentials

hilbert s 6th problem exact and approximate hydrodynamic manifolds for kinetic equations the problem of the derivation of hydrodynamics from the boltz mann equation and related dissipative systems is formulated as the problem of slow invariant manifold in the space of distributions we review a few in stances where such hydrodynamic manifolds were found analytically both as the result of summation of the chapman enskog asymptotic expansion and by the direct solution of the invariance equation these model cases comprising grad s moment systems both linear and nonlinear are studied in depth in order to gain understanding of what can be expected for the boltzmann equa tion particularly the dispersive dominance and saturation of dissipation rate of the exact hydrodynamics in the short wave limit and the viscosity modifica tion at high divergence of the flow velocity are indicated as severe obstacles to the resolution of hilbert s 6th problem furthermore we review the derivation of the approximate hydrodynamic manifold for the boltzmann equation using newton s iteration and avoiding smallness parameters and compare this to the exact solutions additionally we discuss the problem of projection of the boltzmann equation onto the approximate hydrodynamic invariant manifold using entropy concepts finally a set of hypotheses is put forward where we describe open questions and set a horizon for what can be derived exactly or proven about the hydrodynamic manifolds for the boltzmann equation in the future

signed k uniform hypergraphs and tensors abstract a signed k graph is a k graph with an incidence function on each vertex and its incident edge in this paper we extend some concepts of the adjacency and laplacian tensor from k graphs to signed k graphs based on the spectral properties of the real symmetric tensor we investigate the h spectrum of adjacency and laplacian tensors for signed k graphs finally the inverse perron value of any vertex is present some bounds for the inverse perron value are established in terms of some invariants including the number of vertices edges the degree sequence

e cospectral hypergraphs and some hypergraphs determined by their spectra two k uniform hypergraphs are said to be cospectral e cospectral if their adjacency tensors have the same characteristic polynomial e characteristic polynomial a k uniform hypergraph h is said to be determined by its spectrum if there is no non isomorphic k uniform hypergraph cospectral with h in this note we give a method for constructing e cospectral hypergraphs which is similar with godsil mckay switching some hypergraphs are shown to be determined by their spectra

elastic wave scattering and dynamic stress concentrations in cylindrical shells with a circular cutout in this paper based on the theory of elastic wave motion for open cylindrical shell wave scattering and dynamic stress concentrations in open cylindrical shells with a hole are studied by making use of small parameter perturbation methods and boundary integral equation techniques the boundary integral equations and iterative imminent series of scattered waves around the cavity of the cylindrical shell are derived by employing this method the approximately analytical solutions of scattered waves on the edge of cutout are gained the computational formula for getting the dynamic stress concentration factors on the contour of cavity is developed as an example the numerical results of these dynamic stress concentration factors are graphically presented and discussed the analytical methods put forward in the present work have practical significances for solving the problem of elastic wave scattering and dynamic stress concentrations in cylindrical shells with a circular cutout

spectra of cycle and path families of oriented hypergraphs abstract an oriented hypergraph is a hypergraph where each vertex edge incidence is given a label of either 1 or 1 this generalizes signed graphs to a hypergraph setting and simultaneously provides a natural definition for a signed adjacency which is used to define the adjacency and laplacian matrices many properties of these matrices are known but there are no nontrivial families of oriented hypergraphs with their spectrum determined in this paper we define and study hypergraph families that are analogous to cycles and paths in graphs and signed graphs the adjacency and laplacian eigenvalues for some of these families is determined

distance signless laplacian spectral radius of uniform hypergraphs we determine the unique hypergraphs with minimum distance laplacian spectral radius among connected k uniform hypergraphs and k uniform hypertrees respectively we also determine the unique hypergraphs with minimum distance signless laplacian spectral radius among connected k uniform hypergraphs k uniform hypertrees and connected k uniform hypergraphs with given number of pendant edges respectively we propose a graft transformation for uniform hypergraphs that increases the distance laplacian distance signless laplacian respectively spectral radius and as applications we determine the unique power hypertrees with maximum distance laplacian distance signless laplacian respectively spectral radius