competition for a single limiting resource in continuous culture the variable yield model the global asymptotic behavior of solutions of the so called variable yield model of competition between two microbial populations for a single growth limiting nutrient is determined this model also referred to as the variable internal stores or caperon droop model is a generalization of the classical monod model it assumes that the growth rate of a population depends on the amount of nutrient stored by the organism taken to be the same for all organisms rather than the concentration of ambient nutrient in the culture it is shown that competitive exclusion holds the winner being the organism that can grow at the lower ambient nutrient concentration

integral inequalities for concave functions with applications to special functions in this paper we prove refinements extensions and counterparts of known integral inequalities for concave functions furthermore we apply our results to find new inequalities for some special functions and mean values

on a system of reaction diffusion equations arising from competition with internal storage in an unstirred chemostat abstract in this paper we study a system of reaction diffusion equations arising from competition of two microbial populations for a single limited nutrient with internal storage in an unstirred chemostat the conservation principle is used to reduce the dimension of the system by eliminating the equation for the nutrient the reduced system limiting system generates a strongly monotone dynamical system in its feasible domain under a partial order we construct suitable upper lower solutions to establish the existence of positive steady state solutions given the parameters of the reduced system we answer the basic questions as to which species survives and which does not in the spatial environment and determine the global behaviors the primary conclusion is that the survival of species depends on species s intrinsic biological characteristics the external environment forces and the principal eigenvalues of some scalar partial differential equations we also lift the dynamics of the limiting system to the full system

error bounds in approximating the riemann stieltjes integral of c n 1 class integrands and nonsmooth integrators in the present paper we investigate the problem of approximating the riemann stieltjes integral a b f du in the case when the integrand f is n 1 time differentiable n 0 and the derivative f n 1 is continuous on a b while the integrator u is riemann integrable on a b a priory error bounds for different classes of functions are provided

bifurcation analysis on a two neuron system with distributed delays abstract a general two neuron model with distributed delays is studied in this paper its local linear stability is analyzed by using the routh hurwitz criterion if the mean delay is used as a bifurcation parameter we prove that hopf bifurcation occurs for a weak kernel this means that a family of periodic solutions bifurcates from the equilibrium when the bifurcation parameter exceeds a critical value the direction and stability of the bifurcating periodic solutions are determined by the normal form theory and the center manifold theorem some numerical simulations for justifying the theoretical analysis are also given

competition for one resource with internal storage and inhibitor in an unstirred chemostat abstract this paper deals with a mathematical model of two species competing for one resource in an unstirred chemostat with internal storage and inhibitor first the well posedness of the competition model is established second the threshold dynamics of the single population model is determined by the principal eigenvalue of a nonlinear eigenvalue problem third sufficient conditions for the asymptotic instability of trivial and semi trivial steady state solutions are given in terms of the principal eigenvalues of the corresponding nonlinear eigenvalue problems furthermore under these conditions the system is uniformly persistent and there is at least one coexistence equilibrium the main methods used here are the nonlinear eigenvalue problems and uniform persistence theory

new result on the model based biological control of the chemostat abstract we investigate a known competition model of the chemostat with general nonmonotone response functions and distinct removal rates based on the competitive exclusion principle rapaport and harmand 2008 13 established the concept of the so called biological control the proof of the latter result is based on a theorem of li 1998 11 here we first propose a modification of li s theorem and then present an extension of the biological control concept

some lipschitz type inequalities for complex functions abstract in this paper we establish some lipschitz type inequalities for complex functions when some convexity properties for powers of the absolute value of the derivative are assumed the case of functions defined by power series is analyzed some applications in bounding the jensen difference for complex functions are also provided

asymptotically autonomous semiflows chain recurrence and lyapunov functions from the work of c conley it is known that the omega limit set of a precompact orbit of an autonomous semiflow is a chain recurrent set here we improve a result of l markus by showing that the omega limit set of a solution of an asymptotically autonomous semiflow is a chain recurrent set relative to the limiting autonomous semiflow in the special case that there is a lyapunov function for the limiting semiflow sufficient conditions are given for an omega limit set of the asymptotically autonomous semiflow to be contained in a level set of the lyapunov function

inequalities of hadamard s type for lipschitzian mappings and their applications in this paper we give some inequalities of hadamard s type for m lipschitzian functions some applications which are connected with arithmetic mean geometric mean harmonic mean logarithmical mean identric mean etc for two positive numbers are also given

on a nonlocal reaction diffusion advection system modelling the growth of phytoplankton with cell quota structure abstract phytoplankton species in a water column compete for mineral nutrients and light and the existing models usually neglect differences in the nutrient content and the amount of light absorbed of individuals in this current paper we examine a size structured and nonlocal reaction diffusion advection system which describes the dynamics of a single phytoplankton species in a water column where the species depends simply on light for its growth our model is under the assumption that the amount of light absorbed by individuals is proportional to cell size which varies for populations that reproduce by simple division into two equally sized daughters we first establish the existence of a critical death rate and our analysis indicates that the phytoplankton survives if and only if its death rate is less than the critical death rate the critical death rate depends on a general reproductive rate the characteristics of the water column e g turbulent diffusion rate sinking depth cell growth cell division and cell size

two mappings in connection to hadamard s inequalities w are known in the literature as hadamard s inequalities we note that j hadamard was not the first who discovered them as is pointed out by d s mitrinovic and i b laekovic 3 the inequalities 1 are due to c hermite who obtained them in 1883 ten years before j hadamard in this paper by the use of mappings 2 and 4 we shall establish some refinements of 1 for other inequalities in connection to hadamard s result see 1 2 51 where further references are given

ostrowski s type inequalities for some classes of continuous functions of selfadjoint operators in hilbert spaces new ostrowski s type inequalities for some classes of continuous functions of selfadjoint operators in hilbert spaces under suitable assumptions for the involved operators are given

refinements and similar extensions of hermite hadamard inequality for fractional integrals and their applications in this paper we establish some new refinements and similar extensions of hermite hadamard inequality for fractional integrals and present several applications in the beta function

differences between means with bounds from a riemann stieltjes integral an identity for the difference between two integral means is obtained in terms of a riemann stieltjes integral this enables bounds to be procured when the integrand is of bounded variation lipschitzian and monotonic if f is absolutely continuous bounds are also obtained for f lp a b 1 p the usual lebesgue norms this supplements earlier results involving f l a b

ostrowski and trapezoid type inequalities for the stieltjes integral with lipschitzian integrands or integrators some ostrowski and trapezoid type inequalities for the stieltjes integral in the case of lipschitzian integrators for both holder continuous and monotonic integrals are obtained the dual case is also analysed applications for the midpoint rule are pointed out as well

multidimensional hermite hadamard inequalities and the convex order abstract the problem of establishing inequalities of the hermite hadamard type for convex functions on n dimensional convex bodies translates into the problem of finding appropriate majorants of the involved random vector for the usual convex order we present two results of partial generality which unify and extend the most part of the multidimensional hermite hadamard inequalities existing in the literature at the same time that lead to new specific results the first one fairly applies to the most familiar kinds of polytopes the second one applies to symmetric random vectors taking values in a closed ball for a given but arbitrary norm on r n related questions such as estimates of approximation and extensions to signed measures also are briefly discussed

global asymptotic behavior of a chemostat model with discrete delays this paper studies the global asymptotic behavior of an exploitative competition model between n species in a chemostat the model incorporates discrete time delays to describe the delay in the conversion of nutrient consumed to viable biomass and hence includes delays simultaneously in variables of nutrient and species concentrations in the case where only two species are engaged in competition it is shown that competitive exclusion holds for any monotone growth response functions sufficient conditions are also obtained for the model to exhibit competitive exclusion in the n species case in regard to the delay effects on the qualitative outcome of competition it is demonstrated that when the delays are relatively small the predictions of the model are identical with the predictions given by corresponding models without time delays however including large delays in the model may alter the predicted outcome of competition the techniques used also work when different removal rates are permitted an

on euler trapezoid formulae modified versions of the euler trapezoid formula for functions whose derivatives are either functions of bounded variation or lipschitzian functions or functions in l p spaces are given the results are applied to quadrature formulae

hadamard type inequalities for lipschitzian functions in one and two variables with applications abstract in this paper we shall establish some hadamard type inequalities for lipschitzian functions in one and two variables and give several applications for special means

a nonlinear boundary value problem arising from competition in the chemostat a boundary value problem with two parameters for two coupled second order nonlinear ordinary differential equations is considered this problem arises in a study of chemostats without a well mixing hypothesis the existence of a solution in the interior of the positive cone is established via two bifurcations some open questions are pointed out

periodic trajectories of distributed parameter biochemical systems with time delay abstract this paper deals with a model of a biochemical reactor system with distributed parameters and with a time delay in the growth response time delay has been introduced in microbial growth systems to explain the time lapse between the consumption of liquid substrate and its conversion to solid biomass we study here the properties of the resulting system of partial functional differential equations we first prove the existence positivity and a compactness property of the system trajectories we then prove the existence of periodic solutions of the system for large values of the delay numerical simulations illustrate the existence of such solutions

cooperative systems of differential equations with concave nonlinearities on considere le systeme dynamique discret engendre par une application monotone concave t r n r n on etudie ensuite des systemes d equations differentielles cooperatifs periodiques x f t x a non linearites concaves

global dynamics of a mathematical model of competition in the chemostat general response functions and differential death rates a model of exploitative competition of n species in a chemostat for a single essential nonreproducing growth limiting resource is considered s b hsu siam j appl math 34 1978 pp 760 763 applies lasalle s extension theorem of lyapunov stability theory to study the asymptotic behavior of solutions in the special case that the response functions are modeled by michaelis menten dynamics g j butler and g s k wolkowicz siam j appl math 45 1985 pp 138 151 on the other hand allow more general response functions including monotone and nonmonotone functions but their analysis requires the assumption that the death rates of all the species are negligible in comparison with the washout rate and hence can be ignored by means of lyapunov stability theory the global dynamics of the model for a large class of response functions are studied including both monotone and nonmonotone functions though it is not as general as the class studied by butler and wolkowicz and the results in

hadamard type and bullen type inequalities for lipschitzian functions and their applications in this paper we shall establish some hadamard type and bullen type inequalities for lipschitzian functions and give several applications for special means

global stability for infinite delay lotka volterra type systems abstract in this paper sufficient conditions are established for the global stability of the saturated equilibrium of an infinite delay nonautonomous lotka volterra type system the present work is distinguished from previous work principally be allowing the system to be nonautonomous and relaxing the traditional requirement that the undelayed intraspecific competition dominates both the delayed intraspecific competition as well as the interspecific interactions we require the undelayed intraspecific competition to dominate the latter but not the former

numerical algorithms for forward backward stochastic differential equations efficient numerical algorithms are proposed for a class of forward backward stochastic differential equations fbsdes connected with semilinear parabolic partial differential equations as in j douglas jr j ma and p protter ann appl probab 6 1996 pp 940 968 the algorithms are based on the known four step scheme for solving fbsdes the corresponding semilinear parabolic equation is solved by layer methods which are constructed by means of a probabilistic approach the derivatives of the solution u of the semilinear equation are found by finite differences the forward equation is simulated by mean square methods of order 1 2 and 1 corresponding convergence theorems are proved along with the algorithms for fbsdes on a fixed finite time interval we also construct algorithms for fbsdes with random terminal time the results obtained are supported by numerical experiments

on some inequalities of brenner and alzer for concave functions in this paper are given some simple proofs and extensions of some inequalities for concave functions obtained recently by brenner and alzer

stability and bifurcation analysis of a reaction diffusion equation with spatio temporal delay abstract the spatio temporal patterns of a general reaction diffusion equation with non local delay are considered by analyzing the characteristic equations and applying the complex plane theory the stability of the constant steady state and the possible hopf bifurcations are obtained with the mean delay as a bifurcation parameter the absolute stable conditional stable and unstable region can be explicitly divided by the coefficients of the term without delay and the term with non local delay by investigating the effect of any weight function on the dynamics of the system we establish the existence of spatially homogeneous and inhomogeneous bifurcating periodic solutions and the algorithm for determining the direction and stability of hopf bifurcation is derived by computing the normal form on the center manifold the result shows that for the case of strong kernel the average delay may induce the stability switches finally the theoretical results are applied to a diffusive food limited population model numerical simulations show the existence of the orbit connecting the unstable spatially inhomogeneous periodic solutions to stable spatially homogeneous periodic solutions

a probabilistic method for numerical solution of quasi linear parabolic equations abstract based on the work of milstein g n milstein the probability approach to numerical solution of nonlinear parabolic equations numer meth partial differen equat 18 2002 490 522 and makarov r n makarov numerical solution of quasilinear parabolic equations and backward stochastic differential equations russ j numer anal math modell 18 5 2003 397 412 we extend the layer method to be a more accurate numerical scheme than that makarov used moreover some convergence results of the obtained algorithms are proved a numerical example is provided and the numerical results show that our scheme is much better than that used by makarov

hopf bifurcation analysis of a turbidostat model with discrete delay in this contribution the dynamic behaviors of a turbidostat model with discrete delay are investigated with the delay of digestion chosen to be a bifurcation parameter we find that hopf bifurcations occur when the delay crosses some critical values also by employing the normal form theory and the center manifold theorem we determine the type and stability of the bifurcating periodic solutions finally numerical simulations are offered to support our results

inequalities of hadamard s type for lipschitzian mappings in this paper we establish several inequalities of hadamard s type for lipschitzian mappings

the linear and nonlinear diffusion of the competitive lotka volterra model abstract this paper has studied the effects of linear and nonlinear diffusion of the competitive lotka volterra model and has investigated how the linear and nonlinear diffusions lead from the extinction of one species to the persistence or global asymptotic stability of all species this research has important implications in the design of nature reserves

competition in the chemostat global asymptotic behavior of a model with delayed response in growth the asymptotic behavior of solutions of a model for competition between two microorganisms in the chemostat is studied the model incorporates time delays that allow for the cellular components of each competing species to be structured to include unassimilated or stored nutrient sufficient conditions on the parameters of the model that guarantee i competitive exclusion and ii persistence of at least one of the competing organisms are determined the results extend earlier results on the corresponding model for a single species and also extend these to include competition

approximating the riemann stieltjes integral in terms of generalised trapezoidal rules abstract error bounds in approximating the riemann stieltjes integral in terms of some new generalised trapezoidal rules are given applications for approximating the riemann integral of a two function product are provided as well

competition in the chemostat a distributed delay model and its global asymptotic behavior in this paper we propose a two species competition model in a chemostat that uses a distributed delay to model the lag in the process of nutrient conversion and study the global asymptotic behavio

solving forward backward stochastic differential equations explicitly a four step scheme in this paper we investigate the nature of the adapted solutions to a class of forward backward stochastic differential equations sdes for short in which the forward equation is non degenerate we prove that in this case the adapted solution can always be sought in an ordinary sense over an arbitrarily prescribed time duration via a direct four step scheme using this scheme we further prove that the backward components of the adapted solution are determined explicitly by the forward components via the solution of a certain quasilinear parabolic pde system moreover the uniqueness of the adapted solutions over an arbitrary time duration as well as the continuous dependence of the solutions on the parameters can all be proved within this unified framework some special cases are studied separately in particular we derive a new form of the integral representation of the clark haussmann ocone type for functionals or functions of diffusions in which the conditional expectation is no longer needed

new inequalities of ostrowski type for mappings whose derivatives are s convex in the second sense via fractional integrals a new identity similar to an identity proved in alomari et al 2010 15 for fractional integrals is established then by making use of the established identity some new ostrowski type inequalities for riemann liouville fractional integral are established our results have some relationships with the results of alomari et al 2010 proved in 15 and the analysis used in the proofs is simple

some inequalities for differentiable convex mapping with application to weighted trapezoidal formula and higher moments of random variables abstract several inequalities for differentiable convex wright convex and quasi convex mapping are obtained respectively that are connected with the celebrated hermite hadamard integral inequality also some error estimates for weighted trapezoid formula and higher moments of random variables are given

reaction diffusion equations of two species competing for two complementary resources with internal storage abstract this paper examines a system of reaction diffusion equations arising from a mathematical model of two microbial species competing for two complementary resources with internal storage in an unstirred chemostat the governing system can be reduced to a limiting system based on two uncoupled conservation principles one of main technical difficulties in our analysis is the singularities in the reaction terms conditions for persistence of one population and coexistence of two competing populations are derived from eigenvalue problems maximum principle and the theory of monotone dynamical systems

global asymptotic behavior of a chemostat model with two perfectly complementary resources and distributed delay a model of the chemostat involving two species of microorganisms competing for two perfectly complementary growth limiting nutrients is considered the model incorporates distributed time delay in the form of integral differential equations in order to describe the time involved in converting nutrient to biomass the delays are included in the nutrient and species concentrations simultaneously a general class of monotone increasing functions is used to describe nutrient uptake sufficient conditions based on biologically meaningful parameters in the model are given that predict competitive exclusion for certain parameter ranges and coexistence for others we prove that the global asymptotic attractivity of steady states of the model is similar to that of the corresponding model without time delays however our results indicate that when the inherent delays are in fact large ignoring them may result in incorrect predictions

coexistence in a model of competition in the chemostat incorporating discrete delays a model of two species competing for a single nutrient in the chemostat is proposed incorporating general monotone uptake functions with discrete time delays after transforming the model an anal

a system of partial differential equations modeling the competition for two complementary resources in flowing habitats abstract this paper examines a system of reaction diffusion equations arising from a flowing water habitat in this habitat one or two microorganisms grow while consuming two growth limiting complementary essential resources for the single population model the existence and uniqueness of a positive steady state solution is proved furthermore the unique positive solution is globally attracting for the system with regard to nontrivial nonnegative initial values mathematical analysis for the two competing populations is carried out more precisely the long time behavior is determined by using the monotone dynamical system theory when the semi trivial solutions are both unstable it is also shown that coexistence solutions exist by using the fixed point index theory when the semi trivial solutions are both asymptotically stable

global periodic solutions for a differential delay system modeling a microbial population in the chemostat abstract we show that nonconstant periodic solutions of the delay differential system modeling a microbial population in the chemostat s t 1 s t p s t u t u t u t 1 p s t exist whenever the positive equilibrium is unstable

steady state solutions of a reaction diffusion system arising from intraguild predation and internal storage abstract intraguild predation is added to a mathematical model of competition between two species for a single nutrient with internal storage in the unstirred chemostat at first we established the sharp a priori estimates for nonnegative solutions of the system which assure that all of nonnegative solutions belong to a special cone the selection of this special cone enables us to apply the topological fixed point theorems in cones to establish the existence of positive solutions secondly existence for positive steady state solutions of intraguild prey and intraguild predator is established in terms of the principal eigenvalues of associated nonlinear eigenvalue problems by means of the degree theory in the special cone it turns out that positive steady state solutions exist when the associated principal eigenvalues are both negative or both positive

a competition model for two resources in un stirred chemostat abstract this paper studies a un stirred chemostat with two species competing for two growth limiting non reproducing resources we determine the conditions for positive steady states of the two species and then consider the global attractors of the model in addition we obtain the conditions under which the two populations uniformly strongly persist or go to extinction since the diffusion mechanism with homogeneous boundary conditions inhibits the growth of the organism species it can be understood that the coexistence will be ensured by proportionally smaller diffusions for the two species in particular it is found that both instability and bi stability subcases of the two semitrivial steady states are included in the coexistence region the two populations will go to extinction when both possess large diffusion rates if just one of them spreads faster with the other one diffusing slower then the related semitrivial steady state will be globally attracting the techniques used for the above results consist of the degree theory the semigroup theory and the maximum principle

a system of resource based growth models with two resources in the unstirred chemostat abstract models of single species growth in the unstirred chemostat on two growth limiting nonreproducing resources are considered for the case of two complementary resources the existence and uniqueness of a positive steady state solution is established it is also proved that the unique positive solution is globally attracting for the system with regard to nontrivial nonnegative initial values for the case of two substitutable resources the existence of a positive steady state solution is determined for a range of the parameter m n techniques include the maximum principle monotone method and global bifurcation theory the longtime behavior of the corresponding limiting system is given for a range of m n in the special case of m n the uniqueness and global attractivity of the positive steady state solution of the original system is established

on ostrowski type inequalities for stieltjes integrals with absolutely continuous integrands and integrators of bounded variation some ostrowski type inequalities are given for the stieltjes integral where the integrand is absolutely continuous while the integrator is of bounded variation the case when f is convex is explored applications for the mid point rule and a generalised trapezoid type rule are also presented

a companion of ostrowski s inequality for the riemann stieltjes integral a b f t du t where f is of bounded variation and u is of r h holder type and applications some companions of ostrowski s integral inequality for the riemann stieltjes integral a bf t du t where f is assumed to be of bounded variation on a b and u is of r h holder type on a b are proved applications to the approximation problem of the riemann stieltjes integral in terms of riemann stieltjes sums are also pointed out

on a system of reaction diffusion equations arising from competition in an unstirred chemostat a system of reaction diffusion equations modeling two species competition in an unstirred chemostat is considered the asymptotic behavior of solutions is given as a function of the parameters and it is determined when neither one or both competing populations survive techniques include the maximum principle theory of uniform persistence in infinite dimensional dynamical systems and the theory of strongly order preserving semidynamical system

the dynamical systems approach to differential equations bulletin new series of the american mathematical society volume 11 number 1 july 1984 the dynamical systems approach to differential equations by morris w hirsch 1 this harmony that human intelligence believes it discovers in nature does it exist apart from that intelligence no without doubt a reality completely independent of the spirit which conceives it sees it or feels it is an impossibility a world so exterior as that even if it existed would be forever inaccessible to us but what we call objective reality is in the last analysis that which is common to several thinking beings and could be common to all this common part we will see can be nothing but the harmony expressed by mathematical laws h poincare la valeur de la science p 9 ignorance of the roots of the subject has its price no one denies that modern formulations are clear elegant and precise it s just that it s impossible to comprehend how any one ever thought of them m spivak a comprehensive introduction to differential geometry introduction the mathematical subject we call dynamical systems was fathered by poin care developed sturdily under birkhoff and has enjoyed a vigorous new growth for the last twenty years as i try to show in chapter i it is interesting to look at this mathematical development as the natural outcome of a much broader theme which is as old as science itself the universe is a sytem which changes in time every mathematician has a particular way of thinking about mathematics but rarely makes it explicit yet such perspectives in any particular field can be of great value to nonexperts who must apply the results of the field or who want to learn them in some fields the points of view of its practitioners cluster around several comparatively clearly defined concepts and insights dynamical systems theory is one of them the basic concepts go back to poincare the main thrust of chapters i and ii is to explain rather discursively some of received by the editors december 14 1983 1980 mathematics subject classification primary 00a25 00a99 01a45 01a55 01a60 34a40 34c35 34d10 34d30 35b05 35b35 35b40 35b50 35b30 35k55 54h20 58f25 58f40 58f12 58f10 58d25 58d07 46a40 46e10 47h07 47h20 92a15 92a17 secondary 06f30 35b65 58f13 35j60 46e05 92 03 90 03 90a16 supported in part by national science foundation and institute for advanced study 1984 american mathematical society 0273 0979 84 1 00 25 per page

bifurcation analysis of a chemostat model with a distributed delay a chemostat model of a single species feeding on a limiting nutrient supplied at a constant rate is proposed the model incorporates a general nutrient uptake function and a distributed delay the delay indicates that the growth of the species depends on the past concentration of nutrient using the average time delay as a bifurcation parameter it is proven that the model undergoes a sequence of hopf bifurcations stability criteria for the bifurcating periodic solutions are derived it is also found that the periodic solutions become unstable if the dilution rate is increased computer simulations illustrate the results

refinements of hadamard type inequalities for quasi convex functions with applications to trapezoidal formula and to special means in this paper some inequalities of hadamard s type for quasi convex functions are given some error estimates for the trapezoidal formula are obtained applications to some special means are considered

the ostrowski s integral inequality for lipschitzian mappings and applications abstract a generalization of ostrowski s inequality for lipschitzian mappings and applications in numerical analysis and for euler s beta function are given

generalizations of weighted ostrowski type inequalities for mappings of bounded variation and their applications abstract in this paper we establish some generalizations of weighted ostrowski type inequalities

fejer and hermite hadamard type inequalities for superquadratic functions abstract in this paper we use basic properties of superquadratic functions to obtain new inequalities including fejer s type and hermite hadamard type inequalities for superquadratic functions which are also convex we get refinements of known results

stable oscillations in a predator prey model with time lag the lotka volterra model with carrying capacity at the prey and time delay in the equation concerning the predator is considered the time delay is taken into consideration by an integral with the weight function a exp at it is shown that under certain conditions imposed upon the parameters of the system a supercritical hopf bifurcation takes place at a certain value a0 of a and the bifurcating closed paths are orbitally asymptotically stable for values of a below a0

on the generalization of ostrowski and gruss type discrete inequalities abstract the main purpose of this paper is to establish a generalization of discrete inequalities of the ostrowski and gruss type involving two functions the analysis used in the proofs is fairly elementary

numerical algorithms for semilinear parabolic equations with small parameter based on approximation of stochastic equations the probabilistic approach is used for constructing special layer methods to solve the cauchy problem for semilinear parabolic equations with small parameter despite their probabilistic nature these methods are nevertheless deterministic the algorithms are tested by simulating the burgers equation with small viscosity and the generalized kpp equation with a small parameter