zero stability properties of the three ordinate variable stepsize variable formula methods consider the systemy f x y y a x a b y r n wheref is continuous and lipschitzian with respect to the second argument very often linear multistep variable stepsize variable formula methods lm vsvfm s are used to computey k y xk on the points of the grid a x 0 x1 x2 xn b the general lm vsvfm is based on formulae of the following type y k sum limits i 1 s k alpha i bar h k sk y k i sum limits i 0 s k h k i beta i bar h k s k f x k i y k i whereh k xk xk 1 bar h k h k h k 1 ldots h k s k s k k k 1 1 n the coefficients i and i depend on the lasts k 1 stepsizes and on the formula used at stepk only the zero stability properties of some special classes of lm vsvfm s as for example those based on adams formulae were investigated in the literature a class of three ordinate lm vsvfm s is defined in this paper some results concerning the zero stability properties of these methods are proved it is shown that some well known results are simple corollaries of the results found for the three ordinate lm vsvfm s it is easily seen that similar results hold for the corresponding one leg vsvfm s finally the use of the theoretical results in the practical implementations is briefly discussed

a calculus for log convex interference functions the behavior of certain interference coupled multiuser systems can be modeled by means of logarithmically convex log convex interference functions in this paper we show fundamental properties of this framework a key observation is that any log convex interference function can be expressed as an optimum over elementary log convex interference functions the results also contribute to a better understanding of certain quality of service qos tradeoff regions which can be expressed as sublevel sets of log convex interference functions we analyze the structure of the qos region and provide conditions for the achievability of boundary points the proposed framework of log convex interference functions generalizes the classical linear interference model which is closely connected with the theory of irreducible nonnegative matrices perron frobenius theory we discuss some possible applications in robust communication cooperative game theory and max min fairness

neural networks with a redundant representation detecting the undetectable we consider a three layer sejnowski machine and show that features learnt via contrastive divergence have a dual representation as patterns in a dense associative memory of order p 4 the latter is known to be able to hebbian store an amount of patterns scaling as n p ensuremath 1 where n denotes the number of constituting binary neurons interacting p wisely we also prove that by keeping the dense associative network far from the saturation regime namely allowing for a number of patterns scaling only linearly with n while pg2 such a system is able to perform pattern recognition far below the standard signal to noise threshold in particular a network with p 4 is able to retrieve information whose intensity is mathcal o 1 even in the presence of a noise mathcal o sqrt n in the large n limit this striking skill stems from a redundancy representation of patterns which is afforded given the relatively low load information storage and it contributes to explain the impressive abilities in pattern recognition exhibited by new generation neural networks the whole theory is developed rigorously at the replica symmetric level of approximation and corroborated by signal to noise analysis and monte carlo simulations

the worst additive noise under a covariance constraint the maximum entropy noise under a lag p autocorrelation constraint is known by burg s theorem to be the pth order gauss markov process satisfying these constraints the question is what is the worst additive noise for a communication channel given these constraints is it the maximum entropy noise the problem becomes one of extremizing the mutual information over all noise processes with covariances satisfying the correlation constraints r sub 0 r sub p for high signal powers the worst additive noise is gauss markov of order p as expected but for low powers the worst additive noise is gaussian with a covariance matrix in a convex set which depends on the signal power

on the existence of solution of stage equations in implicit runge kutta methods abstract this paper is concerned with the unique solvability of stage equations which arise when implicit runge kutta methods apply to nonlinear stiff systems of differential equations y f t y denoting by a the matrix of coefficients of the runge kutta method and by 2 j the logarithmic norm of the matrix j associated with the l2 norm several authors crouzeix et al bit 23 1983 84 91 hundsdorfer and spijker siam j numer anal 24 1987 583 594 kraaijevanger and schneid numer math 59 1991 129 157 liu and kraaijevanger bit 28 4 1988 825 838 have obtained conditions on a that ensure for a given the unique solvability of stage equations for all stepsize h and stiff system with h 2 f t y

from semidiscrete to fully discrete stability of runge kutta schemes by the energy method the integration of semidiscrete approximations for time dependent problems is encountered in a variety of applications the runge kutta rk methods are widely used to integrate the ode systems which arise in this context resulting in large ode systems called methods of lines these methods of lines are governed by possibly ill conditioned systems with a growing dimension consequently the naive spectral stability analysis based on scalar eigenvalues arguments may be misleading instead we present here a stability analysis of rk methods for well posed semidiscrete approximations based on a general energy method we review the stability question for such rk approximations and highlight its intricate dependence on the growing dimension of the problem in particular we prove the strong stability of general fully discrete rk methods governed by coercive approximations we conclude with two nontrivial examples which demonstrate the versatility of our approach in the context of general systems of convection diffusion equations with variable coefficients a straightforward implementation of our results verify the strong stability of rk methods for local finite difference schemes as well as global spectral approximations since our approach is based on the energy method which is carried in the physical space and since it avoids the von neumann analysis which is carried in the dual fourier space we are able to easily adapt additional extensions due to nonperiodic boundary conditions general geometries etc

capacity regions and optimal power allocation for cdma cellular radio the capacity region of code division multiple access cdma is determined by the set of transmission rates combined with quality of service qos requirements which allow for a feasible power allocation scheme for n mobiles in a cellular network the geometrical and topological properties of the capacity region are investigated in the present paper for the case of unlimited and limited power respectively as a central result we show that the capacity region is convex by breaking the complicated topological structure into characteristic properties of its boundary and interior points each of interest in itself based on these results we furthermore investigate optimal power assignment schemes in the case that the demand of a community of users is infeasible weighted minimax and bayes solutions are explicitly determined as appropriate means to share the capacity of a cellular network in a reasonable and fair way

neural networks retrieving boolean patterns in a sea of gaussian ones restricted boltzmann machines are key tools in machine learning and are described by the energy function of bipartite spin glasses from a statistical mechanical perspective they share the same gibbs measure of hopfield networks for associative memory in this equivalence weights in the former play as patterns in the latter as boltzmann machines usually require real weights to be trained with gradient descent like methods while hopfield networks typically store binary patterns to be able to retrieve the investigation of a mixed hebbian network equipped with both real e g gaussian and discrete e g boolean patterns naturally arises we prove that in the challenging regime of a high storage of real patterns where retrieval is forbidden an additional load of boolean patterns can still be retrieved as long as the ratio between the overall load and the network size does not exceed a critical threshold that turns out to be the same of the standard amit gutfreund sompolinsky theory assuming replica symmetry we study the case of a low load of boolean patterns combining the stochastic stability and hamilton jacobi interpolating techniques the result can be extended to the high load by a non rigorous but standard replica computation argument

essentially optimal explicit runge kutta methods with application to hyperbolic parabolic equations optimal explicit runge kutta methods consider more stages in order to include a particular spectrum in their stability domain and thus reduce time step restrictions this idea so far used mostly for real line spectra is generalized to more general spectra in the form of a thin region in thin regions the eigenvalues may extend away from the real axis into the imaginary plane we give a direct characterization of optimal stability polynomials containing a maximal thin region and calculate these polynomials for various cases semi discretizations of hyperbolic parabolic equations are a relevant application which exhibit a thin region spectrum as a model linear scalar advection diffusion is investigated the second order stabilized explicit runge kutta methods derived from the stability polynomials are applied to advection diffusion and compressible viscous fluid dynamics in numerical experiments due to the stabilization the time step can be controlled solely from the hyperbolic cfl condition even in the presence of viscous fluxes

error probabilities for binary direct sequence spread spectrum communications with random signature sequences binary direct sequence spread spectrum multiple access communications an additive white gaussian noise channel and a coherent correlation receiver are considered an expression for the output of the receiver is obtained for the case of random signature sequences and the corresponding characteristic function is determined the expression is used to study the density function of the multiple access interference and to determine arbitrarily tight upper and lower bounds on the average probability of error the bounds which are obtained without making a gaussian approximation are compared to results obtained using a gaussian approximation the effects of transmitter power the length of the signature sequences and the number of interfering transmitters are illustrated each transmitter is assumed to have the same power although the general approach can accommodate the case of transmitters with unequal powers

increasing the real stability boundary of explicit methods abstract based on the simplest well known integration rules such as the forward euler scheme and the classical runge kutta method an extension is proposed to enlarge the real stability boundary the main characteristic of the resulting schemes is that the computational complexity is hardly increased

stability of semidiscretizations of hyperbolic problems semidiscretization of hyperbolic systems yields systems of ordinary differential equations of first or second order in both cases we investigate the stability of a general class of methods namely the m stage k step methods when applied to linear test equations a map sigma is introduced which allows us at least for constant linear systems to investigate whether a second order system should be integrated directly or should first be reduced to a first order system

performance and interference control in wireless ad hoc and mesh networks a generalized lagrangian approach this paper deals with the problem of power allocation in heterogeneous wireless networks such as wireless ad hoc and mesh networks we propose a general approach to power control with the goal of optimizing some network performance metric subject general constraints at selected links such as constraints on signal to interference ratio or restrictions on received power based on a novel construction of generalized lagrangian function we develop a primal dual iteration that converges to a locally optimal power allocation vector the iteration exhibits linear convergence and has certain advantages in applicability efficiency and complexity when compared to classical primal dual methods of similar convergence using the recently proposed concept of an adjoint network we finally present a feedback scheme which allows for the highly desired decentralized realization of the proposed iteration

the space of interactions in neural network models the typical fraction of the space of interactions between each pair of n ising spins which solve the problem of storing a given set of p random patterns as n bit spin configurations is considered the volume is calculated explicitly as a function of the storage ratio alpha p n of the value kappa 0 of the product of the spin and the magnetic field at each site and of the magnetisation m here m may vary between 0 no correlation and 1 completely correlated the capacity increases with the correlation between patterns from alpha 2 for correlated patterns with kappa 0 and tends to infinity as m tends to 1 the calculations use a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is shown to be locally stable a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist

strong stability preserving explicit linear multistep methods with variable step size strong stability preserving ssp methods are designed primarily for time integration of nonlinear hyperbolic pdes for which the permissible ssp step size varies from one step to the next we develop the first ssp linear multistep methods of order two and three with variable step size and prove their optimality stability and convergence the choice of step size for multistep ssp methods is an interesting problem because the allowable step size depends on the ssp coefficient which in turn depends on the chosen step sizes the description of the methods includes an optimal step size strategy we prove sharp upper bounds on the allowable step size for explicit ssp linear multistep methods and show the existence of methods with arbitrarily high order of accuracy the effectiveness of the methods is demonstrated through numerical examples

equilibrium statistical mechanics of bipartite spin systems the aim of this paper is to give an extensive treatment of bipartite mean field spin systems pure and disordered at first bipartite ferromagnets are investigated and an explicit expression for the free energy is achieved through a new minimax variational principle then via the hamilton jacobi technique the same structure of the free energy is obtained together with the existence of its thermodynamic limit and the minimax principle is connected to a standard max one the same is investigated for bipartite spin glasses by the borel cantelli lemma we obtain the control of the high temperature regime while via the double stochastic stability technique we also obtain the explicit expression of the free energy in the replica symmetric approximation uniquely defined by a minimax variational principle again we also obtain a general result that states that the free energies of these systems are convex linear combinations of their independent one party model counterparts for the sake of completeness we show further that at zero temperature the replica symmetric entropy becomes negative and consequently such a symmetry must be broken the treatment of the fully broken replica symmetry case is deferred to a forthcoming paper as a first step in this direction we start deriving the linear and quadratic constraints to overlap fluctuations

strict convexity of the feasible log sir region the feasible log sir region is defined as a set of all signal to interference ratios sir expressed in logarithmic scale that can be supported in a wireless network by means of power control and with all users being active concurrently recently the feasible log sir region was shown to be a convex set which is a key ingredient in the development of some power control strategies for wireless systems in this paper under the assumption of a noiseless channel we strengthen these results by proving a necessary and sufficient condition for the feasible log sir region to be a strictly convex set the strict convexity property is of interest since it is closely related to the problem of the existence and uniqueness of a so called log sir fair power vector

a receiver centric approach to interference management fairness and outage optimization effective interference management in the multiuser interference channel necessitates that the users form their transmission and interference management decisions in coordination and adapt them to the state of the channel establishing such coordination often facilitated through information exchange is prohibitive in fast varying channels especially when the network size grows this paper focuses on the multiuser gaussian interference channel and offers a receiver centric approach to interference management in this approach the transmitters deploy rate splitting and superposition coding to generate their messages according to independent gaussian codebooks the receivers can freely decode any arbitrary set of interfering messages along with their designated messages in any desired joint or ordered fashion and treat the rest of the interferers as gaussian noise the proposed receiver centric interference management approach is applied to two class of problems outage optimization and fairness constrained rate allocation and constructive proofs are provided to establish the following properties for the proposed approach 1 the optimal set of codebooks to be decoded by each receiver is a local decision made by each receiver based on its local channel state information csi 2 the globally optimal transmission rates are related to locally optimal rates computed by the receivers based on their local information which implies that the transmitters do not require explicit knowledge of the csi and can determine their rates via limited feedback from the receivers and 3 obtaining the optimal interference management strategy at each receiver has controlled complexity

high order difference schemes with reduced dispersion for hyperbolic differential equations abstract we investigate difference schemes for systems of first order hyperbolic differential equations in two space dimensions possessing the following characteristics 1 the spatial discretizations are fourth order accurate 2 the time discretization is of explicit runge kutta type and is also fourth order accurate 3 the scaled stability boundary is approximately 1 2 2 4 the weights in the space discretizations and the runge kutta parameters can be adapted in order to reduce the dispersion of dominant fourier components this method is illustrated by applying it to the shallow water equations simulating the motion of water in a shallow sea due to tidal forces since in such problems the dominant frequencies in the solution are known in advance the method can take fully advantage of the possibility to tune the various parameters to these dominant frequencies

information theoretic upper bounds on the capacity of large extended ad hoc wireless networks we derive an information theoretic upper bound on the rate per communication pair in a large ad hoc wireless network we show that under minimal conditions on the attenuation due to the environment and for networks with a constant density of users this rate tends to zero as the number of users gets large

stepsize conditions for general monotonicity in numerical initial value problems for runge kutta methods and linear multistep methods much attention has been paid in the literature to special nonlinear stability properties indicated by the terms total variation diminishing tvd strong stability preserving ssp and monotonicity stepsize conditions guaranteeing these properties were studied e g by shu and osher j comput phys 77 1988 pp 439 471 gottlieb shu and tadmor siam rev 43 2001 pp 89 112 hundsdorfer and ruuth monotonicity for time discretizations dundee conference report na 217 2003 university of dundee dundee uk 2003 pp 85 94 higueras j sci comput 21 2004 pp 193 223 and siam j numer anal 43 2005 pp 924 948 spiteri and ruuth siam j numer anal 40 2002 pp 469 491 gottlieb j sci comput 25 2005 pp 105 128 and ferracina and spijker siam j numer anal 42 2004 pp 1073 1093 and math comp 74 2005 pp 201 219 in the present paper we obtain a special stepsize condition guaranteeing the above properties for a generic numerical process this condition is best possible in a well defined and natural sense it is applicable to the important class of general linear methods and it can also be used to answer some open questions for methods of which the above stability properties were studied earlier

an extension and analysis of the shu osher representation of runge kutta methods in the context of solving nonlinear partial differential equations shu and osher introduced representations of explicit runge kutta methods which lead to stepsize conditions under which the numerical process is total variation diminishing tvd much attention has been paid to these representations in the literature in general a shu osher representation of a given runge kutta method is not unique therefore of special importance are representations of a given method which are best possible with regard to the stepsize condition that can be derived from them several basic questions are still open notably regarding the following issues 1 the formulation of a simple and general strategy for finding a best possible shu osher representation for any given runge kutta method 2 the question of whether the tvd property of a given runge kutta method can still be guaranteed when the stepsize condition corresponding to a best possible shu osher representation of the method is violated 3 the generalization of the shu osher approach to general possibly implicit runge kutta methods in this paper we give an extension and analysis of the original shu osher representation by means of which the above questions can be settled moreover we clarify analogous questions regarding properties which are referred to in the literature by the terms monotonicity and strong stability preserving ssp

partially implicit bdf2 blends for convection dominated flows in this paper we consider various blends of implicit and explicit time integration schemes based on the well known bdf2 method applied to convection diffusion problems with dominating convection a fully implicit treatment of convection terms is often not very efficient we shall deal with second order schemes that are implicit in the convection terms only locally in space without introducing the internal inconsistencies that are common with many time splitting methods along with implementation aspects of the implicit relations we shall discuss accuracy of the schemes positivity and monotonicity properties

a new achievable rate region for the interference channel a new achievable rate region for the general interference channel which extends previous results is presented and evaluated the technique used is a generalization of superposition coding to the multivariable case a detailed computation for the gaussian channel case clarifies to what extent the new region improves previous ones the capacity of a class of gaussian interference channels is also established

global optimization of explicit strong stability preserving runge kutta methods strong stability preserving runge kutta ssprk methods are a type of time discretization method that are widely used especially for the time evolution of hyperbolic partial differential equations pdes under a suitable stepsize restriction these methods share a desirable nonlinear stability property with the underlying pde e g positivity or stability with respect to total variation this is of particular interest when the solution exhibits shock like or other nonsmooth behaviour a variety of optimality results have been proven for simple ssprk methods however the scope of these results has been limited to low order methods due to the detailed nature of the proofs in this article global optimization software baron is applied to an appropriate mathematical formulation to obtain optimality results for general explicit ssprk methods up to fifth order and explicit low storage ssprk methods up to fourth order throughout our studies allow for the possibility of negative coefficients which correspond to downwind biased spatial discretizations guarantees of optimality are obtained for a variety of third and fourth order schemes where optimality is impractical to guarantee specifically for fifth order methods and certain low storage methods extensive numerical optimizations are carried out to derive numerically optimal schemes as a part of these studies several new schemes arise which have theoretically improved time stepping restrictions over schemes appearing in the recent literature

model problems in numerical stability theory for initial value problems in the past numerical stability theory for initial value problems in ordinary differential equations has been dominated by the study of problems with essentially trivial dynamics whilst this has resulted in a coherent and self contained body of knowledge it has not thoroughly addressed the problems of real interest in applications recently there have been a number of studies of numerical stability for wider classes of problems admitting more complicated dynamics this on going work is unified and possible directions for future work are outlined in particular striking similarities between this new developing stability theory and the classical non linear stability theory are emphasised the classical theories of a b and algebraic stability for runge kutta methods are briefly reviewed and it is emphasised that the classes of equations to which these theories apply linear decay and contractive problems only admit trivial dynamics four other categories of equations gradient dissipative conservative and hamiltonian systems are considered relationships and differences between the possible dynamics in each category which range from multiple competing equilibria to fully chaotic solutions are highlighted and it is stressed that the wide range of possible behaviour allows a large variety of applications runge kutta schemes which preserve the dynamical structure of the underlying problem are sought and indications of a strong relationship between the developing stability theory for these new categories and the classical existing stability theory for the older problems are given algebraic stability in particular is seen to play a central role the effects of error control are considered and multi step methods are discussed briefly finally various open problems are described

small correlation expansions for the inverse ising problem we present a systematic small correlation expansion to solve the inverse ising problem and find a set of couplings and fields corresponding to a given set of correlations and magnetizations couplings are calculated up to the third order in the correlations for generic magnetizations and to the seventh order in the case of zero magnetizations in addition we show how to sum some useful classes of diagrams exactly the resulting expansion outperforms existing algorithms on the sherrington kirkpatrick spin glass model

the capacity of the gaussian interference channel under strong interference corresp the capacity region of a gaussian interference channel with two separate messages is obtained for the case of moderately strong interference it is shown that the region coincides with the one where both messages are required in both receiving terminals

providing quadratic convergence of decentralized power control in wireless networks the method of min max functions this paper addresses the problem of power control in wireless networks with elastic traffic and no central network controller such as ad hoc networks or hybrid mesh networks we propose a novel power allocation iteration prove its local quadratic convergence and design a feedback handshake scheme for a distributed implementation of the iteration the combined feature of quadratic convergence and amenability to decentralized realization makes the algorithm suitable for efficient online application and incorporation in multihop policies to the best of our knowledge a similarly fast convergence and decentralization are not offered by any known power control algorithm the proposed power control concept is designed and analyzed using the powerful framework of convex concave functions and min max functions

stability of explicit time discretizations for solving initial value problems stability regions of explicit linear time discretization methods for solving initial value problems are treated if an integration method needsm function evaluations per time step then we scale the stability region by dividing bym we show that the scaled stability region of a method satisfying some reasonable conditions cannot be properly contained in the scaled stability region of another method bounds for the size of the stability regions for three different purposes are then given for general nonlinear ordinary differential systems for systems obtained from parabolic problems and for systems obtained from hyperbolic problems we also show how these bounds can be approached by high order methods

on monotonicity and boundedness properties of linear multistep methods in this paper an analysis is provided of nonlinear monotonicity and boundedness properties for linear multistep methods instead of strict monotonicity for arbitrary starting values we shall focus on generalized monotonicity or boundedness with runge kutta starting procedures this allows many multistep methods of practical interest to be included in the theory in a related manner we also consider contractivity and stability in arbitrary norms

computation of optimal monotonicity preserving general linear methods monotonicity preserving numerical methods for ordinary differential equations prevent the growth of propagated errors and preserve convex boundedness properties of the solution we formulate the problem of finding optimal monotonicity preserving general linear methods for linear autonomous equations and propose an efficient algorithm for its solution this algorithm reliably finds optimal methods even among classes involving very high order accuracy and that use many steps and or stages the optimality of some recently proposed methods is verified and many more efficient methods are found we use similar algorithms to find optimal strong stability preserving linear multistep methods of both explicit and implicit type including methods for hyperbolic pdes that use downwind biased operators

stepsize restrictions for stability in the numerical solution of ordinary and partial differential equations abstract this paper deals with the stability analysis of one step methods for the numerical solution of initial value problems both stiff ordinary and partial differential equations are included the problem is considered how to restrict the stepsize in the methods in order that they behave stable we review and extend some recent results on this problem that are based on the use of stability regions in the complex plane we focus on differential equations that are essentially more general than the classical test equation u t u t with a complex constant further the emphasis is on stability of the methods with respect to the maximum norm

staggered time integrators for wave equations we consider variations of the adams bashforth backward differentiation and runge kutta families of time integrators to solve systems of linear wave equations on uniform time staggered grids these methods are found to have smaller local truncation errors and to allow larger stable time steps than traditional nonstaggered versions of equivalent orders we investigate the accuracy and stability of these methods analytically experimentally and through the use of a novel root portrait technique

stability of linear multistep methods for nonlinear neutral delay differential equations in banach space this paper is devoted to investigating the nonlinear stability properties of linear multistep methods for the solution to neutral delay differential equations in banach space two approaches to numerically treating the neutral term are considered which allow us to prove several results on numerical stability of linear multistep methods these results provide some criteria for choosing the step size such that the numerical method is stable some examples of application and a numerical experiment which further confirms the main results are given

absolute monotonicity of rational functions occurring in the numerical solution of initial value problems this paper deals with rational functions o z approximating the exponential function exp z related to numerical procedures for solving initial value problems motivated by positivity and contractivity requirements imposed on these numerical procedures we study the greatest nonnegative numberr denoted byr o such that o is absolutely monotonic on r 0 an algorithm for the computation ofr o is presented application of this algorithm yields the valuer o for the well known pade approximations to exp z for some specific values ofm n andp we determine the maximum ofr o when o varies over the class of all rational functions o with degree of the numerator m degree of the denominator n and o z exp z z p 1 forz 0

explicit nordsieck methods with extended stability regions abstract we describe the construction of explicit nordsieck methods of order p and stage order q p with large regions of absolute stability we also discuss error propagation and estimation of local discretization errors the error estimators are derived for examples of general linear methods constructed in this paper some numerical experiments are presented which illustrate the effectiveness of proposed methods

mmse of bad codes we examine codes over the additive gaussian noise channel designed for reliable communication at some specific signal to noise ratio snr and constrained by the permitted minimum mean square error mmse at lower snrs the maximum possible rate is below point to point capacity and hence these are nonoptimal codes alternatively referred to as bad codes we show that the maximum possible rate is the one attained by superposition codebooks moreover the mmse and mutual information behavior as a function of snr for any code attaining the maximum rate under the mmse constraint is known for all snr we also provide a lower bound on the mmse for finite length codes as a function of the error probability of the code

imex extensions of linear multistep methods with general monotonicity and boundedness properties for solving hyperbolic systems with stiff sources or relaxation terms time stepping methods should combine favorable monotonicity properties for shocks and steep solution gradients with good stability properties for stiff terms in this paper we consider implicit explicit imex multistep methods suitable methods will be constructed based on explicit methods with general monotonicity and boundedness properties for hyperbolic equations numerical comparisons are made with several implicit explicit runge kutta methods

optimized strong stability preserving imex runge kutta methods we construct and analyze robust strong stability preserving implicit explicit runge kutta imex rk methods for models of flow with diffusion as they appear in astrophysics and in many other fields where equations with similar structure arise it turns out that besides the optimization of the region of absolute monotonicity some other properties of the methods are crucial for the success of such simulations in particular the models in our focus dictate to also take into account the step size limits associated with dissipativity positivity of the stiff parabolic terms which represent transport by diffusion the uniform convergence with respect to different stiffness properties of those same terms etc furthermore in the literature some other properties like the inclusion of a part of the imaginary axis in the stability region have been argued to be relevant in this paper we construct several new imex rk methods which differ from each other by taking various or even all of these constraints simultaneously into account it is demonstrated for some simple examples as well as for the problem of double diffusive convection that the newly constructed schemes provide a significant computational advantage over other methods from the literature due to their accumulation of different stability properties the optimized imex rk methods obtained in this paper are robust schemes that may also be useful for general models which involve the solution of advection diffusion equations or other transport equations with similar stability requirements

contractivity preserving explicit linear multistep methods we investigate contractivity properties of explicit linear multistep methods in the numerical solution of ordinary differential equations the emphasis is on the general test equation frac d dt u t au t wherea is a square matrix of arbitrary orders 1 the contractivity is analysed with respect to arbitrary norms in thes dimensional space which are not necessarily generated by an inner product for given order and stepnumber we construct optimal multistep methods allowing the use of a maximal stepsize

capacity bounds for the gaussian interference channel the capacity region of the two user gaussian interference channel ic is studied three classes of channels are considered weak one sided and mixed gaussian ics for the weak gaussian ic a new outer bound on the capacity region is obtained that outperforms previously known outer bounds the sum capacity for a certain range of channel parameters is derived for this range it is proved that using gaussian codebooks and treating interference as noise are optimal it is shown that when gaussian codebooks are used the full han kobayashi achievable rate region can be obtained by using the naive han kobayashi achievable scheme over three frequency bands equivalently three subspaces for the one sided gaussian ic an alternative proof for the sato s outer bound is presented we derive the full han kobayashi achievable rate region when gaussian codebooks are utilized for the mixed gaussian ic a new outer bound is obtained that outperforms previously known outer bounds for this case the sum capacity for the entire range of channel parameters is derived it is proved that the full han kobayashi achievable rate region using gaussian codebooks is equivalent to that of the one sided gaussian ic for a particular range of channel parameters

on positivity shape and norm bound preservation of time stepping methods for semigroups abstract we use functional calculus methods to investigate qualitative properties of c 0 semigroups that are preserved by time discretization methods preservation of positivity concavity and other qualitative shape properties which can be described via positivity are treated in a banach lattice framework preservation of contractivity or norm bound of the semigroup is investigated in the banach space setting the use of the hille phillips h p functional calculus instead of the dunford taylor functional calculus allows us to extend fundamental qualitative results concerning time discretization methods and simplify their proofs including results on multi step schemes and variable step sizes since the h p functional calculus is used throughout the paper we present an elementary introduction to it based on the riemann stieltjes integral

a generalization of nash bargaining and proportional fairness to log convex utility sets with power constraints many solutions and concepts in resource allocation and game theory rely on the assumption of a convex utility set in this paper we show that the less restrictive assumption of a logarithmic hidden convexity is sometimes sufficient we consider the problems of nash bargaining and proportional fairness which are closely related we extend the nash bargaining framework to a broader family of log convex sets we then focus on the set of feasible signal to interference plus noise ratios sinrs for the cases of individual power constraints and a sum power constraint under the assumption of log convex interference functions we show how pareto optimality of boundary points depends on the interference coupling between the users finally we provide necessary and sufficient conditions for strict log convexity of the feasible sinr region

stability and boundedness in the numerical solution of initial value problems abstract this paper concerns the theoretical analysis of step by step meth ods for solving initial value problems in ordinary and partial differential equa tions the main theorem of the paper answers a natural question arising in thelinear stability analysis of such methods it guarantees a strong version ofnumerical stability under a stepsize restriction related to the stability regionof the numerical method and to a circle condition on the differential equation the theorem also settles an open question related to the properties total variation diminishing strong stability preserving monotonic and total variation bounded under a monotonicity condition on the forward eulermethod the theorem specifies a stepsize condition guaranteeing boundednessfor linear problems the main theorem is illustrated by applying it to linear multistep methods for important classes of these methods conclusions are thus obtained whichsupplement earlier results in the literature

a sharpened condition for strict log convexity of the spectral radius via the bipartite graph abstract friedland 1981 10 showed that for a nonnegative square matrix a the spectral radius r e d a is a log convex functional over the real diagonal matrices d he showed that for fully indecomposable a log r e d a is strictly convex over d 1 d 2 if and only if d 1 d 2 c i for any c r here the condition of full indecomposability is shown to be replaceable by the weaker condition that a and a a be irreducible which is the sharpest possible replacement condition irreducibility of both a and a a is shown to be equivalent to irreducibility of a 2 and a a which is the condition for a number of strict inequalities on the spectral radius found in cohen et al 1982 8 such two fold irreducibility is equivalent to joint irreducibility of a a 2 a a and aa or in combinatorial terms equivalent to the directed graph of a being strongly connected and the simple bipartite graph of a being connected additional ancillary results are presented

regions of stability equivalence theorems and courant friedichs lewy condition the celebrated cfl condition for discretizations of hyperbolic pdes is shown to be equivalent to some results of jeltsch and nevanlinna concerning regions of stability ofk step m stage linear methods for the integration of odes we characterize the methods for the numerical integration of the model equation ut ux which are weakly stable when the mesh ratio takes the maximum value allowed by the cfl condition we provide new equivalence theorems between stability and convergence which improve on the classical results

accuracy and stability of multistage multistep formulas we continue our research concerning relations between accuracy and stability of time discretizations here we concentrate rather than on the size and shape of the stability region on the obtainable accuracy when the number of parameters in the formulas is large

unconditional contractivity in the maximum norm of diagonally split runge kutta methods we consider the new class of diagonally split runge kutta dsrk methods 1 5 for the standard initial value problem 1 1 by using the theory of p series we derive the order conditions on the method s coefficients stability analysis is also carried out by showing that the method is unconditionally contractive in the maximum norm if and only if the method is based on particular pairs of discrete and continuous runge kutta methods for an effective implementation of the method an iteration scheme is proposed which preserves the contractivity of any iterate

stepsize restrictions for stability of one step methods in the numerical solution of initial value problems this paper deals with the analysis of general one step methods for the numerical solution of initial boundary value problems for stiff ordinary and partial differential equations restrictions on the stepsize are derived that are necessary and sufficient for the rate of error growth in these methods to be of moderate size these restrictions are related to disks contained in the stability region of the method and the errors are measured with arbitrary norms not necessarily generated by an inner product the theory is illustrated in the numerical solution of a diffusion convection problem where the error growth is measured with the maximum norm

contractivity in the numerical solution of initial value problems consider a linear autonomous system of ordinary differential equations with the property that the norm u t of each solutionu t satisfies u t u 0 t 0 we call a numerical process for solving such a system contractive if a discrete version of this property holds for the numerical approximations a givenk step method is said to be unconditionally contractive if for each stepsizeh 0 the numerical process is contractive in this paper a general theory is given which yields necessary and sufficient conditions for unconditional contractivity it turns out that unconditionally contractive methods are subject to an order barrierp 1 further the concept of a contractivity threshold is studied which makes it possible to compare the contractivity behaviour of methods with an orderp 1 as well most theoretical results in this paper are formulated for differential equations in arbitrary banach spaces applications are given to numerical methods for solving ordinary as well as partial differential equations

an efficient algorithm based on splitting for the time integration of the schrodinger equation we present a practical algorithm based on symplectic splitting methods intended for the numerical integration in time of the schrodinger equation when the hamiltonian operator is either time independent or changes slowly with time in the later case the evolution operator can be effectively approximated in a step by step manner first divide the time integration interval in sufficiently short subintervals and then successively solve a schrodinger equation with a different time independent hamiltonian operator in each of these subintervals when discretized in space the schrodinger equation with the time independent hamiltonian operator obtained for each time subinterval can be recast as a classical linear autonomous hamiltonian system corresponding to a system of coupled harmonic oscillators the particular structure of this linear system allows us to construct a set of highly efficient schemes optimized for different precision requirements and time intervals sharp local error bounds are obtained for the solution of the linear autonomous hamiltonian system considered in each time subinterval our schemes can be considered in this setting as polynomial approximations to the matrix exponential in a similar way as methods based on chebyshev and taylor polynomials the theoretical analysis supported by numerical experiments performed for different time independent hamiltonians indicates that the new methods are more efficient than schemes based on chebyshev polynomials for all tolerances and time interval lengths the algorithm we present automatically selects for each time subinterval the most efficient splitting scheme among several new optimized splitting methods for a prescribed error tolerance and given estimates of the upper and lower bounds of the eigenvalues of the discretized version of the hamiltonian operator

distributed utility based power control objectives and algorithms this paper deals with the problem of medium access control mac layer fair power control in a wireless mesh network with an established network topology the notion of mac layer fairness is defined along similar lines as end to end fairness for elastic traffic except that instead of end to end flows mac layer flows are considered being that one hop flows between neighboring nodes in this paper we identify a class of utility functions of link rates that allows for a convex problem formulation the convexity property is a key prerequisite for implementing power control algorithms in practice we present a novel distributed algorithmic solution to the power control problem based on gradient projection methods prove its global convergence and provide sufficient conditions for a geometric convergence rate the main novelty of our scheme lies in the use of the so called adjoint network in such a way that each transmitter can estimate its current update direction from the received signal power this mitigates the problem of global coordination of the transmitters when carrying out gradient projection algorithms in distributed wireless networks the price for this are possible estimation errors so that the proposed scheme is analyzed within the framework of stochastic approximation

accuracy bounds for semidiscretizations of hyperbolic problems bounds are given for the error constant of stable finite difference methods for first order hyperbolic equations in one space dimension which use r downwind and s upwind points in the discretization of the space derivatives and which are of optimal order p min r s 2 r 2 2 s it is known that this order can be obtained by interpolatory methods examples show however that their error constants can be improved

phase diagram of restricted boltzmann machines and generalized hopfield networks with arbitrary priors restricted boltzmann machines are described by the gibbs measure of a bipartite spin glass which in turn can be seen as a generalized hopfield network this equivalence allows us to characterize the state of these systems in terms of their retrieval capabilities both at low and high load of pure states we study the paramagnetic spin glass and the spin glass retrieval phase transitions as the pattern i e weight distribution and spin i e unit priors vary smoothly from gaussian real variables to boolean discrete variables our analysis shows that the presence of a retrieval phase is robust and not peculiar to the standard hopfield model with boolean patterns the retrieval region becomes larger when the pattern entries and retrieval units get more peaked and conversely when the hidden units acquire a broader prior and therefore have a stronger response to high fields moreover at low load retrieval always exists below some critical temperature for every pattern distribution ranging from the boolean to the gaussian case

constructive characterization ofa stable approximations to exp z and its connection with algebraically stable runge kutta methods all rational approximations to exp z of order 2m m denotes the maximal degree of nominator and denominator are given by a closed formula involving real parameters using the theory of order stars 9 necessary and sufficient conditions fora stability respectivelyi stability are given on the basis of this characterization relations between the concepts ofa stability and algebraic stability for implicit runge kutta methods are investigated in particular we can partly prove the conjecture that to any irreduciblea stabler z of oderp 0 there exist algebraically stable runge kutta methods of the same order withr z as stability function

stability of parallel explicit ode methods general purpose numerical methods for ordinary initial value problems are usually found in the class of linear multistage multivalue methods first formulated by j c butcher among these the explicit methods are easiest to implement for this reason there has been considerable research activity devoted to generating methods of this class which utilize independent function evaluations that can be performed in par allel each such group of concurrent function evaluations can be regarded as a stage of the method it is desirable that methods have a large region of stability so that stability dictated restrictions on the stepsize will occur less frequently in order to do a fair comparison of stability regions one should consider the scaled stability region this is defined using not the nominal stepsize but rather the stepsize per unit of work each stage counts as one work unit in the context of parallel computing each stage may involve any number of independent function evaluations very important results on scaled stability regions were obtained by jeltsch and nevanlinna in the context of serial computation they identify a type of method actually a type of alge braic function whose scaled stability region is optimal this means that the region is not a proper subset of some other scaled stability region the distinguished type of algebraic functions is one that satisfies property c which requires that the principal branch of the algebraic function be analytic outside the stability region except for a pole at infinity whose multiplicity equals the number of work units of the method it is shown that by modifying property c the result on optimal stability regions holds also in the context of parallel computa tion an important consequence is that although the use of parallelism enlarges the class of scaled stability regions it does not result in scaled stability regions that properly contain those obtained without parallelism roughly stated parallelism cannot improve the stability of an explicit linear method but it can even for the standard test problem improve the accuracy

preserving stability implicit euler method for nonlinear volterra and neutral functional differential equations in banach space this paper is concerned with the contractivity and asymptotic stability properties of the implicit euler method iem for nonlinear functional differential equations fdes these properties are first analyzed for volterra fdes and then the analysis is extended to the case of neutral fdes nfdes such an extension is particularly important since nfdes are more general and have received little attention in the literature the main result we establish is that the iem with linear interpolation can completely preserve these stability properties of the analytical solution to such fdes

contractivity preserving implicit linear multistep methods we investigate contractivity properties of implicit linear multistep methods in the numerical solution of ordinary differential equations the emphasis is on nonlinear and linear systems d u t f t u t where f satisfies a so called circle condition in an arbitrary norm the results for the two types of systems turn out to be closely related we construct optimal multistep methods of given order and stepnumber which allow the use of a maximal stepsize

numerical integrators for the hybrid monte carlo method we construct numerical integrators for hamiltonian problems that may advantageously replace the standard verlet time stepper within hybrid monte carlo and related simulations past attempts have often aimed at boosting the order of accuracy of the integrator and or reducing the size of its error constants order and error constant are relevant concepts in the limit of vanishing step length we propose an alternative methodology based on the performance of the integrator when sampling from gaussian distributions with not necessarily small step lengths we construct new splitting formulae that require two three or four force evaluations per time step limited proof of concept numerical experiments suggest that the new integrators may provide an improvement on the efficiency of the standard verlet method especially in problems with high dimensionality

to decode the interference or to consider it as noise in this paper the impact of noncoordinated interfering signals on a point to point communication is addressed while the transmitter has no information about the other users messages the receiver has full knowledge of the codebooks of the interfering users and can potentially decode some part of the interference a simple coding strategy is proposed for this channel assuming its own data is decoded successfully the receiver partitions the set of interfering users into two disjoint subsets namely the set of decodable users and the set of nondecodable users then the transmitter s rate is chosen such that the intended signal can be jointly decoded with the set of decodable users it is proved that the proposed strategy achieves the capacity of the additive gaussian channel with gaussian interfering users a polynomial time algorithm is proposed to compute the achievable rate of the scheme this algorithm is based a subroutine which separates the set of interfering users into decodable and nondecodable users in polynomial time the proposed scheme is also applied to the case of user interference channel and some achievable points are characterized by successive maximization of users rates

stepsize restrictions for the total variation diminishing property in general runge kutta methods much attention has been paid in the literature to total variation diminishing tvd numerical processes in the solution of nonlinear hyperbolic differential equations for special runge kutta methods conditions on the stepsize were derived that are sufficient for the tvd property see e g shu and osher j comput phys 77 1988 pp 439 471 and gottlieb and shu math comp 67 1998 pp 73 85 various basic questions are still open regarding the following issues 1 the extension of the above conditions to more general runge kutta methods 2 simple restrictions on the stepsize which are not only sufficient but at the same time necessary for the tvd property and 3 the determination of optimal runge kutta methods with the tvd property in this paper we propose a theory by means of which we are able to clarify the above questions moreover by applying our theory we settle analogous questions regarding the related strong stability preserving ssp property see e g gottlieb shu and

resummed mean field inference for strongly coupled data we present a resummed mean field approximation for inferring the parameters of an ising or a potts model from empirical noisy one and two point correlation functions based on a resummation of a class of diagrams of the small correlation expansion of the log likelihood the method outperforms standard mean field inference methods even when they are regularized the inference is stable with respect to sampling noise contrarily to previous works based either on the small correlation expansion on the bethe free energy or on the mean field and gaussian models because it is mostly analytic its complexity is still very low requiring an iterative algorithm to solve for n auxiliary variables that resorts only to matrix inversions and multiplications we test our algorithm on the sherrington kirkpatrick model submitted to a random external field and large random couplings and demonstrate that even without regularization the inference is stable across the whole phase diagram in addition the calculation leads to a consistent estimation of the entropy of the data and allows us to sample form the inferred distribution to obtain artificial data that are consistent with the empirical distribution

combinatorial characterization of interference coupling in wireless systems we provide a combinatorial characterization of interference coupling in wireless systems with the intent of obtaining a better insight into interference coordination and management we introduce two bipartite graphs namely the power graph and interference graph we utilize these graphs and global dependency matrix containing only binary 0 and 1 entries to capture the effects of interference coupling in communication systems we show that the irreducibility of the global dependency matrix g is related to the connectivity of the power graph and the irreducibility of the matrix ggt is related to the connectivity of the interference graph we prove that for strictly positive and strictly log convex interference functions the irreducibility of the matrices g and ggt are necessary and sufficient conditions for the considered utility sets to be strictly convex in this case there exists a unique optimizer for the problem of maximizing the product of utilities we show that an interference balancing function is strictly log convex if and only if matrices g and ggt are irreducible we provide a simple yet comprehensive combinatorial characterization of interference coupled systems which abstracts away certain complexities of the physical layer

broken replica symmetry bounds in the mean field spin glass model by using a simple interpolation argument in previous work we have proven the existence of the thermodynamic limit for mean field disordered models including the sherrington kirkpatrick model and the derrida p spin model here we extend this argument in order to compare the limiting free energy with the expression given by the parisi ansatz and including full spontaneous replica symmetry breaking our main result is that the quenched average of the free energy is bounded from below by the value given in the parisi ansatz uniformly in the size of the system moreover the difference between the two expressions is given in the form of a sum rule extending our previous work on the comparison between the true free energy and its replica symmetric sherrington kirkpatrick approximation we give also a variational bound for the infinite volume limit of the ground state energy per site

power control with imperfect exchanges and applications to spectrum sharing in various applications the effect of errors in gradient based iterations is of particular importance when seeking saddle points of the lagrangian function associated with constrained convex optimization problems of particular interest here are problems arising in power control applications where network utility is maximized subject to minimum signal to interference plus noise ratio sinr constraints maximum interference constraints maximum received power constraints or simultaneous minimum and maximum sinr constraints especially when the gradient iterations are executed in a distributed fashion imperfect exchanges among the link nodes may result in erroneous gradient vectors in order to assess and cope with such errors two running averages ergodic sequences are formed from the iterates generated by the perturbed saddle point method each with complementary strengths under the assumptions of problem convexity and error boundedness bounds on the constraint violation and the suboptimality per iteration index are derived the two types of running averages are tested on a spectrum sharing problem with minimum and maximum sinr constraints as well as maximum interference constraints

contractivity of waveform relaxation runge kutta iterations and related limit methods for dissipative systems in the maximum norm contractivity properties of runge kutta methods are analyzed with suitable interpolation implemented using waveform relaxation strategy for systems of ordinary differential equations that are dissipative in the maximum norm in general this type of implementation which is quite appropriate in a parallel computing environment improves the stability properties of runge kutta methods as a result of this analysis a new class of methods is determined which is different from runge kutta methods but closely related to them and which combines its high order of accuracy and unconditional contractivity in the maximum norm this is not possible for classical runge kutta methods

stability and accuracy of time discretizations for initial value problems this paper continues earlier work by the same authors concerning the shape and size of the stability regions of general linear discretization methods for initial value problems here the treatment is extended to cover also implicit schemes and by placing the accuracy of the schemes into a more central position in the discussion general method free statements are again obtained more specialized results are additionally given for linear multistep methods and for the taylor series method

mathematical model for studying the sulphur pollution over europe the long range transport of sulphur pollutants in the atmosphere is an important environmental problem a mathematical model consisting of a system of partial differential equations has been developed at the danish agency of environmental protection for studying this phenomenon the model is handled numerically a pseudospectral fourier algorithm is used in the space discretization phase a variable stepsize variable formula method vsvfm based on predictor corrector schemes may be specified during the time discretization part the time integration formulae are especially constructed so that a the predictor corrector schemes have good absolute stability properties on the imaginary axis and b the fundamental properties consistency zero stability and convergence of the global vsvfm are not affected by the changes of the stepsize and or the predictor corrector scheme both the accuracy requirements and the stability requirements are taken into account in the stepsize selection strategy as well as in the formula selection strategy this double check leads to a great efficiency in the numerical treatment of problems arising in atmospheric environments this is demonstrated by many numerical results the question of the reliability of the results obtained in runs of real physical problems is discussed some results obtained in the comparison of calculated results with measurements are reported an illustration where the model is applied in studying the air pollution in the area around denmark due to danish sources is presented

high order linear multistep methods with general monotonicity and boundedness properties we consider linear multistep methods that possess general monotonicity and boundedness properties strict monotonicity in terms of arbitrary starting values for the multistep schemes is only valid for a small class of methods under very stringent step size restrictions this makes them uncompetitive with the strong stability preserving ssp runge kutta methods by relaxing these strict monotonicity requirements a larger class of methods can be considered including many methods of practical interest in this paper we construct linear multistep methods of high order up to six that possess relaxed monotonicity or boundedness properties with optimal step size conditions numerical experiments show that the new schemes perform much better than the classical monotonicity preserving multistep schemes moreover there is a substantial gain in efficiency compared to recently constructed ssp runge kutta ssprk methods

outage minimization and rate allocation for the multiuser gaussian interference channels with successive group decoding we consider a memoryless gaussian interference channel gic where k single antenna users communicate with their respective receivers using gaussian codebooks each receiver employs a successive group decoder with a specified complexity constraint to decode its designated user it is aware of the coding schemes employed by all other users and may choose to decode some or all of them only if it deems that doing so will aid the decoding of its desired user for a gic with predetermined rates for all transmitters we obtain the minimum outage probability decoding strategy at each receiver which satisfies the imposed complexity constraint and reveals the optimal subset of interferers that must be decoded along with the desired user we then consider the rate allocation problem over the gic under successive group decoding and design a sequential rate allocation algorithm which yields a pareto optimal rate allocation and two parallel rate allocation algorithms which yield the symmetric fair rate allocation and the max min fair rate allocation respectively remarkably even though the proposed decoding and rate allocation algorithms use ldquogreedyrdquo or myopic subroutines they achieve globally optimal solutions finally we also propose rate allocation algorithms for a cognitive radio system

a step size selection strategy for explicit runge kutta methods based on lyapunov exponent theory using a stability result for variable time step runge kutta methods applied to nonautonomous real linear scalar test problem that decays exponentially fast a step size selection algorithm is devised the step size selection algorithm is based partly on stability information obtained by estimating the discrete lyapunov exponent of a runge kutta method applied to a nonautonomous linear scalar problem the utility of the approach is illustrated in numerical experiments that demonstrate how the new algorithm performs against a standard accuracy based step size selection strategy

numerical integration of positive linear differential algebraic systems in the simulation of dynamical processes in economy social sciences biology or chemistry the analyzed values often represent non negative quantities like the amount of goods or individuals or the density of a chemical or biological species such systems are typically described by positive ordinary differential equations odes that have a non negative solution for every non negative initial value besides positivity these processes often are subject to algebraic constraints that result from conservation laws limitation of resources or balance conditions and thus the models are differential algebraic equations daes in this work we present conditions under which both these properties the positivity as well as the algebraic constraints are preserved in the numerical simulation by runge kutta or multistep discretization methods using a decomposition approach we separate the dynamic and the algebraic equations of a given linear positive dae to give positivity preserving conditions for each part separately for the dynamic part we generalize the results for positive odes to daes using the solution representation via drazin inverses for the algebraic part we use the consistency conditions of the discretization method to derive conditions under which this part of the approximation overestimates the exact solution and thus is non negative we analyze these conditions for some common runge kutta and multistep methods and observe that for index 1 systems and stiffly accurate runge kutta methods positivity is conditionally preserved under similar conditions as for odes for higher index problems however none of the common methods is suitable

a combinatorial strongly polynomial algorithm for minimizing submodular functions this paper presents a combinatorial polynomial time algorithm for minimizing submodular functions answering an open question posed in 1981 by grotschel lovasz and schrijver the algorithm employs a scaling scheme that uses a flow in the complete directed graph on the underlying set with each arc capacity equal to the scaled parameter the resulting algorithm runs in time bounded by a polynomial in the size of the underlying set and the length of the largest absolute function value the paper also presents a strongly polynomial version in which the number of steps is bounded by a polynomial in the size of the underlying set independent of the function values

order stars and stiff integrators order stars introduced in g wanner e hairer s p norsett order stars and stability theorems bit 18 1978 475 489 have become a fundamental tool for the understanding of order and stability properties of numerical methods for stiff differential equations this survey retraces their discovery and their principal achievements we also sketch some later extensions and describe some recent developments

absolute monotonicity of polynomials occurring in the numerical solution of initial value problems this paper deals with polynomial approximations o x to the exponential function exp x related to numerical procedures for solving initial value problems motivated by positivity and contractivity requirements imposed on these numerical procedures we study the smallest negative argument denoted by r o at which o is absolutely monotonic for given integersp 1 m 1 we determine the maximum ofr o when o varies over the class of all polynomials of a degree m with phi left x right exp left x right mathcal o left x p 1 right left for x to 0 right forx 0

boundedness and strong stability of runge kutta methods in the literature much attention has been paid to runge kutta methods rkms satisfying special nonlinear stability requirements indicated by the terms total variation diminishing tvd strong stability preserving ssp and monotonicity stepsize conditions guaranteeing these properties were derived by shu osher j comput phys 77 1988 pp 439 471 and in numerous subsequent papers these special stability requirements imply essential boundedness properties for the numerical methods among which the property of being total variation bounded unfortunately for many rkms the above special requirements are violated so that one cannot conclude in this way that the methods are total variation bounded in this paper we study stepsize conditions for boundedness directly rather than via the detour of the above special stability properties we focus on stepsize conditions which are optimal in that they are not unnecessarily restrictive we nd that in situations where the special stability properties mentioned above are violated boundedness can be present only within a class of very special rkms as a byproduct our analysis sheds new light on the known theory of monotonicity for rkms we obtain separate results for internal and external monotonicity as well as a new proof of the fundamental relation between monotonicity and kraaijevanger s coecient this proof distinguishes itself from older ones in that it is shorter and more transparent while it requires more simple assumptions on the rkms under consideration

two results concerning the stability of staggered multistep methods staggered spatial node layouts are widely used in the context of solving wave equations when this idea of staggering is extended to time integrators for wave equations one finds methods that offer both better accuracy and less restrictive stability conditions for an equivalent computational cost here we prove two results concerning staggered multistep methods we first prove that staggered backwards differentiation methods are unstable for all orders p ge 5 we then extend dahlquist s first stability barrier to explicit staggered multistep methods