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## ‣ Smoothed analysis of Gaussian elimination

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 60 p.; 2613986 bytes; 2619272 bytes; application/pdf; application/pdf

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We present a smoothed analysis of Gaussian elimination, both with partial pivoting and without pivoting. Let A be any matrix and let A be a slight random perturbation of A. We prove that it is unlikely that A has large condition number. Using this result, we prove it is unlikely that A has large growth factor under Gaussian elimination without pivoting. By combining these results, we bound the smoothed precision needed to perform Gaussian elimination without pivoting. Our results improve the average-case analysis of Gaussian elimination without pivoting performed by Yeung and Chan (SIAM J. Matrix Anal. Appl., 1997). We then extend the result on the growth factor to the case of partial pivoting, and present the first analysis of partial pivoting that gives a sub-exponential bound on the growth factor. In particular, we show that if the random perturbation is Gaussian with variance [sigma]², then the growth factor is bounded by (n/[sigma])[to the power of] (o log n) with very high probability.; by Arvind Sankar.; Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.; Includes bibliographical references (p. 59-60).

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## ‣ Asymptotic analysis of extreme electrochemical transport

Fonte: Massachusetts Institute of Technology
Publicador: Massachusetts Institute of Technology

Tipo: Tese de Doutorado
Formato: 244 p.; 11555675 bytes; 11566560 bytes; application/pdf; application/pdf

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In the study of electrochemical transport processes, experimental exploration currently outpaces theoretical understanding of new phenomena. Classical electrochemical transport theory is not equipped to explain the behavior of electrochemical systems in the extreme operating conditions required by modern devices. In this thesis, we extend the classical theory to examine the response of two electrochemical systems that form the basis for novel electrochemical devices. We first examine the DC response of an electrochemical thin film, such as the separator in a micro-battery, driven by current applied through reactive electrodes. The model system consists of a binary electrolyte between parallel-plate electrodes, each possessing a compact Stern layer which mediates Faradaic reactions with Butler-Volmer kinetics. Our analysis differs from previous studies in two significant ways. First, we impose the full nonlinear, reactive boundary conditions appropriate for electrolytic/galvanic cells.; (cont.) Since surface effects become important for physically small systems, the use of reactive boundary conditions is critical in order to gain insight into the behavior of actual electrochemical thin films that are sandwiched between reactive electrodes...

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## ‣ A confirmatory factor analysis of Attitudes Toward Mathematics Inventory (ATMI)

Fonte: Association of Mathematics Educators
Publicador: Association of Mathematics Educators

Tipo: Artigo de Revista Científica

Publicado em //2013
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#Attitudes toward mathematics#Attitudes Toward Mathematics Inventory (ATMI)#confirmatory factor analysis#Australia

Students’ attitudes toward mathematics have been known to influence students’ participation, engagement, and achievement in mathematics. A variety of instruments have been developed to measure students’ attitudes toward mathematics for example Mathematics Attitude Scale (Aiken, 1974), Fennema-Sherman Mathematics Attitudes Scales (Fennema & Sherman, 1976), and Attitudes Toward Mathematics Inventory (ATMI) (Tapia & Marsh, 1996). The purpose of this paper is to report the validation of the ATMI instrument. It was administered to 699 Year 7 and 8 students in 14 schools in South Australia. The students responded on a five-point Likert scale. Confirmatory factor analysis (CFA) supported the original four-factor correlated structure based on several fit indices. The validation provided evidence that ATMI can be a viable scale to measure students’ attitudes toward mathematics in a South Australian context; Aysha Abdul Majeed, I Gusti Ngurah Darmawan, Peggy Lynch

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## ‣ An investigation into the nature of mathematics textbooks at junior cycle and their role in mathematics education

Fonte: University of Limerick
Publicador: University of Limerick

Tipo: info:eu-repo/semantics/doctoralThesis; all_ul_research; ul_published_reviewed; ul_theses_dissertations

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peer-reviewed; This research study is aimed at improving the quality of the mathematics textbooks
available for junior cycle students. It is widely agreed that there is room for
improvement with regard to the quality of mathematics at both junior and senior
cycle level in Ireland. One such area which can be improved is the e ectiveness of
the resources available in both junior and senior cycle mathematics classrooms.
While the TIMSS report (Valverde et al., 2002) has explored textbooks on an
international scale, minimal research (minor role in TIMSS Report) has been
carried out on Irish mathematics textbooks. Considering the level of responsibility
shouldered by mathematics textbooks, there is an obvious gap in mathematics
education research.
The aim of this study is to investigate the quality of the mathematics textbooks
currently in use at junior secondary school level in Ireland. This is achieved by
investigating, extending and applying suitable methodological tools for textbook
analysis. Ultimately the aim of this research is to improve the quality of teaching
and learning of mathematics at junior cycle level which should feed directly into
improving the quality of mathematics at senior cycle. This will be achieved by
rst measuring the quality of the current junior cycle mathematics textbooks and
then highlighting the role of improved textbooks in students' conceptual under-
standing. At present in Ireland...

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## ‣ Meaning in Classical Mathematics: Is it at Odds with Intuitionism?

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/10/2011
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#Mathematics - Logic#Mathematics - Classical Analysis and ODEs#Mathematics - History and Overview#01A85, 26E35, 03A05, 97A20, 97C30

We examine the classical/intuitionist divide, and how it reflects on modern
theories of infinitesimals. When leading intuitionist Heyting announced that
"the creation of non-standard analysis is a standard model of important
mathematical research", he was fully aware that he was breaking ranks with
Brouwer. Was Errett Bishop faithful to either Kronecker or Brouwer? Through a
comparative textual analysis of three of Bishop's texts, we analyze the
ideological and/or pedagogical nature of his objections to infinitesimals a la
Robinson. Bishop's famous "debasement" comment at the 1974 Boston workshop,
published as part of his Crisis lecture, in reality was never uttered in front
of an audience. We compare the realist and the anti-realist intuitionist
narratives, and analyze the views of Dummett, Pourciau, Richman, Shapiro, and
Tennant. Variational principles are important physical applications, currently
lacking a constructive framework. We examine the case of the Hawking-Penrose
singularity theorem, already analyzed by Hellman in the context of the
Quine-Putnam indispensability thesis.; Comment: 88 pages, to appear in Intellectica 56 (2011), no. 2

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## ‣ A cognitive analysis of Cauchy's conceptions of function, continuity, limit, and infinitesimal, with implications for teaching the calculus

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 07/01/2014
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In this paper we use theoretical frameworks from mathematics education and
cognitive psychology to analyse Cauchy's ideas of function, continuity, limit
and infinitesimal expressed in his Cours D'Analyse. Our analysis focuses on the
development of mathematical thinking from human perception and action into more
sophisticated forms of reasoning and proof, offering different insights from
those afforded by historical or mathematical analyses. It highlights the
conceptual power of Cauchy's vision and the fundamental change involved in
passing from the dynamic variability of the calculus to the modern
set-theoretic formulation of mathematical analysis. This offers a re-evaluation
of the relationship between the natural geometry and algebra of elementary
calculus that continues to be used in applied mathematics, and the formal set
theory of mathematical analysis that develops in pure mathematics and evolves
into the logical development of non-standard analysis using infinitesimal
concepts. It suggests that educational theories developed to evaluate student
learning are themselves based on the conceptions of the experts who formulate
them. It encourages us to reflect on the principles that we use to analyse the
developing mathematical thinking of students...

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## ‣ Numerical Methods and Analysis via Random Field Based Malliavin Calculus for Backward Stochastic PDEs

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/06/2013
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#Mathematics - Probability#Mathematics - Dynamical Systems#Mathematics - Numerical Analysis#Mathematics - Optimization and Control#Mathematics - Statistics Theory#60H35, 65C30, 60H15, 60K37, 60H30

We study the adapted solution, numerical methods, and related convergence
analysis for a unified backward stochastic partial differential equation
(B-SPDE). The equation is vector-valued, whose drift and diffusion coefficients
may involve nonlinear and high-order partial differential operators. Under
certain generalized Lipschitz and linear growth conditions, the existence and
uniqueness of adapted solution to the B-SPDE are justified. The methods are
based on completely discrete schemes in terms of both time and space. The
analysis concerning error estimation or rate of convergence of the methods is
conducted. The key of the analysis is to develop new theory for random field
based Malliavin calculus to prove the existence and uniqueness of adapted
solutions to the first-order and second-order Malliavin derivative based
B-SPDEs under random environments.; Comment: 39 pages

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## ‣ Harmonic analysis on the infinite-dimensional unitary group and determinantal point processes

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Representation Theory#Mathematical Physics#Mathematics - Classical Analysis and ODEs#Mathematics - Combinatorics#Mathematics - Probability

The infinite-dimensional unitary group U(infinity) is the inductive limit of
growing compact unitary groups U(N). In this paper we solve a problem of
harmonic analysis on U(infinity) stated in the previous paper math/0109193. The
problem consists in computing spectral decomposition for a remarkable
4-parameter family of characters of U(infinity). These characters generate
representations which should be viewed as analogs of nonexisting regular
representation of U(infinity).
The spectral decomposition of a character of U(infinity) is described by the
spectral measure which lives on an infinite-dimensional space Omega of
indecomposable characters. The key idea which allows us to solve the problem is
to embed Omega into the space of point configurations on the real line without
2 points. This turns the spectral measure into a stochastic point process on
the real line. The main result of the paper is a complete description of the
processes corresponding to our concrete family of characters. We prove that
each of the processes is a determinantal point process. That is, its
correlation functions have determinantal form with a certain kernel. Our
kernels have a special `integrable' form and are expressed through the Gauss
hypergeometric function.
In simpler situations of harmonic analysis on infinite symmetric group and
harmonic analysis of unitarily invariant measures on infinite hermitian
matrices similar results were obtained in our papers math/9810015...

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## ‣ A quantization of the harmonic analysis on the infinite-dimensional unitary group

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Representation Theory#Mathematics - Classical Analysis and ODEs#Mathematics - Combinatorics#Mathematics - Probability

The present work stemmed from the study of the problem of harmonic analysis
on the infinite-dimensional unitary group U(\infty). That problem consisted in
the decomposition of a certain 4-parameter family of unitary representations,
which replace the nonexisting two-sided regular representation (Olshanski, J.
Funct. Anal., 2003, arXiv:0109193). The required decomposition is governed by
certain probability measures on an infinite-dimensional space \Omega, which is
a dual object to U(\infty). A way to describe those measures is to convert them
into determinantal point processes on the real line, it turned out that their
correlation kernels are computable in explicit form --- they admit a closed
expression in terms of the Gauss hypergeometric function 2-F-1 (Borodin and
Olshanski, Ann. Math., 2005, arXiv:0109194).
In the present work we describe a (nonevident) q-discretization of the whole
construction. This leads us to a new family of determinantal point processes.
We reveal its connection with an exotic finite system of q-discrete orthogonal
polynomials --- the so-called pseudo big q-Jacobi polynomials. The new point
processes live on a double q-lattice and we show that their correlation kernels
are expressed through the basic hypergeometric function 2-\phi-1.
A crucial novel ingredient of our approach is an extended version G of the
Gelfand-Tsetlin graph (the conventional graph describes the Gelfand-Tsetlin
branching rule for irreducible representations of unitary groups). We find the
q-boundary of G...

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## ‣ The essence of the homotopy analysis method

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 31/05/2011
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#Nonlinear Sciences - Exactly Solvable and Integrable Systems#Mathematics - Analysis of PDEs#Mathematics - Classical Analysis and ODEs#Mathematics - Numerical Analysis

The generalized Taylor expansion including a secret auxiliary parameter $h$
which can control and adjust the convergence region of the series is the
foundation of the homotopy analysis method proposed by Liao. The secret of $h$
can't be understood in the frame of the homotopy analysis method. This is a
serious shortcoming of Liao's method. We solve the problem. Through a detailed
study of a simple example, we show that the generalized Taylor expansion is
just the usual Taylor's expansion at different point $t_1$. We prove that there
is a relationship between $h$ and $t_1$, which reveals the meaning of $h$ and
the essence of the homotopy analysis method. As an important example, we study
the series solution of the Blasius equation. Using the series expansion method
at different points, we obtain the same result with liao's solution given by
the homotopy analysis method.; Comment: 5 pages

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## ‣ Comparison of a general series expansion method and the homotopy analysis method

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 31/05/2011
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#Nonlinear Sciences - Exactly Solvable and Integrable Systems#Mathematics - Analysis of PDEs#Mathematics - Classical Analysis and ODEs#Mathematics - Numerical Analysis

A simple analytic tool namely the general series expansion method is proposed
to find the solutions for nonlinear differential equations. By choosing a set
of suitable basis functions $\{e_n(t,t_0)\}_{n=0}^{+\infty}$ such that the
solution to the equation can be expressed by
$u(t)=\sum_{n=0}^{+\infty}c_ne_n(t,t_0)$. In general, $t_0$ can control and
adjust the convergence region of the series solution such that our method has
the same effect as the homotopy analysis method proposed by Liao, but our
method is more simple and clear. As a result, we show that the secret parameter
$h$ in the homotopy analysis methods can be explained by using our parameter
$t_0$. Therefore, our method reveals a key secret in the homotopy analysis
method. For the purpose of comparison with the homotopy analysis method, a
typical example is studied in detail.; Comment: 8 pages

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## ‣ Automated Bifurcation Analysis for Nonlinear Elliptic Partial Difference Equations on Graphs

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 11/10/2010
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#Mathematics - Dynamical Systems#Mathematics - Analysis of PDEs#Mathematics - Group Theory#Mathematics - Numerical Analysis#20C35, 35P10, 65N25

We seek solutions $u\in\R^n$ to the semilinear elliptic partial difference
equation $-Lu + f_s(u) = 0$, where $L$ is the matrix corresponding to the
Laplacian operator on a graph $G$ and $f_s$ is a one-parameter family of
nonlinear functions. This article combines the ideas introduced by the authors
in two papers: a) {\it Nonlinear Elliptic Partial Difference Equations on
Graphs} (J. Experimental Mathematics, 2006), which introduces analytical and
numerical techniques for solving such equations, and b) {\it Symmetry and
Automated Branch Following for a Semilinear Elliptic PDE on a Fractal Region}
wherein we present some of our recent advances concerning symmetry,
bifurcation, and automation fo
We apply the symmetry analysis found in the SIAM paper to arbitrary graphs in
order to obtain better initial guesses for Newton's method, create informative
graphics, and be in the underlying variational structure. We use two modified
implementations of the gradient Newton-Galerkin algorithm (GNGA, Neuberger and
Swift) to follow bifurcation branches in a robust way. By handling difficulties
that arise when encountering accidental degeneracies and higher-dimension we
can find many solutions of many symmetry types to the discrete nonlinear
system. We present a selection of experimental results which demonstrate our
algorithm's capability to automatically generate bifurcation diagrams and
solution graphics starting with only an edgelis of a graph. We highlight
interesting symmetry and variational phenomena.

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## ‣ A Burgessian critique of nominalistic tendencies in contemporary mathematics and its historiography

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - History and Overview#Mathematics - Classical Analysis and ODEs#Mathematics - Logic#01A85, 26E35, 03A05, 97A20, 97C30

We analyze the developments in mathematical rigor from the viewpoint of a
Burgessian critique of nominalistic reconstructions. We apply such a critique
to the reconstruction of infinitesimal analysis accomplished through the
efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy's
foundational work associated with the work of Boyer and Grabiner; and to
Bishop's constructivist reconstruction of classical analysis. We examine the
effects of a nominalist disposition on historiography, teaching, and research.; Comment: 57 pages; 3 figures. Corrected misprints

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## ‣ Idempotent mathematics and interval analysis

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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A brief introduction into Idempotent Mathematics and an idempotent version of
Interval Analysis are presented. Some applications are discussed.; Comment: 22 pages, no figures; to be published in an abridged form in Reliable
Computing (Kluwer); the bibliography is updated in this version

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## ‣ Convex conditions for robust stability analysis and stabilization of linear aperiodic impulsive and sampled-data systems under dwell-time constraints

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Optimization and Control#Computer Science - Systems and Control#Mathematics - Classical Analysis and ODEs#Mathematics - Dynamical Systems

Stability analysis and control of linear impulsive systems is addressed in a
hybrid framework, through the use of continuous-time time-varying discontinuous
Lyapunov functions. Necessary and sufficient conditions for stability of
impulsive systems with periodic impulses are first provided in order to set up
the main ideas. Extensions to stability of aperiodic systems under minimum,
maximum and ranged dwell-times are then derived. By exploiting further the
particular structure of the stability conditions, the results are
non-conservatively extended to quadratic stability analysis of linear uncertain
impulsive systems. These stability criteria are, in turn, losslessly extended
to stabilization using a particular, yet broad enough, class of state-feedback
controllers, providing then a convex solution to the open problem of robust
dwell-time stabilization of impulsive systems using hybrid stability criteria.
Relying finally on the representability of sampled-data systems as impulsive
systems, the problems of robust stability analysis and robust stabilization of
periodic and aperiodic uncertain sampled-data systems are straightforwardly
solved using the same ideas. Several examples are discussed in order to show
the effectiveness and reduced complexity of the proposed approach.; Comment: 12 pages...

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## ‣ Universal algorithms, mathematics of semirings and parallel computations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 07/05/2010
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#Mathematics - Numerical Analysis#Computer Science - Data Structures and Algorithms#Computer Science - Mathematical Software#Computer Science - Neural and Evolutionary Computing#15A80 (Primary), 20M99, 65F99, 65K10, 68N19, 65G30, 68W10, 68N30
(Secondary), 68Q65, 49M99, 12K10

This is a survey paper on applications of mathematics of semirings to
numerical analysis and computing. Concepts of universal algorithm and generic
program are discussed. Relations between these concepts and mathematics of
semirings are examined. A very brief introduction to mathematics of semirings
(including idempotent and tropical mathematics) is presented. Concrete
applications to optimization problems, idempotent linear algebra and interval
analysis are indicated. It is known that some nonlinear problems (and
especially optimization problems) become linear over appropriate semirings with
idempotent addition (the so-called idempotent superposition principle). This
linearity over semirings is convenient for parallel computations.; Comment: 36 pages, 1 figure. To appear in Springer Lecture Notes in
Computational Science and Engineering.

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## ‣ Harmonic analysis of oscillators through standard numerical continuation tools

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 25/06/2010
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#Mathematics - Dynamical Systems#Mathematics - Classical Analysis and ODEs#Mathematics - Numerical Analysis

In this paper, we describe a numerical continuation method that enables
harmonic analysis of nonlinear periodic oscillators. This method is formulated
as a boundary value problem that can be readily implemented by resorting to a
standard continuation package - without modification - such as AUTO, which we
used. Our technique works for any kind of oscillator, including electronic,
mechanical and biochemical systems. We provide two case studies. The first
study concerns itself with the autonomous electronic oscillator known as the
Colpitts oscillator, and the second one with a nonlinear damped oscillator, a
non-autonomous mechanical oscillator. As shown in the case studies, the
proposed technique can aid both the analysis and the design of the oscillators,
by following curves for which a certain constraint, related to harmonic
analysis, is fulfilled.; Comment: 20 pages, 4 figures

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## ‣ Reconstructing initial data using observers : error analysis of the semi-discrete and fully discrete approximations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 27/08/2010
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#Mathematics - Numerical Analysis#Mathematics - Analysis of PDEs#Mathematics - Optimization and Control#35Q93, 35L05, 35J10, 65M22

A new iterative algorithm for solving initial data inverse problems from
partial observations has been recently proposed in Ramdani, Tucsnak and Weiss
[15]. Based on the concept of observers (also called Luenberger observers),
this algorithm covers a large class of abstract evolution PDE's. In this paper,
we are concerned with the convergence analysis of this algorithm. More
precisely, we provide a complete numerical analysis for semi-discrete (in
space) and fully discrete approximations derived using finite elements in space
and finite differences in time. The analysis is carried out for abstract
Schr\"odinger and wave conservative systems with bounded observation (locally
distributed).; Comment: 38 pages, 1 figures

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## ‣ Fundaments of Quaternionic Clifford Analysis III: Fischer Decomposition in Symplectic Harmonic Analysis

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/04/2014
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#Mathematics - Classical Analysis and ODEs#Mathematics - Analysis of PDEs#Mathematics - Complex Variables#35J05 31B05 30G35 22E60

In the framework of quaternionic Clifford analysis in Euclidean space
$\mathbb{R}^{4p}$, which constitutes a refinement of Euclidean and Hermitian
Clifford analysis, the Fischer decomposition of the space of complex valued
polynomials is obtained in terms of spaces of so--called (adjoint) symplectic
spherical harmonics, which are irreducible modules for the symplectic group
Sp$(p)$. Its Howe dual partner is determined to be $\mathfrak{sl}(2,\mathbb{C})
\oplus \mathfrak{sl}(2,\mathbb{C}) = \mathfrak{so}(4,\mathbb{C})$.

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## ‣ Idempotent and tropical mathematics. Complexity of algorithms and interval analysis

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 08/09/2012
Português

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#Mathematics - Numerical Analysis#Computer Science - Data Structures and Algorithms#Mathematics - Rings and Algebras#81Q20, 16S80, 65G99, 68Q25, 15A80, 81S99, 65G30, 12K10, 46L65,
28B10, 28A80, 28A25

A very brief introduction to tropical and idempotent mathematics is
presented. Tropical mathematics can be treated as a result of a dequantization
of the traditional mathematics as the Planck constant tends to zero taking
imaginary values. In the framework of idempotent mathematics usually
constructions and algorithms are more simple with respect to their traditional
analogs. We especially examine algorithms of tropical/idempotent mathematics
generated by a collection of basic semiring (or semifield) operations and other
"good" operations. Every algorithm of this type has an interval version. The
complexity of this interval version coincides with the complexity of the
initial algorithm. The interval version of an algorithm of this type gives
exact interval estimates for the corresponding output data. Algorithms of
linear algebra over idempotent and semirings are examined. In this case, basic
algorithms are polynomial as well as their interval versions. This situation is
very different from the traditional linear algebra, where basic algorithms are
polynomial but the corresponding interval versions are NP-hard and interval
estimates are not exact.; Comment: 26 pages, submitted to the journal "Computers and Mathematics with
Applications". arXiv admin note: substantial text overlap with
arXiv:1001.4247...

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