Meromorphic Functions Sharing Two Values IM with Their Derivatives

Let f be a nonconstant meromorphic function such that ${\overline{N}(r, f) < \lambda T(r, f)}$ , where ${\lambda \in [0, \frac{1}{9})}$ , and let a, b be two distinct finite values. If f and f′ share a, b IM, then ${f \equiv f'}$ .     

Sharing Set and Normal Families of Entire Functions

Let ${\mathcal{F}}$ be a family of holomorphic functions defined in a domain ${\mathcal{D}}$ , let k( ≥ 2) be a positive integer, and let S = {a, b}, where a and b are two distinct finite complex numbers. If for each ${f \in \mathcal{F}}$ , all zeros of f(z) are of multiplicity at least k, and f and f ^(k) share the set S in ${\mathcal{D}}$ , then ${\mathcal{F}}$ is normal in ${\mathcal{D}}$ . As an application, we prove a uniqueness theorem.     

Elliptic Reflection Structures, K-Loop Derivations and Triangle-Inequality

This paper is a part of our general aim to study properties of elliptic and ordered elliptic geometries and then using some of these properties to introduce new concepts and develop their theories. If ${(P,\mathfrak{G}, \equiv,\tau)}$ denotes an elliptic geometry ordered via a separation ?? then there are polar points o and ?? and on the line ${ \overline{K} := \overline{\infty,o}}$ there can be established an operation ??+?? such that ${(\overline{K},+)}$ becomes a commutative group and the map ${ a^+ :\overline{K}\to \overline{K} ; x \mapsto a + x}$ is a motion on ${\overline{K}}$ . The separation ?? induces on ${\overline{K}}$ a cyclic order ?? with [o, e, ??] = 1 such that ${(\overline{K},+,\omega)}$ becomes a cyclic ordered group. For ${a,b \in K := \overline{K} {\setminus}\{\infty\}}$ we set ${a < b :\Longleftrightarrow [a,b,\infty] =1}$ and for all ${a\in K\,a < \infty}$ . Then (K,?<) is a totally ordered set. We show there is a surjective distance function: $$ \lambda : P \times P \to \overline{K}_+ := \{x \in \overline{K}\,|\,o \leq x\leq\infty\}, $$ with ?? ${\lambda(a,b) = \lambda(c,d) \ \Longleftrightarrow (a,b) \equiv (c,d)}$ ??. We prove in the first part of our project like (cf. Gr?ger in Mitt Math Ges Hamburg 11:441?C457, 1987) the following triangle-inequality: (cf. Theorem 8.2). If (a, b, c) is a triangle consisting of pairwise not polar points with ??(a, c), ??(b, c) < e then ??(a, b) ?? ??(a, c) + ??(b, c) < ??.     

Note on Group Distance Magic Graphs G[C 4]

A group distance magic labeling or a ${\mathcal{G}}$ -distance magic labeling of a graph G =  (V, E) with ${|V | = n}$ is a bijection f from V to an Abelian group ${\mathcal{G}}$ of order n such that the weight ${w(x) = \sum_{y\in N_G(x)}f(y)}$ of every vertex ${x \in V}$ is equal to the same element ${\mu \in \mathcal{G}}$ , called the magic constant. In this paper we will show that if G is a graph of order n =  2 ^p (2k + 1) for some natural numbers p, k such that ${\deg(v)\equiv c \mod {2^{p+1}}}$ for some constant c for any ${v \in V(G)}$ , then there exists a ${\mathcal{G}}$ -distance magic labeling for any Abelian group ${\mathcal{G}}$ of order 4n for the composition G[C ₄]. Moreover we prove that if ${\mathcal{G}}$ is an arbitrary Abelian group of order 4n such that ${\mathcal{G} \cong \mathbb{Z}_2 \times\mathbb{Z}_2 \times \mathcal{A}}$ for some Abelian group ${\mathcal{A}}$ of order n, then there exists a ${\mathcal{G}}$ -distance magic labeling for any graph G[C ₄], where G is a graph of order n and n is an arbitrary natural number.     

Semi-Heyting Algebras Term-equivalent to Gödel Algebras

In this paper we investigate those subvarieties of the variety $\mathcal {SH}$ of semi-Heyting algebras which are term-equivalent to the variety $\mathcal L_{\mathcal H}$ of Gödel algebras (linear Heyting algebras). We prove that the only other subvarieties with this property are the variety $\mathcal L^{\rm Com}$ of commutative semi-Heyting algebras and the variety $\mathcal L^{\vee}$ generated by the chains in which a?<?b implies a → b?=?b. We also study the variety $\mathcal C$ generated within $\mathcal{SH}$ by $\mathcal L_{\mathcal H}$ , $\mathcal L_\vee$ and $\mathcal L_{\rm Com}$ . In particular we prove that $\mathcal C$ is locally finite and we obtain a construction of the finitely generated free algebra in $\mathcal C$ .     

{\mathbb{G}_a^ M} degeneration of flag varieties

Let ${\mathcal{F}_\lambda}$ be a generalized flag variety of a simple Lie group G embedded into the projectivization of an irreducible G-module V _λ . We define a flat degeneration ${\mathcal{F}_\lambda^a}$ , which is a ${\mathbb{G}^M_a}$ variety. Moreover, there exists a larger group G ^a acting on ${\mathcal{F}_\lambda^a}$ , which is a degeneration of the group G. The group G ^a contains ${\mathbb{G}^M_a}$ as a normal subgroup. If G is of type A, then the degenerate flag varieties can be embedde‘d into the product of Grassmannians and thus to the product of projective spaces. The defining ideal of ${\mathcal{F}_\lambda}$ is generated by the set of degenerate Plücker relations. We prove that the coordinate ring of ${\mathcal{F}_\lambda^a}$ is isomorphic to a direct sum of dual PBW-graded ${\mathfrak{g}}$ -modules. We also prove that there exists bases in multi-homogeneous components of the coordinate rings, parametrized by the semistandard PBW-tableux, which are analogs of semistandard tableaux.     

Erratic behavior of CAT(0) geodesics under G-equivariant quasi-isometries

We show that, given any connected, compact space ${Z \subset \mathbb{R}^n}$ , there exists a group G acting geometrically on two CAT(0) spaces X and Y, a G-equivariant quasi-isometry ${f\colon X\rightarrow Y}$ , and a geodesic ray c in X such that the closure of f (c), intersected with ${\partial Y}$ , is homeomorphic to Z. This characterizes all homeomorphism types of ??geodesic boundary images?? that arise in this manner.     

On Zero-Sum {\mathbb{Z}_k} -Magic Labelings of 3-Regular Graphs

Let G =  (V, E) be a finite loopless graph and let (A, +) be an abelian group with identity 0. Then an A-magic labeling of G is a function ${\phi}$ from E into A ? {0} such that for some ${a \in A, \sum_{e \in E(v)} \phi(e) = a}$ for every ${v \in V}$ , where E(v) is the set of edges incident to v. If ${\phi}$ exists such that a =  0, then G is zero-sum A-magic. Let zim(G) denote the subset of ${\mathbb{N}}$ (the positive integers) such that ${1 \in zim(G)}$ if and only if G is zero-sum ${\mathbb{Z}}$ -magic and ${k \geq 2 \in zim(G)}$ if and only if G is zero-sum ${\mathbb{Z}_k}$ -magic. We establish that if G is 3-regular, then ${zim(G) = \mathbb{N} - \{2\}}$ or ${\mathbb{N} - \{2,4\}.}$      

Fractional f-Edge Cover Chromatic Index of Graphs

Let G be a graph, and let f be an integer function on V with ${1\leq f(v)\leq d(v)}$ to each vertex ${\upsilon \in V}$ . An f-edge cover coloring is a coloring of edges of E(G) such that each color appears at each vertex ${\upsilon \in V(G)}$ at least f(υ) times. The maximum number of colors needed to f-edge cover color G is called the f-edge cover chromatic index of G and denoted by ${\chi^{'}_{fc}(G)}$ . It is well known that any simple graph G has the f-edge cover chromatic index equal to δ _f (G) or δ _f (G) ? 1, where ${\delta_{f}(G)=\,min\{\lfloor \frac{d(v)}{f(v)} \rfloor: v\in V(G)\}}$ . The fractional f-edge cover chromatic index of a graph G, denoted by ${\chi^{'}_{fcf}(G)}$ , is the fractional f-matching number of the edge f-edge cover hypergraph ${\mathcal{H}}$ of G whose vertices are the edges of G and whose hyperedges are the f-edge covers of G. In this paper, we give an exact formula of ${\chi^{'}_{fcf}(G)}$ for any graph G, that is, ${\chi^{'}_{fcf}(G)=\,min \{\min\limits_{v\in V(G)}d_{f}(v), \lambda_{f}(G)\}}$ , where ${\lambda_{f}(G)=\min\limits_{S} \frac{|C[S]|}{\lceil (\sum\limits_{v\in S}{f(v)})/2 \rceil}}$ and the minimum is taken over all nonempty subsets S of V(G) and C[S] is the set of edges that have at least one end in S.     

Deterministic and stochastic differential inclusions with multiple surfaces of discontinuity

We consider a class of deterministic and stochastic dynamical systems with discontinuous drift f and solutions that are constrained to live in a given closed domain G in ${\mathbb{R}}^{n}$ according to a constraint vector field D(·) specified on the boundary $\partial G$ of the domain. Specifically, we consider equations of the form $\phi = \theta + \eta + u , \quad \dot{\theta}(t) \in F(\phi(t)), \quad \mbox{a.e. } t$ for u in an appropriate class of functions, where η is the “constraining term” in the Skorokhod problem specified by (G, D) and F is the set-valued upper semicontinuous envelope of f. The case $G ={\mathbb{R}}^{n}$ (when there is no constraining mechanism) and u is absolutely continuous corresponds to the well known setting of differential inclusions (DI). We provide a general sufficient condition for uniqueness of solutions and Lipschitz continuity of the solution map, in the form of existence of a Lyapunov set. Here we assume (i) G is convex and admits the representation $G=\cup_i\overline{C_i}$ , where $\{C_i,i\in {\mathbb{I}}\}$ is a finite collection of disjoint, open, convex, polyhedral cones in ${\mathbb{R}}^{n}$ , each having its vertex at the origin; (ii) f =  b +  f ^c is a vector field defined on G such that b assumes a constant value on each of the given cones and f ^c is Lipschitz continuous on G; and (iii) D is an upper semicontinuous, cone-valued vector field that is constant on each face of ?G. We also provide existence results under much weaker conditions (where no Lyapunov set condition is imposed). For stochastic differential equations (SDE) (possibly, reflected) that have drift coefficient f and a Lipschitz continuous (possibly degenerate) diffusion coefficient, we establish strong existence and pathwise uniqueness under appropriate conditions. Our approach yields new existence and uniqueness results for both DI and SDE even in the case $G = {\mathbb{R}}^{n}.$ The work has applications in the study of scaling limits of stochastic networks.     

MULTIPLICITY-FREE PRIMITIVE IDEALS ASSOCIATED WITH RIGID NILPOTENT ORBITS

Let G be a simple algebraic group defined over ?. Let e be a nilpotent element in $ \mathfrak{g} $ = Lie(G) and denote by U ( $ \mathfrak{g} $ , e) the finite W-algebra associated with the pair ( $ \mathfrak{g} $ , e). It is known that the component group Γ of the centraliser of e in G acts on the set ? of all one-dimensional representations of U ( $ \mathfrak{g} $ , e). In this paper we prove that the fixed point set ?^Γ is non-empty. As a corollary, all finite W-algebras associated with $ \mathfrak{g} $ admit one-dimensional representations. In the case of rigid nilpotent elements in exceptional Lie algebras we find irreducible highest weight $ \mathfrak{g} $ -modules whose annihilators in U ( $ \mathfrak{g} $ ) come from one-dimensional representations of U ( $ \mathfrak{g} $ , e) via Skryabin’s equivalence. As a consequence, we show that for any nilpotent orbit $ \mathcal{O} $ in $ \mathfrak{g} $ there exists a multiplicity-free (and hence completely prime) primitive ideal of U ( $ \mathfrak{g} $ ) whose associated variety equals the Zariski closure of $ \mathcal{O} $ in $ \mathfrak{g} $ .     

On a class of smooth Frechet subalgebras of C * -algebras

The paper contributes to understanding the differential structure in a C ^*-algebra. Refining the Banach $(D_p^*)$ -algebras investigated by Kissin and Shulman as noncommutative analogues of the algebra C ^p [a,b] of p-times continuously differentiable functions, we investigate a Frechet $(D_\infty^*)$ -subalgebra $\ensuremath{{\mathcal B}}$ of a C ^*-algebra as a noncommutative analogue of the algebra C ^?∞?[a,b] of smooth functions. Regularity properties like spectral invariance, closure under functional calculi and domain invariance of homomorphisms are derived expressing $\ensuremath{{\mathcal B}}$ as an inverse limit over n of Banach $(D^*_n)$ -algebras. Several examples of such smooth algebras are exhibited.     

The Roles Played by Order of Convexity or Starlikeness and the Bloch Condition in the Extension of Mappings from the Disk to the Ball

Let ${G: \mathbb {C}^{n-1} \rightarrow \mathbb {C}}$ be holomorphic such that G(0)?=?0 and DG(0)?=?0. When f is a convex (resp. starlike) normalized (f(0)?=?0, f??(0)?=?1) univalent mapping of the unit disk ${\mathbb {D}}$ in ${\mathbb {C}}$ , then the extension of f to the Euclidean unit ball ${\mathbb {B}}$ in ${\mathbb {C}^n}$ given by ${\Phi_G(f)(z)=(f(z_1)+G(\sqrt{f^{\prime}(z_1)} \, \hat{z}),\sqrt{f^{\prime}(z_1)}\, \hat{z})}$ , ${\hat{z}=(z_2,\dots,z_n) \in \mathbb {C}^{n-1}}$ , is known to be convex (resp. starlike) if G is a homogeneous polynomial of degree 2 with sufficiently small norm. Conversely, it is known that G cannot have terms of degree greater than 2 in its expansion about 0 in order for ${\Phi_G(f)}$ to be convex (resp. starlike), in general. We examine whether the restriction that f be either convex or starlike of a certain order ${\alpha \in (0,1]}$ allows, in general, for G to contain terms of degree greater than 2 and still have ${\Phi_G(f)}$ maintain the respective geometric property. Related extension results for convex and starlike Bloch mappings are also given.     

The Majorization Theorem for Signless Laplacian Spectral Radii of Connected Graphs

Let ${\pi=(d_{1},d_{2},\ldots,d_{n})}$ and ${\pi'=(d'_{1},d'_{2},\ldots,d'_{n})}$ be two non-increasing degree sequences. We say ${\pi}$ is majorizated by ${\pi'}$ , denoted by ${\pi \vartriangleleft \pi'}$ , if and only if ${\pi\neq \pi'}$ , ${\sum_{i=1}^{n}d_{i}=\sum_{i=1}^{n}d'_{i}}$ , and ${\sum_{i=1}^{j}d_{i}\leq\sum_{i=1}^{j}d'_{i}}$ for all ${j=1,2,\ldots,n}$ . If there exists one connected graph G with ${\pi}$ as its degree sequence and ${c=(\sum_{i=1}^{n}d_{i})/2-n+1}$ , then G is called a c-cyclic graph and ${\pi}$ is called a c-cyclic degree sequence. Suppose ${\pi}$ is a non-increasing c-cyclic degree sequence and ${\pi'}$ is a non-increasing graphic degree sequence, if ${\pi \vartriangleleft \pi'}$ and there exists some t ${(2\leq t\leq n)}$ such that ${d'_{t}\geq c+1}$ and ${d_{i}=d'_{i}}$ for all ${t+1\leq i\leq n}$ , then the majorization ${\pi \vartriangleleft \pi'}$ is called a normal majorization. Let μ(G) be the signless Laplacian spectral radius, i.e., the largest eigenvalue of the signless Laplacian matrix of G. We use C _π to denote the class of connected graphs with degree sequence π. If ${G \in C_{\pi}}$ and ${\mu(G)\geq \mu(G')}$ for any other ${G'\in C_{\pi}}$ , then we say G has greatest signless Laplacian radius in C _π . In this paper, we prove that: Let π and π′ be two different non-increasing c-cyclic (c ≥ 0) degree sequences, G and G′ be the connected c-cyclic graphs with greatest signless Laplacian spectral radii in C _π and C _π', respectively. If ${\pi \vartriangleleft \pi'}$ and it is a normal majorization, then ${\mu(G) < \mu(G')}$ . This result extends the main result of Zhang (Discrete Math 308:3143–3150, 2008).     

Rainbow Connection Number, Bridges and Radius

Let G be a connected graph. The notion of rainbow connection number rc(G) of a graph G was introduced by Chartrand et al. (Math Bohem 133:85–98, 2008). Basavaraju et al. (arXiv:1011.0620v1 [math.CO], 2010) proved that for every bridgeless graph G with radius r, ${rc(G)\leq r(r+2)}$ and the bound is tight. In this paper, we show that for a connected graph G with radius r and center vertex u, if we let D ^r  = {u}, then G has r?1 connected dominating sets ${ D^{r-1}, D^{r-2},\ldots, D^{1}}$ such that ${D^{r} \subset D^{r-1} \subset D^{r-2} \cdots\subset D^{1} \subset D^{0}=V(G)}$ and ${rc(G)\leq \sum_{i=1}^{r} \max \{2i+1,b_i\}}$ , where b _i is the number of bridges in E[D ⁱ , N(D ⁱ )] for ${1\leq i \leq r}$ . From the result, we can get that if ${b_i\leq 2i+1}$ for all ${1\leq i\leq r}$ , then ${rc(G)\leq \sum_{i=1}^{r}(2i+1)= r(r+2)}$ ; if b _i  > 2i + 1 for all ${1\leq i\leq r}$ , then ${rc(G)= \sum_{i=1}^{r}b_i}$ , the number of bridges of G. This generalizes the result of Basavaraju et al. In addition, an example is given to show that there exist infinitely graphs with bridges whose rc(G) is only dependent on the radius of G, and another example is given to show that there exist infinitely graphs with bridges whose rc(G) is only dependent on the number of bridges in G.     

Nielsen coincidence,fixed point and root theories of n-valued maps

Let ${(\phi, \psi)}$ be an (m, n)-valued pair of maps ${\phi, \psi : X \multimap Y}$ , where ${\phi}$ is an m-valued map and ${\psi}$ is n-valued, on connected finite polyhedra. A point ${x \in X}$ is a coincidence point of ${\phi}$ and ${\psi}$ if ${\phi(x) \cap \psi(x) \neq \emptyset}$ . We define a Nielsen coincidence number ${N(\phi : \psi)}$ which is a lower bound for the number of coincidence points of all (m, n)-valued pairs of maps homotopic to ${(\phi, \psi)}$ . We calculate ${N(\phi : \psi)}$ for all (m, n)-valued pairs of maps of the circle and show that ${N(\phi : \psi)}$ is a sharp lower bound in that setting. Specifically, if ${\phi}$ is of degree a and ${\psi}$ of degree b, then ${N(\phi : \psi) = \frac{|an - bm|}{\langle m, n \rangle}}$ , where ${\langle m, n \rangle}$ is the greatest common divisor of m and n. In order to carry out the calculation, we obtain results, of independent interest, for n-valued maps of compact connected Lie groups that relate the Nielsen fixed point number of Helga Schirmer to the Nielsen root number of Michael Brown.     

Generalized Complex Spherical Harmonics, Frame Functions, and Gleason Theorem

Consider a finite dimensional complex Hilbert space ${\mathcal{H}}$ , with ${dim(\mathcal{H}) \geq 3}$ , define ${\mathbb{S}(\mathcal{H}):= \{x\in \mathcal{H} \:|\: \|x\|=1\}}$ , and let ${\nu_\mathcal{H}}$ be the unique regular Borel positive measure invariant under the action of the unitary operators in ${\mathcal{H}}$ , with ${\nu_\mathcal{H}(\mathbb{S}(\mathcal{H}))=1}$ . We prove that if a complex frame function ${f : \mathbb{S}(\mathcal{H})\to \mathbb{C}}$ satisfies ${f \in \mathbb{L}^2(\mathbb{S}(\mathcal{H}), \nu_\mathcal{H})}$ , then it verifies Gleason’s statement: there is a unique linear operator ${A: \mathcal{H} \to \mathcal{H}}$ such that ${f(u) = \langle u| A u\rangle}$ for every ${u \in \mathbb{S}(\mathcal{H}).\,A}$ is Hermitean when f is real. No boundedness requirement is thus assumed on f a priori.     

The Signed Edge-Domatic Number of a Graph

For a nonempty graph G = (V, E), a signed edge-domination of G is a function ${f: E(G) \to \{1,-1\}}$ such that ${\sum_{e'\in N_{G}[e]}{f(e')} \geq 1}$ for each ${e \in E(G)}$ . The signed edge-domatic number of G is the largest integer d for which there is a set {f ₁,f ₂, . . . , f _d } of signed edge-dominations of G such that ${\sum_{i=1}^{d}{f_i(e)} \leq 1}$ for every ${e \in E(G)}$ . This paper gives an original study on this concept and determines exact values for some special classes of graphs, such as paths, cycles, stars, fans, grids, and complete graphs with even order.     

Singular Solutions of the Generalized Dhombres Functional Equation

We consider singular solutions of the functional equation ${f(xf(x)) = \varphi (f(x))}$ where ${\varphi}$ is a given and f an unknown continuous map ${\mathbb R_{+} \rightarrow \mathbb R_{+}}$ . A solution f is regular if the sets ${R_f \cap (0, 1]}$ and ${R_f \cap [1, \infty)}$ , where R _f is the range of f, are ${\varphi}$ -invariant; otherwise f is singular. We show that for singular solutions the associated dynamical system ${({R_f}, \varphi|_{R_f})}$ can have strange properties unknown for the regular solutions. In particular, we show that ${\varphi |_{R_f}}$ can have a periodic point of period 3 and hence can be chaotic in a strong sense. We also provide an effective method of construction of singular solutions.     

The geometric meaning of Zhelobenko operators

Let $ \mathfrak{g} $ be the complex semisimple Lie algebra associated to a complex semisimple algebraic group G, $ \mathfrak{b} $ a Borel subalgebra of $ \mathfrak{g} $ , $ \mathfrak{h}\subset \mathfrak{b} $ the Cartan sublagebra, and N ? G the unipotent subgroup corresponding to the nilradical $ \mathfrak{n}\subset \mathfrak{b} $ . We show that the explicit formula for the extremal projection operator for $ \mathfrak{g} $ obtained by Asherova, Smirnov, and Tolstoy and similar formulas for Zhelobenko operators are related to the existence of a birational equivalence $ N\times \mathfrak{h}\to \mathfrak{b} $ given by the restriction of the adjoint action. Simple geometric proofs of formulas for the “classical” counterparts of the extremal projection operator and of Zhelobenko operators are also obtained.