principal hierarchies of infinite dimensional frobenius manifolds the extended 2d toda lattice abstract we define a dispersionless tau symmetric bihamiltonian integrable hierarchy on the space of pairs of functions analytic inside outside the unit circle with simple poles at 0 respectively which extends the dispersionless 2d toda hierarchy of takasaki and takebe then we construct the deformed flat connection of the infinite dimensional frobenius manifold m 0 introduced by carlet dubrovin and mertens 2011 3 and by explicitly solving the deformed flatness equations we prove that the extended 2d toda hierarchy coincides with principal hierarchy of m 0

on dispersionless lax representations for two primary models the equivalence between bi hamiltonian formulation and lax representation of dispersionless integrable hierarchies associated with two primary models of frobenius manifolds is explicitly verified

block type symmetry of bigraded toda hierarchy in this paper we define orlov schulman s operators ml mr and then use them to construct the additional symmetries of the bigraded toda hierarchy we further show that these additional symmetries form an interesting infinite dimensional lie algebra known as a block type lie algebra whose structure theory and representation theory have recently received much attention in the literature by acting on two different spaces under the weak w constraints we find in particular two representations of this block type lie algebra

the importance of being discrete life always wins on the surface many systems in chemistry biology finance and social sciences present emerging features that are not easy to guess from the elementary interactions of their microscopic individual components in the past the macroscopic behavior of such systems was modeled by assuming that the collective dynamics of microscopic components can be effectively described collectively by equations acting on spatially continuous density distributions it turns out that to the contrary taking into account the actual individual discrete character of the microscopic components of these systems is crucial for explaining their macroscopic behavior in fact we find that in conditions in which the continuum approach would predict the extinction of all of the population respectively the vanishing of the invested capital or the concentration of a chemical substance etc the microscopic granularity insures the emergence of macroscopic localized subpopulations with collective adaptive properties that allow their survival and development in particular it is found that in two dimensions life the localized proliferating phase always prevails

on some classes of discrete polynomials and ordinary difference equations we introduce two classes of discrete polynomials and use them to construct ordinary difference equations admitting a lax representation in terms of these polynomials we also construct lattice integrable hierarchies in their explicit form and show some examples

killer geometries in competing species dynamics we discuss a cellular automata model to study the competition between an emergent better fitted species against an existing majority species the model implements local fights among small group of individuals and a synchronous random walk on a 2d lattice the faith of the system i e the spreading or disappearance of the species is determined by their initial density and fight frequency the initial density of the emergent species has to be higher than a critical threshold for total spreading but this value depends in a non trivial way on the fight frequency below the threshold any better adapted species disappears showing that a qualitative advantage is not enough for a minority to win no strategy is involved but spatial organization turns out to be crucial for instance at minority densities of zero measure some very rare local geometries which occur by chance are found to be killer geometries once set they lead with high probability to the total destruction of the preexisting majority species the occurrence rate of these killer geometries is a function of the system size this model may apply to a large spectrum of competing groups like smoker non smoker opinion forming diffusion of innovation setting of industrial standards species evolution epidemic spreading and cancer growth

ternary hom nambu lie algebras induced by hom lie algebras the need to consider n ary algebraic structures generalizing lie and poisson algebras has become increasingly important in physics and it should therefore be of interest to study the mathematical concepts related to n ary algebras the purpose of this paper is to investigate ternary multiplications as deformations of n lie structures constructed from the binary multiplication of a hom lie algebra a linear twisting map and a trace function satisfying certain compatibility conditions we show that the relation between the kernels of the twisting maps and the trace function plays an important role in this context and provide examples of hom nambu lie algebras obtained using this construction

on multiple fixed sign solutions of a discrete system with hermite boundary conditions abstract we consider the discrete system m u i k p i k u 1 k u 2 k u n k k 0 1 n 1 i n together with hermite boundary conditions j u i k 0 j 0 m 1 1 r 1 i n where r 2 m 1 for 1 r 1rm m and k s are integers such that 0 k1

multiple fixed sign solutions for a system of difference equations with sturm liouville conditions we consider the following system of difference equations d mu i k p i k u 1 k u 2 k u n k 0 k 0 1 n i 1 2 ntogether with sturm liouville boundary conditions d ju i 0 0 0 2 n m 1 z 0 w 0 h 0 d w z w n 1 z d h w 0 by using two different fixed point theorems we develop criteria for the existence of three solutions of the system which are of fixed signs on 0 1 n m examples are also included to illustrate the results obtained

infrequent social interaction can accelerate the spread of a persuasive idea we study the spread of a persuasive new idea through a population of continuous time random walkers in one dimension the idea spreads via social gatherings involving groups of nearby walkers who act according to a biased majority rule after each gathering the group takes on the new idea if more than a critical fraction 1 2 1 2 of them already hold it otherwise they all reject it the boundary of a domain where the new idea has taken hold expands as a traveling wave in the density of new idea holders our walkers move by levy motion and we compute the wave velocity analytically as a function of the frequency of social gatherings and the exponent of the jump distribution when this distribution is sufficiently heavy tailed then counter to intuition the idea can propagate faster if social gatherings are held less frequently when jumps are truncated a critical gathering frequency can emerge which maximizes propagation velocity we explore our model by simulation confirming our analytical results

differential equations in the spectral parameter we determine all the potentialsv x for the schrodinger equation 2 v x k2 such that some family of eigenfunctions satisfies a differential equation in the spectral parameterk of the formb k k o x o for each suchv x we determine the algebra of all possible operatorsb and the corresponding functions x

bispectral algebras of commuting ordinary differential operators we develop a systematic way for constructing bispectral algebras of commuting ordinary differential operators of any rank n it combines and unifies the ideas of duistermaat grunbaum and wilson our construction is completely algorithmic and enables us to obtain all previously known classes or individual examples of bispectral operators the method also provides new broad families of bispectral algebras which may help to penetrate deeper into the problem

the extended bigraded toda hierarchy we generalize the toda lattice hierarchy by considering n m dependent variables we construct roots and logarithms of the lax operator which are uniquely defined operators with coefficients that are series of differential polynomials in the dependent variables and we use them to provide a lax pair definition of the extended bigraded toda hierarchy generalizing 4 using r matrix theory we give the bi hamiltonian formulation of this hierarchy and we prove the existence of a tau function for its solutions finally we study the dispersionless limit and its connection with a class of frobenius manifolds on the orbit space of the extended affine weyl groups of the a series defined by dubrovin and zhang 1998 compos math 111 167

social paradoxes of majority rule voting and renormalization group real space renormalization group ideas are used to study a voting problem in political science a model to construct self directed pyramidal structures from bottom up to the top is presented using majority rules it is shown that a minority and even a majority can be systematically self eliminated from top leadership provided the hierarchy has a minimal number of levels in some cases 70 of the population is found to have zero representation after six hierarchical levels results are discussed with respect to the internal operation of political organizations

existence of triple positive solutions of two point right focal boundary value problems on time scales we consider the following boundary value problem 1 n 1y d n t 1 p 1f t y s n 1 t email protected a b t y d n a 0 email protected email protected p 1 y d n s b 0 email protected email protected n 1 where n 2 1 p n 1 is fixed and t is a time scale by applying fixed point theorems for operators on a cone existence criteria are developed for triple positive solutions of the boundary value problem we also include examples to illustrate the usefulness of the results obtained

r matrices and higher poisson brackets for integrable systems the tri hamiltonian nature of lax equations is revealed starting with an r matrix on an associative algebra g equipped with a trace form there are g compatible poisson brackets with linear quadratic and cubic dependence on the coordinates the invariant functions casimir functions on g are in involution relative to these brackets they yield a hierarchy of integrable tri hamiltonian lax equations the results can be applied to solvable pde s such as the korteweg de vries equation as well as to finite integrable systems such as the toda lattice in these cases the poisson structures considered here turn out to be abstract versions of the first 3 hamiltonian operators of these equations obtained by their well known recursion operators

three positive solutions to a discrete focal boundary value problem abstract we are concerned with the discrete focal boundary value problem 3x t k f x t x a x t2 2x b 1 0 under various assumptions on f and the integers a t2 and b we prove the existence of three positive solutions of this boundary value problem to prove our results we use fixed point theorems concerning cones in a banach space

spatial evolutionary prisoner s dilemma game with three strategies and external constraints the emergency of mutual cooperation is studied in a spatially extended evolutionary prisoner s dilemma game in which the players are located on the sites of cubic lattices for dimensions d51 2 and 3 each player can choose one of the three following strategies cooperation c defection d or tit for tat t during the evolutionary process the randomly chosen players adopt one of their neighboring strategies if the chosen neighbor has a higher payoff moreover an external constraint imposes that the players always cooperate with probability p the stationary state phase diagram is computed by both using generalized mean field approximations and monte carlo simulations nonequilibrium second order phase transitions associated with the extinction of one of the possible strategies are found and the corresponding critical exponents belong to the directed percolation universality class it is shown that externally forcing the collaboration does not always produce the desired result

positive solutions of nonlinear singular third order two point boundary value problem abstract in this paper we are concerned with the existence of single and multiple positive solutions to the nonlinear singular third order two point boundary value problem u t a t f u t 0 0 t 1 u 0 u 0 u 1 0 where is a positive parameter under various assumptions on a and f we establish intervals of the parameter which yield the existence of at least one at least two and infinitely many positive solutions of the boundary value problem by using krasnoselskii s fixed point theorem of cone expansion compression type

a modified galam s model for word of mouth information exchange abstract in this paper we analyze the stochastic model proposed by galam in s galam modelling rumors the no plane pentagon french hoax case physica a 320 2003 571 580 for information spreading in a word of mouth process among agents based on a majority rule using the communications rules among agents defined in the above reference we first perform simulations of the word of mouth process and compare the results with the theoretical values predicted by galam s model some dissimilarities arise in particular when a small number of agents is considered we find motivations for these dissimilarities and suggest some enhancements by introducing a new parameter dependent model we propose a modified galam s scheme which is asymptotically coincident with the original model in the above reference furthermore for relatively small values of the parameter we provide a numerical experience proving that the modified model often outperforms the original one

existence of triple positive solutions for a third order three point boundary value problem in this paper we investigate the existence of triple positive solutions for the nonlinear third order three point boundary value problem u t a t f t u t u t u t 0 0 q 0 1 0 are continuous first green s function for the associated linear boundary value problem is constructed and then by using a fixed point theorem due to avery and peterson we establish results on the existence of triple positive solutions to the boundary value problem

r matrix approach to lattice integrable systems an r matrix formalism is applied to the construction of the integrable lattice systems and their bi hamiltonian structure miura like gauge transformations between the hierarchies are also investigated in the end the ladder of linear maps between generated hierarchies is established and described

existence of positive solutions to second order time scale systems in this paper the author consider the following dynamic system on a measure chain u i d d t f i t u 1 s t u 2 s t u n s t 0 email protected a b satisfying sturm liouville boundary conditions a iu i a email protected iu i d a 0 c iu i s b email protected iu i d s b 0 i 1 2 n some new results are obtained for the existence of single twin and triple positive solutions of the above system by using nonlinear alternative of leray schauder type krasnosel skii s fixed point theorem in a cone and leggett williams fixed point theorem

multiple positive solutions to a third order discrete focal boundary value problem abstract we are concerned with the discrete focal boundary value problem 3 x t k f x t x a x t 2 2 x b 1 0 under various assumptions on f and the integers a t 2 and b we prove the existence of three positive solutions of this boundary value problem to prove our results we will apply a generalization of the leggett williams fixed point theorem

existence of three positive solutions to a second order boundary value problem on a measure chain growth conditions are imposed on f such that the boundary value problem x t f x t t t1 t2 x t1 x t1 0 and x t2 x t2 0 where t1 t2 from a measure chain t has at least three positive solutions by way of the five functionals fixed point theorem

effect of initial concentration and spatial heterogeneity of active agent distribution on opinion dynamics abstract we analyze the effect of spatial heterogeneity in the initial spin distribution on spin dynamics in a three state square lattice divided into spatial cells districts in the spirit of the statistical mechanics of social impact we introduce a dominant influence rule dir according to which in a single update step a chosen node adopts the state determined by the influence of its discussion group formed by the node itself and its neighbors within one or more coordination spheres in contrast to models based on some form of majority rule mr a system governed by the dir is easily trapped in a stable non consensus state if all nodes of the discussion group have the same weight of influence to ensure that a consensus in the dir system is necessarily reached we need to put a stochastic process in the update rule further the stochastic dir model is used as a starting point for understanding the effect of spatial heterogeneity of active agent non zero spin distribution on the exit probabilities initially the positive and negative spins active agents are assigned to some nodes with non uniform spatial distributions while the rest of the nodes remain in the state with spin zero uncommitted voters by varying the relative means and skewness of the initial spin distributions we observe critical behaviors of exit probabilities in finite size systems the critical exponents are obtained by monte carlo simulations the results of numerical simulations are discussed in the context of social dynamics

local and non local multiplicative poisson vertex algebras and differential difference equations we develop the notions of multiplicative lie conformal and poisson vertex algebras local and non local and their connections to the theory of integrable differential difference hamiltonian equations we establish relations of these notions to q deformed w algebras and lattice poisson algebras we introduce the notion of adler type pseudodifference operators and apply them to integrability of differential difference hamiltonian equations

nonlinear poisson structures and r matrices we introduce quadratic poisson structures on lie groups associated with a class of solutions of the modified yang baxter equation and apply them to the hamiltonian description of lax systems the formal analog of these brackets on associative algebras provides second structures for certain integrable equations in particular the integrals of the toda flow on generic orbits are shown to satisfy recursion relations finally we exhibit a third order poisson bracket for which ther matrix approach is feasible

additional symmetries of the extended bigraded toda hierarchy the extended bigraded toda hierarchy ebth is an integrable system satisfied by the total descendant potential of with two orbifold points we construct additional symmetries of the ebth and describe explicitly their action on the lax operator wave operators and tau function of the hierarchy in particular we obtain infinitesimal symmetries of the ebth that act on the tau function as a subalgebra of the virasoro algebra generalizing those of dubrovin and zhang

discretization of nonlinear evolution equations over associative function algebras abstract a general approach is proposed for discretizing nonlinear dynamical systems and field theories on suitable functional spaces defined over a regular lattice of points in such a way that both their symmetry and integrability properties are preserved a class of discrete kdv equations is introduced also new hierarchies of discrete evolution equations of gelfand dickey type are defined

the algebra of q pseudodifferential symbols and the q wkp n algebra in this paper we continue with the program to explore the topography of the space of w type algebras in the present case the starting point is the work of khesin lyubashenko and roger on the algebra of q deformed pseudodifferential symbols and their associated integrable hierarchies the analysis goes on by studying the associated hamiltonian structures for which compact expressions are found the fundamental poisson brackets yield q deformations of wkp and related w type algebras which in specific cases coincide with the ones constructed by frenkel and reshetikhin the construction underlies a continuous correspondence between the hamiltonian structures of the toda lattice and the kp hierarchies

positive solutions of singular third order three point boundary value problem abstract in this paper we investigate the problem of existence of positive solutions for the nonlinear singular third order three point boundary value problem u t a t f t u t 0 0 t 1 u 0 u u 1 0 where is a positive parameter and 1 2 1 is a constant by using a fixed point theorem of cone expansion compression type due to krasnosel skii we establish various results on the existence of single and multiple positive solutions to the boundary value problem under various assumptions on functions f and a we give explicitly the intervals for parameter in which the existence of positive solutions is guaranteed especially we allow the function a t of nonlinear term to have suitable singularities

existence of monotone positive solutions to a third order two point generalized right focal boundary value problem in this paper we are concerned with the third order two point generalized right focal boundary value problem x t f t x t 0 a

gromov witten theory of orbicurves the space of tri polynomials and symplectic field theory of seifert fibrations we compute with symplectic field theory sft techniques the gromov witten theory of mathbb p 1 alpha 1 ldots alpha a i e the complex projective line with a orbifold points a natural subclass of these orbifolds the ones with polynomial quantum cohomology gives rise to a family of polynomial frobenius manifolds and integrable systems of hamiltonian pdes which extend the dispersionless bigraded toda hierarchy carlet the extended bigraded toda hierarchy arxiv preprint arxiv math ph 0604024 we then define a frobenius structure on the spaces of polynomials in three complex variables of the form f x y z xyz p1 x p2 y p3 z which contains as special cases the ones constructed on the space of laurent polynomials dubrovin geometry of 2d topologica field theories integrable systems and quantum groups springer lecture notes in mathematics 1620 120 348 1996 milanov and tseng the space of laurent polynomials mathbb p 1 orbifolds and integrable hierarchies preprint arxiv math 0607012v3 math ag we prove a mirror theorem stating that these frobenius structures are isomorphic to the ones found before for polynomial mathbb p 1 orbifolds finally we link rational sft of seifert fibrations over mathbb p 1 a b c with orbifold gromov witten invariants of the base extending a known result bourgeois a morse bott approach to contact homology ph d dissertation stanford university 2002 valid in the smooth case

positive solutions to a singular third order three point boundary value problem with an indefinitely signed green s function abstract we determine a green s function for the singular three point third order bvp u t a t f t u t 0 t 1 u 0 u 1 u 0 and then we apply the classical krasnosel ski s fixed point theorem on an also new cone the emphasis is mainly that although this bvp does not admit a positive green s function the solution obtained is still positive in our approach we find out that 17 24 1 our technique is a combination of the above fixed point theorem and properties of the corresponding vector field

hirota equations for the extended bigraded toda hierarchy and the total descendent potential of cp1 orbifolds we prove that the hirota quadratic equations of milanov and tseng define an integrable hierarchy which is equivalent to the extended bigraded toda hierarchy ebth in particular this confirms a conjecture that relates the total descendent potential of the orbifold ck m with a tau function of the ebth

on additional symmetries of the kp hierarchy and sato s backlund transformation a short proof is given to the fact that the additional symmetries of the kp hierarchy defined by their action on pseudodifferential operators according to fuchssteiner chen lee lin orlov shulman coincide with those defined by their action on functions as sato s backlund transformations a new simple formula for the generator of additional symmetries is also presented

nontrivial solution of third order nonlinear eigenvalue problems ii abstract in this paper we firstly make some corrections for typos errors on our paper nontrivial solution of third order nonlinear eigenvalue problems appl math comput 176 2006 714 721 and then we supplement some cases which have not been studied in the above paper

a matrix integral solution to two dimensional w p gravity thep th gel fand dickey equation and the string equation l p 1 have a common solution expressible in terms of an integral overn n hermitean matrices for largen the integrand being a perturbation of a gaussian generalizing kontsevich s integral beyond the kdv case it is equivalent to showing that is a vacuum vector for aw generated from the coefficients of the vertex operator this connection is established via a quadratic identity involving the wave function and the vertex operator which is a disguised differential version of the fay identity the latter is also the key to the spectral theory for the two compatible symplectic structures of kdv in terms of the stress energy tensor associated with the virasoro algebra given a differential operator l d p q 2 t d p 2 cdots q p t with d frac partial dx t t 1 t 2 t 3 x equiv t 1 consider the deformation equations1 0 1 begin gathered frac partial l partial t n l n p l n 1 2 n 0 mod p hfill p reduced kp equation hfill end gathered ofl for which there exists a differential operatorp possibly of infinite order such that 0 2 l p 1 string equation in this note we give a complete solution to this problem in section 1 we give a brief survey of useful facts about thei function t the wave function t z solution of t n l n p x andl 1 p z and the corresponding infinitedimensional planev 0 of formal power series inz for largez v 0 span psi t z for all t in mathbb c infty in sato s grassmannian the three theorems below form the core of the paper their proof will be given in subseuqent sections each of which lives on its own right

a fundamentally simple approach to a singular boundary value problem abstract existence results by means of a simple approach for the 3rd order differential equation x t t f t x t x t x t 0 t 1 satisfying the three point boundary value conditions x 0 x x 1 0 are given where 0 1 is a fixed point in contrast to the usual case 1 2 1 f t x y z 0 for any t 0 1 x 0 y z r and the map t is positive on 0 1 and suitable singular at the end points furthermore the obtained solution x t is positive and concave on 0 1 in addition without any monotonicity assumption on the nonlinearity we prove the existence of a sequence of such solutions with lim n x n 0 our principal tools are very simple applications on the plane of the well known from combinatorial topology sperner s lemma in comparison with other classical approaches like fixed point or degree theorems

discrete third order three point right focal boundary value problems we are concerned with the discrete right focal boundary value problem 3x t f t x t 1 x t1 x t2 2x t3 0 and the eigenvalue problem 3x t a t f x t 1 with the same boundary conditions where t1 t2 t3 under various assumptions on f a and we prove the existence of positive solutions of both problems by applying a fixed point theorem

virasoro symmetries of the extended toda hierarchy we prove that the extended toda hierarchy of 1 admits a nonabelian lie algebra of infinitesimal symmetries isomorphic to half of the virasoro algebra the generators lm m 1 of the lie algebra act by linear differential operators onto the tau function of the hierarchy we also prove that the tau function of a generic solution to the extended toda hierarchy is annihilated by a combination of the virasoro operators and the flows of the hierarchy as an application we show that the validity of the virasoro constraints for the cp1 gromov witten invariants and their descendents implies that their generating function is the logarithm of a particular tau function of the extended toda hierarchy