solution to a conjecture on the maximal energy of bipartite bicyclic graphs abstract the energy of a simple graph g denoted by e g is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix let c n denote the cycle of order n and p n 6 6 the graph obtained from joining two cycles c 6 by a path p n 12 with its two leaves let b n denote the class of all bipartite bicyclic graphs but not the graph r a b which is obtained from joining two cycles c a and c b a b 10 and a b 2 mod 4 by an edge in i gutman d vidovic quest for molecular graphs with maximal energy a computer experiment j chem inf sci 41 2001 1002 1005 gutman and vidovic conjectured that the bicyclic graph with maximal energy is p n 6 6 for n 14 and n 16 in x li j zhang on bicyclic graphs with maximal energy linear algebra appl 427 2007 87 98 li and zhang showed that the conjecture is true for graphs in the class b n however they could not determine which of the two graphs r a b and p n 6 6 has the maximal value of energy in b furtula s radenkovic i gutman bicyclic molecular graphs with the greatest energy j serb chem soc 73 4 2008 431 433 numerical computations up to a b 50 were reported supporting the conjecture so it is still necessary to have a mathematical proof to this conjecture this paper is to show that the energy of p n 6 6 is larger than that of r a b which proves the conjecture for bipartite bicyclic graphs for non bipartite bicyclic graphs the conjecture is still open

on tests against one sided hypotheses in some generalized linear models one sided hypotheses arise naturally in many situations when testing against such hypotheses it is desirable to take the available one sided information into account rather than simply applying a two sided test what we expect to gain by applying a one sided test instead of a two sided test is an increase in the power of the test we consider various tests of one sided hypotheses in a class of models that includes generalized linear and cox regression models the tests are likelihood ratio wald score generalized distance and a pearson chi square it is shown that these test statistics are asymptomatically equivalent in terms of local power this is a generalization of the well known corresponding result for two sided alternatives two examples are also discussed they are on 1 testing for interaction in binomial response models and 2 comparison of treatments with ordinal categorical responses

unicyclic graphs with maximal energy abstract let 1 2 n be the eigenvalues of a graph g of order n the energy of g is defined as e g 1 2 n let p n 6 6 be the graph obtained from two copies of c 6 joined by a path p n 10 b n be the class of all bipartite bicyclic graphs that are not the graph obtained from two cycles c a and c b a b 10 and a b 2 mod 4 joined by an edge in this paper we show that p n 6 6 is the graph with maximal energy in b n which gives a partial solution to gutman s conjecture in gutman and vidovic 2001 i gutman d vidovic quest for molecular graphs with maximal energy a computer experiment j chem inf sci 41 2001 1002 1005

inference and testing on the boundary in extended constant conditional correlation garch models we consider inference and testing in extended constant conditional correlation garch models in the case where the true parameter vector is a boundary point of the parameter space this is of particular importance when testing for volatility spillovers in the model the large sample properties of the qmle are derived together with the limiting distributions of the related lr wald and lm statistics due to the boundary problem these large sample properties become nonstandard the size and power properties of the tests are investigated in a simulation study as an empirical illustration we test for no volatility spillovers between foreign exchange rates

bootstrap inference on the boundary of the parameter space with application to conditional volatility models abstract it is a well established fact that with an unknown number of nuisance parameters at the boundary testing a null hypothesis on the boundary of the parameter space is infeasible in practice as the limiting distributions of standard test statistics are non pivotal in particular likelihood ratio statistics have limiting distributions which can be characterized in terms of quadratic forms minimized over cones where the shape of the cones depends on the unknown location of the possibly multiple model parameters not restricted by the null hypothesis we propose to solve this inference problem by a novel bootstrap which we show to be valid under general conditions irrespective of the presence of unknown nuisance parameters on the boundary that is the new bootstrap replicates the unknown limiting distribution of the likelihood ratio statistic under the null hypothesis and is bounded in probability under the alternative the new bootstrap approach which is very simple to implement is based on shrinkage of the parameter estimates used to generate the bootstrap sample toward the boundary of the parameter space at an appropriate rate as an application of our general theory we treat the problem of inference in finite order arch models with coefficients subject to inequality constraints extensive monte carlo simulations illustrate that the proposed bootstrap has attractive finite sample properties both under the null and under the alternative hypothesis

sharp bounds on the distance spectral radius and the distance energy of graphs the d eigenvalues 1 2 p of a graph g are the eigenvalues of its distance matrix d and form the d spectrum of g denoted by specd g the greatest d eigenvalue is called the d spectral radius of g denoted by 1 the d energy ed g of the graph g is the sum of the absolute values of its d eigenvalues in this paper we obtain some lower bounds for 1 and characterize those graphs for which these bounds are best possible we also obtain an upperbound for ed g and determine those maximal d energy graphs

on the asymptotics of constrained m estimation limit theorems for an m estimate constrained to lie in a closed subset of r d are given under two different sets of regularity conditions a consistent sequence of global optimizers converges under chernoff regularity of the parameter set a n consistent sequence of local optimizers converges under clarke regularity of the parameter set in either case the asymptotic distribution is a projection of a normal random vector on the tangent cone of the parameter set at the true parameter value limit theorems for the optimal value are also obtained agreeing with chernoff s result in the case of maximum likelihood with global optimizers

an edge grafting theorem on the energy of unicyclic and bipartite graphs the energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix the edge grafting operation on a graph is certain kind of edge moving between two pendant paths starting from the same vertex in this paper we show how the graph energy changes under the edge grafting operations on unicyclic and bipartite graphs we also give some applications of this result on the comparison of graph energies between unicyclic or bipartite graphs

escape by diffusion from a square well across a square barrier abstract the problem of escape of a particle by diffusion from a square potential well across a square barrier is studied on the basis of the one dimensional smoluchowski equation for the space and time dependent probability distribution for the model potential the smoluchowski equation is solved exactly by a laplace transform with respect to time in the limit of a high barrier the rate of escape is given by an asymptotic result similar to that derived by kramers for a curved well and a curved barrier an approximate analytic formula is derived for the outward time dependent probability current in terms of the width and depth of the well and the width and height of the barrier a similar expression holds for the complete probability distribution

diffusion and convection after escape from a potential well abstract the escape by diffusion of a particle from a potential well in one dimension is strongly influenced by the application of a field in the adjacent half space at long times the probability distribution becomes a uniformly moving and steadily broadening gaussian in this half space the mean time of escape from the well is given by a simple expression in terms of the mean first passage time and the coefficient of the long time tail in the occupation probability of the well in the absence of the field transient effects in space and time are studied in explicit form for a parabolic potential well

brownian dynamics divergence of mobility tensor abstract the mobility tensor for many spheres suspended in a viscous fluid is considered an analytical formula for the divergence of this tensor is derived it is then applied in calculations of long time collective diffusion coefficient of hard sphere suspension by means of brownian dynamics method

asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions abstract large sample properties of the likelihood function when the true parameter value may be on the boundary of the parameter space are described specifically the asymptotic distribution of maximum likelihood estimators and likelihood ratio statistics are derived these results generalize the work of moran 1971 chant 1974 and chernoff 1954 some of chant s results are shown to be incorrect the approach used in deriving these results follows from comments made by moran and chant the problem is shown to be asymptotically equivalent to the problem of estimating the restricted mean of a multivariate gaussian distribution from a sample of size 1 in this representation the gaussian random variable corresponds to the limit of the normalized score statistic and the estimate of the mean corresponds to the limit of the normalized maximum likelihood estimator thus the limiting distribution of the maximum likelihood estimator is the same as the distribution of the projection of the gaussian random v

non standard inference for augmented double autoregressive models with null volatility coefficients abstract this paper considers an augmented double autoregressive dar model which allows null volatility coefficients to circumvent the over parameterization problem in the dar model since the volatility coefficients might be on the boundary the statistical inference methods based on the gaussian quasi maximum likelihood estimation gqmle become non standard and their asymptotics require the data to have a finite sixth moment which narrows the applicable scope in studying heavy tailed data to overcome this deficiency this paper develops a systematic statistical inference procedure based on the self weighted gqmle for the augmented dar model except for the lagrange multiplier test statistic the wald quasi likelihood ratio and portmanteau test statistics are all shown to have non standard asymptotics the entire procedure is valid as long as the data are stationary and its usefulness is illustrated by simulation studies and one real example

testing when a parameter is on the boundary of the maintained hypothesis this paper considers testing problems where several of the standard regularity conditions fail to hold we consider the case where i parameter vectors in the null hypothesis may lie on the boundary of the maintained hypothesis and ii there may be a nuisance parameter that appears under the alternative hypothesis but not under the null the paper establishes the asymptotic null and local alternative distributions of quasi likelihood ratio rescaled quasi likelihood ratio wald and score tests in this case the results apply to tests based on a wide variety of extremum estimators and apply to a wide variety of models examples treated in the paper are i tests of the null hypothesis of no conditional heteroskedasticity in a garch 1 1 regression model and ii tests of the null hypothesis that some random coefficients have variances equal to zero in a random coefficients regression model with possibly correlated random coefficients

transient survival probability of a particle diffusing in a disordered system of perfectly absorbing spherical sinks abstract the transient survival probability of a particle diffusing in a disordered system of perfectly absorbing non overlapping spherical sinks is studied by use of a self consistent cluster expansion for its laplace transform the self consistent cluster expansion which in principle is exact is formulated in two alternative versions corresponding to different ways of handling the instantaneous absorption of particles created initially inside a sink in each version of the theory the cluster expansion is truncated at either the singlet or the pair level so that four different approximations to the exact result are obtained an approximation on the pair level constitutes an improvement to the corresponding singlet approximation it is shown that at moderate volume fraction of the sink system the two pair approximations yield almost identical results for the survival probability all four approximations lead to a band gap in the spectrum of relaxation rates followed by a continuum extending to infinity

escape by diffusion from a parabolic well across a parabolic barrier abstract the problem of escape of a particle by diffusion from a parabolic potential well across a parabolic barrier adjacent to free space is studied on the basis of the one dimensional smoluchowski equation for the space and time dependent probability distribution for the model potential the smoluchowski equation is solved exactly by a laplace transform with respect to time for the initial condition that at time zero the probability distribution is given by a thermal equilibrium distribution in the well in the limit of a high barrier the rate of escape is given by an asymptotic result due to kramers for a potential barrier of moderate height there are significant corrections to the asymptotic result the probability that the particle resides in the well barrier region at time t decays with a 1 t long time tail

on the asymptotics of constrained local m estimators we discuss in this paper asymptotics of locally optimal solutions of maximum likelihood and more generally m estimation procedures in cases where the true value of the parameter vector lies on the boundary of the parameter set s we give a counterexample showing that regularity of s in the sense of clarke is not sufficient for asymptotic equivalence of n consistent locally optimal m estimators we argue further that stronger properties such as so called near convexity or prox regularity of s are required in order to ensure that any two n consistent locally optimal m estimators have the same asymptotics

on tricyclic graphs of a given diameter with minimal energy abstract the energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph let g n d be the class of tricyclic graphs g on n vertices with diameter d and containing no vertex disjoint odd cycles c p c q of lengths p and q with p q 2 mod 4 in this paper we characterize the graphs with minimal energy in g n d

asymptotics of cholesky garch models and time varying conditional betas this paper proposes a new model with time varying slope coefficients our model called char is a cholesky garch model based on the cholesky decomposition of the conditional variance matrix introduced by pourahmadi 1999 in the context of longitudinal data we derive stationarity and invertibility conditions and prove consistency and asymptotic normality of the full and equation by equation qml estimators of this model we then show that this class of models is useful to estimate conditional betas and compare it to the approach proposed by engle 2016 finally we use real data in a portfolio and risk management exercise we find that the char model outperforms a model with constant betas as well as the dynamic conditional beta model of engle 2016

adaptive inference for a semiparametric generalized autoregressive conditional heteroskedasticity model abstract this paper considers a semiparametric generalized autoregressive conditional heteroskedasticity s garch model for this model we first estimate the time varying long run component for unconditional variance by the kernel estimator and then estimate the non time varying parameters in garch type short run component by the quasi maximum likelihood estimator qmle we show that the qmle is asymptotically normal with the parametric convergence rate next we construct a lagrange multiplier test for linear parameter constraint and a portmanteau test for model checking and obtain their asymptotic null distributions our entire statistical inference procedure works for the non stationary data with two important features first our qmle and two tests are adaptive to the unknown form of the long run component second our qmle and two tests share the same efficiency and testing power as those in variance targeting method when the s garch model is stationary

on the minimal energy of graphs abstract the energy of a graph is the sum of the absolute values of the eigenvalues of the graph which is used to approximate the total electron energy of the molecule in this paper we determine the n e graphs with minimal energy for e n 1 and n 2 which is giving a complete solution to the conjecture for e n 1 and e n 2 proposed by caporossi et al in 4 moreover we determine the graphs with the minimal and second minimal energies for n 1 e 3 n 2 3 and the unique graph with minimal energy for 3 n 5 2 e 2 n 4 among all quasi trees with n vertices and e edges respectively

a score test against one sided alternatives abstract a score type statistic t s is introduced for testing h 0 against k 0 and more general one sided hypotheses when nuisance parameters may be present is a vector parameter the main advantages of t s are that it requires estimation of the model only under the null hypothesis and that it is asymptotically equivalent to the likelihood ratio statistic these are precisely the reasons for the popularity of the score tests for testing against two sided alternatives in this sense t s preserves the main attractive features of the classical two sided score test the theoretical results are presented in a general framework where the likelihood based score function is replaced by an estimating function so that the test is applicable even if the exact population distribution is unknown computation of t s is simplified by the fact that it can be computed easily once the corresponding two sided statistic has been computed the relevance and simplicity of t s are illustrated by discussing

dynamic computer simulation of concentrated hard sphere suspensions ii re analysis of mean square displacement data in view of recent theoretical results by cichocki and felderhof we reevaluate previously published simulation data for the mean square displacement of a particle in a hard sphere suspension without hydrodynamic interactions this new evaluation is consistent with the theoretically predicted algebraic long time tails we observe better agreement for dense systems and obtain values for the relaxation time almost twice as large as previously

rotational velocity autocorrelation function of interacting brownian particles recently hagen frenkel and lowe physica a 272 1999 376 have obtained interesting simulation data for the rotational velocity autocorrelation function of a selected particle in a dense suspension of interacting spherical brownian particles although a wide time interval was studied it was found that the correlation function had not yet settled down to its t 5 2 long time tail at the longest time considered in the following analysis the method of pade approximants is applied to the laplace transform of the correlation function a dierent value for the coecient of the long time tail is obtained than found by hagen et al by extrapolation a prediction is made for the behavior of the correlation function outside the range of presently available data c 2001 elsevier science b v all rights reserved

slow dynamics of linear relaxation systems abstract linear relaxation occurring in dielectrics viscoelastic fluids and many other systems is often characterized by a broad continuous spectrum we show that the relaxation behavior may be analyzed effectively by means of n point pade approximants applied in the complex plane of the square root of frequency the method leads to a compact analytic expression for the laplace transform of the relaxation function characterized by a small number of poles and their residues in the square root of frequency plane we study the method in detail for a model of diffusion in three dimensions with a single or double radial potential barrier and demonstrate its use in the analysis of the viscoelastic relaxation spectrum of polyisobutylene

note on unicyclic graphs with given number of pendent vertices and minimal energy abstract for a simple graph g the energy e g is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix let g n p denote the set of unicyclic graphs with n vertices and p pendent vertices in h hua m wang unicyclic graphs with given number of pendent vertices and minimal energy linear algebra appl 426 2007 478 489 hua and wang discussed the graphs that have minimal energy in g n p and determined the minimal energy graphs among almost all different cases of n and p in their work certain case of the values was left as an open problem in which the minimal energy species have to be chosen in two candidate graphs but cannot be determined by comparing of the corresponding coefficients of their characteristic polynomials this paper aims at solving the problem completely by using the well known coulson integral formula

testing the nullity of garch coefficients correction of the standard tests and relative efficiency comparisons this article is concerned with testing the nullity of coefficients in generalized autoregressive conditionally heteroscedastic garch models the problem is nonstandard because the quasi maximum likelihood estimator is subject to positivity constraints this article establishes the asymptotic null and local alternative distributions of wald score and quasi likelihood ratio tests efficiency comparisons under fixed alternatives are considered two cases of special interest are tests of the null hypothesis of one coefficient equal to zero and tests of the null hypothesis of no conditional heteroscedasticity finally the proposed approach is used in the analysis of financial data and suggests reconsidering the preeminence of garch 1 1 among garch models

transitions between wells and escape by diffusion in a one dimensional potential landscape abstract the time dependence of the occupation probabilities of neighboring wells due to diffusion in one dimension is formulated in terms of a set of generalized rate equations describing transitions between neighboring wells and escape across a final barrier the equations contain rate coefficients memory coefficients and a long time coefficient characterizing the amplitude of long time decay on a more microscopic level the stochastic process is described by a smoluchowski equation for the one dimensional probability distribution a numerical procedure is presented which allows calculation of the transport coefficients in the set of generalized rate equations on the basis of the smoluchowski equation