numeric and mesh algorithms for the coxeter spectral study of positive edge bipartite graphs and their isotropy groups we develop algorithmic techniques for the coxeter spectral analysis of the class ubigr n of connected loop free positive edge bipartite graphs d with n 2 vertices i e signed graphs in particular we present numerical and graphical algorithms allowing us a computer search in the study of such graphs d by means of their gram matrix email protected d the complex spectrum specc email protected c of the coxeter matrix cox d email protected email protected email protected d t r and the geometry of weyl orbits in the set mor d d of matrix morsifications email protected m n z of a simply laced dynkin diagram email protected a n d n e 6 e 7 e 8 associated with d and mesh root systems of type email protected our algorithms construct the coxeter gram polynomials cox d t z t and mesh geometries of root orbits of small connected loop free positive edge bipartite graphs d we apply them to the study of the following coxeter spectral analysis problem does email protected email protected hold i e the email protected email protected d arez congruent for any pair of connected positive loop free edge bipartite graphs d d inubigr nsuch thatspecc d specc d the problem if any square integer matrix email protected m n z is z congruent with its transpose a t r is also discussed we present a solution for graphs in ubigr n with email protected 6

matrices a such that as 1r ra with rk i abstract we study matrices a c n n such that a s 1 r r a where r k i n and s k are nonnegative integers with k 2 such matrices are called r s 1 k potent matrices the s 0 case corresponds to matrices such that a r a r 1 with r k i n and is studied using spectral properties of the matrix r for s 1 various characterizations of the class of r s 1 k potent matrices and relationships between these matrices and other classes of matrices are presented

on sesquilinear forms over fields with involution abstract let k be a field of characteristic 2 with an involution a matrix a is split if there is a change of variables q such that q t aq consists of two complementary diagonal blocks we classify all matrices that do not split as a consequence we obtain a new proof for the following result given a square matrix a there is a matrix s such that s t as a t and s s i

pairs of hermitian and skew hermitian quaternionic matrices canonical forms and their applications canonical forms are described for pairs of quaternionic matrices or equivalently matrix pencils where each one of the matrices is either hermitian or skew hermitian under strict equivalence and simultaneous congruence several applications are developed including structure of selfadjoint and skew adjoint matrices with respect to a regular sesquilinear quaternion valued form or with respect to a regular skew sesquilinear form another application gives canonical forms for quaternionic matrices under congruence

a reference free cuscore chart for dynamic mean change detection and a unified framework for charting performance comparison to detect and estimate nonconstant time varying mean shifts statistical process control spc tools such as the cumulative score cuscore and generalized likelihood ratio test glrt charts have recently been proposed however their efficiency is based on previous and exact knowledge of a reference pattern in this article a reference free cuscore rfcuscore chart is proposed that can trace and detect dynamic mean changes quickly without knowing the reference pattern in addition a unified framework that contains most of the control charts is presented and applied for a theoretical comparison of the rfcuscore cuscore glrt and cusum charts in detecting dynamic mean changes moreover numerical simulations and a real example are used to illustrate and verify the results both theoretical analysis and numerical results show that the rfcuscore chart performs not only robustly but also quickly in detecting both small and large dynamic mean changes

asymptotic optimality of running consensus in testing binary hypotheses consensus in sensor networks is a procedure to corroborate the local measurements of the sensors with those of the surrounding nodes and leads to a final agreement about a common value that in detection applications represents the decision statistic as the amount of collected data increases the convergence toward the final statistic is ruled by suitable scaling laws and the question arises if the asymptotic large sample properties of a detection statistic are retained when this statistic is approximated via consensus algorithms we investigate the asymptotic properties of running consensus detectors both under the neyman pearson paradigm fixed number of data and in the sequential case an appropriate asymptotic framework is developed and exact theoretical results are provided showing the asymptotic optimality of the running consensus detector in addition numerical experiments are performed to address nonasymptotic scenarios

the solution of the equation xa axt 0 and its application to the theory of orbits abstract describe how to find the general solution of the matrix equation xa ax t 0 with a c n n which allows us to determine the dimension of its solution space this result has immediate applications in the theory of congruence orbits of matrices in c n n because the set xa ax t x c n n is the tangent space at a to the congruence orbit of a hence the codimension of this orbit is precisely the dimension of the solution space of xa ax t 0 as a consequence we also determine the generic canonical structure of matrices under the action of congruence all these results can be directly extended to palindromic pencils a a t

equivalence and normal forms of bilinear forms abstract we present an alternative account of the problem of classifying and finding normal forms for arbitrary bilinear forms beginning from basic results developed by riehm our solution to this problem hinges on the classification of indecomposable forms and in how uniquely they fit together to produce all other forms we emphasize the use of split forms i e those bilinear forms such that the minimal polynomial of the asymmetry of their non degenerate part splits over ground field rather than restricting the field to be algebraically closed in order to obtain the most explicit results without resorting to the classification of hermitian symmetric and quadratic forms we merely require that the underlying field be quadratically closed

distributed nonlinear estimation over unbalanced directed networks this paper investigates distributed nonlinear parameter estimation in unbalanced multi agent networks where individual agents sequentially make local nonlinear noisy measurements of the true but unknown parameter and aim at estimating the parameter consistently by sharing their measurements with their neighbors via an arbitrarily predesigned directed communication topology to this end a distributed nonlinear least squares estimator termed as dnls mathcal ab is proposed by using row and column stochastic matrices associated with the communication topology under the strong connectivity of the communication topology and several mild assumptions on the sensing model functions it is shown that the dnls mathcal ab estimator leads to consistent parameter estimates if the stepsize is chosen small enough furthermore the performance of the designed algorithm in terms of the asymptotic mean and asymptotic covariance of the error sequence between the weighted estimate sequence and the true value of the parameter is also analyzed and shown to be consistent with that of one widely used centralized estimator finally a numerical example is given to validate the proposed algorithm

convergence in a quasilinear parabolic equation with time almost periodic boundary conditions abstract consider the problem u t a u x u x x f u x x 1 t 0 u x 1 t g t t 0 where g t is an almost periodic function we prove the following convergence results if a time global solution u is bounded then it converges to an almost periodic solution which is unique up to space shift stable and asymptotically stable if u goes to infinity with positive lower average speed then it converges to an almost periodic traveling wave which is unique up to space shift stable asymptotically stable and has positive average speed

recursive distributed detection for composite hypothesis testing nonlinear observation models in additive gaussian noise this paper studies recursive composite hypothesis testing in a network of sparsely connected agents the network objective is to test a simple null hypothesis against a composite alternative concerning the state of the field modeled as a vector of continuous unknown parameters determining the parametric family of probability measures induced on the agents observation spaces under the hypotheses specifically under the alternative hypothesis each agent sequentially observes an independent and identically distributed time series consisting of a nonlinear function of the true but unknown parameter corrupted by gaussian noise whereas under the null they obtain noise only two distributed recursive generalized likelihood ratio test type algorithms of the consensus innovations form are proposed namely mathcal ciglrt l and mathcal ciglrt nl in which the agents estimate the underlying parameter and in parallel also update their test decision statistics by simultaneously processing the latest local sensed information and information obtained from neighboring agents for mathcal ciglrt nl for a broad class of nonlinear observation models and under a global observability condition algorithm parameters which ensure asymptotically decaying probabilities of errors probability of miss and probability of false detection are characterized for mathcal ciglrt l a linear observation model is considered and upper bounds on large deviations decay exponent for the error probabilities are obtained

convergence in a quasilinear parabolic equation with neumann boundary conditions abstract consider the problem u t a u x u x x f u x x 1 t 0 u x 1 t k t u 1 t t 0 where k are smooth functions which are periodic in both t and u brunovsk et al proved in their paper brunovsk et al 1992 8 that if a time global solution u is bounded then it converges to a periodic solution we prove that if u is unbounded from above then it converges to a periodic traveling wave v x t c t in case k k t or k k u where v is a time periodic function and c 0 in addition the periodic traveling wave is unique up to space shifts or time shifts it is stable and asymptotically stable the average traveling speed c and the instantaneous speed v t c are also studied

canonical matrices of isometric operators on indefinite inner product spaces abstract we give canonical matrices of a pair a b consisting of a nondegenerate form b and a linear operator a satisfying b ax ay b x y on a vector space over f in the following cases f is an algebraically closed field of characteristic different from 2 or a real closed field and b is symmetric or skew symmetric f is an algebraically closed field of characteristic 0 or the skew field of quaternions over a real closed field and b is hermitian or skew hermitian with respect to any nonidentity involution on f these classification problems are wild if b may be degenerate we use a method that admits to reduce the problem of classifying an arbitrary system of forms and linear mappings to the problem of classifying representations of some quiver this method was described in v v sergeichuk classification problems for systems of forms and linear mappings math ussr izv 31 3 1988 481 501

on the structure of loop free non negative edge bipartite graphs abstract we continue the study of a class of signed graphs called finite connected loop free edge bipartite graphs bigraphs for short started in simson 2013 33 and continued in g siorek et al 2016 14 and simson zaj c 2017 38 in this paper we present an algorithmic approach to the study of non negative bigraphs with n r 2 vertices of corank r 1 that is bigraphs with the symmetric gram matrix g 1 2 g g t r m n r 1 2 z positive semi definite of rank n 1 where g m n r z is the non symmetric gram matrix of one of the main results of this paper are theorem 2 1 theorem 2 17 asserting that every such a bigraph with n r 2 vertices can be obtained from a corank zero i e positive connected loop free bigraph with n 1 vertices by an r point extension procedure u 1 u r u 1 u r along the r 1 roots u 1 u r z n of the positive definite gram form q z n z v v g v t r it is also shown that the extension procedure yields a combinatorial algorithm allowing us to obtain inductively all connected loop free non negative corank r 1 bigraphs with n r 2 vertices from the corank zero bigraphs with n 1 vertices

recurrent traveling waves in a two dimensional saw toothed cylinder and their average speed abstract we study a curvature dependent motion of plane curves in a two dimensional infinite cylinder with spatially undulating boundary the law of motion is given by v a where v is the normal velocity of the curve is the curvature and a is a positive constant the boundary undulation is assumed to be almost periodic or more generally recurrent in a certain sense we first introduce the definition of recurrent traveling waves and establish a necessary and sufficient condition for the existence of such traveling waves we then show that the traveling wave is asymptotically stable if it exists next we show that a regular traveling wave has a well defined average speed if the boundary shape is strictly ergodic finally we study what we call virtual pinning which means that the traveling wave propagates over the entire cylinder with zero average speed such a peculiar situation can occur only in non periodic environments and never occurs if the boundary undulation is periodic

congruences of a square matrix and its transpose abstract it is known that any square matrix a over any field is congruent to its transpose at stas for some nonsingular s moreover s can be chosen such that s2 i that is s can be chosen to be involutory we show that a and at are congruent over any field f of characteristic not two with involution a a the involution can be the identity a t s t as for some nonsingular s moreover s can be chosen such that s s i that is s can be chosen to be coninvolutory the short and simple proof is based on sergeichuk s canonical form for congruence math ussr izvestiya 31 3 1988 481 it follows that any matrix a over f can be represented as a eb in which e is coninvolutory and b is symmetric

distributed detection via gaussian running consensus large deviations asymptotic analysis we study by large deviations analysis the asymptotic performance of gaussian running consensus distributed detection over random networks in other words we determine the exponential decay rate of the detection error probability with running consensus at each time step each sensor averages its decision variable with the neighbors decision variables and accounts on the fly for its new observation we show that 1 when the rate of network information flow the speed of averaging is above a threshold then gaussian running consensus is asymptotically equivalent to the optimal centralized detector i e the exponential decay rate of the error probability for running consensus equals the chernoff information and 2 when the rate of information flow is below a threshold running consensus achieves only a fraction of the chernoff information rate we quantify this achievable rate as a function of the network rate of information flow simulation examples demonstrate our theoretical findings on the behavior of running consensus detection over random networks

symplectic spaces and pairs of symmetric and nonsingular skew symmetric matrices under congruence abstract let f be a field of characteristic not 2 and let a b be a pair of n n matrices over f in which a is symmetric and b is skew symmetric a canonical form of a b with respect to congruence transformations s t a s s t b s was given by sergeichuk 1988 25 up to classification of symmetric and hermitian forms over finite extensions of f we obtain a simpler canonical form of a b if b is nonsingular such a pair a b defines a quadratic form on a symplectic space that is on a vector space with scalar product given by a nonsingular skew symmetric form as an application we obtain known canonical matrices of quadratic forms and hamiltonian operators on real and complex symplectic spaces

almost periodic and almost automorphic solutions of linear differential difference equations without favard s separation condition ii abstract the well known favard s theorem states that the linear differential equation 1 x a t x f t with bohr almost periodic coefficients admits at least one bohr almost periodic solution if it has a bounded solution the main assumption in this theorem is the separation among bounded solutions of homogeneous equations 2 x b t x where b h a b b t lim n a t t n if there are bounded solutions which are non separated sometimes almost periodic solutions do not exist r johnson r ortega and m tarallo v zhikov and b levitan in this paper we prove that linear differential equation 1 with levitan almost periodic coefficients has a unique levitan almost periodic solution if it has at least one bounded solution and the bounded solutions of the homogeneous equation 3 x a t x are homoclinic to zero i e lim t t 0 for all bounded solutions of 3 if the coefficients of 1 are bohr almost periodic and all bounded solutions of all limiting equations 2 are homoclinic to zero then eq 1 admits a unique almost automorphic solution the analogue of this result for difference equations is also given we study the problem of existence of bohr levitan almost periodic solutions of eq 1 in the framework of general non autonomous dynamical systems cocycles

information bounds and quickest change detection in decentralized decision systems the quickest change detection problem is studied in decentralized decision systems where a set of sensors receive independent observations and send summary messages to the fusion center which makes a final decision in the system where the sensors do not have access to their past observations the previously conjectured asymptotic optimality of a procedure with a monotone likelihood ratio quantizer mlrq is proved in the case of additive gaussian sensor noise if the signal to noise ratios snr at some sensors are sufficiently high this procedure can perform as well as the optimal centralized procedure that has access to all the sensor observations even if all snrs are low its detection delay will be at most pi 2 1ap 57 larger than that of the optimal centralized procedure next in the system where the sensors have full access to their past observations the first asymptotically optimal procedure in the literature is developed surprisingly the procedure has the same asymptotic performance as the optimal centralized procedure although it may perform poorly in some practical situations because of slow asymptotic convergence finally it is shown that neither past message information nor the feedback from the fusion center improves the asymptotic performance in the simplest model

discrete denoising with shifts we introduce s dude a new algorithm for denoising discrete memoryless channel dmc corrupted data the algorithm which generalizes the recently introduced dude discrete universal denoiser aims to compete with a genie that has access in addition to the noisy data also to the underlying clean data and that can choose to switch up to m times between sliding window denoisers in a way that minimizes the overall loss when the underlying data form an individual sequence we show that the s dude performs essentially as well as this genie provided that m is sublinear in the size of the data when the clean data are emitted by a piecewise stationary process we show that the s dude achieves the optimum distribution dependent performance provided that the same sublinearity condition is imposed on the number of switches to further substantiate the universal optimality of the s dude we show that when the number of switches is allowed to grow linearly with the size of the data any sequence of scheme s fails to compete in the above sense using dynamic programming we derive an efficient implementation of the s dude which has complexity time and memory growing linearly with the data size and the number of switches m preliminary experimental results are presented suggesting that s dude has the capacity to improve on the performance attained by the original dude in applications where the nature of the data abruptly changes in time or space as is often the case in practice

a regularization algorithm for matrices of bilinear and sesquilinear forms abstract over a field or skew field f with an involution a a possibly the identity involution each singular square matrix a is congruent to a direct sum s as b j n 1 j n p 1 n 1 n p inwhich s is nonsingular and s s t b is nonsingular and is determined by a up to congruence and the ni by ni singular jordan blocks j n i and their multiplicities are uniquely determined by a we give a regularization algorithm that needs only elementary row operations to construct such a decomposition if f c respectively f r we exhibit a regularization algorithm that uses only unitary respectively real orthogonal transformations and a reduced form that can be achieved via a unitary congruence or congruence respectively a real orthogonal congruence the selfadjoint matrix pencil a a is decomposed by our regularization algorithm into the direct sum s a a s b b j n 1 j n 1 j n p j n p with selfadjoint summands

classification of sesquilinear forms with the first argument on a subspace or a factor space abstract let v be a vector space over a field or skew field f and let u be its subspace we study the canonical form problem for bilinear or sesquilinear forms u v f v u v f and linear mappings u v v u v u v v v u we solve it over f c and reduce it over all f to the canonical form problem for ordinary linear mappings w w and bilinear or sesquilinear forms w w f moreover we give an algorithm that realizes this reduction the algorithm uses only unitary transformations if f c which improves its numerical stability for linear mapping this algorithm can be derived from the algorithm by nazarova et al l a nazarova a v roiter v v sergeichuk v m bondarenko application of modules over a dyad for the classification of finite p groups possessing an abelian subgroup of index p and of pairs of mutually annihilating operators zap nauchn sem leningrad otdel mat inst steklov lomi 28 1972 69 92 translation in j soviet math 3 5 1975 636 654

almost periodic and almost automorphic solutions of linear differential equations the paper is dedicated to the study problem of existence of almost periodic respectively almost automorphic recurrent solutions of linear differential equations u t a t y u t f t y y y in banach space with almost periodic respectively almost automorphic recurrent coefficients where y r is a dynamical system on the metric space y in particular we prove that if the operator a y is positive and auto adjoint then for the equation 1 u t a t y u t 0 one of the following alternative is fulfilled i there exists a complete trajectory of 1 with constant positive norm ii the trivial solution of equation 1 is uniformly asymptotically stable this problem we investigate in the framework of general linear non autonomous dynamical system we apply our general results also for the functional differential equations and difference equations

change of the congruence canonical form of 2 by 2 and 3 by 3 matrices under perturbations and bundles of matrices under congruence we construct the hasse diagrams g2 and g3 for the closure ordering on the sets of congruence classes of 2 2 and 3 3 complex matrices in other words we construct two directed graphs whose vert

non parametric mean curvature evolution with boundary conditions soit un domaine borne de r n n 2 de frontiere lisse et soit a l operateur de surface minimale donne par a d i a i p a i p i 1 p 2 1 2 on considere des surfaces dont la vitesse en direction de leur normale unite est egale a la courbure moyenne on etudie l equation d evolution qui en resulte u 1 du 2 1 2 au 0 dans 0 t

a diffusion process and its applications to detecting a change in the drift of brownian motion abstract the purpose of the present paper is to make a quantitative comparison of the shiryayev roberts and page procedures we do this in the context of continuous time in order to use the machinery of diffusion processes to perform explicitly certain calculations which seem impossible in discrete time although the continuous time results are not especially good approximations to the corresponding quantities in discrete time they provide very useful comparative information on which to base selection of a stopping rule this paper is organized as follows the shiryayev roberts process is defined and shown to be a novel diffusion process with some surprising properties we also specify more precisely the basis for our comparison of the two procedures and give the results of some elementary calculations these developments contain an asymptotic evaluation we define a modification of our basic procedure and give an asymptotic evaluation of its average run length numerical comparisons and a discussion of their significance are contained next our conclusions are roughly these in simple situations where the two procedures can be directly compared neither seems dramatically superior to the other however the shiryayev roberts procedure is more easily adapted to complex circumstances and consequently warrants additional study

on the strong maximum principle and the large time behavior of generalized mean curvature flow with the neumann boundary condition abstract a generalized mean curvature flow is considered in a cylindrical domain under the right angle boundary condition its large time behavior is studied by analyzing the level set equation of the flow when the initial hypersurface is not necessarily graph like for a convex domain it is shown that the solution of the level set equation converges to a function whose level sets are perpendicular to the lateral boundary of the domain as time tends to infinity when the initial data is constant outside some bounded set for this purpose a version of the strong maximum principle is established for the level set minimal surface equation although the equation is degenerate

second order asymptotic optimality in multisensor sequential change detection a generalized multisensor sequential change detection problem is considered in which a number of possibly correlated sensors monitor an environment in real time the joint distribution of their observations is determined by a global parameter vector and at some unknown time there is a change in an unknown subset of components of this parameter vector the goal is to detect the change as soon as possible while controlling the rate of false alarms we establish the second order asymptotic optimality with respect to lorden s criterion of various generalizations of the cusum rule that is we show that their additional expected worst case detection delay relative to the one that could be achieved if the affected subset was known remains bounded as the rate of false alarm goes to 0 for any possible subset of affected components this general framework incorporates the traditional multisensor setup in which only an unknown subset of sensors is affected by the change the latter problem has a special structure which we exploit in order to obtain feasible representations of the proposed schemes we present the results of a simulation study where we compare the proposed schemes with scalable detection rules that are only first order asymptotically optimal finally in the special case that the change affects exactly one sensor we consider the scheme that runs in parallel the local cusum rules and study the problem of specifying the local thresholds

miniversal deformations of matrices under congruence and reducing transformations arnold 1971 1 constructed a miniversal deformation of a square complex matrix under similarity that is a simple normal form to which not only a given square matrix a but all matrices b close to it can be reduced by similarity transformations that smoothly depend on the entries of b we give miniversal deformations of matrices of sesquilinear forms that is of square complex matrices under congruence and construct an analytic reducing transformation to a miniversal deformation analogous results for matrices under congruence were obtained by dmytryshyn futorny and sergeichuk 2012 11

sequential multi sensor change point detection we develop a mixture procedure to monitor parallel streams of data for a change point that affects only a subset of them without assuming a spatial structure relating the data streams to one another observations are assumed initially to be independent standard normal random variables after a change point the observations in a subset of the streams of data have nonzero mean values the subset and the post change means are unknown the procedure we study uses stream specific generalized likelihood ratio statistics which are combined to form an overall detection statistic in a mixture model that hypothesizes an assumed fraction p 0 of affected data streams an analytic expression is obtained for the average run length arl when there is no change and is shown by simulations to be very accurate similarly an approximation for the expected detection delay edd after a change point is also obtained numerical examples are given to compare the suggested procedure to other procedures for unstructured problems and in one case where the problem is assumed to have a well defined geometric structure finally we discuss sensitivity of the procedure to the assumed value of p 0 and suggest a generalization

sequential detection of a change in a normal mean when the initial value is unknown three stopping rules are proposed to detect a change in a normal mean when the initial value of the mean is unknown but an estimate obtained from a training sample is available asymptotic approximations are given for the average run length when there is no change under certain hypotheses about the length of time before the change occurs and the magnitude of the change we obtain asymptotic approximations for the expected delay in detection in terms of the corresponding expected delay in the much simpler case of a known initial value the results of a monte carlo experiment supplement our asymptotic theory to yield some general conclusions about the relative merits of the three stopping rules and guidelines for choosing the size of the training sample

optimal designs for dose finding studies with an active control summary dose finding studies often compare several doses of a new compound with a marketed standard treatment as an active control in the past however research has focused mostly on experimental designs for placebo controlled dose finding studies to the best of our knowledge optimal designs for dose finding studies with an active control have not been considered so far as the statistical analysis for an active controlled dose finding study can be formulated in terms of a mixture of two regression models the related design problem is different from what has been investigated before in the literature we present a rigorous approach to the problem of determining optimal designs for estimating the smallest dose achieving the same treatment effect as the active control we determine explicitly the locally optimal designs for a broad class of models employed in such studies we also discuss robust design strategies and determine related bayesian and standardized minimax optimal designs we illustrate the results by investigating alternative designs for a clinical trial which has recently appeared in a consulting project of one of the authors

congruence and conjunctivity of matrices to their adjoints abstract every square matrix over a field f is involutorily congruent over f to its transpose and hence each such matrix is the product of a symmetric matrix and an involutory matrix over f in the usual complex case every matrix which is conjunctive with its adjoint conjugate transpose is involutorily conjunctive with its adjoint and hence is the product of a hermitian matrix and an involutory matrix furthermore every such matrix is conjunctive with a real matrix these three conditions on a matrix 1 being conjunctive with its adjoint 2 being involutorily conjunctive with its adjoint and 3 being conjunctive with a real matrix are studied in the more general context of a field f with involution and it is shown in general that 3 implies 2 that 2 implies 3 if char f 2 a 2 2 counterexample exists for each f with char f 2 and that 1 does not in general imply 2 a 2 2 counterexample in the complexification of the rational field is presented the problem of deciding which matrices satisfy 2 is equivalent even in this general context to the problem of deciding which pairs of self adjoint hermitian matrices are involutorily conjunctive for the general 2 2 case the three conditions are characterized in terms of norms

sequential changepoint detection in quality control and dynamical systems after a brief survey of a large variety of sequential detection procedures that are widely scattered in statistical references on quality control and engineering references on fault detection and signal processing we study some open problems concerning these procedures and introduce a unified theory of sequential changepoint detection this theory leads to a class of sequential detection rules which are not too demanding in computational and memory requirements for on line implementation and yet are nearly optimal under several performance criteria

coxeter spectral classification of almost tp critical one peak posets using symbolic and numeric computations following an idea of spectral graph theory we give several characterizations of finite one peak posets i with almost p critical quadratic tits form q i zi z by applying linear algebra methods combinatorial algorithms an old idea of s a ovsienko and recent results by v m bondarenko and m v stepochkina our study is mainly motivated by an important application of the quadratic tits form of a poset in constructing linear algebra invariants that measure a geometric complexity of nazarova roiter matrix problems over a field k in particular our study is inspired by a well known result of ju a drozd explained in introduction one of our main results asserts that given a finite poset i 1 n t such that is its unique maximal element and q i is almost p critical we have i the form q i is non negative and the subgroup kerq i v zn 1 q i v 0 of zi zn 1 is generated by a vector h h1 hn h with h1 0 hn 0 ii 5 i 9 and the coxeter polynomial coxi t z t coincides with the coxeter polynomial f i t of a uniquely determined euclidean diagram i d 4 e 6 e 7 e 8 iii the z bilinear tits form b i zn 1 zn 1 z of i is z equivalent to the gram z bilinear form b i zn 1 zn 1 z of the euclidean diagram i iv there is a z invertible matrix c such that c itr ctr c i c and c2 e where c i mn 1 z is the tits matrix of i and v i is one of the posets l1 l132 listed in the paper in particular we get in an alternative way the list of p critical posets t in the form t i presented by v m bondarenko and m v stepochkina 6 and we determine their coxeter euclidean types there are 115 one peak posets i l1 l115 with q i p critical and 17 exceptional posets i l116 l132 with q i almost p critical but not p critical our lists are obtained by using computer algebra tools mainly symbolic computation in maple and numeric computation in c

multiphase thermomechanics with interfacial structure 2 evolution of an isothermal interface paper 1 1988 g 1 of this series began an investigation whose goal is a thermomechanics of two phase continua based on gibbs s notion of a sharp phase interface endowed with thermomechanical structure in that paper a balance law balance of capillary forces was introduced and then applied in conjunction with suitable statements of the first two laws of thermodynamics the chief results are thermodynamic restrictions on constitutive equations exact and approximate free boundary conditions at the interface and a hierarchy of free boundary problems the simplest versions of these problems the mullins sekerka problems are essentially the classical stefan problem with the free boundary condition u 0 for the temperature replaced by the condition u h k where k is the curvature of the free boundary and h 0 is a material constant this dependence on curvature renders the problem difficult and apart from numerical studies involving linearization stability there are almost no supporting theoretical results

classification of linear mappings between indefinite inner product spaces abstract let a u v be a linear mapping between vector spaces u and v over a field or skew field f with symmetric or skew symmetric or hermitian forms b u u f and c v v f we classify the triples a b c if f is r or c or the skew field of quaternions h we also classify the triples a b c up to classification of symmetric forms and hermitian forms if the characteristic of f is not 2

skew symmetric matrix polynomials and their smith forms we characterize the smith form of skew symmetric matrix polynomials over an arbitrary field f showing that all elementary divisors occur with even multiplicity restricting the class of equivalence transformations to unimodular congruences a smith like skew symmetric canonical form for skew symmetric matrix polynomials is also obtained these results are used to analyze the eigenvalue and elementary divisor structure of matrices expressible as products of two skew symmetric matrices as well as the existence of structured linearizations for skew symmetric matrix polynomials by contrast with other classes of structured matrix polynomials e g alternating or palindromic polynomials every regular skew symmetric matrix polynomial is shown to have a structured strong linearization while there are singular skew symmetric polynomials of even degree for which a structured linearization is impossible for each odd degree we develop a skew symmetric companion form that uniformly provides a structured linearization for every regular and singular skew symmetric polynomial of that degree finally the results are applied to the construction of minimal symmetric factorizations of skew symmetric rational matrices

a gram classification of non negative corank two loop free edge bipartite graphs we continue the coxeter spectral study of finite connected loop free edge bipartite graphs with m 2 3 vertices a class of signed graphs started in simson 2013 49 by means of the non symmetric gram matrix g mm 2 z of its symmetric gram matrix g 12 g g tr mm 2 12z the gram quadratic form q zm 2 z and the coxeter spectrum specc c i e the complex spectrum of the coxeter matrix cox g g tr gl m 2 z in the present paper we study non negative edge bipartite graphs of corank two in the sense that the symmetric gram matrix g mm 2 z of is positive semi definite of rank m 1 one of our aims is to get a complete classification of all connected corank two loop free edge bipartite graphs with m 2 3 vertices up to the weak gram z congruence z where z means that g btr g b for some b mm 2 z such that det b 1 by one vertex extensions of the simply laced euclidean diagrams a m m 1 d m m 4 e 6 e 7 e 8 we construct a family of connected loop free corank two diagrams a m 2 d m 2 e 6 2 e 7 2 e 8 2 called simply extended euclidean diagrams such that they classify all connected corank two loop free edge bipartite graphs with m 2 3 vertices up to the weak gram z congruence z a structure of connected corank two loop free edge bipartite graphs is described it is shown that every such contains a connected positive edge bipartite subgraph that is z congruent with a simply laced dynkin diagram dyn called the dynkin type of such that is a two point extension u w of along two roots u w of the positive definite gram form q zm z this yields a combinatorial algorithm u w u w allowing us to construct all connected corank two loop free edge bipartite graphs with m 2 3 vertices and d dyn from the triples u w where is positive of the dynkin type d and u w are roots of the positive definite gram form q zm z

similarity of some dual operators in characteristic two abstract let k be a field of characteristic two with involution x x let v be a finite dimensional hermitian space over k and v v a linear operator whose dual is 1 we prove that 1 1 where is an isometry and 2 1 if e is a given subspace of v then can also be chosen to stabilize e but the equality 1 1 is only true modulo a combination of certain bracket operators as a corollary we solve the following congruence problem given a square matrix a over k there is a non singular matrix s satisfying a t s t as and s s 1

distributed linear parameter estimation asymptotically efficient adaptive strategies this paper considers the problem of distributed adaptive linear parameter estimation in multiagent inference networks local sensing model information is only partially available at the agents and interagent communication is assumed to be unpredictable the paper develops a generic mixed time scale stochastic procedure consisting of simultaneous distributed learning and estimation in which the agents adaptively assess their relative observation quality over time and fuse the innovations accordingly under rather weak assumptions on the statistical model and the interagent communication it is shown that by properly tuning the consensus potential with respect to the innovation potential the asymptotic information rate loss incurred in the learning process may be made negligible as such it is shown that the agent estimates are asymptotically efficient in that their asymptotic covariance coincides with that of a centralized estimator the inverse of the centralized fisher information rate for gaussian

stratification of moduli spaces of lie algebras similar matrices and bilinear forms abstract in this paper the authors apply a stratification of moduli spaces of complex lie algebras to analyzing the moduli spaces of n n matrices under scalar similarity and bilinear forms under the cogredient action for similar matrices we give a complete description of a stratification of the space by some very simple projective orbifolds of the form p n g where g is a subgroup of the symmetric group n 1 acting on p n by permuting the projective coordinates for bilinear forms we give a similar stratification up to dimension 4

topological classification of sesquilinear forms reduction to the nonsingular case abstract two sesquilinear forms c m c m c and c n c n c are called topologically equivalent if there exists a homeomorphism c m c n i e a continuous bijection whose inverse is also a continuous bijection such that x y x y for all x y c m r a horn and v v sergeichuk in 2006 constructed a regularizing decomposition of a square complex matrix a that is a direct sum s a s r j n 1 j n p in which s and r are nonsingular and each j n i is the n i by n i singular jordan block in this paper we prove that and are topologically equivalent if and only if the regularizing decompositions of their matrices coincide up to permutation of the singular summands j n i and replacement of r c r r by a nonsingular matrix r c r r such that r and r are the matrices of topologically equivalent forms c r c r c analogous results for bilinear forms over c and over r are also obtained

the heat equation shrinks embedded plane curves to round points soit c 0 s 1 r 2 une courbe lisse plongee dans le plan alors c s 1 0 t r 2 existe en satisfaisant c t k n ou k est la courbure de c et n est son vecteur unite normal entrant c t est lisse pour tout t il converge vers un point quand t t et sa forme limite quand t t est un cercle rond avec convergence dans norme c

approximate tail probabilities for the maxima of some random fields abstract hogan and siegmund 1986 adapt the method developed by pickands 1969 qualls and watanabe 1973 and bickel and rosenblatt 1973 to obtain explicit large deviation approximations for the maxima of several gaussian random fields arising in statistics using a special argument for one particular case they suggest a heuristic second order approximation for that case and they show by a monte carlo experiment that the second order approximation frequently gives considerably better numerical results the purpose of this paper is to show that the method developed by woodroofe 1976 1982 for problems in one dimensional time can be adapted to study maxima of random fields overall it involves simpler computations than the previous method and consequently seems potentially capable of delivering a genuine second order approximation should one seem desirable

efficient scalable schemes for monitoring a large number of data streams the sequential changepoint detection problem is studied in the context of global online monitoring of a large number of independent data streams we are interested in detecting an occurring event as soon as possible but we do not know when the event will occur nor do we know which subset of data streams will be affected by the event a family of scalable schemes is proposed based on the sum of the local cumulative sum cusum statistics from each individual data stream and is shown to asymptotically minimize the detection delays for each and every possible combination of affected data streams subject to the global false alarm constraint the usefulness and limitations of our asymptotic optimality results are illustrated by numerical simulations and heuristic arguments the appendices contain a probabilistic result on the first epoch to simultaneous record values for multiple independent random walks

convergence asymptotic periodicity and finite point blow up in one dimensional semilinear heat equations abstract we consider the initial value problem for the semilinear heat equation u t u xx f u t 0 x l t 0 under the dirichlet the neumann or the periodic boundary conditions we show that each solution whether it exists globally for t 0 or blows up in a finite time possesses an asymptotic profile in a certain sense and tends to this profile as time increases in the special case where f u t t f u t for some t 0 among other things the above statement is interpreted as saying that any bounded global solution converges as t to a time t periodic solution having some specific spatial structure in the case where the solution blows up in a finite time say at t t 0 assuming simply that f is a smooth function satisfying some growth conditions and that the initial data is a nonconstant bounded function we prove that the blow up set is a finite set and that lim t t 0 u x t x exists with being a smooth function having at most finitely many singular points

opuc cmv matrices and perturbations of measures supported on the unit circle abstract let us consider a hermitian linear functional defined on the linear space of laurent polynomials with complex coefficients in the literature canonical spectral transformations of this functional are studied the aim of this research is focused on perturbations of hermitian linear functionals associated with a positive borel measure supported on the unit circle some algebraic properties of the perturbed measure are pointed out in a constructive way we discuss the corresponding sequences of orthogonal polynomials as well as the connection between the associated verblunsky coefficients then the structure of the matrices of the perturbed linear functionals which is the main tool for the comparison of their corresponding cmv matrices is deeply analyzed from the comparison between different cmv matrices other families of perturbed verblunsky coefficients will be considered we introduce a new matrix named fundamental matrix that is a tridiagonal symmetric unitary matrix containing basic information about the family of orthogonal polynomials however we show that it is connected to another family of orthogonal polynomials through the takagi decomposition

multiscale change point inference summary we introduce a new estimator the simultaneous multiscale change point estimator smuce for the change point problem in exponential family regression an unknown step function is estimated by minimizing the number of change points over the acceptance region of a multiscale test at a level the probability of overestimating the true number of change points k is controlled by the asymptotic null distribution of the multiscale test statistic further we derive exponential bounds for the probability of underestimating k by balancing these quantities will be chosen such that the probability of correctly estimating k is maximized all results are even non asymptotic for the normal case on the basis of these bounds we construct asymptotically honest confidence sets for the unknown step function and its change points at the same time we obtain exponential bounds for estimating the change point locations which for example yield the minimax rate up to a log term finally the simultaneous multiscale change point estimator achieves the optimal detection rate of vanishing signals as n even for an unbounded number of change points we illustrate how dynamic programming techniques can be employed for efficient computation of estimators and confidence regions the performance of the multiscale approach proposed is illustrated by simulations and in two cutting edge applications from genetic engineering and photoemission spectroscopy

structure and a coxeter dynkin type classification of corank two non negative posets inspired by famous results of drozd 1974 13 nazarova ro ter 1975 42 and bondarenko stepochkina 2005 8 on the importance of finite posets i with the non negative tits quadratic form q i zi z in the representation theory we study the finite non negative posets i of corank two that is such posets i that the symmetric gram matrix gi 12 ci citr mi q of i is positive semi definite of corank two where ci mi z is the incidence matrix of i a structure of such posets is described in the paper it is shown that every such a poset i can be obtained in a natural way from a corank zero poset j and a pair of roots of j following main ideas of the coxeter spectral analysis of posets we associate with any connected corank two poset i the incidence quadratic form qi zi z the coxeter matrix coxi ci ci tr the coxeter polynomial coxi t det t e coxi z t its complex coxeter spectrum specci the coxeter number ci n the reduced coxeter number c i n the incidence defect homomorphism i zi kerqi zi and a simply laced dynkin diagram dyni in particular we study the question when the coxeter spectrum specci of i together with dyni determines the incidence matrix ci of i hence the poset i uniquely up to a z congruence by applying linear algebra techniques and a combinatorial computer aided analysis of posets we prove that the question has an affirmative answer for connected corank two posets i with at most 15 elements a complete list of such posets is presented and their coxeter type invariants are computed it follows that i 6 there is only one such a poset i of cardinality 6 and 8 such posets of cardinality 7 up to the poset duality i iop the existence of a i mesh translation quiver ri kerqi i on the set ri kerqi is discussed where ri v zi qi v 1 is the set of roots of qi

matrices that are self congruent only via matrices of determinant one abstract ocovic and szechtman d ocovic f szechtman characterization of bilinear spaces with unimodular isometry group proc amer math soc 133 2005 2853 2863 considered a vector space v endowed with a bilinear form they proved that all isometries of v over a field f of characteristic not 2 have determinant 1 if and only if v has no orthogonal summands of odd dimension the case of characteristic 2 was also considered their proof is based on riehm s classification of bilinear forms coakley et al e s coakley f m dopico c r johnson matrices with orthogonal groups admitting only determinant one linear algebra appl 428 2008 796 813 gave another proof of this criterion over r and c using thompson s canonical pairs of symmetric and skew symmetric matrices for congruence let m be the matrix of the bilinear form on v we give another proof of this criterion over f using our canonical matrices for congruence and obtain necessary and sufficient conditions involving canonical forms of m for congruence of m t m for equivalence and of m t m if m is nonsingular for similarity

one peak posets with positive quadratic tits form their mesh translation quivers of roots and programming in maple and python by applying linear algebra and computer algebra tools we study finite posets with positive quadratic tits form our study is motivated by applications of matrix representations of posets in representation theory where a matrix representation of a partially ordered set t p1 pn with a partial order means a block matrix m m1 m2 mn over a field k of size d d1 dn determined up to all elementary row transformations elementary column transformations within each of the substrips m1 m2 mn and additions of linear combinations of columns of mi to columns of mj if pi pj drozd 10 proves that t has only a finite number of direct sum indecomposable representations if and only if its quadratic tits form q x1 xn x x12 xn2 x 2 pi pjxixj x x1 xn is weakly positive i e q a1 an a 0 for all non zero vectors a1 an a with integral non negative coefficients in this case there exists an indecomposable representation m of size d d1 dn if and only if d1 dn d is a root of q i e q d1 dn d 1 bondarenko and stepochkina 8 give a list of posets t with positive tits form consisting of four infinite series and 108 posets up to duality in the paper we construct this list in an alternative way by applying computational algorithms implemented in maple and python moreover given any poset t of the list we show that a the coxeter polynomial coxi t of the poset i t obtained from t by adding a unique maximal element called a peak is a coxeter polynomial f i t of a simply laced dynkin diagram i an dn e6 e7 e8 and b the set of orbits of the set rq of integral roots of q admits a mesh translation quiver structure rq of a cylinder shape where is the coxeter transformation of i t in the sense of drozd 10

parabolic equations for curves on surfaces part i curves with p integrable curvature this is the first of a two part paper in which we develop a theory of parabolic equations for curves on surfaces which can be applied to the so called curve shortening of flow by mean curvature problem as well as to a number of models for phase transitions in two dimensions we introduce a class of equations for which the initial value problem is solvable for initial data with p integrable curvature and we also give estimates for the rate at which the p norms of the curvature must blow up if the curve becomes singular in finite time a detailed discussion of the way in which solutions can become singular and a method for continuing the solution through a singularity will be the subject of the second part

canonical matrices of bilinear and sesquilinear forms abstract canonical matrices are given for i bilinear forms over an algebraically closed or real closed field ii sesquilinear forms over an algebraically closed field and over real quaternions with any nonidentity involution and iii sesquilinear forms over a field f of characteristic different from 2 with involution possibly the identity up to classification of hermitian forms over finite extensions of f the canonical matrices are based on any given set of canonical matrices for similarity over f a method for reducing the problem of classifying systems of forms and linear mappings to the problem of classifying systems of linear mappings is used to construct the canonical matrices this method has its origins in representation theory and was devised in v v sergeichuk classification problems for systems of forms and linear mappings math ussr izv 31 1988 481 501

the heat equation shrinking convex plane curves soient m et m des varietes de riemann et f m m une application reguliere si m est une courbe convexe plongee dans le plan r 2 l equation de la chaleur contracte m a un point

miniversal deformations of matrices of bilinear forms arnold v i arnold on matrices depending on parameters russian math surveys 26 2 1971 29 43 constructed miniversal deformations of square complex matrices under similarity that is a simp

canonical forms for complex matrix congruence and congruence abstract canonical forms for congruence and congruence of square complex matrices were given by horn and sergeichuk in linear algebra appl 389 2004 347 353 based on sergeichuk s paper math ussr izvestiya 31 3 1988 481 501 which employed the theory of representations of quivers with involution we use standard methods of matrix analysis to prove directly that these forms are canonical our proof provides explicit algorithms to compute all the blocks and parameters in the canonical forms we use these forms to derive canonical pairs for simultaneous congruence of pairs of complex symmetric and skew symmetric matrices as well as canonical forms for simultaneous congruence of pairs of complex hermitian matrices

a multivariate change point model for statistical process control multivariate statistical process control spc carries out ongoing checks to ensure that a process is in control these checks on the process are traditionally done by t2 multivariate cusum and multivariate exponentially weighted moving average control charts these traditional spc charts assume that the in control true parameters are known exactly and use these assumed true values to set the control limits the reality however is that true parameter values are seldom if ever known exactly rather they are commonly estimated from a phase i sample it is increasingly recognized that this phase i study needs to involve large samples if the parameter estimates are to provide run behavior matching that of the known parameter situation but apart from the general undesirability of large and thus expensive studies preliminary to actual charting some industrial settings have a paucity of relevant data for estimating the process parameters an attractive alternative to traditional charting methods when mon

congruence of multilinear forms abstract let f u u k g v v k be two n linear forms with n 2 on finite dimensional vector spaces u and v over a field k we say that f and g are symmetrically equivalent if there exist linear bijections 1 n u v such that f u 1 u n g i 1 u 1 i n u n for all u 1 u n u and each reordering i 1 i n of 1 n the forms are said to be congruent if 1 n let f and g be symmetrically equivalent we prove that i if k c then f and g are congruent ii if k r f f 1 f s 0 g g 1 g r 0 and all summands f i and g j are nonzero and direct sum indecomposable then s r and after a suitable reindexing f i is congruent to g i

glrt based spectrum sensing for cognitive radio with prior information we consider the spectrum sensing problem in cognitive radio networks we offer a framework for optimal joint detection and parameter estimation when the secondary users have only a small number of signal samples we discuss the finite sample optimality of the generalized likelihood ratio test glrt and derive the corresponding glrt spectrum sensing algorithms by exploiting the statistics of the received signal and the prior information on the channel noise as well as the data signal an iterative glrt sensing algorithm and a simple non iterative glrt sensing algorithm are developed for slow and fast fading channels respectively with the latter also serving as an approximate sensing method for slow fading channels the proposed techniques are also extended for spectrum sensing in orthogonal frequency division multiple access ofdma systems and in multiple input multiple output mimo systems it is seen that the proposed simple non iterative fast fading glrt sensing algorithm offers the best performance in all systems under considerations including slow fading channels fast fading channels ofdma systems and mimo systems and it significantly outperforms several state of the art spectrum sensing methods in these systems when there is noise uncertainty

tridiagonal canonical matrices of bilinear or sesquilinear forms and of pairs of symmetric skew symmetric or hermitian forms abstract tridiagonal canonical forms of square matrices under congruence or congruence pairs of symmetric or skew symmetric matrices under congruence and pairs of hermitian matrices under congruence are given over an algebraically closed field of characteristic different from 2

using the generalized likelihood ratio statistic for sequential detection of a change point we study sequential detection of a change point using the generalized likelihood ratio statistic for the special case of detecting a change in a normal mean with known variance we give approximations to the average run lengths and compare our procedure to standard cusum tests and combined cusum shewhart tests several examples indicating extensions to problems involving multiple parameters are discussed

homogenization limit of a parabolic equation with nonlinear boundary conditions abstract consider the parabolic equation e u t a u x u x x f u x 1 x 1 t 0 with nonlinear boundary conditions u x 1 t g u 1 t e nbc u x 1 t g u 1 t e where e 0 is a parameter g is a function which takes values near its supremum frequently each almost periodic function is a special example of g we consider a time global solution u e of e nbc and show that its homogenization limit as e 0 is the solution of e with linear boundary conditions lbc x 1 t sup g x 1 t sup g provided moves upward monotonically when g is almost periodic lou preprint 21 obtained the unique almost periodic traveling wave u e of e nbc this paper proves that the homogenization limit of u e is a classical traveling wave of e lbc

a square matrix is congruent to its transpose for any matrix x let xdenote its transpose we show that if a is an n by n matrix over afi eldk t hena and aare congruent over k i e pap afor some p gln k