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Quantitative theory of ordinary differential equations and tangential Hilbert 16th problem
 DEPARTMENT OF MATHEMATICS, WEIZMANN INSTITUTE OF SCIENCE, P.O.B. 26, REHOVOT 76100, ISRAEL EMAIL ADDRESS: YAKOV@WISDOM.WEIZMANN.AC.IL WWW
, 2001
"... These highly informal lecture notes aim at introducing and explaining several closely related problems on zeros of analytic functions defined by ordinary differential equations and systems of such equations. The main incentive for this study was its potential application to the tangential Hilbert 1 ..."
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These highly informal lecture notes aim at introducing and explaining several closely related problems on zeros of analytic functions defined by ordinary differential equations and systems of such equations. The main incentive for this study was its potential application to the tangential Hilbert 16th problem on zeros of complete Abelian integrals. The exposition consists mostly of examples illustrating various phenomena related to this problem. Sometimes these examples give an insight concerning the proofs, though the complete exposition of the latter is mostly relegated to separate expositions.
Implementing WS1S via Finite Automata
"... It has long been known that WS1S is decidable through the use of finite automata. However, since the worst case running time has been proven to grow extremely quickly, few have explored the implementation of the algorithm. In this paper we describe some of the points of interest that have come up wh ..."
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It has long been known that WS1S is decidable through the use of finite automata. However, since the worst case running time has been proven to grow extremely quickly, few have explored the implementation of the algorithm. In this paper we describe some of the points of interest that have come up while coding and running the algorithm. These points include the data structures used as wekk as the special properties of the automata, which we can exploit to perform minimization very quickly in certain cases. We also present some data that enable us to gain insight into how the algorithm performs in the average case, both on random inputs ans on inputs that come from the use of Presburger Arithmetic (which can be converted to WS1S) in compiler optimization.
Gödel on computability
"... Around 1950, both Gödel and Turing wrote papers for broader audiences. 1 Gödel drew in his 1951 dramatic philosophical conclusions from the general formulation of his second incompleteness theorem. These conclusions concerned the nature of mathematics and the human mind. The general formulation of t ..."
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Around 1950, both Gödel and Turing wrote papers for broader audiences. 1 Gödel drew in his 1951 dramatic philosophical conclusions from the general formulation of his second incompleteness theorem. These conclusions concerned the nature of mathematics and the human mind. The general formulation of the second theorem was explicitly based on Turing’s 1936 reduction of finite procedures to machine computations. Turing gave in his 1954 an understated analysis of finite procedures in terms of Post production systems. This analysis, prima facie quite different from that given in 1936, served as the basis for an exposition of various unsolvable problems. Turing had addressed issues of mentality and intelligence in contemporaneous essays, the best known of which is of course Computing machinery and intelligence. Gödel’s and Turing’s considerations from this period intersect through their attempt, on the one hand, to analyze finite, mechanical procedures and, on the other hand, to approach mental phenomena in a scientific way. Neuroscience or brain science was an important component of the latter for both: Gödel’s remarks in the Gibbs Lecture as well as in his later conversations with Wang and Turing’s Intelligent Machinery can serve as clear evidence for that. 2 Both men were convinced that some mental processes are not mechanical, in the sense that Turing machines cannot mimic them. For Gödel, such processes were to be found in mathematical experience and he was led to the conclusion that mind is separate from matter. Turing simply noted that for a machine or a brain it is not enough to be converted into a universal (Turing) machine in order to become intelligent: “discipline”, the characteristic
Uniqueness and symmetry in problems of optimally dense packings
"... We analyze the general problem of determining optimally dense packings, in a Euclidean or hyperbolic space, of congruent copies of some fixed finite set of bodies. We are strongly guided by examples of aperiodic tilings in Euclidean space and a detailed analysis of a new family of examples in the hy ..."
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We analyze the general problem of determining optimally dense packings, in a Euclidean or hyperbolic space, of congruent copies of some fixed finite set of bodies. We are strongly guided by examples of aperiodic tilings in Euclidean space and a detailed analysis of a new family of examples in the hyperbolic plane. Our goal is to understand qualitative features of such optimum density problems, in particular the appropriate meaning of the uniqueness of solutions, and the role of symmetry in classfying optimally dense packings.
Constants of Weitzenböck derivations and invariants of unipotent transformations acting on relatively free algebras
 J. Algebra
"... Abstract. In commutative algebra, a Weitzenböck derivation is a nonzero triangular linear derivation of the polynomial algebra K[x1,..., xm] in several variables over a field K of characteristic 0. The classical theorem of Weitzenböck states that the algebra of constants is finitely generated. (This ..."
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Abstract. In commutative algebra, a Weitzenböck derivation is a nonzero triangular linear derivation of the polynomial algebra K[x1,..., xm] in several variables over a field K of characteristic 0. The classical theorem of Weitzenböck states that the algebra of constants is finitely generated. (This algebra coincides with the algebra of invariants of a single unipotent transformation.) In this paper we study the problem of finite generation of the algebras of constants of triangular linear derivations of finitely generated (not necessarily commutative or associative) algebras over K assuming that the algebras are free in some sense (in most of the cases relatively free algebras in varieties of associative or Lie algebras). In this case the algebra of constants also coincides with the algebra of invariants of some unipotent transformation. The main results are the following: 1. We show that the subalgebra of constants of a factor algebra can be lifted to the subalgebra of constants. 2. For all varieties of associative algebras which are not nilpotent in Lie sense the subalgebras of constants of the relatively free algebras of rank ≥ 2 are not finitely generated. 3. We describe the generators of the subalgebra of constants for all factor algebras K〈x, y〉/I modulo a GL2(K)invariant ideal I. 4. Applying known results from commutative algebra, we construct classes of automorphisms of the algebra generated by two generic 2 × 2 matrices. We obtain also some partial results on relatively free Lie algebras. 1.
Local identification of scalar hybrid models with tree structure
, 2004
"... Local identification of scalar hybrid models with tree structure 1 Standard modeling approaches, for example in chemical engineering, suffer from two principal difficulties: the curse of dimension and a lack of extrapolability. We propose an approach via structured hybrid models to resolve both issu ..."
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Local identification of scalar hybrid models with tree structure 1 Standard modeling approaches, for example in chemical engineering, suffer from two principal difficulties: the curse of dimension and a lack of extrapolability. We propose an approach via structured hybrid models to resolve both issues. For simplicity we consider reactor models which can be written as a treelike composition of scalar inputoutput functions uj. The vertices j of the finite tree structure represent known or unknown subprocesses of the overall process. Known processes are modeled by whitebox functions uj; unknown processes are represented by black boxes uj. Oriented edges of the tree indicate composition of the inputoutput relations uj in a feed forward structure. The tree structure of a mixture of black and white boxes constitutes what we call a structured hybrid model (SHM). Under certain assumptions on differentiability, genericity, and monotonicity, we provide an inductive algorithm which uniquely identifies all black boxes in the SHM, up to a trivial scaling calibration between adjacent black boxes. Our result does not require any extra measurements interior to the SHM. Instead, we only require global, overall inputoutput data, clustered along a ddimensional data base of inputs. The dimension d need not exceed the maximal input dimension of any individual black box in the SHM. Compared to the total input dimension of the reactor, which may be much higher than d, this dimension reduction effectively avoids the curse of dimension. Moreover, our unique identification of all black boxes accomodates a reliable global extrapolation, far beyond the original data base, to inputregions of full dimension. We illustrate our results with a model of an industrial continuous polymerization plant. 1
Discrete Quantum Walks and Quantum Image Processing
, 2005
"... ... Processing. Our work is a contribution within the field of quantum computation from the perspective of a computer scientist. With the purpose of finding new techniques to develop quantum algorithms, there has been an increasing interest in studying Quantum Walks, the quantum counterparts of clas ..."
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... Processing. Our work is a contribution within the field of quantum computation from the perspective of a computer scientist. With the purpose of finding new techniques to develop quantum algorithms, there has been an increasing interest in studying Quantum Walks, the quantum counterparts of classical random walks. Our work in quantum walks begins with a critical and comprehensive assessment of those elements of classical random walks and discrete quantum walks on undirected graphs relevant to algorithm development. We propose a model of discrete quantum walks on an infinite line using pairs of quantum coins under different degrees of entanglement, as well as quantum walkers in different initial state configurations, including superpositions of corresponding basis states. We have found that the probability distributions of such quantum walks have particular forms which are different from the probability distributions of classical random walks. Also, our numerical results show that the symmetry properties of quantum walks with entangled coins have a nontrivial relationship with corresponding initial states and evolution operators. In addition, we have studied the properties of the entanglement generated between walkers, in a
1 Science for Global Sustainability Toward a New Paradigm
"... This paper provides a context for the Dahlem Workshop on “Earth Systems Science and Sustainability. ” We begin by characterizing the contemporary epoch of Earth history in which humanity has emerged as a major — and uniquely selfreflexive — geological force. We turn next to the extraordinary revolu ..."
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This paper provides a context for the Dahlem Workshop on “Earth Systems Science and Sustainability. ” We begin by characterizing the contemporary epoch of Earth history in which humanity has emerged as a major — and uniquely selfreflexive — geological force. We turn next to the extraordinary revolution in our
Editor: R. de la Llave UNIQUENESS AND SYMMETRY IN PROBLEMS OF OPTIMALLY DENSE PACKINGS
"... Abstract. Part of Hilbert’s eighteenth problem is to classify the symmetries of the densest packings of bodies in Euclidean and hyperbolic spaces, for instance the densest packings of balls or simplices. We prove that when such a packing problem has a unique solution up to congruence then the soluti ..."
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Abstract. Part of Hilbert’s eighteenth problem is to classify the symmetries of the densest packings of bodies in Euclidean and hyperbolic spaces, for instance the densest packings of balls or simplices. We prove that when such a packing problem has a unique solution up to congruence then the solution must have cocompact symmetry group, and we prove that the densest packing of unit disks in the Euclidean plane is unique up to congruence. We also analyze some densest packings of polygons in the hyperbolic plane. I.
Introducción a la computación cuántica
, 2007
"... La mecánica cuántica es la rama de la física que describe el comportamiento de la naturaleza a escalas muy pequeñas (por ejemplo, el comportamiento de los átomos). La teoría de la computación se encarga de estudiar si un problema es susceptible de ser resuelto utilizando una computadoa, así como la ..."
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La mecánica cuántica es la rama de la física que describe el comportamiento de la naturaleza a escalas muy pequeñas (por ejemplo, el comportamiento de los átomos). La teoría de la computación se encarga de estudiar si un problema es susceptible de ser resuelto utilizando una computadoa, así como la cantidad de recursos (tiempo, energía) que se debe invertir en caso de existir solución. En consecuencia, la computación cuántica hace uso de la mecánica cuántica con el objetivo de incrementar nuestra capacidad computacional para el procesamiento de información y solución de problemas. Por otra parte, la teoría de la información cuántica estudia los métodos, capacidades y límites que las leyes de la física imponen en la transmisión y recuperación de información. El estudio formal de la computación cuántica comenzó con las preguntas que Richard Feynman planteó sobre dos temas: 1) la posibilidad de simular sistemas cuánticos, y 2) las leyes de la física que caracterizan al proceso de calcular [92, 93]. A partir de ese trabajo, la computación cuántica ha avanzado a paso firme; por ejemplo, se ha definido formalmente la estructura de una computadora cuántica [3], se han encontrado resultados espectaculares como el algoritmo de Shor [4] (capaz de factorizar un número entero muy largo en tiempo razonable utilizando una computadora cuántica [4, 7]) y el algoritmo de Grover [5] (este algoritmo encuentra elementos en conjuntos desordenados de forma más eficiente que cualquier algoritmo posible ejecutado en computadoras convencionales [5, 7]), y se ha diseñado una teoría y práctica de la criptografía usando las propiedades de la física cuántica [6]. En el futuro mediato, la computación cuántica tendrá gran impacto en la industria de la computación y el desarrollo de protocolos de criptografía y seguridad computacional [12–14]. La computación y la información cuánticas representan un reto teórico y experimental por la cantidad y complejidad de problemas a resolver. A pesar de dichos retos, los avances realizados hasta ahora permiten ya pensar en aplicaciones de esta disciplina en áreas del conocimiento tales como la