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 c...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.展开更多
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-semil...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).展开更多
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 approx...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.展开更多
基金supported by the Key R&D Programs of Zhejiang Province(Nos.2023C01224 and 2022C01220)the National Natural Science Foundation of China(No.61702458)+1 种基金Yun Zhang was partially supported by Zhejiang Province Public Welfare Technology Application Research(No.LGG22F020009)Key Lab of Film and TV Media Technology of Zhejiang Province(No.2020E10015).
文摘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.
基金Project supported by the National Natural Science Foundation of China.
文摘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).
文摘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.
