Tietze's extension theorem in fuzzy topological spaces

This paper discusses the conceptual difficulties of generalizing standard topological terms of L-fuzzy topological terms. In particular, a theory of relative topologies and relative functions for L-fuzzy topological spaces is developed. The extension of a relative L-fuzzy continuous function into the fuzzy unit interval is defined. The equivalence of L-fuzzy continuous functions and monotone families of open sets is proved, and this equivalence is exploited to establish a fuzzy version of Tietze's Extension Theorem. Finally, a partial converse to the theorem is proved.     

Extensions from the Sobolev Spaces H 1 Satisfying Prescribed Dirichlet Boundary Conditions

Extensions from H ¹(_P) into H ¹() (where _P ) are constructed in such a way that extended functions satisfy prescribed boundary conditions on the boundary of . The corresponding extension operator is linear and bounded.     

Simple birational extensions of the polynomial algebra

The Abhyankar-Sathaye Problem asks whether any biregular embedding can be rectified, that is, whether there exists an automorphism such that is a linear embedding. Here we study this problem for the embeddings whose image is given in by an equation , where and . Under certain additional assumptions we show that, indeed, the polynomial is a variable of the polynomial ring (i.e., a coordinate of a polynomial automorphism of ). This is an analog of a theorem due to Sathaye (1976) which concerns the case of embeddings . Besides, we generalize a theorem of Miyanishi (1984) giving, for a polynomial as above, a criterion for when .

     

The Representation Theorem of onto Isometric Mappings between Two Unit Spheres of l^1（Γ） Type Spaces and The Application to the Isometric Extension Problem

In this paper, we first derive the representation theorem of onto isometric mappings in the unit spheres of l^1（Γ） type spaces, and then conchlde that such mappings can be extended to the whole space as real linear isometries by using a previous result of the author.     

Characteristic Modules of Dual Extensions and Groebner Bases

Let C be a finite dimensional directed algebra over an algebraically closed field k and A=A(C) the dual extension of C. The characteristic modules of A are constructed explicitly for a class of directed algebras, which generalizes the results of Xi. Furthermore, it is shown that the characteristic modules of dual extensions of a certain class of directed algebras admit the left Groebner basis theory in the sense of E. L. Green.     

Single-valued extension property at the points of the approximate point spectrum

A localized version of the single-valued extension property is studied at the points which are not limit points of the approximate point spectrum, as well as of the surjectivity spectrum. In particular, we shall characterize the single-valued extension property at a point in the case that λ_oI−T is of Kato type. From this characterizations we shall deduce several results on cluster points of some distinguished parts of the spectrum.     

On a Generalization of Nikolskij's Extension Theorem in the Case of Two Variables

A modification of the Nikolskij extension theorem for functions from Sobolev spaces H ^k() is presented. This modification requires the boundary to be only Lipschitz continuous for an arbitrary k however, it is restricted to the case of two-dimensional bounded domains.     

A bilinear extension inequality in two dimensions

We provide sharp decay estimates for circular averages of a certain bilinear extension operator on .     

Analyticity of Smooth Eigenfunctions and Spectral Analysis of the Gauss Map

We provide a sufficient condition of analyticity of infinitely differentiable eigenfunctions of operators of the form Uf(x)=a(x,y)f(b(x,y))(dy) acting on functions (evolution operators of one-dimensional dynamical systems and Markov processes have this form). We estimate from below the region of analyticity of the eigenfunctions and apply these results for studying the spectral properties of the Frobenius–Perron operator of the continuous fraction Gauss map. We prove that any infinitely differentiable eigenfunction f of this Frobenius–Perron operator, corresponding to a non-zero eigenvalue admits a (unique) analytic extension to the set . Analyzing the spectrum of the Frobenius–Perron operator in spaces of smooth functions, we extend significantly the domain of validity of the Mayer and Röpstorff asymptotic formula for the decay of correlations of the Gauss map.     

Relaxation of Chain Extension of Amorphous PET Fiber Above the Glass Transition Temperature

The average relaxation time for segmental orientation is considerably shorter than that for chain extension1,2, due to the difference of the ability to move between segments and chains. So in certain time intervals, an almost total relaxation of the segmental orientation can take place with little relaxation of global chain extension. Under the controlled conditions1,3, high global chain orientation but nearly random segmental orientation (GOLR) state can be achieved. Through the hot shrinka…