decroissance exponentielle du noyau de la chaleur sur la diagonale i we give examples based upon large deviation s theory where the heat kernel of a degenerate diffusion has an exponential decay over the diagonal using malliavin calculus we give conditions for a more generalized heat kernel to have an exponential decay over the diagonal we give lower bound in some particular case by using the bismut s condition

stability and b convergence properties of multistep runge kutta methods this paper continues earlier work by the same author concerning the stability and b convergence properties of multistep runge kutta methods for the numerical solution of nonlinear stiff initial value problems in a hilbert space a series of sufficient conditions and necessary conditions for a multistep runge kutta method to be algebraically stable diagonally stable b or optimally b convergent are established by means of which six classes of high order algebraically stable and b convergent multistep runge kutta methods are constructed in a unified pattern these methods include the class constructed by burrage in 1987 as special case and most of them can be regarded as extension of the gauss radauia radauiia and lobattoiiic runge kutta methods we find that the classes of multistep runge kutta methods constructed in the present paper are superior in many respects to the corresponding existing one step runge kutta schemes

order results for implicit runge kutta methods applied to stiff systems an extensive order analysis is given for the local and global errors of certain classes of runge kutta methods applied to general nonlinear stiff initial value problems it turns out that there are three order levels for linear problems y ay the classical orders apply for general nonlinear problems however two types of order reductions occur there a distinction has to be made between stiff problems where only f y is large and stiff problems where also some of the other derivatives f t f tt f yy cdots are large for these two classes of problems order results can be guaranteed which differ only by one power of the stepsize parameter h the order levels for both classes of problems are considerably lower than the respective conventional orders these order results are optimal in the sense that higher orders would contradict numerical observations made with particular initial value problems

stationary distribution of a stochastic predator prey model with distributed delay and general functional response abstract in this paper we study a stochastic predator prey model with distributed delay and general functional response we transform the model with weak kernel case into an equivalent system through the linear chain technique since the diffusion matrix is degenerate the uniform ellipticity condition does not hold the markov semigroup theory is used to derive the existence of a unique stable stationary distribution we verify the densities of the distributions of the positive solutions can converge in l 1 to an invariant density

quadratic chabauty for modular curves and modular forms of rank one in this paper we provide refined sufficient conditions for the quadratic chabauty method on a curve x to produce an effective finite set of points containing the rational points x mathbb q with the condition on the rank of the jacobian of x replaced by condition on the rank of a quotient of the jacobian plus an associated space of chow heegner points we then apply this condition to prove the effective finiteness of x mathbb q for any modular curve x x 0 n or x mathrm ns n of genus at least 2 with n prime the proof relies on the existence of a quotient of their jacobians whose mordell weil rank is equal to its dimension and at least 2 which is proven via analytic estimates for orders of vanishing of l functions of modular forms thanks to a kolyvagin logachev type result

environmental variability in a stochastic epidemic model abstract in this paper we investigate the stochastic dynamics of a simple epidemic model incorporating the mean reverting ornstein uhlenbeck process analytically and numerically we define two threshold parameters the stochastic demographic reproduction number r d s and the stochastic basic reproduction number r 0 s to utilize in identifying the stochastic extinction and persistence of the disease we find that the stochastic disease dynamics can be determined by the environment fluctuations which measured by the intensity of volatility and the speed of reversion the larger intensity of volatility or the smaller speed of reversion can suppress the outbreak of the disease the smaller intensity of volatility or the the higher speed of reversion can enhance the outbreak of the disease furthermore via numerical simulations we find that the stochastic model has an endemic stationary distribution which leads to the stochastic persistence of the disease our results show that mean reverting process is a well established way of introducing stochastic environmental noise into biologically realistic population dynamic models

on the modularity of ℚ curves a q curve is an elliptic curve over a number field k which is geometrically isogenous to each of its galois conjugates ribet 16 asked whether every q curve is modular and showed that a positive answer would follow from serre s conjecture on mod p galois representations we answer ribet s question in the affirmative subject to certain local conditions at 3 msc classification 11g18 14g35 14h52

the fermat type equations 2 or 3 solved through ℚ curves we solve the diophantine equations x 5 y 5 dz p with d 2 3 for a set of prime numbers of density 1 4 1 2 respectively the method consists in relating a possible solution to another diophantine equation and solving the later by using q curves and a generalized modular technique as in work of ellenberg and dieulefait jimenez along with some new techniques for eliminating newforms

high order algebraically stable multistep runge kutta methods in this paper we examine some of the stability properties of a special class of multivalue methods called in k burrage order properties of multivalue methods ima j numer anal submitted multistep runge kutta methods and construct families of algebraically stable methods of arbitrarily high order

on relations between jacobians of certain modular curves abstract we confirm a conjecture of l merel h darmon and l merel j reine angew math 490 1997 81 100 describing a certain relation between the jacobians of various quotients of x p in terms of specific correspondences the method of proof involves reducing this conjecture to a question about certain z gl 2 f p module homomorphisms which is in turn answered by exhibiting some peculiar relations in a double coset algebra

on the contractivity and convergence of general linear methods on semi infinite intervals the strict contractivity and the convergence of general linear methods on the classes of strictly dissipative and dissipative differential systems regarding some inner product are analyzed new convergence and contractivity results of the methods on semi infinite intervals are provided for the case of strictly dissipative problems some applications of the main results to the class of runge kutta multistep methods are supplied

qualitative analysis of a stochastic ratio dependent predator prey system a stochastic ratio dependent predator prey model is investigated in this paper by the comparison theorem of stochastic equations and ito s formula we obtain the global existence of a positive unique solution of the ratio dependent model besides a condition for species to be extinct is given and a persistent condition is established we also conclude that both the prey population and the ratio dependent function are stable in time average in the end numerical simulations are carried out to confirm our findings

stability of markov semigroups and applications to parabolic systems abstract a new theorem for asymptotic stability of markov semigroups is proved this result is applied to semigroups generated by parabolic systems describing the evolution of densities of two state diffusion processes

fermat type equations of signature 13 13 p via hilbert cuspforms in this paper we prove that for p 13649 equations of the form x 13 y 13 cz p have no non trivial primitive solutions a b c such that 13 not mid c for an infinite family of values for c our method consists on relating a solution a b c to the previous equation to a solution a b c 1 of another diophantine equation with coefficients in mathbb q sqrt 13 then we attach to a b c 1 a frey curve e a b defined over mathbb q sqrt 13 that is not a mathbb q curve we prove a modularity result of independent interest for certain elliptic curves over totally real abelian number fields satisfying some local conditions at 3 this theorem in particular implies modularity of e a b this enables us to use level lowering results and apply the modular approach via hilbert cuspforms over mathbb q sqrt 13 to prove the non existence of a b c 1 and consequently of a b c

on the unique solvability of the runge kutta equations we consider the existence of a unique solution to the systems of equations that arise when we apply a runge kutta method to a stiff nonlinear system of differential equationsu f t u withf satisfying a one sided lipschitz condition with constant for any given product h whereh denotes the step size we present algebraic conditions on the runge kutta matrixa which are necessary and sufficient for unique solvability of the equations as a second topic we consider the question whether the solution to the system of equations is stable with respect to perturbations known as bsi stability for this property also necessary and sufficient algebraic conditions ona are presented

recipes to fermat type equations of the form x r y r cz p we describe a strategy to attack infinitely many fermat type equations of signature r r p where r ge 7 is a fixed prime and p is a prime allowed to vary indeed to a solution a b c of x r y r cz p we will attach several frey curves e e a b defined over totally real subfields of mathbb q zeta r we prove modularity of all the frey curves and the exsitence of a constant constant m r depending only on r such that for all p m r the representations bar rho e p are absolutely irreducible along the way we also prove modularity of certain elliptic curves that are semistable at all v mid 3 finally we illustrate our methods by proving arithmetic statements about equations of signature 7 7 p among which we emphasize that using a multi frey technique we show there is some constant m such that if p m then the equation x 7 y 7 3z p has no non trivial primitive solutions

a stochastic sirs epidemic model with infectious force under intervention strategies abstract in this paper we extend a classical sirs epidemic model with the infectious forces under intervention strategies from a deterministic framework to a stochastic differential equation sde one through introducing random fluctuations the value of our study lies in two aspects mathematically by using the markov semigroups theory we prove that the reproduction number r 0 s can be used to govern the stochastic dynamics of sde model if r 0 s 1 under mild extra conditions the sde system has a disease free absorbing set which means the extinction of disease with probability one if r 0 s 1 under mild extra conditions it has an endemic stationary distribution which leads to the stochastical persistence of the disease epidemiologically we find that random fluctuations can suppress disease outbreak which can provide us some useful control strategies to regulate disease dynamics

a stochastic sirs epidemic model with nonlinear incidence rate a stochastic sirs model with ratio dependent incidence rate is developed the global dynamics of the deterministic model is shown the stochastic dynamics of the sde model is given the existence of a unique stationary distribution of the sde model is displayed in this paper we investigate the global dynamics of a general sirs epidemic model with a ratio dependent incidence rate and its corresponding stochastic differential equation version for the deterministic model we show that the basic reproduction number r0 determines whether there is an endemic outbreak or not if r0 1 the disease persists for the stochastic model we show that its related reproduction number r0s can determine whether there is a unique disease free stationary distribution or a unique endemic stationary distribution in addition we provide analytic results regarding the stochastic boundedness and permanence extinction one of the most interesting findings is that random fluctuations introduced in our stochastic model can suppress disease outbreak which can provide us some useful control strategies to regulate disease dynamics

dynamics of the stochastic leslie gower predator prey system with randomized intrinsic growth rate abstract this paper investigates the stochastic leslie gower predator prey system with randomized intrinsic growth rate existence of a unique global positive solution is proved firstly then we obtain the sufficient conditions for permanence in mean and almost sure extinction of the system furthermore the stationary distribution is derived based on the positive equilibrium of the deterministic model which shows the population is not only persistent but also convergent by time average under some assumptions finally we illustrate our conclusions through two examples

galois representations attached to q curves and the generalized fermat equation a 4 b 2 c p we prove that the equation a 4 b 2 c p has no solutions in coprime positive integers when p 211 the main step is to show that for all sufficiently large primes p every q curve over an imaginary quadratic field k with a prime of potentially multiplicative reduction greater than 6 has a surjective mod p galois representation the bound on p depends on k and the degree of the isogeny between e and its galois conjugate but is independent of the choice of e the proof of this theorem combines geometric arguments due to mazur momose darmon and merel with an analytic estimate of the average special values of certain l functions