sum of two squares

Abstract. Using Howe duality we compute explicitly Kostant-type homology groups for a wide class of representations of the infinite-dimensional Lie superalgebra b gl ∞| ∞ and its classical subalgebras at positive integral levels. We also obtain Kostant-type homology formulas for the Lie algebra b gl ∞ at negative integral levels. We further construct resolutions in terms of generalized Verma modules for these representations. Contents

The construction of asymptotically distribution free time series model specification tests using as statistics the estimated residual autocorrelations is considered from a general view point. We focus our attention on Box-Pierce type tests based on the sum of squares of a few estimated residual autocorrelations. This type of tests belongs to the class defined by quadratic forms of weighted residual autocorrelations, where weights are suitably transformed resulting in asymptotically distribution free tests. The weights can be optimally chosen to maximize the power function when testing in the direction of local alternatives. The optimal test in this class against MA, AR or Bloomfield alternatives is a Box-Pierce type test based on the sum of squares of a few transformed residual autocorrelations. Such transformations are, in fact, the recursive residuals in the projection of the residual autocorrelations on a certain score function.

This paper describes a Matlab-based interactive tool focused on the modeling and control of nonlinear systems using a Takagi-Sugeno (T-S) fuzzy approach. Noticeably the modeling of dynamical nonlinear systems plays a fundamental role in the field of automatic control, since it allows to apply well-established analysis and control synthesis tools based on the Lyapunov theory. The interactive tool presented in this work is dedicated to aid control practitioners to easily derive a fuzzy state-space representation which is locally equivalent to the original nonlinear system and also to obtain a controller meeting desired closed-loop requirements. Keywords— Nonlinear systems, T-S Fuzzy Systems, Modeling, Control Synthesis.

A new adaptive orthogonal search (AOS) algorithm is proposed for model subset selection and nonlinear system identification. Model structure detection is a key step in any system identification problem. This consists of selecting significant model terms from a redundant dictionary of candidate model terms, and determining the model complexity (model length or model size). The final objective is to produce a parsimonious model that can well capture the inherent dynamics of the underlying system. In the new AOS algorithm, a modified generalized cross-validation criterion, called the adjustable prediction error sum of squares (APRESS), is introduced and incorporated into a forward orthogonal search procedure. The main advantage of the new AOS algorithm is that the mechanism is simple and the implementation is direct and easy, and more importantly it can produce efficient model subsets for most nonlinear identification problems.

A new adaptive orthogonal search (AOS) algorithm is proposed for model subset selection and nonlinear system identification. Model structure detection is a key step in any system identification problem. This consists of selecting significant model terms from a redundant dictionary of candidate model terms, and determining the model complexity (model length or model size). The final objective is to produce a parsimonious model that can well capture the inherent dynamics of the underlying system. In the new AOS algorithm, a modified generalized cross-validation criterion, called the adjustable prediction error sum of squares (APRESS), is introduced and incorporated into a forward orthogonal search procedure. The main advantage of the new AOS algorithm is that the mechanism is simple and the implementation is direct and easy, and more importantly it can produce efficient model subsets for most nonlinear identification problems.

A method for the simultaneous co-registration and georeferencing of multiple 3D pointclouds and associated intensity information is proposed. It is a generalization of the 3D surface matching problem. The simultaneous co-registration provides for a strict solution to the problem, as opposed to sequential pairwise registration. The problem is formulated as the Least Squares matching of overlapping 3D surfaces. The parameters of 3D transformations of multiple surfaces are simultaneously estimated, using the Generalized GaussMarkoff model, minimizing the sum of squares of the Euclidean distances among the surfaces. An observation equation is written for each surface-to-surface correspondence. Each overlapping surface pair contributes a group of observation equations to the design matrix. The parameters are introduced into the system as stochastic variables, as a second type of (fictitious) observations. This extension allows to control the estimated parameters. Intensity information is...

We present an algorithm for the least squares matching of overlapping 3D surfaces. It estimates the transformation parameters between two or more fully 3D surfaces, using the Generalized Gauss-Markoff model, minimizing the sum of squares of the Euclidean distances between the surfaces. This formulation gives the opportunity of matching arbitrarily oriented 3D surfaces simultaneously, without using explicit tie points. Besides the mathematical model of the procedure, we discuss the computational aspects. We give practical examples to demonstrate the method.

The automatic co-registration of point clouds, representing 3D surfaces, is a relevant problem in 3D modeling. This multiple registration problem can be defined as a surface matching task. We treat it as least squares matching of overlapping surfaces. The surface may have been digitized/sampled point by point using a laser scanner device, a photogrammetric method or other surface measurement techniques. Our proposed method estimates the transformation parameters of one or more 3D search surfaces with respect to a 3D template surface, using the Generalized Gauss-Markoff model, minimizing the sum of squares of the Euclidean distances between the surfaces. This formulation gives the opportunity of matching arbitrarily oriented 3D surface patches. It fully considers 3D geometry. Besides the mathematical model and execution aspects we address the further extensions of the basic model. We also show how this method can be used for curve matching in 3D space and matching of curves to surfaces. Some practical examples based on the registration of close-range laser scanner and photogrammetric point clouds are presented for the demonstration of the method. This surface matching technique is a generalization of the least squares image matching concept and offers high flexibility for any kind of 3D surface correspondence problem, as well as statistical tools for the analysis of the quality of final matching results.

We present fast new algorithms for evaluating trees with respect to least squares and minimum evolution (ME), the most commonly used criteria for inferring phylogenetic trees from distance data. The new algorithms include an optimal O(N 2 ) time algorithm for calculating the edge (branch or internode) lengths on a tree according to ordinary or unweighted least squares (OLS); an O(N 3 ) time algorithm for edge lengths under weighted least squares (WLS) including the Fitch-Margoliash method; and an optimal O(N 4 ) time algorithm for generalized least-squares (GLS) edge lengths (where N is the number of taxa in the tree). The ME criterion is based on the sum of edge lengths. Consequently, the edge lengths algorithms presented here lead directly to O(N 2 ), O(N 3 ), and O(N 4 ) time algorithms for ME under OLS, WLS, and GLS, respectively. All of these algorithms are as fast as or faster than any of those previously published, and the algorithms for OLS and GLS are the fastest possible (with respect to order of computational complexity). A major advantage of our new methods is that they are as well adapted to multifurcating trees as they are to binary trees. An optimal algorithm for determining path lengths from a tree with given edge lengths is also developed. This leads to an optimal O(N 2 ) algorithm for OLS sums of squares evaluation and corresponding O(N 3 ) and O(N 4 ) time algorithms for WLS and GLS sums of squares, respectively. The GLS algorithm is time-optimal if the covariance matrix is already inverted. The speed of each algorithm is assessed analytically-the speed increases we calculate are confirmed by the dramatic speed increases resulting from their implementation in PAUP* 4.0. The new algorithms enable far more extensive tree searches and statistical evaluations (e.g., bootstrap, parametric bootstrap, or jackknife) in the same amount of time. Hopefully, the fast algorithms for WLS and GLS will encourage the use of these criteria for evaluating trees and their edge lengths (e.g., for approximate divergence time estimates), since they should be more statistically efficient than OLS.

A facility has to be located within a given region taking two criteria of equity and efficiency into account. Equity is sought by minimizing the inequality in the inhabitant-facility distances, as measured by the sum of the absolute differences between all pairs of squared Euclidean distances from inhabitants to the facility. This measure meets the Pigou-Dalton condition of transfers, and can easily be minimized. Efficiency is measured through optimizing the sum of squared inhabitant-facility distances, either to be minimized or maximized for an attracting or repellent facility respectively. Geometric localization results are obtained for the whole set of Pareto optimal solutions for each of the two resulting bicriteria problems within a convex polygonal region. A polynomial procedure is developed to obtain the full bicriteria plot, both trade-off curves and the corresponding efficient sets.

The use of curve fitting for the analysis and interpretation of voltammetric data obtained while working with micro electrodes is discussed as a useful exercise for introducing students to the principle of problem solving using least-squares curve-fitting techniques. The advantages associated with this approach to data processing over the approach where the limiting current (i L) alone is used are discussed and its limitations are highlighted. This technique was applied to the determination of unknown concentrations of ferrocyanide and the most satisfactory recovery of concentrations was found when both the va1ues of the formal potential (EO') and concentration ( C) were varied to match the experimental results with an equation characterizing the current potential curve for a reversible couple. In this case recoveries of 100% ± 5% were obtained for the concentration range 5 X 10--4 to 1 x 10M . It was also found chat Solver was unable to fit the equation when the sum of squared ...

In this paper, estimation of the linear regression model is made by ordinary least squares method and the partially linear regression model is estimated by penalized least squares method using smoothing spline. Then, it is investigated that differences and similarity in the sum of squares related for linear regression and partial linear regression models (semi-parametric regression models). It is denoted that the sum of squares in linear regression is reduced to sum of squares in partial linear regression models. Furthermore, we indicated that various sums of squares in the linear regression are similar to different deviance statements in partial linear regression. In addition to, coefficient of the determination derived in linear regression model is easily generalized to coefficient of the determination of the partial linear regression model. For this aim, it is made two different applications. A simulated and a real data set are considered to prove the claim mentioned here. In this way, this study is supported with a simulation and a real data example.

In this paper, estimation of the linear regression model is made by ordinary least squares method and the partially linear regression model is estimated by penalized least squares method using smoothing spline. Then, it is investigated that differences and similarity in the sum of squares related for linear regression and partial linear regression models (semi-parametric regression models). It is denoted that the sum of squares in linear regression is reduced to sum of squares in partial linear regression models. Furthermore, we indicated that various sums of squares in the linear regression are similar to different deviance statements in partial linear regression. In addition to, coefficient of the determination derived in linear regression model is easily generalized to coefficient of the determination of the partial linear regression model. For this aim, it is made two different applications. A simulated and a real data set are considered to prove the claim mentioned here. In this way, this study is supported with a simulation and a real data example.

In recent years using symmetry has proven to be a very useful tool to simplify computations in semidefinite programming. This dissertation examines the possibilities of exploiting discrete symmetries in three contexts: In SDP-based relaxations for polynomial optimization, in testing positivity of symmetric polynomials, and combinatorial optimization. In these contexts the thesis provides new ways for exploiting symmetries and thus deeper insight in the paradigms behind the techniques and studies a concrete combinatorial optimization question.In den letzten Jahren hat sich die Nutzung von Symmetrien in Anwendungen der semidefiniten Optimierung als vorteilhaft erwiesen. Die Arbeit untersucht Möglichkeiten der Nutzung diskreter Symmetrien im Kontext dreier Problemfelder: In der polynomiellen Optimierung, beim Positivitätstest symmetrischer Polynome und in der kombinatorischen Optimierung. Die Arbeit präsentiert hierzu neue Zugänge, ermöglicht damit neue Einsichten in die zugrunde liege...

Polya's fundamental enumeration theorem and some results from Williamson's generalized setup of it are proved in terms of Schur- Macdonald's theory (S-MT) of "invariant matrices". Given a permutation group W ≤ Sd and a one-dimensional characterof W, the polynomial functor Fcorresponding via S-MT to the induced monomial representation U� = ind S d W (�) of Sd, is studied. It turns out that the characteristic ch(F�) is the weighted inventory of some set J(�) of W-orbits in the integer-valued hypercube (0, ∞)d. The elements of J(�) can be distinguished among all W-orbits by a maximum property. The identity ch(F�) = ch(U�) of both characteristics is a consequence of S-MT, and is equivalent to a result of Williamson. Polya's theorem can be obtained from the above identity by the specialization � = 1W, where 1W is the unit character of W.

Polya's fundamental enumeration theorem is generalized in terms of Schur-Macdonald's theory (S-MT) of invariant matrices. Given a permutation group $W\leq S_d$ and a one-dimensional character $\chi$ of $W$, the polynomial functor $F_\chi$ corresponding via S-MT to the induced monomial representation $U_\chi = ind_W^{S_d}(\chi)$ of $S_d$, is studied. It turns out that the characteristic $ch(F_\chi)$ is the weighted inventory of some set $J(\chi)$ of $W$-orbits in the integer-valued hypercube $[0,\infty)^d$. The elements of $J(\chi) can be distinguished among all $W$-orbits by a maximum property. The identity $ch(F_\chi) = ch(U_\chi)$ of both characteristics is a consequence of S-MT. Polya's theorem can be obtained from the above identity by specialization $\chi=1_W$, where $1_W$ is the unit character of $W$.

The notion of exact bisimulation equivalence for nondeterministic discrete systems has recently resulted in notions of exact bisimulation equivalence for continuous and hybrid systems. In this paper, we establish the more robust notion of approximate bisimulation equivalence for nondeterministic nonlinear systems. This is achieved by requiring that a distance between system observations starts and remains, close, in the presence of nondeterministic system evolution. We show that approximate bisimulation relations can be characterized using a class of functions called bisimulation functions. For nondeterministic nonlinear systems, we show that conditions for the existence of bisimulation functions can be expressed in terms of Lyapunov-like inequalities, which for deterministic systems can be computed using recent sum-of-squares techniques. Our framework is illustrated on a safety verification example.

Multilinear regression is applied when experimenters wish to investigate the relationship between a block of predictor variables (X), whose values are fixed by the experimenter, and one or more responses (Y), measured at each experiment. The objective is to find a mathematical equation relating X and Y, by means of regression coefficients. Regression analysis is said to give good results when a squared multiple regression coefficient R 2 close to 1 is obtained and when the sum of squares of residuals (differences between experimental value of the response and computed value) is small. However, it must be noted that a bad choice of the experiments renders the obtained regression equation meaningless. In this paper an example of correct and incorrect use of multilinear regression is presented in detail the quality of the coefficients and the goodness of the prediction depend on the experimental design, and the value of R 2 gives no information at all about them. A set of criteria useful to judge the quality of an experimental plan, before carrying out any experiment, is proposed.

For evaluating the performance of color-difference equations, several goodness-of-fit measures were proposed in the past, such as Pearson's correlation coefficient (r), the performance factor PF=3, and the recently proposed standardized residual sum of squares (STRESS) measure. The STRESS shares its main advantage, which is the possibility to statistically test performance differences, with the correlation coefficient. We show, by mathematical analysis supported by instructive numerical examples, that the STRESS has no meaningful interpretation in this regression analysis context. In addition, we present objections to the use of the STRESS for evaluating colordifference equations. Therefore, we recommend using the correlation coefficient in combination with a graphical and diagnostics analysis to ensure proper application as with any statistical technique.

The Rado-Horn Theorem gives a characterization of those sets of vectors which can be written as the union of a fixed number of linearly independent sets. In this paper we study the redundant case. We show that then the span of the vectors can be written as the direct sum of a subspace which directly fails the Rado-Horn criteria and a subspace for which the Rado-Horn criteria hold. As a corollary, we characterize those sets of vectors which, after the deletion of a fixed number of vectors, can be written as the finite union of linearly independent sets.

m, be polynomials on R n and S := {x ∈ R n | g i (x) = 0, i = 1,. .. , l, h j (x) ≥ 0, j = 1,. .. , m}. This paper proposes a method for finding the global infimum of the polynomial f on the semialgebraic set S via sum of squares relaxation over its truncated tangency variety, even in the case where the polynomial f does not attain its infimum on S. Under a constraint qualification condition, it is demonstrated that: (i) The infimum of f on S and on its truncated tangency variety coincide; and (ii) A sums of squares certificate for nonnegativity of f on its truncated tangency variety. These facts imply that we can find a natural sequence of semidefinite programs whose optimal values converge, monotonically increasing to the infimum of f on S.

In this paper, we extend Bai and Perron's (1998, Econometrica, p.47-78) framework for multiple break testing to linear models estimated via Two Stage Least Squares (2SLS). Within our framework, the break points are estimated simultaneously with the regression parameters via minimization of the residual sum of squares on the second step of the 2SLS estimation. We establish the consistency of the resulting estimated break point fractions. We show that various F-statistics for structural instability based on the 2SLS estimator have the same limiting distribution as the analogous statistics for OLS considered by Bai and Perron (1998). This allows us to extend Bai and Perron's (1998) sequential procedure for selecting the number of break points to the 2SLS setting. Our methods also allow for structural instability in the reduced form that has been identified a priori using data-based methods. As an empirical illustration, our methods are used to assess the stability of the New Keynesian Phillips curve.

$\mathbb B$-convexity was defined in [7] as a suitable Kuratowski-Painlevé upper limit of linear convexities over a finite dimensional Euclidean vector space. Excepted in the special case where convex sets are subsets of $\mathbb R^n_ +$, $\mathbb B$-convexity was not defined with respect to a given explicit algebraic structure. This is done in that paper, which proposes an extension of $\mathbb B$-convexity to the whole Euclidean vector space. An unital idempotent and non-associative magma is defined over the real set and an extended $n$-ary operation is introduced. Along this line, the existence of the Kuratowski-Painlevé limit of the convex hull of two points over $\mathbb R^n$ is shown and an explicit extension of $\mathbb B$-convexity is proposed.

Estimates of molecular divergence times when rates of evolution vary require the assumption of a model of rate change. Brownian motion is one such model, and since rates cannot become negative, a log Brownian model seems appropriate. Divergence time estimates can then be made using weighted least squares penalties. As sequences become long, this approach effectively becomes equivalent to penalized likelihood or Bayesian approaches. Different forms of the least squares penalty are considered to take into account correlation due to shared ancestors. It is shown that a scale parameter is also needed since the sum of squares changes with the scale of time. Errors or uncertainty on fossil calibrations, may be folded in with errors due to the stochastic nature of Brownian motion and ancestral polymorphism, giving a total sum of squares to be minimized. Applying these methods to placental mammal data the estimated age of the root decreases from 125 to about 94 mybp. However, multiple fossil calibration points and relative molecular divergence times inflate the sum of squares more than expected. If fossil data are also bootstrapped, then the confidence interval for the root of placental mammals varies widely from ~70 to 130 mybp. Such a wide interval suggests that more and better fossil calibration data is needed and/or better models of rate evolution are needed and/or better molecular data are needed. Until these issues are thoroughly investigated, it is premature to declare either the old molecular dates frequently obtained (e.g. > 110 mybp) or the lack of identified placental fossils in the Cretaceous, more indicative of when crown-group placental mammals evolved.

(1980) have developed an alternating least-squares method TUCKALS-3 as a solution for Tucker's three-way principal components model. The present paper offers some additional features of their method. Starting from a reanalysis of Tucker's problem in terms of a rank-constrained regression problem, it is shown that the fitted sum of squares in TUCKALS-3 can be partitioned according to elements of each mode of the three-way data matrix. An upper bound to the total fitted sum of squares is derived. Finally, a special case of TUCKALS-3 is related to the Carroll/l-Iarshman CANDECOMP/PARAFAC model.

We prove recursive formulas for sums of squares and sums of triangular numbers in terms of sums of divisors functions and we give a variety of consequences of these formulas. Intermediate applications include statements about positivity of the coefficients of some infinite products.

We prove recursive formulas for sums of squares and sums of triangular numbers in terms of sums of divisors functions and we give a variety of consequences of these formulas. Intermediate applications include statements about positivity of the coefficients of some infinite products.

Resumo -Neste trabalho foram estudadas as distribuições dos quadrados médios na análise de variância com as freqüências alélicas em amostras de indivíduos extraídas em populações haplóides, procurando avaliar o uso do teste F para testar a hipótese de nulidade do coeficiente de coancestria. Foi demonstrado que as expressões da soma de quadrados entre populações e da soma de quadrados entre indivíduos dentro de populações divididas pelas respectivas esperanças de quadrados médios possuem distribuições de χ 2 aproximadas, indicando que o quociente entre os quadrados médios na análise de variância tem distribuição F aproximada. Um estudo de simulação foi realizado comparando-se os valores experimentais simulados com os da distribuição teórica de F, procurando-se avaliar a validade das aproximações sugeridas. Os resultados mostraram que o teste F pode ser usado para testar a hipótese de nulidade do coeficiente de coancestria quando se trabalha com pelo menos cinco populações apresentando freqüência alélica média entre 0,3 e 0,7 utilizando-se no mínimo 50 indivíduos. Termos para indexação: ascendência, haploidia, genética de população, análise estatística.

Resumo -Neste trabalho foram estudadas as distribuições dos quadrados médios na análise de variância com as freqüências alélicas em amostras de indivíduos extraídas em populações haplóides, procurando avaliar o uso do teste F para testar a hipótese de nulidade do coeficiente de coancestria. Foi demonstrado que as expressões da soma de quadrados entre populações e da soma de quadrados entre indivíduos dentro de populações divididas pelas respectivas esperanças de quadrados médios possuem distribuições de χ 2 aproximadas, indicando que o quociente entre os quadrados médios na análise de variância tem distribuição F aproximada. Um estudo de simulação foi realizado comparando-se os valores experimentais simulados com os da distribuição teórica de F, procurando-se avaliar a validade das aproximações sugeridas. Os resultados mostraram que o teste F pode ser usado para testar a hipótese de nulidade do coeficiente de coancestria quando se trabalha com pelo menos cinco populações apresentando freqüência alélica média entre 0,3 e 0,7 utilizando-se no mínimo 50 indivíduos. Termos para indexação: ascendência, haploidia, genética de população, análise estatística.

Neste trabalho foram estudadas as distribuições dos quadrados médios na análise de variância com as freqüências alélicas em amostras de indivíduos extraídas em populações haplóides, procurando avaliar o uso do teste F para testar a hipótese de nulidade do coeficiente de coancestria. Foi demonstrado que as expressões da soma de quadrados entre populações e da soma de quadrados entre indivíduos dentro de populações divididas pelas respectivas esperanças de quadrados médios possuem distribuições dechi² aproximadas, indicando que o quociente entre os quadrados médios na análise de variância tem distribuição F aproximada. Um estudo de simulação foi realizado comparando-se os valores experimentais simulados com os da distribuição teórica de F, procurando-se avaliar a validade das aproximações sugeridas. Os resultados mostraram que o teste F pode ser usado para testar a hipótese de nulidade do coeficiente de coancestria quando se trabalha com pelo menos cinco populações apresentando freqüê...

Moisture sorption isotherms of milled switchgrass (Panicum virgatum) were developed to provide guidelines for proper storage and drying of biomass. Samples were stored for a minimum of 8 days inside incubators set at 15°C, 25°C, or 35°C at different relative humidities, ranging from 16% to 88%. Equilibrium moisture content (EMC) data were fitted with the Henderson model and modified versions of the Chung-Pfost, Guggenheim-Anderson-deBoer (GAB), Halsey, Henderson, and Oswin models. The modified GAB model (R 2 = 0.99; SE = 0.58% d.b.) followed by the modified Henderson model (R 2 = 0.99; SE = 0.68% d.b.) produced randomized residuals and the best fit for milled switchgrass. The results have practical applications in the handling, drying, storage, and transportation of switchgrass to ensure that quality deterioration and spoilage by microorganisms are minimized.

Analysis of reliability data plays an important role in the maintenance decision making process. The accurate estimation of residual life in components and systems can be a great asset when planning the preventive replacement of components on machines. Artificial intelligence is a field that has rapidly developed over the last twenty years and practical applications have been found in many diverse areas. The use of such methods in the maintenance field have however not yet been fully explored.

Since the time of Gauss, it has been generally accepted that 2 -methods of combining observations by minimizing sums of squared errors have significant computational advantages over earlier 1 -methods based on minimization of absolute errors advocated by Boscovich, Laplace and others. However, 1 -methods are known to have significant robustness advantages over 2 -methods in many applications, and related quantile regression methods provide a useful, complementary approach to classical least-squares estimation of statistical models. Combining recent advances in interior point methods for solving linear programs with a new statistical preprocessing approach for 1 -type problems, we obtain a 10-to 100-fold improvement in computational speeds over current (simplex-based) 1 -algorithms in large problems, demonstrating that 1 -methods can be made competitive with 2 -methods in terms of computational speed throughout the entire range of problem sizes. Formal complexity results suggest that 1 -regression can be made faster than least-squares regression for n sufficiently large and p modest.

In this note we analyze the relationship between one-step ahead prediction errors and interpolation errors in time series. We obtain an expression of the prediction errors in terms of the interpolation errors and then we show that minimizing the sum of squares of the one step-ahead standardized prediction errors is equivalent to minimizing the sum of squares of standardized interpolation errors.

Multivariate hypothesis testing in studies of vegetation is likely to be hindered by unrealistic assumptions when based on conventional statistical methods. This can be overcome by randomization tests. In this paper, the accuracy and power of a MANOVA randomization test are evaluated for one and two factors with interaction with simulated data from three distributions. The randomization test is based on the partitioning of sum of squares computed from Euclidean distances. In one-factor designs, sample size and variance inequality were evaluated. The results showed a high level of accuracy. The power curve was higher with normal distribution, lower with uniform, intermediate with lognormal and was sensitive to variance inequality. In two-factor designs, three methods of permutations and two statistics were compared. The results showed that permutation of the residuals with F pseudo is accurate and can give good power for testing the interaction and restricted permutation for testing main factors.

This paper describes a Matlab-based interactive tool focused on the modeling and control of nonlinear systems using a Takagi-Sugeno (T-S) fuzzy approach. Noticeably the modeling of dynamical nonlinear systems plays a fundamental role in the field of automatic control, since it allows to apply well-established analysis and control synthesis tools based on the Lyapunov theory. The interactive tool presented in this work is dedicated to aid control practitioners to easily derive a fuzzy state-space representation which is locally equivalent to the original nonlinear system and also to obtain a controller meeting desired closed-loop requirements. Keywords— Nonlinear systems, T-S Fuzzy Systems, Modeling, Control Synthesis.

The use of multivariate spectrophotometric calibration for the simultaneous determination of several active components and excipients in ophthalmic solutions is presented. The resolution of five-component mixtures of phenylephrine, chloramphenicol, antipyrine, methylparaben and thimerosal has been accomplished by using partial least-squares (PLS-1) and a variant of the so-called hybrid linear analysis (HLA). Notwithstanding the presence of a large number of components and their high degree of spectral overlap, they have been determined simultaneously with high accuracy and precision, with no interference, rapidly and without resorting to extraction procedures using non aqueous solvents. A simple and fast method for wavelength selection in the calibration step is presented, based on the minimisation of the predicted error sum of squares (PRESS) calculated as a function of a moving spectral window.

In this paper, we present results for testing main, simple and interaction effects in heteroscedastic two factor MANOVA models. In particular, we suggest modifications to the MANOVA sum of squares and cross product matrices to account for heteroscedasticity. Based on these modified matrices, we define some multivariate test statistics and derive their asymptotic distributions under non-normality for the null as well as non-null cases. Derivation of these results relies on the perturbation method and limit theorems for independently distributed random matrices. Based on the asymptotic distributions, we devise small sample approximations for the quantiles of the null distributions. The numerical accuracy of the large sample as well as small sample approximations are favorable. A real data set from a Smoking Cessation Trial is analyzed to illustrate the application of the methods.

Longitudinal stresses due to combined horizontal and vertical bending moments in ships, corresponding to a return period of 20 years, are estimated by linear response analysis. In principle, the stress should be obtained by combining the stress in all sea states that can occur over a long-term period. A method to determine the desired long-term extreme stress by considering only a few short-term sea states is presented. The sea states have a certain probability of occurrence, and are each identified by a contour line in the (H s , T p)-plane. This approach makes it possible to estimate the extreme loads on the vessel in a practical and accurate manner. Moreover, it is shown that the long-term stress can be estimated by combining the individual long-term extreme stresses due to vertical and horizontal bending moments by using the sum-of-squares approach and accounting for the correlation between stresses. It was found that the correlation coefficient can be taken as the largest of the ones calculated along the contour line. It is shown that this correlation coefficient can even be approximated by the normalized phase angle at the wave length where the dominant response has its peak value. A comparison with the results obtained using well-known combination rules is presented. While linear analysis has been used here, it is believed that the approach can be generalized to stresses with nonlinear behavior, and hence represent a significant improvement in calculation efficiency.

Three cores from two crater lakes in central Italy (Albano and Nemi) spanning the last ca. 11 kyr BP are discussed here. They were analysed for organic matter, dry density, algal and photosynthetic bacteria pigments, diatoms and Cladocera. These palaeolimnological records show great internal consistency in their response to distinct, but interrelated , aspects of the lakecatchment system, and reflect major changes in these systems particularly in lake productivity. The early Holocene was characterised by very high productivity at both sites. Trends in productivity are similar for many proxies in the Holocene records of Lake Albano and Lake Nemi. Trophic levels throughout much of the period of human activity (ca. 4000 yr BP onward) are lower but have remained relatively high and stable at both lakes. It is only in the last few decades that inferred productivity levels have exceeded those of the early Holocene. Statistical analysis (CONstrained Incremental Sum of Square cluster analysis, CONISS) showed five distinct periods during the Holocene characterised by major variations in the abundance and community structure of biological remains and geochemistry. A comparison of results from the different taxonomic groups has enabled the reconstruction of phases of increased productivity and the recognition of signals of anthropogenic impact on the lakes and their catchments. From published pollen diagrams, nine major deforestation events are recognised during the Holocene. These events, in particular the one which occurred in both lakes at 1800 yr BP and identifies the Roman Period, plus another at 3000-3500 yr BP related to agricultural activity during the Bronze Age, produced profound changes in the pelagic community structure of diatoms and Cladocera, which in some cases were even more marked than the accelerated eutrophication seen in recent times. The mid-Holocene climatic transition is also clearly evident and is characterised by a strong decrease in the concentrations of the biological proxies. Most of the observed lacustrine environmental changes are almost synchronous in the two lakes indicating the importance of regional rather than local changes.

Lovász and Schrijver [9] have constructed semidefinite relaxations for the stable set polytope of a graph G = (V, E) by a sequence of lift-and-project operations; their procedure finds the stable set polytope in at most α(G) steps, where α(G) is the stability number of G. Two other hierarchies of semidefinite bounds for the stability number have been proposed by Lasserre [4],[5] and by de Klerk and Pasechnik [3], which are based on relaxing nonnegativity of a polynomial by requiring the existence of a sum of squares decomposition. The hierarchy of Lasserre is known to converge in α(G) steps as it refines the hierarchy of Lovász and Schrijver, and de Klerk and Pasechnik conjecture that their hierarchy also finds the stability number after α(G) steps. We prove this conjecture for graphs with stability number at most 8 and we show that the hierarchy of Lasserre refines the hierarchy of de Klerk and Pasechnik.

Multi-category classification methods were used to detect SNP-mortality associations in broilers. The objective was to select a subset of whole genome SNPs associated with chick mortality. This was done by categorizing mortality rates and using a filter-wrapper feature selection procedure in each of the classification methods evaluated. Different numbers of categories (2, 3, 4, 5 and 10) and three classification algorithms (naïve Bayes classifiers, Bayesian networks and neural networks) were compared, using early and late chick mortality rates in low and high hygiene environments. Evaluation of SNPs selected by each classification method was done by predicted residual sum of squares and a significance test-related metric. A naïve Bayes classifier, coupled with discretization into two or three categories generated the SNP subset with greatest predictive ability. Further, an alternative categorization scheme, which used only two extreme portions of the empirical distribution of mortality rates, was considered. This scheme selected SNPs with greater predictive ability than those chosen by the methods described previously. Use of extreme samples seems to enhance the ability of feature selection procedures to select influential SNPs in genetic association studies.

This paper proposes a compositional method for verifying the safety of a dynamical system, given as an interconnection of subsystems. The safety verification is conducted by the use of the barrier certificate method; hence, the contribution of this paper is to show how to obtain compositional conditions for safety verification. We show how to formulate the verification problem, as a composition of coupled subproblems, each given for one subsystem. Furthermore, we show how to find the compositional barrier certificates via linear and sum of squares programming problems. The proposed method makes it possible to verify the safety of higher dimensional systems, than the method for centrally computed barrier certificates. This is demonstrated by verifying the safety of an emergency shutdown of a wind turbine.