on the number of quintic fields we show that the number of quintic number fields whose discriminant does not exceed x in absolute value is bounded by a constant times x1 e for any e 0 this may be compared with the conjecture that the number of such fields is asymptotic to a constant times x as x tends to infinity

the concomitants of a prehomogeneous vector space abstract the classical method of radial separation of variables is applied to gain some information about the concomitants also known as relatively equivariant polynomial maps or covariants of a class of prehomogeneous vector spaces particular attention is payed to concomitants that are themselves prehomogeneous the results are applied to classify all prehomogeneous concomitants of several examples of interest in arithmetic

secondary terms in counting functions for cubic fields we prove the existence of secondary terms of order x5 6 in the davenport heilbronn theorems on cubic fields and 3 torsion in class groups of quadratic fields for cubic fields this confirms a conjecture of datskovsky wright and roberts we also prove a variety of generalizations including to arithmetic progressions where we discover a curious bias in the secondary term roberts s conjecture has also been proved independently by bhargava shankar and tsimerman in contrast to their work our proof uses the analytic theory of zeta functions associated to the space of binary cubic forms developed by shintani and datskovsky wright

levels of distribution for sieve problems in prehomogeneous vector spaces in our companion paper 28 we developed an efficient algebraic method for computing the fourier transforms of certain functions defined on prehomogeneous vector spaces over finite fields and we carried out these computations in a variety of cases here we develop a method based on fourier analysis and algebraic geometry which exploits these fourier transform formulas to yield level of distribution results in the sense of analytic number theory such results are of the shape typically required for a variety of sieve methods as an example of such an application we prove that there are gg frac x log x quartic fields whose discriminant is squarefree bounded above by x and has at most eight prime factors

colliding dissipative pulses the shooting manifold abstract we study multi pulse solutions in excitable media under the assumption that a single pulse is asymptotically stable we show that there is a well defined shooting manifold consisting of two pulses traveling towards each other in phase space the two dimensional manifold is a graph over the manifold of linear superpositions of two pulses located at x 1 and x 2 with x 1 x 2 1 it is locally invariant under the dynamics of the reaction diffusion system and uniformly asymptotically attracting with asymptotic phase the main difficulty in the proof is the fact that the linearization at the leading order approximation is strongly non autonomous since pulses approach each other with speed of order one

on the orthogonalised reverse path method for nonlinear system identification abstract the problem of obtaining the underlying linear dynamic compliance matrix in the presence of nonlinearities in a general multi degree of freedom mdof system can be solved using the conditioned reverse path crp method introduced by richards and singh 1998 journal of sound and vibration 213 4 pp 673 708 the crp method also provides a means of identifying the coefficients of any nonlinear terms which can be specified a priori in the candidate equations of motion although the crp has proved extremely useful in the context of nonlinear system identification it has a number of small issues associated with it one of these issues is the fact that the nonlinear coefficients are actually returned in the form of spectra which need to be averaged over frequency in order to generate parameter estimates the parameter spectra are typically polluted by artefacts from the identification of the underlying linear system which manifest themselves at the resonance and anti resonance frequencies a further problem is associated with the fact that the parameter estimates are extracted in a recursive fashion which leads to an accumulation of errors the first minor objective of this paper is to suggest ways to alleviate these problems without major modification to the algorithm the results are demonstrated on numerically simulated responses from mdof systems in the second part of the paper a more radical suggestion is made to replace the conditioned spectral analysis which is the basis of the crp method with an alternative time domain decorrelation method the suggested approach the orthogonalised reverse path orp method is illustrated here using data from simulated single degree of freedom sdof and mdof systems

entire solutions of time periodic fisher kpp equation on the half line abstract we study the entire solutions of the fisher kpp equation of the form u t u x x f t u on 0 with boundary condition u t 0 0 where f t u is a fisher kpp type of nonlinearity t periodic in t we obtain two types of entire solutions i for any c 2 a with a 1 t 0 t f u s 0 d s we obtain an entire solution u c t x connecting the periodic traveling wave solution c t x c t at t with the periodic solution v t x at t ii the second type of entire solution u t x connecting the solution of the initial problems of ode t f t at t and v t x at t with properties u x x 0 and u t p t where p t is the periodic solution of t f t

entire solutions of time periodic bistable reaction advection diffusion equations in infinite cylinders this paper is concerned with the existence and some qualitative properties of entire solutions for a reaction advection diffusion equation in infinite cylinders with time periodic bistable nonlinearity here an entire solution means a solution defined in the whole space and for all time t by the comparison principle coupled with the supersolution and subsolution technique it is proved that there exists an entire solution furthermore it is shown that such an entire solution is unique and liapunov stable unlike the reaction diffusion equation without advection the lack of symmetry between increasing and decreasing traveling fronts caused by the advection affects the construction of supersolutions and subsolutions

some entire solutions to the competitive reaction diffusion system abstract this manuscript focuses on the study of the classical lotka volterra competitive reaction diffusion system morita and tachibana 2009 19 proved the existence of entire solutions of this system they used traveling front solutions the lower bound of the proportion between two components of which is positive to construct these entire solutions in order to reasonably explain this positive lower bound we investigate it based on the eigenvalues of the linearized system and show some sufficient and necessary conditions particularly if the parameters in the system satisfy some conditions then the positive lower bound spontaneously holds when the ratio is zero we also construct new entire solutions based on the super sub solution method the comparison theorem and a pair of exact traveling front solutions in addition another kind of entire solutions is revealed by the combination of traveling front solutions their reflects and the solutions of the diffusion free system

pulsating type entire solutions of monostable reaction advection diffusion equations in periodic excitable media abstract we establish the existence of pulsating type entire solutions of reaction advection diffusion equations with monostable nonlinearities in a periodic framework here the nonlinearities include the classic kpp case the pulsating type entire solutions are defined in the whole space and for all time t r by studying a pulsating traveling front connecting a constant unstable stationary state to a stable stationary state which is allowed to be a positive function we proved that there exist pulsating type entire solutions behaving as two pulsating traveling fronts coming from both directions and approaching each other the key techniques are to characterize the asymptotic behavior of the solutions as t in terms of appropriate subsolutions and supersolutions

entire solutions of reaction advection diffusion equations with bistable nonlinearity in cylinders this paper deals with entire solutions and the interaction of traveling wave fronts of bistable reaction advection diffusion equation with infinite cylinders assume that the equation admits three equilibria two stable equilibria 0 and 1 and an unstable equilibrium it is well known that there are different wave fronts connecting any two of those three equilibria by considering a combination of any two of those different traveling wave fronts and constructing appropriate subsolutions and supersolutions we establish three different types of entire solutions finally we analyze a model for shear flows in cylinders to illustrate our main results

entire solutions in bistable reaction diffusion equations with nonlocal delayed nonlinearity this paper is concerned with entire solutions for bistable reaction diffusion equations with nonlocal delay in one dimensional spatial domain here the entire solutions are defined in the whole space and for all time t r assuming that the equation has an increasing traveling wave solution with nonzero wave speed and using the comparison argument we prove the existence of entire solutions which behave as two traveling wave solutions coming from both ends of the x axis and annihilating at a finite time furthermore we show that such an entire solution is unique up to space time translations and is liapunov stable a key idea is to characterize the asymptotic behavior of the solutions as t in terms of appropriate subsolutions and supersolutions in order to illustrate our main results two models of reaction diffusion equations with nonlocal delay arising from mathematical biology are considered

travelling fronts and entire solutions of the fisher kpp equation in ℝn this paper is devoted to time global solutions of the fisher kpp equation in n where f is a c 2 concave function on 0 1 such that f 0 f 1 0 and f 0 on 0 1 it is well known that this equation admits a finite dimensional manifold of planar travelling fronts solutions by considering the mixing of any density of travelling fronts we prove the existence of an infinite dimensional manifold of solutions in particular there are infinite dimensional manifolds of nonplanar travelling fronts and radial solutions furthermore up to an additional assumption a given solution u can be represented in terms of such a mixing of travelling fronts

higher composition laws iv the parametrization of quintic rings rst three parts of this series we considered quadratic cubic and quartic rings i e rings free of ranks 2 3 and 4 over z respectively and found that various algebraic structures involving these rings could be completely parametrized by the integer orbits of an appropriate group representation on a vector space these orbit results are summarized in table 1 in particular the theories behind the parametrizations of quadratic cubic and quartic rings noted in items 2 9 and 13 of table 1 were seen to closely parallel the classical developments of the solutions to the quadratic cubic and quartic equations respectively despite the quintic having been shown to be unsolvable nearly two centuries ago by abel it turns out there still remains much to be said regarding the integral theory of the quintic although a solution naturally still is not possible we show in this article that it is nevertheless possible to completely parametrize quintic rings indeed a theory just as complete as in the quadratic cubic and quartic cases exists also in the case of the quintic in fact we present here a unied theory of ring parametrizations which includes the cases n 2

an entire solution of a bistable parabolic equation on r with two colliding pulses we consider semilinear parabolic equations of the form equation a ut uxx f u x r t i where i 0 i 0 or i i solutions defined for all x t r2 x t r2 are referred to as entire solutions assuming that f c1 r f c1 r is of a bistable type with stable constant steady states 0 and 0 0 we show the existence of an entire solution u x t u x t of the following form for t t u t u t has two humps or pulses one near the other near as t increases the humps move toward the origin x 0x 0 eventually colliding and forming a one hump final shape of the solution with respect to the locally uniform convergence the solution u t u t is a heteroclinic orbit connecting the stable steady state 0 to the unstable ground state of the equation uxx f u 0uxx f u 0 we find the solution u as the limit of a sequence of threshold solutions of the cauchy problem for equation a

time periodic traveling wave solutions for periodic advection reaction diffusion systems abstract we study the existence uniqueness and asymptotic stability of time periodic traveling wave solutions to a class of periodic advection reaction diffusion systems under certain conditions we prove that there exists a maximal wave speed c such that for each wave speed c c there is a time periodic traveling wave connecting two periodic solutions of the corresponding kinetic system it is shown that such a traveling wave is unique modulo translation and is monotone with respect to its co moving frame coordinate we also show that the traveling wave solutions with wave speed c c are asymptotically stable in certain sense in addition we establish the nonexistence of time periodic traveling waves with speed c c

front propagation into unstable states universal algebraic convergence towards uniformly translating pulled fronts fronts that start from a local perturbation and propagate into a linearly unstable state come in two classes pulled fronts and pushed fronts the term pulled front expresses that these fronts are pulled along by the spreading of linear perturbations about the unstable state accordingly their asymptotic speedv equals the spreading speed of perturbations whose dynamics is governed by the equations linearized about the unstable state the central result of this paper is the analysis of the convergence of asymptotically uniformly traveling pulled fronts towardsv we show that when such fronts evolve from sufficiently steep initial conditions which initially decay faster than e x forx 1 t hey have auniversal relaxation behavioras timet 1 the velocity of a pulled front always relaxes algebraically like v t d v 3 2t c 3 p d d 2 t 3 2 c o 1 t 2 the parametersv and d are determined through a saddle point analysis from the equation of motion linearized about the unstable invaded state this front velocity is independent of the precise value of the front amplitude which one tracks to measure the front position the interior of the front is essentially slaved to the leading edge and develops universally as x t d8v t x rt dt 0 v t 0 c o 1 t 2 where8v x vt is a uniformly translating front solution with velocity v v our result which can be viewed as a general center manifold result for pulled front propagation is derived in detail for the well known nonlinear diffusion equation of type t d 2 x c 3 where the invaded unstable state is d 0 even for this simple case the subdominant t 3 2 term extends an earlier result of bramson our analysis is then generalized to more general sets of partial differential equations with higher spatial or temporal derivatives to pdes with memory kernels and also to difference equations such as those that occur in numerical finite difference codes our universal result for pulled fronts thus implies independence i of the level curve which is used to track the front position ii of the precise nonlinearities iii of the precise form of the linear operators in the dynamical equation and iv of the precise initial conditions as long as they are sufficiently steep the only remainders of the explicit form of the dynamical equation are the nonlinear solutions 8v and the three saddle point parameters v and d as our simulations confirm all our analytical predictions in every detail it can be concluded that we have a complete analytical understanding of the propagation mechanism and relaxation behavior of pulled fronts if they are uniformly translating for t 1 an immediate consequence of the slow algebraic relaxation is that the standard moving boundary approximation breaks down for weakly curved pulled fronts in two or three dimensions in addition to our main result for pulled fronts we also discuss the propagation and convergence of fronts emerging from

entire solutions in a time delayed and diffusive epidemic model this paper is concerned with entire solutions in the following time delayed and diffusive epidemic model email protected tu 1 x t d email protected u 1 x t a 1 1u 1 x t a 1 2u 2 x t email protected tu 2 x t d email protected u 2 x t a 2 2u 2 x t gu 1x email protected where d i 0 i 1 2 using comparison arguments we establish the existence and its properties of entire solutions for the above system here an entire solution means a classical solution defined for all email protected r

periodic traveling curved fronts in reaction diffusion equation with bistable time periodic nonlinearity abstract this paper is concerned with the existence and stability of periodic traveling curved fronts for reaction diffusion equations with time periodic bistable nonlinearity in two dimensional space by constructing supersolution and subsolution we prove the existence of periodic traveling wave fronts furthermore we show that the front is globally stable

identification of dynamic characteristics of nonlinear joint based on the optimum equivalent linear frequency response function abstract identification of dynamic characteristics of local nonlinearities has been aimed in this paper the spirit of the identification method is established on optimum equivalent linear frequency response function oelf dynamic behavior of nonlinear elements in system is extracted from oelf using two different techniques the first technique is direct identification method dim in which no pre assumed model is considered for the nonlinearity s behavior and the second technique is model based identification method mim the second technique is introduced with two different formulations in order to take into account the practical limits due to the inaccessibility of nonlinearity location and or indeterminability of degree of freedom dynamic characteristics of common nonlinearity mechanisms like cubic stiffness pure slip and stick slip have been identified using the proposed technique and it has been shown that although the proposed identification technique is simple it does not require any sophisticated measurement hardwares and techniques as required by most of the identification methods proposed so far also the relation of this technique to harmonic balance method is discussed

convergence in time periodic quasilinear parabolic equations in one space dimension abstract we consider time periodic quasilinear parabolic equations in the domain t x r 2 0 x r t t 0 where the right boundary r t of the spatial interval is a positive function which might be periodic or asymptotically periodic or a function tending to infinity or infinity we show that in the first case that is r t is a periodic function any bounded solution of the equations converges as t to a periodic one in the other three cases any positive bounded solution converges as t to a nonnegative periodic one using such a result we study the long time dynamics of the initial boundary value problem on the half line as well as the stefan free boundary problem of a general heterogeneous reaction diffusion equation also we use the convergence result to study the long time dynamics of the initial boundary value problem for a time periodic mean curvature flow equation

entire solutions originating from multiple fronts of an epidemic model with nonlocal dispersal and bistable nonlinearity abstract this paper is concerned with the entire solutions of a nonlocal dispersal epidemic model which arises from the spread of fecally orally transmitted diseases under bistable assumptions it is well known that this model has three different types of traveling wave fronts the annihilating front and merging front entire solutions originating from two fronts of the system have also been constructed in 38 we first prove the uniqueness liapunov stability and continuous dependence on shift parameters of annihilating front entire solutions obtained in 38 a positive time derivative estimate for such entire solution is also obtained then we establish the existence of two different types of entire solutions merging three different fronts furthermore we show that these entire solutions are global lipschitz continuous with respect to the spatial variable x to the best of our knowledge it is the first time that the entire solutions originating from three fronts of diffusion systems have been constructed

identification of a continuous structure with a geometrical non linearity part i conditioned reverse path method particular effort has been spent in the field of identification of multi degree of freedom non linear systems the newly developed methods permit the structural analyst to consider increasingly complex systems the aim of this paper and a companion paper is to study by means of two methods a continuous non linear system consisting of an experimental cantilever beam with a geometrical non linearity in this paper part i the ability of the conditioned reverse path method which is a frequency domain technique to identify the behaviour of this structure is assessed the companion paper part ii is devoted to the application of proper orthogonal decomposition which is an updating technique to the test example

higher composition laws ii on cubic analogues of gauss composition in our first article 2 we developed a new view of gauss composition of binary quadratic forms which led to several new laws of composition on various other spaces of forms moreover we showed that the groups arising from these composition laws were closely related to the class groups of orders in quadratic number fields while the spaces underlying those composition laws were closely related to certain exceptional lie groups in this paper our aim is to develop analogous laws of composition on certain spaces of forms so that the resulting groups yield information on the class groups of orders in cubic fields that is we wish to obtain genuine cubic analogues of gauss composition the fundamental object in our treatment of quadratic composition 2 was the space of 2 x 2 x 2 cubes of integers in particular gauss composition arose from the three different ways of slicing a cube a into two 2 x 2 matrices mi ni i 1 2 3 each such pair mi ni gives rise to a binary quadratic form qa x y qi x y defined by qi x y det mix niy the cube law of 2 declares that as a ranges over all cubes the sum of qi q2 q3 is zero it was shown in 2 that the cube law gives a law of addition on binary quadratic forms that is equivalent to gauss composition various other invariant theoretic constructions using the space of 2 x 2 x 2 cubes led to several new composition laws on other spaces of forms furthermore we showed that each of these composition laws gave rise to groups that are closely related to the class groups of orders in quadratic fields based on the quadratic case described above our first inclination for the cubic case might be to examine 3 x 3 x 3 cubes of integers a 3 x 3 x 3 cube c can be sliced in three different ways into three 3 x 3 matrices li mi ni i 1 2 3 we may therefore obtain from c three ternary cubic forms fl x y z f2 x y z f3 x y z defined by

quotients of some prehomogeneous vector spaces abstract we give quotients v g spec c v g for some prehomogeneous vector spaces g h v

entire solutions for a multi type sis nonlocal epidemic model in r or z abstract in this paper we study entire solutions for a multi type sis nonlocal epidemic model in r or z the existence and asymptotic behavior of spatially independent solutions are first established some new entire solutions are then constructed by combining traveling wave fronts with different speeds and a spatially independent solution from the viewpoint of epidemiology the results provide some new spread ways of the epidemic

existence and uniqueness of entire solutions for a reaction diffusion equation abstract we consider entire solutions of u t u xx f u i e solutions that exist for all x t r 2 where f 0 f 1 0 f 0 in particular we are interested in the entire solutions which behave as two opposite wave fronts of positive speed s approaching each other from both sides of the x axis and then annihilating in a finite time in the case f 1 0 we show that such entire solution exists and is unique up to space time translations in the case f 1 0 we derive two families of such entire solutions in the first family one cannot be any space time translation of the other yet all entire solutions in the second family only differ by a space time translation

entire solutions of nonlocal dispersal equations with monostable nonlinearity in space periodic habitats abstract this paper is concerned with the new types of entire solutions other than traveling wave solutions of nonlocal dispersal equations with monostable nonlinearity in space periodic habitats we first establish the existence and properties of spatially periodic solutions connecting two steady states then new types of entire solutions are constructed by combining the rightward and leftward pulsating traveling fronts with different speeds and a spatially periodic solution finally for a class of special heterogeneous reaction we further establish the uniqueness of entire solutions and the continuous dependence of such an entire solution on parameters such as wave speeds and the shifted variables in other words we build a five dimensional manifold of solutions and the traveling wave solutions are on the boundary of the manifold

nonlinear structural identification by the reverse path spectral method abstract when dealing with nonlinear dynamical systems it is important to have efficient accurate and reliable tools for estimating both the linear and nonlinear system parameters from measured data an approach for nonlinear system identification widely studied in recent years is reverse path this method is based on broad band excitation and treats the nonlinear terms as feedback forces acting on an underlying linear system parameter estimation is performed in the frequency domain using conventional multiple input multiple output or multiple input single output techniques this paper presents a generalized approach to apply the method of reverse path on continuous mechanical systems with multiple nonlinearities the method requires few spectral calculations and is therefore suitable for use in iterative processes to locate and estimate structural nonlinearities the proposed method is demonstrated in both simulations and experiments on continuous nonlinear mechanical structures the results show that the method is effective on both simulated as well as experimental data

asymptotic behavior of traveling fronts and entire solutions for a periodic bistable competition diffusion system abstract this paper is concerned with a time periodic competition diffusion system u t u x x u r 1 t a 1 t u b 1 t v t 0 x r v t d v x x v r 2 t a 2 t u b 2 t v t 0 x r where u t x and v t x denote the densities of two competing species d 0 is some constant r i t a i t and b i t are t periodic continuous functions under suitable conditions it has been confirmed by bao and wang 2013 2 that this system admits periodic traveling fronts connecting two stable semi trivial t periodic solutions p t 0 and 0 q t associated to the corresponding kinetic system in the present work we first investigate the asymptotic behavior of periodic bistable traveling fronts with non zero speeds at infinity by a dynamical approach combined with the two sided laplace transform method with these asymptotic properties we then obtain some key estimates as a result by applying the super and subsolutions techniques as well as the comparison principle we establish the existence and various qualitative properties of the so called entire solutions defined for all time and the whole space which provides some new spreading ways other than periodic traveling waves for two strongly competing species interacting in a heterogeneous environment

generation and propagation of interfaces for reaction diffusion equations abstract this paper is concerned with the asymptotic behavior as e 0 of the solution ue of the reaction diffusion equation in r n r u t 1 2 0 where is the derivative of a bistable potential we show that if the initial data u 0 has values in both domains of attraction of the potential then an interface will develop in a short time o e 2 ln e we also show that if the wells of the potential are of equal depth then this interface will propagate with normal velocity equal to its mean curvature our result is valid as long as the interface remains smooth if the initial interface is compact and n 2 then the interface will disappear in a finite time but not if n 1 in case the depths of the wells are not equal then in order to obtain get reasonable results we must work on the scaled time s t e slower time scale in this scale we show that the interface moves with a constant speed proportional to the difference of the depths of the two wells along the normal towards the domain of the deeper well this result is valid for all s 0 and does not actually depend on the regularity of the interface we also extend the above results to the homogeneous second initial boundary value problem in case the depths of the wells are equal and the initial interface is orthogonal to the boundary we prove that the interface moves with normal velocity equal to its mean curvature provided that there is a family of hypersurfaces which moves according to their mean curvature and intersect the boundary orthogonally

prehomogeneous vector spaces and field extensions letk be an infinite field of characteristic not equal to 2 3 5 in this paper we construct a natural map from the set of orbits of certain prehomogeneous vector spaces to the set of isomorphism classes of galois extensions ofk which are splitting fields of equations of certain degrees and prove that the inverse image of this map corresponds bijectively with conjugacy classes of galois homomorphisms

identification of multi degree of freedom non linear systems under random excitations by the reverse path spectral method abstract conventional frequency response estimation methods such as the h 1 and h 2 methods often yield measured frequency response functions which are contaminated by the presence of non linearities and hence make it difficult to extract underlying linear system properties to overcome this deficiency a new spectral approach for identifying multi degree of freedom non linear systems is introduced which is based on a reverse path formulation as available in the literature for single degree of freedom non linear systems certain modifications are made in this article for a multi degree of freedom reverse path formulation that utilizes multiple input multiple output data from non linear systems when excited by gaussian random excitations conditioned h c 1 and h c 2 frequency response estimates now yield the underlying linear properties without contaminating effects from the non linearities once the conditioned frequency response functions have been estimated the non linearities which are described by analytical functions are also identified by estimating the coefficients of these functions identification of the local or distributed non linearities which exist at or away from the excitation locations is possible the new spectral approach is successfully tested on several example systems which include a three degree of freedom system with an asymmetric non linearity a three degree of freedom system with distributed non linearities and a five degree of freedom system with multiple non linearities and multiple excitations

d4 modular forms in this paper we study modular forms on two simply connected groups of type d 4 over q one group g s is a globally split group of type d 4 viewed as the group of isotopies of the split rational octonions the other g c is the isotopy group of the rational nonsplit octonions we study automorphic forms on g s in analogy to the work of gross gan and savin on g 2 namely we study automorphic forms whose component at infinity corresponds to a quaternionic discrete series representation we study automorphic forms on g c using gross s formalism of algebraic modular forms finally we follow work of gan savin gross rallis and others to study an exceptional theta correspondence connecting modular forms on g c and g s this can be thought of as an octonionic generalization of the jacquet langlands correspondence

a new derivation of the frequency response function matrix for vibrating non linear systems abstract frequency response function matrices relate the inputs and the outputs of structural dynamic systems if a system is linear the frequency response function matrix is the same for any combination or types of inputs over the entire operating range furthermore the frequency response matrix of a linear vibrating system is a simple combination of temporal and spatial characteristics the modal frequencies modal vectors and modal scale factors when a system is non linear the inputs interact through an exchange of energy between the linear and non linear elements in the system no general combination of the temporal and spatial non linear characteristics has to date been proposed to describe these linear non linear interactions this article introduces a unifying perspective of non linearities as internal feedback forces that act together with the external forces to generate the response of the non linear system this perspective of the non linearities is spatial in nature and leads to two simple but conceptually powerful relationships between the frequency response function matrix of a non linear multiple degree of freedom system and its linear counterpart several single and multiple degree of freedom systems are used to demonstrate the use and interpretation of these relationships the broad implication of the new input output frequency response representation for both linear and non linear systems are also addressed in particular the merits of the spatial perspective of non linear systems and the new frequency response relationships are stated in the context of linear and non linear system characterization and identification one implication is that these relationships suggest there is an input output dependent temporal spatial modal decomposition of the frequency response function matrix for non linear systems

orbit closure diagram for the space of quadruples of quinary alternating forms abstract we construct the hasse diagram for the closure ordering of the orbits of the irreducible reduced prehomogeneous vector space of quadruples of quinary alternating forms in five variables

entire solutions in nonlocal dispersal equations with bistable nonlinearity abstract we consider entire solutions of nonlocal dispersal equations with bistable nonlinearity in one dimensional spatial domain a two dimensional manifold of entire solutions which behave as two traveling wave solutions coming from both directions is established by an increasing traveling wave front with nonzero wave speed furthermore we show that such an entire solution is unique up to space time translations and liapunov stable a key idea is to characterize the asymptotic behaviors of the solutions as t in terms of appropriate subsolutions and supersolutions we have to emphasize that a lack of regularizing effect occurs

how many rational points does a random curve have a large part of modern arithmetic geometry is dedicated to or motivated by the study of rational points on varieties for an elliptic curve over q the set of rational points forms a finitely generated abelian group the ranks of these groups when ranging over all elliptic curves are conjectured to be evenly distributed between rank 0 and rank 1 with higher ranks being negligible we will describe these conjectures and discuss some results on bounds for average rank highlighting the recent work of bhargava and shankar

higher composition laws i a new view on gauss composition and quadratic generalizations two centuries ago in his celebrated work disquisitiones arithmeticae of 1801 gauss laid down the beautiful law of composition of integral binary quadratic forms which would play such a critical role in number theory in the decades to follow even today two centuries later this law of composition still remains one of the primary tools for understanding and computing with the class groups of quadratic orders it is hence only natural to ask whether higher analogues of this composition law exist that could shed light on the structure of other algebraic number rings and fields this article forms the first of a series of four articles in which our aim is precisely to develop such higher composition laws in fact we show that gauss s law of composition is only one of at least fourteen composition laws of its kind which yield information on number rings and their class groups in this paper we begin by deriving a general law of composition on 2 2 2 cubes of integers from which we are able to obtain gauss s composition law on binary quadratic forms as a simple special case in a manner reminiscent of the group law on plane elliptic curves we also obtain from this composition law on 2 2 2 cubes four further new laws of composition these laws of composition are defined on 1 binary cubic forms 2 pairs of binary quadratic forms 3 pairs of quaternary alternating 2 forms and 4 senary six variable alternating 3 forms more precisely gauss s theorem states that the set of sl2 z equivalence classes of primitive binary quadratic forms of a given discriminant d has an inherent group structure the five other spaces of forms mentioned above including the space of 2 2 2 cubes also possess natural actions by special linear groups over z and certain products thereof we prove that just like gauss s space of binary quadratic forms each of these group actions has the following remarkable properties first each of these six spaces possesses only a single polynomial invariant for the corresponding group action which we call the discriminant this discriminant invariant is found to take only values that

asymptotic behavior of traveling fronts and entire solutions for a nonlocal monostable equation abstract this paper is concerned with the asymptotic behavior of traveling fronts and entire solutions for a nonlocal monostable equation the asymptotic behavior of traveling fronts is obtained using ikehara s theorem by the sub super solution method the existence of entire solutions is obtained they behave as two opposite wave fronts approaching each other from either side of the x axis and then annihilating in a finite time by an entire solution we mean one which is defined over the whole space for all time t r

entire solutions for nonlocal dispersal equations with spatio temporal delay monostable case abstract this paper deals with entire solutions for a general nonlocal dispersal monostable equation with spatio temporal delay i e solutions that are defined in the whole space and for all time t r we first derive a particular model for a single species and show how such systems arise from population biology then we construct some new types of entire solutions other than traveling wave solutions and equilibrium solutions of the equation under consideration with quasi monotone and non quasi monotone nonlinearities various qualitative properties of the entire solutions are also investigated in particular the relationship between the entire solutions and the traveling wave fronts which they originated is considered our main arguments are based on the comparison principle the method of super and sub solutions and the construction of auxiliary control systems

entire solutions of diffusive lotka volterra system abstract this work is concerned with the existence of entire solutions of the diffusive lotka volterra competition system on the real line we prove the existence of some entire solutions that are asymptotic as t to a traveling wave consisting of a single species depending on system parameters these entire solutions evolve into two ore more stacked invasion waves as t our results cover both the weak and strong competition case in the weak competition case the existence of a class of entire solutions that forms a 4 dimensional manifold is proved

entire solutions in monostable reaction advection diffusion equations in infinite cylinders abstract this paper is concerned with entire solutions of a monostable reaction advection diffusion equation in infinite cylinders without the condition f u f 0 by constructing a quasi invariant manifold we prove that there exist two classes of entire solutions furthermore we show that one class of such entire solutions is unique up to space and time translation

bistable traveling waves around an obstacle we consider traveling waves for a nonlinear diffusion equation with a bistable or multistable nonlinearity the goal is to study how a planar traveling front interacts with a compact obstacle that is placed in the middle of the space n as a first step we prove the existence and uniqueness of an entire solution that behaves like a planar wave front approaching from infinity and eventually reaching the obstacle this causes disturbance on the shape of the front but we show that the solution will gradually recover its planar wave profile and continue to propagate in the same direction leaving the obstacle behind whether the recovery is uniform in space is shown to depend on the shape of the obstacle 2008 wiley periodicals inc

a modular compactification of m_ 1 n from a_ infty structures we show that a certain moduli space of minimal a structures coincides with the modular compactification m1 n n 1 of m1 n constructed by smyth in 22 in addition we describe these moduli spaces and the universal curves over them by explicit equations prove that they are normal and gorenstein show that their picard groups have no torsion and that they have rational singularities if and only if n 11

spatial truncation in models of non linear vibrating systems abstract vibrating systems usually have an infinite number of degrees of freedom d o f since a finite number of measurement d o f can only capture certain deformation patterns the spatial characteristics of vibrating systems are only partially observed experimentally this research examines the effects of spatial truncation on non linear system identification it is demonstrated that truncation produces frequency correlated noise which cannot be extracted completely with traditional techniques the goal of this article is to provide insight and practical recommendations for diagnosing and or compensating for errors due to spatial truncation in multiple d o f systems the article demonstrates that non linear dynamic systems should be instrumented with sufficient sensors model properties should be used in addition to the input output data and additional temporal data should be collected to help diagnose errors due to spatial truncation

frequency domain arx model and multi harmonic frf estimators for non linear dynamic systems abstract non linear dynamic systems respond at frequencies other than the excitation frequency however standard frequency response function estimators for linear systems do not accommodate this harmonic distortion a new multi harmonic frequency response function estimator that utilizes discrete frequency models for non linear systems is introduced here the multi harmonic estimator relates the frequency response at each frequency to the input and output spectra within a given frequency band in the same way that autoregressive exogenous input models relate inputs and outputs at particular samples in the time domain overdetermined least mean squares calculations are used to minimize model error throughout a frequency band rather than at a single frequency as in the corresponding linear estimators the resulting multi harmonic frequency response function models are non parametric e g vary with amplitude when linear functions are used and parametric when non linear functions are used a new sensitive indicator for experimentally characterizing non linearity is introduced

a phase plane discussion of convergence to travelling fronts for nonlinear diffusion the paper is concerned with the asymptotic behavior as t of solutions u x t of the equation u t u xx f left u right 0 x in left infty infty right in the case f 0 f 1 0 with f u non positive for u 0 sufficiently close to zero and f u non negative for u 1 sufficiently close to 1 this guarantees the uniqueness but not the existence of a travelling front solution u u x ct u 0 u and it is shown in essence that solutions with monotonic initial data converge to a translate of this travelling front if it exists and to a stacked combination of travelling fronts if it does not the approach is to use the monotonicity to take u and t as independent variables and p ux as the dependent variable and to apply ideas of sub and super solutions to the diffusion equation for p

pulsating fronts and front like entire solutions for a reaction advection diffusion competition model in a periodic habitat abstract this paper is devoted to the study of pulsating fronts and pulsating front like entire solutions for a reaction advection diffusion model of two competing species in a periodic habitat under certain assumptions the competition system admits a leftward and a rightward pulsating fronts in the bistable case in this work we construct some other types of entire solutions by interacting the leftward and rightward pulsating fronts some of these entire solutions behave as the two pulsating fronts approaching each other from both sides of the x axis which turn out to be unique and liapunov stable 2 dimensional manifolds of solutions furthermore the leftward and rightward pulsating fronts are on the boundary of these 2 dimensional manifolds the others behave as the two pulsating fronts propagating from one side of the x axis the faster one then invades the slower one as t these kinds of pulsating front like entire solutions then provide some new spreading ways other than pulsating fronts for two strongly competing species interacting in a heterogeneous habitat

entire solutions of the kpp equation this paper deals with the solutions defined for all time of the kpp equation ut uxx f u 0 0 f 0 1 0i n 0 1 and f 0 s f 0 0 in 0 1 this equation admits infinitely many traveling wave type solutions increasing or decreasing in x i t also admits solutions that depend only on t in this paper we build four other manifolds of solutions one is 5 dimensional one is 4 dimensional and two are 3 dimensional some of these new solutions are obtained by considering two traveling waves that come from both sides of the real axis and mix furthermore the traveling wave solutions are on the boundary of these four manifolds c 1999 john wiley sons inc

entire solutions in monostable reaction diffusion equations with delayed nonlinearity abstract entire solutions for monostable reaction diffusion equations with nonlocal delay in one dimensional spatial domain are considered a comparison argument is employed to prove the existence of entire solutions which behave as two traveling wave solutions coming from both directions some new entire solutions are also constructed by mixing traveling wave solutions with heteroclinic orbits of the spatially averaged ordinary differential equations and the existence of such a heteroclinic orbit is established using the monotone dynamical systems theory key techniques include the characterization of the asymptotic behaviors of solutions as t in term of appropriate subsolutions and supersolutions two models of reaction diffusion equations with nonlocal delay arising from mathematical biology are given to illustrate main results

existence and stability of time periodic traveling waves for a periodic bistable lotka volterra competition system abstract this paper is concerned with the time periodic lotka volterra competition diffusion system u t d 1 t u x x u r 1 t a 1 t u b 1 t v v t d 2 t v x x v r 2 t a 2 t u b 2 t v x r t 0 where d i t r i t a i t b i t c 2 r are t periodic functions 0 1 i 1 2 under certain conditions the system admits two stable semi trivial periodic solutions p t 0 and 0 q t and a unique coexistence periodic solution u 0 t v 0 t which is unstable and satisfies 0 u 0 t p t and 0 v 0 t q t for t 0 t in this paper we prove that the system admits a time periodic traveling wave solution u x t v x t u x c t t v x c t t connecting two periodic solutions p t 0 and 0 q t as x where c is the wave speed by using a dynamical method we show that the time periodic traveling wave solution u x c t t v x c t t is asymptotically stable and unique modulo translation for front like initial values