Symmetrization of quasi-regular patterns with periodic tilting of regular polygons

Computer-generated aesthetic patterns arewidely used as design materials in various fields. Themost common methods use fractals or dynamicalsystems as basic tools to create various patterns. Toenhance aesthetics and controllability, some researchershave introduced symmetric layouts along with thesetools. One popular strategy employs dynamical systemscompatible with symmetries that construct functionswith the desired symmetries. However, these aretypically confined to simple planar symmetries. Theother generates symmetrical patterns under theconstraints of tilings. Although it is slightly moreflexible, it is restricted to small ranges of tilingsand lacks textural variations. Thus, we proposed anew approach for generating aesthetic patterns bysymmetrizing quasi-regular patterns using general kuniformtilings. We adopted a unified strategy toconstruct invariant mappings for k-uniform tilings thatcan eliminate texture seams across the tiling edges.Furthermore, we constructed three types of symmetriesassociated with the patterns: dihedral, rotational, andreflection symmetries. The proposed method can beeasily implemented using GPU shaders and is highlyefficient and suitable for complicated tiling with regularpolygons. Experiments demonstrated the advantages of our method over state-of-the-art methods in terms offlexibility in controlling the generation of patterns withvarious parameters as well as the diversity of texturesand styles.

E-IDEAL QUASI-REGULAR SEMIGROUPS

This paper defines a kind of quasi-regular semigroups and the so-called E-ideal quasiregular semigroups and gives some of its characteristics and its two special cases (E-left regular quasi-regular semigroups, E-semilattice quasi-regular semigroups), thus estabishing a structure theorem for it, and as corollaries, obtaining a construction for a left regular band and the known construction for bands (Petrich, 1967).

MOSCO CONVERGENCE OFQUASI-REGULAR DIRICHLET FORMS

A sufficient condition for the Mosco limit of a sequence of quasi-regular Dirichlet forms to be quasi-regular is given. In particular, a Dirichlet form is a quasi-regular Dirchlet form if and only if its Yosida approximation sequency satisfies the conditon. Furthermore, conditions for the Mosco limit of a sequence of symmetric (strictly strong) local quasi-regular Dirichlet forms to be (strictly strong) local are also presented. This paper extends the results of [1] from regular Dirichlet space to quasi-regular Dirichlet space.

Some new results on planar harmonic mappings

This survey paper contains some new results on the Landau theorem, Bloch theorem and Schwarz-Pick lemma for planar harmonic mappings.

Construction of Markov process with Markov resolvent on Lp(E,m)

Lei E be a Hausdorff topological space and let A(E) be its Borel σ-field.Let m be a σ-finitc measure on (E,A(E)).A necessary and sufficient condition for a Markov resolvent on Lp(E,m ) to be associated with an m-tight m-special standard process (with state space E) is given.Furthermore some new examples which do not belong to the framework of Dirichlet space are also given.

Weak convergence of Dirichlet processes

Weak convergence of Markov processes is studied by means of Dirichlet forms and two theorems for weak convergence of Hunt processes on general metric spaces are established.As applications,examples for weak convergence of symmetric or non symmetric Dirichlet processes on finite and infinite spaces are given.

Characterisations of semigroups by the properties of their picture fuzzy bi-ideals

In this article,a new generalisation of fuzzy bi-ideals and intuitionsitic fuzzy bi-ideals of a semigroup considered so called picture fuzzy bi-ideals no a semigroup.It is well known that the intra-regular semigroups play an essential role in studying the structure,especially the decomposition,of semigroups.The purpose of this paper is to deal with the algebraic structure of semigroups by applying picture fuzzy set theory.As an application of our results we get characterisations of intra-regular regular semigroups in terms of picture fuzzy bi-ideals.We prove that a semigroup is both regular and intra-regular if and only if every picture fuzzy bi-ideal on S is idempotent.

A NOTE ON BEURLING-DENY FORMULAE IN INFINITE DIMENSIONAL SPACES

The uniqueness of the Beurling-Deny first formula in quasi-regular Dirichlet spaces is verified in terms of the strictly strong local property. An extension of the Beurling-Deny second formula is obtained in infinite dimensional spaces.