modified schmidt games and diophantine approximation with weights abstract we show that the sets of weighted badly approximable vectors in r n are winning sets of certain games which are modifications of games introduced by w m schmidt in 1966 the latter winning property is stable with respect to countable intersections and is shown to imply full hausdorff dimension

badly approximable points on planar curves and a problem of davenport let mathcal c be two times continuously differentiable curve in mathbb r 2 with at least one point at which the curvature is non zero for any i j geqslant 0 with i j 1 let mathbf bad i j denote the set of points x y in mathbb r 2 for which max vert qx vert 1 i vert qy vert 1 j c q for all q in mathbb n here c c x y is a positive constant our main result implies that any finite intersection of such sets with mathcal c has full hausdorff dimension this provides a solution to a problem of davenport dating back to the sixties

rising relative fluctuation as a warning indicator of discontinuous transitions in symbiotic metapopulations abstract the long time limit behavior of the stochastic lotka volterra model of a symbiotic metapopulation subjected to generalized verhulst self regulation is considered the influence of a time variable environment on the carrying capacities of subpopulations is modeled as a periodic deterministic part and a symmetric dichotomous noise relying on the mean field approach it is established that at certain parameter regimes the mean field average subpopulations size exhibits hysteresis in respect to the noise correlation time manifested in the appearance of colored noise induced discontinuous transitions especially it is shown that the relative fluctuation of the subpopulation sizes exhibits accelerated increase prior to abrupt transitions of the metapopulation state moreover in certain cases the autocorrelation function of the population sizes demonstrates anticorrelation at some values of the lag time

effect of multiplicative noise on stationary stochastic process an open system that can be analyzed using the langevin equation with multiplicative noise is considered the stationary state of the system results from a balance of deterministic damping and random pumping simulated as noise with controlled periodicity the dependence of statistical moments of the variable that characterizes the system on parameters of the problem is studied a nontrivial decrease in the mean value of the main variable with an increase in noise stochasticity is revealed applications of the results in several physical chemical biological and technical problems of natural and humanitarian sciences are discussed

badly approximable points on planar curves and winning abstract for any i j 0 with i j 1 let bad i j denote the set of points x y r 2 such that max q x 1 i q y 1 j c q for some positive constant c c x y and all q n we show that bad i j c is winning in the sense of schmidt games for a large class of planar curves c namely everywhere non degenerate planar curves and straight lines satisfying a natural diophantine condition this strengthens recent results solving a problem of davenport from the sixties in short within the context of davenport s problem the winning statement is best possible furthermore we obtain the inhomogeneous generalisations of the winning results for planar curves and lines and also show that the inhomogeneous form of bad i j is winning for two dimensional schmidt games

double stochastic resonance in an insect ecosystem with time delays abstract considering additive and multiplicative periodic signals we have investigated the stochastic resonance sr phenomenon in an insect ecosystem with two different types of time delays at which the insect dynamics is assumed to be disturbed by both additive and multiplicative noises the signal to noise ratio snr is calculated when two different weak periodic signals are added to the insect system it is shown from the snr that the additive and multiplicative periodic signals can cause the sr and double sr in the case of time delays in the stochastic and deterministic force respectively a detailed study of the snr depending on various noise characteristics is performed and the main results about sr are compared for two different cases of time delays

a particular solution trefftz method for non linear poisson problems in heat and mass transfer this paper proposes a computational procedure based on the trefftz method for the solution of non linear poisson problems the problem is solved by finding an approximate particular solution to the poisson equation and using boundary collocation to solve the resulting laplace equation this method results in a global collocation and hence eliminates the need for discretization of the domain in both two and three dimensions the solution procedure proposed earlier in the literature based upon the above method has been reformulated for increased computational efficiency a quasi newton iteration method along with a new heuristic for source point location is used for efficient convergence of the numerical scheme the efficacy of the new formulation has been demonstrated for two classes of problems viz the thermal explosion problem and the diffusion reaction problem in a partially wetted catalyst pellet a brief error analysis of the method as well as future research directions is presented

bounded geodesics of riemann surfaces and hyperbolic manifolds we study the set of bounded geodesics of hyperbolic manifolds for general riemann surfaces and for hyperbolic manifolds with some finiteness assumption on their geometry we determine its hausdorff dimension some applications to diophantine approximation are included

application of global optimization and radial basis functions to numerical solutions of weakly singular volterra integral equations a novel approach to the numerical solution of weakly singular volterra integral equa tions is presented using the cm multiquadric mq radial basis function rbf expansion rather than the more traditional finite difference finite element or polynomial spline schemes to avoid the collocation procedure that is usually ill conditioned we used a global minimization procedure combined with the method of successive approximations that utilized a small finite set of mq basis functions accurate solutions of weakly singular volterra integral equations are obtained with the minimal number of mq basis functions the expansion and optimization procedure was terminated whenever the global errors were less than 5 lo 2002 elsevier science ltd all rights reserved

jarnik type inequalities it is well known due to jarnik 19 that the set badr of badly approximable numbers is of hausdorff dimension one if badr c denotes the subset of x badr for which the approximation constant c x c then jarnik was in fact more precise and gave nontrivial lower and upper bounds of the hausdorff dimension of badr c in terms of the parameter c 0 our aim is to determine simple conditions on a framework which allow to extend jarnik s inequality to further examples among the applications we discuss the set badrn of badly approximable vectors in r with weights r and the set of geodesics in the hyperbolic space h which avoid a suitable collection of convex sets

relaxation dynamics in the presence of pulse multiplicative noise sources with different correlation properties the relaxation dynamics of a system described by a langevin equation with pulse multiplicative noise sources with different correlation properties is considered the solution of the corresponding fokker planck equation is derived for gaussian white noise moreover two pulse processes with regulated periodicity are considered as a noise source the dead time distorted poisson process and the process with fixed time intervals which is characterized by an infinite correlation time we find that the steady state of the system is dependent on the correlation properties of the pulse noise an increase of the noise correlation causes the decrease of the mean value of the solution at the steady state the analytical results are in good agreement with the numerical ones

cantor winning sets and their applications abstract we introduce and develop a class of cantor winning sets that share the same amenable properties as the classical winning sets associated to schmidt s game these include maximal hausdorff dimension invariance under countable intersections with other cantor winning sets and invariance under bi lipschitz homeomorphisms it is then demonstrated that a wide variety of badly approximable sets appearing naturally in the theory of diophantine approximation fit nicely into our broad reaching framework as applications of this phenomenon we answer several previously open questions including some related to the mixed littlewood conjecture and approximation by multiplicative semigroups of integers

diophantine approximation and badly approximable sets let x d be a metric space and d a compact subspace of x which supports a non atomic finite measure m we consider natural classes of badly approximable subsets of loosely speaking these consist of points in which stay clear of some given set of points in x the classical set bad of badly approximable numbers in the theory of diophantine approximation falls within our framework as do the sets bad i j of simultaneously badly approximable numbers under various natural conditions we prove that the badly approximable subsets of have full hausdorff dimension applications of our general framework include those from number theory classical complex p adic and formal power series and dynamical systems iterated function schemes rational maps and kleinian groups

stability in a system subject to noise with regulated periodicity the stability of a simple dynamical system subject to multiplicative one side pulse noise with hidden periodicity is investigated both analytically and numerically the stability analysis is based on the exact result for the characteristic functional of the renewal pulse process the influence of the memory effects on the stability condition is analyzed for two cases i the dead time distorted poissonian process and ii the renewal process with pareto distribution we show that for fixed noise intensity the system can be stable when the noise is characterized by high periodicity and unstable at low periodicity

badly approximable points on manifolds this paper is motivated by two problems in the theory of diophantine approximation namely davenport s problem regarding badly approximable points on submanifolds of a euclidean space and schmidt s problem regarding the intersections of the sets of weighted badly approximable points the problems have been recently settled in dimension two but remain open in higher dimensions in this paper we develop new techniques that allow us to tackle them in full generality the techniques rest on lattice points counting and a powerful quantitative result of bernik kleinbock and margulis the main theorem of this paper implies that any finite intersection of the sets of weighted badly approximable points on any analytic nondegenerate submanifold of mathbb r n has full dimension one of the consequences of this result is the existence of transcendental real numbers badly approximable by algebraic numbers of any bounded degree

an application of mean escape time and metapopulation on forestry catastrophe insurance abstract a forestry catastrophe insurance model due to forestry pest infestations and disease epidemics is developed by employing metapopulation dynamics and statistics properties of mean escape time met the probability of outbreak of forestry catastrophe loss and the catastrophe loss payment time with met are respectively investigated forestry loss data in china is used for model simulation experimental results are concluded as 1 the model with analytical results is shown to be a better fit 2 within the condition of big area of patches and structure of patches high system factor low extinction rate high multiplicative noises and additive noises with a high cross correlated strength range an outbreak of forestry catastrophe loss or catastrophe loss payment due to forestry pest infestations and disease epidemics could occur 3 an optimal catastrophe loss payment time met due to forestry pest infestations and disease epidemics can be identified by taking proper value of multiplicative noises and limits the additive noises on a low range of value and cross correlated strength at a high range of value

trichotomous noise controlled signal amplification in a generalized verhulst model abstract the long time limit of the probability distribution and statistical moments for a population size are studied by means of a stochastic growth model with generalized verhulst self regulation the effect of variable environment on the carrying capacity of a population is modeled by a multiplicative three level markovian noise and by a time periodic deterministic component exact expressions for the moments of the population size have been calculated it is shown that an interplay of a small periodic forcing and colored noise can cause large oscillations of the mean population size the conditions for the appearance of such a phenomenon are found and illustrated by graphs implications of the results on models of symbiotic metapopulations are also discussed particularly it is demonstrated that the effect of noise generated amplification of an input signal gets more pronounced as the intensity of symbiotic interaction increases

multiplicatively badly approximable numbers and generalised cantor sets abstract let p be a prime number the p adic case of the mixed littlewood conjecture states that l i m i n f q q q p q 0 for all r we show that with the additional factor of log q log log q the statement is false indeed our main result implies that the set of for which l i m i n f q q log q log log q q p q 0 is of full dimension the result is obtained as an application of a general framework for cantor sets developed in this paper

a contribution to the systematics of stochastic volatility models abstract we systematically compare several classes of stochastic volatility models of stock market fluctuations we show that the long time return distribution is either gaussian or develops a power law tail while the short time return distribution has generically a stretched exponential form but can also assume an algebraic decay in the family of models which we call garch type the intermediate regime is found in the exponential ornstein uhlenbeck process we also calculate the decay of the autocorrelation function of volatility

on a problem in simultaneous diophantine approximation schmidt s conjecture for any i j 0 with i j 1 let bad i j denote the set of points x y 2 r 2 for which maxfkqxk 1 i kqyk 1 j g c q for all q 2 n here c c x y is a positive constant our main result implies that any nite intersection of such sets has full dimension this settles a conjecture of wolfgang m schmidt in the theory of simultaneous diophantine approximation

the ergodic theory of shrinking targets to any dynamical system equipped with a metric we associate a class of well approximable sets in the case of an expanding rational map of the riemann sphere acting on its julia set we estimate and in some cases compute the hausdorff dimension of the associated well approximable sets the methods used show a clear link between distortion properties and the type of results obtained in this paper via ergodic theory and ubiquity

solution and control of pde via global optimization methods abstract based on the concept of equivalent solutions not to be confused with approximations to the exact solution a new consideration is given to ill posed 1 2 and overdetermined pde problems and to problems with nonexistent solutions 3 then a new method based on full global optimization techniques is developed for solution and control of processes described by partial differential equations the ideas are illustrated by examples and a case study is presented in comparison with the quasi reversibility method 4

intrinsic diophantine approximation on manifolds general theory we investigate the question of how well points on a nondegenerate k dimensional submanifold m subseteq mathbb r d can be approximated by rationals also lying on m establishing an upper bound on the intrinsic dirichlet exponent for m we show that relative to this exponent the set of badly intrinsically approximable points is of full dimension and the set of very well intrinsically approximable points is of zero measure our bound on the intrinsic dirichlet exponent is phrased in terms of an explicit function of k and d which does not seem to have appeared in the literature previously it is shown to be optimal for several particular cases the requirement that the rationals lie on m distinguishes this question from the more common context of ambient diophantine approximation on manifolds and necessitates the development of new techniques our main tool is an analogue of the simplex lemma for rationals lying on m which provides new insights on the local distribution of rational points on nondegenerate manifolds

exact probability distribution for the bernoulli malthus verhulst model driven by a multiplicative colored noise we report an exact result for the calculation of the probability distribution of the bernoulli malthus verhulst model driven by a multiplicative colored noise we study the conditions under which the probability distribution of the malthus verhulst model can exhibit a transition from a unimodal to a bimodal distribution depending on the value of a critical parameter also we show that the mean value of x t in the latter model always approaches asymptotically the value 1

random walks on homogeneous spaces and diophantine approximation on fractals we extend results of y benoist and j f quint concerning random walks on homogeneous spaces of simple lie groups to the case where the measure defining the random walk generates a semigroup which is not necessarily zariski dense but satisfies some expansion properties for the adjoint action using these dynamical results we study diophantine properties of typical points on some self similar fractals in mathbb r d as examples we show that for any self similar fractal mathcal k subseteq mathbb r d satisfying the open set condition for instance any translate or dilate of cantor s middle thirds set or of a koch snowflake almost every point with respect to the natural measure on mathcal k is not badly approximable furthermore almost every point on the fractal is of generic type which means in the one dimensional case that its continued fraction expansion contains all finite words with the frequencies predicted by the gauss measure we prove analogous results for matrix approximation and for the case of fractals defined by mobius transformations

application of the multiquadric method for numerical solution of elliptic partial differential equations abstract we have used the multiquadric mq approximation scheme for the solution of elliptic partial differential equations with dirichlet and or neumann boundary conditions the scheme has the advantage of using the data points in arbitrary locations with an arbitrary ordering two dimensional laplace poisson and biharmonic equations describing the various physical processes have been taken as the test examples the agreement is found to be very good between the computed and exact solutions the method also provides an excellent approximation with a curved boundary

on a problem of k mahler diophantine approximation and cantor sets let k denote the middle third cantor set and mathcal a 3 n n 0 1 2 ldots given a real positive function let w mathcal a psi denote the set of real numbers x in the unit interval for which there exist infinitely many p q in mathbb z times mathcal a such that x p q q the analogue of the hausdorff measure version of the duffin schaeffer conjecture is established for w mathcal a psi cap k one of the consequences of this is that there exist very well approximable numbers other than liouville numbers in k an assertion attributed to k mahler explicit examples of irrational numbers satisfying mahler s assertion are also given

a note on badly approximabe sets in projective space recently ghosh and haynes j reine angew math 712 39 50 2016 proved a khintchine type result for the problem of diophantine approximation in certain projective spaces in this note we complement their result by observing that a jarnik type result also holds for badly approximable points in real projective space in particular we prove that the natural analogue in projective space of the classical set of badly approximable numbers has full hausdorff dimension when intersected with certain compact subsets of real projective space furthermore we also establish an analogue of khintchine s theorem for convergence relating to intrinsic approximation of points in these compact sets

some issues on interpolation matrices of locally scaled radial basis functions abstract radial basis function interpolation on a set of scattered data is constructed from the corresponding translates of a basis function which is conditionally positive definite of order m 0 with the possible addition of a polynomial term in many applications the translates of a basis function are scaled differently in order to match the local features of the data such as the flat region and the data density then a fundamental question is the non singularity of the perturbed interpolation n n matrix in this paper we provide some counter examples of the matrices which become singular for n 3 although the matrix is always non singular when n 2 one interesting feature is that a perturbed matrix can be singular with rather small perturbation of the scaling parameter