sum of two squares

Seven wood sorption models available in the literature were parameterized for prediction of equilibrium moisture content for temperatures up to 220ºC. The criterion used to evaluate the models is the uncorrected sum of squares of differences between the predicted equilibrium moisture content and experimental values published in literature for temperatures between 0ºC and 160ºC. The results show that the Malmquist model gave the best overall fit to experimental data for the relative humidities considered (40%, 52%, 65%, 75% and 85%). However, the García model performs better for temperatures between 110ºC and 155ºC at a relative humidity of 85%. Therefore, the choice of the sorption model depends on the specific hygrothermal conditions required by the user.

We initiate a direction for proving lower bounds on the size of non-commutative arithmetic circuits. This direction is based on a connection between lower bounds on the size of non-commutative arithmetic circuits and a problem about commutative degree-four polynomials, the classical sum-of-squares problem: find the smallest n n such that there exists an identity ( 0.1 ) ( x 1 2 + x 2 2 + ⋯ + x k 2 ) ⋅ ( y 1 2 + y 2 2 + ⋯ + y k 2 ) = f 1 2 + f 2 2 + ⋯ + f n 2 , \begin{equation*} (0.1)\quad \quad (x_1^2+x_2^2+\cdots + x_k^2)\cdot (y_1^2+y_2^2+\cdots + y_k^2)= f_{1}^{2}+f_{2}^{2}+\dots +f_{n}^{2} , \quad \quad \end{equation*} where each f i = f i ( X , Y ) f_{i} = f_i(X,Y) is a bilinear form in X = { x 1 , … , x k } X=\{x_{1},\dots ,x_{k}\} and Y = { y 1 , … , y k } Y=\{y_{1},\dots , y_{k}\} . Over the complex numbers, we show that a sufficiently strong superlinear lower bound on n n in (0.1), namely, n ≥ k 1 + ϵ n\geq k^{1+\epsilon } with ϵ > 0 \epsilon >0 , implies an exponenti...

The minimum sum-of-squared error clustering problem is shown to be a concave continuous optimization problem whose every local minimum solution must be integer. We characterize its local minima. A procedure of moving from a fractional solution to a better integer solution is given. Then we adapt Tuy's convexity cut method to find a global optimum of the minimum sum-of-squared error clustering problem. We prove that this method converges in finite steps to a global minimum. Promising numerical examples are reported. 1 Introduction. Clustering (or cluster analysis) is one of the basic tools in data analysis. In this paper, we consider clustering based on minimum within-group sum-ofsquared error criterion. Many early studies on minimum sum-of-squared error clustering (or MSSC in brief) were focused on the well-known K-means algorithm and its variants (see for a survey). Usually, these methods can only reach a local solution, not a global minimum of the distortion function. From a theoretical viewpoint, the minimum sum-of-squared error

The minimum-sum-of-squared error clustering (MSSC) is one of the most intuitive and popular clustering algorithms. In this paper, we first show that MSSC can be equivalently cast as a concave minimization problem. To find the global solution of MSSC, we construct a procedure to move from a fractional solution of the relaxed minimization problem to an integer solution with improved objective value. Then we adapt Tuy's convexity cut method, a powerful algorithm for global concave optimization, to find a global optimum to MSSC. Promising numerical examples are reported.

O caminho percorrido foi longo e grandes foram as dificuldades encontradas. Entretanto, com a colaboração de algumas pessoas, foi possível concluir este trabalho. Relaciono, a seguir, as pessoas a quem apresento meus mais sinceros agradecimentos: À Profa. Thelma Sáfadi, grande amiga, pelos indiscutíveis ensinamentos transmitidos e pela sua primorosa co-orientação. Ao prof. Luiz Gonzaga de Castro Jr., orientador, grande amigo, que confiou imensamente no trabalho. Ao Washington Silva, econometrista de inquestionável competência, que contribuiu com fortes sugestões para a conclusão desta dissertação. A BM&F -Bolsa de Mercadorias e Futuros de São Paulo, que me franqueou a base de dados e as publicações essenciais à realização deste trabalho.

The theory of pseudo circle packings is developed. It generalizes the theory of circle packings. It allows the realization of almost any graph embedding by a geometric structure of circles. The corresponding Thurston's relaxation mapping is defined and is used to prove the existence and the rigidity of the pseudo circle packing. It is shown that iterates of this mapping, starting from an arbitrary point, converge to its unique positive fixed point. The coordinates of this fixed point give the radii of the packing. A key property of the relaxation mapping is its superadditivity. The proof of that is reduced to show that a certain real polynomial in four variables and of degree 20 is always nonnegative. This in turn is proved by using recently developed algorithms from real algebraic geometry. Another important ingredient in the development of the theory is the use of nonnegative matrices and the corresponding Perron-Frobenius theory.

Mobile sensor networks are a great source of data. By collecting data with mobile sensor nodes from individuals in a user community, e.g. using their smartphones, we can learn global information such as traffic congestion patterns in the city, location of key community facilities, and locations of gathering places. Can we publish and run queries on mobile sensor network databases without disclosing information about individual nodes? Differential privacy is a strong notion of privacy which guarantees that very little will be learned about individual records in the database, no matter what the attackers already know or wish to learn. Still, there is no practical system applying differential privacy algorithms for clustering points on real databases. This paper describes the construction of small coresets for computing k-means clustering of a set of points while preserving differential privacy. As a result, we give the first k-means clustering algorithm that is both differentially private, and has an approximation error that depends sub-linearly on the data's dimension d. Previous results introduced errors that are exponential in d. We implemented this algorithm and used it to create differentially private location data from GPS tracks. Specifically our algorithm allows clustering GPS databases generated from mobile nodes, while letting the user control the introduced noise due to privacy. We provide experimental results for the system and algorithms, and compare them to existing techniques. To the best of our knowledge, this is the first practical system that enables differentially private clustering on real data.

Moving our hands smoothly is essential to execute ordinary tasks, such as carrying a glass of water without spilling. Past studies have revealed a natural tendency to generate smooth trajectories when moving the hand from one point to another in free space. Here we provide a new perspective on movement smoothness by showing that smoothness is also enforced when the hand maintains contact with a curved surface. Maximally smooth motions over curved surfaces occur along geodesic lines that depend on fundamental features of the surface, such as its radius and center of curvature. Subjects were requested to execute movements of the hand while in contact with a virtual sphere that they could not see. We found that with practice, subjects tended to move their hand along smooth trajectories, near geodesic pathways joining start to end positions, to reduce contact forces with constrained boundary, variance of contact force, tangential velocity profile error and sum of square jerk along the time span of movement. Furthermore, after practicing movements in a region of the sphere, subjects executed near-geodesic movements, less contact forces, less contact force variance, less tangential velocity profile error and less sum of square jerk in a different region. These findings suggest that the execution of smooth movements while the hand is in contact with a surface is a means for extracting information about the surface's geometrical features.

Adem and Reichstein introduced the ideal of truncated symmetric polynomials to present the permutation invariant subring in the cohomology of a finite product of projective spaces. Building upon their work, I describe a generating set of the ideal of truncated symmetric polynomials in arbitrary positive characteristic, and offer a conjecture for minimal generators.

In this work we show that among all n-vertex graphs with edge or vertex connectivity k, ), the join of K k , the complete graph on k vertices, with the disjoint union of K 1 and K n-k-1 , is the unique graph with maximum sum of squares of vertex degrees. This graph is also the unique n-vertex graph with edge or vertex connectivity k whose hyper-Wiener index is minimum.

Complementarity problems may be formulated as nonlinear systems of equations with non-negativity constraints. The natural merit function is the sum of squares of the components of the system. Sufficient conditions are established which guarantee that stationary points are solutions of the complementarity problem. Algorithmic consequences are discussed.

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.

Microaggregation for Statistical Disclosure Control (SDC) is a family of methods to protect microdata from individual identification. SDC seeks to protect microdata in such a way that can be published and mined without providing any private information that can be linked to specific individuals. The aim of SDC is to modify the original microdata in such a way that the modified data and the original data are similar. Microaggregation works by partitioning the microdata into groups, also called clusters of at least k records and then replacing the records in each group with the centroid of the group. In this work we introduce a new microaggregation method, where the centroid is considered as median. The new method guarantees that the microaggregated data and the original data are similar by using statistical test. Another contribution of this work is that we propose a distance metric, called absolute deviation from median (ADM) to evaluate the amount of mutual information among records in microdata. We showed that ADM is always less than the most commonly used measure of distortion called sum of squares of errors (SSE) for any dataset. Thus ADM causes least information loss and can be used as a measure of information loss for a microaggregated microdata set.

Joint regression analysis (JRA) is a popular method for analyzing genotype × environment (G × E) interactions, but multivariate techniques such as AMMI (additive main effects and multiplicative interaction) analysis have been recently advocated. The objective of this study was to investigate and compare empirically the effectiveness of the two techniques under differing environmental diversity and numbers of environments, and when log‐transformed data were used for JRA. I analyzed grain yield data from three seasons of a regional bread wheat (Triticum aestivum L.) yield trial, grown at 30 to 40 sites. Sites were split into irrigated‐high rainfall and rainfed‐low rainfall groups. Three equal‐size samples differing in environmental diversity and three similar diversity samples differing in numbers of sites were also formed. Both raw and log10‐transformed data were used for JRA. The fitting mode of the MATMODEL program (version 2.0) was used on raw data for AMMI analysis. Percentages o...

This paper addresses a new method to overcome the unmeasurable premise variables in TS nonlinear discrete time systems for observer design. It is known that a TS system can be obtained directly from a nonlinear one by using the sector nonlinear transformation in a given compact set of the state space. However, this procedure often leads to TS systems having premise variables depending on one or more unmeasurable states. Clearly, the observer design for such a class of TS systems is more difficult than designing observers for TS systems having measured or known premise variables. The objective of this paper is to use the immersion approach before using the sector nonlinear transformation in order to obtain a TS system with measured premise variables. In addition the proposed immersion technique leads to the use of an augmented state vector which first components are directly the state variables of the original system avoiding the computation of an inverse transformation. Example results are provided in order to illustrate the proposed approach.

Abstract. The paper deals with a method for determining a switching combination of several local linear models using only the knowledge of the inputoutput data. The method is a direct optimisation of the sum of square errors between measured output and predicted model output; in this procedure, the originality is based on the choice of an adapted criterion with particularly weights. Keywords-component. Data, signal, classification, segmentation, switching.

Sugarcane (Saccharum sp.) is one of the most important crops in Brazil. The high demand for sugarcane-derived products has stimulated the expansion of sugarcane cultivation in recent years, exploring different environments. The adaptability and the phenotypic stability of sugarcane genotypes in the Minas Gerais state, Brazil, were evaluated based on the additive main effects and multiplicative interaction (AMMI) method. We evaluated 15 genotypes (13 clones and two checks: RB867515 and RB72454) in nine environments. The average of two cuttings for the variable tons of pol per hectare (TPH) measure was used to discriminate genotypes. Besides the check RB867515 (20.44 t ha -1 ), the genotype RB987935 showed a high average TPH (20.71 t ha -1 ), general adaptability and phenotypic stability, and should be suitable for cultivation in the target region. The AMMI method allowed for easy visual identification of superior genotypes for each set of environments.

This work focuses on the computation of a candidate Lyapunov function for a Low Earth Orbit satellite which is actuated using only magnetorquers. A satellite having only electromagnetic actuation is not controllable when the magnetic moment produced by the magnetorquers is parallel to the geomagnetic field. Further, the dynamics of the system are periodic due to the periodic nature of the geomagnetic field. Previously, a locally stable Proportional-Derivative control has been designed for such a satellite. In this work, we have found a polynomial candidate Lyapunov function for the resultant closed loop system using Sum-of-Squares (SoS) polynomials and Putinar's Positivstellensatz. Unlike previous applications of SoS techniques on rigid bodies, the kinematics have been defined using unit quaternions. The unit quaternions have a well-known advantage of being a singularity free representation of attitude kinematics with only a single constraint. The unit quaternion constraint has been ensured using semialgebraic sets. Furthermore, special emphasis has been placed on the verification of the candidate Lyapunov function and we have simulated the closed loop system with the candidate Lyapunov function.

Genetic analysis of yield and morphological traits has been carried out in Coffea arabica from a half-diallel including the parental lines. The trial was established in west Cameroon with completely randomized single-tree plots. Observations included yield (four years), stem diameter, height and number of primaries. General combining abilities (GCA) and specific combining abilities (SCA) as heritabilities and genetic correlations were estimated. A significant SCA variance was observed for all the traits. Morphological traits, stem diameter, plant height and number of primaries, were genetically correlated to the yield. The hybrids were, on average, better performing than lines. There was no clear relationship between performance of lines and their general combining ability. Contribution of the seven lines to the SCA sum of squares was shown to be unequal for all the studied traits, one parent (Java) being far the most interactive. This variation of interactivity seemed to be related...

Climate change studies based only on raw long-term data are potentially flawed due to the many breaks introduced from non-climatic sources, consequently quality controlled and homogenised climate data is desirable for basing climate related decision making on. Seasonal cycles of precipitation in Ireland and the UK are projected to become more marked as the climate changes, and regional extremes in summer dry spells and winter precipitation have been recorded in recent years. Therefore to analyse and monitor the evolution of precipitation patterns across Ireland, quality controlled and homogenous climate series are needed.

Magic, a key quantum resource beyond entanglement, remains poorly understood in terms of its structure and classification. In this paper, we demonstrate a striking connection between high-dimensional symmetric lattices and quantum magic states. By mapping vectors from the E 8 , BW 16 , and E 6 lattices into Hilbert space, we construct and classify stabiliser and maximal magic states for two-qubit, three-qubit and one-qutrit systems. In particular, this geometric approach allows us to construct, for the first time, closed-form expressions for the maximal magic states in the three-qubit and one-qutrit systems, and to conjecture their total counts. In the three-qubit case, we further classify the extremal magic states according to their entanglement structure. We also examine the distinctive behaviour of one-qutrit maximal magic states with respect to Clifford orbits. Our findings suggest that deep algebraic and geometric symmetries underlie the structure of extremal magic states.

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We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several complex variables and modern operator theory. The second part of the survey focuses on recently discovered connections between real algebraic geometry and optimization as well as polynomials in matrix variables and some control theory problems. These new applications have prompted a series of recent studies devoted to the structure of positivity and convexity in a free * -algebra, the appropriate setting for analyzing inequalities on polynomials having matrix variables. We sketch some of these developments, add to them and comment on the rapidly growing literature.

Solution methods for the minimum sum-of-squares clustering (MSSC) problem are analyzed and developed in this paper. Based on the DCA (Difference-of-Convex functions Algorithms) in DC programming and recently established qualitative properties of the MSSC problem, we suggest several improvements of the incremental algorithms of Ordin and Bagirov and of Bagirov. Properties of the new algorithms are obtained and preliminary numerical tests of those on real-world databases are shown. Finite convergence, convergence, and the rate of convergence of solution methods for the MSSC problem are presented for the first time in our paper. This Part 1 is devoted to the incremental heuristic clustering algorithm of Ordin and Bagirov and the modified version proposed herein.

This study aims to present a comparative analysis of the Bayesian regularization backpropagation and Levenberg–Marquardt training algorithms in neural networks for the metrics prediction of damaged archaeological artifacts, of which the state of conservation is often fragmented due to different reasons, such as ritual, use wear, or post-depositional processes. The archaeological artifacts, specifically laminar blanks (so-called blades), come from different sites located in the Southern Levant that belong to the Pre-Pottery B Neolithic (PPNB) (10,100/9500–400 cal B.P.). This paper shows the entire procedure of the analysis, from its normalization of the dataset to its comparative analysis and overfitting problem resolution.

In many quality control studies the performance of a product or process is usually characterized by a singleresponse variable However, in some applications of quality control, the performance of a product or aprocess can be best characterized by a linear relationship between a response variable and one or more explanatory variables(Noorossana et al., 2011). But, when more than oneexplanatory variables are involved in the profile it may indicate the presence of high collinearity among explanatory variables which is called multicollinearity(Gujarati et al., 2012). It should be noted that if the multicollinearity is neglected during the profile monitoring, then the designed control charts applied in phase II lack the sufficient effectiveness in detecting shifts or out of control signals. In this paper, the effect of the multicollinearity on the monitoring of multiple linear profiles has been studied andsome new type ofExponentially Weighted Moving Average (EWMA) control chartsfor Intercept, Slopes and Mean Squared Error (MSE) byusing Ridge Regression Estimators (RRE) have been proposed in order to provide the solution for multicollinearity. An application of wind tunnel data by NASA Langley Research Centre was used for monitoringprofiles based on proposed EWMA control charts for Intercept, Slopes and MSE. The performance of the proposed EWMA control charts have been evaluated on theAverage Run Length (ARL) criterion,the results indicated that the proposed EWMA control charts obtained from RRE for Intercept, Slopes and MSEoutperformthe existing control charts obtained fromOrdinary Least Squared (OLS) estimator.

Let K be a totally real number field with Galois closure L. We prove that if Moreover, our argument is constructive and generalizes to the case of commutative K-algebras. This result gives a partial resolution to a question of Sturmfels on the algebraic degree of certain semidefinite programming problems.

Suppose f is a real univariate polynomial of degree D with exactly 4 monomial terms. We present an algorithm, with complexity polynomial in log D on average (relative to the stable log-uniform measure), for counting the number of real roots of f . The best previous algorithms had complexity super-linear in D. We also discuss connections to sums of squares and A-discriminants, including explicit obstructions to expressing positive definite sparse polynomials as sums of squares of few sparse polynomials. Our key tool is the introduction of efficiently computable chamber cones, bounding regions in coefficient space where the number of real roots of f can be computed easily. Much of our theory extends to n-variate (n + 3)-nomials.

Let K be a totally real number field with Galois closure L. We prove that if Moreover, our argument is constructive and generalizes to the case of commutative K-algebras. This result gives a partial resolution to a question of Sturmfels on the algebraic degree of certain semidefinite programming problems.

Assessing the adaptability and stability of promising rice genotypes is one of the important steps for accurate evaluation. This study determined the genotype × environment interaction (GEI) and stability performance of 12 promising rice genotypes in four environments during 2009 Aman season. The experiment used randomized complete block design with three replications. Yield stability and adaptability of yield performance were analyzed by combined analysis and additive main effects and multiplicative interaction (AMMI) model. The environment, genotype main effects, and the GEI were all highly significant (P<0.001). The study indicated that the tested genotypes, such as BRRHA G1 (5.47 tha -1 ), G2 (5.68 tha -1 ), G3 (6.29 tha -1 ) and G4 (5.27 tha -1 ) had higher average yields, which indicated these genotypes adapted to favourable environments (E1 and E3). Whereas the environment, E3 could be regarded as a more stable site for high yielding hybrid rice improvement than the other locations. Based on AMMI biplot analysis, genotypes BRRI1A/BRRI827R (G1), IR58025A/BRRI10R (G2), BRRI 10A/BRRI 10R (G3) and BRRI hybrid dhan1 (G4) have higher average mean yields with high main (additive) effects and positive IPCA1 score, among them BRRI 10A/BRRI10R (G3) being the overall best. Locations E1 and E3 could be regarded as a good selection site for rice hybrid improvement due to stable yields.

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.

We present a solution of the operator-valued Schur-function realization problem on the right-half plane by developing the corresponding de Branges-Rovnyak canonical conservative simple functional model. This model corresponds to the closely connected unitary model in the disk setting, but we work the theory out directly in the righthalf plane, which allows us to exhibit structure which is absent in the disk case. A main feature of the study is that the connecting operator is unbounded, and so we need to make use of the theory of well-posed continuous-time systems. In order to strengthen the classical uniqueness result (which states uniqueness up to unitary similarity), we introduce non-invertible intertwinements of system nodes.

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&#39;) 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&#39;s fundamental enumeration theorem and some results from Williamson&#39;s generalized setup of it are proved in terms of Schur- Macdonald&#39;s theory (S-MT) of &quot;invariant matrices&quot;. 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&#39;s theorem can be obtained from the above identity by the specialization � = 1W, where 1W is the unit character of W.