Classification and syzygies of smooth projective varieties with 2-regular structure sheaf

The geometric and algebraic properties of smooth projective varieties with 1-regular structure sheaf are well understood, and the complete classification of these varieties is a classical result. The aim of this paper is to study the next case: smooth projective varieties with 2-regular structure sheaf. First, we give a classification of such varieties using adjunction mappings. Next, under suitable conditions, we study the syzygies of section rings of those varieties to understand the structure of the Betti tables, and show a sharp bound for Castelnuovo–Mumford regularity.     

Algebraic atomistic lattices of quasivarieties

The Gorbunov-Tumanov conjecture on the structure of lattices of quasivarieties is proved true for the case of algebraic lattices. Namely, for an algebraic atomistic lattice L, the following conditions are equivalent: (1) L is represented as L_q(K) for some algebraic quasivariety K; (2) L is represented as S_Λ (A) for some algebraic lattice A which satisfies the minimality condition and nearly satisfies the maximality conditions; (3) L is a coalgebraic lattice admitting an equaclosure operator. Supported by RFFR grants Nos. 96-01-01525 and 96-0-000976, and by DFG grant No. 436 (RUS) 113/2670. Translated from Algebra i Logika, Vol. 36, No. 4, pp. 363–386, July–August, 1997.     

Coalgebraic logic

We present a generalization of modal logic to logics which are interpreted on coalgebras of functors on sets. The leading idea is that infinitary modal logic contains characterizing formulas. That is, every model-world pair is characterized up to bisimulation by an infinitary formula. The point of our generalization is to understand this on a deeper level. We do this by studying a fragment of infinitary modal logic which contains the characterizing formulas and is closed under infinitary conjunction and an operation called Δ. This fragment generalizes to a wide range of coalgebraic logics. Each coalgebraic logic is determined by a functor on sets satisfying a few properties, and the formulas of each logic are interpreted on coalgebras of that functor. Among the logics obtained are the fragment of infinitary modal logic mentioned above as well as versions of natural logics associated with various classes of transition systems, including probabilistic transition systems. For most of the interesting cases, there is a characterization result for the coalgebraic logic determined by a given functor. We then apply the characterization result to get representation theorems for final coalgebras in terms of maximal elements of ordered algebras. The end result is that the formulas of coalgebraic logics can be viewed as approximations to the elements of a final coalgebra.     

双正交小波滤波器簇代数结构及构造

研究了一类向量多项式两种特殊分解结构,由此引进了与双正交小波滤波器簇相应的多相向量概念,分析了多相向量分解代数结构,得到了在低通滤波器给定条件下,满足任意阶可和规则的对偶低通滤波器构造方法.分析并证明了双正交滤波器簇对应多相向量至多具有的3种代数分解结构,根据其分解的形式得到了双正交小波基构造的新方法,该方法便于双正交小波构造计算机程序化.     

Quantified functional analysis and seminormed spaces: A dual adjunction

In this work we investigate the natural algebraic structure that arises on dual spaces in the context of quantified functional analysis. We show that the category of absolutely convex modules is obtained as the category of Eilenberg-Moore algebras induced by the dualization functor [−,R] on locally convex approach spaces. We also establish a dual adjunction between the latter category and the category of seminormed spaces.     

Parameters identification and optimal control in pharmacokinetics

We first introduce some main problems in pharmacokinetics and then we propose methods for their resolution. Of course, identification problem is considered. A numerical efficient method is given. It brings back to the resolution of a succession of linear algebraic systems. In the case of non unicity we suggest an approach based on the adjunction of a criterion to minimize. The last part studies the optimal injection of a drug associated to an optimization criterion. An explicit solution can be found for a n-compartments linear system.     

Composition générale des relations

A general scheme of composition of a finite sequence of heterogeneous relations in sets of finite arities is built; all known compositions of relations are obtained as special cases. Some of the corresponding algebraic structures in the class of relations are explicited. The general composition of relations with respect to a given scheme is presented as an appropriate sequence of binary compositions.

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Presented by F. E. J. Linton.     

On Mathematical Modelling of Time-Related Musical Structures

In this article, the first steps towards mathematical modelling of time-related musical structures are taken, and the algebraic structure of musical time relations is elaborated starting from a perceptive point of view. A basic characterization of fundamental properties of perceived time relations and their interpretations regarding musical context are given, and some mathematical properties of the proposed definitions are examined. Stemming from musical motivation, a category is found whose objects are finite strict (partially) ordered sets and whose morphisms are weakly monotone and reflect the strict order of the codomain. The category is found to have initial and terminal objects, equalizers, and coequalizers but fails to have binary products or coproducts.     

General affine adjunctions,Nullstellensätze,and dualities

We introduce and investigate a category-theoretic abstraction of the standard “system-solution” adjunction in affine algebraic geometry. We then look further into these geometric adjunctions at different levels of generality, from syntactic categories to (possibly infinitary) equational classes of algebras. In doing so, we discuss the relationships between the dualities induced by our framework and the well-established theory of concrete dual adjunctions. In the context of general algebra we prove an analogue of Hilbert's Nullstellensatz, thereby achieving a complete characterisation of the fixed points on the algebraic side of the adjunction.     

Binary Relations on Ternary Semihypergroups

Ternary semihypergroups are algebraic structures with one associative hyperoperation. The main propose of this article is to study binary relations on ternary semihypergroups and study some basic properties of compatible relations on them. In particular, we analyze the ternary hypergroup associated with a binary relation.     

Trees, set compositions and the twisted descent algebra

We first show that increasing trees are in bijection with set compositions, extending simultaneously a recent result on trees due to Tonks and a classical result on increasing binary trees. We then consider algebraic structures on the linear span of set compositions (the twisted descent algebra). Among others, a number of enveloping algebra structures are introduced and studied in detail. For example, it is shown that the linear span of trees carries an enveloping algebra structure and embeds as such in an enveloping algebra of increasing trees. All our constructions arise naturally from the general theory of twisted Hopf algebras.     

Universal algebra in a closed category

The development of finitary universal algebra is carried out in a suitable closed category called a π-category. The π-categories are characterized by their completeness and cocompleteness and some product-colimit commutativities. We establish the existence of left adjoints to algebraic functors, completeness and cocompleteness of algebraic categories, a structure-semantics adjunction, a characterization theory for algebraic categories and the existence of the theory generated by a presentation. The conditions on the closed category are sufficiently weak to be satisfied by any (complete and cocomplete) cartesian closed category, semi-additive category, commutatively algebraic category and also the categories of semi-normed spaces, normed spaces and Banach spaces.     

Towards higher topology

We categorify the adjunction between locales and topological spaces, this amounts to an adjunction between (generalized) bounded ionads and topoi. We show that the adjunction is idempotent. We relate this adjunction to the Scott adjunction, which was discussed from a more categorical point of view in [21]. We hint that 0-dimensional adjunction inhabits the categorified one.     

Lie algebraic structure of akns matrix system

For an AKNS matrix system, Lie algebraic structure and its mastersymmetry are obtained by a purely algebraic approach; and by using the reduced technique, two similar algebraic structures for MKdV and KdV matrix systems are given.This project is supported by the National Education Foundation of China.     

Geometric combinatorial algebras: cyclohedron and simplex

In this paper we report on results of our investigation into the algebraic structure supported by the combinatorial geometry of the cyclohedron. Our new graded algebra structures lie between two well known Hopf algebras: the Malvenuto–Reutenauer algebra of permutations and the Loday–Ronco algebra of binary trees. Connecting algebra maps arise from a new generalization of the Tonks projection from the permutohedron to the associahedron, which we discover via the viewpoint of the graph associahedra of Carr and Devadoss. At the same time, that viewpoint allows exciting geometrical insights into the multiplicative structure of the algebras involved. Extending the Tonks projection also reveals a new graded algebra structure on the simplices. Finally this latter is extended to a new graded Hopf algebra with basis all the faces of the simplices.     

线性逻辑相空间的分层结构

本文研究了相空间的代数结构,定义了多层相空间,给出了一个相空间的最小子空间的性质.并据此讨论了线性重言式的某些特点.     

A Galois-Theoretic Approach to the Covering Theory of Quandles

The purpose of this article is to clarify the relationship between the algebraic notion of quandle covering introduced by M. Eisermann and the categorical notion of covering arising from Galois theory. A crucial role is played by the adjunction between the variety of quandles and its subvariety of trivial quandles.     

转动系统相对论性动力学方程的代数结构与Poisson积分

研究转动相对论系统动力学方程的代数结构,得到了完整保守转动相对论系统与特殊非完整转动相对论系统动力学方程具有Lie代数结构;一般完整转动相对论系统、一般非完整转动相对论系统动力学方程具有Lie容许代数结构。并给出转动相对论系统动力学方程的Poisson积分。     

Completeness for μ-calculi: A coalgebraic approach

We set up a generic framework for proving completeness results for variants of the modal mu-calculus, using tools from coalgebraic modal logic. We illustrate the method by proving two new completeness results: for the graded mu-calculus (which is equivalent to monadic second-order logic on the class of unranked tree models), and for the monotone modal mu-calculus.Besides these main applications, our result covers the Kozen–Walukiewicz completeness theorem for the standard modal mu-calculus, as well as the linear-time mu-calculus and modal fixpoint logics on ranked trees. Completeness of the linear-time mu-calculus is known, but the proof we obtain here is different and places the result under a common roof with Walukiewicz' result.Our approach combines insights from the theory of automata operating on potentially infinite objects, with methods from the categorical framework of coalgebra as a general theory of state-based evolving systems. At the interface of these theories lies the notion of a coalgebraic modal one-step language. One of our main contributions here is the introduction of the novel concept of a disjunctive basis for a modal one-step language. Generalizing earlier work, our main general result states that in case a coalgebraic modal logic admits such a disjunctive basis, then soundness and completeness at the one-step level transfer to the level of the full coalgebraic modal mu-calculus.     

Non-isospectral flows of noncommutative differential-difference KP equation

We present master symmetries of noncommutative differential-difference KP equation by considering Sato approach, where the field variables are defined over associative algebras. The Lie algebraic structures of generalized and master symmetries are given. They form a Virasoro Lie algebraic structure.