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ARTIN BRAID GROUPS AND HOMOTOPY GROUPS
"... Abstract. We study the Brunnian subgroups and the boundary Brunnian subgroups of the Artin braid groups. The general higher homotopy groups of the sphere are given by mirror symmetric elements in the quotient groups of the Artin braid groups modulo the boundary Brunnian braids, as well as given as a ..."
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Abstract. We study the Brunnian subgroups and the boundary Brunnian subgroups of the Artin braid groups. The general higher homotopy groups of the sphere are given by mirror symmetric elements in the quotient groups of the Artin braid groups modulo the boundary Brunnian braids, as well as given as a summand of the center of the quotient groups of Artin pure braid groups modulo boundary Brunnian braids. These results give some deep and fundamental connections between the braid groups and the general higher homotopy groups of spheres. 1.
Spectral estimation theory: Beyond linear but before Bayesian
 JOURNAL OF THE OPTICAL SOCIETY OF AMERICA A, (IN
, 2003
"... Most coloracquisition devices capture spectral signals by acquiring only three samples, critically undersampling the spectral information. We analyze the problem of estimating highdimensional spectral signals from lowdimensional device responses. We begin with the theory and geometry of linear es ..."
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Cited by 4 (0 self)
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Most coloracquisition devices capture spectral signals by acquiring only three samples, critically undersampling the spectral information. We analyze the problem of estimating highdimensional spectral signals from lowdimensional device responses. We begin with the theory and geometry of linear estimation methods. These methods use linear models to characterize the likely input signals and reduce the number of estimation parameters. Next, we introduce two submanifold estimation methods. These methods are based on the observation that for many data sets the deviation between the signal and the linear estimate is systematic; the methods incorporate knowledge of these systematic deviations to improve upon linear estimation methods. We describe the geometric intuition of these methods and evaluate the submanifold method on hyperspectral image data.
Towards Hilbert's 24th Problem: Combinatorial Proof Invariants
, 2006
"... Proofs Without Syntax [37] introduced polynomialtime checkable combinatorial proofs for classical propositional logic. This sequel approaches Hilbert’s 24 th Problem with combinatorial proofs as abstract invariants for sequent calculus proofs, analogous to homotopy groups as abstract invariants for ..."
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Cited by 3 (0 self)
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Proofs Without Syntax [37] introduced polynomialtime checkable combinatorial proofs for classical propositional logic. This sequel approaches Hilbert’s 24 th Problem with combinatorial proofs as abstract invariants for sequent calculus proofs, analogous to homotopy groups as abstract invariants for topological spaces. The paper lifts a simple, strongly normalising cut elimination from combinatorial proofs to sequent calculus, factorising away the mechanical commutations of structural rules which litter traditional syntactic cut elimination. Sequent calculus fails to be surjective onto combinatorial proofs: the paper extracts a semantically motivated closure of sequent calculus from which there is a surjection, pointing towards an abstract combinatorial refinement of Herbrand’s theorem.
Extending the concept of genus to dimension n
 Proc. Amer. Math. Soc
, 1981
"... Abstract. Some graphtheoretical tools are used to introduce the concept of "regular genus " §(M„), for every closed ndimensional PLmanifold M„. Then it is proved that the regular genus of every surface equals its genus, and that the regular genus of every 3manifold Af j equals its Heegaard genus ..."
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Abstract. Some graphtheoretical tools are used to introduce the concept of "regular genus " §(M„), for every closed ndimensional PLmanifold M„. Then it is proved that the regular genus of every surface equals its genus, and that the regular genus of every 3manifold Af j equals its Heegaard genus, if M3 is orientable, and twice its Heegaard genus, if M3 is nonorientable. A geometric approach, and some applications in dimension four are exhibited. 1. Definitions and notations. Let T = (V, E) be a regular multigraph of degree n + 1, y: E> A „ = { / G Z0 < i < n] an (n + l)linecolouring of T [Ha, p. 133]. Such a pair (I \ y) is said to be an (n + l)coloured graph. For every subset © of A„, r9 will denote the subgraph (V, y~\9>)); further, for every colour c G A„, c will denote the set A „ —{c}. A 2cell imbedding i: r  —> F [W, p. 40] of an (n + l)coloured graph (I \ y) on a closed surface F is called a regular imbedding if its regions are bounded by 2coloured cycles. Moreover, i is called a stronglyregular imbedding if there exists a cyclic permutation e = (e0,..., e„) of A„, so that each region is bounded by a
Generalized EulerPoincaré Theorem
"... The EulerPoincaré Theorem relates the numbers of vertices, V, edges E, faces F, cells C, etc, of graphs, polygons, polyhedra, and even higherdimensional polytopes. It can be presented in many different ways. For a single 3dimensional polyhedral body without any holes, Euler [1] originally stated i ..."
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The EulerPoincaré Theorem relates the numbers of vertices, V, edges E, faces F, cells C, etc, of graphs, polygons, polyhedra, and even higherdimensional polytopes. It can be presented in many different ways. For a single 3dimensional polyhedral body without any holes, Euler [1] originally stated it as: V + F = E + 2. (1) Poincaré [2] extended the formulation to such a body in Ddimensional space:
ADVANCES IN COMPUTATIONAL MECHANICS Via Graph Theory
, 2006
"... The aim of the present work is two fold. In one hand it shows to mathematicians how the apparently pure mathematical concepts can be applied to the efficient solution of problems in structural mechanics. In the other hand it illustrates to engineers the important role of mathematical concepts for th ..."
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The aim of the present work is two fold. In one hand it shows to mathematicians how the apparently pure mathematical concepts can be applied to the efficient solution of problems in structural mechanics. In the other hand it illustrates to engineers the important role of mathematical concepts for the solution of engineering problems. In this paper a number of applications of graph theory in structural mechanics are presented. These applications simplify the analysis of structures and make their optimal analysis feasible. For each case, the main problem is stated and then the formulation together with illustrative examples is presented.
3.1.
"... Abstract. Simplicial and ∆structures of configuration spaces are investigated. New connections between the homotopy groups of the 2sphere and the braid groups are given. The higher homotopy groups of the 2 sphere are shown to be derived groups of the braid groups over the 2sphere. Moreover the h ..."
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Abstract. Simplicial and ∆structures of configuration spaces are investigated. New connections between the homotopy groups of the 2sphere and the braid groups are given. The higher homotopy groups of the 2 sphere are shown to be derived groups of the braid groups over the 2sphere. Moreover the higher homotopy groups of the 2sphere are shown to be isomorphic to the
unknown title
, 2012
"... Poincaré and the idea of a group In many different fields of mathematics and physics Poincaré found many uses for the idea of a group, but not for group theory. He used the idea in his work on automorphic functions, in number theory, in his epistemology, Lie theory (on the socalled Campbell–Baker–H ..."
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Poincaré and the idea of a group In many different fields of mathematics and physics Poincaré found many uses for the idea of a group, but not for group theory. He used the idea in his work on automorphic functions, in number theory, in his epistemology, Lie theory (on the socalled Campbell–Baker–Hausdorff and Poincaré–Birkhoff–Witt theorems), in physics (where he introduced the Lorentz group), in his study of the domains of complex functions of several variables, and in his pioneering study of 3manifolds. However, as a general rule, he seldom appealed to deep results in group theory, and developed no more structural analysis of any group than was necessary to solve a problem. It was usually enough for him that there is a group, or that there are different groups. In this article Jeremy Gray gives a brief history on Poincaré’s group idea. It is wellknown that between 1880 and 1884 Poincaré brought together in a completely unexpected way the subjects of complex function theory, linear differential equations, Riemann surfaces, and nonEuclidean geometry (see, for example, [42] and [7]). Cauchy’s approach to complex function theory and the theory of differential equations were mainstream topics in the education of a French mathematician at the time, but Riemann surfaces were not, largely because Riemann’s way of thinking was not congenial to Charles Hermite, who dominated the scene in the
ON NETWORK MODELS AND THE SYMBOLIC SOLUTION OF NETWORK EQUATIONS
"... This paper gives an overview of the formulation and solution of network equations, with emphasis on the historical development of this area. Networks are mathematical models. The three ingredients of network descriptions are discussed. It is shown how the network equations of onedimensional multip ..."
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This paper gives an overview of the formulation and solution of network equations, with emphasis on the historical development of this area. Networks are mathematical models. The three ingredients of network descriptions are discussed. It is shown how the network equations of onedimensional multiport networks can be formulated and solved symbolically. If necessary, the network graph is modified so as to obtain an admittance representation for all kinds of multiports. Ndimensional networks are defined as graphs with the algebraic structure of Ndimensional vectors. In civil engineering, framed structures in two and three spatial dimensions can be modeled as 3dimensional or 6dimensional networks. The separation of geometry from topology is a characteristic feature of such networks.
École Doctorale Mathématiques et Sciences et Technologies de l’Information et de la Communication Thèse
"... A study of some morphological operators in simplicial complex spaces Une étude de certains opérateurs morphologiques dans les complexes simpliciaux Thèse dirigée par Laurent NAJMAN Coencadré par Jean COUSTY ..."
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A study of some morphological operators in simplicial complex spaces Une étude de certains opérateurs morphologiques dans les complexes simpliciaux Thèse dirigée par Laurent NAJMAN Coencadré par Jean COUSTY