conewise linear systems non zenoness and observability conewise linear systems are dynamical systems in which the state space is partitioned into a finite number of nonoverlapping polyhedral cones on each of which the dynamics of the system is described by a linear differential equation this class of dynamical systems represents a large number of piecewise linear systems most notably linear complementarity systems with the p property and their generalizations to affine variational systems which have many applications in engineering systems and dynamic optimization the challenges of dealing with this type of hybrid system are due to two major characteristics mode switchings are triggered by state evolution and states are constrained in each mode in this paper we first establish the absence of zeno states in such a system based on this fundamental result we then investigate and relate several state observability notions short time and t time or finite time local global observability for the short time observability notions constructive finitely verifiable algebraic both sufficient and necessary conditions are derived due to their long time mode transitional behavior which is very difficult to predict only partial results are obtained for the t time observable states nevertheless we completely resolve the t time local observability for the bimodal conewise linear system for finite t and provide numerical examples to illustrate the difficulty associated with the long time observability

shift in the velocity of a front due to a cutoff we consider the effect of a small cutoff ensuremath varepsilon on the velocity of a traveling wave in one dimension simulations done over more than ten orders of magnitude as well as a simple theoretical argument indicate that the effect of the cutoff ensuremath varepsilon is to select a single velocity that converges when ensuremath varepsilon ensuremath rightarrow 0 to the one predicted by the marginal stability argument for small ensuremath varepsilon the shift in velocity has the form k mathrm ln ensuremath varepsilon ensuremath 2 and our prediction for the constant k agrees very well with the results of our simulations a very similar logarithmic shift appears in more complicated situations in particular in finite size effects of some microscopic stochastic systems our theoretical approach can also be extended to give a simple way of deriving the shift in position due to initial conditions in the fisher kolmogorov or similar equations

tracing the equilibrium path beyond compound critical points a unified presentation of some popular continuation procedures used in the non linear finite element analysis of structural mechanics is introduced an extension of the elliptical constraint equation proposed by crisfield is given it is shown that in the proposed procedure real roots can always be obtained in solving the iterative change of the load parameter updated weighting factors are introduced in the constraint equation in order to get better convergence characteristics in the case when localized deformations occur for bifurcation points a modification of rheinboldt s branching procedure is presented post critical response after limit and bifurcation points is determined in some numerical examples

stability analysis and control of rigid body systems with impacts and friction many critical tasks in robotics such as locomotion or manipulation involve collisions between a rigid body and the environment or between multiple bodies methods based on sums of squares sos for numerical computation of lyapunov certificates are a powerful tool for analyzing the stability of continuous nonlinear systems and can additionally be used to automatically synthesize stabilizing feedback controllers here we present a method for applying sums of squares verification to rigid bodies with coulomb friction undergoing discontinuous inelastic impact events the proposed algorithm explicitly generates lyapunov certificates for stability positive invariance and safety over admissible non penetrating states and contact forces we leverage the complementarity formulation of contact which naturally generates the semialgebraic constraints that define this admissible region the approach is demonstrated on multiple robotics examples including simple models of a walking robot a perching aircraft and control design of a balancing robot

optimization of lyapunov invariants in verification of software systems the paper proposes a control theoretic framework for verification of numerical software systems and puts forward software verification as an important application of control and systems theory the idea is to transfer lyapunov functions and the associated computational techniques from control systems analysis and convex optimization to verification of various software safety and performance specifications these include but are not limited to absence of overflow absence of division by zero termination in finite time absence of dead code and certain user specified assertions central to this framework are lyapunov invariants these are properly constructed functions of the program variables and satisfy certain properties analogous to those of lyapunov functions along the execution trace the search for the invariants can be formulated as a convex optimization problem if the associated optimization problem is feasible the result is a certificate for the specification

a natural classification of vibration modes of polygonal ducts based on group theoretic analysis in this work the natural frequencies and mode shapes of polygonal ducts are obtained by a finite element analysis via a group theoretic approach we show that the group theoretic techniques provide a natural setting for the solution of the problem via optimum utilization of the symmetry in the problem the simplifications and insights due to the inherent symmetry is intuitively obvious in simple symmetric structures however a systematic exploitation of this symmetry in more complicated structures is less obvious and is possible perhaps only via useful theorems from group representation theory and the associated projection operator theory we illustrate all the main steps with the help of an example of dihedral groups this example is directly applicable to the problem addressed in this paper the group theoretic approach splits up the original problem into independent subproblems and thereby affects significant computational savings more importantly the approach captures the role of symmetry in the problem and provides insights which are otherwise not obvious the multiplicity of the natural frequencies arising out of the inherent symmetry is determined a priori and neatly separated into different subproblems this suggests a neat physical classification of the mode shapes on a symmetry basis which is unique to this approach the values of the natural frequencies and the mode shapes for triangular square and pentagonal ducts are shown to be in good agreement with the existing results reported in the literature the classification of mode shapes on this symmetry basis is used in the analysis of the polygonal ducts with number of sides varying up to 16

criteria for copositive matrices abstract some finite criteria for copositive copositive plus and strictly copositive matrices are proposed and compared with existing determinantal tests the basic mathematical tool is principal pivoting

strongly regular differential variational systems a differential variational system is defined by an ordinary differential equation ode parameterized by an algebraic variable that is required to be a solution of a finite dimensional variational inequality containing the state variable of the system this paper addresses two system theoretic topics for such a nontraditional nonsmooth dynamical system namely non zenoness and local observability of a given state satisfying a blanket strong regularity condition for the former topic which is of contemporary interest in the study of hybrid systems we extend the results in our previous paper where we have studied zeno states and switching times in a linear complementarity system lcs as a special case of the differential variational inequality dvi the lcs consists of a linear time invariant ode and a linear complementarity problem the extension to a nonlinear complementarity system ncs with analytic inputs turns out to be non trivial as we need to use the lie derivatives of analytic functions in order to arrive at an expansion of the solution trajectory near a given state further extension to a differential variational inequality is obtained via its equivalent karush kuhn tucker formulation for the second topic which is classical in system theory we use the non zenoness result and the recent results in a previous paper pertaining to the b differentiability of the solution operator of a nonsmooth ode to obtain a sufficient condition for the short time local observability of a given strongly regular state of an ncs refined sufficient conditions and necessary conditions for local observability of the lcs satisfying the p property are obtained

positive invariance of constrained affine dynamics and its applications to hybrid systems and safety verification motivated by long time dynamic analysis of hybrid systems and safety verification problems this paper addresses fundamental positive invariance issues of an affine dynamical system on a general polyhedron and their applications necessary and sufficient algebraic conditions are established for the existence of a positively invariant set of an affine system on a polyhedron using the tools of lexicographic relation and long time oscillatory dynamic analysis a linear program based algorithm is proposed to verify these conditions and its computational complexity is analyzed the positive invariance results are applied to obtain an explicit characterization of global switching behaviors of piecewise affine systems further the positive invariance techniques developed in this paper are exploited to show the decidability of safety verification of a class of affine dynamics on semialgebraic sets

differential variational inequalities this paper introduces and studies the class of differential variational inequalities dvis in a finite dimensional euclidean space the dvi provides a powerful modeling paradigm for many applied problems in which dynamics inequalities and discontinuities are present examples of such problems include constrained time dependent physical systems with unilateral constraints differential nash games and hybrid engineering systems with variable structures the dvi unifies several mathematical problem classes that include ordinary differential equations odes with smooth and discontinuous right hand sides differential algebraic equations daes dynamic complementarity systems and evolutionary variational inequalities conditions are presented under which the dvi can be converted either locally or globally to an equivalent ode with a lipschitz continuous right hand function for dvis that cannot be so converted we consider their numerical resolution via an euler time stepping procedure which involves the solution of a sequence of finite dimensional variational inequalities borrowing results from differential inclusions dis with upper semicontinuous closed and convex valued multifunctions we establish the convergence of such a procedure for solving initial value dvis we also present a class of dvis for which the theory of dis is not directly applicable and yet similar convergence can be established finally we extend the method to a boundary value dvi and provide conditions for the convergence of the method the results in this paper pertain exclusively to systems with index not exceeding two and which have absolutely continuous solutions

a group theoretic approach to the linear free vibration analysis of shells with dihedral symmetry this paper deals with a group theoretic approach to the finite element analysis of linear free vibrations of shells with dihedral symmetry examples of such shell structures are cylindrical shells conical shells shells with circumferential stiffeners corrugated shells spherical shells etc the group theoretic approach is used to exploit the inherent symmetry in the problem for vibration analysis the group theoretic results give the correct symmetry adapted basis for the displacement field the stiffness matrix k and the mass matrix m are identically block diagonalized in this basis the generalized linear eigenvalue problem of free vibration gets split into independent subproblems due to this block diagonalization the simo element is used in the finite element formulation of the shell equilibrium equations numerical results for natural frequencies and natural modes of vibration of several dihedral shell structures are presented the results are shown to be in very good agreement with those reported in the literature the computational advantages and physical insights due to the group theoretic approach are also discussed

a group theoretic approach to the global bifurcation analysis of an axially compressed cylindrical shell abstract the accurate prediction of the buckling load of thin shell structures is an important yet elusive goal it is particularly important in the aerospace industry where thin shell members are commonly used as structural elements due to a lack of adequate analytical results current practices in industry put heavy reliance on experimental testing and empirical data to supplement theoretical analysis see d bushnell computerized buckling analysis of shells martinus nijhoff dordrecht 1985 this paper focuses on recent results of the group theoretic approach to a numerical global postbuckling analysis of a perfect axially compressed cylindrical shell with built in end conditions the built in end conditions obviate the existence of a trivial membrane solution branch the example of an axially compressed cylindrical shell was chosen because it is well known that for thin shells the primary axisymmetric solution branch is riddled with closely spaced symmetry breaking bifurcation points in a numerical arc length continuation scheme the close proximity of the bifurcation points on the primary path manifests itself in severe ill conditioning of the tangent stiffness matrix group theory helps one systematically find an optimal set of basis vectors or symmetry modes which reflect the symmetry of a given solution path the immediate payoff in using these symmetry modes as basis vectors is that the tangent stiffness matrix block diagonalizes and the numerical ill conditioning is avoided thus an efficient and accurate technique for computing solution branches of a specific type and a subsequent diagnosis for symmetry breaking bifurcations is made relatively simple understanding the global behavior of the perfect structure is crucial in identifying critical imperfections and will ultimately diminish the heavy reliance on expensive experimental verification

global stability analysis of fluid flows using sum of squares this paper introduces a new method for proving global stability of uid ows through the construction of lyapunov functionals for nite dimensional approximations of uid systems we show how one can exploit recently developed optimization methods based on sum of squares decomposition to construct a polynomial lyapunov function we then show how these methods can be extended to innite dimensional navier stokes systems using robust optimization techniques crucially this extension requires only the solution of innite dimension al linear eigenvalue problems and nite dimensional sum of squares optimization problems we further show that subject to minor technical constraints a general polynomial lyapunov function is always guaranteed to provide better results than the classical energy methods in determining a lower bound on the maximum reynolds number for which a ow is globally stable if the ow does remain globally stable for reynolds numbers at least slightly beyond the energy stability limit such polynomial functions can be searched for eciently using the sos technique we propose

group theoretic exploitations of symmetry in computational solid and structural mechanics the use of group theory in simplifying the study of problems involving symmetry is a well established approach in various branches of physics and chemistry and major applications in these areas date back more than 70 years within the engineering disciplines the search for more systematic and more efficient strategies for exploiting symmetry in the computational problems of solid and structural mechanics has led to the development of group theoretic methods over the past 40 years this paper reviews the advances made in the application of group theory in areas such as bifurcation analysis vibration analysis and finite element analysis and summarizes the various implementation procedures currently available illustrative examples of typical solution procedures are drawn from recent work of the author it is shown how the group theoretic approach through the characteristic vector space decomposition enables considerable simplifications and reductions in computational effort to be achieved in many cases group theoretic considerations also allow valuable insights on the behaviour or properties of a system to be gained before any actual calculations are carried out copyright 2009 john wiley sons ltd

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 procedure for the positive definiteness of forms of even order a positivity method for forms of even order is reported it is shown that testing for positivity of such forms can be reduced to a test for positive definiteness of parametric matrices special emphasis is placed on quartic and sixth order forms examples are given to illustrate the proposed criterion

post critical analysis of structures with a nonlinear pre buckling state in the presence of imperfections in the context of the geometrically nonlinear analysis of structures by continuation methods a technique for the determination of the imperfection sensitivity is carried out the technique is based on galerkin s reduction of equilibrium equations in the subspace of extrapolation branches emanating from the bifurcation points the pre and post critical behaviour in the neighbourhood of bifurcation points however is well approximated provided that extrapolations have a large convergence domain to this end an efficient asymptotic representation of the primary and secondary equilibrium paths and procedures that overcome the ill conditioning of the systems defined in the critical points is used here

some computational aspects of the stability analysis of nonlinear structures abstract the perturbation method and the continuation method are the two most popular techniques for the solution of finite element equations that describe instability phenomena this paper presents a summary of the principles involved and describes some trends in the development of these techniques

on the decomposition of generalized eigenproblems for the free vibration analysis of cyclically symmetric finite element models free vibration analysis is a major part of any dynamic analysis natural frequencies and related mode shapes may be obtained from free vibration analysis as the solutions of generalized eigenproblems although the eigensolutions of large scale structures require large computational efforts these solutions may be achieved simply for symmetric structures we present an efficient method for the decomposition of generalized eigenproblems related to finite element models with cyclic symmetry having nodes at the axis of symmetry into eigensubproblems with significantly smaller dimensions this decomposition is obtained by block diagonalization of a matrix with a special pattern known as a block circulant using the concept of the kronecker product and similarity transformations the proposed method is applied to three finite element models discretized by triangular and four node quadrilateral plate and shell elements and its efficiency accuracy and simplicity are evaluated copyright 2009 john wiley sons ltd

lower bound for the number of iterations in semidefinite hierarchies for the cut polytope hierarchies of semidefinite relaxations for 0 1 polytopes have been constructed by lasserre 2001a and by lovaisz and schrijver 1991 the cut polytope of a graph onn nodes can be expressed as a projection of such a semidefinite relaxation after at mostn steps we show that n 2 iterations are needed for finding the cut polytope of the complete graphk n

recent improvements in the complexity of the effective nullstellensatz we bring up to date the estimates on the complexity of the effective nullstellensatz and the membership problem

a comparison of the sherali adams lov 341sz schrijver and lasserre relaxations for 0 1 programming sherali and adams sa90 lov asz and schrijver ls91 and recently lasserre las01b have proposed lift and project methods for constructing hierarchies of successive linear or semidefinite relaxations of a 0 1 polytope p subseteq or n converging to p in n steps lasserre s approach uses results about representations of positive polynomials as sums of squares and the dual theory of moments we present the three methods in a common elementary framework and show that the lasserre construction provides the tightest relaxations of p as an application this gives a direct simple proof for the convergence of the lasserre s hierarchy we describe applications to the stable set polytope and to the cut polytope

effect of selection on ancestry an exactly soluble case and its phenomenological generalization we consider a family of models describing the evolution under selection of a population whose dynamics can be related to the propagation of noisy traveling waves for one particular model that we shall call the exponential model the properties of the traveling wave front can be calculated exactly as well as the statistics of the genealogy of the population one striking result is that for this particular model the genealogical trees have the same statistics as the trees of replicas in the parisi mean field theory of spin glasses we also find that in the exponential model the coalescence times along these trees grow like the logarithm of the population size a phenomenological picture of the propagation of wave fronts that we introduced in a previous work as well as our numerical data suggest that these statistics remain valid for a larger class of models while the coalescence times grow like the cube of the logarithm of the population size

controlling a class of nonlinear systems on rectangles in this paper we focus on a particular class of nonlinear affine control systems of the form xdot f x bu where the drift f is a multi affine vector field i e affine in each state component the control distribution b is constant and the control u is constrained to a convex set for such a system we first derive necessary and sufficient conditions for the existence of a multiaffine feedback control law keeping the system in a rectangular invariant we then derive sufficient conditions for driving all initial states in a rectangle through a desired facet in finite time if the control constraints are polyhedral we show that all these conditions translate to checking the feasibility of systems of linear inequalities to be satisfied by the control at the vertices of the state rectangle this work is motivated by the need to construct discrete abstractions for continuous and hybrid systems in which analysis and control tasks specified in terms of reachability of sets of states can be reduced to searches on finite graphs we show the application of our results to the problem of controlling the angular velocity of an aircraft with gas jet actuators

on the calculation of limit and bifurcation points in stability problems of elastic shells abstract numerical methods for the computation of singular points of nonlinear equations g u 0 are discussed where and are real parameters simple and double limit points are treated in some detail and numerical algorithms are presented and applied to elastic shell stability problems the case of simple symmetry breaking bifurcation points is also treated with applications to nonsymmetric bifurcation from axisymmetric states of deformation of shells of revolution

equation free modelling of evolving diseases coarse grained computations with individual based models we demonstrate how direct simulation of stochastic individualbased models can be combined with continuum numericalanalysis techniques to study the dynamics of evolving diseases sidestepping the necessity of obtaining explicit populationlevel models the approach analyses the unavailable in closed form coarse macroscopic equations estimating the necessary quantities through appropriately initialized short bursts of individualbased dynamic simulation we illustrate this approach by analysing a stochastic and discrete model for the evolution of disease agents caused by point mutations within individual hosts building up from classical susceptibleinfectedrecovered and susceptibleinfectedrecoveredsusceptible models our example uses a onedimensional lattice for variant space and assumes a finite number of individuals macroscopic computational tasks enabled through this approach include stationarystate computation coarse projective integration parametric continuation and stability analysis

on linear systems containing strict inequalities this paper deals with systems of an arbitrary possibly infinite number of both weak and strict linear inequalities we analyze the existence of solutions for such kind of systems and show that the large class of convex sets which admit this kind of linear representations i e the so called evenly convex sets enjoys most of the well known properties of the subclass of the closed convex sets we also show that it is possible to obtain geometrical information on these sets from a given linear representation finally we discuss the theory and methods for those linear optimization problems which contain strict inequalities as constraints

comments on a procedure for the positive definiteness of forms of even order hasan and hasan ieee trans automat control vol 41 p 615 17 1996 give a necessary and sufficient condition for the positive definiteness of a quadratic programming problem the purpose of this paper is to point out that the main result in the above mentioned paper by is erroneous

cones of matrices and set functions and 0 1 optimization it has been recognized recently that to represent a polyhedron as the projection of a higher dimensional but simpler polyhedron is a powerful tool in polyhedral combinatorics a general method is developed to construct higher dimensional polyhedra or in some cases convex sets whose projection approximates the convex hull of 0 1 valued solutions of a system of linear inequalities an important feature of these approximations is that one can optimize any linear objective function over them in polynomial time in the special case of the vertex packing polytope a sequence of systems of inequalities is obtained such that the first system already includes clique odd hole odd antihole wheel and orthogonality constraints in particular for perfect and many other graphs this first system gives the vertex packing polytope for various classes of graphs including t perfect graphs it follows that the stable set polytope is the projection of a polytope with a polynomial number of facets an extension o

computational methods for bifurcation problems with symmetries with special attention to steady state and hopf bifurcation points we show how group theoretical methods can be employed to utilize the symmetry of a bifurcation problem in numerical computations we extend the approach by werner 1988 by presenting methods for the detection of bifurcation points and the computation of multiple hopf points the essential numerical point is the utilization of certain reduced instead of full systems involving appropriate subgroups of the underlying symmetry group the group theoretical tool is an a priori knowledge of the interaction of certain subgroups 0 and of at possibly multiple steady state or hopf bifurcation points minimal 0 breaking bifurcation we introduce a bifurcation graph which shows graphically this a priori information its edges represent possible symmetry breaking bifurcations our analysis follows the lines of golubitsky stewart and schaeffer 1988 but it is aimed to numerical applications we have chosen a 4 box brusselator model in order to explain our notions and ideas and to discuss the numerical procedure

sharp effective nullstellensatz the usual proofs of this result however give no information about the g1 s for instance they give no bound on their degrees this question was first considered by g hermann h she used elimination theory to get a bound on the degree of the gi s which was doubly exponential in the number of variables her results were later improved in mw and in th all these produce bounds that are doubly exponential in the number of variables a major breakthrough was achieved by brownawell b1 who proved the following result

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

semi infinite programming using high degree polynomial interpolants and semidefinite programming in a common formulation of semi infinite programs the infinite constraint set is a requirement that a function parametrized by the decision variables is nonnegative over an interval if this function is sufficiently closely approximable by a polynomial or a rational function then the semi infinite program can be reformulated as an equivalent semidefinite program which in turn can be solved with interior point methods very efficiently to high accuracy on the other hand solving this semidefinite program is challenging if the polynomials involved are of high degree due to numerical difficulties and bad scaling arising both from the polynomial approximations and from the fact that the semidefinite programming constraints coming from the sum of squares representation of nonnegative polynomials are badly scaled we combine polynomial function approximation techniques and polynomial programming to overcome these numerical difficulties using sum of squares interpolants specifically it is shown that the c

approximation of the stability number of a graph via copositive programming lovasz and schrijver siam j optim 1 1991 pp 166 190 showed how to formulate increasingly tight approximations of the stable set polytope of a graph by solving semidefinite programs sdps of increasing size lift and project method in this paper we present a similar idea we show how the stability number can be computed as the solution of a conic linear program lp over the cone of copositive matrices subsequently we show how to approximate the copositive cone ever more closely via a hierarchy of linear or semidefinite programs of increasing size liftings the latter idea is based on recent work by parrilo structured semidefinite programs and semi algebraic geometry methods in robustness and optimization ph d thesis california institute of technology pasadena ca 2000 in this way we can compute the stability number alpha g of any graph g v e after at most alpha g 2 successive liftings for the lp based approximations one can compare this to the n alpha g 1 bound for the lp based lift and project scheme of lovasz and schrijver our approach therefore requires fewer liftings for families of graphs where alpha g o sqrt n we show that the first sdp based approximation for alpha g in our series of increasingly tight approximations coincides with the vartheta function of schrijver ieee trans inform theory 25 1979 pp 425 429 we further show that the second approximation is tight for complements of triangle free graphs and for odd cycles

stability and performance verification of optimization based controllers this paper presents a method to verify closed loop properties of optimization based controllers for deterministic and stochastic constrained polynomial discrete time dynamical systems the closed loop properties amenable to the proposed technique include global and local stability performance with respect to a given cost function both in a deterministic and stochastic setting and the mathcal l 2 gain the method applies to a wide range of practical control problems for instance a dynamical controller e g a pid plus input saturation model predictive control with state estimation inexact model and soft constraints or a general optimization based controller where the underlying problem is solved with a fixed number of iterations of a first order method are all amenable to the proposed approach the approach is based on the observation that the control input generated by an optimization based controller satisfies the associated karush kuhn tucker kkt conditions which provided all data is polynomial are a system of polynomial equalities and inequalities the closed loop properties can then be analyzed using sum of squares sos programming

robust non zenoness of piecewise affine systems with applications to linear complementarity systems piecewise affine systems pass constitute an important class of nonsmooth switching dynamical systems subject to state dependent mode transitions arising from control and dynamic optimization a fundamental issue in dynamics analysis of switching systems pertains to the possible occurrence of infinitely many switchings in finite time referred to as the zeno behavior there has been a growing interest in characterization of zeno free switching systems different from the recent non zeno analysis of switching systems the present paper studies non zeno properties of pass subject to system parameter and or initial state perturbations inspired by sensitivity and uncertainty analysis of pass specifically by exploiting the geometry of polyhedral subdivisions and dynamical system techniques this paper establishes a uniform bound on the number of mode switchings for a family of lipschitz pass under mild uniform conditions on system parameters and associated polyhedral subdivisions this result is employed t

evolution on a smooth landscape we study in detail a recently proposed simple discrete model for evolution on smooth landscapes an asymptotic solution of this model for long times is constructed we find that the dynamics of the population is governed by correlation functions that although being formally down by powers ofn the population size nonetheless control the evolution process after a very short transient the long time behavior can be found analytically since only one of these higher order correlators the two point function is relevant we compare and contrast the exact findings derived herein with a previously proposed phenomenological treatment employing mean field theory supplemented with a cutoff at small population density finally we relate our results to the recently studied case of mutation on a totally flat landscape

branch switching for pitchfork bifurcation in nonlinear finite element method a general branch switching approach is proposed based on pseudo arc length continuation in the nonlinear finite element method fem to trace the branch solution curve at the pitchfork bifurcation point in parameter space by this approach branch direction can be determined without derivatives of tangent stiffness matrix this approach is proved mathematically and is inserted into a general fem code stability of equilibrium corresponding to each solution curve is estimated by the lagrange dirichlet criteria as an example the whole process of the elastic mode jumping for rectangular thin plates in which there are many pitchfork bifurcations in parameter space in loading and unloading is simulated by the improved fem code the numerical results are identical with that of an experiment done by previous researchers

improvement of the scaled corrector method for bifurcation analysis using symmetry exploiting block diagonalization in the nonlinear bifurcation analysis for large scaled structures the standard eigenanalysis of the tangent stiffness matrix yields important information but at the same time demands a large amount of computational cost the scaled corrector method was developed as a numerically efficient eigenanalysis free bifurcation analysis strategy which exploits byproducts of the numerical iteration for path tracing this method however has a problem in its accuracy especially when eigenvalues are nearly or exactly coincidental as a remedy for this we propose a new bifurcation analysis method through the implementation of bifurcation mechanism of a symmetric structure into the scaled corrector method the bifurcation mode is accurately approximated by decomposing a scaled corrector vector into a number of vectors by means of block diagonalization method in group theoretic bifurcation theory and in turn by choosing the predominant one among these vectors in order to demonstrate the usefulness of this method it is applied to the bifurcation analysis of reticulated regular hexagonal truss domes to compute accurately the locations of double bifurcation points and nearly coincidental bifurcation points and associated bifurcation modes

maximum block improvement and polynomial optimization in this paper we propose an efficient method for solving the spherically constrained homogeneous polynomial optimization problem the new approach has the following three main ingredients first we establish a block coordinate descent type search method for nonlinear optimization with the novelty being that we accept only a block update that achieves the maximum improvement hence the name of our new search method maximum block improvement mbi convergence of the sequence produced by the mbi method to a stationary point is proved second we establish that maximizing a homogeneous polynomial over a sphere is equivalent to its tensor relaxation problem thus we can maximize a homogeneous polynomial function over a sphere by its tensor relaxation via the mbi approach third we propose a scheme to reach a kkt point of the polynomial optimization provided that a stationary solution for the relaxed tensor problem is available numerical experiments have shown that our new method works very efficiently for a majority of the test instances that we have experimented with the method finds the global optimal solution at a low computational cost

a framework for worst case and stochastic safety verification using barrier certificates this paper presents a methodology for safety verification of continuous and hybrid systems in the worst case and stochastic settings in the worst case setting a function of state termed barrier certificate is used to certify that all trajectories of the system starting from a given initial set do not enter an unsafe region no explicit computation of reachable sets is required in the construction of barrier certificates which makes it possible to handle nonlinearity uncertainty and constraints directly within this framework in the stochastic setting our method computes an upper bound on the probability that a trajectory of the system reaches the unsafe set a bound whose validity is proven by the existence of a barrier certificate for polynomial systems barrier certificates can be constructed using convex optimization and hence the method is computationally tractable some examples are provided to illustrate the use of the method

some computational aspects of a group theoretic finite element approach to the buckling and postbuckling analyses of plates and shells of revolution abstract progress in the application of group theoretic methods to the nonlinear bifurcation analysis of symmetric structures has provided new hope in understanding the global postbuckling behavior of symmetric thin shell structures however what is still lacking in the computational mechanics community is a general purpose fe code with a group theoretic capability designed to take optimal advantage of existing symmetry this paper addresses some of the important issues associated with developing such a code the long term goal being to provide structural analysts with new numerical tools for doing careful global analyses of imperfection sensitive structures which may eventually lead to more accurate predictions of the maximum load carrying capacity of thin imperfection sensitive shell structures

the efficient computation of sparse jacobian matrices using automatic differentiation this paper is concerned with the efficient computation of sparse jacobian matrices of nonlinear vector maps using automatic differentiation ad specifically we propose the use of a graph coloring technique bicoloring to exploit the sparsity of the jacobian matrix j and thereby allow for the efficient determination of j using ad software we analyze both a direct scheme and a substitution process we discuss the results of numerical experiments indicating significant practical potential of this approach

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

analysis of evolution through competitive selection recent studies of in vitro evolution of dna via protein binding indicate that the evolution behavior is qualitatively different in different parameter regimes i here present a general theory that is valid for a wide range of parameters and which reproduces and extends previous results specifically the mean field theory of a general translation invariant model can be reduced to the basic diffusion equation with a dynamic boundary condition the simple analytical form yields both quantitatively accurate predictions and valuable insight into the principles involved

exact block diagonalization of large eigenvalue problems for structures with symmetry we consider large eigenvalue problems for skeletal structures with symmetry we present an algorithm based upon a novel combination of group theoretic ideas and substructuring techniques that block diagonalizes such systems exactly and efficiently the procedure requires only the structural matrices of a repeating substructure together with the symmetry modes which are obtained from symmetry considerations alone we first present a simple paradigmatic example and then follow with several non trivial applications involving large lattice structures

a branching random walk seen from the tip we show that all the time dependent statistical properties of the rightmost points of a branching brownian motion can be extracted from the traveling wave solutions of the fisher kpp equation the distribution of all the distances between the rightmost points has a long time limit which can be understood as the delay of the fisher kpp traveling waves when the initial condition is modified the limiting measure exhibits the surprising property of superposability the statistical properties of the distances between the rightmost points of the union of two realizations of the branching brownian motion shifted by arbitrary amounts are the same as those of a single realization we discuss the extension of our results to more general branching random walks

on invariant polyhedra of continuous time linear systems this note presents some conditions of existence of positively invariant polyhedra for linear continuous time systems these conditions are first described algebraically then interpreted on the basis of the system eigenstructure then a simple state feedback placement method is described for solving some linear regulation problems under constraints

the computation of symmetry breaking bifurcation points we consider symmetry breaking simple singular points of a parameter dependent nonlinear equation g lambda x 0 such points are intimately connected to pitchfork bifurcation points and we sh

observability analysis of conewise linear systems via directional derivative and positive invariance techniques belonging to the broad framework of hybrid systems conewise linear systems clss form a class of lipschitz piecewise linear systems subject to state triggered mode switchings motivated by state estimation of nonsmooth switched systems this paper exploits directional derivative and positive invariance techniques to characterize finite time and long time local observability of a general cls for the former observability notion directional derivative results are developed from the simple switching property and they yield improved observability conditions for the latter notion we focus on the case where a nominal trajectory has finitely many switchings in order to characterize long time behaviors of the cls necessary and sufficient conditions are obtained for the interior of a positively invariant cone by employing these conditions we establish connections between finite time and long time local observability underlying positive invariance properties are unveiled

analytic approach to the evolutionary effects of genetic exchange we present an approximate analytic study of our previously introduced model of evolution including the effects of genetic exchange this model is motivated by the process of bacterial transformation we solve for the velocity the rate of increase of fitness as a function of the fixed population size n we find the velocity increases with ln n eventually saturating at an n which depends on the strength of the recombination process the analytical treatment is seen to agree well with direct numerical simulations of our model equations

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

bifurcation hierarchy of symmetric structures abstract a group theoretic method for the analysis of bifurcation behavior of regular polygonal symmetric structures is described possible bifurcation paths and points of these structures are categorized in terms of dihedral and cyclic groups which express the symmetry in particular we offer a complete description of those double bifurcation points which occur due to group symmetry the type the number and the stability of bifurcation paths branching at these points are determined by deriving bifurcation equations the existence of a potential function plays a substantial role for the existence of bifurcation paths as a result of these all possible bifurcation process can be known a priori as a natural consequence of bifurcation hierarchy before actual numerical analysis an implementation of this method in numerical analysis shows its validity and usability

on the combinatorial and algebraic complexity of quantifier elimination in this paper a new algorithm for performing quantifier elimination from first order formulas over real closed fields in given this algorithm improves the complexity of the asymptotically fastest algorithm for this problem known to this data a new feature of this algorithm is that the role of the algebraic part the dependence on the degrees of the imput polynomials and the combinatorial part the dependence on the number of polynomials are sparated another new feature is that the degrees of the polynomials in the equivalent quantifier free formula that is output are independent of the number of input polynomials as special cases of this algorithm new and improved algorithms for deciding a sentence in the first order theory over real closed fields and also for solving the existential problem in the first order theory over real closed fields are obtained

a copositive formulation for the stability number of infinite graphs in the last decade copositive formulations have been proposed for a variety of combinatorial optimization problems for example the stability number independence number in this paper we generalize this approach to infinite graphs and show that the stability number of an infinite graph is the optimal solution of some infinite dimensional copositive program for this we develop a duality theory between the primal convex cone of copositive kernels and the dual convex cone of completely positive measures we determine the extreme rays of the latter cone and we illustrate this theory with the help of the kissing number problem

brief paper local stability analysis using simulations and sum of squares programming the problem of computing bounds on the region of attraction for systems with polynomial vector fields is considered invariant subsets of the region of attraction are characterized as sublevel sets of lyapunov functions finite dimensional polynomial parametrizations for lyapunov functions are used a methodology utilizing information from simulations to generate lyapunov function candidates satisfying necessary conditions for bilinear constraints is proposed the suitability of lyapunov function candidates is assessed solving linear sum of squares optimization problems qualified candidates are used to compute invariant subsets of the region of attraction and to initialize various bilinear search strategies for further optimization we illustrate the method on small examples from the literature and several control oriented systems

survey paper set invariance in control the properties of positively invariant sets are involved in many different problems in control theory such as constrained control robustness analysis synthesis and optimization in this paper we provide an overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems

reachability and control synthesis for piecewise affine hybrid systems on simplices in this paper we consider the synthesis of control laws for piecewise affine hybrid systems on simplices the construction is based on the solution to the control to facet problem at the continuous level and on dynamic programming at the discrete level the construction is given as an explicit algorithm using only linear algebra and reach set computations for automata no numerical integration is required the method is conservative in that it may fail to find a control law where one exists but one cannot hope for a sharp algorithm for control synthesis since reachability for piecewise affine hybrid systems is undecidable

the speed of evolution in large asexual populations we consider an asexual biological population of constant size n evolving in discrete time under the influence of selection and mutation beneficial mutations appear at rate u and their selective effects s are drawn from a distribution g s after introducing the required models and concepts of mathematical population genetics we review different approaches to computing the speed of logarithmic fitness increase as a function of n u and g s we present an exact solution of the infinite population size limit and provide an estimate of the population size beyond which it is valid we then discuss approximate approaches to the finite population problem distinguishing between the case of a single selection coefficient g s s s b and a continuous distribution of selection coefficients analytic estimates for the speed are compared to numerical simulations up to population sizes of order 10300

a model with darwinian dynamics on a rugged landscape we discuss the population dynamics with selection and random diffusion keeping the total population constant in a fitness landscape associated with constraint satisfaction a paradigm for difficult optimization problems we obtain a phase diagram in terms of the size of the population and the diffusion rate with a glass phase inside which the dynamics keeps searching for better configurations and outside which deleterious mutations spoil the performance the phase diagram is analogous to that of dense active matter in terms of temperature and drive

a complete characterization of the gap between convexity and sos convexity our first contribution in this paper is to prove that three natural sum of squares sos based sufficient conditions for convexity of polynomials via the definition of convexity its first order characterization and its second order characterization are equivalent these three equivalent algebraic conditions henceforth referred to as sos convexity can be checked by semidefinite programming whereas deciding convexity is np hard if we denote the set of convex and sosconvex polynomials in n variables of degree d with cn d and cn d respectively then our main contribution is to prove that cn d cn d if and only if n 1 or d 2 or n d 2 4 we also present a complete characterization for forms homogeneous polynomials except for the case n d 3 4 which is joint work with g blekherman and is to be published elsewhere our result states that the set cn d of convex forms in n variables of degree d equals the set cn d of sos convex forms if and only if n 2 or d 2 or n d 3 4 to prove these results we present in particular explicit examples of polynomials in c2 6 c2 6 and c3 4 c3 4 and forms in c3 6 c3 6 and c4 4 c4 4 and a general procedure for constructing forms in cn d 2 cn d 2 from nonnegative but not sos forms in n variables and degree d although for disparate reasons the remarkable outcome is that convex polynomials resp forms are sos convex exactly in cases where nonnegative polynomials resp forms are sums of squares as characterized by hilbert

diffusion reproduction processes interesting physics emerges from studying a population of reproducing individuals each can be regarded as a random walker but it can either duplicate or die novel features of the collective behavior are quite surprising if individuals reproduce or die freely the life expectation is proportional to the size of the population and if it is kept constant the center of mass moves in space as if it were a single walker i e the diffusion constant is independent of population s size biology inspired interactions are also considered

positivity and optimization for semi algebraic functions we describe algebraic certificates of positivity for functions belonging to a finitely generated algebra of borel measurable functions with particular emphasis on algebras generated by semi algebraic functions in this case the standard global optimization problem with constraints given by elements of the same algebra is reduced via a natural change of variables to the better understood case of polynomial optimization a collection of simple examples and numerical experiments complement the theoretical parts of the article

temporal logic analysis of gene networks under parameter uncertainty the lack of precise numerical information for the values of biological parameters severely limits the development and analysis of models of genetic regulatory networks to deal with this problem we propose a method for the analysis of genetic regulatory networks under parameter uncertainty we consider models based on piecewise multiaffine differential equations dynamical properties expressed in temporal logic and intervals for the values of uncertain parameters the problem is then either to guarantee that the system satisfies the expected properties for every possible parameter value the corresponding parameter set is then called valid or to find valid subsets of a given parameter set the proposed method uses discrete abstractions and model checking and allows for efficient search of the parameter space however the abstraction process creates spurious behaviors in the abstract systems along which time does not progress consequently the verification of liveness properties expressing that something will eventually happen and implicitly assuming progress of time often fails a solution to this second problem is proposed using the notion of transient regions this approach has been implemented in a tool for robust verification of gene networks and applied to the tuning of a synthetic network built in e coli

the evolution of the minimum degree ordering algorithm over the past fifteen years the implementation of the minimum degree algorithm has received much study and many important enhancements have been made to it this paper describes these various enhancements their historical development and some experiments showing how very effective they are in improving the execution time of the algorithm a shortcoming is also presented that exists in all of the widely used implementations of the algorithm namely that the quality of the ordering provided by the implementations is surprisingly sensitive to the initial ordering for example changing the input ordering can lead to an increase or decrease of as much as a factor of three in the cost of the subsequent numerical factorization this sensitivity is caused by the lack of an effective tie breaking strategy and the authors experiments illustrate the importance of developing such a strategy