a geometrically based criterion to avoid infimum gaps in optimal control abstract in optimal control theory infimum gap means a non zero difference between the infimum values of a given minimum problem and an extended problem obtained by embedding the original family v of controls in a larger family w for some embeddings like standard convex relaxations or impulsive extensions the normality of an extended minimizer has been shown to be sufficient for the avoidance of infimum gaps a natural issue is then the search of a general hypothesis under which the criterium normality implies no gap holds true we prove that this criterium is actually valid as soon as v is abundant in w without any convexity assumption on the extended dynamics abundance which was introduced by j warga in a convex context and was later generalized by b kaskosz strengthens density the latter being not sufficient for the mentioned criterium to hold true

the nilpotent centralizer method for spectrally arbitrary patterns this paper introduces a new criterion for an n n sign pattern respectively zero nonzero pattern a to be spectrally arbitrary that is to have the property that for each monic real polynomial r x of degree n there exists a matrix with sign pattern respectively zero nonzero pattern a that has r x as its characteristic polynomial to date analytic properties of a certain polynomial map associated with a have been used to prove that a is spectrally arbitrary in this paper we derive a method that uses the algebraic structure of a certain nilpotent matrix to show that a sign pattern respectively zero nonzero pattern is spectrally arbitrary

potentially nilpotent patterns and the nilpotent jacobian method a nonzero pattern is a matrix with entries in 0 a pattern is potentially nilpotent if there is some nilpotent real matrix with nonzero entries in precisely the entries indicated by the pattern we develop ways to construct some potentially nilpotent patterns including some balanced tree patterns we explore the index of some of the nilpotent matrices constructed and observe that some of the balanced trees are spectrally arbitrary using the nilpotent jacobian method inspired by an argument of pereira nilpotent matrices and spectrally arbitrary sign patterns electron j linear algebra 16 2007 235 we uncover a feature of the nilpotent jacobian method in particular we show that if n is the nilpotent matrix employed in this method to show that a pattern is a spectrally arbitrary pattern then n must have full index

dynamic programming for nonlinear systems driven by ordinaryand impulsive controls a dynamic programming approach is considered for a class of minimum problems with impulses the minimization domain consists of trajectories satisfying an ordinary differential equation whose right hand side depends not only on a measurable control v but also on a second control u and on its time derivative dot u for this reason the control u and the differential equation are called impulsive the value function of the considered minimum problem turns out to depend on the time the state the u variable and the variation allowed to the impulsive control it is shown that the value function satisfies in a generalized sense a dynamic programming equation dpe which is obtained from a dynamic programming principle involving space time trajectories moreover the value function is the unique map solving equation dpe satisfying either an inequality condition or a supersolution condition at each point of the boundary incidentally this extends a result by barron jensen and menaldi nonlinear anal 21 1993 pp 241 268 where the impulsive control is scalar monotone and the corresponding vector field is independent of the state variable next a maximum principle is proved and the well known relationship between adjoint variables and value function is suitably extended to impulsive control systems a fully elaborated example concludes the paper

symmetric sign pattern matrices that require unique inertia abstract a sign pattern matrix is a matrix whose entries are from the set 0 the purpose of this paper is to characterize symmetric sign patterns that require unique inertia that is all the real symmetric matrices with the given sign pattern must have the same inertia further some constructions to obtain sign patterns that require unique inertia are provided sign patterns corresponding to some special graphs are also considered finally extensions to complex sign patterns are mentioned

measure driven differential inclusions measure driven differential inclusions arise when we attempt to derive necessary conditions of optimality for optimal impulsive control problems with nonsmooth data we introduce the concept of a robust solution to a measure driven inclusion which extends to a multifunction setting interpretations of solutions to measure driven differential equations provided by dal maso and rampazzo and others closure properties of sets of robust solutions are established and notions of relaxation investigated implications for optimality conditions for impulsive control problems are pursued in a companion paper

on the intersection of a clarke cone with a boltyanskii cone we provide an example of two closed sets s 1 s 2 subset r 4 such that s 1 cap s 2 0 yet at the origin a boltyanskii tangent cone c 1 to s 1 and the clarke tangent cone c 2 to s 2 are strongly transversal this settles a question originally proposed by h sussmann

the minimal critical sets of refined inertias for 3 3 full sign patterns abstract a sign pattern matrix a is a matrix whose entries are from the set 0 if no entry of a is zero then a is called a full sign pattern the inertia of a real matrix a is the ordered triple n n n 0 in which n n and n 0 are the numbers of its eigenvalues counting multiplicities with positive negative and zero real parts respectively the refined inertia of a is the ordered 4 tuple n n n z 2 n p where n z resp 2 n p is the number of zero resp nonzero pure imaginary eigenvalues of a an n n sign pattern a is an inertially arbitrary pattern iap if it allows all possible inertias similarly a is a refined inertially arbitrary pattern riap if it allows all possible refined inertias a proper subset s of the set of all possible inertias resp refined inertia of real matrices of order n is called a critical set of inertias resp critical set of refined inertias for a family f of sign patterns of order n if for every a f s i a ensures that a is inertially arbitrary resp s ri a ensures that a is refined inertially arbitrary s is called a minimal critical set of inertias resp minimal critical set of refined inertias for f if no proper subset of s is a critical set of inertias resp refined inertias for f in this paper all minimal critical sets of inertias and refined inertias for 3 3 full sign patterns are identified

pontryagin s maximum principle for constrained impulsive control problems abstract necessary conditions in the form of pontryagin s maximum principle are derived for impulsive control problems with mixed constraints a new mathematical concept of impulsive control is introduced as a requirement for the consistency of the impulsive framework additionally this control concept enables the incorporation of the engineering needs to consider conventional control action while the impulse develops the regularity assumptions under which the maximum principle is proved are weaker than those in the known literature ekeland s variational principle and lebesgue s discontinuous time variable change are used in the proof the article also contains an example showing how such impulsive controls could be relevant in actual applications

nested sequences of principal minors and potential stability abstract sufficient conditions are determined for an n by n zero nonzero pattern to allow a nested sequence of nonzero principal minors a method is then given to sign such a pattern so that it allows a nested sequence of k by k principal minors with sign 1 k for 1 k n it is a classical result that a sign pattern with such a sequence is potentially stable thus classes of potentially stable sign patterns can be identified a necessary and sufficient condition for a certain tree sign pattern class to be potentially stable is also given

new classes of spectrally arbitrary ray patterns an n n ray pattern matrix s is said to be spectrally arbitrary if for every monic nth degree polynomial f with coefficients from c there is a complex matrix in the ray pattern class of s such that its characteristic polynomial is f in this article we give new classes of spectrally arbitrary ray pattern matrices

testing for co integration in vector autoregressions with non stationary volatility abstract many key macroeconomic and financial variables are characterized by permanent changes in unconditional volatility in this paper we analyse vector autoregressions with non stationary unconditional volatility of a very general form which includes single and multiple volatility breaks as special cases we show that the conventional rank statistics computed as in johansen 1988 johansen 1991 are potentially unreliable in particular their large sample distributions depend on the integrated covariation of the underlying multivariate volatility process which impacts on both the size and power of the associated co integration tests as we demonstrate numerically a solution to the identified inference problem is provided by considering wild bootstrap based implementations of the rank tests these do not require the practitioner to specify a parametric model for volatility or to assume that the pattern of volatility is common to or independent across the vector of series under analysis the bootstrap is shown to perform very well in practice

positive feedback may sometimes promote stability abstract some sufficient conditions are given for a matrix to be potentially stable these are used to furnish some counter intuitive examples of stability behavior

spectrally and inertially arbitrary sign patterns we introduce some n by n sign patterns which allow for arbitrary spectrum and hence also arbitrary inertia consequently we demonstrate that some known inertially arbitrary patterns are in fact spectrally arbitrary we demonstrate that all inertially arbitrary patterns of order 3 are spectrally arbitrary and classify all spectrally arbitrary patterns of order 3 we illustrate that in general the class of spectrally arbitrary patterns is distinct from the inertially arbitrary patterns and present some observations about inertially arbitrary patterns

a combinatorial converse to the perron frobenius theorem abstract a sign pattern requires allows the perron property if every some matrix with that sign pattern has its special radius among its eigenvalues we characterize those sign patterns that require the perron property which in a sense provides a converse to the classical perron frobenius theorem also a large class of patterns that allow the perron property is identified but a complete characterization remains an open problem

sets of refined inertias of zero nonzero patterns abstract a set h n of refined inertias for zero nonzero patterns is introduced that is analogous to the set h n previously considered for sign patterns for n 3 and 4 a complete characterization of irreducible zero nonzero patterns that allow or require h n is given and each zero nonzero pattern that allows h n has a signing that allows h n in contrast for n 5 a family of irreducible zero nonzero patterns is given that allows h n but for which no one signing allows h n

spectrally arbitrary zero nonzero patterns and field extensions abstract an n n matrix pattern is said to be spectrally arbitrary over a field f provided for every monic polynomial p t of degree n with coefficients from f there exists a matrix with entries from f in the given pattern that has characteristic polynomial p t let e f k be an extension of fields it is natural to ask whether a pattern that is spectrally arbitrary over f must also be spectrally arbitrary over e or k in this article it is shown that if f is dense in k and k is a complete metric space then any spectrally arbitrary or relaxed spectrally arbitrary pattern over f is relaxed spectrally arbitrary over k it is also established that if e is an algebraically closed subfield of a field f then any spectrally arbitrary pattern over f is spectrally arbitrary over e the 2n conjecture and the superpattern conjecture are explored over fields other than the real numbers in particular examples are provided to show that the superpattern conjecture is false over the field with 3 elements

the generalized solutions of nonlinear optimization problems with impulse control the optimal control problem described by a nonlinear ordinary differential equation is considered with unbounded controls the aim of this paper is to provide a representation of generalized discontinuous solutions in terms of differential equations with a measure by using the method of discontinuous time change the problem with nonlinear dependence on the unbounded controls can be reduced to the simpler one with bounded controls these results are applied to solving the problem of generalized discontinuous solution representation in the dynamic system with impulse type controls

tests for parameter instability and structural change with unknown change point this paper considers tests for parameter instability and structural change with unknown change point the results apply to a wide class of parametric models that are suitable for estimation by generalized method of moments procedures the asymptotic distributions of the test statistics considered here are nonstandard because the change point parameter only appears under the alternative hypothesis and not under the null the tests considered here are shown to have nontrivial asymptotic local power against all alternatives for which the parameters are nonconstant the tests are found to perform quite well in a monte carlo experiment reported elsewhere copyright 1993 by the econometric society

necessary conditions for optimal impulsive control problems necessary conditions of optimality in the form of a maximum principle are derived for a class of optimal control problems certain of whose controls are represented by measures and whose state trajectories are functions of bounded variation state trajectories are interpreted as robust solutions of the dynamic equations a concept of solutions which takes account of the interaction between the measure control and the state variables during the jumps the maximum principle which is derived improves on earlier optimality conditions for problems of this nature by allowing nonsmooth data measurable time dependence and a possibly time varying constraint set for the conventional controls

a proof of the tn conjecture centralizers jacobians and spectrally arbitrary sign patterns we prove the so called tn conjecture for every real monic polynomial p x of degree n 2 there exists an n by n matrix with sign pattern tn 0 0 0 0 0 0 0 0 whose characteristic polynomial is p x the proof converts the problem of determining the nonsingularity of a certain jacobi matrix to the problem of proving the non existence of a nonzero matrix b that commutes with a nilpotent matrix with sign pattern tn and has zeros in positions 1 1 and j 1 j for j 2 n 1

inertially arbitrary 2r 1 diagonal sign patterns the set of all eigenvalues counting multiplicities of a matrix a is denoted by a and the inertia of a is the ordered triple i a i a i a i0 a in which i a i a and i0 a are the numbers of eigenvalues with positive negative and zero real parts respectively an n n sign pattern s sij has sij 0 and the qualitative class of s is q s a aij mn r sign aij sijforalli j the inertia of s is the set of ordered triples i s i a a q s an n n sign pattern s is an inertially arbitrary pattern iap if n1 n2 n3 i s for each nonnegative triple n1 n2 n3 with n1 n2 n3 n consider the n n 2r 1 diagonal sign pattern sn r with positive entry i j for 1 j i r 1 or i j n negative entry i j for 1 i j r 1 or i j 1 and zero entry otherwise j h drew et al proved that sn 2 is an iap for 2 n 7 and conjectured that sn 2 is an iap for n 8 gao et al proved that sn n is an iap for n 2 in this paper it is proved that n k k 0 i sn r for 0 k n 2 r n and that sn n 1 is an iap for n 3

spectrally arbitrary patterns over rings with unity abstract a zero nonzero pattern a is a matrix with entries from the set 0 a square pattern a is spectrally arbitrary over r a commutative ring with unity if for each n th degree monic polynomial f x r x there exists a matrix a over r with pattern a such that the characteristic polynomial p a x f x a pattern a is relaxed spectrally arbitrary over r if for each n th degree monic polynomial f x r x there exists a matrix a over r with either pattern a or a subpattern of a such that the characteristic polynomial p a x f x we evaluate how the structure of rings affects how we determine if a pattern is spectrally arbitrary we consider whether a pattern a that is spectrally arbitrary over r is spectrally arbitrary or relaxed spectrally arbitrary over another commutative ring s with unity in particular we establish that a pattern that is spectrally arbitrary over z is relaxed spectrally arbitrary over z m for all m z and spectrally arbitrary over q similarly a pattern that is spectrally arbitrary over the p adic integers is spectrally arbitrary over the p adic numbers

the potentially stable tree sign patterns for dimensions less than five the problem of characterizing potentially stable sign pattern matrices remains unsolved even for tree patterns we present here in tabular form all irreducible n by n potentially stable sign patterns whose undirected graphs are trees for n 5 in hopes that this may be useful to others in formulating and evaluating conjectures on the subject some background and a descriptio of methodology which includes some new theoretical ideas are also included

optimization and controllability without differentiability assumptions let mathcal x and mathcal y be real normed vector spaces k subset mathcal x convex and compact c subset mathcal y a convex body mathcal u subset k and phi phi k to mathbb r m times mathcal y a function that can be appropriately approximated by functions phi i phi i whose compositions with linear maps of small finite dimensional simplices are c 1 we derive sufficient conditions for phi to be locally controllable on mathcal u subject to the restriction phi u in c and obtain as a corollary corresponding necessary conditions for a related restricted minimum these conditions are formulated in terms of directional derivate containers which are a type of set valued derivatives of phi phi and they improve on and extend previously obtained results they are used elsewhere to obtain new conditions for controllability and restricted minimum in nonsmooth optimal control problems defined by differential or functional integral equatio

on the potential stability of star sign pattern matrices abstract a sign pattern matrix a is called potentially stable if there exists a real matrix b q a such that b is stable a combinatorially symmetric sign pattern matrix a is called tree star sign pattern if its graph g a is a tree star in this paper a characterization for a star sign pattern matrix being potentially stable is presented

some nilpotent tridiagonal matrices with a special sign pattern abstract we prove that for each n 2 there is a nilpotent n n tridiagonal matrix satisfying a the super diagonal is positive b the sub diagonal is negative c the diagonal is zero except that the 1 1 position is negative and the n n position is positive

sign properties of metzler matrices with applications abstract several results about sign properties of metzler matrices are obtained it is first established that checking the sign stability of a metzler sign matrix can be either characterized in terms of the hurwitz stability of the unit sign matrix in the corresponding qualitative class or in terms the negativity of the diagonal elements of the metzler sign matrix and the acyclicity of the associated directed graph similar results are obtained for the case of metzler block matrices and metzler mixed matrices the latter being a class of metzler matrices containing both sign and real type entries the problem of assessing the sign stability of the convex hull of a finite and summable family of metzler matrices is also solved and a necessary and sufficient condition for the existence of common lyapunov functions for all the matrices in the convex hull is obtained the concept of sign stability is then generalized to the concept of ker b sign stability a problem that arises in the analysis of certain jump markov processes a sufficient condition for the ker b sign stability of metzler sign matrices is obtained and formulated using inverses of sign matrices and the concept of l matrices several applications of the results are discussed in the last section

on spiking models for synaptic activity and impulsive differential equations we illustrate the problems that can arise in writing differential equations that include dirac delta functions to model equations with state dependent impulsive forcing specifically difficulties arise in the interpretation of the products of distributions with discontinuous functions we suggest several methods to resolve these ambiguities such as using limiting sequences and asymptotic analysis with applications of the results given for discrete maps these suggestions are applied to a popular model describing synaptic connections in the brain

potentially nilpotent and spectrally arbitrary even cycle sign patterns abstract an n n sign pattern s n is potentially nilpotent if there is a real matrix having sign pattern s n and characteristic polynomial x n a new family of sign patterns c n with a cycle of every even length is introduced and shown to be potentially nilpotent by explicitly determining the entries of a nilpotent matrix with sign pattern c n these nilpotent matrices are used together with a jacobian argument to show that c n is spectrally arbitrary i e there is a real matrix having sign pattern c n and characteristic polynomial x n i 1 n 1 i i x n i for any real i some results and a conjecture on minimality of these spectrally arbitrary sign patterns are given

l1 limit solutions for control systems abstract for a control cauchy problem x f t x u v 1 m g x u x a x on an interval a b we propose a notion of limit solution x verifying the following properties i x is defined for l 1 impulsive inputs u and for standard bounded measurable controls v ii in the commutative case i e when g g 0 for all 1 m x coincides with the solution one can obtain via a change of coordinates that makes the g simultaneously constant iii x subsumes former concepts of solution valid for the generic noncommutative case in particular when u has bounded variation we investigate the relation between limit solutions and single valued graph completion solutions furthermore we prove consistency with the classical caratheodory solution when u and x are absolutely continuous even though some specific problems are better addressed by means of special representations of the solutions we believe that various theoretical issues call for a unified notion of trajectory for instance this is the case of optimal control problems possibly with state and endpoint constraints for which no extra assumptions like e g coercivity bounded variation commutativity are made in advance

unit root tests with a break in innovation variance abstract it is shown that an abrupt change in the innovation variance of an integrated process can generate spurious rejections of the unit root null hypothesis in routine applications of dickey fuller tests we develop and investigate modified test statistics based on unit root tests of perron for a time series with a changing level or changing intercept and slope which are applicable when there is a change in innovation variance of an unknown magnitude at an unknown location

sign patterns requiring a unique inertia abstract a sign pattern requires a unique inertia if every real matrix in the sign pattern class has the same inertia several sufficient or necessary conditions are given for a sign pattern to require a unique inertia it is proved that a sign pattern requires a unique inertia if and only if it requires a unique refined inertia all sign patterns of orders 2 and 3 that require a unique inertia are characterized if the underlying graph of a sign pattern is a tree then it is shown that any skew symmetric sign pattern requires a unique inertia and a linear time algorithm is given to determine whether or not a symmetric sign pattern requires a unique inertia

spectrally arbitrary patterns reducibility and the 2n conjecture for n 5 a sign pattern z a matrix whose entries are elements of 0 is spectrally arbitrary if for any self conjugate spectrum there is a real matrix with sign pattern z having the given spectrum spectrally arbitrary sign patterns were introduced in 5 where it was incorrectly stated that if a sign pattern z is reducible and each of its irreducible components is a spectrally arbitrary sign pattern then z is a spectrally arbitrary sign pattern and it was conjectured that the converse is true as well we present counterexamples to both of these statements in 2 it was conjectured that any n n spectrally arbitrary sign pattern must have at least 2n nonzero entries we establish that this conjecture is true for 5 5 sign patterns we also establish analogous results for nonzero patterns

bordering for spectrally arbitrary sign patterns abstract we develop a matrix bordering technique that can be applied to an irreducible spectrally arbitrary sign pattern to construct a higher order spectrally arbitrary sign pattern this technique generalizes a recently developed triangle extension method we describe recursive constructions of spectrally arbitrary patterns using our bordering technique and show that a slight variation of this technique can be used to construct inertially arbitrary sign patterns

inertially arbitrary sign patterns with no nilpotent realization abstract an n by n sign pattern s is inertially arbitrary if each ordered triple n1 n2 n3 of nonnegative integers with n1 n2 n3 n is the inertia of some real matrix in q s the sign pattern class of s if every real monic polynomial of degree n having a positive coefficient of xn 2 is the characteristic polynomial of some matrix in q s then it is shown that s is inertially arbitrary a new family of irreducible sign patterns g 2 k 1 k 2 is presented and proved to be inertially arbitrary but not potentially nilpotent and thus not spectrally arbitrary the well known nilpotent jacobian method cannot be used to prove that g 2 k 1 is inertially arbitrary since g 2 k 1 has no nilpotent realization in order to prove that q g 2 k 1 allows each inertia with n3 1 a realization of g 2 k 1 with only zero eigenvalues except for a conjugate pair of pure imaginary eigenvalues is identified and used with the implicit function theorem matrices in q g 2 k 1 with inertias having n3 0 are constructed by a recursive procedure from those of lower order some properties of the coefficients of the characteristic polynomial of an arbitrary matrix having certain fixed inertias are derived and are used to show that g 5 and g 7 are minimal inertially arbitrary sign patterns

algebraically positive matrices abstract we introduce the concept of algebraically positive matrices and investigate some basic properties including a characterization the index of algebraic positivity and sign patterns that allow or require this property we also pose two open problems

reparametrizations and approximate values of integrals of the calculus of variations abstract we prove an approximation result that implies the non occurrence of the lavrentiev phenomenon

refined inertias of strongly connected orientations of the petersen graph abstract the 18 non isomorphic strongly connected orientations of the petersen graph give rise to matrix patterns in which nonzero entries can be taken to be strictly positive of arbitrary sign or of fixed sign the allowed refined inertias in which the number of zero eigenvalues is split from others on the imaginary axis are considered for each such pattern each nonnegative pattern is shown to have unique refined inertia determined by the number of required zero eigenvalues for zero nonzero patterns a complete list of allowed refined inertias is given for each orientation one particular sign pattern is presented that allows only two distinct refined inertias out of a possible 161 for a sign pattern of order 10

estimating and testing linear models with multiple structural changes this paper develops the statistical theory for testing and estimating multiple change points in regression models the rate of convergence and limiting distribution for the estimated parameters are obtained several test statistics are proposed to determine the existence as well as the number of change points a partial structural change model is considered the authors study both fixed and shrinking magnitudes of shifts in addition the models allow for serially correlated disturbances mixingales an estimation strategy for which the location of the breaks need not be simultaneously determined is discussed instead the authors method successively estimates each break point

the minimum number of nonzeros in a spectrally arbitrary ray pattern abstract a ray pattern a is spectrally arbitrary if given any monic polynomial r x of order n with complex coefficients there exists b q r a such that the characteristic polynomial of b is r x in 1 the authors first presented that a family of n n irreducible ray patterns with exactly 3n nonzeros is spectrally arbitrary then they proved that every n n irreducible spectrally arbitrary ray pattern has at least 3 n 1 nonzeros so as pointed in 1 the minimum number of nonzeros in an n n irreducible spectrally arbitrary ray pattern is either 3n or 3 n 1 in this paper we provide an irreducible spectrally arbitrary ray pattern of order n with 3 n 1 nonzeros so the minimum number of nonzeros in an n n irreducible spectrally arbitrary ray pattern is 3 n 1

spectrally arbitrary star sign patterns abstract an n n sign pattern s n is spectrally arbitrary if for any given real monic polynomial g x of degree n there is a real matrix having sign pattern s n and characteristic polynomial g x all n n star sign patterns that are spectrally arbitrary and all minimal such patterns are characterized this subsequently leads to an explicit characterization of all n n star sign patterns that are potentially nilpotent it is shown that any super pattern of a spectrally arbitrary star sign pattern is also spectrally arbitrary

a nondegenerate maximum principle for the impulse control problem with state constraints in this article a free time impulsive control problem with state constraints and equality and inequality constraints on the trajectory endpoints is considered a weakened maximum principle is obtained for problems with data measurable in the time variable being the time transversality conditions deduced with the help of some extra convexity assumption on the state constraints in the case of smooth problems a nondegenerate maximum principle is derived by using a penalty function method

spectrally arbitrary pattern extensions abstract a matrix pattern is often either a sign pattern with entries in 0 or more simply a nonzero pattern with entries in 0 a matrix pattern a is spectrally arbitrary if for any choice of a real matrix spectrum there is a real matrix having the pattern a and the chosen spectrum we describe a graphical technique a triangle extension for constructing spectrally arbitrary patterns out of some known lower order spectrally arbitrary patterns these methods provide a new way of viewing some known spectrally arbitrary patterns as well as providing many new families of spectrally arbitrary patterns we also demonstrate how the technique can be applied to certain inertially arbitrary patterns to obtain larger inertially arbitrary patterns we then provide an additional extension method for zero nonzero patterns patterns with entries in 0

on the spectra of close to schwarz matrices several inverse eigenvalue problems for tridiagonal matrices are discussed in particular it is shown that for every set s of n complex numbers which is symmetric with respect to the real axis there exists a symmetric in modulus tridiagonal n n matrix a which has a nonnegative super diagonal a nonpositive subdiagonal all diagonal elements zero except for a11 which is nonnegative and ann which is nonpositive and with spectrum s 2002 elsevier science inc all rights reserved

spectrally arbitrary patterns abstract an n n sign pattern matrix s is an inertially arbitrary pattern iap if each nonnegative triple n 1 n 2 n 3 with n 1 n 2 n 3 n is the inertia of a matrix with sign pattern s analogously s is a spectrally arbitrary pattern sap if for any given real monic polynomial r x of order n there is a matrix with sign pattern s and characteristic polynomial r x focusing on tree sign patterns consider the n n tridiagonal sign pattern t n with each superdiagonal entry positive each subdiagonal entry negative the 1 1 entry negative the n n entry positive and every other entry zero it is conjectured that t n is an iap by constructing matrices a n with pattern t n it is proved that t n allows any inertia with n 3 0 1 2 n 1 n for all n 2 this leads to a proof of the conjecture for n 5 the truth of the conjecture is extended to n 7 by showing the stronger result that t n is a sap the proof of this latter statement involves finding a matrix a n with pattern t n that is nilpotent further questions about patterns that are saps and iaps are considered

spectrally arbitrary ray patterns abstract an n n ray pattern a is said to be spectrally arbitrary if for every monic nth degree polynomial f x with coefficients from c there is a matrix in the pattern class of a such that its characteristic polynomial is f x in this article the authors extend the nilpotent jacobi method for sign patterns to ray patterns establishing a means to show that an irreducible ray pattern and all its superpatterns are spectrally arbitrary they use this method to establish that a particular family of n n irreducible ray patterns with exactly 3 n nonzeros is spectrally arbitrary they then show that every n n irreducible spectrally arbitrary ray pattern has at least 3 n 1 nonzeros

inertia sets of two classes of symmetric sign patterns abstract a sign pattern is a matrix whose entries are from the set 0 for a symmetric sign pattern a of order n the inertia set of a is the set of inertias of all real symmetric matrices with the same sign pattern as a the purpose of this paper is to characterize the inertia sets of symmetric star and nonnegative symmetric tridiagonal sign patterns

spurious rejections by dickey fuller tests in the presence of a break under the null abstract it is well known that if a series is generated by a process that is stationary around a broken trend conventional dickey fuller tests can have very low power in this paper the converse phenomenon is studied and illustrated suppose that the true generating process is integrated of order one but with a break then it is shown that if the break occurs early in the series routine application of standard dickey fuller tests can lead to a very serious problem of spurious rejection of the unit root null hypothesis

sparse inertially arbitrary patterns an n by n sign pattern a is a matrix with entries in 0 an n by n nonzero pattern a is a matrix with entries in 0 where represents a nonzero entry a pattern a is inertially arbitrary if for every set of nonnegative integers n1 n2 n3 with n1 n2 n3 n there is a real matrix with pattern a having inertia n1 n2 n3 we explore how the inertia of a matrix relates to the signs of the coefficients of its characteristic polynomial and describe the inertias allowed by certain sets of polynomials this information is useful for describing the inertia of a pattern and can help show a pattern is inertially arbitrary britz et al t britz j j mcdonald d d olesky p van den driessche minimal spectrally arbitrary sign patterns siam j matrix anal appl 26 2004 257 271 conjectured that irreducible spectrally arbitrary patterns must have at least 2n nonzero entries we demonstrate that irreducible inertially arbitrary patterns can have less than 2n nonzero entries

companion matrix patterns abstract companion matrices especially the frobenius companion matrices are used in algorithms for finding roots of polynomials and are also used to find bounds on eigenvalues of matrices in 2003 fiedler introduced a larger class of companion matrices that includes the frobenius companion matrices one property of the combinatorial pattern of these companion matrices is that up to diagonal similarity they uniquely realize every possible spectrum of a real matrix we characterize matrix patterns that have this property and consequently introduce more companion matrix patterns we observe that each fiedler companion matrix is permutationally similar to a unit hessenberg matrix we provide digraph characterizations of the classes of patterns described and in particular all sparse companion matrices noting that there are companion matrices that are not sparse

minimal time sequential batch reactors with bounded and impulse controls for one or more species we consider the optimal control problem of feeding in minimal time a tank where several species compete for a single resource with the objective being to reach a given level of the resource we allow controls to be bounded measurable functions of time plus possible impulses for the one species case we show that the immediate one impulse strategy filling the whole reactor with one single impulse at the initial time is optimal when the growth function is monotonic for nonmonotonic growth functions with one maximum we show that a particular singular arc strategy precisely defined in section 3 is optimal these results extend and improve former ones obtained for the class of measurable controls only for the two species case with monotonic growth functions we give conditions under which the immediate one impulse strategy is optimal we also give optimality conditions for the singular arc strategy at a level that depends on the initial condition to be optimal the possibility for the immediate one impulse strategy to be nonoptimal while both growth functions are monotonic is a surprising result and is illustrated with the help of numerical simulations

low rank perturbations and the spectrum of a tridiagonal sign pattern abstract the n by n tridiagonal sign pattern t n has every superdiagonal entry positive every subdiagonal entry negative the 1 1 entry negative the n n entry positive and every other diagonal entry zero inertia and spectral results for matrices a n having the sign pattern t n are proved using new techniques on low rank perturbations it is also shown by using maple that for 8 n 16 t n allows any spectrum these results extend those previously in the literature and strengthen the conjecture that t n allows any spectrum for all values of n in addition bounds on the algebraic multiplicity of an eigenvalue of a low rank perturbation of a general matrix are obtained

three remarks on spectra of zero nonzero patterns abstract using standard techniques from combinatorics model theory and algebraic geometry we prove generalized versions of several basic results in the theory of spectrally arbitrary matrix patterns also we point out a counterexample to a conjecture proposed recently by mcdonald and melvin

sign stability of positive markov jump linear systems abstract this paper investigates stability of positive markov jump linear systems pmjlss in the absence of a numerical realization it considers the situation when only signs and not magnitudes of the entries of the subsystem matrices and the markov transition matrices are known the result is an analysis of a qualitative notion of stability known as sign stability although the notions of sign stability of pmjlss are natural extensions of standard stochastic stability concepts such as exponential almost sure mean and mean square stability the sign stability notions are proven equivalent which is not the case for their corresponding standard concepts moreover for irreducible markov chains the particular structure of the markov chain is shown to have no bearing on sign stability

potentially nilpotent sign pattern matrices abstract a sign pattern is said to be potentially nilpotent if it allows nilpotence in this paper a number of qualitative necessary or sufficient conditions for a sign pattern to allow nilpotence are established the sign patterns that allow nilpotence of index 2 are investigated for orders up to three potentially nilpotent sign patterns are characterized potentially nilpotent tree sign patterns are also explored

necessary conditions of optimality for vector valued impulsive control problems abstract a vector valued impulsive control problem is considered whose dynamics defined by a differential inclusion are such that the vector fields associated with the singular term do not satisfy the so called frobenius condition a concept of robust solution based on a new reparametrization procedure is adopted in order to derive necessary conditions of optimality these conditions are obtained by taking a limit of those for an appropriate sequence of auxiliary standard optimal control problems approximating the original one an example to illustrate the nature of the new optimality conditions is provided

qualitative stability of linear systems abstract this paper provides a finitely computable graph theoretic answer to the following question concerning linear dynamical systems when given only the signs of entries or 0 in a real square matrix a can one be certain that all positive trajectories of the system ax are bounded matrices having such sign patterns are called sign quasistable with bounded replaced by convergent to the origin the matrices are called sign stable and were fully described in earlier papers however when a s digraph has several strong components so that the system is actually a hierarchy of subsystems and when some of those subsystems fail to be sign stable the recognition of sign quasistability is a very delicate matter by means of certain graph color tests it is possible to identify the system variables that are capable for some choice of matrix entry magnitudes and initial conditions of emitting nonzero constant

sign patterns allowing nilpotence of index 3 suppose p is a property referring to a real matrix we say that a sign pattern a allows p if there exists at least one matrix with the same sign pattern as a that has the property p in this paper we study sign patterns allowing nilpotence of index 3 four methods for constructing sign patterns that allow nilpotence of index 3 are obtained all tree sign patterns that allow nilpotence of index 3 are characterized sign patterns of order 3 that allow nilpotence are identified

minimizers that are not also relaxed minimizers relaxation is a widely used regularization procedure in optimal control involving the replacement of velocity sets by their convex hulls to ensure the existence of a minimizer it can be an important step in the construction of suboptimal controls for the original unrelaxed optimal control problem which may not have a minimizer based on obtaining a minimizer for the relaxed problem and approximating it in some cases the infimum cost of the unrelaxed problem is strictly greater than the infimum cost over relaxed state trajectories we need to identify such situations because then the above procedure fails the noncoincidence of these two infima leads also to a breakdown of the dynamic programming method because typically solving the hamilton jacobi equation yields the minimum cost of the relaxed not the original optimal control problem following on from earlier work by warga we explore the relation between on the one hand noncoincidence of the minimum cost of the optimal control and its re

sign patterns that require or allow particular refined inertias abstract the refined inertia ri a of a real n n matrix a is the ordered 4 tuple n n n z 2 n p where n resp n n z 2 n p is the number of positive resp negative zero nonzero pure imaginary eigenvalues of a let h n 0 n 0 0 0 n 2 0 2 2 n 2 0 0 an n n sign pattern s n requires h n if h n ri a a has sign pattern s n and allows h n if h n ri a a has sign pattern s n sign patterns that require or allow h n are investigated for small values of n and for patterns with all diagonal entries negative examples are given relating these concepts to hopf bifurcation in dynamical systems

moving constraints as stabilizing controls in classical mechanics the paper analyzes a lagrangian system which is controlled by directly assigning some of the coordinates as functions of time by means of frictionless constraints in a natural system of coordinates the equations of motion contain terms which are linear or quadratic with respect to time derivatives of the control functions after reviewing the basic equations we explain the significance of the quadratic terms related to geodesics orthogonal to a given foliation we then study the problem of stabilization of the system to a given point by means of oscillating controls this problem is first reduced to theweak stability for a related convex valued differential inclusion then studied by lyapunov functions methods in the last sections we illustrate the results by means of various mechanical examples

allow problems concerning spectral properties of sign pattern matrices a survey abstract an n n sign pattern matrix has entries in 0 this paper surveys the following problems concerning spectral properties of sign pattern matrices sign patterns that allow all possible spectra spectrally arbitrary sign patterns sign patterns that allow all inertias inertially arbitrary sign patterns sign patterns that allow nilpotency potentially nilpotent sign patterns and sign patterns that allow stability potentially stable sign patterns relationships between these four classes of sign patterns are given and several open problems are identified

combinatorial matrix analysis an overview abstract we survey recent work in some components of combinatorial matrix analysis including qualitative matrix theory matrix completion problems bounding of matrix parameters the role of the longest simple circuit and attenuation of matrix products some definitional thoughts historical perspectives and unification challenges etc are offered the style is informal following the talk on which this writeup is based

sign patterns that require repeated eigenvalues abstract motivated by the question of which sign patterns allow a diagonalizable matrix we relate a number of properties to that of requiring repeated eigenvalues these are then used to make several observations about sign patterns that allow diagonalizability the only barrier to diagonalizability is a required nontrivial jordan structure associated with zero we note that the question of sign patterns that require diagonalizability is also open

maximum likelihood estimation of multiple change points summary maximum likelihood estimation of the locations of changes in sequences of independent categorical random variables when there are known lower bounds on the lengths of the sub sequences between the change points is discussed a method is developed which finds one of the maximum likelihood solutions the problem of estimating the number of changed segments is discussed and some numerical results about the precision of the estimates of the locations of changes are presented the method also allows the boundary distributions for the changed segments to be different from the distribution for the central region of the changed segments an application to the prediction of protein helical regions is presented