Instability of equilibrium points of some Lagrangian systems

In this work we show that, if L is a natural Lagrangian system such that the k-jet of the potential energy ensures it does not have a minimum at the equilibrium and such that its Hessian has rank at least n - 2, then there is an asymptotic trajectory to the associated equilibrium point and so the equilibrium is unstable. This applies, in particular, to analytic potentials with a saddle point and a Hessian with at most 2 null eigenvalues. The result is proven for Lagrangians in a specific form, and we show that the class of Lagrangians we are interested can be taken into this specific form by a subtle change of spatial coordinates. We also consider the extension of this results to systems subjected to gyroscopic forces. (C) 2008 Elsevier Inc. All rights reserved.

Spectral flow and iteration of closed semi-Riemannian geodesics

We introduce the notion of spectral flow along a periodic semi-Riemannian geodesic, as a suitable substitute of the Morse index in the Riemannian case. We study the growth of the spectral flow along a closed geodesic under iteration, determining its asymptotic behavior.

The first group (co)homology of a group G with coefficients in some G-modules

Let G be a group. We give some formulas for the first group homology and cohomology of a group G with coefficients in an arbitrary G-module (Z) over tilde. More explicit calculations are done in the special cases of free groups, abelian groups and nilpotent groups. We also perform calculations for certain G-module M, by reducing it to the case where the coefficient is a G-module (Z) over tilde. As a result of the well known equalities H-1(X, M) = H-1(pi(1)(X), M) and H-1(X, M) = H-1(pi(1) (X), M), for any G-module M, we are able to calculate the first homology and cohomology groups of topological spaces with certain local system of coefficients.

A matrix formula for the skewness of maximum likelihood estimators

We give a general matrix formula for computing the second-order skewness of maximum likelihood estimators. The formula was firstly presented in a tensorial version by Bowman and Shenton (1998). Our matrix formulation has numerical advantages, since it requires only simple operations on matrices and vectors. We apply the second-order skewness formula to a normal model with a generalized parametrization and to an ARMA model. (c) 2010 Elsevier B.V. All rights reserved.

Covariance matrix formula for Birnbaum-Saunders regression models

The Birnbaum-Saunders regression model is commonly used in reliability studies. We derive a simple matrix formula for second-order covariances of maximum-likelihood estimators in this class of models. The formula is quite suitable for computer implementation, since it involves only simple operations on matrices and vectors. Some simulation results show that the second-order covariances can be quite pronounced in small to moderate sample sizes. We also present empirical applications.

A note on influence diagnostics in nonlinear mixed-effects elliptical models

This paper provides general matrix formulas for computing the score function, the (expected and observed) Fisher information and the A matrices (required for the assessment of local influence) for a quite general model which includes the one proposed by Russo et al. (2009). Additionally, we also present an expression for the generalized leverage on fixed and random effects. The matrix formulation has notational advantages, since despite the complexity of the postulated model, all general formulas are compact, clear and have nice forms. (C) 2010 Elsevier B.V. All rights reserved.

The 3-colored Ramsey number of even cycles

Denote by R(L, L, L) the minimum integer N such that any 3-coloring of the edges of the complete graph on N vertices contains a monochromatic copy of a graph L. Bondy and Erdos conjectured that when L is the cycle C(n) on n vertices, R(C(n), C(n), C(n)) = 4n - 3 for every odd n > 3. Luczak proved that if n is odd, then R(C(n), C(n), C(n)) = 4n + o(n), as n -> infinity, and Kohayakawa, Simonovits and Skokan confirmed the Bondy-Erdos conjecture for all sufficiently large values of n. Figaj and Luczak determined an asymptotic result for the `complementary` case where the cycles are even: they showed that for even n, we have R(C(n), C(n), C(n)) = 2n + o(n), as n -> infinity. In this paper, we prove that there exists n I such that for every even n >= n(1), R(C(n), C(n), C(n)) = 2n. (C) 2009 Elsevier Inc. All rights reserved.

Local power of some tests in exponential family nonlinear models

In this paper we obtain asymptotic expansions up to order n(-1/2) for the nonnull distribution functions of the likelihood ratio, Wald, score and gradient test statistics in exponential family nonlinear models (Cordeiro and Paula, 1989), under a sequence of Pitman alternatives. The asymptotic distributions of all four statistics are obtained for testing a subset of regression parameters and for testing the dispersion parameter, thus generalising the results given in Cordeiro et al. (1994) and Ferrari et al. (1997). We also present Monte Carlo simulations in order to compare the finite-sample performance of these tests. (C) 2010 Elsevier B.V. All rights reserved.

Thermo-solvatochromism of merocyanine polarity probes - What are the consequences of increasing probe lipophilicity through annelation?

The question raised in the title has been answered by comparing the solvatochromism of two series of polarity probes, the lipophilicities of which were increased either by increasing the length of an alkyl group (R) attached to a fixed pyridine-based structure or through annelation (i.e., by fusing benzene rings onto a central pyridine-based structure). The following novel solvatochromic probes were synthesized: 2,6-dibromo-4-[(E)-2-(1-methylquinolinium-4-yl)ethenyl]-phenolate (MeQMBr(2)) and 2,6-dibromo-4-[(E)-2-(1-methyl-acridinium-4- yl) ethenyl)]phenolate (MeAMBr(2) The solvatochromic behavior of these probes, along with that of 2,6dibromo-4-[(E)-2-(1-methylpyridinium-4-yl)ethenyl]phenol-ate(MePMBr(2)) was analyzed in terms of increasing probe lipophilicity, through annelation. Values of the empirical solvent polarity scale [E(T)(MePMBr(2))] in kcalmol(-1) correlated linearly with ET(30), the corresponding values for the extensively employed probe 2,6-diphenyl-4-(2,4,6-triphenylpyridinium-1-yl)phenolate (RB). On the other hand, the nonlinear correlations of ET(MeQMBr(2)) or ET(MeAMBr(2)) with E(T)(30) are described by second-order polynomials. Possible reasons for this behavior include: i) self-aggregation of the probe, ii) photoinduced cis/trans isomerization of the dye, and iii) probe structure- and solvent-dependent contributions of the quinonoid and zwitterionic limiting formulas to the ground and excited states of the probe. We show that mechanisms (i) and (ii) are not operative under the experimental conditions employed; experimental evidence (NMR) and theoretical calculations are presented to support the conjecture that the length of the central ethenylic bond in the dye increases in the order MeAMBr(2) > MeQMBr(2) > MePMBr(2), That is, the contribution of the zwitterionic limiting formula predominates for the latter probe, as is also the case for RB, this being the reason for the observed linear correlation between the ET(MePMBr2) and the ET(30) scales. The effect of increasing probe lipophilicity on solvatochromic behavior therefore depends on the strategy employed. Increasing the length of R affects solvatochromism much less than annelation, because the former structural change hardly perturbs the energy of the intramolecular charge-transfer transition responsible for solvatochromism. The thermo-solvatochromic behavior (effect of temperature on solvatochromism) of the three probes was studied in mixtures of water with propanol and/or with DMSO. The solvation model used explicitly considers the presence of three ""species"" in the system: bulk solution and probe solvation shell [namely, water (W), organic solvent (Solv)], and solvent-water hydrogen-bonded aggregate (Solv-W). For aqueous propanol, the probe is efficiently solvated by Solv-W; the strong interaction of DMSO with W drastically decreases the efficiency of Solv-W in solvating the probe, relative to its precursor solvents. Temperature increases resulted in desolvation of the probes, due to the concomitant reduction in the structured characters of the components of the binary mixtures.

Intrachain Energy Migration to Weak Charge-Transfer State in Polyfluorene End-Capped with Naphthalimide Derivative

Polyfluorene end-capped with N-(2-benzothiazole)-1 8-naphthalimide (PF-BNI) is a highly fluorescent material with fluorescence emission modulated by solvent polarity Its low energy excited state is assigned as a mixed configuration state between the singlet S(1) of the fluorene backbone (F) with the charge transfer (CI) of the end group BNI The triexponential fluorescence decays of PF-BNI were associated with fast energy migration to form an intrachain charge-transfer (ICCT) state polyfluorene backbone decay and ICCT deactivation Time-resolved fluorescence anisotropy exhibited biexponential relaxation with a fast component of 12-16 ps in addition to a slow one in the range 0 8-1 4 ns depending on the solvent showing that depolarization occurs from two different processes energy migration to form the ICCT state and slow rotational diffusion motion of end segments at a longer time Results from femtosecond transient absorption measurements agreed with anisotropy decay and showed a decay component of about 16 ps at 605 nm in PF BNI ascribed to the conversion of S(1) to the ICCT excited state From the ratio of asymptotic and initial amplitudes of the transient absorption measurement the efficiency of intrachain ICCT formation is estimated in 0 5 which means that on average, half of the excited state formed in a BNI-(F)(n)-BNI chain with n = 32 is converted to its low energy intrachain charge-transfer (ICCT) state

Residual-based test for Nonlinear Cointegration with application in PPPs

Nested by linear cointegration first provided in Granger (1981), the definition of nonlinear cointegration is presented in this paper. Sequentially, a nonlinear cointegrated economic system is introduced. What we mainly study is testing no nonlinear cointegration against nonlinear cointegration by residual-based test, which is ready for detecting stochastic trend in nonlinear autoregression models. We construct cointegrating regression along with smooth transition components from smooth transition autoregression model. Some properties are analyzed and discussed during the estimation procedure for cointegrating regression, including description of transition variable. Autoregression of order one is considered as the model of estimated residuals for residual-based test, from which the teststatistic is obtained. Critical values and asymptotic distribution of the test statistic that we request for different cointegrating regressions with different sample sizes are derived based on Monte Carlo simulation. The proposed theoretical methods and models are illustrated by an empirical example, comparing the results with linear cointegration application in Hamilton (1994). It is concluded that there exists nonlinear cointegration in our system in the final results.

Testing common nonlinear features in vector nonlinear autoregressive models

This paper studies a special class of vector smooth-transition autoregressive (VSTAR) models that contains common nonlinear features (CNFs), for which we proposed a triangular representation and developed a procedure of testing CNFs in a VSTAR model. We first test a unit root against a stable STAR process for each individual time series and then examine whether CNFs exist in the system by Lagrange Multiplier (LM) test if unit root is rejected in the first step. The LM test has standard Chi-squared asymptotic distribution. The critical values of our unit root tests and small-sample properties of the F form of our LM test are studied by Monte Carlo simulations. We illustrate how to test and model CNFs using the monthly growth of consumption and income data of United States (1985:1 to 2011:11).

Testing linear cointegration against smooth-transition cointegration

This paper studies a smooth-transition (ST) type cointegration. The proposed ST cointegration allows for regime switching structure in a cointegrated system. It nests the linear cointegration developed by Engle and Granger (1987) and the threshold cointegration studied by Balke and Fomby (1997). We develop F-type tests to examine linear cointegration against ST cointegration in ST-type cointegrating regression models with or without time trends. The null asymptotic distributions of the tests are derived with stationary transition variables in ST cointegrating regression models. And it is shown that our tests have nonstandard limiting distributions expressed in terms of standard Brownian motion when regressors are pure random walks, while have standard asymptotic distributions when regressors contain random walks with nonzero drift. Finite-sample distributions of those tests are studied by Monto Carlo simulations. The small-sample performance of the tests states that our F-type tests have a better power when the system contains ST cointegration than when the system is linearly cointegrated. An empirical example for the purchasing power parity (PPP) data (monthly US dollar, Italy lira and dollar-lira exchange rate from 1973:01 to 1989:10) is illustrated by applying the testing procedures in this paper. It is found that there is no linear cointegration in the system, but there exits the ST-type cointegration in the PPP data.

Localization in splitting of matter waves

In this paper we present an analysis of how matter waves, guided as propagating modes in potential structures, are split under adiabatic conditions. The description is formulated in terms of localized states obtained through a unitary transformation acting on the mode functions. The mathematical framework results in coupled propagation equations that are decoupled in the asymptotic regions as well before as after the split. The resulting states have the advantage of describing propagation in situations, for instance matter-wave interferometers, where local perturbations make the transverse modes of the guiding potential unsuitable as a basis. The different regimes of validity of adiabatic propagation schemes based on localized versus delocalized basis states are also outlined. Nontrivial dynamics for superposition states propagating through split potential structures is investigated through numerical simulations. For superposition states the influence of longitudinal wave-packet extension on the localization is investigated and shown to be accurately described in quantitative terms using the adiabatic formulations presented here.

Testing Seasonal Unit Roots in Data at Any Frequency, an HEGY approach

This paper generalizes the HEGY-type test to detect seasonal unit roots in data at any frequency, based on the seasonal unit root tests in univariate time series by Hylleberg, Engle, Granger and Yoo (1990). We introduce the seasonal unit roots at first, and then derive the mechanism of the HEGY-type test for data with any frequency. Thereafter we provide the asymptotic distributions of our test statistics when different test regressions are employed. We find that the F-statistics for testing conjugation unit roots have the same asymptotic distributions. Then we compute the finite-sample and asymptotic critical values for daily and hourly data by a Monte Carlo method. The power and size properties of our test for hourly data is investigated, and we find that including lag augmentations in auxiliary regression without lag elimination have the smallest size distortion and tests with seasonal dummies included in auxiliary regression have more power than the tests without seasonal dummies. At last we apply the our test to hourly wind power production data in Sweden and shows there are no seasonal unit roots in the series.