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23
Remarks on the Apolynomial of a Knot.
 J. Knot Theory Ramifications
"... This paper reviews the two variable polynomial invariant of knots defined using representations of the fundamental group of the knot complement into SLC. The slopes of the sides of the Newton polygon of this polynomial are boundary slopes of incompressible surfaces in the knot complement. The po ..."
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Cited by 42 (2 self)
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This paper reviews the two variable polynomial invariant of knots defined using representations of the fundamental group of the knot complement into SLC. The slopes of the sides of the Newton polygon of this polynomial are boundary slopes of incompressible surfaces in the knot complement. The polynomial also contains information about which surgeries are cyclic, and about the shape of the cusp when the knot is hyperbolic. We prove that at least some mutants have the same polynomial, and that most untwisted doubles have nontrivial polynomial. We include several open questions.
Graphical models and exponential families
 In Proceedings of the 14th Annual Conference on Uncertainty in Arti cial Intelligence (UAI98
, 1998
"... We provide a classification of graphical models according to their representation as subfamilies of exponential families. Undirected graphical models with no hidden variables are linear exponential families (LEFs), directed acyclic graphical models and chain graphs with no hidden variables, includin ..."
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Cited by 22 (1 self)
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We provide a classification of graphical models according to their representation as subfamilies of exponential families. Undirected graphical models with no hidden variables are linear exponential families (LEFs), directed acyclic graphical models and chain graphs with no hidden variables, including Bayesian networks with several families of local distributions, are curved exponential families (CEFs) and graphical models with hidden variables are stratified exponential families (SEFs). An SEF is a finite union of CEFs satisfying a frontier condition. In addition, we illustrate how one can automatically generate independence and nonindependence constraints on the distributions over the observable variables implied by a Bayesian network with hidden variables. The relevance of these results for model selection is examined. 1
Anisotropic Branching Random Walks On Homogeneous Trees
, 1999
"... Symmetric branching random walk on a homogeneous tree exhibits a weak survival phase: For parameter values in a certain interval, the population survives forever with positive probability, but, with probability one, eventually vacates every finite subset of the tree. In this phase, particle trail ..."
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Cited by 19 (3 self)
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Symmetric branching random walk on a homogeneous tree exhibits a weak survival phase: For parameter values in a certain interval, the population survives forever with positive probability, but, with probability one, eventually vacates every finite subset of the tree. In this phase, particle trails must converge to the geometric boundary\Omega of the tree. The random subset of the boundary consisting of all ends of the tree in which the population survives, called the limit set of the process, is shown to have Hausdorff dimension no larger than one half the Hausdorff dimension of the entire geometric boundary. Moreover, there is strict inequality at the phase separation point between weak and strong survival except when the branching random walk is isotropic. It is further shown that in all cases there is a distinguished probability measure supported by\Omega such that the Hausdorff dimension of "\Omega , where\Omega is the set of \Gammageneric points of \Omega...
Random walks on regular languages and algebraic systems of generating functions
 In Algebraic methods in statistics and probability (Notre Dame, IN, 2000), volume 287 of Contemp. Math
, 2001
"... Abstract. A random walk on a regular language is a Markov chain on the set of all finite words from a finite alphabet A whose transition probabilities obey the following rules: (1) Only the last two letters of a word may be modified in one jump, and at most one letter may be adjoined or deleted. (2) ..."
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Cited by 14 (2 self)
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Abstract. A random walk on a regular language is a Markov chain on the set of all finite words from a finite alphabet A whose transition probabilities obey the following rules: (1) Only the last two letters of a word may be modified in one jump, and at most one letter may be adjoined or deleted. (2) Probabilities of modification, deletion, and/or adjunction depend only on the last two letters of the current word. Special cases include (a) reflecting random walks on the nonnegative integers; (b) LIFO queues; (c) finiterange random walks on homogeneous trees; and (d) random walks on the modular group P SL2(Z). It is shown that the n−step transition probabilities of a random walk on a regular language must obey one of three different types of power laws. The analysis is based on the study of an algebraic system of generating functions related to the Green’s function. 1.
ON PSEUDOSPECTRA OF MATRIX POLYNOMIALS AND THEIR BOUNDARIES
"... Abstract. In the first part of this paper (Sections 2–4), the main concern is with the boundary of the pseudospectrum of a matrix polynomial and, particularly, with smoothness properties of the boundary. In the second part (Sections 5–6), results are obtained concerning the number of connected compo ..."
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Abstract. In the first part of this paper (Sections 2–4), the main concern is with the boundary of the pseudospectrum of a matrix polynomial and, particularly, with smoothness properties of the boundary. In the second part (Sections 5–6), results are obtained concerning the number of connected components of pseudospectra, as well as results concerning matrix polynomials with multiple eigenvalues, or the proximity to such polynomials. 1.
Complexity and Real Computation: A Manifesto
 International Journal of Bifurcation and Chaos
, 1995
"... . Finding a natural meeting ground between the highly developed complexity theory of computer science with its historical roots in logic and the discrete mathematics of the integers and the traditional domain of real computation, the more eclectic less foundational field of numerical analysis ..."
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Cited by 12 (0 self)
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. Finding a natural meeting ground between the highly developed complexity theory of computer science with its historical roots in logic and the discrete mathematics of the integers and the traditional domain of real computation, the more eclectic less foundational field of numerical analysis with its rich history and longstanding traditions in the continuous mathematics of analysis presents a compelling challenge. Here we illustrate the issues and pose our perspective toward resolution. This article is essentially the introduction of a book with the same title (to be published by Springer) to appear shortly. Webster: A public declaration of intentions, motives, or views. k Partially supported by NSF grants. y International Computer Science Institute, 1947 Center St., Berkeley, CA 94704, U.S.A., lblum@icsi.berkeley.edu. Partially supported by the LettsVillard Chair at Mills College. z Universitat Pompeu Fabra, Balmes 132, Barcelona 08008, SPAIN, cucker@upf.es. P...
Representation Theory and the Apolynomial of a Knot
 Chaos, Solitons & Fractals 9
, 1998
"... this article. The idea for understanding degenerations geometrically is roughly the following. Let G be a finitely generated group and (l/V, dw) a metric space. Consider a sequence, p, of representations of G into the group, Isom(W), of isometries of W. We will say that this sequence blows up if th ..."
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Cited by 7 (0 self)
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this article. The idea for understanding degenerations geometrically is roughly the following. Let G be a finitely generated group and (l/V, dw) a metric space. Consider a sequence, p, of representations of G into the group, Isom(W), of isometries of W. We will say that this sequence blows up if there is an element g of G such that the sequence p(g) cannot be conjugated to lie in any compact subset of Isom(W). Suppose that it is possible to linearly rescale the metric on W to obtain a sequence of metric spaces, V, which converge in some sense to a limiting metric space W. We want to do this in such a way that the actions p, on the rescaled spaces V have a subsequence that converges to an interesting action, , by isometries on . The particular case we have in mind is that W = ]I]I 3 and W is some sort of tree
Formal Orthogonality On An Algebraic Curve
 Annals Numer. Math
, 1993
"... Formal orthogonality on an algebraic variety for polynomials is defined and studied. Examples are given. Vector orthogonal polynomials of dimension d are proved to be orthogonal with respect to points on the unit circle. ..."
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Cited by 7 (4 self)
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Formal orthogonality on an algebraic variety for polynomials is defined and studied. Examples are given. Vector orthogonal polynomials of dimension d are proved to be orthogonal with respect to points on the unit circle.