discontinuous inverse eigenvalue problems on considere des problemes de sturm liouville inverses dans lesquels les fonctions propres ont une discontinuite dans un point interieur on montre que si le potentiel est connu sur la moitie de l intervalle et si une condition limite est donnee alors le potentiel et l autre condition limite sont determines de facon unique par les valeurs propres

inverse wave scattering with discontinuous wave speed the inverse scattering problem on the line is studied for the generalized schrodinger equation d2 dx2 k2h x 2 q x where h x is a positive piecewise continuous function with positive limits h as x this equation in the frequency domain describes the wave propagation in a nonhomogeneous medium where q x is the restoring force and 1 h x is the variable wave speed changing abruptly at various interfaces a related riemann hilbert problem is formulated and the associated singular integral equation is obtained and proved to be uniquely solvable the solution of this integral equation leads to the recovery of h x in terms of the scattering data consisting of q x a reflection coefficient either of h and the bound state energies and norming constants some explicitly solved examples are provided

examples of underdetermined scattering problems the determination of the shape of a scatterer by non destructive methods such as scattering experiments raises the question of indetermination or ambiguity problems for the scatterer s shape depending on the given scattering data these ambiguity problems are discussed here by means of numerical constructions of equivalent scattering problems i e we determined scatterers that produced the same scattering amplitude in given conditions fixed energies fixed incident angles and or directions of receivers this construction is given in the context of a generalized scattering theory that takes into account impedance discontinuities inside the scatterer we started with a scattering problem defined by a discontinuous curve of arbitrary shape and defined by boundary conditions then a circular curve with appropriate boundary conditions was determined such that these two scattering problems yielded the same scattering amplitude within the given conditions we also present numerical results for two particular cases of the generalized scattering theory calculated by means of the nystrom method we used these results to verify that the equivalence obtained in the born approximation holds for the exact scattering amplitudes

on the riemann hilbert problem for the one dimensional schrodinger equation a matrix riemann hilbert problem associated with the one dimensional schrodinger equation is considered and the existence and uniqueness of its solutions are studied the solution of this riemann hilbert problem yields the solution of the inverse scattering problem for a larger class of potentials than the usual faddeev class some examples of explicit solutions of the riemann hilbert problem are given and the connection with ambiguities in the inverse scattering problem is established

a contribution to the topological classification of homogeneous vector fields in r3 in this paper we study the topological classification problem for homogeneous polynomial vector fields of degree k 2 in three variables in 1960 markus 7 see also 6 classified topologically the quadratic homogeneous vector fields in two variables subsequent works relative to the differentiable classification of these vector fields can be found in date 4 newton lo samardzija 13 vulpe and sibirskii 14 let us now consider a homogeneous vector field q q q2 q of degree k 2 in r3 with an isolated singularity at 0 e r3 i e all the qi i 1 2 3 are homogeneous polynomials of degree k with an isolated common zero at 0 e r3 the vector field defined as qr q i x qi x2q2 x x3 q f x x 2 x2 x3 e s2 is tangent to the 2 sphere s2 and for any regular orbit y of qr the set

elastic waves in discontinuous media three dimensional scattering this report contains an exact study of elastic wave propagation and its scattering in discontinuous media where hard reflectors are onionlike sets of surfaces in order to reformulate the problem as a finite set of boundary integral equations the wave motion between reflectors is represented by means of elastic potentials which involve vectorial densities on the surfaces in the external medium an outgoing asymptotic condition generalizes the silver muller and the sommerfeld condition to the case of coupled waves s and p waves moving with different velocities the uniqueness of the green s function which guarantees the uniqueness of the direct problem solution is proven for any incident wave and arbitrary number of surfaces the transmission and scattering problems are studied with and without the simplification obtained by assuming constant poisson ratios according to the parameter ranges the equations which are obtained are well posed either as second kind fredholm equations or because the

inverse problems for nonabsorbing media with discontinuous material properties one dimensional electromagnetic and elastic inverse problems are formulated for media with discontinuous material properties in addition an impedence mismatch between source and medium is allowed the measured data for either problem is shown to generate a reflection kernel which is used in the solution of the inverse problem the solution algorithm itself is a time domain technique which is a special case of previously obtained results for absorbing media

percolation and epidemics in a two dimensional small world percolation on two dimensional small world networks has been proposed as a model for the spread of plant diseases in this paper we give an analytic solution of this model using a combination of generating function methods and high order series expansion our solution gives accurate predictions for quantities such as the position of the percolation threshold and the typical size of disease outbreaks as a function of the density of shortcuts in the small world network our results agree with scaling hypotheses and numerical simulations for the same model

stability properties of autonomous homogeneous polynomial differential systems abstract a geometrical approach is used to derive a generalized characteristic value problem for dynamic systems described by homogeneous polynomials it is shown that a nonlinear homogeneous polynomial system possesses eigenvectors and eigenvalues quantities normally associated with a linear system these quantities are then employed in studying stability properties the necessary and sufficient conditions for all forms of stabilities characteristic of a two dimensional system are provided this result together with the classical theorem of frommer completes a stability analysis for a two dimensional homogeneous polynomial system

an exponential formula for polynomial vector fields ii lie series exponential substituition and rooted trees abstract grobner s lie series and the exponential formula provide different explicit formulas for the flow generated by a finite dimensional polynomial vector field the present paper gives 1 a generalization of the lie series in case of non commuting variables called exponential substituition 2 a structural understanding of the three formulas and their mutual relationships in terms of rooted trees and 3 as a byproduct new results on the enumeration coding and statistics of different kinds of rooted trees

the surviving rate of planar graphs let g be a connected graph with n 2 vertices let k 1 be an integer suppose that a fire breaks out at a vertex v of g a firefighter starts to protect vertices at each time interval the firefighter protects k vertices not yet on fire at the end of each time interval the fire spreads to all the unprotected vertices that have a neighbor on fire let sn k v denote the maximum number of vertices in g that the firefighter can save when a fire breaks out at vertex v the k surviving rate r k g of g is defined to be v v g sn k v n 2 which is the average proportion of saved vertices in this paper we show that every planar graph g with minimum degree d satisfies r 4 g 311 if d 5 r 4 g 319 if d 4 and r 4 g 19 if d 3 this improves a result in w wang s finbow p wang the surviving rate of an infected network theoret comput sci 411 2010 3651 3660

on sturm liouville operators with discontinuity conditions inside an interval abstract in this study transformation operator some properties of kernel properties of spectral characteristics and also uniquiness theorems for the inverse problems are studied for a class of sturm liouville differential operators which have discontinuity conditions inside a finite interval

moduli of stability for germs of homogeneous vectorfields on r3 abstract in this paper we will show that there are an uncountable number of homogeneous quadratic vectorfields on r 3 so that no two of these vectorfields are topologically equivalent

extended integrability and bi hamiltonian systems the current notion of integrability of hamiltonian systems was fixed by liouville in a famous 1855 paper it describes systems in a 2k dimensional phase space whose trajectories are dense on tori q or wind on toroidal cylinders m q m within liouville s construction the dimension q cannot exceed k and is the main invariant of the system in this paper we generalize liouville integrability so that trajectories can be dense on tori k of arbitrary dimensions q 1 2k 1 2k and an additional invariant v 2 q k v 2 q 2 can be recovered the main theorem classifies all k k 1 2 canonical forms of hamiltonian systems that are integrable in a newly defined broad sense an integrable physical problem having engineering origin is presented the notion of extended compatibility of two poisson structures is introduced the corresponding bi hamiltonian systems are shown to be integrable in the broad sense

three parameter complex hadamard matrices of order 6 a three parameter family of complex hadamard matrices of order 6 is presented it significantly extends the set of closed form complex hadamard matrices of this order and in particular contains all previously described one and two parameter families as subfamilies

examples of quaternionic and kahler structures in hamiltonian models of nearly geostrophic flow we study 2 forms on phase spaces of hamiltonian models of nearly geostrophic flows a quaternionic structure is identified and the complex part of a symplectic representation of this structure corresponds to an elliptic monge ampere equation the real part is an invariant kahler structure

exact solution of site and bond percolation on small world networks we study percolation on small world networks which has been proposed as a simple model of the propagation of disease the occupation probabilities of sites and bonds correspond to the susceptibility of individuals to the disease and the transmissibility of the disease respectively we give an exact solution of the model for both site and bond percolation including the position of the percolation transition at which epidemic behavior sets in the values of the critical exponents governing this transition the mean and variance of the distribution of cluster sizes disease outbreaks below the transition and the size of the giant component epidemic above the transition

the jacobian conjecture reduction of degree and formal expansion of the inverse introduction i the jacobian conjecture 1 statement of the jacobian problem first observations 2 some history of the jacobian conjecture 3 faulty proofs 4 the use of stabilization and of formal methods ii the reduction theorem 1 notation 2 statement of the reduction theorem 3 reduction to degree 3 4 proof of the reduction theorem 5 r linearization and unipotent reduction 6 nilpotent rank 1 jacobians iii the formal inverse 1 notation 2 abhyankar s inversion formula 3 the terms gj 4 the tree expansion g lt a t lfptf 5 calculations references

transition functions and moduli of stability for 3 dimensional homogeneous vector fields with a hyperbolic blowing up abstract in the first part of the paper we introduce the normal transition function for saddle connections of planar diffeomorphisms it is a positive multiple of the usual transition function but in its definition we do not need c 1 linearizing coordinates among other nice properties it is found to be analytic when the diffeomorphism is the second part of the paper deals with the existence of a modulus of stability for germs on r 3 of homogeneous vector fields with a hyperbolic blowing up we show that inside a specific class of examples the modulus occurs for a sufficiently high degree

on a generalized hilbert problem the problem analyzed is to find functions f meromorphic in c respectively with values that are linear operators on a banach space and such that their boundary values on r satisfy the equation f f where the operator valued function as well as the positions of the poles of f and the ranges of their residues are given uniqueness results are obtained under certain conditions an index is proved to exist and the determination of f is reduced to the solution of a generalization of marchenko s fundamental equation the results are applied to inverse scattering and inverse spectral problems

inverse scattering theory inverse scattering series method for one dimensional non compact support potential the reversion of the born neumann series of the lippmann schwinger equation is one of the standard ways to solve the inverse acoustic scattering problem one limitation of the current inversion methods based on the reversion of the born neumann series is that the velocity potential should have compact support however this assumption cannot be satisfied in certain cases especially in seismic inversion based on the idea of distorted wave scattering we explore an inverse scattering method for velocity potentials without compact support the strategy is to decompose the actual medium as a known single interface reference medium which has the same asymptotic form as the actual medium and a perturbative scattering potential with compact support after introducing the method to calculate the green s function for the known reference potential the inverse scattering series and volterra inverse scattering series are derived for the perturbative potential analytical and numerical examples demonstrate the feasibility and effectiveness of this method besides to ensure stability of the numerical computation the lanczos averaging method is employed as a filter to reduce the gibbs oscillations for the truncated discrete inverse fourier transform of each order our method provides a rigorous mathematical framework for inverse acoustic scattering with a non compact support velocity potential

coherent states in quaternionic quantum mechanics we develop perelomov s coherent states formalism to include the case of a quaternionic hilbert space we find that because of the closure requirement an attempted quaternionic generalization of the special nilpotent or weyl group reduces to the normal complex case for the case of the compact group su 2 however coherent states can be formulated using the quaternionic half integer spin matrices of finkelstein jauch and speiser giving a nontrivial quaternionic analog of coherent states

inverse scattering for a layered acoustic medium using the first order equations of motion the inverse scattering problem for a layered acoustic medium is considered from the first order differential equations of motion resulting in a vector formulation of the problem and using a vector form of the schrodinger inverse scattering methods the result is a vector marchenko equation the differentiability constraints on the acoustic impedance are somewhat relaxed compared to the more standard approach of beginning with the wave equation the solution for plane waves at normal incidence is given along with a good approximate solution which is easily obtainable and takes into account transmission losses not included in the normal wkbj born approximation a new solution for extracting separately the velocity and density of the medium using the reflection response for two different angles of incidence is given which involves a nonlinear integral equation to relate the apparent traveltimes to depth

on modeling discontinuous media three dimensional scattering a realistic model for the propagation of scalar waves in a medium must take into account discontinuities of the medium parameters which may act as hard reflectors discontinuities of their gradient which may act as soft reflectors and continuous variations of parameters the direct scattering problem is presented here the mixed potential impedance equation that characterizes the model depends on two arbitrary parameters one of them would be the potential if the equation reduces to the schrodinger equation the other one would be related to the impedance if the equation describes more classical waves after the mathematical tools are constructed green s functions etc a rigorous three dimensional scattering theory is described it encompasses the quantum three dimensional scattering theory and the theory of acoustical scattering for instance by systems of regular surfaces of arbitrary shape the main integral equations of the quantum scattering theory are generalized scattering amplitudes due to reflectors and scattering amplitudes due to diffuse scattering after reflectors have been taken into account are defined and constructed born and quadratic approximations are discussed the explicit formulas corresponding to the scattering by discontinuities and those corresponding to diffuse scattering are not reducible to each other exactly i e unless filtering and errors are allowed the results can also be used to described rigorously the three dimensional scattering by the wave equation in the frequency domain and in particular the response to an impulsive localized source further generalizations are in progress

stability of vector fields on r3 determined by the first non vanishing jet soit m k l espace des champs vectoriels x sur r 3 tels que j k 1 x 0 0 et j k x 0 0 dote de la c k topologie soit m k l espace des champs vectoriels x m k tel que l explosion de x satisfait 3 conditions soit x m k alors il existe un voisinage u de x dans m k tel que tout y u est c 0 equivalent en 0 a x les elements de m k sont localement stables dans m k

enumerations of lattice animals and trees we have developed an improved algorithm that allows us to enumerate the number of site animals on the square lattice up to size 46 we also calculate the number of lattice trees up to size 44 and the radius of gyration of both lattice animals and trees up to size 42 analysis of the resulting series yields an improved estimate 4 062570 8 for the growth constant of lattice animals and 0 3 795254 8 for the growth constant of trees and confirms to a very high degree of certainty that both the animal and tree generating functions have a logarithmic divergence analysis of the radius of gyration series yields the estimate 0 64115 5 for the size exponent

quaternionic quantum field theory we show that a quaternionic quantum field theory can be formulated when the numbers of bosonic and fermionic degrees of freedom are equal and the fermions as well as the bosons obey a second order wave equation the theory takes the form of either a functional integral with quaternion imaginary lagrangian or a schrodinger equation and transformation theory for quaternion valued wave functions with a quaternion imaginary hamiltonian the connection between the two formulations is developed in detail and many related issues including the breakdown of the correspondence principle and the hilbert space structure are discussed

inverse scattering for discontinuous impedance schrodinger operators a model example we discuss the possibility of applying the classical reconstruction procedure in the inverse scattering problems for schr odinger operators in impedance form with a non smooth impedance we illustrate on a simple model example that the formal application of the classical algorithm gives a wrong result whereas an approach based on regularization leads in the limit to a correct reconstruction of the impedance function

exact behavior of jost functions at low energy for schrodinger operators with central potential q r and angular momentum l the behavior of the jost function fl k as k 0 is investigated it is assumed that 0dr 1 r q r where 1 situations where q is integrable with 1 2 but not with 2 are of particular interest for potentials satisfying q r q0r 2 e 0 e 1 and l 0 the leading behavior of f0 k and the phase shift 0 k as k 0 is derived also comments are made on the differentiability properties of the jost solutions with respect to the variable k at k 0 for 1 levinson s theorem is proved thereby clarifying some questions raised recently by newton j math phys 27 2720 1986

the surviving rate of an outerplanar graph for the firefighter problem let g be a connected graph with n 2 vertices let k 1 be an integer suppose that a fire breaks out at a vertex v of g a firefighter starts to protect vertices at each time interval the firefighter protects k vertices not yet on fire at the end of each time interval the fire spreads to all the unprotected vertices that have a neighbour on fire let sn k v denote the maximum number of vertices in g that the firefighter can save when a fire breaks out at vertex v the k surviving rate r k g of g is defined to be v v g sn k v n 2 which is the average proportion of saved vertices in this paper we investigate the surviving rate of outerplanar graphs g with n 2 vertices the main results are as follows 1 lim n email protected 5 g 1 and 2 r 1 g 4381 53n 3n 2 if n 8 and r 1 g 13 if n 2 which improves the result in l cai w wang the surviving rate of a graph for the firefighter problem siam j discrete math 23 2009 1814 1826

inverse schro dinger scattering on the line with partial knowledge of the potential the one dimensional schrodinger equation is considered when the potential and its first moment are absolutely integrable the potential is uniquely constructed in terms of the scattering data consisting of the reflection coefficient from the right left and the knowledge of the potential on the right left half line of the real axis hence neither the bound state energies nor the bound state norming constants are needed to determine the potential and in fact these are uniquely determined by the scattering data an explicit example is provided also considered are two inverse scattering problems for a generalized schrodinger equation with two potentials when the potential to be recovered is partially known

some properties of the k dimensional lyness map this paper is devoted to study some properties of the k dimensional lyness map f x1 xk x2 xk a ki 2xi x1 our main result presents a rational vector field that gives a lie symmetry for f this vector field is used for k 5 to give information about the nature of the invariant sets under f when k is odd we also present a new as far as we know first integral for ff which allows us to deduce in a very simple way several properties of the dynamical system generated by f in particular for this case we prove that except on a given codimension one algebraic set none of the positive initial conditions can be a periodic point of odd period

scattering and inverse scattering in one dimensional nonhomogeneous media the wave propagation in a one dimensional nonhomogeneous medium is considered where the wave speed and the restoring force depend on location in the frequency domain this is equivalent to the schrodinger equation d2 dx2 k2 k2p x q x with an added potential proportional to energy the scattering and bound state solutions of this equation are studied and the properties of the scattering matrix are obtained the inverse scattering problem of recovering the restoring force when the wave speed and the scattering data are known are also solved

one dimensional quaternion homogeneous polynomial differential equations in this paper we study the dynamics of real four dimensional polynomial differential equations of the forms q aqkq m or q qkaqm where q a are quaternions q is the conjugate of q and k and m are non negative integers in many parts of the work the quaternion structure is used along the proofs

factorization of scattering matrices due to partitioning of potentials in one dimensional schrodinger type equations the one dimensional schrodinger equation and two of its generalizations are considered as they arise in quantum mechanics wave propagation in a nonhomogeneous medium and wave propagation in a nonconservative medium where energy may be absorbed or generated generically the zero energy transmission coefficient vanishes when the potential is nontrivial but in the exceptional case this coefficient is nonzero resulting in tunneling through the potential it is shown that any nontrivial exceptional potential can always be fragmented into two generic pieces furthermore any nontrivial potential generic or exceptional can be fragmented into generic pieces in infinitely many ways the results remain valid when dirac delta functions are included in the potential and other coefficients are added to the schrodinger equation for such schrodinger equations factorization formulas are obtained that relate the scattering matrices of the fragments to the scattering matrix of the full problem

on the polynomial vector fields on let x be a polynomial vector field of degree n on m if m rm then the dynamics and the algebraic geometric properties of the vector fields x s have been studied intensively mainly for the case m r2 and specially when n 2 in this last case more than one thousand of papers have been published if m s2 there are several papers dedicated to the study of the homogeneous polynomial vector field of degree n on s2 mainly if n 2 but there are very few results on the non homogeneous polynomial vector fields of degree n on s2 in this paper we shall try to full a little this gap

quaternionic valued ordinary differential equations ii coinciding sectors abstract we give some sufficient conditions for the existence of at least one periodic solution of the quaternionic polynomial equations in some cases we are able to prove uniqueness of periodic solutions inside some sectors of h and detect solutions heteroclinic to the periodic ones

holomorphic structures in hydrodynamical models of nearly geostrophic flow we study complex structures arising in hamiltonian models of nearly geostrophic flows in hydrodynamics in many of these models an elliptic monge ampere equation defines the relationship between a balanced velocity field defined by a constraint in the hamiltonian formalism and the materially conserved potential vorticity elliptic monge ampere operators define an almost complex structure and in this paper we show that a natural extension of the so called geostrophic momentum transformation of semi geostrophic theory which has a special importance in theoretical meteorology defines kahler and special kahler structures on phase space furthermore analogues of the geostrophic momentum coordinates are shown to be special lagrangian coordinates under conditions which depend upon the physical approximations under consideration certain duality properties of the operators are studied within the framework of the kahler geometry

enumerating 2d percolation series by the finite lattice method theory this paper describes a transfer matrix algorithm for enumeration of series of interest in percolation on the square lattice it allows efficient generation of both low temperature and high temperature expansions as well as the combinatorially interesting enumeration of undirected animals by area or perimeter with moments of the other property

discontinuous media and underdetermined scattering problems the problem of ambiguities in trying to determine a shape by means of scattering experiments with one or a few illuminating angles and all directions of receivers is discussed by means of numerical experiments the model equation gives a good representation of scattering of scalar waves which can taken into account impedance discontinuities inside the scatterer physical problems include for instance acoustical waves in media where the density and the lame parameter lambda may vary continuously everywhere except across a finite number of smooth surfaces through which they jump

a type of moduli for saddle connections of planar diffeomorphisms on considere un diffeomorphisme f sur une variete a 2 dimensions m ayant une paire de points fixes hyperboliques p et q de type selle de sorte que l une des composantes connexes de w u p p est une composante connexe de w s q q q on etudie les conjugaisons hors de la connexion selle on introduit la notion de conjugaison pincee

a quaternionic structure in the three dimensional euler and ideal magneto hydrodynamics equations by considering the three dimensional incompressible euler equations a 4 vector is constructed out of a combination of scalar and vector products of the vorticity and the vortex stretching vector u s the evolution equation for can then be cast naturally into a quaternionic riccati equation this is easily transformed into a quaternionic zero eigenvalue schrodinger equation whose potential depends on the hessian matrix of the pressure with minor modifications this system can alternatively be written in complex notation an infinite set of solutions of scalar zero eigenvalue schrodinger equations has been found by adler and moser which are discussed in the context of the present problem similarly when the equations for ideal magneto hydrodynamics mhd are written in elsasser variables a pair of 4 vectors are governed by coupled

handbook of differential equations chapter headings definitions and concepts transformations exact analytical methods exact methods for odes exact methods for pdes approximate analytical methods numerical methods concepts numerical methods for odes numerical methods for pdes list of tables list of programs list of figures subject index

integral equation methods for the inverse problem with discontinuous wave speed the recovery of the coefficient h x in the one dimensional generalized schrodinger equation d2 dx2 k2h x 2 q x where h x is a positive piecewise continuous function with positive limits h as x is studied the large k asymptotics of the wave functions and the scattering coefficients are analyzed a factorization formula is given expressing the total scattering matrix as a product of simpler scattering matrices using this factorization an algorithm is presented to obtain the discontinuities in h x and h x h x in terms of the large k asymptotics of the reflection coefficient when there are no bound states it is shown that h x is recovered from an appropriate set of scattering data by using the solution of a singular integral equation and the unique solvability of this integral equation is established an equivalent marchenko integral equation is derived and is shown to be uniquely solvable the unique recovery of h x from the solution of this marchenko equation is presented some expli

the projective vector field of a kind of three dimensional quasi homogeneous system on s2 abstract in this paper we study the projective vector field q t of a three dimensional quasi homogeneous system with weight 1 1 3 and degree 2 3 2 projective vector fields q t of this kind are classified into two types for one type q t has no closed orbit and at most eight singularities which lead to a global topology of the three dimensional system for the other type q t has at most ten singularities in addition we show a relationship between q t and a lienard system of this type for both of them we obtain some conditions for the existence of limit cycles

inverse scattering in 1 d nonhomogeneous media and recovery of the wave speed the inverse scattering problem for the 1 d schrodinger equation d2 dx2 k2 k2p x q x is studied this equation is equivalent to the 1 d wave equation with speed 1 1 p x in a nonhomogeneous medium where q x acts as a restoring force when q x is integrable with a finite first moment p x 1 and bounded below and satisfies two integrability conditions p x is recovered uniquely when the scattering data and q x are known some explicitly solved examples are provided

hopf algebraic structure of families of trees in this paper we describe hopf algebras which are associated with certain families of trees these hopf algebras originally arose in a natural fashion one of the authors 5 was investigating data structures based on trees which could be used to efficiently compute certain differential operators given data structures such as trees which can be multiplied and which act as higherorder derivations on an algebra one expects to find a hopf algebra of some sort we were pleased to find that not only was there a hopf algebra associated with these data structures but that it could be used to give new proofs of enumerations of such objects as rooted trees and ordered rooted trees previous work applying hopf algebras to combinatorial objects such as 10 13 or 14 has concerned itself with algebraic structures on polynomial algebras and on partially ordered sets rather than on trees themselves

the 2 surviving rate of planar graphs without 4 cycles let g be a connected graph with n 2 vertices suppose that a fire breaks out at a vertex v of g a firefighter starts to protect vertices at each time interval the firefighter protects two vertices not yet on fire at the end of each time interval the fire spreads to all the unprotected vertices that have a neighbour on fire let sn 2 v denote the maximum number of vertices in g that the firefighter can save when a fire breaks out at vertex v the surviving rate r 2 g of g is defined to be v v g sn 2 v n 2 which is the average proportion of saved vertices in this paper we show that if g is a planar graph with n 2 vertices and without 4 cycles then r 2 g 176

random graphs with arbitrary degree distributions and their applications recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the poisson degree distributions that have been widely studied in the past in this paper we develop in detail the theory of random graphs with arbitrary degree distributions in addition to simple undirected unipartite graphs we examine the properties of directed and bipartite graphs among other results we derive exact expressions for the position of the phase transition at which a giant component first forms the mean component size the size of the giant component if there is one the mean number of vertices a certain distance away from a randomly chosen vertex and the average vertex vertex distance within a graph we apply our theory to some real world graphs including the worldwide web and collaboration graphs of scientists and fortune 1000 company directors we demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world while in others there is a measurable discrepancy between theory and reality perhaps indicating the presence of additional social structure in the network that is not captured by the random graph

scaling and percolation in the small world network model in this paper we study the small world network model of watts and strogatz which mimics some aspects of the structure of networks of social interactions we argue that there is one nontrivial length scale in the model analogous to the correlation length in other systems which is well defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero giving a normal critical point in this limit this length scale governs the crossover from large to small world behavior in the model as well as the number of vertices in a neighborhood of given radius on the network we derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex vertex distance on the network and using series expansion and pad e approximants find an approximate analytic form for the scaling function we calculate the effective dimension of small world graphs and show that this dimension varies as a function of the length scale on which it is measured in a manner reminiscent of multifractals we also study the problem of site percolation on small world networks as a simple model of disease propagation and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms in epidemiological terms the point at which an epidemic occurs the typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph all our analytic results are confirmed by extensive numerical simulations of the model

the 2 surviving rate of planar graphs without 6 cycles the firefighter problem models the spread of a fire or virus through a network and the k surviving rate of g r k g is the expected proportion of nodes k fire fighters per turn can save in this paper we show that if g is a planar graph with at least two vertices and having no 6 cycles then r 2 g 185

quaternionic valued ordinary differential equations the riccati equation abstract we give some sufficient conditions for the existence of at least two periodic solutions of the quaternionic riccati equation in some cases we are able to give a full description of dynamics and detect solutions heteroclinic to the periodic ones we also provide an example of the riccati equation without periodic solutions which appears in the euler vorticity dynamics

h 2 reducible complex hadamard matrices of order 6 complex hadamard matrices h of order 6 are characterized in a novel manner according to the presence absence of order 2 hadamard submatrices it is shown that if there exists one such submatrix h is equivalent to a hadamard matrix where all the nine submatrices are hadamard the ensuing subset of h 2 reducible complex hadamard matrices is more general than might be thought and significantly includes all the up till now described one and two parameter families of order 6 a known isolated matrix and most numerically generated matrices fall outside the subset

non stabilisable jets of diffeomorphisms in r2 and of vector fields in r3 we essentially prove two results firstly that for any x germ of a coo vector field on r2 with a certain specific 9 jet there is a germ of a c diffeomorphism f with j f o j x o x denoting the flow of x such that f and x are not c0 conjugate theorem 1 and secondly that whenever x is a germ of a coo vector field on r3 whose 9 jet belongs to a certain class of 9 jets there can be found a germ of a coo vector field y on r3 with jo x 0 jo y 0 such that x and y are not c0 equivalent theorem 2 the second result implies the existence of jets of vector fields in r3 which are not stabilisable for the notion of c0 equivalence the question of the existence of such jets was raised by thom in 1970 and very soon takens found examples in rn with n 2 4 ta ta3 the first result which we use in proving the second also implies the existence of jets of diffeomorphisms in r2 which are not stabilisable for the notion of c conjugacy the diffeomorphisms having such a jet contain hyperbolic or saddle sectors with a different topological structure than the hyperbolic sectors occurring for vector fields in finite codimension in these sectors the singularity scheme of a nice desingularisation after blowing up contains certain successions of semi hyperbolic singularities these cases had not been treated in d r r and our example now shows that the problems we encountered in d r r in trying to add this case were not technical but essential our examples provide germs of coo diffeomorphisms in r2 which are infinitely tangent to the time l mapping of a vector field of finite codimension and not c embeddable in the flow of a vector field of finite codimension this is definitely not possible in r germs of orientable coo diffeomorphisms of finite codimension are coo embeddable in a flow ta4 even the germs of diffeomorphisms infinitely tangent to the identity and with isolated fixed point show up in a c flow but they do not necessarily show up in a c 2 flow se possibly as

on modeling discontinuous media one dimensional approximations distorted wave born series are derived for the impedance equation a 1 d dx a d dx k2 p 0 where a is a piecewise differentiable function the first and second orders are given explicitly they hold for impedances a which include discontinuities before and or after perturbation so that the problem could not be reduced to the schrodinger one the results are used for discussing homogenizations currently practiced in modeling discontinuous media

exact results and scaling properties of small world networks we study the distribution function for minimal paths in small world networks we derive its general form which we verify numerically and also discuss its scaling properties using the general form of this distribution function we derive exact expressions for the average minimal distance l and its variance sigma 2 finally we study the limit of large system sizes and obtain some approximate results

solving nonlinear equations from higher order derivations in linear stages for all polynomials a r of degree less than or equal to d and all higher order derivations e e s e 1 and f f s f 1 where 1 s and 1 i m of length s less or equal to r the reason for wanting such derivations e1 em is simple given several derivations f1 fm it is useful to have derivations e1 em which are good local approximations to the fi and which are easy to compute with notice that since the right hand sides of equations are known while the left hand sides involve the unspecified coefficients of the polynomial functions the equations are equivalent to a system of nonlinear algebraic equations involving the coefficients it is the purpose of this paper to give an algorithm which solves such a system when a solution exists in fact as the title suggests this algorithm will work by solving a sequence of r linear systems

an exponential formula for polynomial vector fields abstract the well known matrix exponentiation for linear first order systems of ode is generalised to polynomial vector fields the substitution process of our exponential formula is closely related to the usual iteration of polynomial mappings

the surviving rate of an infected network let g be a connected network let k 1 be an integer suppose that a vertex v of g becomes infected a program is then installed on k nodes not yet infected afterwards the virus spreads to all its unprotected neighbors in each time interval the virus and the network administrator take turns until the virus can no longer spread further let sn k v denote the maximum number of vertices in g the network administrator can save when a virus infects v the k surviving rate r k g of g is defined to be the average value v v g sn k v n 2 in particular we write r g email protected 1 g in this paper we first use a probabilistic method to show that almost all networks have k surviving rate arbitrarily close to 0 then we prove the following results 1 r g 235 for a planar network g of girth at least 9 2 r 2 g 116 for a series parallel network g and 3 r 2 d 1 g 25d for a d degenerate network g

periodic solutions of homogeneous equations abstract stability conditions for periodic spiral solutions of homogeneous systems in the nonautonomous and the autonomous case are derived as well as a concept of lyapunov functions and a bendixson criterion