shift and torsion contact problems for arbitrary axisymmetric normal stress distributions abstract in this paper we analyze the contact interaction of axisymmetric particles subject to a subsequent application of a constant normal load and a tangential or rotational force a rigorous solution to the frictional contact problem is given by the known jager theorem that presents a relationship between shear and normal stress distributions provided the latter is an exact solution to a normal contact problem however in the case of strong loading when the normal displacement reaches a value of 5 10 of the spheres diameter exact solutions for the normal problem are absent some model concepts exist instead for instance the rod model describes strong normal loading of spheres as a sort of combination of the hertz problem weak loading of spheres with a compression of a pair of confined cylinder of the same radius as the hertz contact spot here we propose a method that is based on considerations similar to the jager theorem but is appropriate for any model or empirical normal stress distribution the resulting integral representations describe both shearing and torsion of prestressed particles with axisymmetric profiles rolling of particles as well as plasticity and adhesion of the particles material are not considered we also analyze the asymptotic behavior of the integral representations for weak and strong strains the obtained general solutions allow us to use the method of memory diagrams in order to calculate the reaction of the system on arbitrarily varying tangential or rotational actions

a two component generalization of the reduced ostrovsky equation and its integrable semi discrete analogue in the present paper we propose a two component generalization of the reduced ostrovsky vakhnenko equation whose differential form can be viewed as the short wave limit of a two component degasperis procesi dp equation they are integrable due to the existence of lax pairs moreover we have shown that the two component reduced ostrovsky equation can be reduced from an extended bkp hierarchy with negative flow through a pseudo 3 reduction and a hodograph reciprocal transform as a by product its bilinear form and n soliton solution in terms of pfaffians are presented one and two soliton solutions are provided and analyzed in the second part of the paper we start with a modified bkp hierarchy which is a backlund transformation of the above extended bkp hierarchy an integrable semi discrete analogue of the two component reduced ostrovsky equation is constructed by defining an appropriate discrete hodograph transform and dependent variable transformations in particular the backward difference form of above semi discrete two component reduced ostrovsky equation gives rise to the integrable semi discretization of the short wave limit of a two component dp equation their n soliton solutions in terms of pffafians are also provided

the cauchy problem for a generalized two component short pulse system with high order nonlinearities abstract this paper is concerned with the cauchy problem of a generalized short pulse system in the periodic setting local well posedness of this system is established in the besov space with s max 3 2 1 1 p it is then shown that the data to solution map is not uniformly continuous from b l r s b l r s to c 0 t b l r s c 0 t b l r s finally we provide a blow up result of the solution

computing tight upper bounds on the algebraic connectivity of certain graphs a generalized bethe tree is a rooted unweighted tree in which vertices at the same level have the same degree let b be a generalized bethe tree the algebraic connectivity of the generalized bethe tree b a tree obtained from the union of b and a tree t isomorphic to a subtree of b such that the root vertex of t is the root vertex of b a tree obtained from the union of r generalized bethe trees joined at their respective root vertices a graph obtained from the cycle cr by attaching b by its root to each vertex of the cycle and a tree obtained from the path pr by attaching b by its root to each vertex of the path is the smallest eigenvalue of a special type of symmetric tridiagonal matrices in this paper we first derive a procedure to compute a tight upper bound on the smallest eigenvalue of this special type of matrices finally we apply the procedure to obtain a tight upper bound on the algebraic connectivity of the above mentioned graphs

on the τ functions of the degasperis procesi equation the degasperis procesi dp equation is investigated from the point of view of determinant pfaffian identities the reciprocal link between the dp equation and the pseudo 3 reduction of the c two dimensional toda system is used to construct the n soliton solution of the dp equation the n soliton solution of the dp equation is presented in the form of pfaffian through a hodograph reciprocal transformation the bilinear equations the identities between determinants and pfaffians and the functions of the dp equation are obtained from the pseudo 3 reduction of the c two dimensional toda system

complex short pulse and coupled complex short pulse equations abstract in the present paper we propose a complex short pulse equation and a coupled complex short equation to describe ultra short pulse propagation in optical fibers they are integrable due to the existence of lax pairs and infinite number of conservation laws furthermore we find their multi soliton solutions in terms of pfaffians by virtue of hirota s bilinear method one and two soliton solutions are investigated in details showing favorable properties in modeling ultra short pulses with a few optical cycles especially same as the coupled nonlinear schrodinger equation there is an interesting phenomenon of energy redistribution in soliton interactions it is expected that for the ultra short pulses the complex and coupled complex short pulses equation will play the same roles as the nonlinear schrodinger equation and coupled nonlinear schrodinger equation

improved bounds for the largest eigenvalue of trees let t be a tree with vertex set v let dv denotes the degree of v v let max dv v v let u v such that du let k eu 1 where eu is the excentricity of u for j 1 2 k 2 let j maxdv distv u j we prove that 1t maxmax2 j k 2 j 1 j j 1 1 1 1 1 and 1t maxmax2 j k 2 j 1 j 1 1 1 1 where 1 t and 1 t are the largest eigenvalue of the laplacian matrix and adjacency matrix of t respectively these bounds give better results than those obtained in d stevanovic linear algebra appl 360 2003 35 42 except if 1

an explicit formula for eigenvalues of bethe trees and upper bounds on the largest eigenvalue of any tree a bethe tree bd k is a rooted unweighted of k levels in which the root vertex has degree equal to d the vertices at level j 2 j k 1 have degree equal to d 1 and the vertices at level k are the pendant vertices in this paper we first derive an explicit formula for the eigenvalues of the adjacency matrix of bd k moreover we give the corresponding multiplicities next we derive an explicit formula for the simple nonzero eigenvalues among them the largest eigenvalue of the laplacian matrix of bd k finally we obtain upper bounds on the largest eigenvalue of the adjacency matrix and of the laplacian matrix of any tree t these upper bounds are given in terms of the largest vertex degree and the radius of t and they are attained if and only if t is a bethe tree

periodicity of grover walks on generalized bethe trees abstract we focus on the periodicity of the grover walk on the generalized bethe tree which is a rooted tree such that in each level the vertices have the same degree since the grover walk is induced by the underlying graph its properties depend on the graph in this paper we say that the graph induces periodic grover walks if and only if there exists k n such that the k th power of the time evolution operator becomes the identity operator our aim is to characterize such graphs we give the perfect characterizations of the generalized bethe trees which induce periodic grover walks

propagation of ultra short optical pulses in cubic nonlinear media abstract we derive a partial differential equation that approximates solutions of maxwell s equations describing the propagation of ultra short optical pulses in nonlinear media and which extends the prior analysis of alterman and rauch phys lett a 264 2000 390 diffractive nonlinear geometric optics for short pulses preprint 2002 we discuss non rigorously conditions under which this approximation should be valid but the main contributions of this paper are 1 an emphasis on the fact that the model equation for short pulse propagation may depend on the details of the optical susceptibility in the wavelength regime under consideration 2 a numerical comparison of solutions of this model equation with solutions of the full nonlinear partial differential equation 3 a local well posedness result for the model equation and 4 a proof that in contrast to the nonlinear schrodinger equation which models slowly varying wavetrains this equation has no smooth pulse solutions which propagate with fixed shape and speed

a robust numerical integrator for the short pulse equation near criticality abstract the short pulse equation was introduced by schafer wayne 2004 for modeling the propagation of ultrashort optical pulses while it can describe a wide range of solutions its ultrashort pulse solutions with a few cycles which the conventional nonlinear schrodinger equation does not possess have drawn much attention in such a region the solutions can become quite close to a singularity and thus existing numerical methods cease to work stably or even if they do they require high computational cost in this paper we propose a robust numerical integration method which is obtained by combining a hodograph transformation and some structure preserving methods the resulting scheme successfully works even when the singularity occurs and thus gives a highly robust method for resolving near singular solutions of interest it is confirmed by numerical experiments and some new insights about derivative blow up are obtained

the spectra of a graph obtained from copies of a generalized bethe tree abstract we generalize the concept of a bethe tree as follows we say that an unweighted rooted tree is a generalized bethe tree if in each level the vertices have equal degree if b k is a generalized bethe tree of k levels then we characterize completely the eigenvalues of the adjacency matrix and laplacian matrix of a graph b k r obtained from the union of r copies of b k and the cycle c r connecting the r vertex roots moreover we give results on the multiplicity of the eigenvalues on the spectral radii and on the algebraic conectivity

multi soliton multi breather and higher order rogue wave solutions to the complex short pulse equation abstract in the present paper we are concerned with the general analytic solutions to the complex short pulse csp equation including soliton breather and rogue wave solutions with the aid of a generalized darboux transformation we construct the n bright soliton solution in a compact determinant form the n breather solution including the akhmediev breather and a general higher order rogue wave solution the first and second order rogue wave solutions are given explicitly and analyzed the asymptotic analysis is performed rigorously for both the n soliton and the n breather solutions all three forms of the analytical solutions admit either smoothed cusped or looped type ones for the csp equation depending on the parameters it is noted that due to the reciprocal hodograph transformation the rogue wave solution to the csp equation can be a smoothed cusponed or a looped one which is different from the rogue wave solution found so far

a note on the integer eigenvalues of the laplacian matrix of a balanced binary tree abstract let l b k be the laplacian matrix of an unweighted balanced binary tree b k of k levels in this note we show that 1 a 1 is an eigenvalue of l b k with multiplicity l 0 k 2 4 2 k 4l 2 if k is not a multiple of 4 or 1 l 0 k 2 4 2 k 4l 2 if k is a multiple of 4 where k 2 4 is the greatest integer not exceeding k 2 4 1 b 3 is an eigenvalue of l b k if and only if k is even 2 c 5 is an eigenvalue of l b k if and only if k is a multiple of 4

integrable discretizations and numerical simulation for a modified coupled integrable dispersionless equation abstract in this paper we propose an integrable semi discrete and full discrete analogues of the modified coupled integrable dispersionless mcid equation that was derived recently as a two component generalization of cid equation the key of the integrable discretization is to find the relation between the mcid equation and the sine gordon equation we complete the integrable discrete analogues based on the discrete sine gordon equation numerical simulation of one soliton two soliton and breather solution are conducted based on the integrable semi discrete scheme we also present the casorati determinant solution in parametric form of n soliton solutions for the semi discrete analogue of the mcid equation we show that the continuum limit of the full discrete mcid equation leads to the semi discrete mcid equation

applications of recurrence relations for the characteristic polynomials of bethe trees a bethe tree of k levels b k d is a rooted tree such that the root vertex has degree d the vertices from level 2 to k 1 have degree d 1 and the vertices at level k are leaves in this paper we obtain a recurrence relation for the characteristic polynomial and the laplacian characteristic polynomial of bethe trees as an application we prove that there are no integral bethe trees except for the star b 2 n 2 and we obtain a recurrence relation for the laplacian energy like invariant of bethe trees

long time asymptotics for the short pulse equation abstract in this paper we analyze the long time behavior of the solution of the initial value problem ivp for the short pulse sp equation as the sp equation is a completely integrable system which posses a wadati konno ichikawa wki type lax pair we formulate a 2 2 matrix riemann hilbert problem to this ivp by using the inverse scattering method since the spectral variable k is the same order in the wki type lax pair we construct the solution of this ivp parametrically in the new scale y t whereas the original scale x t is given in terms of functions in the new scale in terms of the solution of this riemann hilbert problem however by employing the nonlinear steepest descent method of deift and zhou for oscillatory riemann hilbert problems we can get the explicit leading order asymptotic of the solution of the short pulse equation in the original scale x t as time t goes to infinity

the short pulse hierarchy we study a new hierarchy of equations containing the short pulse equation which describes the evolution of very short pulses in nonlinear media and the elastic beam equation which describes nonlinear transverse oscillations of elastic beams under tension we show that the hierarchy of equations is integrable we obtain the two compatible hamiltonian structures we construct an infinite series of both local and nonlocal conserved charges a lax description is presented for both systems for the elastic beam equations we also obtain a nonstandard lax representation

method of memory diagrams for mechanical frictional contacts subject to arbitrary 2d loading abstract in this paper we present a solution to the contact problem for two elastic bodies with friction loaded by an arbitrarily varying force in 2d the general statement of the problem concerns the following two important challenges on one hand the consideration of an arbitrary contact geometry implies that a multitude of individual arbitrarily shaped contact zones appears which can split and merge as a result of changes in the normal loading on the other hand the existence of friction invokes the presence of a tangential force and consequently of a complicated traction distribution inside each of the contact zones the complexity of these traction distributions results from the fact that generally each contact spot contains stick and slip zones with various boundary conditions defined by the previous tangential contact interaction forces the presence of two classes of boundary conditions makes the problem memory dependent and hysteretic in spite of this by using both conventional and original methods of contact mechanics it is possible to drastically simplify the problem first of all the geometric difficulty residing in the description of arbitrary multiple contacts can be successfully dealt with by applying the reduced friction principle that expresses the tangential solution through the normal loading curve for both normal and tangential forces kept constant the principle is valid for a broad range of geometries and makes it possible to replace an arbitrary contact geometry by an equivalent system of two axisymmetric bodies secondly the hysteretic character of the equivalent axisymmetric problem can be addressed through the method of memory diagrams which replaces the complex traction distribution in the contact zone by a simple internal functional dependency comprising the same memory information following any change of loading the internal memory function evolves in accordance to a definite procedure that is based on the force balance equation written for small increments of the normal and tangential forces in this paper we formulate the method prove its validity and illustrate its numerical implementation with some simulated examples of hysteretic load displacement curves these obtained results might be of use for applications related to solids with cracks and contacts showing acoustical nonlinearity and for any modeling requiring physics based boundary conditions to describe and predict the effect of mechanical contacts with friction

a self adaptive moving mesh method for the camassa holm equation a self adaptive moving mesh method is proposed for the numerical simulations of the camassa holm equation it is an integrable scheme in the sense that it possesses the exact n soliton solution it is named a self adaptive moving mesh method because the non uniform mesh is driven and adapted automatically by the solution once the non uniform mesh is evolved the solution is determined by solving a tridiagonal linear system due to these two superior features of the method several test problems give very satisfactory results even if by using a small number of grid points

on the spectra of some graphs like weighted rooted trees abstract let g be a weighted rooted graph of k levels such that for j 2 k 1 each vertex at level j is adjacent to one vertex at level j 1 and all edges joining a vertex at level j with a vertex at level j 1 have the same weight where the weight is a positive real number 2 if two vertices at level j are adjacent then they are adjacent to the same vertex at level j 1 and all edges joining two vertices at level j have the same weight 3 two vertices at level j have the same degree 4 there is not a vertex at level j adjacent to others two vertices at the same level we give a complete characterization of the eigenvalues of the laplacian matrix and adjacency matrix of g they are the eigenvalues of leading principal submatrices of two nonnegative symmetric tridiagonal matrices of order k k and the roots of some polynomials related with the characteristic polynomial of the referred submatrices by application of the above mentioned results we derive an upper bound on the largest eigenvalue of a graph defined by a weighted tree and a weighted triangle attached by one of its vertices to a pendant vertex of the tree

local well posedness and blow up phenomena of the generalized short pulse equation in this paper we investigate the cauchy problem of the generalized short pulse equation in periodic domain we first establish the local well posedness of the equation in hs s s 2 by taking advantage of the kato method and present some useful conservation laws then we obtain the boundedness of the solution by virtue of the conservation laws finally we give a precise blow up criterion for 0 or 0 and get several blow up results provided the initial data satisfies some certain conditions

the generalized cattaneo partial slip plane contact problem i theory the cattaneo problem is considered for a general plane contact between elastically similar materials i e a monotonically increasing tangential load starting from zero with normal loading held fixed instead of the classical argument on the displacement field in the stick zone of cattaneo solution we attack the problem implicitly from the governing integral equations in the stick zones after discussing and solving the full stick case we move to the more realistic for finite friction case of partial slip we show that upon isolating the effect of full sliding the equalities and inequalities governing the corrective solution for the corrective shearing tractions in the stick zone are exactly the same as those governing the solution of the normal contact problem with a lower load but the same rotation as the actual one this analogy permits us to deduce several general properties and gives a general procedures for solving partial slip cattaneo problems as frictionless normal indentation ones therefore the general solutions for single multiple and periodic contacts is given a comprehensive set of explicit results is given in the part ii of the paper

locating the eigenvalues of trees abstract e give an o n method that computes for any tree t and interval how many eigenvalues of t lie within the interval our method is based on sylvester s law of inertia we use our algorithm to show that the nonzero eigenvalues of a caterpillar are simple it follows that caterpillars having b back nodes where b 2 1 are not integral we also show that among the regular caterpillars c b k formed by adjoining k legs to each of b back nodes all positive roots are in the interval k 1 k 2 and c b k is not integral if b 2

localization of the grover walks on spidernets and free meixner laws a spidernet is a graph obtained by adding large cycles to an almost regular tree and considered as an example having intermediate properties of lattices and trees in the study of discrete time quantum walks on graphs we introduce the grover walk on a spidernet and its one dimensional reduction we derive an integral representation of the n step transition amplitude in terms of the free meixner law which appears as the spectral distribution as an application we determine the class of spidernets which exhibit localization our method is based on quantum probabilistic spectral analysis of graphs

upper bound on randic energy of some graphs abstract applying the cauchy schwarz inequality we obtain a sharp upper bound on the randic energy of a bipartite graph and of graphs whose adjacency matrix is partitioned into blocks with constant row sum

an integrable coupled short pulse equation an integrable coupled short pulse csp equation is proposed for the propagation of ultra short pulses in optical fibers based on two sets of bilinear equations to a two dimensional toda lattice linked by a backlund transformation and an appropriate hodograph transformation the proposed csp equation is derived meanwhile its n soliton solutions are given by the casorati determinant in a parametric form the properties of one and two soliton solutions are investigated in detail same as the short pulse equation the two soliton solution turns out to be a breather type if the wave numbers are complex conjugate we also illustrate an example of soliton breather interaction

explicit construction of the eigenvectors and eigenvalues of the graph laplacian on the cayley tree abstract a generalized fourier analysis on arbitrary graphs calls for a detailed knowledge of the eigenvectors of the graph laplacian using the symmetries of the cayley tree we recursively construct the family of eigenvectors with exponentially growing eigenspaces associated with eigenvalues in the lower part of the spectrum the spectral gap decays exponentially with the tree size for large trees the eigenvalues and eigenvectors obey recursion relations which arise from the nested geometry of the tree

on the sth laplacian eigenvalue of trees of order st 1 abstract let 1 g 2 g n g 0 be the laplacian eigenvalues of a simple undirected graph g let s 2 and t 2 be integers and let t s t be the rooted tree of three levels and order st 1 such that the vertex root has degree s the vertices in level 2 have degree t and the s t 1 pendants vertices are in level 3 we prove that s t s t max s t t is a tree of order st 1 1 2 t 1 t 2 2 t 3 this result solves a conjecture due to shao et al in j y shao l zhang x y yuan on the second laplacian eigenvalue of trees of odd order linear algebra appl 419 2006 475 485

on the characteristic polynomial of a special class of graphs and spectra of balanced trees abstract let h be a simple graph with n vertices and g be a sequence of n rooted graphs g 1 g 2 g n godsil and mckay c d godsil b d mckay a new graph product and its spectrum bull austral math soc 18 1978 21 28 defined the rooted product h g of h by g by identifying the root vertex of g i with the ith vertex of h and determined the characteristic polynomial of h g in this paper we prove a general result on the determinants of some special matrices and as a corollary determine the characteristic polynomials of adjacency and laplacian matrices of h g rojo and soto o rojo r soto the spectra of the adjacency matrix and laplacian matrix for some balanced trees linear algebra appl 403 2005 97 117 computed the characteristic polynomials and the spectrum of adjacency and laplacian matrices of a class of balanced trees as an application of our results we obtain their conclusions by a simple method

the spectra of the adjacency matrix and laplacian matrix for some balanced trees abstract let t be an unweighted rooted tree of k levels such that in each level the vertices have equal degree let dk j 1 denotes the degree of the vertices in the level j we find the eigenvalues of the adjacency matrix and of the laplacian matrix of t they are the eigenvalues of principal submatrices of two nonnegative symmetric tridiagonal matrices of order k k the codiagonal entries for both matrices are d j 1 2 j k 1 and d k while the diagonal entries are zeros in the case of the adjacency matrix and dj 1 j k in the case of the laplacian matrix moreover we give some results concerning to the multiplicity of the above mentioned eigenvalues

on exact solutions of the schafer wayne short pulse equation wki eigenvalue problem we consider a new equation recently found by sch afer and wayne hereafter named schshort pulse equation swspe describing the propagation of an ultrashort pulse in nonlinear media using some vanishing boundary conditions we construct and discuss the n soliton solutions to the previous equation by means of the wadati konno ichikawa wki method which is arguably more direct than the map through the sine gordon equation investigated much earlier by sakovich and sakovich 2005 j phys soc japan 74 239 2006 j phys a math gen 39 l361 we particularly focus our attention on the two soliton solution as an application as a result the collision process for two soliton solutions with similar amplitudes exhibits very different behaviour from the case when the amplitudes are dissimilar

the spectrum of the laplacian matrix of a balanced binary tree abstract let l b k be the laplacian matrix of an unweighted balanced binary tree b k of k levels we prove that spectrum of l b k is l b k j 1 k 1 t j s k where for 1 j k 1 t j is the j j principal submatrix of the tridiagonal k k matrix s k s k 1 2 0 0 2 3 2 0 2 0 3 2 0 0 2 2 we derive that the multiplicity of each eigenvalue of t j 1 j k 1 as an eigenvalue of l b k is at least 2 k j 1 finally for each t j using some results in electron j linear algebra 6 2000 62 we obtain lower and upper bounds for its smallest eigenvalue and an upper bound for its largest eigenvalue in particular we give upper bounds for the second largest eigenvalue and for the largest eigenvalue of l b k

spectra of weighted generalized bethe trees joined at the root abstract a generalized bethe tree is a rooted tree in which vertices at the same distance from the root have the same degree let b i 1 i m be a set of trees such that for i 1 2 m 1 b i is a generalized bethe tree of k i levels 2 the vertices of b i at the level j have degree d i k i j 1 for j 1 2 k i and 3 the edges of b i joining the vertices at the level j with the vertices at the level j 1 have weight w i k i j for j 1 2 k i 1 let v b i 1 i m be the tree obtained from the union of the trees b i joined at their respective root vertices we give a complete characterization of the eigenvalues of the laplacian and adjacency matrices of v b i 1 i m moreover we derive results concerning their multiplicities in particular we characterize the spectral radii the algebraic conectivity and the second largest laplacian eigenvalue

quantum walk algorithm for element distinctness we use quantum walks to construct a new quantum algorithm for element distinctness and its generalization for element distinctness the problem of finding two equal items among n given items we get an o n 2 3 query quantum algorithm this improves the previous o n 3 4 quantum algorithm of buhrman et al siam j comput 34 2005 pp 1324 1330 and matches the lower bound of aaronson and shi j acm 51 2004 pp 595 605 we also give an o n k k 1 query quantum algorithm for the generalization of element distinctness in which we have to find k equal items among n items

casorati determinant solution for the relativistic toda lattice equation the relativistic toda lattice equation is decomposed into three toda systems the toda lattice itself backlund transformation of toda lattice and discrete time toda lattice it is shown that the solutions of the equation are given in terms of the casorati determinant by using the casoratian technique the bilinear equations of toda systems are reduced to the laplace expansion form for determinants the n soliton solution is explicitly constructed in the form of the casorati determinant

discrete analogues of the generalized coupled integrable dispersionless equations we study the integrable discretization of the generalized coupled integrable dispersionless equations two semi discrete and one fully discrete versions of the system are given via hirota s bilinear method soliton solutions for the derived discrete systems are also presented

integrable discretizations for the short wave model of the camassa holm equation the link between the short wave model of the camassa holm equation sche and bilinear equations of the two dimensional toda lattice equation is clarified the parametric form of the n cuspon solution of the sche in casorati determinant is then given based on the above finding integrable semi discrete and full discrete analogues of the sche are constructed the determinant solutions of both semi discrete and fully discrete analogues of the sche are also presented

periodicity of grover walks on complete graphs with self loops abstract we investigate a periodicity of grover walk on complete graphs with a self loop at each vertex we study an evolution matrix which steps forward a state of probability amplitude vector by using algebraic method then we find a periodic behaviour that the probability amplitude vector at each vertex gets back to initial state after some steps it is shown that grover walk on complete graphs on n vertices with a self loop at each vertex is periodic with period 2n

an integrable semi discretization of the camassa holm equation and its determinant solution an integrable semi discretization of the camassa holm ch equation is presented the keys of its construction are bilinear forms and determinant structure of solutions of the ch equation determinant formulas of n soliton solutions of the continuous and semi discrete camassa holm equations are presented based on determinant formulas we can generate multi soliton multi cuspon and multi soliton cuspon solutions numerical computations using the integrable semi discrete camassa holm equation are performed it is shown that the integrable semi discrete camassa holm equation gives very accurate numerical results even in the cases of cuspon cuspon and soliton cuspon interactions the numerical computation for an initial value condition which is not an exact solution is also presented

five diagonal matrices and zeros of orthogonal polynomials on the unit circle it is shown that monic orthogonal polynomials on the unit circle are the characteristic polynomials of certain five diagonal matrices depending on the schur parameters this result is achieved through the study of orthogonal laurent polynomials on the unit circle more precisely it is a consequence of the five term recurrence relation obtained for these orthogonal laurent polynomials and the one to one correspondence established between them and the orthogonal polynomials on the unit circle as an application some results relating the behaviour of the zeros of orthogonal polynomials and the location of schur parameters are obtained

a quantum walk induced by hoffman graphs and its periodicity abstract in order to define the discrete time quantum walks we divide the vertex set of a graph by two partitions and give a time evolution operator by a product of two local unitary operators on each partition in this paper we make such partitions from a hoffman graph which gives generalization of line graphs and study a staggered walk we first show that the square of the time evolution operator of the grover walk is unitarily equivalent to that of the staggered walk on the generalized line graph induced from a hoffman graph furthermore we focus on the periodicity of quantum walks and show that the staggered walk on the generalized line graph is periodic whenever the grover walk on the original graph is periodic by using the unitary equivalence this work is the first trial to develop the relation between quantum walks and hoffman graphs

detecting anomalous activity on networks with the graph fourier scan statistic we consider the problem of deciding based on a single noisy measurement at each vertex of a given graph whether the underlying unknown signal is constant over the graph or there exists a cluster of vertices with anomalous activation this problem is relevant to several applications such as surveillance disease outbreak detection biomedical imaging environmental monitoring etc since the activations in these problems often tend to be localized to small groups of vertices in the graphs we model such activity by a class of signals that are elevated over a possibly disconnected cluster with low cut size relative to its size we analyze the corresponding generalized likelihood ratio glr statistics and relate it to the problem of finding a sparsest cut in the graph we develop a convex relaxation of the glr statistic based on spectral graph theory which we call the graph fourier scan statistic gfss in our main theoretical result we show that the performance of the gfss depends explicitly on the spectral properties of the graph to assess the optimality of the gfss we prove an information theoretic lower bound for the detection of anomalous activity on graphs because the gfss requires the specification of a tuning parameter we develop an adaptive version of the gfss using these results we are able to characterize in a very explicit form the performance of the gfss on a few notable graph topologies we demonstrate that the gfss can efficiently detect a simulated arsenic contamination in groundwater

integrable semi discretizations of the reduced ostrovsky equation based on our previous work on the reduced ostrovsky equation j phys a math theor 45 355203 we construct its integrable semi discretizations since the reduced ostrovsky equation admits two alternative representations one being its original form the other the differentiated form the short wave limit of the degasperis procesi equation two semi discrete analogues of the reduced ostrovsky equation are constructed possessing the same n loop soliton solution the relationship between these two versions of semi discretizations is also clarified

integrable semi discretization of a multi component short pulse equation in the present paper we mainly study the integrable semi discretization of a multi component short pulse equation first we briefly review the bilinear equations for a multi component short pulse equation proposed by matsuno j math phys 52 123702 2011 and reaffirm its n soliton solution in terms of pfaffians then by using a backlund transformation of the bilinear equations and defining a discrete hodograph reciprocal transformation an integrable semi discrete multi component short pulse equation is constructed meanwhile its n soliton solution in terms of pfaffians is also proved

on the aα spectra of trees abstract let g be a graph with adjacency matrix a g and let d g be the diagonal matrix of the degrees of g for every real 0 1 define the matrix a g as a g d g 1 a g this paper gives several results about the a matrices of trees in particular it is shown that if t is a tree of maximal degree then the spectral radius of a t satisfies the tight inequality a t 2 1 1 which implies previous bounds of godsil lovasz and stevanovic the proof is deduced from some new results about the a matrices of bethe trees and generalized bethe trees in addition several bounds on the spectral radius of a of general graphs are proved implying tight bounds for paths and bethe trees

integrable discretizations of the short pulse equation in this paper we propose integrable semi discrete and full discrete analogues of the short pulse sp equation the key construction is the bilinear form and determinant structure of solutions of the sp equation we also give the determinant formulas of n soliton solutions of the semi discrete and full discrete analogues of the sp equations from which the multi loop and multi breather solutions can be generated in the continuous limit the full discrete sp equation converges to the semi discrete sp equation and then to the continuous sp equation based on the semi discrete sp equation an integrable numerical scheme i e a self adaptive moving mesh scheme is proposed and used for the numerical computation of the short pulse equation

on the smallest eigenvalues of the line graphs of some trees abstract in this paper we study the characteristic polynomials of the line graphs of generalized bethe trees we give an infinite family of such graphs sharing the same smallest eigenvalue our family generalizes the family of coronas of complete graphs discovered by cvetkovic and stevanovic

solitary wave solutions of the short pulse equation an exact nonsingular solitary wave solution of the schafer wayne short pulse equation is derived from the breather solution of the sine gordon equation by means of a transformation between these two integrable equations

improved bounds for the laplacian energy of bethe trees abstract for k d 2 a bethe tree is a rooted tree with k levels which the root vertex has degree d the vertices from level 2 to k 1 have degree d 1 and the vertices at the level k are pendent vertices so et al using a theorem by ky fan have obtained both upper and lower bounds for the laplacian energy of bipartite graphs we shall employ the above mentioned theorem to obtain new and improved bounds for the laplacian energy in the case of bethe trees