a new mixed finite element for the stokes and elasticity problems the stokes problem is approximated by a mixed finite element method using a new finite el ement which has properties analogous to the finite volume methods namely the local conservation of the momentum and the mass estimates of optimal order are derived for the errors in the velocity the pressure and the gradient of the velocity this new finite element also works for the elasticity problem and all estimates are valid uniformly with respect to the compressibility finally some numerical results for the incompressible navier stokes equations are presented

matrix sum of squares relaxations for robust semi definite programs we consider robust semi definite programs which depend polynomially or rationally on some uncertain parameter that is only known to be contained in a set with a polynomial matrix inequality description on the basis of matrix sum of squares decompositions we suggest a systematic procedure to construct a family of linear matrix inequality relaxations for computing upper bounds on the optimal value of the corresponding robust counterpart with a novel matrix version of putinar s sum of squares representation for positive polynomials on compact semi algebraic sets we prove asymptotic exactness of the relaxation family under a suitable constraint qualification if the uncertainty region is a compact polytope we provide a new duality proof for the validity of putinar s constraint qualification with an a priori degree bound on the polynomial certificates finally we point out the consequences of our results for constructing relaxations based on the so called full block s procedure which allows to apply recently developed tests in order to computationally verify the exactness of possibly small sized relaxations

a branched chain approach to fractal time abstract a branched chain description of fractal time is suggested within its framework the long time tails are the result of a stochastic renormalization procedure the jumps are grouped into blocks of random size the blocks of blocks etc the size of a block being described by a branching chain process the block formation ends up after a random number of steps according to a scale invariant probability law the general scaled form of a multi state waiting time density function is derived as expected this function has a long tail for large times the shlesinger hughes time scaling physica a 109 1981 597 is recovered as a particular case of our approach the fluctuation analysis outlines an essential feature of the theory due to jump clustering the correlation functions have also long tails however unlike the case of true chain reactions the system has the tendency to smooth the fluctuations

semidefinite programming relaxations for semialgebraic problems a hierarchy of convex relaxations for semialgebraic problems is introduced for questions reducible to a finite number of polynomial equalities and inequalities it is shown how to construct a complete family of polynomially sized semidefinite programming conditions that prove infeasibility the main tools employed are a semidefinite programming formulation of the sum of squares decomposition for multivariate polynomials and some results from real algebraic geometry the techniques provide a constructive approach for finding bounded degree solutions to the positivstellensatz and are illustrated with examples from diverse application fields

a random walk approach to spiral motion the paper deals with random spiral motion described in polar coordinates the trajectory of a particle is described by the distance r from a given point as a function of the angle theta relative to a given direction a stochastic description of the spiral shapes generated by the random variations of r and theta is suggested based on the following assumptions a the ray r is made up of additive contributions corresponding to certain jump processes b the angle phi between two jumps is a random variable selected from a given probability law g phi d phi with finite or infinite moments c the contributions rho 1 rho 2 of the different jumps to the ray r are independent random variables selected from a probability law p rho d rho with finite moments an expression for the generating functional of a random spiral r theta is given in the form of an infinite series which can be used to evaluate the stochastic properties of the ray r the asymptotic properties of the random spiral shapes depend on the function g phi d phi if all moments of the angle between two jumps exist and are finite the average shape is a linear archimedean spiral r theta approximately theta as theta to infinity and the dispersion of the ray increases linearly with the angle delta r2 theta approximately theta as theta to infinity if g phi has a long tail g phi approximately phi 1 h as phi to infinity with 1 h 0 the average shape is a nonlinear fractal archimedean spiral r theta approximately theta h as phi to infinity and the fluctuations of the ray have an intermittent behaviour delta r2 theta approximately theta 2h as phi to infinity a complete analysis is possible in a markovian like case for which the angle between two jumps is exponentially distributed in this case a closed expression tor the generating functional is available and all cumulants of the ray can be computed exactly the mth cumulant which expresses the correlations among the rays at different angles theta 1 theta m is proportional to the minimum angle min theta 1 theta m and the intermittent behaviour is missing an alternative stochastic description is suggested based on the assumption that the number of jumps in a given angle interval is distributed according to an inhomogeneous poisson law this model is also analytically tractable it is less restrictive in the sense that the average shape corresponds to a broader class of spirals including the linear and the fractal archimedean and the logarithmic ones however it cannot be used to describe the intermittent behaviour

polynomially parameter dependent lyapunov functions for robust stability of polytopic systems an lmi approach in this note robust stability of state space models with respect to real parametric uncertainty is considered specifically a new class of parameter dependent quadratic lyapunov functions for establishing stability of a polytope of matrices is introduced i e the homogeneous polynomially parameter dependent quadratic lyapunov functions hpd qlfs the choice of this class which contains parameter dependent quadratic lyapunov functions whose dependence on the uncertain parameters is expressed as a polynomial homogeneous form is motivated by the property that a polytope of matrices is stable if and only there exists an hpd qlf the main result of the note is a sufficient condition for determining the sought hpd qlf which amounts to solving linear matrix inequalities lmis derived via the complete square matricial representation csmr of homogeneous matricial forms and the lyapunov matrix equation numerical examples are provided to demonstrate the effectiveness of the proposed approach

a discontinuous petrov galerkin method with lagrangian multipliers for second order elliptic problems we present a discontinuous petrov galerkin dpg method for the finite element discretization of second order elliptic boundary value problems the novel approach emanates from a one element weak formulation of the differential problem this procedure which is typical of discontinuous galerkin dg methods is based on introducing variables defined in the interior and on the boundary of the element the interface variables are suitable lagrangian multipliers that enforce interelement continuity of the solution and of its normal derivative thus providing the proper connection between neighboring elements the internal variables can be eliminated in favor of the interface variables using static condensation to end up with a system of reduced size in the sole lagrangian multipliers a stability and convergence analysis of the novel formulation is carried out and its connection with mixed hybrid and dg methods is explored numerical tests on several benchmark problems are included to validate the convergence performance and the flux conservation properties of the dpg method

the scenario approach to robust control design this paper proposes a new probabilistic solution framework for robust control analysis and synthesis problems that can be expressed in the form of minimization of a linear objective subject to convex constraints parameterized by uncertainty terms this includes the wide class of np hard control problems representable by means of parameter dependent linear matrix inequalities lmis it is shown in this paper that by appropriate sampling of the constraints one obtains a standard convex optimization problem the scenario problem whose solution is approximately feasible for the original usually infinite set of constraints i e the measure of the set of original constraints that are violated by the scenario solution rapidly decreases to zero as the number of samples is increased we provide an explicit and efficient bound on the number of samples required to attain a priori specified levels of probabilistic guarantee of robustness a rich family of control problems which are in general hard to solve in a deterministically robust sense is therefore amenable to polynomial time solution if robustness is intended in the proposed risk adjusted sense

generating functional approach to space and time dependent colored noise a stochastic model for time and space dependent colored noise is suggested based on the following assumptions 1 the noise field is generated by a random number of point events corresponding to a correlated point process 2 the contributions of the different events to the noise field are additive each contribution being a random function selected from the same probability density functional an analytical treatment of the noise field described by these assumptions is possible the generating functional of the noise field as well as the corresponding cumulants can be computed exactly all cumulants are explicitly evaluated when the contribution of an event is given by a diffusion equation a detailed analysis of the asymptotic behavior is made for time homogeneous and translationally invariant processes a gaussian random field colored in space and time emerges for very frequent independent events of very small intensities provided that the central moments of the contribution of an event to the noise field are finite and fast decreasing in space the correlations among events lead to corrections of the gauss limit law however the cumulants of the random fields are also finite if the central moments of an event are slowly decreasing in space a different type of asymptotic behavior occurs the cumulants of the noise field become infinite and the resulting field is described by a non gaussian stable law of the l evy type the theory may be applied to the study of stochastic gravitational fluctuations in galactic systems to the analysis of concentration fluctuations for diffusion processes in disordered systems and for the analysis of the influence of environmental fluctuations in continuum mechanics and electrodynamics

long range anticorrelations and non gaussian behavior of the heartbeat we find that the successive increments in the cardiac beat to beat interals of healthy subjects display scale invariant long range anticorrelations up to 10 4 heart beats furthermore we find that the histogram for the heartbeat intervals increments is well described by a l evy stable distribution for a group of subjects with severe heart disease we find that the distribution is unchanged but the long range correlations vanish therefore the different scaling behavior in health and disease must relate to the underlying dynamics of the heartbeat

finite volume methods for two dimensional incompressible flows with complex boundaries abstract the paper discusses possible ways of developing accurate and efficient finite volume methods for complex incompressible flows the issues of boundary fitted grid generation choice of coupled or uncoupled procedures cartesian or grid aligned velocity components staggered or non staggered variable arrangement discretization methods solution algorithms and convergence acceleration by multigrid methods are addressed an example of a finite volume method for general two dimensional flows is described in some detail and applications of this method to a variety of flows are presented

robust convex optimization we study convex optimization problems for which the data is not specified exactly and it is only known to belong to a given uncertainty set u yet the constraints must hold for all possible values of the data from u the ensuing optimization problem is called robust optimization in this paper we lay the foundation of robust convex optimization in the main part of the paper we show that if u is an ellipsoidal uncertainty set then for some of the most important generic convex optimization problems linear programming quadratically constrained programming semidefinite programming and others the corresponding robust convex program is either exactly or approximately a tractable problem which lends itself to efficientalgorithms such as polynomial time interior point methods

a region dividing technique for constructing the sum of squares approximations to robust semidefinite programs in this technical note we present a novel approach to robust semidefinite programs of which coefficient matrices depend polynomially on uncertain parameters the approach is based on approximation with the sum of squares polynomials but in contrast to the conventional sum of squares approach the quality of approximation is improved by dividing the parameter region into several subregions the optimal value of the approximate problem converges to that of the original problem as the resolution of the division becomes finer an advantage of this approach is that an upper bound on the approximation error can be explicitly obtained in terms of the resolution of the division a numerical example on polynomial optimization is presented to show usefulness of the present approach

robust solutions to uncertain semidefinite programs in this paper we consider semidefinite programs sdps whose data depend on some unknown but bounded perturbation parameters we seek robust solutions to such programs that is solutions which minimize the worst case objective while satisfying the constraints for every possible value of parameters within the given bounds assuming the data matrices are rational functions of the perturbation parameters we show how to formulate sufficient conditions for a robust solution to exist as sdps when the perturbation is full our conditions are necessary and sufficient in this case we provide sufficient conditions which guarantee that the robust solution is unique and continuous holder stable with respect to the unperturbed problem s data the approach can thus be used to regularize ill conditioned sdps we illustrate our results with examples taken from linear programming maximum norm minimization polynomial interpolation and integer programming

on the gap between positive polynomials and sos of polynomials this note investigates the gap existing between positive polynomials and sum of squares sos of polynomials which affects several analysis and synthesis tools in control systems based on polynomial sos relaxations and about which almost nothing is known in particular a matrix characterization of the pns that is the positive homogeneous forms that are not sos is proposed which allows to show that any pns is the vertex of an unbounded cone of pns moreover a complete parametrization of the set of pns is introduced

on two dimensional fractional brownian motion and fractional brownian random field as a generalization of one dimensional fractional brownian motion 1dfbm we introduce a class of two dimensional self similar strongly correlated random walks whose variance scales with power law we report analytical results on the statistical size and shape and segment distribution of its trajectory in the limit of large n the relevance of these results to polymer theory is discussed we also study the basic properties of a second generalization of 1dfbm the two dimensional fractional brownian random field 2dfbrf it is shown that the product of two 1dfbms is the only 2dfbrf which satisfies the self similarity defined by sinai

a dual mixed hybrid formulation for fluid mechanics problems mathematical analysis and application to semiconductor process technology in this paper we propose a dual mixed hybrid formulation capable of treating under a unified framework both compressible and incompressible problems in continuum mechanics a theoretical analysis of the method is carried out and optimal error estimates are derived for both mixed and hybrid variables the potentialities of the novel approach are exploited in the computation of the stress field required in the simulation of the thermal oxidation process in semiconductor technology the numerical formulation is validated both on model problems in continuum mechanics and on a realistic example of the thermal oxidation process in a local oxidation structure

global optimization with polynomials and the problem of moments we consider the problem of finding the unconstrained global minimum of a real valued polynomial p x mathbb r n to mathbb r as well as the global minimum of p x in a compact set k defined by polynomial inequalities it is shown that this problem reduces to solving an often finite sequence of convex linear matrix inequality lmi problems a notion of karush kuhn tucker polynomials is introduced in a global optimality condition some illustrative examples are provided

a new family of mixed finite elements in ir 3 we introduce two families of mixed finite element on conforming inh div and one conforming inh curl these finite elements can be used to approximate the stokes system

evolution towards ergodic behavior of stationary fractal random processes with memory application to the study of long range correlations of nucleotide sequences in dna abstract the possible occurrence of ergodic behavior for large times is investigated in the case of stationary random processes with memory it is shown that for finite times the time average of a state function is generally a random variable and thus two types of cumulants can be introduced for the time average and for the statistical ensemble respectively in the limit of infinite time a transition from the random to the deterministic behavior of the time average may occur resulting in an ergodic behavior the conditions of occurrence of this transition are investigated by analyzing the scaling behavior of the cumulants of the time average a general approach for the computation of these cumulants is developed explicit computations are presented both for short and long memory in the particular case of separable stationary processes for which the cumulants of a statistical ensemble can be factorized into products of functions depending on binary time differences in both cases the ergodic behavior emerges for large times provided that the cumulants of a statistical ensemble decrease to zero as the time differences increase to infinity the analysis leads to the surprising conclusion that the scaling behavior of the cumulants of the time average is relatively insensitive to the type of memory considered both for short and long memory the cumulants of the time average obey inverse different from zero for large time differences then the time averaage is random even as the length of the total time interval tends to infinity and the ergodic behavior no longer holds the theory is applied to the study of long range correlations of nucleotide sequences in dna in this case the length t of a sequence of nucleotides plays the role of the time variable a proportionality relationship is established between the cumulants of the pyrimidine excess in a sequence of length t and the cumulants of the time length average of the probability of occurrence of a pyrimidine it is shown that the statistical analysis of the dna data presented in the literature is consistent with the occurrence of the ergodic behavior for large lengths the implications of the approach to the analysis of the large time behavior of stochastic cellular automata and of fractional brownian motion are also investigated

a new mixed formulation for elasticity we propose a new mixed variational formulation for the equations of linear elasticity it does not require symmetric tensors and consequently is easy to discretize by adapting mixed finite elements developed for scalar second order elliptic equations

analogs of renormalization group transformations in random processes abstract we review the properties of a real space renormalization group transformation of the free energy including the existence of oscillatory terms multiplying the non analytic part of the free energy we then construct stochastic processes which incorporate into probability distributions the features of the free energy scaling equation the essential information is obtainable from the scaling equation and a direct solution for a probability is not necessary these random processes are shown to be generated directly from cantor sets in a spatial representation the ensuing random process exhibits a transition between gaussian and fractal behavior in the fractal regime the trajectories will in an average sense form self similar clusters in a temporal representation the random process exhibits a transition between an asymptotically constant renewal rate and fractal behavior the fractal regime represents a frozen state with only transient effects allowed and is related to charge transport in glasses

computation of an optimal communication schedule in a nonlinear networked control system using sum of squares abstract in this paper we consider the design of a control optimal communication schedule in nonlinear networked control with nonlinear affine plant systems we focus here on periodic communication schedules in control systems with time triggered control communication and assume the existence of predesigned controllers which need to be optimally integrated via the control communication system for this we formulate a quadratic cost function for the nonlinear affine plant the system and the cost are first discretized using an exact mixed discretization approach which allows to lift the problem from the continuous discrete domain into a purely discrete time representation periodicity due to the fixed communication schedule is dealt with the discrete lifting technique to allow for the computation of the cost nonlinearities are approximated in terms of a taylor series which allows the use of the sum of squares framework once a cost function of the networked control system for a given fixed communication schedule is defined it is then possible to compare and choose those sequences that minimize this cost for optimal controller integration

on tractable approximations of uncertain linear matrix inequalities affected by interval uncertainty we present efficiently verifiable sufficient conditions for the validity of specific np hard semi infinite systems of linear matrix inequalities lmis arising from lmis with uncertain data and demonstrate that these conditions are tight up to an absolute constant factor in particular we prove that given an n times n interval matrix cal u rho a mid a ij a ij leq rho c ij one can build a computable lower bound accurate within the factor pi over 2 on the supremum of those rho for which all instances of cal u rho share a common quadratic lyapunov function we then obtain a similar result for the problem of quadratic lyapunov stability synthesis finally we apply our techniques to the problem of maximizing a homogeneous polynomial of degree 3 over the unit cube

a renormalization approach to a class of exponential random processes with application to the bronchial tree abstract a new stochastic renormalization approach for multi step decay phenomena is developed a simple physical interpretation of the renormalization method is suggested it consists in grouping successions of variable numbers of decay events into blocks the law of probability multiplication leads to an exponential random process y yx1 x2 0 where both the exponents x1 x2 and the basis y0 are random variables the physically consistent solution of the model corresponds to a super strong renormalization regime the dependence of the moments of the decay parameter on the number of decay steps q can be exactly determined it consists in a linear superposition of inverse power laws in q modulated by periodic functions in in q having different periods the theory is applied to the renormalization of the bronchial tree our computation shows that the lung structure is very tolerant to fluctuations this result supports the mechanism of morphogenesis of fractal biological organs suggested by west ann biomed eng 18 1990 135 the possibilities of application to the scattering phenomena in one dimensional disordered media are also investigated