The Endotopism Semigroups of a Partial Equivalence Relation

Siberian Mathematical Journal - We study the endotopism semigroups of a nonreflexive partial equivalence. We describe the existence conditions for six types of endotopisms and...     

Graphs realised by r.e. equivalence relations

We investigate dependence of recursively enumerable graphs on the equality relation given by a specific r.e. equivalence relation on ω. In particular we compare r.e. equivalence relations in terms of graphs they permit to represent. This defines partially ordered sets that depend on classes of graphs under consideration. We investigate some algebraic properties of these partially ordered sets. For instance, we show that some of these partial ordered sets possess atoms, minimal and maximal elements. We also fully describe the isomorphism types of some of these partial orders.     

Weak Solutions of Semilinear PDEs in Sobolev Spaces and Their Probabilistic Interpretation via the FBSDEs

Abstract

The article is devoted to representation of weak solutions (in Sobolev sense) of degenerate parabolic partial differential equations through forward-backward stochastic differential equations. Before, we prove a weak version of a norm equivalence result.     

保等价部分变换半群的变种半群上的正则元

在现有的保等价部分变换半群的基础上,引入了一个新的运算,得出保等价部分变换半群的变种半群的概念,利用格林关系及幂等元的正则性,讨论了这类半群中元素的正则性,给出了保等价部分变换半群的变种半群中一个元是正则元的充要条件     

Homological equivalence relations on an algebraic curve

We study the collection of homological equivalence relations on a fixed curve. We construct a moduli space for pairs consisting of a curve of genus g and a homological equivalence relation of degree n on the curve, and a classifying set for homological equivalence relations of degree n on a fixed curve, modulo automorphisms of the curve. We identify a special type of homological equivalence relations, and we characterize the special homological equivalence relations in terms of the existence of elliptic curves in the Jacobian of the curve.     

The Stone-Čech compactification of a partial frame via ideals and cozero elements

A partial frame is a meet-semilattice in which certain designated subsets are required to have joins, and finite meets distribute over these. The designated subsets are specified by means of a so-called selection function, denoted by S ; these partial frames are called S-frames.

We construct free frames over S-frames using appropriate ideals, called S-ideals. Taking S-ideals gives a functor from S-frames to frames. Coupled with the functor from frames to S-frames that takes S-Lindelöf elements, it provides a category equivalence between S-frames and a non-full subcategory of frames. In the setting of complete regularity, we provide the functor taking S-cozero elements which is right adjoint to the functor taking S-ideals. This adjunction restricts to an equivalence of the category of completely regular S-frames and a full subcategory of completely regular frames. As an application of the latter equivalence, we construct the Stone-? ech compactification of a completely regular S-frame, that is, its compact coreflection in the category of completely regular S-frames.

A distinguishing feature of the study of partial frames is that a small collection of axioms of an elementary nature allows one to do much that is traditional at the level of frames or locales and of uniform or nearness frames. The axioms are sufficiently general to include as examples of partial frames bounded distributive lattices, σ-frames, κ-frames and frames.     

Equivalence relations of AF-algebra extensions

In this paper, we consider equivalence relations of C*-algebra extensions and describe the relationship between the isomorphism equivalence and the unitary equivalence. We also show that a certain group homomorphism is the obstruction for these equivalence relations to be the same.     

A confinal family of equivalence relations and Borel ideals generating them

An increasing θ₁-sequence of Borel equivalence relations on a Polish space that is cofinal (in the sense of Borel reducibility) in the family of all Borel equivalence relations is defined as a development of Rosendal’s construction. It is proved that equivalence relations from this sequence are generated by explicitly defined Borel ideals.     

A transfer algorithm for hierarchical clustering

Transfer algorithms are usually used to optimize an objective function that is defined on the set of partitions of a finite set X. In this paper we define an equivalence relation ? on the set of fuzzy equivalence relations on X and establish a bijection from the set of hierarchies on X to the set of equivalence classes with respect to ?. Thus, hierarchies can be identified with fuzzy equivalence relations and the transfer algorithm can be modified in order to optimize an objective function that is defined on the set of hierarchies on X.     

On the product of two tolerances

Summary In the present note we study the product (composition) of two tolerances; some known results concerning the product of two equivalence relations are generalized to that of two tolerances and new results are obtained via an analysis of possible interactions among them.

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Sunto Nella presente nota si studia il prodotto (composizione) di due tolleranze; alcuni noti risultati sul prodotto di due relazioni di equivalenza vengono generalizzati a quello di due tolleranze, e nuovi risultati sono ottenuti in base ad una analisi delle possibili interazioni tra di esse.

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Partially supported by a grant of italian M.P.I.     

Flows on graphs applied to diagonal similarity and diagonal equivalence for matrices

Three equivalence relations are considered on the set of n × n matrices with elements in F₀, an abelian group with absorbing zero adjoined. They are the relations of diagonal similarity, diagonal equivalence, and restricted diagonal equivalence. These relations are usually considered for matrices with elements in a field. But only multiplication is involved. Thus our formulation in terms of an abelian group with o is natural. Moreover, if F is chosen to be an additive group, diagonal similarity is characterized in terms of flows on the pattern graph of the matrices and diagonal equivalence in terms of flows on the bipartie graph of the matrices. For restricted diagonal equivalence a pseudo-diagonal of the graph must also be considered. When no pseudo-diagonal is present, the divisibility properties of the group F play a role. We show that the three relations are characterized by cyclic, polygonal, and pseudo-diagonal products for multiplicative F. Thus, our method of reducing propositions concerning the three equivalence relations to propositions concerning flows on graphs, provides a unified approach to problems previously considered independently, and yields some n, w or improved results. Our consideration of cycles rather than circuits eliminates certain restrictions (e.g., the complete reducibility of the matrices) which have previously been imposed. Our results extend theorems in Engel and Schneider [5], where however the group F is permitted to be non-commutative.     

Minimization algorithm of fuzzy finite automata

In this paper, a class of fuzzy finite automata corresponding to the Mealy type of ordinary automata is formulated, and also two types of statewise equivalence relations are introduced. From the equivalence relations, a minimal form is defined and a minimization algorithm of the Mealy type of fuzzy finite automata is obtained.     

Some applications of the Adams-Kechris technique

We analyze the technique used by Adams and Kechris (2000) to obtain their results about Borel reducibility of countable Borel equivalence relations. Using this technique, we show that every equivalence relation is Borel reducible to the Borel bi-reducibility of countable Borel equivalence relations. We also apply the technique to two other classes of essentially uncountable Borel equivalence relations and derive analogous results for the classification problem of Borel automorphisms.

     

Bisimulations for fuzzy automata

Bisimulations have been widely used in many areas of computer science to model equivalence between various systems, and to reduce the number of states of these systems, whereas uniform fuzzy relations have recently been introduced as a means to model the fuzzy equivalence between elements of two possible different sets. Here we use the conjunction of these two concepts as a powerful tool in the study of equivalence between fuzzy automata. We prove that a uniform fuzzy relation between fuzzy automata A and B is a forward bisimulation if and only if its kernel and co-kernel are forward bisimulation fuzzy equivalence relations on A and B and there is a special isomorphism between factor fuzzy automata with respect to these fuzzy equivalence relations. As a consequence we get that fuzzy automata A and B are UFB-equivalent, i.e., there is a uniform forward bisimulation between them, if and only if there is a special isomorphism between the factor fuzzy automata of A and B with respect to their greatest forward bisimulation fuzzy equivalence relations. This result reduces the problem of testing UFB-equivalence to the problem of testing isomorphism of fuzzy automata, which is closely related to the well-known graph isomorphism problem. We prove some similar results for backward-forward bisimulations, and we point to fundamental differences. Because of the duality with the studied concepts, backward and forward-backward bisimulations are not considered separately. Finally, we give a comprehensive overview of various concepts on deterministic, nondeterministic, fuzzy, and weighted automata, which are related to bisimulations.     

Stochastic partial differential equations in bounded domains with Dirichlet boundary conditions

In this paper we prove the existence of a unique solution for a class of stochastic parabolic partial differential equations in bounded domains, with Dirichlet boundary conditions. The main tool is an equivalence result, provided by the stochastic characteristics method, between the stochastic equations under investigation and a class of deterministic parabolic equations with moving boundaries, depending on random coefficients. We show the existence of the solution to this last problem, thus providing a solution to the former.     

Notes on R1-ideals in partial abelian monoids

In the present paper, we deal with a class of R ₁-ideals of cancellative positive partial abelian monoids (CPAMs). We prove that, for I being an R ₁-ideal of a CPAM P, P/I is a CPAM. The lattice of congruence relations associated with R ₁-ideals is a sublattice of the lattice of all equivalence relations. Finally, we prove that an intersection of two Riesz ideals is a Riesz ideal and that the lattice of Riesz ideals is a sublattice of the lattice of all ideals. Received March 19, 1999; accepted in final form December 16, 1999.     

Rings Defined by Extensions

In this article, we study subrings of the Ext-algebra of a graded module over a graded ring R. We show these subrings can be defined by equivalence relations on exact sequences over the ring. In particular, the shriek ring, R ^!, and the even part of an Ext-algebra of a d-Koszul algebra can be defined by equivalence relations on exact sequences.     

On Schauder equivalence relations

We study Schauder equivalence relations, which are Borel equivalence relations induced by actions of Banach spaces with Schauder bases. Firstly, we show that and are minimal Schauder equivalence relations. Then, we prove that neither of them is Borel reducible to the quotient where T is the Tsirelson space. This implies that they cannot form a basis for the Schauder equivalence relations. In addition, we apply an argument of Farah to show that every basis for the Schauder equivalence relations, if such exist, has to be of cardinality .     

Global observer design for nonlinear systems

This paper is a geometric study of the global observer design for nonlinear systems. Using the theory of foliations, we derive necessary and sufficient conditions for global exponential observers for nonlinear systems under some assumptions. Our proof for these necessary and sufficient conditions for global exponential observers if via defining two equivalence relations known as horizontal and vertical equivalence relations, and constructing two foliations known as horizontal and vertical foliations from these equivalence relations. Finally, as a corollary of our global theorem, we derive necessary and sufficient conditions for local exponential observers of critically Lyapunov stable nonlinear systems.