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"Theoretical mathematics”: Toward a cultural synthesis of mathematics and theoretical physics
 BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY
, 1993
"... Is speculative mathematics dangerous? Recent interactions between physics and mathematics pose the question with some force: traditional mathematical norms discourage speculation, but it is the fabric of theoretical physics. In practice there can be benefits, but there can also be unpleasant and de ..."
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Cited by 40 (1 self)
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Is speculative mathematics dangerous? Recent interactions between physics and mathematics pose the question with some force: traditional mathematical norms discourage speculation, but it is the fabric of theoretical physics. In practice there can be benefits, but there can also be unpleasant and destructive consequences. Serious caution is required, and the issue should be considered before, rather than after, obvious damage occurs. With the hazards carefully in mind, we propose a framework that should allow a healthy and positive role for speculation.
Problems in the Steenrod algebra
 Bull. London Math. Soc
, 1998
"... This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development ..."
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Cited by 29 (1 self)
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This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development of the Steenrod algebra and its connections to the various topics indicated below. Contents 1 Historical background 4
Higher fundamental functors for simplicial sets, Cahiers Topologie Géom
 Diff. Catég
"... Abstract. An intrinsic, combinatorial homotopy theory has been developed in [G3] for simplicial complexes; these form the cartesian closed subcategory of simple presheaves in!Smp, the topos of symmetric simplicial sets, or presheaves on the category!å of finite, positive cardinals. We show here how ..."
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Cited by 12 (8 self)
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Abstract. An intrinsic, combinatorial homotopy theory has been developed in [G3] for simplicial complexes; these form the cartesian closed subcategory of simple presheaves in!Smp, the topos of symmetric simplicial sets, or presheaves on the category!å of finite, positive cardinals. We show here how this homotopy theory can be extended to the topos itself,!Smp. As a crucial advantage, the fundamental groupoid Π1:!Smp = Gpd is left adjoint to a natural functor M1: Gpd =!Smp, the symmetric nerve of a groupoid, and preserves all colimits – a strong van Kampen property. Similar results hold in all higher dimensions. Analogously, a notion of (nonreversible) directed homotopy can be developed in the ordinary simplicial topos Smp, with applications to image analysis as in [G3]. We have now a homotopy ncategory functor ↑Πn: Smp = nCat, left adjoint to a nerve Nn = nCat(↑Πn(∆[n]), –). This construction can be applied to various presheaf categories; the basic requirements seem to be: finite products of representables are finitely presentable and there is a representable 'standard interval'.
Computing homotopy types using crossed ncubes of groups
 in Adams Memorial Symposium on Algebraic Topology
, 1992
"... Dedicated to the memory of Frank Adams ..."
Mimetic framework on curvilinear quadrilaterals of arbitrary order
, 2011
"... Abstract. In this paper higher order mimetic discretizations are introduced which are firmly rooted in the geometry in which the variables are defined. The paper shows how basic constructs in differential geometry have a discrete counterpart in algebraic topology. Generic maps which switch between t ..."
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Cited by 10 (5 self)
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Abstract. In this paper higher order mimetic discretizations are introduced which are firmly rooted in the geometry in which the variables are defined. The paper shows how basic constructs in differential geometry have a discrete counterpart in algebraic topology. Generic maps which switch between the continuous differential forms and discrete cochains will be discussed and finally a realization of these ideas in terms of mimetic spectral elements is presented, based on projections for which operations at the finite dimensional level commute with operations at the continuous level. The two types of orientation (inner and outerorientation) will be introduced at the continuous level, the discrete level and the preservation of orientation will be demonstrated for the new mimetic operators. The onetoone correspondence between the continuous formulation and the discrete algebraic topological setting, provides a characterization of the oriented discrete boundary of the domain. The Hodge decomposition at the continuous, discrete and finite dimensional level will be presented. It appears to be a main ingredient of the structure in this framework.
A unified formula for Steenrod operations in flag manifolds
, 2005
"... A unified formula for Steenrod operations in flag manifolds ..."
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Cited by 10 (4 self)
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A unified formula for Steenrod operations in flag manifolds
A Brouwer fixed point theorem for graph endomorphisms
, 2012
"... We prove a Lefschetz formula L(T) = ∑ x∈F iT (x) for graph endomorphisms T: G → G, where G is a general finite simple graph and F is the set of simplices fixed by T. The degree iT (x) of T at the simplex x is defined as (−1) dim(x) sign(T x), a graded sign of the permutation of T restricted to the ..."
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Cited by 9 (7 self)
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We prove a Lefschetz formula L(T) = ∑ x∈F iT (x) for graph endomorphisms T: G → G, where G is a general finite simple graph and F is the set of simplices fixed by T. The degree iT (x) of T at the simplex x is defined as (−1) dim(x) sign(T x), a graded sign of the permutation of T restricted to the simplex. The Lefschetz number L(T) is defined similarly as in the continuum as L(T) = ∑ k (−1)ktr(Tk), where Tk is the map induced on the k’th cohomology group Hk (G) of G. The theorem can be seen as a generalization of the NowakowskiRival fixed edge theorem [26]. A special case is the identity map T, where the formula reduces to the EulerPoincaré formula equating the Euler characteristic with the cohomological Euler characteristic. The theorem assures that if L(T) is nonzero, then T has a fixed clique. A special case is the discrete Brouwer fixed point theorem for graphs: if T is a graph endomorphism of a connected graph G, which is starshaped in the sense that only the zero’th cohomology group is nontrivial, like for connected trees or triangularizations of star shaped Euclidean domains, then there is clique x which is fixed by T. If A is the automorphism group of a graph, we look at the average Lefschetz number L(G). We prove that this is the Euler characteristic of the chain G/A and especially an integer. We also show that as a consequence of the Lefschetz formula, the zeta function ζT (z) = exp ( ∑∞ n=1 L(T n) zn) is a product n of two dynamical zeta functions and therefore has an analytic continuation as a rational function. This explicitly computable product formula involves the dimension and the signature of prime orbits.
Dynamic coverage verification in mobile sensor networks via switched higher order Laplacians
 in Robotics: Science & Systems
, 2007
"... Abstract — In this paper, we study the problem of verifying dynamic coverage in mobile sensor networks using certain switched linear systems. These switched systems describe the flow of discrete differential forms on timeevolving simplicial complexes. The simplicial complexes model the connectivity ..."
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Cited by 9 (1 self)
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Abstract — In this paper, we study the problem of verifying dynamic coverage in mobile sensor networks using certain switched linear systems. These switched systems describe the flow of discrete differential forms on timeevolving simplicial complexes. The simplicial complexes model the connectivity of agents in the network, and the homology groups of the simplicial complexes provides information about the coverage properties of the network. Our main result states that the asymptotic stability the switched linear system implies that every point of the domain covered by the mobile sensor nodes is visited infinitely often, hence verifying dynamic coverage. The enabling mathematical technique for this result is the theory of higher order Laplacian operators, which is a generalization of the graph Laplacian used in spectral graph theory and continuoustime consensus problems.
Mixed mimetic spectral element method for Stokes flow: a pointwise divergencefree solution
 Journal Computational Physics
"... Abstract. In this paper we apply the recently developed mimetic discretization method to the mixed formulation of the Stokes problem in terms of vorticity, velocity and pressure. The mimetic discretization presented in this paper and in [50] is a higherorder method for curvilinear quadrilaterals an ..."
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Cited by 8 (4 self)
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Abstract. In this paper we apply the recently developed mimetic discretization method to the mixed formulation of the Stokes problem in terms of vorticity, velocity and pressure. The mimetic discretization presented in this paper and in [50] is a higherorder method for curvilinear quadrilaterals and hexahedrals. Fundamental is the underlying structure of oriented geometric objects, the relation between these objects through the boundary operator and how this defines the exterior derivative, representing the grad, curl and div, through the generalized Stokes theorem. The mimetic method presented here uses the language of differential kforms with kcochains as their discrete counterpart, and the relations between them in terms of the mimetic operators: reduction, reconstruction and projection. The reconstruction consists of the recently developed mimetic spectral interpolation functions. The most important result of the mimetic framework is the commutation between differentiation at the continuous level with that on the finite dimensional and discrete level. As a result operators like gradient, curl and divergence are discretized exactly. For Stokes flow, this implies a pointwise divergencefree solution. This is confirmed using a set of test cases on both Cartesian and curvilinear meshes. It will be shown that the method converges optimally for all admissible boundary conditions. 1.
An index formula for simple graphs
, 2012
"... Abstract. We prove that any odd dimensional geometric graph G = (V, E) has zero curvature everywhere. To do so, we prove that for every injective function f on the vertex set V of a simple graph the index formula 1 [1 − 2 χ(S(x))/2 − χ(Bf (x))] = (if (x) + i−f (x))/2 = jf (x) holds, where if (x) is ..."
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Cited by 7 (7 self)
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Abstract. We prove that any odd dimensional geometric graph G = (V, E) has zero curvature everywhere. To do so, we prove that for every injective function f on the vertex set V of a simple graph the index formula 1 [1 − 2 χ(S(x))/2 − χ(Bf (x))] = (if (x) + i−f (x))/2 = jf (x) holds, where if (x) is a discrete analogue of the index of the gradient vector field ∇f and where Bf (x) is a graph defined by G and f. The PoincaréHopf formula ∑ x jf (x) = χ(G) allows so to express the Euler characteristic χ(G) of G in terms of smaller dimensional graphs defined by the unit sphere S(x) and the ”hypersurface graphs ” Bf (x). For odd dimensional geometric graphs, Bf (x) is a geometric graph of dimension dim(G)−2 and jf (x) = −χ(Bf (x))/2 = 0 implying χ(G) = 0 and zero curvature K(x) = 0 for all x. For even dimensional geometric graphs, the formula becomes jf (x) = 1−χ(Bf (x))/2 and allows with PoincaréHopf to write the Euler characteristic of G as a sum of the Euler characteristic of smaller dimensional graphs. The same integral geometric index formula also is valid for compact Riemannian manifolds M if f is a Morse function, S(x) is a sufficiently small geodesic sphere around x and Bf (x) = S(x) ∩ {y  f(y) = f(x)}. 1.