The moduli space of sheaves and a generalization of MacMahon’s formula

M. Vuletic has recently found a two-parameter generalization of MacMahon’s formula. In this paper we show that the coefficients in her formula are the Betti numbers of certain subvarieties in the moduli space of sheaves on the projective plane.     

The γ-vector of a barycentric subdivision

We prove that the γ-vector of the barycentric subdivision of a simplicial sphere is the f-vector of a balanced simplicial complex. The combinatorial basis for this work is the study of certain refinements of Eulerian numbers used by Brenti and Welker to describe the h-vector of the barycentric subdivision of a boolean complex.     

The interior operation in a ψ -density topology

We prove that for an arbitrary setA ⊂ ℝ its interior in aψ-density topology equalsA ∩ φ^β(B), whereB is a measurable kernel ofA andβ is some countable ordinal. Moreover, eachβ, 1≤β<Ω, realizes the interior ofA for someA εS.     

The ω-limit set of a graph map

Let G be a graph and be continuous. Denote by P(f), , ω(f) and Ω(f) the set of periodic points, the closure of the set of periodic points, ω-limit set and non-wandering set of f, respectively. In this paper we show that: (1) v∈ω(f) if and only if v∈P(f) or there exists an open arc L=(v,w) contained in some edge of G such that every open arc U=(v,c)⊂L contains at least 2 points of some trajectory; (2) v∈ω(f) if and only if every open neighborhood of v contains at least r+1 points of some trajectory, where r is the valence of v; (3) ; (4) if , then x has an infinite orbit.     

Separability of the space of continuous functions that are continuous in Λ-variation

The Shao-Sablin index of a Λ-sequence Λ=(λi) is defined by . The main result of the paper states that the Banach space CΛBV of continuous functions of bounded Λ-variation with the standard Λ-variation norm is separable if and only if SΛ<2. Also, ΛBV=ΛBVc if and only if SΛ<2, where ΛBVc denotes the space of functions continuous in Λ-variation. A number of corollaries is drawn, and one of them being that the Garsia-Sawyer class GS is a dense subset of the Banach space HBV of functions of bounded harmonic variation.     

Some algebraic identities for the α-permanent

The permanent of a matrix is a linear combination of determinants of block diagonal matrices which are simple functions of the original matrix. To prove this, we first show a more general identity involving α-permanents: for arbitrary complex numbers α and β, we show that the α-permanent of any matrix can be expressed as a linear combination of β-permanents of related matrices. Some other identities for the α-permanent of sums and products of matrices are shown, as well as a relationship between the α-permanent and general immanants. We conclude with some discussion and a conjecture for the computational complexity of the α-permanent, and provide some numerical illustrations.     

γ-Robust facility relocation problem

In this paper, we consider relocating facilities, where we have demand changes in the network. Relocations are performed by closing some of the existing facilities from low demand areas and opening new ones in newly emerging areas. However, the actual changes of demand are not known in advance. Therefore, different scenarios with known probabilities are used to capture such demand changes. We develop a mixed integer programming model for facility relocation that minimizes the expected weighted distance while making sure that relative regret for each scenario is no greater than γ. We analyzed the problem structure and developed a Lagrangian Decomposition Algorithm (LDA) to expedite the solution process. Numerical experiments are carried out to show the performance of LDA against the exact solution method.     

The structure of analytic τ-sheaves

Let R be a complete discrete valuation -algebra whose residue field is algebraic over , and let K denote its fraction field. In this paper, we study the structure of τ-sheaves M without good reduction on the curve , seen as a rigid analytic space. One motivation is the Tate uniformization theorem for t-motives of Drinfeld modules, which we want to extend to general τ-sheaves. On the other hand, we are interested in the action of inertia on a generic Tate module T_?(M) of M.For a given τ-sheaf M on , we prove the existence of a maximal model for M on , an R-model of , and, over a finite separable extension R′ of R, of nondegenerate models for M.We prove the following ‘semistability’ theorem: there exists a finite extension K′ of K, a nonempty open subscheme C′⊂C, and a filtration

     

The minimal displacement problem in the space l ∞

We give a lower bound for the minimal displacement characteristic in the space l ^∞.     

The linear flows in the space of Krichever–Lax matrices over an algebraic curve

Krichever (Commun Math Phys 229(2):229–269, 2002) invented the space of matrices parametrizing the cotangent bundle of moduli space of stable vector bundles over a compact Riemann surface, which is named as the Hitchin system after the investigation (Hitchin, Duke Math J 54(1):91–114, 1987). We study a necessary and sufficient condition for the linearity of flows on the space of Krichever–Lax matrices in a Lax representation in terms of cohomological classes using the similar technique and analysis from the work by Griffiths (Am J Math 107(6):1445–1484, 1985).      

Continuous mappings on subspaces of products with the κ-box topology

Much of General Topology addresses this issue: Given a function f∈C(Y,Z) with Y⊆Y^′ and Z⊆Z^′, find , or at least , such that ; sometimes Z=Z^′ is demanded. In this spirit the authors prove several quite general theorems in the context Y^′=κ(XI)=_∏i∈IXi in the κ-box topology (that is, with basic open sets of the form _∏i∈IUi with Ui open in Xi and with Ui≠Xi for <κ-many i∈I). A representative sample result, extending to the κ-box topology some results of Comfort and Negrepontis, of Noble and Ulmer, and of Hušek, is this.

Theorem. Letω?κ?α (that means: κ<α, and[β<αandλ<κ]⇒βλ<α) with α regular,be a set of non-empty spaces with eachd(Xi)<α,π[Y]=XJfor each non-emptyJ⊆Isuch that|J|<α, and the diagonal in Z be the intersection of <α-many regular-closed subsets ofZ×Z. Then (a) Y is pseudo-(α,α)-compact, (b) for everyf∈C(Y,Z)there isJ∈[I]^<αsuch thatf(x)=f(y)wheneverxJ=yJ, and (c) every such f extends to.     

On the Cauchy problem of a weakly dissipative μ-Hunter–Saxton equation

In this paper, we study the Cauchy problem of a weakly dissipative μ-Hunter–Saxton equation. We first establish the local well-posedness for the weakly dissipative μ-Hunter–Saxton equation by Kato's semigroup theory. Then, we derive the precise blow-up scenario for strong solutions to the equation. Moreover, we present some blow-up results for strong solutions to the equation. Finally, we give two global existence results to the equation.     

The σ-isotypic decomposition and the σ-index of reversible-equivariant systems

This work is concerned with dynamical systems in presence of symmetries and reversing symmetries. We describe a construction process of subspaces that are invariant by linear Γ-reversible-equivariant mappings, where Γ is the compact Lie group of all the symmetries and reversing symmetries of such systems. These subspaces are the σ-isotypic components, first introduced by Lamb and Roberts in (1999) [10] and that correspond to the isotypic components for purely equivariant systems. In addition, by representation theory methods derived from the topological structure of the group Γ, two algebraic formulae are established for the computation of the σ-index of a closed subgroup of Γ. The results obtained here are to be applied to general reversible-equivariant systems, but are of particular interest for the more subtle of the two possible cases, namely the non-self-dual case. Some examples are presented.     

Distributing inspections in space and time – Proposed solution of a difficult problem

There are several identical facilities in which precious or dangerous material is processed or stored. Since parts of this material may be diverted by some manager or employee of these facilities or since failures in the processing of the material may occur, an authorized organization inspects these facilities regularly at the beginning and at the end of some reference time interval. In order to shorten the time required for detecting such an illegal activity or failures, in addition some interim inspections are performed in these facilities during the reference time interval. The optimal distribution of these interim inspections in space and time poses considerable analytical problems since adversary strategies have to be taken into account. So far only special cases have been analysed successfully, but these results lead to a conjecture for the solution of the general case which is surprisingly simple in view of the complexity of this inspection problem.     

A Kähler structure on the moduli space of isometric maps of a circle into Euclidean space

We study the spaces and and _Lip of smooth (resp. non-degenerate Lipschitz) isometric maps of a circle into Euclidean space modulo orientation preserving Euclidean motions. We prove that and _Lip are infinite dimensional Kähler manifolds. In particular, they are complex Fréchet (resp. Banach) manifolds. This is proved by an infinite dimensional version of the Kirwan, Kempf-Ness Theorem [Kir84], [KN78], [Nes84] relating symplectic quotients to holomorphic quotients, applied to the action ofPSL ₂() on the free loop space ofS ².Oblatum 15-X-1994 & 5-VII-1995This research was supported in part by NSF grant DMS-92-05154.This research was partially supported by AFOSR grant F49620-92-J-0093.     

The ω-Turing degrees

In this paper we initiate the study of the ω-Turing reducibility between sequences of sets of natural numbers. We shall prove that the induced degree structure is an extension of the structure of the Turing degrees and that the two structures are closely connected, but different enough. Further we shall prove some definability results for the local theory of the newly defined structure.     

On the nonexistence of a relation between σ-left-porosity and σ-right-porosity

Given an arbitrarily weak notion of left-〈f〉$〈f〉$-porosity and an arbitrarily strong notion of right-〈g〉$〈g〉$-porosity, we construct an example of closed subset of R$\mathbb{R}$ which is not σ  -left-〈f〉$〈f〉$-porous and is right-〈g〉$〈g〉$-porous. We also briefly summarize the relations between three different definitions of porosity controlled by a function; we then observe that our construction gives the example for any combination of these definitions of left-porosity and right-porosity.