Vanishing of a certain kind of Vassiliev invariants of 2-knots

In knot theory, Vassiliev's 1-knot invariants are defined in a combinatorial way as finite type invariants. By a natural generalization of the combinatorial definition, one has a certain family of 2-knot invariants, which should be called finite type 2-knot invariants. They form a subspace of the whole space of ``Vassiliev 2-knot invariants'. In this paper we prove that it is 1-dimensional.

     

A Ramsey property for graph invariants

We consider the problem of which graph invariants have a certain property relating to Ramsey's theorem. Invariants which have this property are called Ramsey functions. We examine properties of chains of graphs associated with Ramsey functions. Methods are developed which enable one to prove that a given invariant is not a Ramsey function. Results for several familiar invariants are presented.     

Asymptotic invariants constructed from generic initial ideals

Generic initial ideals (gins for short) were systematically introduced by Galligo in 1974 under the name of Grauert invariants since they appeared apparently first in works of Grauert and Hironaka. Ever since they have been of interest in commutative algebra and indirectly in algebraic geometry. Recently, Mayes in a series of articles associated with gins of graded families of ideals geometric objects called limiting shapes. The construction resembles that of Okunkov bodies but there are some differences as well. This work is motivated by Mayes articles and explores the connections between gins, limiting shapes, and some asymptotic invariants of homogeneous ideals which are associated with the gins, for example, asymptotic regularity, Waldschmidt constant and some new invariants, which seem relevant from geometric point of view.     

分子刚度对膜锚定受体-配体结合动力学影响的理论与模拟研究

人体内大部分生物学过程都离不开细胞黏附.细胞黏附行为主要由锚定于细胞膜上的特异性分子（又称受体和配体）的结合动力学关系来决定.已有研究表明，特异性分子的结合关系受外力及细胞膜波动等多种因素影响.然而，特异性分子刚度对细胞膜锚定受体 配体结合关系的影响机制仍不清楚.近期关于新冠病毒强传染力的研究表明，特异性黏附分子刚度对病毒与细胞结合具有重要影响.该文通过建立生物膜黏附的粗粒度模型，借助分子模拟和理论分析来研究分子刚度在黏附中的作用.结果表明，始终存在一个最佳膜间距及最佳分子刚度值，使得黏附分子亲和力和结合动力学参数达到最大值.这项研究不仅能加深人们对细胞黏附的认知，还有助于指导药物设计、疫苗研发等.     

Monoidal Morita invariants for finite group algebras

Two Hopf algebras are called monoidally Morita equivalent if module categories over them are equivalent as linear monoidal categories. We introduce monoidal Morita invariants for finite-dimensional Hopf algebras based on certain braid group representations arising from the Drinfeld double construction. As an application, we show, for any integer n, the number of elements of order n is a monoidal Morita invariant for finite group algebras. We also describe relations between our construction and invariants of closed 3-manifolds due to Reshetikhin and Turaev.     

Complete system of two-dimensional invariants for formulation of triangular finite element of composite shells

The paper describes a system of invariants of symmetric two-dimensional tensors defined on a plane or a surface. The system comprises the well-known first and second invariants and a new quantity called the combined invariant of two tensors. The focus is on the expression for the invariants in terms of normal components of the tensors determined in three different directions on the surface. The system of invariants is used to construct a triangular finite element for geometrically nonlinear analysis of shear deformable anisotropic shells subject to the Reissner–Mindlin assumptions. The relations obtained allow one to readily determine the strain energy of the element for the normal components of the stress and strain tensors in the direction of the element edges. Numerical examples are given to demonstrate some nonlinear capabilities of the element.     

Orthogonal and bounded orthogonal spectral representations

The concept of an orthogonal spectral representation (OTSR) of a Hilbert spaceH relative to a spectral measureE(.) is introduced and it is shown that every Hilbert space admits an OTSR relative to a given spectral measure. Apart from the various results obtained about OTSRs, the principal result of Allan Brown (1974) is deduced as an easy consequence of this study. A new complete system of unitary invariants called the “equivalence of OTSRs”, is given for spectral measures. Two special types of OTSRs called “BOTSR” and “COBOTSR” are introduced and characterized respectively in terms of the “GCGS-property” and “CGS-property” of the associated spectral measure. Various complete systems of unitary invariants are given for spectral measures with the GCGS-property. Finally, the Wecken-Plesner-Rohlin theorem on hermitian operators with simple spectra is generalized to arbitrary spectral measures.     

Complexity of a knot invariant

The algebraic structures called quandles constitute a complete invariant for tame knots. However, determining when two quandles are isomorphic is an empirically hard problem, so there is some dissatisfaction with quandles as knot invariants. We have confirmed this apparent difficulty, showing within the framework of Borel reducibility that the general isomorphism problem for quandles is as complex as possible. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Chamber Structure for Some Equivariant Relative Gromov–Witten Invariants of ℙ<Superscript>1</Superscript> in Genus 0

In this paper, we study genus 0 equivariant relative Gromov-Witten invariants of P¹ whose corresponding relative stable maps are totally ramified over one point. For fixed number of marked points, we show that such invariants are piecewise polynomials in some parameter space. The parameter space can then be divided into polynomial domains, called chambers. We determine the difference of polynomials between two neighbouring chambers. In some special chamber, which we called the totally negative chamber, we show that such a polynomial can be expressed in a simple way. The chamber structure here shares some similarities to that of double Hurwitz numbers.     

Homology of perfect complexes

It is proved that the sum of the Loewy lengths of the homology modules of a finite free complex F over a local ring R is bounded below by a number depending only on R. This result uncovers, in the structure of modules of finite projective dimension, obstructions to realizing R as a closed fiber of some flat local homomorphism. Other applications include, as special cases, uniform proofs of known results on free actions of elementary abelian groups and of tori on finite CW complexes. The arguments use numerical invariants of objects in general triangulated categories, introduced here and called levels. They allow one to track, through changes of triangulated categories, homological invariants like projective dimension, as well as structural invariants like Loewy length. An intermediate result sharpens, with a new proof, the New Intersection Theorem for commutative algebras over fields. Under additional hypotheses on the ring R stronger estimates are proved for Loewy lengths of modules of finite projective dimension.     

Collective coordinates and the mechanism for conformational transitions of complex molecules

Reduction of dimensionality is crucial for the deeper understanding of the mechanism for large-amplitude conformational transitions of complex molecules. By taking up a six-atomcluster as an illustrative example, we present a general methodology to understand conformational transitions of molecules in terms of the low-dimensional dynamics of molecular gyration radii. The dynamics of gyration radii is generally governed by the interplay between the ordinary potential force and a dynamical force called the internal centrifugal force. We show that the internal centrifugal force can be more important than the original potential barrier and gives rise to a new dynamical barrier that truly dominates the conformational transitions of the system. This kind of dynamical effect should be crucially important in a wide class of molecular reaction dynamics. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Self-bounded controlled invariants versus stabilizability

Self-bounded controlled and self-hidden conditioned invariant subspaces, recently introduced by the authors for a more direct and neat handling of some fundamental concepts of the geometric approach, such as controllability subspaces, are proved in this paper to be very useful tools also in dealing with synthesis problems with stability requirements.Definitions concerning stability of invariants and stabilizability of controlled invariants, simple and self-bounded, are first presented and discussed. In particular, it is shown that a more straightforward definition for controlled invariant stabilizability allows a simpler development of the theory, Then, some fundamental results relating self-boundedness to stabilizability are derived. For the sake of completeness, all statements are dualized to conditioned invariants, simple and self-hidden.     

Topological Persistence for Circle-Valued Maps

We study circle-valued maps and consider the persistence of the homology of their fibers. The outcome is a finite collection of computable invariants which answer the basic questions on persistence and in addition encode the topology of the source space and its relevant subspaces. Unlike persistence of real-valued maps, circle-valued maps enjoy a different class of invariants called Jordan cells in addition to bar codes. We establish a relation between the homology of the source space and of its relevant subspaces with these invariants and provide a new algorithm to compute these invariants from an input matrix that encodes a circle-valued map on an input simplicial complex.     

Type II Matrices and Their Bose-Mesner Algebras

Type II matrices were introduced in connection with spin models for link invariants. It is known that a pair of Bose-Mesner algebras (called a dual pair) of commutative association schemes are naturally associated with each type II matrix. In this paper, we show that type II matrices whose Bose-Mesner algebras are imprimitive are expressed as so-called generalized tensor products of some type II matrices of smaller sizes. As an application, we give a classification of type II matrices of size at most 10 except 9 by using the classification of commutative association schemes.     

Newtonian Mechanics and the Sixth Form∗ Syllabus

It is shown that a class of planar conjugated hydrocarbon molecules can be modelled in terms of a simple graph and the related adjacency matrix. Various physical and chemical properties are related to properties of the graphs. The resonance energy is related to such simple invariants as the total number of CC links and the number of Kekulé structures. Rules for enumerating these structures are described.     

Complex systems in drug research: II. The ligand—active site—water confluence as a complex system

Three partners are involved in any pharmacological phenomenon examined at the molecular level, namely a functional biomacromolecule (e.g., receptor or enzyme), a ligand molecule whose binding to the active site of the macromolecule triggers a response, and water which acts as a structural component and as a solvent. Some of the molecular events involved in initial steps of pharmacological responses are examined here as emergent properties of the triad active site-ligand-water, a complex system seldom viewed as such. For example, the exquisite selectivity and efficiency of long- and short-range recognition of ligands may rest on more than simple random encounters. The emergent property of function suggests the possibility that active sites are maintained near criticality by low-levels of endogenous ligands.     

Generalized normal matrices

A matrix A ∈ Mn（C） is called generalized normal provided that there is a positive definite Hermite matrix H such that HAH is normal. In this paper, these matrices are investigated and their canonical form, invariants and relative properties in the sense of congruence are obtained.     

On incidence structures of nonsingular points and hyperbolic lines of ovoids in finite orthogonal spaces

We study the point-line incidence structures of nonsingular points and hyperbolic secant lines associated with ovoids in finite orthogonal spaces. We show that these incidence structures frequently produce partial linear spaces and the parameters of the bipartite graphs (called ovoidal graphs) associated with these structures produce simple and effective isomorphism invariants to distinguish non-isomorphic ovoids. We prove explicit formulas for these isomorphism invariants for a number of infinite families of 2-transitive ovoids.     

Extension of one-dimensional proximity regions to higher dimensions

Proximity regions (and maps) are defined based on the relative allocation of points from two or more classes in an area of interest and are used to construct random graphs called proximity catch digraphs (PCDs) which have applications in various fields. The simplest of such maps is the spherical proximity map which gave rise to class cover catch digraph (CCCD) and was applied to pattern classification. In this article, we note some appealing properties of the spherical proximity map in compact intervals on the real line, thereby introduce the mechanism and guidelines for defining new proximity maps in higher dimensions. For non-spherical PCDs, Delaunay tessellation (triangulation in the real plane) is used to partition the region of interest in higher dimensions. We also introduce the auxiliary tools used for the construction of the new proximity maps, as well as some related concepts that will be used in the investigation and comparison of these maps and the resulting PCDs. We provide the distribution of graph invariants, namely, domination number and relative density, of the PCDs and characterize the geometry invariance of the distribution of these graph invariants for uniform data and provide some newly defined proximity maps in higher dimensions as illustrative examples.