The classification of (m, \rho)-quasi-Einstein manifolds

If there exist a smooth function f on $(M^n, g)$ and three real constants $m,\rho ,\lambda $ ( $0<m\le \infty $ ) such that $$\begin{aligned} R_{ij}+f_{ij}-\frac{1}{m}f_if_j=(\rho R+\lambda ) g_{ij}, \end{aligned}$$ we call $(M^n,g)$ a $(m,\rho )$ -quasi-Einstein manifold. Here $R_{ij}$ is the Ricci curvature and R is the scalar curvature of the metric g, respectively. This is a special case of the so-called generalized quasi-Einstein manifold which was a natural generalization of gradient Ricci solitons associated with the Hamilton’s Ricci flow. In this paper, we first obtain some rigidity results for compact $(m,\rho )$ -quasi-Einstein manifolds. Then, we give some classifications under the assumption that the Bach tensor of $(M^n,g)$ is flat.     

Norm Estimates of the Pre-Schwarzian Derivatives for \alpha -Spiral-Like Functions of Order \rho

For a constant $\alpha \in (-\frac{\pi }{2},\frac{\pi }{2})$ and $0\!\le \!\rho \!<\!1,$ we define the set of all $\alpha $ -spiral-like functions of order $\rho $ consisting of functions $f$ that are univalent on the unit disk and satisfy the condition $ Re\left(e^{-i\alpha }\frac{zf^{\prime }(z)}{f(z)}\right)>\rho \cos \alpha $ for any point $z$ in the unit disk. In the present paper, we shall give the best estimate for the norm of the pre-Schwarzian derivative ${\text{ T}}_f(z)=f^{\prime \prime }(z)/f^{\prime }(z)$ where $||T_f||= \sup (1-|z|^2)|T_f(z)|$ .     

Some graftings of complex projective structures with Schottky holonomy

Let ${\mathcal{G} ^{*}(S, \rho)}$ be the graph whose vertices are marked complex projective structures with holonomy ${\rho}$ and whose edges are graftings from one vertex to another. If ${\rho}$ is quasi-Fuchsian, a theorem of Goldman implies that ${\mathcal{G} ^{*}(S, \rho)}$ is connected. If ${\rho ( \pi _{1}(S))}$ is a Schottky group Baba has shown that ${\mathcal{G}(S, \rho)}$ (the corresponding graph for unmarked structures) is connected. For the case that ${\rho ( \pi _{1}(S))}$ is a Schottky group, this paper provides formulae for the composition of graftings in a basic setting. Using these formulae, one can construct an infinite number of (standard) projective structures which can be grafted to a common structure. Furthermore, one can construct pairs of projective structures which can be connected by grafting in an infinite number of ways.     

Existence of positive periodic solutions for second-order functional differential equations

In this paper, we show the existence of positive $T$ -periodic solutions of second-order functional differential equations $u^{\prime \prime }(t)-\rho ^2u(t)+\lambda g(t)f(u(t-\tau (t)))=0,\ \ t\in \mathbb R , $ where $\rho >0$ is a constant, $g\in C(\mathbb R ,[0,\infty ))$ , $\tau \in C(\mathbb R ,\mathbb R )$ are $T$ -periodic functions, $f\in C([0,\infty ),[0,\infty ))$ and $\lambda $ is a positive parameter. Our approach based on global bifurcation theorem.     

Triangles and groups via cevians

For a given triangle T and a real number ρ we define Ceva’s triangle ${\mathcal{C}_{\rho}(T)}$ to be the triangle formed by three cevians each joining a vertex of T to the point which divides the opposite side in the ratio ρ: (1 – ρ). We identify the smallest interval ${\mathbb{M}_T \subset \mathbb{R}}$ such that the family ${\mathcal{C}_{\rho}(T), \rho \in \mathbb{M}_T}$ , contains all Ceva’s triangles up to similarity. We prove that the composition of operators ${\mathcal{C}_\rho, \rho \in \mathbb{R}}$ , acting on triangles is governed by a certain group structure on ${\mathbb{R}}$ . We use this structure to prove that two triangles have the same Brocard angle if and only if a congruent copy of one of them can be recovered by sufficiently many iterations of two operators ${\mathcal{C}_\rho}$ and ${\mathcal{C}_\xi}$ acting on the other triangle.     

Toric partial density functions and stability of toric varieties

Let $(L, h)\rightarrow (X, \omega )$ denote a polarized toric Kähler manifold. Fix a toric submanifold $Y$ and denote by $\hat{\rho }_{tk}:X\rightarrow \mathbb {R}$ the partial density function corresponding to the partial Bergman kernel projecting smooth sections of $L^k$ onto holomorphic sections of $L^k$ that vanish to order at least $tk$ along $Y$ , for fixed $t>0$ such that $tk\in \mathbb {N}$ . We prove the existence of a distributional expansion of $\hat{\rho }_{tk}$ as $k\rightarrow \infty $ , including the identification of the coefficient of $k^{n-1}$ as a distribution on $X$ . This expansion is used to give a direct proof that if $\omega $ has constant scalar curvature, then $(X, L)$ must be slope semi-stable with respect to $Y$ (cf. Ross and Thomas in J Differ Geom 72(3): 429–466, 2006). Similar results are also obtained for more general partial density functions. These results have analogous applications to the study of toric K-stability of toric varieties.     

Stability results for the N-dimensional Schiffer conjecture via a perturbation method

Given a eigenvalue $\mu _{0m}^2$ of $-\Delta $ in the unit ball $B_1$ , with Neumann boundary conditions, we prove that there exists a class $\mathcal{D}$ of $C^{0,1}$ -domains, depending on $\mu _{0m} $ , such that if $u$ is a no trivial solution to the following problem $ \Delta u+\mu u=0$ in $\Omega , u=0$ on $\partial \Omega $ , and $ \int \nolimits _{\partial \Omega }\partial _{\mathbf{n}}u=0$ , with $\Omega \in \mathcal{D}$ , and $\mu =\mu _{0m}^2+o(1)$ , then $\Omega $ is a ball. Here $\mu $ is a eigenvalue of $-\Delta $ in $\Omega $ , with Neumann boundary conditions.     

Collapsibility and Vanishing of Top Homology in Random Simplicial Complexes

Let $\Delta _{n-1}$ denote the $(n-1)$ -dimensional simplex. Let $Y$ be a random $d$ -dimensional subcomplex of $\Delta _{n-1}$ obtained by starting with the full $(d-1)$ -dimensional skeleton of $\Delta _{n-1}$ and then adding each $d$ -simplex independently with probability $p=\frac{c}{n}$ . We compute an explicit constant $\gamma _d$ , with $\gamma _2 \simeq 2.45$ , $\gamma _3 \simeq 3.5$ , and $\gamma _d=\Theta (\log d)$ as $d \rightarrow \infty $ , so that for $c < \gamma _d$ such a random simplicial complex either collapses to a $(d-1)$ -dimensional subcomplex or it contains $\partial \Delta _{d+1}$ , the boundary of a $(d+1)$ -dimensional simplex. We conjecture this bound to be sharp. In addition, we show that there exists a constant $\gamma _d< c_d <d+1$ such that for any $c>c_d$ and a fixed field $\mathbb{F }$ , asymptotically almost surely $H_d(Y;\mathbb{F }) \ne 0$ .     

Central limit theorems for hyperbolic spaces and Jacobi processes on [0,\infty [

We present a unified approach to a couple of central limit theorems for the radial behavior of radial random walks on hyperbolic spaces as well as for time-homogeneous Markov chains on $[0,\infty [$ whose transition probabilities are defined in terms of Jacobi convolutions. The proofs of all central limit theorems are based on corresponding limit results for the associated Jacobi functions $\varphi _{\lambda }^{(\alpha ,\beta )}$ . In particular, we consider the limit $\alpha \rightarrow \infty $ , the limit $\varphi _{i\rho -n\lambda }^{(\alpha ,\beta )}(t/n)$ for $n\rightarrow \infty $ , and the behavior of the Jacobi function $\varphi _{i\rho -\lambda }^{(\alpha ,\beta )}(t)$ for small $\lambda $ . The proofs of all these limit results are based on the known Laplace integral representation for Jacobi functions. Parts of the results are known, other improve known ones, and other are new.     

Conditional Reducibility of Certain Unbounded Nonnegative Hamiltonian Operator Functions

Let J and ${{\mathfrak{J}}}$ be operators on a Hilbert space ${{\mathcal{H}}}$ which are both self-adjoint and unitary and satisfy ${J{\mathfrak{J}}=-{\mathfrak{J}}J}$ . We consider an operator function ${{\mathfrak{A}}}$ on [0, 1] of the form ${{\mathfrak{A}}(t)={\mathfrak{S}}+{\mathfrak{B}}(t)}$ , ${t \in [0, 1]}$ , where ${\mathfrak{S}}$ is a closed densely defined Hamiltonian ( ${={\mathfrak{J}}}$ -skew-self-adjoint) operator on ${{\mathcal{H}}}$ with ${i {\mathbb{R}} \subset \rho ({\mathfrak{S}})}$ and ${{\mathfrak{B}}}$ is a function on [0, 1] whose values are bounded operators on ${{\mathcal{H}}}$ and which is continuous in the uniform operator topology. We assume that for each ${t \in [0,1] \,{\mathfrak{A}}(t)}$ is a closed densely defined nonnegative (=J-accretive) Hamiltonian operator with ${i {\mathbb{R}} \subset \rho({\mathfrak{A}}(t))}$ . In this paper we give sufficient conditions on ${{\mathfrak{S}}}$ under which ${{\mathfrak{A}}}$ is conditionally reducible, which means that, with respect to a natural decomposition of ${{\mathcal{H}}}$ , ${{\mathfrak{A}}}$ is diagonalizable in a 2×2 block operator matrix function such that the spectra of the two operator functions on the diagonal are contained in the right and left open half planes of the complex plane. The sufficient conditions involve bounds on the resolvent of ${{\mathfrak{S}}}$ and interpolation of Hilbert spaces.     

Diophantine approximation with sign constraints

Let $\alpha $ and $\beta $ be real numbers such that $1$ , $\alpha $ and $\beta $ are linearly independent over $\mathbb {Q}$ . A classical result of Dirichlet asserts that there are infinitely many triples of integers $(x_0,x_1,x_2)$ such that $|x_0+\alpha x_1+\beta x_2| < \max \{|x_1|,|x_2|\}^{-2}$ . In 1976, Schmidt asked what can be said under the restriction that $x_1$ and $x_2$ be positive. Upon denoting by $\gamma \cong 1.618$ the golden ratio, he proved that there are triples $(x_0,x_1,x_2) \in \mathbb {Z}^3$ with $x_1,x_2>0$ for which the product $|x_0 + \alpha x_1 + \beta x_2| \max \{|x_1|,|x_2|\}^\gamma $ is arbitrarily small. Although Schmidt later conjectured that $\gamma $ can be replaced by any number smaller than $2$ , Moshchevitin proved very recently that it cannot be replaced by a number larger than $1.947$ . In this paper, we present a construction of points $(1,\alpha ,\beta )$ showing that the result of Schmidt is in fact optimal. These points also possess strong additional Diophantine properties that are described in the paper.     

Hypercentralizing generalized skew derivations on left ideals in prime rings

Let $\mathcal{R }$ be a prime ring of characteristic different from $2, \mathcal{Q }_r$ the right Martindale quotient ring of $\mathcal{R }, \mathcal{C }$ the extended centroid of $\mathcal{R }, \mathcal{I }$ a nonzero left ideal of $\mathcal{R }, F$ a nonzero generalized skew derivation of $\mathcal{R }$ with associated automorphism $\alpha $ , and $n,k \ge 1$ be fixed integers. If $[F(r^n),r^n]_k=0$ for all $r \in \mathcal{I }$ , then there exists $\lambda \in \mathcal{C }$ such that $F(x)=\lambda x$ , for all $x\in \mathcal{I }$ . More precisely one of the following holds: (1) $\alpha $ is an $X$ -inner automorphism of $\mathcal{R }$ and there exist $b,c \in \mathcal{Q }_r$ and $q$ invertible element of $\mathcal{Q }_r$ , such that $F(x)=bx-qxq^{-1}c$ , for all $x\in \mathcal{Q }_r$ . Moreover there exists $\gamma \in \mathcal{C }$ such that $\mathcal{I }(q^{-1}c-\gamma )=(0)$ and $b-\gamma q \in \mathcal{C }$ ; (2) $\alpha $ is an $X$ -outer automorphism of $\mathcal{R }$ and there exist $c \in \mathcal{Q }_r, \lambda \in \mathcal{C }$ , such that $F(x)=\lambda x-\alpha (x)c$ , for all $x\in \mathcal{Q }_r$ , with $\alpha (\mathcal{I })c=0$ .     

Asymptotics for Some Combinatorial Characteristics of the Convex Hull of a Poisson Point Process in the Clifford Torus

Let $\mathcal P _\lambda $ be a homogeneous Poisson point process of rate $\lambda $ in the Clifford torus $T^2\subset \mathbb E ^4$ . Let $(f_0, f_1, f_2, f_3)$ be the $f$ -vector of conv $\,\mathcal P _\lambda $ and let $\bar{v}$ be the mean valence of a vertex of the convex hull. Asymptotic expressions for $\mathsf E \, f_1$ , $\mathsf E \, f_2$ , $\mathsf E \, f_3$ and $\mathsf E \, \bar{v}$ as $\lambda \rightarrow \infty $ are proved in this paper.     

On the images of horodisks under holomorphic self-maps of the unit disk

Suppose that f is a holomorphic self map of the unit disk ${\mathbb{D}}$ . Recently several monotonicity results related to the image of smaller disks under f have been proved. These results extend the classical Schwarz lemma in various ways. We prove analogous monotonicity results in the context of Julia’s boundary Schwarz lemma. A horodisk is a disk internally tangent to the unit circle. For positive ${\lambda}$ , we denote by ${H_{\lambda}}$ the disk of radius ${\lambda/(1\,+\,\lambda)}$ centered at the point ${1/(1\,+\,\lambda)}$ . This is a horodisk that touches the unit circle at the point 1. Suppose that f(1) = 1 (in the sense of radial limit) and denote by ${f^{\prime}(1)}$ the angular derivative. By Julia’s lemma ${f(H_{\lambda})\,\subset H_{{\lambda}f^{\prime}(1)}}$ . Let ${\Psi_f(\lambda)\,=\,\inf\,\{\rho > 0 : f(H_{\lambda}) \subset H_\rho\}}$ . We show that the function ${\Psi_f(\lambda)/\lambda}$ is a decreasing function of ${\lambda}$ and that ${\lim_{\lambda\,\to\,0+} \Psi_f(\lambda)/\lambda = f^\prime(1)}$ . This result implies that the constant ${f^\prime(1)}$ in Julia’s lemma is the best possible.     

Geodesic Ricci solitons on unit tangent sphere bundles

In the paper, (Abbassi and Kowalski, Ann Glob Anal Geom, 38: 11–20, 2010) the authors study Einstein Riemannian $g$ natural metrics on unit tangent sphere bundles. In this study, we equip the unit tangent sphere bundle $T_1 M$ of a Riemannian manifold $(M,g)$ with an arbitrary Riemannian $g$ natural metric $\tilde{G}$ and we show that if the geodesic flow $\tilde{\xi }$ is the potential vector field of a Ricci soliton $(\tilde{G},\tilde{\xi },\lambda )$ on $T_1M,$ then $(T_1M,\tilde{G})$ is Einstein. Moreover, we show that the Reeb vector field of a contact metric manifold is an infinitesimal harmonic transformation if and only if it is Killing. Thus, we consider a natural contact metric structure $(\tilde{G}, \tilde{\eta }, \tilde{\varphi }, \tilde{\xi })$ over $T_1 M$ and we show that the geodesic flow $\tilde{\xi }$ is an infinitesimal harmonic transformation if and only if the structure $(\tilde{G}, \tilde{\eta }, \tilde{\varphi },\tilde{\xi })$ is Sasaki $\eta $ -Einstein. Consequently, we get that $(\tilde{G},\tilde{\xi }, \lambda )$ is a Ricci soliton if and only if the structure $(\tilde{G}, \tilde{\eta }, \tilde{\varphi }, \tilde{\xi })$ is Sasaki-Einstein with $\lambda = 2(n-1) >0.$ This last result gives new examples of Sasaki–Einstein structures.     

Boundaries of Disk-Like Self-affine Tiles

Let $T:= T(A, \mathcal{D })$ T : = T ( A , D ) be a disk-like self-affine tile generated by an integral expanding matrix $A$ A and a consecutive collinear digit set $\mathcal{D }$ D , and let $f(x)=x^{2}+px+q$ f ( x ) = x 2 + px + q be the characteristic polynomial of $A$ A . In the paper, we identify the boundary $\partial T$ ? T with a sofic system by constructing a neighbor graph and derive equivalent conditions for the pair $(A,\mathcal{D })$ ( A , D ) to be a number system. Moreover, by using the graph-directed construction and a device of pseudo-norm $\omega $ ω , we find the generalized Hausdorff dimension $\dim _H^{\omega } (\partial T)=2\log \rho (M)/\log |q|$ dim H ω ( ? T ) = 2 log ρ ( M ) / log | q | where $\rho (M)$ ρ ( M ) is the spectral radius of certain contact matrix $M$ M . Especially, when $A$ A is a similarity, we obtain the standard Hausdorff dimension $\dim _H (\partial T)=2\log \rho /\log |q|$ dim H ( ? T ) = 2 log ρ / log | q | where $\rho $ ρ is the largest positive zero of the cubic polynomial $x^{3}-(|p|-1)x^{2}-(|q|-|p|)x-|q|$ x 3 ? ( | p | ? 1 ) x 2 ? ( | q | ? | p | ) x ? | q | , which is simpler than the known result.