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

Fonte: Universidade Federal do Rio Grande do Sul
Publicador: Universidade Federal do Rio Grande do Sul

Tipo: Dissertação
Formato: application/pdf

POR

Relevância na Pesquisa

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#Interval mathematics#Analise : Intervalos#Intervals#Aritmetica intervalar#Equações lineares#Linear equations systems (SELAS)#Intervals methods for resolution of linear systems#Interval applied library

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...

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## Solving linear equation systems using parallel processing

Fonte: Universidade do Minho
Publicador: Universidade do Minho

Tipo: Conferência ou Objeto de Conferência

Publicado em //1995
ENG

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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.

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/01/2011

Relevância na Pesquisa

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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...

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## Definability of linear equation systems over groups and rings

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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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...

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 08/12/2005

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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...

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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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...

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 08/09/2010

Relevância na Pesquisa

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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...

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 29/12/2012
RU

Relevância na Pesquisa

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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

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## Dark energy cosmology with generalized linear equation of state

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/08/2002

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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...

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/10/2001

Relevância na Pesquisa

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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...

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## Slowly rotating fluid balls with linear equation of state

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 13/12/2006

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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

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## A Linear Equation for Wilson Loops

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 18/05/2010

Relevância na Pesquisa

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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

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## Decentralized gradient algorithm for solution of a linear equation

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/09/2015

Relevância na Pesquisa

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#Computer Science - Systems and Control#Computer Science - Distributed, Parallel, and Cluster Computing

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

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## On the linear equation method for the subduction problem in symmetric groups

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Relevância na Pesquisa

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We focus on the tranformation matrices between the standard Young-Yamanouchi
basis of an irreducible representation for the symmetric group S_n and the
split basis adapted to the direct product subgroups S_{n_1} \times S_{n-n_1} .
We introduce the concept of subduction graph and we show that it conveniently
describes the combinatorial structure of the equation system arisen from the
linear equation method. Thus we can outline an improved algorithm to solve the
subduction problem in symmetric groups by a graph searching process. We
conclude observing that the general matrix form for multiplicity separations,
resulting from orthonormalization, can be expressed in terms of Sylvester
matrices relative to a suitable inner product in the multiplicity space.; Comment: 13 pages, 2 figures, iopart class; reference added

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## Analog Computing for Real-Time Solution of Time-Varying Linear Equations

Fonte: Institute of Electrical and Electronics Engineers (IEEE Inc)
Publicador: Institute of Electrical and Electronics Engineers (IEEE Inc)

Tipo: Conference paper

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#Keywords: Computer simulation#Iterative methods#Linear equations#Optimization#Real time systems#Recurrent neural networks#Time varying systems#VLSI circuits#Implicit recurrent neural network (IRNN) model#Real-time computation#Sylvester equations

An implicit recurrent neural network model (IRNN) is proposed in this paper for solving on-line time-varying linear equations. Such a neural network can be implemented as analog circuits or VLSI. Excellent convergent properties have been revealed by careful theoretical analysis. In the specific case where the linear equation is obtained from a time-varying Sylvester equation, the proposed IRNN model coincides with some existing recurrent neural networks reported in recent literature, where simulation examples have been reported to demonstrate the effectiveness and efficiency.

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