Total curvature and L 2 harmonic 1-forms on complete submanifolds in space forms

Let M ⁿ be an n-dimensional complete noncompact oriented submanifold with finite total curvature, i.e., ${\int_M(|A|^2-n|H|^2)^{\frac n2} < \infty}$ , in an (n + p)-dimensional simply connected space form N ^n+p (c) of constant curvature c, where |H| and |A|² are the mean curvature and the squared length of the second fundamental form of M, respectively. We prove that if M satisfies one of the following: (i) n ≥ 3, c = 0 and ${\int_M|H|^n < \infty}$ ; (ii) n ≥ 5, c = ?1 and ${|H| < 1-\frac{2}{\sqrt n}}$ ; (iii) n ≥ 3, c = 1 and |H| is bounded, then the dimension of the space of L ² harmonic 1-forms on M is finite. Moreover, in the case of (i) or (ii), M must have finitely many ends.     

On finite groups with exactly seven element centralizers

For a finite groupG, #Cent(G) denotes the number of centralizers of its elements. A groupG is calledn-centralizer if #Cent(G) =n, and primitive n-centralizer if $\# Cent(G) = \# Cent\left( {\frac{G}{{Z(G)}}} \right) = n$ . The first author in [1], characterized the primitive 6-centralizer finite groups. In this paper we continue this problem and characterize the primitive 7-centralizer finite groups. We prove that a finite groupG is primitive 7-centralizer if and only if $\frac{G}{{Z(G)}} \cong D_{10} $ orR, whereR is the semidirect product of a cyclic group of order 5 by a cyclic group of order 4 acting faithfully. Also, we compute#Cent(G) for some finite groups, using the structure ofG modulu its center.     

Lower bounds on the number of maximal subgroups in a finite group

For a finite group G, let m(G) denote the set of maximal subgroups of G and π(G) denote the set of primes which divide |G|. When G is a cyclic group, an elementary calculation proves that |m(G)| = |π(G)|. In this paper, we prove lower bounds on |m(G)| when G is not cyclic. In general, ${|m(G)| \geq |\pi(G)|+p}$ | m ( G ) | ≥ | π ( G ) | + p , where ${p \in \pi(G)}$ p ∈ π ( G ) is the smallest prime that divides |G|. If G has a noncyclic Sylow subgroup and ${q \in \pi(G)}$ q ∈ π ( G ) is the smallest prime such that ${Q \in {\rm syl}_q(G)}$ Q ∈ syl q ( G ) is noncyclic, then ${|m(G)| \geq |\pi(G)|+q}$ | m ( G ) | ≥ | π ( G ) | + q . Both lower bounds are best possible.     

Bounds on the 2-Rainbow Domination Number of Graphs

A 2-rainbow domination function of a graph G is a function f that assigns to each vertex a set of colors chosen from the set {1, 2}, such that for any ${v\in V(G), f(v)=\emptyset}$ implies ${\bigcup_{u\in N(v)}f(u)=\{1,2\}.}$ The 2-rainbow domination number γ _r2(G) of a graph G is the minimum ${w(f)=\Sigma_{v\in V}|f(v)|}$ over all such functions f. Let G be a connected graph of order |V(G)| = n ≥ 3. We prove that γ _r2(G) ≤ 3n/4 and we characterize the graphs achieving equality. We also prove a lower bound for 2-rainbow domination number of a tree using its domination number. Some other lower and upper bounds of γ _r2(G) in terms of diameter are also given.     

Oscillations of difference equations with several deviated arguments

We find the solutions ${f,g,h \colon G \to X,\,\varphi \colon G \to \mathbb{K}}$ of each of the functional equation $$\sum\limits_{\lambda \in K} f(x + \lambda y) = |K| \varphi (y) g(x) + |K|h(y), \quad x, y \in G$$ , where (G, + ) is an abelian group, K is a finite, abelian subgroup of the automorphism group of G, X is a linear space over the field ${\mathbb{K} \in \{ \mathbb{R},\mathbb{C}\}}$ .     

On an inverse problem to Frobenius’ theorem

Let G be a finite group and e a positive integer dividing |G|, the order of G. Denoting \({L_e(G)=\{x\in G|\,x^e=1\}}\) , Frobenius proved that |L _e (G)| = ke for a positive integer k ≥ 1. In this paper, we give a complete classification of finite groups G with |L _e (G)| ≤ 2e for every e dividing |G|.     

The limitations of the Poincaré inequality for Grušin type operators

We examine the validity of the Poincaré inequality for degenerate, second-order, elliptic operators H in divergence form on \({L_2(\mathbf{R}^{n}\times \mathbf{R}^{m})}\) . We assume the coefficients are real symmetric and \({a_1H_\delta\geq H\geq a_2H_\delta}\) for some \({a_1,a_2>0}\) where H _δ is a generalized Gru?in operator, $$H_\delta=-\nabla_{x_1}\,|x_1|^{\left(2\delta_1,2\delta_1'\right)} \,\nabla_{x_1}-|x_1|^{\left(2\delta_2,2\delta_2'\right)} \,\nabla_{x_2}^2.$$ Here \({x_1 \in \mathbf{R}^n,\; x_2 \in \mathbf{R}^m,\;\delta_1,\delta_1'\in[0,1\rangle,\;\delta_2,\delta_2'\geq0}\) and \({|x_1|^{\left(2\delta,2\delta'\right)}=|x_1|^{2\delta}}\) if \({|x_1|\leq 1}\) and \({|x_1|^{\left(2\delta,2\delta'\right)}=|x_1|^{2\delta'}}\) if \({|x_1|\geq 1}\) . We prove that the Poincaré inequality, formulated in terms of the geometry corresponding to the control distance of H, is valid if n ≥ 2, or if n = 1 and \({\delta_1\vee\delta_1'\in[0,1/2\rangle}\) but it fails if n = 1 and \({\delta_1\vee\delta_1'\in[1/2,1\rangle}\) . The failure is caused by the leading term. If \({\delta_1\in[1/2, 1\rangle}\) , it is an effect of the local degeneracy \({|x_1|^{2\delta_1}}\) , but if \({\delta_1\in[0, 1/2\rangle}\) and \({\delta_1'\in [1/2,1\rangle}\) , it is an effect of the growth at infinity of \({|x_1|^{2\delta_1'}}\) . If n = 1 and \({\delta_1\in[1/2, 1\rangle}\) , then the semigroup S generated by the Friedrichs’ extension of H is not ergodic. The subspaces \({x_1\geq 0}\) and \({x_1\leq 0}\) are S-invariant, and the Poincaré inequality is valid on each of these subspaces. If, however, \({n=1,\; \delta_1\in[0, 1/2\rangle}\) and \({\delta_1'\in [1/2,1\rangle}\) , then the semigroup S is ergodic, but the Poincaré inequality is only valid locally. Finally, we discuss the implication of these results for the Gaussian and non-Gaussian behaviour of the semigroup S.     

The Monotone Catenary Degree of Krull Monoids

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor. The monotone catenary degree c _mon (H) of H is the smallest integer m with the following property: for each ${a \in H}$ and each two factorizations z, z′ of a with length |z| ≤  |z′|, there exist factorizations z = z ₀, ... ,z _k  = z′ of a with increasing lengths—that is, |z ₀| ≤  ... ≤  |z _k |—such that, for each ${i \in [1,k]}$ , z _i arises from z _i-1 by replacing at most m atoms from z _i-1 by at most m new atoms. Up to now there was only an abstract finiteness result for c _mon (H), but the present paper offers the first explicit upper and lower bounds for c _mon (H) in terms of the group invariants of G.     

On the Schröder equation and iterative sequences of C r diffeomorphisms in {\mathbb{R}^{N}} space

Let ${U \subset \mathbb{R}^{N}}$ be a neighbourhood of the origin and a function ${F:U\rightarrow U}$ be of class C ^r , r ≥ 2, F(0) = 0. Denote by F ⁿ the n-th iterate of F and let ${0<|s_1|\leq \cdots \leq|s_N| <1 }$ , where ${s_1, \ldots , s_N}$ are the eigenvalues of dF(0). Assume that the Schröder equation ${\varphi(F(x))=S\varphi(x)}$ , where S: = dF(0) has a C ² solution φ such that dφ(0) = id. If ${\frac{log|s_1|}{log|s_N|} <2 }$ then the sequence {S ^?n F ⁿ (x)} converges for every point x from the basin of attraction of F to a C ² solution φ of (1). If ${2\leq\frac{log|s_1|}{log|s_N|} }$ then this sequence can be diverging. In this case we give some sufficient conditions for the convergence and divergence of the sequence {S ^?n F ⁿ (x)}. Moreover, we show that if F is of class C ^r and ${r>\big[\frac{log|s_1|}{log|s_N|} \big ]:=p \geq 2}$ then every C ^r solution of the Schröder equation such that dφ(0) = id is given by the formula $$\begin{array}{ll}\varphi (x)={\lim\limits_{n \rightarrow \infty}} (S^{-n}F^n(x) + {\sum\limits _{k=2}^{p}} S^{-n}L_k (F^n(x))),\end{array}$$ where ${L_k:\mathbb{R}^{N} \rightarrow \mathbb{R}^{N}}$ are some homogeneous polynomials of degree k, which are determined by the differentials d ^(j) F(0) for 1 < j ≤  p.     

Characterizations of Strength Extremal Graphs

With graphs considered as natural models for many network design problems, edge connectivity κ′(G) and maximum number of edge-disjoint spanning trees τ(G) of a graph G have been used as measures for reliability and strength in communication networks modeled as graph G (see Cunningham, in J ACM 32:549–561, 1985; Matula, in Proceedings of 28th Symposium Foundations of Computer Science, pp 249–251, 1987, among others). Mader (Math Ann 191:21–28, 1971) and Matula (J Appl Math 22:459–480, 1972) introduced the maximum subgraph edge connectivity \({\overline{\kappa'}(G) = {\rm max} \{\kappa'(H) : H {\rm is} \, {\rm a} \, {\rm subgraph} \, {\rm of} G \}}\) . Motivated by their applications in network design and by the established inequalities $$\overline{\kappa'}(G) \ge \kappa'(G) \ge \tau(G),$$ we present the following in this paper:

1. For each integer k > 0, a characterization for graphs G with the property that \({\overline{\kappa'}(G) \le k}\) but for any edge e not in G, \({\overline{\kappa'}(G + e) \ge k+1}\) .
2. For any integer n > 0, a characterization for graphs G with |V(G)| = n such that κ′(G) = τ(G) with |E(G)| minimized.

     

On the Edge Connectivity of Direct Products with Dense Graphs

Let ${\kappa^{\prime}(G)}$ be the edge connectivity of G and G × H the direct product of G and H. Let H be any graph with minimal degree ${\delta(H)>|V(H)|/2}$ . We prove that for any graph ${G, \kappa^{\prime}(G\times H)=\textup{min}\{2\kappa^{\prime}(G)|E(H)|,\delta(G)\delta(H)\}}$ . In addition, the structure of minimum edge cuts is described. As an application, we present a necessary and sufficient condition for G × K _n (n ≥ 3) to be super edge connected.     

On reconstructing separable reducedp-groups with a given socle

Let \(\bar B^* \) be a separable reduced (abelian)p-group which is torsion complete. We ask whether for \(G \subseteq \bar B^* \) there is \(H \subseteq _{pr} \bar B^* ,H[p] = G[p]\) ,H[p]=G[p],H not isomorphic toG. IfG is the sum of cyclic groups or is torsion complete, the answer is easily no. For otherG, we prove that the answer is yes assuming G.C.H. Even without G.C.H. the answer is yes if the density character ofG is equal to Min _n<ω|p ⁿG|, i.e., $$\mathop {Min}\limits_{n< \omega } |p^n G| = \mathop {Min}\limits_m \mathop \Sigma \limits_{n > m} |(p^n G)[p]/(p^{n + 1} G)[p]|$$ Of course, instead of two non-isomorphic we can get many, but we do not deal much with this.     

One-radius results for supermedian functions on {\mathbb R^d} , d ≤ 2

A classical result states that every lower bounded superharmonic function on ${\mathbb{R}^{2}}$ is constant. In this paper the following (stronger) one-circle version is proven. If ${f : \mathbb{R}^{2} \to (-\infty,\infty]}$ is lower semicontinuous, lim inf_|x|→∞ f (x)/ ln |x| ≥ 0, and, for every ${x \in \mathbb{R}^{2}}$ , ${1/(2\pi) \int_0^{2\pi} f(x + r(x)e^{it}) \, dt \le f(x)}$ , where ${r : \mathbb{R}^{2} \to (0,\infty)}$ is continuous, ${{\rm sup}_{x \in \mathbb{R}^{2}} (r(x) - |x|) < \infty},$ , and ${{\rm inf}_{x \in \mathbb{R}^{2}} (r(x)-|x|)=-\infty}$ , then f is constant. Moreover, it is shown that, assuming r ≤ c| · | + M on ${\mathbb{R}^d}$ , d ≤ 2, and taking averages on ${\{y \in \mathbb{R}^{d} : |y-x| \le r(x)\}}$ , such a result of Liouville type holds for supermedian functions if and only if c ≤ c ₀, where c ₀ = 1, if d = 2, whereas 2.50 < c ₀ < 2.51, if d = 1.     

A Berry-Esseen type inequality for convex bodies with an unconditional basis

Suppose X = (X ₁, . . . , X _n ) is a random vector, distributed uniformly in a convex body ${K \subset \mathbb R^n}$ . We assume the normalization ${\mathbb E X_i^2 = 1}$ for i = 1, . . . , n. The body K is further required to be invariant under coordinate reflections, that is, we assume that (±X ₁, . . . , ±X _n ) has the same distribution as (X ₁, . . . , X _n ) for any choice of signs. Then, we show that $$ \mathbb E \left( \, |X| - \sqrt{n} \, \right)^2 \leq C^2,$$ where C  ≤  4 is a positive universal constant, and | · | is the standard Euclidean norm in ${\mathbb R^n}$ . The estimate is tight, up to the value of the constant. It leads to a Berry-Esseen type bound in the central limit theorem for unconditional convex bodies.     

Asymptotics for a nonlinear integral equation with a generalized heat kernel

This paper is concerned with a nonlinear integral equation $$(P)\qquad u(x, t)=\int_{{\bf R}^N}G(x-y, t)\varphi(y){d}y+\int_0^t \int_{{\bf R}^N}G(x-y, t-s)f(y, s:u){d}y{d}s, \quad $$ where N ≥  1, \({\varphi \in L^\infty({\bf R}^N) \cap L^1({\bf R}^N,(1+|x|^K){d}x)}\) for some K ≥  0. Here, G = G(x,t) is a generalization of the heat kernel. We are interested in the asymptotic expansions of the solution of (P) behaving like a multiple of the integral kernel G as \({t \to \infty}\) .     

Multiplicity of solutions to a nonlinear elliptic problem with nonlinear boundary conditions

We study the problem $$\left\{\begin{array}{ll}\Delta_p u = |u|^{q-2}u, & \quad x \in \Omega ,\\ |\nabla u|^{p-2} \frac{\partial u}{\partial \nu}= \lambda |u|^{p-2}u, &\quad x \in \partial \Omega, \end{array}\right.$$ where \({\Omega \subset \mathbb{R}^N}\) is a bounded smooth domain, \({\nu}\) is the outward unit normal at \({\partial \Omega}\) and \({\lambda > 0}\) is regarded as a bifurcation parameter. When p = 2 and in the superlinear regime q > 2, we show existence of n nontrivial solutions for all \({\lambda > \lambda_n}\) , \({\lambda_n}\) being the n-th Steklov eigenvalue. It is proved in addition that bifurcation from the trivial solution takes place at all \({\lambda_n}\) ’s. Similar results are obtained in the sublinear case 1 < q < 2. In this case, bifurcation from infinity takes place in those \({\lambda_n}\) with odd multiplicity. Partial extensions of these features are shown in the nonlinear diffusion case \({p \neq 2}\) and related problems under spatially heterogeneous reactions are also addressed.     

Multiplicity of periodic solutions to symmetric delay differential equations

By applying the method based on the usage of the equivariant gradient degree introduced by G?ba (1997) and the cohomological equivariant Conley index introduced by Izydorek (2001), we establish an abstract result for G-invariant strongly indefinite asymptotically linear functionals showing that the equivariant invariant ${\omega(\nabla \Phi)}$ , expressed as that difference of the G-gradient degrees at infinity and zero, contains rich numerical information indicating the existence of multiple critical points of ${\Phi}$ exhibiting various symmetric properties. The obtained results are applied to investigate an asymptotically linear delay differential equation $$x\prime = - \nabla f \big ({x \big (t - \frac{\pi}{2} \big )} \big ), \quad x \in V \qquad \quad (*)$$ (here ${f : V \rightarrow \mathbb{R}}$ is a continuously differentiable function satisfying additional assumptions) with Γ-symmetries (where Γ is a finite group) using a variational method introduced by Guo and Yu (2005). The equivariant invariant ${\omega(\nabla \Phi) = n_{1}({\bf H}_{1}) + n_{2}({\bf H}_{2}) + \cdots + n_{m}({\bf H}_{m})}$ in the case n _k ≠ 0 (for maximal twisted orbit types (H _k )) guarantees the existence of at least |n _k | different G-orbits of periodic solutions with symmetries at least (H _k). This result generalizes the result by Guo and Yu (2005) obtained in the case without symmetries. The existence of large number of nonconstant periodic solutions for (*) (classified according to their symmetric properties) is established for several groups Γ, with the exact value of ${\omega(\,\nabla \Phi)}$ evaluated.     

On a class of singular elliptic problems with the perturbed Hardy–Sobolev operator

In this paper, firstly, we investigate a class of singular eigenvalue problems with the perturbed Hardy–Sobolev operator, and obtain some properties of the eigenvalues and the eigenfunctions. (i.e. existence, simplicity, isolation and comparison results). Secondly, applying these properties of eigenvalue problem, and the linking theorem for two symmetric cones in Banach space, we discuss the following singular elliptic problem $$\left\{\begin{array}{ll}-\Delta_{p}u-a(x)\frac{|u|^{p-2}u}{|x|^{p}}= \lambda \eta(x)|u|^{p-2}u+ f(x,u) \quad x \in \Omega, \\ u =0 \quad\quad\quad\quad\quad\quad\quad x\in\partial \Omega, \end{array} \right.$$ where ${a(x)=(\frac{n-p}{p})^{p}q(x),}$ if 1 < p < n, ${a(x)=(\frac{n-1}{n})^{n} \frac{q(x)}{({\rm log}\frac{R}{|x|})^{n}},}$ if p = n, and prove the existence of a nontrivial weak solution for any ${\lambda \in \mathbb{R}.}$      

Finite-state self-similar actions of nilpotent groups

Let G be a finitely generated torsion-free nilpotent group and ${\phi:H\rightarrow G}$ be a surjective homomorphism from a subgroup H < G of finite index with trivial ${\phi}$ -core. For every choice of coset representatives of H in G there is a faithful self-similar action of the group G associated with ${(G, \phi)}$ . We are interested in what cases all these actions are finite-state and in what cases there exists a finite-state self-similar action for ${(G, \phi)}$ . These two properties are characterized in terms of the Jordan normal form of the corresponding automorphism $\widehat{\phi}$ of the Lie algebra of the Mal’cev completion of G.     

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.