On higher heine-stieltjes polynomials

Take a linear ordinary differential operator $\mathfrak{d}\left( z \right) = \sum\nolimits_{i = 1}^k {Q_i \left( z \right)\frac{{d^i }} {{dz^i }}}$\mathfrak{d}\left( z \right) = \sum\nolimits_{i = 1}^k {Q_i \left( z \right)\frac{{d^i }} {{dz^i }}} with polynomial coefficients and set r = max _i=1,…,k(deg Q _i (z) − i). If d(z) satisfies the conditions: (i) r ≥ 0 and (ii) deg Q _k (z) = k + r, we call it a non-degenerate higher Lamé operator. Following the classical examples of E. Heine and T. Stieltjes we initiated in [13] the study of the following multiparameter spectral problem: for each positive integer n find polynomials V (z) of degree at most r such that the equation

On the maximum modulus principle for polynomials in a quasidisk

LetG⊂C be a quasidisk,K ⊂ G be a compact set, andp _n be a non-constant complex polynomial of degree at mostn. We establish the inequality whereα _n < 0 depends onn, K, and the geometrical structure of ϖG.     

A linear functional equation with a singularity at a fixed point

Summary. Local solutions of the functional equation¶¶z^k f( z) = ?_k=1ⁿG_k( z) f( s_kz ) +g( z) z{^\kappa} \phi \left( z\right) =\sum_{k=1}^nG_k\left( z\right) \phi \left( s_kz \right) +g\left( z\right) ¶with k > 0 \kappa > 0 and | s_k| \gt 1 \left| s_k\right| \gt 1 are considered. We prove that the equation is solvable if and only if a certain system of k \kappa conditions on G_k (k = 1, 2, ... , n) and g is fulfilled.     

Diffusion approximation for the components in critical inhomogeneous random graphs of rank 1.

Consider the random graph on n vertices 1,…,n. Each vertex i is assigned a type x_i with x₁,…,x_n being independent identically distributed as a nonnegative random variable X. We assume that EX³< ∞. Given types of all vertices, an edge exists between vertices i and j independent of anything else and with probability \begin{align*}\min \{1, \frac{x_ix_j}{n}\left(1+\frac{a}{n^{1/3}} \right) \}\end{align*}. We study the critical phase, which is known to take place when EX² = 1. We prove that normalized by n^‐2/3the asymptotic joint distributions of component sizes of the graph equals the joint distribution of the excursions of a reflecting Brownian motion with diffusion coefficient \begin{align*}\sqrt{{\textbf{ E}}X{\textbf{ E}}X^3}\end{align*}and drift \begin{align*}a-\frac{{\textbf{ E}}X^3}{{\textbf{ E}}X}s\end{align*}. In particular, we conclude that the size of the largest connected component is of order n^2/3. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 43, 486–539, 2013     

On generalized shift bases for the wiener disc algebra

LetW(D) denote the set of functionsf(z)=Σ _n=0 ^∞ A _n Z ⁿ a _nzⁿ for which Σ_n=0 ^∞|a _n |<+∞. Given any finite set lcub;f _i (z)rcub; _i=1 ⁿ inW(D) the following are equivalent: (i) The generalized shift sequence lcub;f ₁(z)z ^kn ,f ₂(z)z ^kn+1, …,f _n (z)z ^(k+1)n−1rcub; _k=0 ^∞ is a basis forW(D) which is equivalent to the basis lcub;z ^m rcub; _m=0 ^∞ . (ii) The generalized shift sequence is complete inW(D), (iii) The function has no zero in |z|≦1, wherew=e ^2πiti /n.     

Star number and star arboricity of a complete multigraph

Let G be a multigraph. The star number s(G) of G is the minimum number of stars needed to decompose the edges of G. The star arboricity sa(G) of G is the minimum number of star forests needed to decompose the edges of G. As usual λK _n denote the λ-fold complete graph on n vertices (i.e., the multigraph on n vertices such that there are λ edges between every pair of vertices). In this paper, we prove that for n ⩾ 2

On the distribution of powers of a complex number

Let α be a complex number of modulus strictly greater than 1, and let ξ ≠ 0 and ν be two complex numbers. We investigate the distribution of the sequence ξ α ⁿ  + ν, n = 0, 1, 2, . . . , modulo ${\mathbb{Z}[i],}Let α be a complex number of modulus strictly greater than 1, and let ξ ≠ 0 and ν be two complex numbers. We investigate the distribution of the sequence ξ α ⁿ  + ν, n = 0, 1, 2, . . . , modulo \mathbbZ[i],{\mathbb{Z}[i],} where i=?{-1}{i=\sqrt{-1}} and \mathbbZ[i]=\mathbbZ+i\mathbbZ{\mathbb{Z}[i]=\mathbb{Z}+i\mathbb{Z}} is the ring of Gaussian integers. For any z ? \mathbbC,{z\in \mathbb{C},} one may naturally call the quantity z modulo \mathbbZ[i]{\mathbb{Z}[i]} the fractional part of z and write {z} for this, in general, complex number lying in the unit square S:={z ? \mathbbC:0 ￡ \mathfrakR(z),\mathfrakJ(z) < 1 }{S:=\{z\in\mathbb{C}:0\leq \mathfrak{R}(z),\mathfrak{J}(z) <1 \}}. We first show that if α is a complex non-real number which is algebraic over \mathbbQ{\mathbb{Q}} and satisfies |α| > 1 then there are two limit points of the sequence {ξ α ⁿ  +ν}, n = 0, 1, 2, . . . , which are ‘far’ from each other (in terms of α only), except when α is an algebraic integer whose conjugates over \mathbbQ(i){\mathbb{Q}(i)} all lie in the unit disc |z| ≤  1 and x ? \mathbbQ(a,i).{\xi\in\mathbb{Q}(\alpha,i).} Then we prove a result in the opposite direction which implies that, for any fixed a ? \mathbbC{\alpha\in\mathbb{C}} of modulus greater than 1 and any sequence z_n ? \mathbbC,n=0,1,2,...,{z_n\in\mathbb{C},n=0,1,2,\dots,} there exists x ? \mathbbC{\xi \in \mathbb{C}} such that the numbers ξ α ⁿ −z _n , n = 0, 1, 2, . . . , all lie ‘far’ from the lattice \mathbbZ[i]{\mathbb{Z}[i]}. In particular, they all can be covered by a union of small discs with centers at (1+i)/2+\mathbbZ[i]{(1+i)/2+\mathbb{Z}[i]} if |α| is large.     

Triangle‐free subgraphs in the triangle‐free process

Consider the triangle‐free process, which is defined as follows. Start with G(0), an empty graph on n vertices. Given G(i ‐ 1), let G(i) = G(i ‐ 1) ∪{g(i)}, where g(i) is an edge that is chosen uniformly at random from the set of edges that are not in G(i ? 1) and can be added to G(i ‐ 1) without creating a triangle. The process ends once a maximal triangle‐free graph has been created. Let H be a fixed triangle‐free graph and let X_H(i) count the number of copies of H in G(i). We give an asymptotically sharp estimate for ??(X_H(i)), for every \begin{align*}1 \ll i \le 2^{-5} n^{3/2} \sqrt{\ln n}\end{align*}, at the limit as n →∞. Moreover, we provide conditions that guarantee that a.a.s. X_H(i) = 0, and that X_H(i) is concentrated around its mean.© 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011     

On Bernstein–Walsh-type lemmas in regions of the complex plane

Let G ì \mathbb C G \subset {\mathbb C} be a finite region bounded by a Jordan curve L: = ?G L: = \partial G , let W: = \textext[`(G)] \Omega : = {\text{ext}}\bar{G} (with respect to [`(\mathbb C)] {\overline {\mathbb C}} ), $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} $ \Delta : = \left\{ {z:\left| z \right| > 1} \right\} , and let w = F(z) w = \Phi (z) be a univalent conformal mapping of Ω onto Δ normalized by $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 $ \Phi \left( \infty \right) = \infty, \;\Phi '\left( \infty \right) > 0 . By A _p (G); p > 0; we denote a class of functions f analytic in G and satisfying the condition

Para-orthogonal Laurent polynomials and associated sequences of rational functions

On the space, , of Laurent polynomials (L-polynomials) we consider a linear functional which is positive definite on (0, ) and is defined in terms of a given bisequence, { _k } _– . Two sequences of orthogonal L-polynomials, {Q _n (z) ₀ and , are constructed which span in the order {1,z ^–1,z,z ^–2,z ²,...} and {1,z,z ^–1,z ²,z ^–2,...} respectively. Associated sequences of L-polynomials {P _n (z) ₀ , and are introduced and we define rational functions , wherew is a fixed positive number. The partial fraction decomposition and integral representation of,M _n (z, w) are given and correspondence of {M _n (z, w)} is discussed. We get additional solutions to the strong Stieltjes moment problem from subsequences of {M _n (z, w)}. In particular when { _k } _– is a log-normal bisequence, {M _2n (z, w)} and {M _2n+1 (z, w)} yield such solutions.Research supported in part by the National Science Foundation under Grant DMS-9103141.     

Inequalities for absolute maxima of a polynomial and its derivative

Suppose z ₁, z ₂, ... z _n are complex numbers with absolute values more than 1 and Arg z _j Arg z _k for j k where Arg w stands for the argument of the complex number w in [0,2). In this note we show that

Projective Freeness of Algebras of Real Symmetric Functions

Let \mathbb Dⁿ:={z=(z₁,?, z_n) ? \mathbb Cⁿ:|z_j| < 1,   j=1,?, n}{\mathbb {D}^n:=\{z=(z_1,\ldots, z_n)\in \mathbb {C}^n:|z_j| < 1, \;j=1,\ldots, n\}}, and let [`(\mathbbD)]ⁿ{\overline{\mathbb{D}}^n} denote its closure in \mathbb Cⁿ{\mathbb {C}^n}. Consider the ring

Algebraic independence of Mahler functions

Suppose that f₁, ?, f_mf_1, \ldots , f_m satisfy functional equations of type¶¶ f_i(z^d) = P_i(z, f_i(z))     or     f_i(z) = P_i(z, f_i(z^d))f_i({z^d}) = P_i(z, f_i(z)) \quad {or} \quad f_i(z) = P_i(z, f_i({z^d})) ¶for i = 1, ?, mi = 1, \ldots , m, an integer d > 1 and polynomials P_i ? \Bbb C (z)[ y]P_i \in \Bbb C (z)[ {y}] with pairwise distinct partial degrees deg_y( P₁), ?, deg_y( P_m)\deg _y( {P_1}), \ldots , \deg _y( {P_m}). Generalizing a result of Keiji Nishioka and using an idea of Kumiko Nishioka we show, that f₁, ?, f_mf_1, \ldots , f_m can only be algebraically dependent over \Bbb C (z)\Bbb C (z), if there is an index k ? { 1, ?, m}\kappa \in \{ {1, \ldots , m}\} such that f_kf_{\kappa } is rational.     

On Classes of Functions Related to Starlike Functions with Respect to Symmetric Conjugate Points Defined by a Fractional Differential Operator

Let A denote the class of analytic functions f, in the open unit disk E = {z : |z| < 1}, normalized by f(0) = f′(0) − 1 = 0. In this paper, we introduce and study the class ST^n,a_l,m(h){ST^{n,\alpha}_{\lambda,m}(h)} of functions f ? A{f\in A}, with \fracD^n,a_l f_m(z)z 1 0{\frac{D^{n,\alpha}_\lambda f_m(z)}{z}\neq 0}, satisfying

Classification of the centers, of their cyclicity and isochronicity for two classes of generalized quintic polynomial differential systems

In this paper we classify the centers localized at the origin of coordinates, the cyclicity of their Hopf bifurcation and their isochronicity for the polynomial differential systems in \mathbbR²{\mathbb{R}^2} of degree d that in complex notation z = x + i y can be written as

Description of Simple Exceptional Sets in the Unit Ball

For z B ⁿ, the boundary of the unit ball in . If the exceptional set for f. In this note we give a tool for describing such sets. Moreover we prove that if Eis a G and F subset of the projective (n– 1)-dimensional space then there exists a holomorphic function fin the unit ball B ⁿso that E(f) = E.     

A Quantitative Version of the Non-Abelian Idempotent Theorem

Suppose that G is a finite group and f is a complex-valued function on G. f induces a (left) convolution operator from L ²(G) to L ²(G) by g ? f *g{g \mapsto f \ast g} where

A sufficient condition for graphs to be weaklyk-linked

We consider graphs, which are finite, undirected, without loops and in which multiple edges are possible. For each natural numberk letg(k) be the smallest natural numbern, so that the following holds:LetG be ann-edge-connected graph and lets ₁,...,s _k,t ₁,...,t _k be vertices ofG. Then for everyi {1,..., k} there existsa pathP _i froms _i tot _i, so thatP ₁,...,P _k are pairwise edge-disjoint. We prove      

Schwache Lösungen des Frankl-Morawetz-Problems im ℝ3

In a bounded simple connected region G ? ?³ we consider the equation $$L\left[ u \right]: = k\left( z \right)\left( {u_{xx} + u_{yy} } \right) + u_{zz} + d\left( {x,y,z} \right)u = f\left( {x,y,z} \right)$$ where k(z)? 0 whenever z ? 0.G is surrounded forz≥0 by a smooth surface Γ₀ with S:=Γ₀ ? {(x,y,z)|=0} and forz<0 by the characteristic \(\Gamma _2 :---(x^2 + y^2 )^{{\textstyle{1 \over 2}}} + \int\limits_z^0 {(---k(t))^{{\textstyle{1 \over 2}}} dt = 0} \) and a smooth surface Γ₁ which intersect the planez=0 inS and where the outer normal n=(n_x, n_y, n_z) fulfills \(k(z)(n_x^2 + n_y^2 ) + n_z^2 |_{\Gamma _1 } > 0\) . Under conditions on Γ₁ and the coefficientsk(z), d(x,y,z) we prove the existence of weak solutions for the boundary value problemL[u]=f inG with \(u|_{\Gamma _0 \cup \Gamma _1 } = 0\) . The uniqueness of the classical solution for this problem was proved in [1].     

Truncated Complex Moment Problems with a $ \widetilde{zz} $ Relation

We solve the truncated complex moment problem for measures supported on the variety K o \mathcal{K}\equiv { z ? \in C: z [(z)\tilde]\widetilde{z} = A+Bz+C [(z)\tilde]\widetilde{z} +Dz ² ,D 1 \neq 0}. Given a doubly indexed finite sequence of complex numbers g o g⁽²ⁿ⁾:g₀₀,g₀₁,g₁₀,?,g_0,2n,g_1,2n-1,?,g_2n-1,1,g_2n,0 \gamma\equiv\gamma^{(2n)}:\gamma_{00},\gamma_{01},\gamma_{10},\ldots,\gamma_{0,2n},\gamma_{1,2n-1},\ldots,\gamma_{2n-1,1},\gamma_{2n,0} , there exists a positive Borel measure m\mu supported in K \mathcal{K} such that g_ij=ò[`(z)]<sup>i</sup>z<sup>j</sup> dm (0 ￡ 1+j ￡ 2n) \gamma_{ij}=\int\overline{z}^{i}z^{j}\,d\mu\,(0\leq1+j\leq2n) if and only if the moment matrix M(n)( g\gamma ) is positive, recursively generated, with a column dependence relation Z [(Z)\tilde]\widetilde{Z} = A1+BZ +C [(Z)\tilde]\widetilde{Z} +DZ <sup>2</sup>, and card V(g) 3\mathcal{V}(\gamma)\geq rank M(n), where V(g)\mathcal{V}(\gamma) is the variety associated to g \gamma . The last condition may be replaced by the condition that there exists a complex number g<sub>n,n+1</sub> \gamma_{n,n+1} satisfying g<sub>n+1,n</sub> o [`(g)]_n,n+1=Ag_n,n-1+Bg_n,n+Cg_n+1,n-1+Dg_n,n+1 \gamma_{n+1,n}\equiv\overline{\gamma}_{n,n+1}=A\gamma_{n,n-1}+B\gamma_{n,n}+C\gamma_{n+1,n-1}+D\gamma_{n,n+1} . We combine these results with a recent theorem of J. Stochel to solve the full complex moment problem for K \mathcal{K} , and we illustrate the connection between the truncated and full moment problems for other varieties as well, including the variety z ^k = p(z, [(Z)\tilde] \widetilde{Z} ), deg p < k.