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## Métodos intervalares para a resolução de sistemas de equações lineares; Interval methods for resolution of linear equation systems

Holbig, Carlos Amaral
Tipo: Dissertação Formato: application/pdf
POR
Relevância na Pesquisa
56.12%
O estudo dos métodos intervalares é importante para a resolução de sistemas de equações lineares, pois os métodos intervalares produzem resultados dentro de limites confiáveis (do intervalo solução) e provam a existência ou não existência de soluções, portanto produzem resultados confiáveis, o que os métodos pontuais podem não proporcionar. Outro aspecto a destacar é o campo de utilizando de sistemas de equações lineares em problemas das engenharias e outras ciências, o que mostra a aplicabilidade desses métodos e por conseguinte a necessidade de elaboração de ferramentas que possibilitem a implementação desses métodos intervalares. O objetivo deste trabalho não é a elaboração de novos métodos intervalares, mas sim o de realizar uma descrição e implementação de alguns dos métodos intervalares encontrados na bibliografia pesquisada. A versão intervalar dos métodos pontuais não é simples e o calculo por métodos intervalares pode ser dispendioso, uma vez que se está tratando com vetores e matrizes de intervalos. A implementação dos métodos intervalares são foi possível graças a existência de ferramentas, como o compilador Pascal-XSC, que incorpora as suas características aspectos importantes como a aritmética intervalar...

## Solving linear equation systems using parallel processing

Tipo: Conferência ou Objeto de Conferência
Relevância na Pesquisa
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This paper shows the abilities of the parallel processing in the solution of linear equation systems. The solution of linear equation systems is one of the most time consuming task in the analysis of the structural problem in civil engineering. This is more evident in finite element analysis because the solving phase spends almost the whole time of the analysis. To solve this time consuming it is proposed the use of the parallel processing in the solution of the equation systems. The Gaussian elimination method, the Cholesky factorization method and the Conjugate Gradient iterative method were chosen. For these methods it was analysed the sequential time, the parallel time, the speedup and the efficiency of the parallel algorithm relatively at the sequential algorithm. Parallel times are gotten for 2 to 16 processors because this work was developed in a parallel computer with 16 transputers IMS T800- 20 every one with 2 MBytes of RAM.

## The Einstein's linear equation of a space-time with a homogeneous section of low dimension

Martinez-Morales, Jose L.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46%
The Einstein's linear equation of a small perturbation in a space-time with a homogeneous section of low dimension, is studied. For every harmonic mode of the horizon, there are two solutions which behave differently at large distance $r$. In the basic mode, the behavior of one of the solutions is ${{(-{r^2}+{t^2})}^{\frac{1-n}{2}}}$ where $n$ is dimension of space. These solutions occur in an integral form. In addition, a main statement of the article is that a field in a black hole decays at infinity according to a universal law. An example of such a field is an eigentensor of the Einstein's linear operator that corresponds to an eigenvalue different from Zero. Possible applications to the stability of black holes of high dimension are discussed. The analysis we present is of a small perturbation of space-time. The perturbation analysis of higher order will appear in a sequel. We determine perturbations of space-time in dimension 1+$n\ge$ 4 where the system of equations is simplified to the Einstein's linear equation, a tensor differential equation. The solutions are some integral transformations which in some cases reduce to explicit functions. We perform some perturbation analysis and we show that there exists no perturbation regular everywhere outside the event horizon which is well behaved at the spatial infinity. This confirms the uniqueness of vacuum space-time within the perturbation theory framework. Our strategy for treating the stability problem is applicable to other space-times of high dimension with a cosmological constant different from Zero.; Comment: 8 pages...

## Definability of linear equation systems over groups and rings

Dawar, Anuj; Kopczynski, Eryk; Holm, Bjarki; Grädel, Erich; Pakusa, Wied
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46%
Motivated by the quest for a logic for PTIME and recent insights that the descriptive complexity of problems from linear algebra is a crucial aspect of this problem, we study the solvability of linear equation systems over finite groups and rings from the viewpoint of logical (inter-)definability. All problems that we consider are decidable in polynomial time, but not expressible in fixed-point logic with counting. They also provide natural candidates for a separation of polynomial time from rank logics, which extend fixed-point logics by operators for determining the rank of definable matrices and which are sufficient for solvability problems over fields. Based on the structure theory of finite rings, we establish logical reductions among various solvability problems. Our results indicate that all solvability problems for linear equation systems that separate fixed-point logic with counting from PTIME can be reduced to solvability over commutative rings. Moreover, we prove closure properties for classes of queries that reduce to solvability over rings, which provides normal forms for logics extended with solvability operators. We conclude by studying the extent to which fixed-point logic with counting can express problems in linear algebra over finite commutative rings...

## Towards a New Global QCD Analysis: Solution to the Non-Linear Equation at Arbitrary Impact Parameter

~Gotsman, E.; ~Kozlov, M.; ~Levin, E.; Maor, U.; ~Naftali, E.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.93%
A numerical solution is presented for the non-linear evolution equation that governs the dynamics of high parton density QCD. It is shown that thesolution falls off as $e^{-b/R}$ at large values of the impact parameter $b$. The power-like tail of the amplitude appears in impact parameter distributions only after the inclusion of dipoles of size larger than the target, a configuration for which the non-linear equation is not valid. The value, energy and impact parameterof the saturation scale $Q_s(y=\ln(1/x),b)$) are calculated both for fixed and running QCD coupling cases. It is shown that the solution exhibits geometrical scaling behaviour. The radius of interaction increases as the rapidity in accordance with the Froissart theorem. The solution we obtain differs from previous attempts, where an anzatz for $b$ behaviour was made. The solutions for running and fixed $\as$ differ. For running $\as$ we obtain a larger radius of interaction (approximately twice as large), a steeper rapidity dependence, and a larger value of the saturation scale.; Comment: 26 pages with 19 figures in eps.files

## Cosmo-dynamics and dark energy with non-linear equation of state: a quadratic model

Ananda, Kishore N.; Bruni, Marco
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.87%
We investigate the general relativistic dynamics of Robertson-Walker models with a non-linear equation of state (EoS), focusing on the quadratic case P = P_0 + \alpha \rho + \beta \rho^2. This may be taken to represent the Taylor expansion of any arbitrary barotropic EoS, P(\rho). With the right combination of P_0, \alpha and \beta, it serves as a simple phenomenological model for dark energy, or even unified dark matter. Indeed we show that this simple model for the EoS can produce a large variety of qualitatively different dynamical behaviors that we classify using dynamical systems theory. An almost universal feature is that accelerated expansion phases are mostly natural for these non-linear EoS's. These are often asymptotically de Sitter thanks to the appearance of an effective cosmological constant. Other interesting possibilities that arise from the quadratic EoS are closed models that can oscillate with no singularity, models that bounce between infinite contraction/expansion and models which evolve from a phantom phase, asymptotically approaching a de Sitter phase instead of evolving to a "Big Rip". In a second paper we investigate the effects of the quadratic EoS in inhomogeneous and anisotropic models, focusing in particular on singularities.; Comment: 25 pages...

## Relativistic stars with a linear equation of state: analogy with classical isothermal spheres and black holes

Chavanis, Pierre-Henri
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.97%
We complete our previous investigation concerning the structure and the stability of "isothermal" spheres in general relativity. This concerns objects that are described by a linear equation of state $P=q\epsilon$ so that the pressure is proportional to the energy density. In the Newtonian limit $q\to 0$, this returns the classical isothermal equation of state. We consider specifically a self-gravitating radiation (q=1/3), the core of neutron stars (q=1/3) and a gas of baryons interacting through a vector meson field (q=1). We study how the thermodynamical parameters scale with the size of the object and find unusual behaviours due to the non-extensivity of the system. We compare these scaling laws with the area scaling of the black hole entropy. We also determine the domain of validity of these scaling laws by calculating the critical radius above which relativistic stars described by a linear equation of state become dynamically unstable. For photon stars, we show that the criteria of dynamical and thermodynamical stability coincide. Considering finite spheres, we find that the mass and entropy as a function of the central density present damped oscillations. We give the critical value of the central density, corresponding to the first mass peak...

## Non-linear equation: energy conservation and impact parameter dependence

Kormilitzin, Andrey; Levin, Eugene
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.97%
In this paper we address two questions: how energy conservation affects the solution to the non-linear equation, and how impact parameter dependence influences the inclusive production. Answering the first question we solve the modified BK equation which takes into account energy conservation. In spite of the fact that we used the simplified kernel, we believe that the main result of the paper: the small ($\leq 40%$) suppression of the inclusive productiondue to energy conservation, reflects a general feature. This result leads us to believe that the small value of the nuclear modification factor is of a non-perturbative nature. In the solution a new scale appears $Q_{fr} = Q_s \exp(-1/(2 \bas))$ and the production of dipoles with the size larger than $2/Q_{fr}$ is suppressed. Therefore, we can expect that the typical temperature for hadron production is about $Q_{fr}$ ($T \approx Q_{fr}$). The simplified equation allows us to obtain a solution to Balitsky-Kovchegov equation taking into account the impact parameter dependence. We show that the impact parameter ($b$) dependence can be absorbed into the non-perturbative $b$ dependence of the saturation scale. The solution of the BK equation, as well as of the modified BK equation without $b$ dependence...

## A special non-linear equation of mathematical physics - the equation of the curvilinear electromagnetic wave

Kyriakos, Alexander G.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.01%
A special non-linear equation of curvilinear electromagnetic wave is presented. The particularity of this equation lies in the fact that in matrix form it is mathematically equivalent to the Dirac electron equation. It is shown that the solution of this is the motion of the plane-polarized electromagnetic wave on a circular trajectory. It is also shown that such twirled wave can be considered as a massive charge particle with a spin of one half, similar to electron. The non-linear equation and its Lagrangian of these EM particles are founded.; Comment: 17 pages, 4 figures, corrected the language

## The lower estimate for wandering rate of solution to a linear equation in terms of its frequency

Tikhomirov, Anastasia
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.87%
This research article compares two characteristics of solutions of linear differential equations of the third order with variable coefficients. It appears that there is a lower estimate for wandering rate of solution to a linear equation in terms of its frequency.; Comment: in Russian

## Dark energy cosmology with generalized linear equation of state

Babichev, E.; Dokuchaev, V.; Eroshenko, Yu.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.03%
Dark energy with the usually used equation of state $p=w\rho$, where $w=const<0$ is hydrodynamically unstable. To overcome this drawback we consider the cosmology of a perfect fluid with a linear equation of state of a more general form $p=\alpha(\rho-\rho_0)$, where the constants $\alpha$ and $\rho_0$ are free parameters. This non-homogeneous linear equation of state provides the description of both hydrodynamically stable ($\alpha>0$) and unstable ($\alpha<0$) fluids. In particular, the considered cosmological model describes the hydrodynamically stable dark (and phantom) energy. The possible types of cosmological scenarios in this model are determined and classified in terms of attractors and unstable points by the using of phase trajectories analysis. For the dark energy case there are possible some distinctive types of cosmological scenarios: (i) the universe with the de Sitter attractor at late times, (ii) the bouncing universe, (iii) the universe with the Big Rip and with the anti-Big Rip. In the framework of a linear equation of state the universe filled with an phantom energy, $w<-1$, may have either the de Sitter attractor or the Big Rip.; Comment: 12 pages, 11 figures, typos corrected, references added

## On cosmological evolution with the Lambda-term and any linear equation of state

Silbergleit, A. S.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.91%
Recent observational indications of an accelerating universe enhance the interest in studying models with a cosmological constant. We investigate cosmological expansion (FRW metric) with $\Lambda>0$ for a general linear equation of state $p=w\rho$, $w>-1$, so that the interplay between cosmological vacuum and quintessence is allowed, as well. Four closed-form solutions (flat universe with any $w$, and $w=1/3$, $-1/3, -2/3$) are given, in a proper compact representation. Various estimates of the expansion are presented in a general case when no closed-form solutions are available. For the open universe a simple relation between solutions with different parameters is established: it turns out that a solution with some $w$ and (properly scaled) $\Lambda$ is expressed algebraically via another solution with special different values of these parameters. The expansion becomes exponential at large times, and the amplitude at the exponent depends on the parameters. We study this dependence in detail, deriving various representations for the amplitude in terms of integrals and series. The closed-form solutions serve as benchmarks, and the solution transformation property noted above serves as a useful tool. Among the results obtained, one is that for the open universe with relatively small cosmological constant the amplitude is independent of the equation of state. Also...

## A linear equation for Minkowski sums of polytopes relatively in general position

Fukuda, Komei; Weibel, Christophe
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.87%
The objective of this paper is to study a special family of Minkowski sums, that is of polytopes relatively in general position. We show that the maximum number of faces in the sum can be attained by this family. We present a new linear equation that is satisfied by f-vectors of the sum and the summands. We study some of the implications of this equation.; Comment: 10 pages, accepted by Europ. J. Combinatorics

## Cosmological Evolution with \Lambda-Term and Any Linear equation of State

Silbergleit, A. S.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.91%
Recent observational indications of an accelerating universe enhance the interest in studying models with a cosmological constant. We investigate cosmological expansion (FRW metric) with $\Lambda>0$ for a general linear equation of state $p=w\rho$, $w>-1$, so that the interplay between cosmological vacuum and quintessence is allowed, as well. Four closed-form solutions (flat universe with any $w$, and $w=1/3$, $-1/3, -2/3$) are given, of which the last one appears to be new. For the open universe a simple relation between solutions with different parameters is established: it turns out that a solution with some $w$ and (properly scaled) $\Lambda$ is expressed algebraically via another solution with special different values of these parameters. The expansion becomes exponential at large times, and the amplitude at the exponent depends on the parameters. We study this dependence in detail, deriving various representations for the amplitude in terms of integrals and series. The closed-form solutions serve as benchmarks, and the solution transformation property noted above serves as a useful tool. Among the results obtained, one is that for the open universe with relatively small cosmological constant the amplitude is independent of the equation of state. Also...

## Slowly rotating fluid balls with linear equation of state

Fodor, Gyula
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.93%
Slowly rotating perfect fluid balls with regular center and asymptotically flat exterior are considered to second order in the rotation parameter. The necessary condition for being Petrov type D is given for general perfect fluid matter. As a special case, fluids with a linear equation of state are considered. Using a power series expansion at the regular center, it is shown that the Petrov D condition is inconsistent with the linear equation of state assumption.; Comment: To make the 2002 conference paper more accessible, 12 pages

## A Linear Equation for Wilson Loops

Olesen, Poul
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
46.01%
The Makeenko-Migdal loop equation is non-linear and first order in the area derivative, but we show that for simple loops in QCD$_2$ it is possible to reformulate this equation as a linear equation with second order derivatives. This equation is a bound state Schr\"odinger equation with a three dimensional Coulomb potential. Thus, loop dynamics leads to a surprising new picture of confinement, where this phenomenon is due to a (bound state) localization in loop space, with the Wilson loops decaying exponentially outside a characteristic radius.; Comment: 6 pages. Some comments added

## Effective Resistances, Statistical Leverage, and Applications to Linear Equation Solving

Drineas, Petros; Mahoney, Michael W.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.92%
Recent work in theoretical computer science and scientific computing has focused on nearly-linear-time algorithms for solving systems of linear equations. While introducing several novel theoretical perspectives, this work has yet to lead to practical algorithms. In an effort to bridge this gap, we describe in this paper two related results. Our first and main result is a simple algorithm to approximate the solution to a set of linear equations defined by a Laplacian (for a graph $G$ with $n$ nodes and $m \le n^2$ edges) constraint matrix. The algorithm is a non-recursive algorithm; even though it runs in $O(n^2 \cdot \polylog(n))$ time rather than $O(m \cdot polylog(n))$ time (given an oracle for the so-called statistical leverage scores), it is extremely simple; and it can be used to compute an approximate solution with a direct solver. In light of this result, our second result is a straightforward connection between the concept of graph resistance (which has proven useful in recent algorithms for linear equation solvers) and the concept of statistical leverage (which has proven useful in numerically-implementable randomized algorithms for large matrix problems and which has a natural data-analytic interpretation).; Comment: 16 pages

## Decentralized gradient algorithm for solution of a linear equation

Anderson, Brian D. O.; Mou, Shaoshuai; Morse, A. Stephen; Helmke, Uwe
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
45.99%
The paper develops a technique for solving a linear equation $Ax=b$ with a square and nonsingular matrix $A$, using a decentralized gradient algorithm. In the language of control theory, there are $n$ agents, each storing at time $t$ an $n$-vector, call it $x_i(t)$, and a graphical structure associating with each agent a vertex of a fixed, undirected and connected but otherwise arbitrary graph $\mathcal G$ with vertex set and edge set $\mathcal V$ and $\mathcal E$ respectively. We provide differential equation update laws for the $x_i$ with the property that each $x_i$ converges to the solution of the linear equation exponentially fast. The equation for $x_i$ includes additive terms weighting those $x_j$ for which vertices in $\mathcal G$ corresponding to the $i$-th and $j$-th agents are adjacent. The results are extended to the case where $A$ is not square but has full row rank, and bounds are given on the convergence rate.; Comment: 10 pages

Chilla, Vincenzo