liouvillian solutions of n th order homogeneous linear differential equations be a linear differential operator with coefficients in f a finite algebraic extension of q x we shall show that one can find in a finite number of steps a basis for the vector space of liouvillian solutions of l y 0 i e those solutions which can be built up from the rational functions by algebraic operations taking exponentials and by integration see section 2 for a precise definition in particular we show how to decide if all solution of l y 0 are liouvillian or if there are any solutions which are liouvillian our algorithm in conjunction with the algorithm in 8 also allows one to determine the algebraic relationships among the liouvillian solutions of l y 0 and in particular determine if all solutions are algebraic c f 111 for second order linear homogeneous equations over q x the problem of determining if all solutions are algebraic functions was considered by fuchs klein and schwarz but none of these mathematicians seems to have presented a complete decision procedure building on the work of klein baldassarri and dwork 21 have given such a procedure baldassarri 11 has extended this to consider linear homogeneous equations whose coefficients are algebraic functions for third order equations painleve and boulanger gave a procedure which in effect reduced the problem of finding algebraic solutions of l y 0 to the problem of effectively bounding the torsion of the jacobian variety of a given curve a complete procedure for deciding if all solutions of an n th order homo

eigenfunctions of the laplacian on rotationally symmetric manifolds eigenfunctions of the laplacian on a negatively curved rotationally symmetric manifold m rn ds2 ds2 dr2 f r 2d 2 are constructed explicitly under the assumption that an integral of f r converges this integral is the same one which gives the existence of nonconstant harmonic functions on m

nonequilibrium phase transition due to isolation of communities we introduce a simple model of a growing system with m competing communities the model corresponds to the phenomenon of defeats suffered by social groups living in isolation a nonequilibrium phase transition is observed when at critical time t c the first isolated cluster occurs in the one dimensional system the volume of the new phase i e the number of the isolated individuals increases with time as z ensuremath sim t 3 for a large number of possible communities the critical density of filled space is equal to ensuremath rho c m n 1 3 where n is the system size a similar transition is observed for erd ifmmode mbox h o else h o fi s r enyi random graphs and barab asi albert scale free networks analytical results are in agreement with numerical simulations

dirichlet conditions at infinity for parabolic and elliptic equations we investigate existence and uniqueness of bounded solutions of parabolic equations with unbounded coefficients in rn r under specific assumptions we establish existence of solutions satisfying prescribed conditions at infinity depending on the direction along which infinity is approached moreover the large time behavior of such solutions is addressed we consider also elliptic equations in rn with similar conditions at infinity

rational group decision making a random field ising model at t 0 abstract a modified version of a finite random field ising ferromagnetic model in an external magnetic field at zero temperature is presented to describe group decision making fields may have a non zero average a postulate of minimum inter individual conflicts is assumed interactions then produce a group polarization along one very choice which is however randomly selected a small external social pressure is shown to have a drastic effect on the polarization individual bias related to personal backgrounds cultural values and past experiences are introduced via quenched local competing fields they are shown to be instrumental in generating a larger spectrum of collective new choices beyond initial ones in particular compromise is found to results from the existence of individual competing bias conflict is shown to weaken group polarization the model yields new psychosociological insights about consensus and compromise in groups

the dirichlet problem at infinity for manifolds of negative curvature on considere l existence de fonctions harmoniques bornees sur des varietes simplement connexes n n de courbure negative

martin compactification of a complete surface with negative curvature in this paper we consider the martin compactification associated with the operator mathcal l delta 1 of a complete non compact surface sigma 2 with negative curvature in particular we investigate positive eigenfunctions with eigenvalue one of the laplace operator delta of sigma 2 and prove a uniqueness result such eigenfunctions are unique up to a positive multiplicative constant if they vanish on the part of the geometric boundary s infty sigma 2 of sigma 2 where the curvature is bounded above by a negative constant and satisfy some growth estimate on the other part of s infty sigma 2 where the curvature approaches zero this uniqueness result plays an essential role in our recent paper cao and he infinitesimal rigidity of steady gradient ricci soliton in dimension three arxiv 1412 2714 2014 preprint in which we prove an infinitesimal rigidity theorem for deformations of certain three dimensional collapsed gradient steady ricci soliton with a non trivial killing vector field

differential galois theory of linear difference equations we present a galois theory of difference equations designed to measure the differential dependencies among solutions of linear difference equations with this we are able to reprove holder s theorem that the gamma function satisfies no polynomial differential equation and are able to give general results that imply for example that no differential relationship holds among solutions of certain classes of q hypergeometric equations

some demographic crashes seen as phase transitions abstract the purpose of this paper is the application of a usual method of statistical mechanics the renormalization based on wilson s recursive relations in order to study the critical behavior of a social index namely the live births per 1000 population the drastic decreases of this index on certain periods have the specific features of the phase transitions as they follow approximately power laws and also they lead to the complete change of the population age structure the values of the critical exponents that are obtained by fitting the experimental data referring to some east european countries are in agreement with the value resulting from the theoretical approach thus showing the universality of the power law behavior in the vicinity of the critical points for complex social systems

some counterexamples to separation of variables abstract among the other things a linear inhomogenous algebraic partial differential equation with integer coefficients is written down that has no differentially algebraic solutions such an equation has no solution that can be generated by a general purpose analog computer i e differential analyzer

the dirichlet problem at infinity for a negatively curved manifold on considere une n variete de riemann complete m qui est connexe et simplement connexe et a des courbures sectionnelles k bornees par 2 constantes finies negatives on etudie l existence de fonctions harmoniques bornees non constantes pour ces varietes

stochastic and boltzmann like models for behavioral changes and their relation to game theory abstract in the last decade stochastic models have shown to be very useful for quantitative modelling of social processes here a configurational master equation for the description of behavioral changes by pair interactions of individuals is developed three kinds of social pair interactions are distinguished avoidance processes compromising processes and imitative processes computational results are presented for a special case of imitative processes the competition of two equivalent strategies they show a phase transition that describes the self organization of a behavioral convention this phase transition is further analyzed by examining the equations for the most probable behavioral distribution which are boltzmann like equations special cases of boltzmann like equations do not obey the h theorem and have oscillatory or even chaotic solutions a suitable taylor approximation leads to the so called game dynamical equations also known as selection mutation equations in the theory of evolution

opinion evolution on a ba scaling network abstract in this paper the dynamics of opinion formation is investigated based on a ba barabasi albert scale free network using a majority minority rule governed by parameter q as the value of q is smoothly varied a phase transition occurs between an ordered phase and a disordered one by performing extensive monte carlo simulations we show that the phase transition is dependent on the system size as well as on m the number of edges added at each time step during the growth of the ba scaling network additionally some theoretical analysis is given based on mean field theory by neglecting fluctuations and correlations it is observed that the theoretical results coincide with results from simulations especially for very large m

opinion formation model with strong leader and external impact a mean field approach abstract we study a model of opinion formation based on the theory of social impact and the concept of cellular automata the case is considered when two strong agents influence the group a strong leader and an external social impact acting uniformly on every individual there are two basic stationary states of the system cluster of the leader s adherents and unification of opinions in the deterministic limit the variation of parameters like the leader s strength or external impact can change the size of the cluster or when they reach some critical values make the system jump to another phase for a certain range of parameters multistability and hysteresis phenomena are observed in the presence of noise social temperature the rapid changes can be regarded as the first order phase transitions when both agents are in a kind of balance a second order transition and critical behaviour can be observed another kind of noise induced transitions are the inverses flips of the unified group opinion due to random flips of the leader s opinion analytical results obtained within a mean field approximation are well reproduced in computer simulations

new classes of parameterized differential galois groups this paper is on the inverse parameterized differential galois problem we show that surprisingly many groups do not occur as parameterized differential galois groups over k x even when k is algebraically closed we then combine the method of patching over fields with a suitable version of galois descent to prove that certain groups do occur as parameterized differential galois groups over k t x this class includes linear differential algebraic groups that are generated by finitely many unipotent elements and also semisimple connected linear algebraic groups

existence of parameterized picard vessiot extensions over fields with algebraically closed constants abstract the purpose of this short note is to establish the existence of parameterized picard vessiot extensions for systems of linear difference differential equations over difference differential fields with an algebraically closed field of constants

phase transitions as a persistent feature of groups with leaders in models of opinion formation a class of models of opinion formation based on the concept of cellular automata and the theory of social impact is studied in particular the case when a strong leader and external impact are present the rapid changes of the opinion distribution with a continuous change of a system parameter which was previously observed for the model with geometric structure prove to be present also for much larger class of mutual interaction architectures we study random connections with dierent probability distributions the theoretical results obtained in the framework of mean eld approximation are conrmed by the numerical simulations of the model c 2000 elsevier science b v all rights reserved

on the modelling and simulation of the competition for a secession under media influence by active particles methods and functional subsystems decomposition this paper deals with the development of a mathematical model for complex socio economical systems where external actions play a key role the aim of the paper is to show the emergence of collective behaviors or macroscopic trends from individual based interactions where agents are identified by functional subsystems the approach is based on the methods of the mathematical kinetic theory for active particles which describes the evolution of large systems of interacting entities which are carriers of specific functions in our specific application socio political activities subjected to the influence of the media

generalized differential galois theory a galois theory of differential fields with parameters is developed in a manner that generalizes kolchin s theory it is shown that all connected differential algebraic groups are galois groups of some appropriate differential field extension

on a new gap phenomenon in riemannian geometry abstract this note announces some theorems that show that the flat metric of euclidean space has no nearby nonflat complete metric with the property that its curvature keeps a sign and is uniformly small in a sense made precise here

calculating differential galois groups of parametrized differential equations with applications to hypertranscendence the main motivation of our work is to create an efficient algorithm that decides hypertranscendence of solutions of linear differential equations via the parameterized and differential galois theories to achieve this we expand the representation theory of linear differential algebraic groups and develop new algorithms that calculate unipotent radicals of parameterized differential galois groups for differential equations whose coefficients are rational functions berman and singer presented an algorithm calculating the differential galois group for differential equations without parameters whose differential operator is a composition of two completely reducible differential operators we use their algorithm as a part of our algorithm as a result we find an effective criterion for the algebraic independence of the solutions of parameterized differential equations and all of their derivatives with respect to the parameter

brownian motion and dirichlet problems at infinity we discuss angular convergence of riemannian brownian motion on a cartan hadamard manifold and show that the dirichlet problem at infinity for such a manifold is uniquely solvable under the curvature conditions ce 2 ar x km x a2 vspace 0 5pt 0 and cr x 2 km x 1 r x 2 2 2 respectively

linear algebraic groups as parameterized picard vessiot galois groups we show that a linear algebraic group is the galois group of a parameterized picard vessiot extension of k x x 1 for certain differential fields k if and only if its identity component has no one dimensional quotient as a linear algebraic group

division of labor as the result of phase transition the emergence of labor division in a multi agent system is analyzed by the method of statistical physics we consider a system consisting of n homogeneous agents their behaviors are determined by the returns from their production using the metropolis method in statistical physics which in this model can be regarded as a kind of uncertainty in decision making we constructed a master equation to describe the evolution of the agent s distribution we introduce an earning function including learning by doing to describe the effect of technical progress and a formula for competitive cooperation and we also introduce two order parameters to characterize the division behavior the model gives us the following interesting results 1 as a result of long term evolution the system can reach a steady state 2 when the parameters exceed a critical point the division of labor emerges as the result of phase transition 3 although the technical progress decides whether or not phase transition occurs the critical point is strongly affected by the competitive cooperation from the above physical model and the corresponding results we can get a more deeply understanding about the labor division

a galois theoretic proof of the differential transcendence of the incomplete gamma function abstract we give simple necessary and sufficient conditions for the t transcendence of the solutions to a parameterized second order linear differential equation of the form 2 y x 2 p y x 0 where p f x is a rational function in x with coefficients in a t field f our criteria imply in particular the t transcendence of the incomplete gamma function t x generalizing a result of johnson reinhart and rubel this result is also an important part of an efficient algorithm to compute the parameterized picard vessiot group of an arbitrary parameterized second order linear differential equation over f x

parameterized picard vessiot extensions and atiyah extensions abstract generalizing atiyah extensions we introduce and study differential abelian tensor categories over differential rings by a differential ring we mean a commutative ring with an action of a lie ring by derivations in particular these derivations act on a differential category a differential tannakian theory is developed the main application is to the galois theory of linear differential equations with parameters namely we show the existence of a parameterized picard vessiot extension and therefore the galois correspondence for many differential fields with possibly non differentially closed fields of constants that is fields of functions of parameters other applications include a substantially simplified test for a system of linear differential equations with parameters to be isomonodromic which will appear in a separate paper this application is based on differential categories developed in the present paper and not just differential algebraic groups and their representations

asymptotic dirichlet problems for harmonic functions on riemannian manifolds we define the asymptotic dirichlet problem and give a sufficient condition for solving it this proves an existence of nontrivial bounded harmonic functions on certain classes of noncompact complete riemannian manifolds 0 introduction in this paper we will prove the existence of nonconstant bounded harmonic functions on certain classes of noncompact riemannian manifolds by defining and solving an asymptotically defined dirichlet problem for harmonic functions the motivation comes from the classical uniformization theorem of riemann surfaces which says that a simply connected riemann surface is biholo morphic to the riemann sphere s2 the complex plane c or the unit disk d this is a geometric theorem but its original proof due to koebe relies heavily on function theory the function theoretic interpretation of this theorem is the following among the noncompact simply connected surfaces c is characterized by the fact that it admits no nonconstant bounded harmonic functions and d by that it admits nonconstant bounded harmonic functions the geometric aspect of the uniformization theorem could be roughly stated as follows let m be a simply connected riemann surface equipped with a complete riemannian metric with gaussian curvature km if km c 0 then m is biholo morphic to s2 if km c 0 then m is biholomorphic to d if km is close to zero then m is biholomorphic to c the following precise version is due to greene and wu gw1 p 120