Morse theory and conjugacy classes of finite subgroups

We construct a CAT(0) group containing a finitely presented subgroup with infinitely many conjugacy classes of finite-order elements. Unlike previous examples (which were based on right-angled Artin groups) our ambient CAT(0) group does not contain any rank 3 free abelian subgroups. We also construct examples of groups of type F _n inside mapping class groups, Aut(), and Out() which have infinitely many conjugacy classes of finite-order elements.      

Morse theory, Milnor fibers and minimality of hyperplane arrangements

Through the study of Morse theory on the associated Milnor fiber, we show that complex hyperplane arrangement complements are minimal. That is, the complement of any complex hyperplane arrangement has the homotopy type of a CW-complex in which the number of -cells equals the -th betti number. Combining this result with recent work of Papadima and Suciu, one obtains a characterization of when arrangement complements are Eilenberg-MacLane spaces.

     

Simplicial collapsibility,discrete Morse theory,and the geometry of nonpositively curved simplicial complexes

Understanding the conditions under which a simplicial complex collapses is a central issue in many problems in topology and combinatorics. Let K be a finite simplicial complex of dimension three or less endowed with the piecewise Euclidean geometry given by declaring edges to have unit length, and satisfying the property that every 2-simplex is a face of at most two 3-simplices in K. Our main result is that if |K| is nonpositively curved [in the sense of CAT(0)] then K simplicially collapses to a point. The main tool used in the proof is Forman’s discrete Morse theory, a combinatorial analog of the classical smooth theory developed in the 1920s. A key ingredient in our proof is a combinatorial analog of the fact that a minimal surface in has nonpositive Gauss curvature.      

Topological vector spaces admissible in economic equilibrium theory

In models of economic equilibrium in markets with infinitely many commodities, the commodity space is an ordered topological vector space endowed with additional structure. In the present paper, we consider ordered topological vector spaces which are admissible (for equilibrium analysis) in the sense that every economy which is reasonably well behaved posesses an equilibrium. It turns out that this condition may be characterized in terms of topology and order. This characterization implies that the commodity space has the structure of a Kakutani space.     

Morse theory for some lower-C 2 functions in finite dimension

Using inf-regularization methods, we prove that Morse inequalities hold for some lower-C ² functions. For this purpose, we first recall some properties of the class of lower-C ² functions and of their Moreau-Yosida approximations. Then, we establish, under some qualification conditions on the critical points, that it is possible to define a Morse index for a lower-C ² functionf. This index is preserved by the Moreau-Yosida approximation process. We prove in particular that the Moreau-Yosida approximations are twice continuolusly differentiable around such a critical point which is shown to be a strict local minimum of the restriction off and of its approximations to some affine space. In a last step, Morse inequalities are written for Moreau-Yosida approximations and with the aid of deformation retractions we prove that these inequalities also hold for some lower-C ² functions.     

On the Laplacian vector fields theory in domains with rectifiable boundary

Given a domain Ω in ?³ with rectifiable boundary, we consider main integral, and some other, theorems for the theory of Laplacian (sometimes called solenoidal and irrotational, or harmonic) vector fields paying a special attention to the problem of decomposing a continuous vector field, with an additional condition, u on the boundary Γof Ω ? ?³ into a sum u = u⁺+u^? were u^± are boundary values of vector fields which are Laplacian in Ω and its complement respectively. Our proofs are based on the intimate relations between Laplacian vector fields theory and quaternionic analysis for the Moisil–Theodorescu operator. Copyright © 2006 John Wiley & Sons, Ltd.     

Morse index and stability of elliptic Lagrangian solutions in the planar three-body problem

We illustrate a new way to study the stability problem in celestial mechanics. In this paper, using the variational nature of elliptic Lagrangian solutions in the planar three-body problem, we study the relation between Morse index and its stability via Maslov-type index theory of periodic solutions of Hamiltonian system. For elliptic Lagrangian solutions we get an estimate of the algebraic multiplicity of unit eigenvalues of its monodromy matrix in terms of the Morse index, which is the key to understand the stability problem. As a special case, we provide a criterion to spectral stability of relative equilibrium.     

Open core and small groups in dense pairs of topological structures

Dense pairs of geometric topological fields have tame open core, that is, every definable open subset in the pair is already definable in the reduct. We fix a minor gap in the published version of van den Dries's seminal work on dense pairs of o-minimal groups, and show that every definable unary function in a dense pair of geometric topological fields agrees with a definable function in the reduct, off a small definable subset, that is, a definable set internal to the predicate.For certain dense pairs of geometric topological fields without the independence property, whenever the underlying set of a definable group is contained in the dense-codense predicate, the group law is locally definable in the reduct as a geometric topological field. If the reduct has elimination of imaginaries, we extend this result, up to interdefinability, to all groups internal to the predicate.     

Marino—Prodi perturbation type results and Morse indices of minimax critical points for a class of functionals in Banach spaces

In this work, we study a class of Euler functionals defined in Banach spaces, associated with quasilinear elliptic problems involving p-Laplace operator (p > 2). First we obtain perturbation results in the spirit of the remarkable paper by Marino and Prodi (Boll. U.M.I. (4) 11(Suppl. fasc. 3): 1–32, 1975), using the new definition of nondegeneracy given in (Ann. Inst. H. Poincaré: Analyse Non Linéaire. 2:271–292, 2003). We also extend Morse index estimates for minimax critical points, introduced by Lazer and Solimini (Nonlinear Anal. T.M.A. 12:761–775, 1988) in the Hilbert case, to our Banach setting. Mathematics Subject Classification (1991) 58E05, 35B20, 35J60, 35J70     

Some new results in multiplicative and additive Ramsey theory

There are several notions of largeness that make sense in any semigroup, and others such as the various kinds of density that make sense in sufficiently well-behaved semigroups including and . It was recently shown that sets in which are multiplicatively large must contain arbitrarily large geoarithmetic progressions, that is, sets of the form , as well as sets of the form . Consequently, given a finite partition of , one cell must contain such configurations. In the partition case we show that we can get substantially stronger conclusions. We establish some combined additive and multiplicative Ramsey theoretic consequences of known algebraic results in the semigroups and , derive some new algebraic results, and derive consequences of them involving geoarithmetic progressions. For example, we show that given any finite partition of there must be, for each , sets of the form together with , the arithmetic progression , and the geometric progression in one cell of the partition. More generally, we show that, if is a commutative semigroup and a partition regular family of finite subsets of , then for any finite partition of and any , there exist and such that is contained in a cell of the partition. Also, we show that for certain partition regular families and of subsets of , given any finite partition of some cell contains structures of the form for some .

     

Modulability and duality of certain cones in pluripotential theory

Let p>0, and let Ep denote the cone of negative plurisubharmonic functions with finite pluricomplex p-energy. We prove that the vector space δEp=Ep−Ep, with the vector ordering induced by the cone Ep is σ-Dedekind complete, and equipped with a suitable quasi-norm it is a non-separable quasi-Banach space with a decomposition property with control of the quasi-norm. Furthermore, we explicitly characterize its topological dual. The cone Ep in the quasi-normed space δEp is closed, generating, and has empty interior.     

A stability theory of sandwich structural elements 1. Analysis of the current state and a refined classification of buckling forms

The main stages of development of the stability theory of sandwich structural elements are considered. The mechanism of their stability loss is revealed using the experimental data and theoretical solutions obtained on the basis of refined statements of problems. A classification of all possible forms of stability loss is given within the limits of continuum representation of load-bearing layers and the core of these structures.Center for Study of Dynamics and Stability, Tupolev Kazan State Technical University, Kazan, Tatarstan, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 6, pp. 707–716, November–December, 1999.     

KCC theory and its application in a tumor growth model

In this study, we consider the stability of tumor model by using the standard differential geometric method that is known as Kosambi‐Cartan‐Chern (KCC) theory or Jacobi stability analysis. In the KCC theory, we describe the time evolution of tumor model in geometric terms. We obtain nonlinear connection, Berwald connection and KCC invariants. The second KCC invariant gives the Jacobi stability properties of tumor model. We found that the equilibrium points are Jacobi unstable for positive coordinates. We also discussed the time evolution of components of deviation tensor and the behavior of deviation vector near the equilibrium points.     

Enumerating disjunctions and conjunctions of paths and cuts in reliability theory

Let G=(V,E) be a (directed) graph with vertex set V and edge (arc) set E. Given a set P of source-sink pairs of vertices of G, an important problem that arises in the computation of network reliability is the enumeration of minimal subsets of edges (arcs) that connect/disconnect all/at least one of the given source-sink pairs of P. For undirected graphs, we show that the enumeration problems for conjunctions of paths and disjunctions of cuts can be solved in incremental polynomial time. Furthermore, under the assumption that P consists of all pairs within a given vertex set, we also give incremental polynomial time algorithm for enumerating all minimal path disjunctions and cut conjunctions. For directed graphs, the enumeration problem for cut disjunction is known to be NP-complete. We extend this result to path conjunctions and path disjunctions, leaving open the complexity of the enumeration of cut conjunctions. Finally, we give a polynomial delay algorithm for enumerating all minimal sets of arcs connecting two given nodes s₁ and s₂ to, respectively, a given vertex t₁, and each vertex of a given subset of vertices T₂.     

A finitization of Littlewood's Tauberian theorem and an application in Tauberian remainder theory

In this paper we study Littlewood's Tauberian theorem from a proof theoretic perspective. We first use the Dialectica interpretation to produce an equivalent, finitary formulation of the theorem, and then carry out an analysis of Wielandt's proof to extract concrete witnessing terms. We argue that our finitization can be viewed as a generalized Tauberian remainder theorem, and we instantiate it to produce two concrete remainder theorems as a corollary, in terms of rates of convergence and rates metastability, respectively. We rederive the standard remainder estimate for Littlewood's theorem as a special case of the former.     

Buckling forms of homogeneous and sandwich plates in pure shear in tangential directions

The mathematical problem of the plane shear buckling form (BF) of sandwich plates and plates homogeneous across the thickness in pure shear is considered. The solution to this problem is compared with the solution to the problem of a flexural BF which is realized in these plates with the formation of oblique waves. It is established that, in the case of plates homogeneous across the thickness, the critical loads corresponding to the plane shear BF are maximum for isotropic ones. In real one-layer structural elements manufactured both of isotropic homogeneous and orthotropic composite materials, these critical loads cannot be reached since they exceed considerably the critical loads for the flexural BFs with oblique waves. The critical loads corresponding to the two BFs are comparable only for relatively thick plates. However, the plane shear BF can be realized in sandwich plates earlier than the flexural one even if the plates are thin. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 2, pp. 215–228, 2000.