Uniformization of Sierpiński carpets in the plane

Let S _i , i∈I, be a countable collection of Jordan curves in the extended complex plane \(\widehat{\mathbb{C}}\) that bound pairwise disjoint closed Jordan regions. If the Jordan curves are uniform quasicircles and are uniformly relatively separated, then there exists a quasiconformal map \(f\colon\widehat{\mathbb{C}}\rightarrow\widehat{\mathbb{C}}\) such that f(S _i ) is a round circle for all i∈I. This implies that every Sierpiński carpet in \(\widehat{\mathbb{C}}\) whose peripheral circles are uniformly relatively separated uniform quasicircles can be mapped to a round Sierpiński carpet by a quasisymmetric map.     

Packing coloring of Sierpiński-type graphs

The packing chromatic number \(\chi _{\rho }(G)\) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into sets \(V_i\), \(i\in \{1,\ldots ,k\}\), where each \(V_i\) is an i-packing. In this paper, we consider the packing chromatic number of several families of Sierpiński-type graphs. While it is known that this number is bounded from above by 8 in the family of Sierpiński graphs with base 3, we prove that it is unbounded in the families of Sierpiński graphs with bases greater than 3. On the other hand, we prove that the packing chromatic number in the family of Sierpiński triangle graphs \(ST^n_3\) is bounded from above by 31. Furthermore, we establish or provide bounds for the packing chromatic numbers of generalized Sierpiński graphs \(S^n_G\) with respect to all connected graphs G of order 4.     

Julia set of λ exp(<Emphasis Type="Italic">z</Emphasis>)/<Emphasis Type="Italic">z</Emphasis> with real parameters λ

In this paper, we investigate the Julia set of the family λ exp(z)/z with real parameters λ. We look for what values of real parameters λ such that the Julia set of λ exp(z)/z does not coincide with the whole plane, and thus gives a complete classification for real parameters, which is similar to Jang’s result of a family of transcendental entire functions. Moreover, We also discuss the shape and size of Fatou sets and Julia sets of λ exp(z)/z with real parameters λ when the Julia sets are not the whole plane.     

Equivalent Semi-Norms of Non-Local Dirichlet Forms on the Sierpiński Gasket and Applications

We construct equivalent semi-norms of non-local Dirichlet forms on the Sierpiński gasket and apply these semi-norms to a convergence problem and a trace problem. We also construct explicitly a sequence of non-local Dirichlet forms with jumping kernels equivalent to |x ? y|^?α?β that converges exactly to local Dirichlet form.     

On two theorems of Sierpiński

A theorem of Sierpiński says that every infinite set Q of reals contains an infinite number of disjoint subsets whose outer Lebesgue measure is the same as that of Q. He also has a similar theorem involving Baire property. We give a general theorem of this type and its corollaries, strengthening classical results.     

Normal functions and a class of associated boundary functions

Letμ′ be the family of non-empty closed subsets of the Riemann sphere and Λ the family of continuous curves λ with values in the unit disk and lim _t→1 |λ(t)|=1. A meromorphic functionf in |z|<1 induces a mapping\(\hat f\) from Λ intoμ′ by setting\(\hat f\left( \lambda \right)\) equal to the cluster set off on λ. The authors show that if\(\hat f\) is continuous then existence of an asymptotic value ate ^iθ implies the existence of an angular limit. Further if the spherical derivative off iso(1/(1?|z|)) then\(\hat f\) is constant on every open disk in the space Λ.     

Boundedness,univalence and quasiconformal extension of Robertson functions

This article contains several results for λ-Robertson functions, i.e., analytic functions f defined on the unit disk ? satisfying f(0) = f′(0) ? 1 = 0 and Re e ^?iλ {1 + zf″(z)/f′(z)} > 0 in ? where λ ∈ (?π/2, π/2). We will discuss about conditions for boundedness and quasiconformal extension of Robertson functions. In the last section we provide another proof of univalence for Robertson functions by using the theory of Löwner chains.     

Coefficient estimates of negative powers and inverse coefficients for certain starlike functions

For ?1≤B<A≤1, let \(\mathcal {S}^{*}(A,B)\) denote the class of normalized analytic functions \(f(z)= z+{\sum }_{n=2}^{\infty }a_{n} z^{n}\) in |z|<1 which satisfy the subordination relation z f ^′(z)/f(z)?(1 + A z)/(1 + B z) and Σ^?(A,B) be the corresponding class of meromorphic functions in |z|>1. For \(f\in \mathcal {S}^{*}(A,B)\) and λ>0, we shall estimate the absolute value of the Taylor coefficients a _n (?λ,f) of the analytic function (f(z)/z)^?λ . Using this we shall determine the coefficient estimate for inverses of functions in the classes \(\mathcal {S}^{*}(A,B)\) and Σ^?(A,B).     

Some Properties of a Singular Integral Equation with Non-Carleman Shift

We study the distribution of roots of the function W(z) = 1?λl^?z cot(πz), z ∈ C, where l ∈ (0, 1) and λ ∈ C are parameters. The results are applied to the problem on the solvability and number of solutions of a singular integral equation in various function spaces.     

Cichoń’s diagram for uncountable cardinals

We develop a version of Cichoń’s diagram for cardinal invariants on the generalized Cantor space 2 ^κ or the generalized Baire space κ ^κ , where κ is an uncountable regular cardinal. For strongly inaccessible κ, many of the ZFC-results about the order relationship of the cardinal invariants which hold for ω generalize; for example, we obtain a natural generalization of the Bartoszyński–Raisonnier–Stern Theorem. We also prove a number of independence results, both with < κ-support iterations and κ-support iterations and products, showing that we consistently have strict inequality between some of the cardinal invariants.     

A note on the quantization for probability measures with respect to the geometric mean error

We study the quantization with respect to the geometric mean error for probability measures μ on \({\mathbb{R}^d}\) for which there exist some constants C, η > 0 such that \({\mu(B(x,\varepsilon))\leq C\varepsilon^\eta}\) for all ε > 0 and all \({x\in\mathbb{R}^d}\) . For such measures μ, we prove that the upper quantization dimension \({\overline{D}(\mu)}\) of μ is bounded from above by its upper packing dimension and the lower one \({\underline{D}(\mu)}\) is bounded from below by its lower Hausdorff dimension. This enables us to calculate the quantization dimension for a large class of probability measures which have nice local behavior, including the self-affine measures on general Sierpiński carpets and self-conformal measures. Moreover, based on our previous work, we prove that the upper and lower quantization coefficient for a self-conformal measure are both positive and finite.     

On power deformations of univalent functions

For an analytic function f (z) on the unit disk |z| < 1 with f (0) = f′(0) ? 1 = 0 and f (z) ≠ 0, 0 < |z| < 1, we consider the power deformation f _c (z) = z(f (z)/z) ^c for a complex number c. We determine those values c for which the operator \({f \mapsto f_c}\) maps a specified class of univalent functions into the class of univalent functions. A little surprisingly, we will see that the set is described by the variability region of the quantity zf′(z)/ f (z), |z| < 1, for most of the classes that we consider in the present paper. As an unexpected by-product, we show boundedness of strongly spirallike functions.     

Meromorphic Solutions for a Class of Differential Equations and Their Applications

In this note, we study the admissible meromorphic solutions for algebraic differential equation fⁿf' + P_n?1(f) = R(z)e^α(z), where P_n?1(f) is a differential polynomial in f of degree ≤ n ? 1 with small function coefficients, R is a non-vanishing small function of f, and α is an entire function. We show that this equation does not possess any meromorphic solution f(z) satisfying N(r, f) = S(r, f) unless P_n?1(f) ≡ 0. Using this result, we generalize a well-known result by Hayman.     

On the dynamics of a family of generated renormalization transformations

We study the family of renormalization transformations of the generalized d  -dimensional diamond hierarchical Potts model in statistical mechanic and prove that their Julia sets and non-escaping loci are always connected, where d?2$d?2$. In particular, we prove that their Julia sets can never be a Sierpiński carpet if the parameter is real. We show that the Julia set is a quasicircle if and only if the parameter lies in the unbounded capture domain of these models. Moreover, the asymptotic formula of the Hausdorff dimension of the Julia set is calculated as the parameter tends to infinity.     

On the rank of odd hyper-quasi-polynomials

Given any nonzero entire function g: ? → ?, the complex linear space F(g) consists of all entire functions f decomposable as f(z + w)g(z - w)=φ₁(z)ψ₁(w)+???+ φ_n(z)ψ_n(w) for some φ₁, ψ₁, …, φ_n, ψ_n: ? → ?. The rank of f with respect to g is defined as the minimum integer n for which such a decomposition is possible. It is proved that if g is an odd function, then the rank any function in F(g) is even.     

Solution of Functional Equations Related to Elliptic Functions

Functional equations of the form f(x + y)g(x ? y) = Σ _j=1 ⁿ α _j (x)β _j (y) as well as of the form f₁(x + z)f₂(y + z)f₃(x + y ? z) = Σ _j=1 ^m φ _j (x, y)ψ _j (z) are solved for unknown entire functions f, g,α _j , β _j : ? → ? and f₁, f₂, f₃, ψ _j : ? → ?, φ _j : ?² → ? in the cases of n = 3 and m = 4.     

Sharpened forms of the generalized Schwarz inequality on the boundary

In this paper, a boundary version of the Schwarz inequality is investigated. We obtain more general results at the boundary. If we know the second coefficient in the expansion of the function f(z) = 1 + c_pz^p + c_{p + 1}z^{p + 1}…, then we obtain new inequalities of the Schwarz inequality at boundary by taking into account c_{p + 1} and zeros of the function f(z) ? 1. The sharpness of these inequalities is also proved.     

Composite meromorphic functions and normal families

In this paper, we study the normality of families of meromorphic functions. We prove the result: Let α(z) be a holomorphic function and \({\mathcal{F}}\) a family of meromorphic functions in a domain D, P(z) be a polynomial of degree at least 3. If P ○ f(z) and P ○ g(z) share α(z) IM for each pair \({f(z),g(z)\in \mathcal{F}}\) and one of the following conditions holds: (1) P(z) ? α(z ₀) has at least three distinct zeros for any \({z_{0}\in D}\); (2) There exists \({z_{0}\in D}\) such that P(z) ? α(z ₀) has at most two distinct zeros and α(z) is nonconstant. Assume that β ₀ is a zero of P(z) ? α(z ₀) with multiplicity p and that the multiplicities l and k of zeros of f(z) ? β ₀ and α(z) ? α(z ₀) at z ₀, respectively, satisfy k ≠ lp, for all \({f(z)\in\mathcal{F}}\). Then \({\mathcal{F}}\) is normal in D. In particular, the result is a kind of generalization of the famous Montel criterion.