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26
Smoothness of stationary subdivision on irregular meshes
 Constructive Approximation
, 1998
"... We derive necessary and sufficient conditions for tangent plane and C kcontinuity of stationary subdivision schemes near extraordinary vertices. Our criteria generalize most previously known conditions. We introduce a new approach to analysis of subdivision surfaces based on the idea of the univers ..."
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Cited by 22 (1 self)
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We derive necessary and sufficient conditions for tangent plane and C kcontinuity of stationary subdivision schemes near extraordinary vertices. Our criteria generalize most previously known conditions. We introduce a new approach to analysis of subdivision surfaces based on the idea of the universal surface. Any subdivision surface can be locally represented as a projection of the universal surface, which is uniquely defined by the subdivision scheme. This approach provides us with a more intuitive geometric understanding of subdivision near extraordinary vertices. AMS MOS classification: 65D10, 65D17, 68U05
C k Continuity of Subdivision Surfaces
, 1996
"... Stationary subdivision is an important tool for generating smooth freeform surfaces for CAGD and computer graphics. One of the challenges in construction of subdivision schemes for arbitrary meshes is to guarantee that the limit surface will have smooth regular parameterization in a neighborhood of ..."
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Cited by 16 (0 self)
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Stationary subdivision is an important tool for generating smooth freeform surfaces for CAGD and computer graphics. One of the challenges in construction of subdivision schemes for arbitrary meshes is to guarantee that the limit surface will have smooth regular parameterization in a neighborhood of any point. First results in this direction were obtained only recently. In this paper we derive necessary and sufficient criteria for C k continuity that generalize and extend most known conditions. We create a general mathematical framework that can be used for analysis of more general types of schemes. Finally, we prove a degree estimate for C k continuous polynomial schemes generalizing an estimate of Reif [20] and give a practical sufficient condition for smoothness. 1 Introduction The main application of subdivision in computer graphics and CAGD is generation of smooth or piecewise smooth surfaces. Given an initial mesh, subdivision computes a sequence of refined meshes convergi...
Understanding Mathematical Discourse
 Dialogue. Amsterdam University
, 1999
"... Discourse Understanding is hard. This seems to be especially true for mathematical discourse, that is proofs. Restricting discourse to mathematical discourse allow us, however, to study the subject matter in its purest form. This domain of discourse is rich and welldefined, highly structured, offers ..."
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Cited by 7 (6 self)
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Discourse Understanding is hard. This seems to be especially true for mathematical discourse, that is proofs. Restricting discourse to mathematical discourse allow us, however, to study the subject matter in its purest form. This domain of discourse is rich and welldefined, highly structured, offers a welldefined set of discourse relations and forces/allows us to apply mathematical reasoning. We give a brief discussion on selected linguistic phenomena of mathematical discourse, and an analysis from the mathematician’s point of view. Requirements for a theory of discourse representation are given, followed by a discussion of proofs plans that provide necessary context and structure. A large part of semantics construction is defined in terms of proof plan recognition and instantiation by matching and attaching. 1
Lattice Points on Circles and the Discrete Velocity Model for the Boltzmann Equation
, 2004
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Generating Series of the Trace Group
 Developments in Language Theory, volume 2710 of LNCS
, 2003
"... Introduction A trace group (monoid) is the quotient of a free group (monoid) by relations of commutation between some pairs of generators. Trace monoids are often used in computer science to model the occurrence of events in concurrent systems, see [8] and the references therein. Trace groups have ..."
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Cited by 4 (1 self)
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Introduction A trace group (monoid) is the quotient of a free group (monoid) by relations of commutation between some pairs of generators. Trace monoids are often used in computer science to model the occurrence of events in concurrent systems, see [8] and the references therein. Trace groups have been studied from several viewpoints (and under various names), see for instance [9, 10, 17]. An important motivation is that trace goups can `approximate' braid groups [17]. In this article, we prove an analog for the trace group of the Mobius inversion formula for the trace monoid (Cartier and Foata [3]).
Verifying Textbook Proofs
 INT. WORKSHOP ON FIRSTORDER THEOREM PROVING (FTP'98), TECHNICAL REPORT E1852GS981
, 1998
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Orbit Counting Far From Hyperbolicity
"... We find a Prime Orbit Theorem and an analogue of Mertens ’ Theorem of analytic number theory for maps of interest in the field of dynamics. The maps studied are examples of SInteger dynamical systems and are built as isometric extensions of hyperbolic maps. The results we obtain bear the same relat ..."
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Cited by 3 (1 self)
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We find a Prime Orbit Theorem and an analogue of Mertens ’ Theorem of analytic number theory for maps of interest in the field of dynamics. The maps studied are examples of SInteger dynamical systems and are built as isometric extensions of hyperbolic maps. The results we obtain bear the same relationship to those known in a hyperbolic setting as Tchebyshev’s Theorem does to the Prime Number Theorem. For the systems closest to hyperbolicity (those for which the set S is finite) the arguments proceed essentially by comparing orbitcounting problems for the S−integer system to the same problem for the hyperbolic base system. For the systems furthest from hyperbolicity (those for which the set S is cofinite) different and more direct methods are used. The S − integer systems are constructed from arithmetic data and in this thesis both characteristic zero and positive characteristic examples are studied.
Some abelian invariants of 3manifolds
 Rev. Roumaine Math. Pures Appl
"... Some invariants for closed orientable 3manifolds are defined using a series of representations of the symplectic groups and the theory of Heegaard splittings. They are natural extensions of the U(1) ChernSimonsWitten invariants. These representations come from the functional equation satisfied by ..."
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Cited by 2 (2 self)
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Some invariants for closed orientable 3manifolds are defined using a series of representations of the symplectic groups and the theory of Heegaard splittings. They are natural extensions of the U(1) ChernSimonsWitten invariants. These representations come from the functional equation satisfied by the theta functions of level k. We analyze the values of these invariants for lens spaces.
Towards the Mechanical Verification of Textbook Proofs
"... Our goal is to implement a program for the machine verification of textbook proofs. We study the task from both the linguistics and automated reasoning perspective and give an indepth analysis for a sample textbook proof. We propose a framework for natural language proof understanding that extends ..."
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Cited by 1 (1 self)
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Our goal is to implement a program for the machine verification of textbook proofs. We study the task from both the linguistics and automated reasoning perspective and give an indepth analysis for a sample textbook proof. We propose a framework for natural language proof understanding that extends and integrates stateoftheart technologies from Natural Language Processing (Discourse Representation Theory) and Automated Reasoning (Proof Planning) in a novel and promising way, having the potential to initiate progress in both of these disciplines.
A DRTbased approach for formula parsing in textbook proofs
 IN THIRD INTERNATIONAL WORKSHOP ON COMPUTATIONAL SEMANTICS (IWCS3
, 1999
"... Knowledge is essential for understanding discourse. Generally, this has to be common sense knowledge and therefore, discourse understanding is hard. For the understanding of textbook proofs, however, only a limited quantity of knowledge is necessary. In addition, we have gained something very essent ..."
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Cited by 1 (1 self)
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Knowledge is essential for understanding discourse. Generally, this has to be common sense knowledge and therefore, discourse understanding is hard. For the understanding of textbook proofs, however, only a limited quantity of knowledge is necessary. In addition, we have gained something very essential: inference. A prerequisite for parsing textbook proofs is to being able to parse formulae that occur in these proofs. Parsing formulae alone in the empty context is trivial. But within the context of textbook proofs the task soon gets complex. Several kinds of references from the text to parts or sets of terms and formulae have to be handled. We describe some of the linguistic phenomena that occur in mathematical texts. The focus is on our treatment of term reference which is embedded in the DRT.