Dirac’s Theorem on Simplicial Matroids

We introduce the notion of k-hyperclique complexes, i.e., the largest simplicial complexes on the set [n] with a fixed k-skeleton. These simplicial complexes are a higher-dimensional analogue of clique (or flag) complexes (case k = 2) and they are a rich new class of simplicial complexes. We show that Dirac’s theorem on chordal graphs has a higher-dimensional analogue in which graphs and clique complexes get replaced, respectively, by simplicial matroids and k-hyperclique complexes. We prove also a higher-dimensional analogue of Stanley’s reformulation of Dirac’s theorem on chordal graphs.      

(Bi-)Cohen–Macaulay Simplicial Complexes and Their Associated Coherent Sheaves

ABSTRACT

Via the BGG correspondence, a simplicial complex Δ on [n] is transformed into a complex of coherent sheaves on P ^n?1. We show that this complex reduces to a coherent sheaf ? exactly when the Alexander dual Δ* is Cohen–Macaulay.

We then determine when both Δ and Δ* are Cohen–Macaulay. This corresponds to ? being a locally Cohen–Macaulay sheaf.

Lastly, we conjecture for which range of invariants of such Δ's it must be a cone, and show the existence of such Δ's which are not cones outside of this range.     

Ramanujan’s inverse elliptic arc approximation

We suggest a continued fraction origin to Ramanujan’s approximation to $(\frac{a-b}{a+b})^{2}$ in terms of the arc length of an ellipse with semiaxes a and b.     

Baxter’s difference systems and orthogonal rational functions

Rational functions orthogonal on the unit circle are considered beginning with their recurrence formulas. Various summability conditions are imposed on the recurrence coefficients and the asymptotics of the solutions are studied and the orthogonality measure is recovered. The techniques developed by Baxter and Benzaid and Lutz are used.     

Equivalent condition for approximately Cohen–Macaulay complexes

We give a necessary and sufficient condition for a simplicial complex to be approximately Cohen–Macaulay. Namely it is approximately Cohen–Macaulay if and only if the ideal associated to its Alexander dual is componentwise linear and generated in two consecutive degrees. This completes the result of J. Herzog and T. Hibi who proved that a simplicial complex is sequentially Cohen–Macaulay if and only if the ideal associated to its Alexander dual is componentwise linear.     

On Baxter’s difference systems

We study the asymptotics of solutions of a difference system introduced by Baxter by using the general method for the asymptotic representation of such solutions due to Benzaid and Lutz. Some results of Tauberian type are obtained in the case when the spectral parameter belongs to the unit circle.     

A generalization of k-Cohen–Macaulay simplicial complexes

For a positive integer k and a non-negative integer t, a class of simplicial complexes, to be denoted by k-CM _t , is introduced. This class generalizes two notions for simplicial complexes: being k-Cohen–Macaulay and k-Buchsbaum. In analogy with the Cohen–Macaulay and Buchsbaum complexes, we give some characterizations of CM _t (=1?CM _t ) complexes, in terms of vanishing of some homologies of its links, and in terms of vanishing of some relative singular homologies of the geometric realization of the complex and its punctured space. We give a result on the behavior of the CM _t property under the operation of join of two simplicial complexes. We show that a complex is k-CM _t if and only if the links of its non-empty faces are k-CM _t?1. We prove that for an integer s≤d, the (d?s?1)-skeleton of a (d?1)-dimensional k-CM _t complex is (k+s)-CM _t . This result generalizes Hibi’s result for Cohen–Macaulay complexes and Miyazaki’s result for Buchsbaum complexes.     

New integrable systems with a quartic integral and new generalizations of Kovalevskaya’s and Goriatchev’s cases

In his paper [1], one of us has introduced a method for constructing integrable conservative two-dimensional mechanical systems, on Riemannian 2D spaces, whose second integral is a polynomial in the velocities. This method was applied successfully in [2] to construction of systems admitting a cubic integral and in [3, 4] and [5] to cases of a quartic integral. The present work is devoted to construction of new integrable systems with a quartic integral. The potential is assumed to have the structure

Jacobson’s lemma for the generalized Drazin inverse

0truemm0truemm We study properties of elements in a ring which admit the generalized Drazin inverse. It is shown that the element 1-ab is generalized Drazin invertible if and only if so is 1-ba and a formula for the generalized Drazin inverse of 1-ba in terms of the generalized Drazin inverse and the spectral idempotent of 1-ab is provided. Further, recent results relating to the Drazin index can be recovered from our theorems.     

Webb’s conjecture for fusion systems

We give a short and conceptual proof of Webb’s conjecture. Our methods are general enough to prove an analogue of the conjecture for saturated fusion systems. The author was supported by an EPSRC grant EP/D506484/1.     

Clifford’s theorem for coherent systems

Let C be an algebraic curve of genus g ≥ 2. We prove an analogue of Clifford’s theorem for coherent systems on C and some refinements using results of Re and Mercat. Received: 27 July 2007     

Quillen’s stratification for fusion systems

The purpose of this note is to provide a reference for the fact that the proof of Quillen’s stratification for finite group cohomology carries over to fusion system. As in the case of Quillen’s stratification for block varieties, the proof is similar to the usual proof for group cohomology except for the use of fusion stable bisets, whose existence is due to Broto et al.     

Conservation laws and Hamilton’s equations for systems with long-range interaction and memory

Using the fact that extremum of variation of generalized action can lead to the fractional dynamics in the case of systems with long-range interaction and long-term memory function, we consider two different applications of the action principle: generalized Noether’s theorem and Hamiltonian type equations. In the first case, we derive conservation laws in the form of continuity equations that consist of fractional time–space derivatives. Among applications of these results, we consider a chain of coupled oscillators with a power-wise memory function and power-wise interaction between oscillators. In the second case, we consider an example of fractional differential action 1-form and find the corresponding Hamiltonian type equations from the closed condition of the form.     

Generalizations of Darbo’s fixed point theorem and solvability of integral and differential systems

In this paper, we establish some new generalizations of Darbo’s fixed point theorem for multivalued mappings. Moreover, we prove the existence of solutions for a class of integral equations by Darbo’s fixed point theorem and the existence of solutions for a class of differential inclusions using a generalization of Darbo’s fixed point theorem.     

Coefficient inverse problem for Poisson’s equation in a cylinder

The inverse problem of determining the coefficient on the right-hand side of Poisson’s equation in a cylindrical domain is considered. The Dirichlet boundary value problem is studied. Two types of additional information (overdetermination) can be specified: (i) the trace of the solution to the boundary value problem on a manifold of lower dimension inside the domain and (ii) the normal derivative on a portion of the boundary. (Global) existence and uniqueness theorems are proved for the problems. The study is performed in the class of continuous functions whose derivatives satisfy a Hölder condition.