Conley pairs in geometry—Lusternik–Schnirelmann theory and more

Firstly, we wish to motivate that Conley pairs, realized via Salamon’s definition (Salamon, 1990), are rather useful building blocks in geometry: Initially we met Conley pairs in an attempt to construct Morse filtrations of free loop spaces (Weber, 2017). From this fell off quite naturally, firstly, an alternative proof (Weber, 2016) of the cell attachment theorem in Morse theory (Milnor, 1963) and, secondly, some ideas (Majer and Weber, 2015) how to try to organize the closures of the unstable manifolds of a Morse–Smale gradient flow as a CW decomposition of the underlying manifold. Relaxing non-degeneracy of critical points to isolatedness we use these Conley pairs to implement the gradient flow proof of the Lusternik–Schnirelmann Theorem (Lusternik and Schnirelmann, 1934) proposed in Bott’s survey (Bott, 1982).Secondly, we shall use this opportunity to provide an exposition of Lusternik–Schnirelmann (LS) theory based on thickenings of unstable manifolds via Conley pairs. We shall cover the Lusternik–Schnirelmann Theorem (Lusternik and Schnirelmann, 1934), cuplength, subordination, the LS refined minimax principle, and a variant of the LS category called ambient category.     

Lusternik—Schnirelmann定理的推广

本文根据Brouwer映射度的理论和微分拓扑的基本方法推广了Lusternik-Schnirelmann定理。     

Lower bounds¶for the relative Lusternik–Schnirelmann category

We give estimates of numerical homotopy invariants of the pair (X,X×S ^p ) in terms of homotopy invariants of X. More precisely, we prove that σ ^{p +1} cat(X) + 1 ≤ cat(X,X×S ^p }), that and that e(X,X×S< ^p )=e(X)+1, where σ ^{p +1} cat is the (relative) σ category of Vandembroucq and e is the (relative) Toomer invariant. The proof is based on an extension of Milnor's construction of the classifying space of a topological group to a relative setting (due to Dold and Lashof). Received: 14 October 1998 / Revised version: 5 November 1999     

The image inH 2(Q 3;ℝ) of the set of closed 2-forms with preassigned kernel

If (P²ⁿ, ω) is a symplectic manifold and Q³ is its orientable closed submanifold such that ω/Q³≠0, then there arises a one-dimensional distribution ￡=(ω/Q³). We study the dependence of ω in a neighborhood of Q³ and of [ω] ɛH ²(Q ³,ℝ) on ￡. Bibliography: 13 titles. Published inZapiski Nauchnykh Seminarov POMI, Vol. 234, 1996, pp. 125–136.     

Unified theory for abstract parabolic boundary problems—a semigroup approach

This paper presents and abstract semigroup formulation ofparabolic boundary value problems. Smoothness of solutions, represented by a semigroup formula, is the primary object of discussion. The generality of our approach enables us to treat in a unified manner theregularity of solutions to parabolic equations for a large variety of nonhomogeneous boundary value problems. In particular, the approach presented here allows us to translate known regularity results of the elliptic theory directly into regularity results for the parabolic solutions. On the one hand, our theory recaptures known regularity results of the parabolic solutions over smooth spatial domains. On the other hand, however, our theory also covers the case of conical spatial domains, for which the standard assumption ofC -boundaries is violated by suitable application of recent relevant results of elliptic theory for such domains. In the concluding section, an application of our general theory to a boundary control problem with a quadratic performance index is presented.Research supported in part under Grant no. 78-3550, AFOSR, Applied Math Division, USAF.     

Calculations in conformal theory needed for verifying the Alday—Gaiotto—Tachikawa hypothesis

Explicitly verifying the Alday—Gaiotto—Tachikawa (AGT) relation between the conformal blocks controlled by the WN symmetry and U(N) Nekrasov functions requires knowing the Shapovalov matrix and various triple correlators for W-algebra descendants. We collect the simplest expressions of this type for N = 3 and for the two lowest descendant levels together with the detailed derivations, which can now be computerized and used in more general studies of conformal blocks and AGT relations at higher levels.     

Twisting theory for weak Hopf algebras

The main aim of this paper is to study the twisting theory of weak Hopf algebras and give an equivalence between the （braided） monoidal categories of weak Hopf bimodules over the original and the twisted weak Hopf algebra to generalize the result from Oeckl （2000）.     

Blaschke-Süss type theorems for closed plane curves

A C¹-class of plane closed curves is considered, which contains all ovals and rosettes. This class is divided into some subclasses for which greatest lower bounds of numbers of antipodal and arc-antipodal sets are determined.     

Two closed forms for the Apostol–Bernoulli polynomials

In this note, we shall obtain two closed forms for the Apostol–Bernoulli polynomials.     

Brill–Noether theory for cyclic covers

We recall that the Brill–Noether Theorem gives necessary and sufficient conditions for the existence of a ${\mathfrak{g}}_d^r$. Here we consider a general n-fold, étale, cyclic cover $p:\stackrel{?}{C}\to C$ of a curve C of genus g and investigate for which numbers $r,d$ a ${\mathfrak{g}}_d^r$ exists on $\stackrel{?}{C}$. For $r=1$ this means computing the gonality of $\stackrel{?}{C}$. Using degeneration to a special singular example (containing a Castelnuovo canonical curve) and the theory of limit linear series for tree-like curves we show that the Plücker formula yields a necessary condition for the existence of a ${\mathfrak{g}}_d^r$ which is only slightly weaker than the sufficient condition given by the results of Laksov and Kleimann [24], for all $n,r,d$.     

Optimality theory for semi-infinite linear programming ∗

This paper presents an exhaustive approach to optimality theory in semi-infinite linear programming, placing a special emphasis on generality. After surveying optimality conditions for general problems, a detailed analysis is made of problems in which the coefficients are continuous functions of a parameter which varies on a compact set, adopting a feasible directions approach. Lastly, the case of analytical coefficients over an interval is considered in some detail.     

Kneser— Hecke— operators in coding theory

The Kneser-Hecke-operator is a linear operator defined on the complex vector space spanned by the equivalence classes of a family of self-dual codes of fixed length. It maps a linear self-dual codeC over a finite field to the formal sum of the equivalence classes of those self-dual codes that intersectC in a codi-mension 1 subspace. The eigenspaces of this self-adjoint linear operator may be described in terms of a coding-theory analogue of the Siegel Φ-operator.     

From microscopic theory to macroscopic theory——symmetries and order parameters of rigid molecules

Density functional theory is used to describe the phase behaviors of rigid molecules. The construction of the kernel function is discussed. Excluded-volume potential is calculated for two types of molecules with C _2v symmetry. Molecular symmetries lead to the symmetries of the kernel function and the density function, enabling a reduction of configuration space. By approximating the kernel function with a polynomial, the system can be fully characterized by some moments corresponding to the form of the kernel function. The symmetries of the kernel function determine the form of the polynomial, while the coefficients are determined by the temperature and molecular parameters. The analysis of the impact of coefficients helps us to choose independent variables in the moments as order parameters. Combining the analysis and some simulation results, we propose a minimal set of order parameters for bent-core molecules.