best constants in martingale version of rosenthal s inequality the following generalization of rosenthal s inequality was proved by burkholder a 1 p s f p d ast p leq f ast p leq b p s f p d ast p for all martingales f n it is known that a p grows like sqrt p as p rightarrow infty in this paper we prove that the growth rate of b p as p rightarrow infty is p ln p

non negative solutions for a class of non positone problems in the recent past many results have been established on non negative solutions to boundary value problems of the form where 0 f 0 0 positone problems in this paper we consider the impact on the non negative solutions when f 0 f u to be convex to guarantee uniqueness of positive solutions and f u to be appropriately concave for multiple positive solutions this is in contrast to the case of positone problems where the roles of convexity and concavity were interchanged to obtain similar results we further establish the existence of non negative solutions with interior zeros which did not exist in positone problems

on s shaped bifurcation curves for multiparameter positone problems abstract we study the existence of multiple positive solutions to the two point boundary value problem u x f u x o u 0 0 u 1 u 1 where 0 0 here f is a smooth function such that f 0 on 0 r for some 0 r in particular we consider the case when f is initially convex and then concave we discuss sufficient conditions for the existence of at least three positive solutions for a certain range independent of of we apply our results to the nonlinearity f u exp cu c u c 4 which arises in combustion theory and to the nonlinearity f u u exp c 1 u 0 fixed c 4 4 which arises in chemical reactor theory

on s shaped bifurcation curves where sz c r n 2 1 is bounded with smooth boundary aq readers interested in the derivation of the problem are referred to the work by bebernen l particularly section 2 relevant work on the analysis of this problem has been done by brown et al 2 shivaji 3 61 hastings and mcleod 7 sattinger 8 parks 9 and tam lo clearly u 0 is a solution of 1 l when 2 0 it follows from the contraction mapping theorem that l l always has a solution for a sufficiently small moreover let

a theorem on s shaped bifurcation curve for a positone problem with convex concave nonlinearity and its applications to the perturbed gelfand problem abstract we study the bifurcation curve and exact multiplicity of positive solutions of the positone problem u x f u 0 1 x 1 u 1 u 1 0 where 0 is a bifurcation parameter f c 2 0 satisfies f 0 0 and f u 0 for u 0 and f is convex concave on 0 under a mild condition we prove that the bifurcation curve is s shaped on the u plane we give an application to the perturbed gelfand problem u x exp a u a u 0 1 x 1 u 1 u 1 0 where a 0 is the activation energy parameter we prove that if a a 4 166 the bifurcation curve is s shaped on the u plane our results improve those in s h wang on s shaped bifurcation curves nonlinear anal 22 1994 1475 1485 and p korman y li on the exactness of an s shaped bifurcation curve proc amer math soc 127 1999 1011 1020

on the rate of convergence in wasserstein distance of the empirical measure let mu n be the empirical measure associated to a n sample of a given probability distribution mu on mathbb r d we are interested in the rate of convergence of mu n to mu when measured in the wasserstein distance of order p 0 we provide some satisfying non asymptotic l p bounds and concentration inequalities for any values of p 0 and d ge 1 we extend also the non asymptotic l p bounds to stationary rho mixing sequences markov chains and to some interacting particle systems

existence of a double s shaped bifurcation curve with six positive solutions for a combustion problem abstract we study the bifurcation curve of positive solutions of the combustion problem with nonlinear boundary conditions given by u x exp u u 0 x 1 u 0 0 u 1 u 1 1 u 1 1 u 1 u 1 1 u 1 0 where 0 is called the frank kamenetskii parameter or ignition parameter 0 is the activation energy parameter u x is the dimensionless temperature and the reaction term exp u u is the temperature dependence obeying the simple arrhenius reaction rate law we prove rigorously that for 1 6 459 for some constant 1 the bifurcation curve is double s shaped on the u plane and the problem has at least six positive solutions for a certain range of positive we give rigorous proofs of some computational results of goddard ii shivaji and lee j goddard ii r shivaji e k lee a double s shaped bifurcation curve for a reaction diffusion model with nonlinear boundary conditions bound value probl 2010 art id 357542 23 pp

s shaped bifurcation curves for nonlinear two parameter problems abstract we consider the nonlinear eigenvalue problem u t 1 u t u t 2 u t p u t 0 t i 1 2 1 2 u 1 2 u 1 2 0 where p 2 is a constant 0 and 0 are parameters this equation was introduced by crandall and rabinowitz 1973 as the typical model which develops the s shaped bifurcation curve when p 3 and 0 1 it is known that if p 2 and 0 1 then there exists a unique solution u r c 2 i with u where 0 is a given constant furthermore is s shaped in this paper we establish the asymptotic formulas for as 0 1 2 when 0 1 where 1 and 2 are two turning points of and is the unique positive solution to the algebraic equation 1 x x 2 x p 0 we also characterize the asymptotic shape of u t as by establishing the asymptotic behavior of u q as when 0 1 is a fixed constant

curves of positive solutions of boundary value problems on time scales abstract let t r be a time scale with a inf t b sup t we consider the nonlinear boundary value problem 2 p t u t q t u t f t u t on t 4 u a u b 0 where r 0 and f t r r satisfies the conditions f t 0 t t r f t f t t t r we prove a strong maximum principle for the linear operator defined by the left hand side of 1 and use this to show that for every solution u of 1 2 u is positive on t a b in addition we show that there exists max 0 possibly max such that if 0 max then 1 2 has a unique solution u while if max then 1 2 has no solution the value of max is characterised as the principal eigenvalue of an associated weighted eigenvalue problem in this regard we prove a general existence result for such eigenvalues for problems with general nonnegative weights

maximal inequalities for partial sums of rho mixing sequences a rosenthal type inequality for maximum partial sums of p mixing sequences is obtained applications to the complete convergence and almost sure summability of partial sums are also discussed

on stabilizability and sampling for infinite dimensional systems some questions either motivated by or related to time sampling regarding the stabilizability of infinite dimensional or distributed parameter systems are considered in particular these include the characteristics of those systems which are finite rank stabilizable and the preservation of stabilizability under sampling a pathological class of systems which when sampled may be rendered unstabilizable by arbitrarily small perturbations in the sampling rate is identified the approach taken in establishing these results is based upon a spectral decomposition of the infinitesimal generator of the underlying open loop semigroup the results obtained apply to control systems with dynamics described by broad classes of partial and functional differential equations that arise frequently in applications

existence and multiplicity of positive solutions for classes of singular elliptic pdes we consider the boundary value problem u g u u in u 0on where rn is a bounded domain is a nonnegative function in l such that 0 on some subset of of positive measure and g 0 r is continuous we establish the existence of three positive solutions when g 0 0 positone the graph of s 1g s is roughly s shaped and 0 we also prove that there exists at least one positive solution when g 0 0 and 0 1 we employ the method of sub super solutions to prove our results

rosenthal type inequalities for the maximum of partial sums of stationary processes and examples the aim of this paper is to propose new rosenthal type inequalities for moments of order higher than 2 of the maximum of partial sums of stationary sequences including martingales and their generalizations as in the recent results by peligrad et al proc amer math soc 135 2007 541 550 and rio j theoret probab 22 2009 146 163 the estimates of the moments are expressed in terms of the norms of projections of partial sums the proofs of the results are essentially based on a new maximal inequality generalizing the doob maximal inequality for martingales and dyadic induction various applications are also provided

s shaped bifurcation curves in ecosystems abstract we consider the existence of multiple positive solutions to the steady state reaction diffusion equation with dirichlet boundary conditions of the form u u u 2 k c u 2 1 u 2 x u 0 x here u div u is the laplacian of u 1 is the diffusion coefficient k and c are positive constants and r n is a smooth bounded region with in c 2 this model describes the steady states of a logistic growth model with grazing in a spatially homogeneous ecosystem it also describes the dynamics of the fish population with natural predation in this paper we discuss the existence of multiple positive solutions leading to the occurrence of an s shaped bifurcation curve we prove our results by the method of sub supersolutions

central limit theorems for additive functionals of markov chains central limit theorems and invariance principles are obtained for additive functionals of a stationary ergodic markov chain say s n g x 1 g x n where e g x 1 0 and e g x 1 2 the conditions imposed restrict the moments of g and the growth of the conditional means e s n x 1 no other restrictions on the dependence structure of the chain are required when specialized to shift processes the conditions are implied by simple integral tests involving g

on the classification and evolution of bifurcation curves for a one dimensional prescribed curvature problem with nonlinearity exp aua u abstract we study the classification and evolution of bifurcation curves of positive solutions u for the one dimensional prescribed curvature problem u x 1 u x 2 exp a u a u l x l u l u l 0 where 0 is a bifurcation parameter and l a 0 are two evolution parameters we prove that on u plane for 0 a 36 17 2 118 the bifurcation curve is shaped while for a 36 17 the bifurcation curve is shaped or reversed e like shaped in particular for a a 4 107 the bifurcation curve is i shaped if l 0 small enough and ii reversed e like shaped if l is large enough

proof of a conjecture for the one dimensional perturbed gelfand problem from combustion theory we study global bifurcation curves and the exact multiplicity of positive solutions for the two point boundary value problem arising in combustion theory left begin array l u prime prime x lambda exp left frac au a u right 0 quad 1 x 1 u 1 u 1 0 end array right where lambda 0 is the frank kamenetskii parameter and a 0 is the activation energy parameter we prove that there exists a critical bifurcation value a0 approx 4 069 such that on the lambda u infty plane the bifurcation curve is s shaped for a a 0 and is monotone increasing for 0 a leqq a 0 that is we prove the long standing conjecture for the one dimensional perturbed gelfand problem we also study in the a lambda left vert u right vert infty space the shape and structure of the bifurcation surface

remarks on an s shaped bifurcation curve abstract we study the non negative solutions of the boundary value problem u exp u u x u 0 x where 0 0 r n is bounded with smooth boundary this problem arises in the theory of combustion we study the estimates on the supremum norm of the solutions and estimates on the critical values of

asymptotic behavior of solutions for finite difference equations of reaction diffusion abstract the method of upper lower solutions and the associated existence comparison theorem for continuous reaction diffusion equations are extended to a finite difference system which is a discrete version of the continuous problem of either parabolic type or elliptic type under appropriate boundary condition it is shown by using the existence comparison theorem for the discrete system that for a certain class of initial functions the corresponding time dependent solution converges monotonically to a steady state solution this monotone convergence leads to the stability property of the steady state solution as well as an estimate of the stability region in the framework of the finite difference system numerical results are given for an enzyme substrate reaction model where multiple steady state solutions exist for a certain range of physical parameters in this situation the time dependent solution converges either to the maximal solution or to the minimal solution depending on the corresponding initial function

uniqueness of positive solutions for a class of elliptic boundary value problems on demontre l unicite des solutions non negatives du probleme aux valeurs limites u f u pour x u 0 pour x ou 0 et est un domaine borne de r n a frontiere lisse

persistence and bifurcation of degenerate solutions abstract we consider a nonlinear equation f e u 0 where f is a differentiable mapping from r r x to y and x y are banach spaces when e varies from a fixed e e 0 bifurcation occurs to the solution curve s u s we study the degenerate solutions of the equation and we obtain several bifurcation theorems on the degenerate solutions which can be applied in many nonlinear problems to obtain precise global bifurcation diagrams

martingale approximations for sums of stationary processes approximations to sums of stationary and ergodic sequences by martingales are investigated necessary and sufficient conditions for such sums to be asymptotically normal conditionally given the past up to time 0 are obtained it is first shown that a martingale approximation is necessary for such normality and then that the sums are asymptotically normal if and only if the approximating martingales satisfy a lindeberg feller condition using the explicit construction of the approximating martingales a central limit theorem is derived for the sample means of linear processes the conditions are not sufficient for the functional version of the central limit theorem this is shown by an example and a slightly stronger sufficient condition is given

global bifurcation of steady state solutions in this paper we study the bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation a f u 0 on the interval l 0 or 0 for some nonzero function c c a for the neumann problem we use entirely different techniques to prove that s is never critical for any cubic polynomial j this implies at once that the neumann problem can have at most one nonconstant solution having a given number of maxima or minima this solution is necessarily strongly nondegenerate in the sense that zero is not contained in the spectrum of the linearized operator see 2 for the dirichlet problem the situation is far more complicated and the bifurcation diagrams undergo qualitative changes depending on the positions of the roots off for example fig 1 shows the bifurcation diagram for the positive solutions for two different cubic functions of the form f u a u u b u c it is interesting to note however that in every case we study there are at most three solutions to the dirichlet problem for each

existence and multiplicity results for a class of semilinear elliptic equations abstract we study the existence and multiplicity of nonnegative solutions as well as the behavior of corresponding parameter dependent branches to the equation u 1 u u m u n in a bounded domain r n endowed with the zero dirichlet boundary data where 0 m 1 and n 0 when 0 the obtained solutions can be seen as steady states of the corresponding reaction diffusion equation describing a model of isothermal autocatalytic chemical reaction with termination in addition to the main new results we formulate a few relevant conjectures

classification and evolution of bifurcation curves for the one dimensional perturbed gelfand equation with mixed boundary conditions abstract we study the classification and evolution of bifurcation curves of positive solutions for the one dimensional perturbed gelfand equation with mixed boundary conditions given by u x exp a u a u 0 0 x 1 u 0 0 u 1 c 0 we prove that for positive a a 0 0 501 and c 0 the bifurcation curve is strictly increasing on the u plane and there exists a positive 0 such that the problem has no positive solution for 0 0 and exactly one positive solution for 0 while for a a 1 4 107 there exists c 1 c 1 a 1 057 such that on the u plane i when 0 c c 1 the bifurcation curve is s shaped and the problem has at least three positive solutions for some range of positive ii when c c 1 the bifurcation curve is shaped and the problem has at least two positive solutions for some range of positive

inequalities for absolutely regular sequences application to density estimation this paper investigates the problem of density estimation for absolutely regular observations in a first part we state two important results a new variance inequality and a rosenthal type inequality this allows us to study the p integrated risk p 2 of a large class of density estimators including kernel or projection estimators under the summability condition on the mixing coefficients k 0 k 1 p 2 k the rates obtained are those known to be optimal in the independent setting

classes of reaction diffusion equations where a parameter influences the equation as well as the boundary condition abstract we study positive solutions to steady state reaction diffusion equations of the form u f u u u 0 where 0 is a bounded domain in r n n 1 with smooth boundary u is the outward normal derivative of u c 0 is strictly increasing such that 0 0 and f c 2 0 r 0 with 0 r 0 if r 0 we assume f c 2 0 r 0 with f r 0 0 and f s 0 for s r 0 and if r 0 we assume lim s f s 0 and lim s f s s 0 sublinear at note here that the parameter influences both the equation and the boundary condition we discuss existence nonexistence multiplicity and uniqueness results for the cases when a f 0 0 b f 0 0 and c f 0 0 we obtain existence and multiplicity results by the method of sub super solutions and uniqueness results by comparison principles and a priori estimates

stabilization of distributed parameter systems by piecewise polynomial control in this paper we consider the stabilization of continuous time distributed parameter systems by piecewise polynomial controls we obtain necessary and sufficient conditions for stabilization by piecewise polynomial controls

almost sure invariance principle for dynamical systems by spectral methods we prove the almost sure invariance principle for stationary d valued random processes with very precise dimension independent error terms solely under a strong assumption concerning the characteristic functions of these processes this assumption is easy to check for large classes of dynamical systems or markov chains using strong or weak spectral perturbation arguments

exact multiplicity of solutions and s shaped bifurcation curves for the p laplacian perturbed gelfand problem in one space variable abstract we study exact multiplicity of positive solutions and the bifurcation curve of the p laplacian perturbed gelfand problem from combustion theory p u x exp a u a u 0 1 x 1 u 1 u 1 0 where p 1 p y y p 2 y p u is the one dimensional p laplacian 0 is the frank kamenetskii parameter u x is the dimensionless temperature and the reaction term f u exp a u a u is the temperature dependence obeying the arrhenius reaction rate law we find explicitly a a p 0 such that if the activation energy a a then the bifurcation curve is s shaped in the u plane more precisely there exist 0 such that the problem has exactly three positive solutions for exactly two positive solutions for and and a unique positive solution for 0 and

a theorem on reversed s shaped bifurcation curves for a class of boundary value problems and its application abstract we study the bifurcation curves of positive solutions of the boundary value problem u x f e u x 0 1 x 1 u 1 u 1 0 where f e u g u e h u e r is a bifurcation parameter and functions g h c 0 c 2 0 satisfy five hypotheses presented herein assuming these hypotheses on fixed g and h we prove that the bifurcation curve is reverse s shaped on the e u plane that is the bifurcation curve has exactly two turning points at some points e u e and e u e such that e e and u e u e in addition we prove that e 0 thus the exact number of positive solutions can be precisely determined by the values of e and e we give an application to the two parameter bifurcation problem u x 1 u 2 e u 3 0 1 x 1 u 1 u 1 0 where e are two positive bifurcation parameters some new results are obtained

evolution of bifurcation curves for semipositone problems when nonlinearities develop multiple zeroes we discuss existence uniqueness and multiplicity results on positive solutions to u x email protected u x 0 for email protected 0 1 u 0 0 u 1 where l 0 is a constant and u is a smooth function satisfying 0 0 semipositone in particular we follow a class of nonlinearities u as they evolve from having one positive zero to three positive zeros and compare the corresponding bifurcation curves we establish that the bifurcation curve splits into two connected components in the critical case when u develops the second zero

exact multiplicity of positive solutions for concave convex and convex concave nonlinearities abstract this note gives an unified treatment of the exact multiplicity results for both s shaped and reversed s shaped bifurcation for positive solutions of the two point problem u f u 0 for 1 x 1 u 1 u 1 0 depending on a positive parameter for both concave convex and convex concave nonlinearities f u

new dependence coefficients examples and applications to statistics to measure the dependence between a real valued random variable x and a algebra we consider four distances between the conditional distribution function of x given and the distribution function of x the coefficients obtained are weaker than the corresponding mixing coefficients and may be computed in many situations in particular we show that they are well adapted to functions of mixing sequences iterated random functions and dynamical systems starting from a new covariance inequality we study the mean integrated square error for estimating the unknown marginal density of a stationary sequence we obtain optimal rates for kernel estimators as well as projection estimators on a well localized basis under a minimal condition on the coefficients using recent results we show that our coefficients may be also used to obtain various exponential inequalities a concentration inequality for lipschitz functions and a berry esseen type inequality

remark on a sufficient condition for multiplicity in a model in chemical reactor theory we consider the model u l s u exp k l u x e 0 1 u 0 0 u 1 which arises in chemical reactor theory here l s and k are positive constants it is known that if there exists an a e 0 s such that h a email protected f s ds a2 f a 0 then there exists a range of l for which there exists at least three nonnegative solutions and using numerical evaluation it was conjectured that such values of a exist in this paper we give an analytical proof of this conjecture

a new maximal inequality and invariance principle for stationary sequences we derive a new maximal inequality for stationary sequences under a martingale type condition introduced by maxwell and woodroofe ann probab 28 2000 713 724 then we apply it to establish the donsker invariance principle for this class of stationary sequences a markov chain example is given in order to show the optimality of the conditions imposed

a simplified proof of a conjecture for the perturbed gelfand equation from combustion theory abstract for the perturbed gelfand s equation on the unit ball in two dimensions y du and y lou 5 proved that the curve of positive solutions is exactly s shaped for sufficiently small values of the secondary parameter we present a simplified proof and some extensions this problem is prominent in combustion theory see e g the book of j bebernes and d eberly 1

global bifurcation and exact multiplicity of positive solutions for a positone problem with cubic nonlinearity and their applications we study the global bifurcation and exact multiplicity of positive solutions of u x lambda f epsilon u 0 1 f epsilon u epsilon u 3 sigma u 2 tau u rho where lambda epsilon 0 are two bifurcation parameters and sigma rho 0 tau 0 are constants by developing some new time map techniques we prove the global bifurcation of bifurcation curves for varying epsilon 0 more precisely we prove that for any sigma rho 0 tau 0 there exists epsilon 0 such that on the lambda parallel to u parallel to infinity plane the bifurcation curve is s shaped for 0 epsilon we also prove the global bifurcation of bifurcation curves for varying lambda 0 thus we are able to determine the exact number of positive solutions by the values of epsilon and lambda we give an application to prove a long standing conjecture for global bifurcation of positive solutions for the problem u x lambda epsilon u 3 u 2 u 1 0 1 u 1 u 1 0 which was studied by crandall and rabinowitz arch rational mech anal 52 1973 p 177 in addition we give an application to prove a conjecture of smoller and wasserman j differential equations 39 1981 p 283 lines 2 3 on the maximum number of positive solutions of a positone problem

on the continuous dependence with respect to sampling of the linear quadratic regulator problem for distributed parameter systems the convergence of solutions to the discrete or sampled time linear quadratic regulator problem and associated riccati equation for infinite dimensional systems to the solutions to the corresponding continuous time problem and equation as the length of the sampling interval the sampling rate tends toward zero infinity is established both the finite and infinite time horizon problems are studied in the finite time horizon case strong continuity of the operators that define the control system and performance index together with a stability and consistency condition on the sampling scheme are required for the infinite time horizon problem in addition the sampled systems must be stabilizable and detectable uniformly with respect to the sampling rate classes of systems for which this condition can be verified are discussed results of numerical studies involving the control of a heat diffusion equation a hereditary or delay system and a flexible beam are presented and discussed

stability and convergence properties of forced infinite dimensional discrete time lur e systems incremental stability and convergence properties for forced infinite dimensional discrete time lur e systems are addressed lur e systems have a linear and nonlinear component and arise as the fe

proof of a conjecture for the perturbed gelfand equation from combustion theory abstract in this paper we prove that when the domain b is a two dimensional ball for all small positive e the solution curve u of the perturbed gelfand equation u eu 1 eu with u b 0 is exactly s shaped this settles a long standing conjecture on the exact number of solutions to this equation

numerical approximation for the infinite dimensional discrete time optimal linear quadratic regulator problem an abstract approximation framework is developed for the finite and infinite horizon discrete time linear quadratic regulator problems for systems whose state dynamics are described by a linear semigroup of operators on an infinite dimensional hilbert space the schemes included in the framework yield finite dimensional approximations to the linear state feedback gains which determine the optimal control law convergence arguments are given examples involving hereditary and parabolic systems and the vibration of a flexible beam are considered spline based finite element schemes for these classes of problems together with numerical results are presented and discussed

convergence rates of the strong law for stationary mixing sequences in this note we estimate the rate of convergence in marcinkiewicz zygmung strong law for partial sumss n of strong stationary mixing sequences of random variables the results improve the corresponding ones obtained by tze leung lai 1977 and christian hipp 1979

stability of infinite dimensional sampled data systems suppose that a static state feedback stabilizes a continuous time linear infinite dimensional control system we consider the following question if we construct a sampled data controller by applying an idealized sample and hold process to a continuous time stabilizing feedback will this sampled data controller stabilize the system for all sufficiently small sampling times here the state space x and the control space u are hilbert spaces the system is of the form x t ax t bu t where a is the generator of a strongly continuous semigroup on x and the continuous time feedback is u t fx t the answer to the above question is known to be yes if x and u are finite dimensional spaces in the infinite dimensional case if f is not compact then it is easy to find counterexamples therefore we restrict attention to compact feedback we show that the answer to the above question is yes if b is a bounded operator from u into x moreover if b is unbounded we show that the answer yes remains correct provided that the semigroup generated by a is analytic we use the theory developed for static state feedback to obtain analogous results for dynamic output feedback control

multiple positive solutions of the one dimensional p laplacian in this work we investigate the existence of positive solutions of the p laplacian using the quadrature method we prove the existence of multiple solutions of the one dimensional p laplacian for 0 and determine their exact number for 0 2005 elsevier inc all rights reserved

asymptotic length of bifurcation curves related to inverse bifurcation problems abstract we consider the nonlinear eigenvalue problem u t f u t 0 u t 0 t i 1 1 u 1 u 1 0 where f u u g u and 0 is a parameter our interest is the asymptotic length l g of the bifurcation curve g u 0 by the notion of l g we will propose a new framework of inverse bifurcation problems precisely we consider whether it is possible to characterize the unknown nonlinear term g u by l g if g u g 1 u u 2 sin u or g 2 u u 1 2 sin u then the bifurcation curves g i i 1 2 are continuous functions of 0 and g i 2 4 as furthermore they intersect the line 2 4 infinitely many times as in this paper by establishing the precise asymptotic formula for g i for 1 we obtain the precise asymptotic length l g i of g i for 1 then we consider the set g c 2 r g 2 4 as and study the existence and non existence of g whose asymptotic formula for l g is equal to those of l g i up to the second term

exact multiplicity of solutions and s shaped bifurcation curve for a class of semilinear elliptic equations abstract the set of steady state solutions to a reaction diffusion equation modeling an autocatalytic chemical reaction is completely determined when the reactor has spherical geometry and the spatial dimension is n 1 or 2 for any reaction order or n 3 for subcritical reaction order bifurcation approach and analysis of linearized problems are used to establish exact multiplicity and precise global bifurcation diagram of positive steady states

critical behaviour of nonlinear elliptic boundary value problems suggested by exothermic reactions we consider a class of semilinear elliptic boundary value problems depending on a parameter which arise in the theory of combustion based on the results in another paper by the same author a rigorous quantitative connection is shown between the solution set of the boundary value problem and that of a simple scalar equation the semenov approximation

strong invariance principles for dependent random variables we establish strong invariance principles for sums of stationary and ergodic processes with nearly optimal bounds applications to linear and some nonlinear processes are discussed strong laws of large numbers and laws of the iterated logarithm are also obtained under easily verifiable conditions

semilinear elliptic boundary value problems in chemical reactor theory abstract this paper is devoted to the study of semilinear elliptic boundary value problems arising in chemical reactor theory which obey the simple arrhenius rate law and newtonian cooling we prove that ignition and extinction phenomena occur in the stable steady temperature profile at some critical values of a dimensionless heat evolution rate