Antimatroids and Balanced Pairs

We generalize the $\frac{1}{3}$ – $\frac{2}{3}$ conjecture from partially ordered sets to antimatroids: we conjecture that any antimatroid has a pair of elements x,y such that x has probability between $\frac{1}{3}$ and $\frac{2}{3}$ of appearing earlier than y in a uniformly random basic word of the antimatroid. We prove the conjecture for antimatroids of convex dimension two (the antimatroid-theoretic analogue of partial orders of width two), for antimatroids of height two, for antimatroids with an independent element, and for the perfect elimination antimatroids and node search antimatroids of several classes of graphs. A computer search shows that the conjecture is true for all antimatroids with at most six elements.     

Invariant submanifolds of Sasakian space forms

In the present study, we consider isometric immersions ${f : M \rightarrow \tilde{M}(c)}$ of (2n + 1)-dimensional invariant submanifold M ²ⁿ⁺¹ of (2m + 1) dimensional Sasakian space form ${\tilde{M}^{2m+1}}$ of constant ${ \varphi}$ -sectional curvature c. We have shown that if f satisfies the curvature condition ${\overset{\_}{R}(X, Y) \cdot \sigma =Q(g, \sigma)}$ then either M ²ⁿ⁺¹ is totally geodesic, or ${||\sigma||^{2}=\frac{1}{3}(2c+n(c+1)),}$ or ${||\sigma||^{2}(x) > \frac{1}{3}(2c+n(c+1)}$ at some point x of M ²ⁿ⁺¹. We also prove that ${\overset{\_ }{R}(X, Y)\cdot \sigma = \frac{1}{2n}Q(S, \sigma)}$ then either M ²ⁿ⁺¹ is totally geodesic, or ${||\sigma||^{2}=-\frac{2}{3}(\frac{1}{2n}\tau -\frac{1}{2}(n+2)(c+3)+3)}$ , or ${||\sigma||^{2}(x) > -\frac{2}{3}(\frac{1}{2n} \tau (x)-\frac{1}{2} (n+2)(c+3)+3)}$ at some point x of M ²ⁿ⁺¹.     

An application of maximum principle to space-like hypersurfaces with constant mean curvature in anti-de Sitter space

In this paper, we study complete hypersurfaces with constant mean curvature in anti-de Sitter space ${H^{n+1}_1(-1)}$ . we prove that if a complete space-like hypersurface with constant mean curvature ${x:\mathbf M\rightarrow H^{n+1}_1(-1) }$ has two distinct principal curvatures ??, ??, and inf|?? ? ??|?>?0, then x is the standard embedding ${ H^{m} (-\frac{1}{r^2})\times H^{n-m} ( -\frac{1}{1 - r^2} )}$ in anti-de Sitter space ${ H^{n+1}_1 (-1) }$ .     

Isolated boundary singularities of semilinear elliptic equations

Given a smooth domain ${\Omega\subset\mathbb{R}^N}$ such that ${0 \in \partial\Omega}$ and given a nonnegative smooth function ?? on ???, we study the behavior near 0 of positive solutions of ???u?=?u ^q in ?? such that u =? ?? on ???\{0}. We prove that if ${\frac{N+1}{N-1} < q < \frac{N+2}{N-2}}$ , then ${u(x)\leq C |x|^{-\frac{2}{q-1}}}$ and we compute the limit of ${|x|^{\frac{2}{q-1}} u(x)}$ as x ?? 0. We also investigate the case ${q= \frac{N+1}{N-1}}$ . The proofs rely on the existence and uniqueness of solutions of related equations on spherical domains.     

A sharp equivalence between H ∞ functional calculus and square function estimates

Let (T _t ) _{t?≥ 0} be a bounded analytic semigroup on L ^p (Ω), with 1?<?p?<?∞. Let ?A denote its infinitesimal generator. It is known that if A and A ^* both satisfy square function estimates ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{\frac{1}{2}} T_t(x)\vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^p} \lesssim \|x\|_{L^p}}$ and ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{*\frac{1}{2}} T_t^*(y) \vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^{p^\prime}} \lesssim \|y\|_{L^{p^\prime}}}$ for ${x\in L^p(\Omega)}$ and ${y\in L^{p^\prime}(\Omega)}$ , then A admits a bounded ${H^{\infty}(\Sigma_\theta)}$ functional calculus for any ${\theta>\frac{\pi}{2}}$ . We show that this actually holds true for some ${\theta<\frac{\pi}{2}}$ .     

Symplectic mean curvature flow in CP 2

Let Σ be an immersed symplectic surface in CP ² with constant holomorphic sectional curvature k > 0. Suppose Σ evolves along the mean curvature flow in CP ². In this paper, we show that the symplectic mean curvature flow exists for long time and converges to a holomorphic curve if the initial surface satisfies ${|A|^2 \leq \lambda|H|^2 + \frac{2\lambda-1}{\lambda}k}$ and ${\cos\alpha\geq\sqrt{\frac{7\lambda-3}{3\lambda}}\left(\frac{1}{2} < \lambda\leq\frac{2}{3}\right) {\rm or} |A|^2\leq \frac{2}{3}|H|^2+\frac{4}{5}k\cos\alpha\, {\rm and} \cos\alpha\geq 1-\varepsilon}$ , for some ${\varepsilon}$ .     

Classification of hypersurfaces with constant M?bius curvature in {\mathbb{S}^{m+1}}

Let ${x: M^{m} \rightarrow \mathbb{S}^{m+1}}$ be an m-dimensional umbilic-free hypersurface in an (m?+?1)-dimensional unit sphere ${\mathbb{S}^{m+1}}$ , with standard metric I?= dx · dx. Let II be the second fundamental form of isometric immersion x. Define the positive function ${\rho=\sqrt{\frac{m}{m-1}}\|II-\frac{1}{m}tr(II)I\|}$ . Then positive definite (0,2) tensor ${\mathbf{g}=\rho^{2}I}$ is invariant under conformal transformations of ${\mathbb{S}^{m+1}}$ and is called M?bius metric. The curvature induced by the metric g is called M?bius curvature. The purpose of this paper is to classify the hypersurfaces with constant M?bius curvature.     

Extension of a theorem of Shi and Tam

In this note, we prove the following generalization of a theorem of Shi and Tam (J Differ Geom 62:79–125, 2002): Let (Ω, g) be an n-dimensional (n ≥ 3) compact Riemannian manifold, spin when n?>?7, with non-negative scalar curvature and mean convex boundary. If every boundary component Σ _i has positive scalar curvature and embeds isometrically as a mean convex star-shaped hypersurface ${{\hat \Sigma}_i \subset \mathbb{R}^n}$ , then $$ \int\limits_{\Sigma_i} H \ d \sigma \le \int\limits_{{\hat \Sigma}_i} \hat{H} \ d {\hat \sigma} $$ where H is the mean curvature of Σ _i in (Ω, g), ${\hat{H}}$ is the Euclidean mean curvature of ${{\hat \Sigma}_i}$ in ${\mathbb{R}^n}$ , and where d σ and ${d {\hat \sigma}}$ denote the respective volume forms. Moreover, equality holds for some boundary component Σ _i if, and only if, (Ω, g) is isometric to a domain in ${\mathbb{R}^n}$ . In the proof, we make use of a foliation of the exterior of the ${\hat \Sigma_i}$ ’s in ${\mathbb{R}^n}$ by the ${\frac{H}{R}}$ -flow studied by Gerhardt (J Differ Geom 32:299–314, 1990) and Urbas (Math Z 205(3):355–372, 1990). We also carefully establish the rigidity statement in low dimensions without the spin assumption that was used in Shi and Tam (J Differ Geom 62:79–125, 2002).     

Meromorphic Functions That Share One Finite Value CM or IM with Their k-th Derivative

In this paper we shall prove that if a non-constant meromorphic f and its k-th derivative f ^(k) (k ≥ 2) share the value ${a\not= 0,\infty\; CM}$ (IM) and if ${\bar{N}(r,\frac{1}{f})=S(r,f)\;\left(\bar{N}\left(r,\frac{1}{f}\right)+\bar{N}\left(r,\frac{1}{f^{(k)}}\right)=S(r,f)\right)}$ , then ${f \equiv f^{(k)}}$ . These results extend the results in Al-Khaladi (J Al-Anbar Univ Pure Sci 3:69–73, 2009).     

Divisibility of class numbers of imaginary quadratic function fields by a fixed odd number

In this paper we find a new lower bound on the number of imaginary quadratic extensions of the function field $\mathbb{F}_{q}(x)$ whose class groups have elements of a fixed odd order. More precisely, for q, a power of an odd prime, and g a fixed odd positive integer ≥?3, we show that for every ε?>?0, there are $\gg q^{L(\frac{1}{2}+\frac{3}{2(g+1)}-\epsilon)}$ polynomials $f \in \mathbb{F}_{q}[x]$ with $\deg f=L$ , for which the class group of the quadratic extension $\mathbb{F}_{q}(x, \sqrt{f})$ has an element of order g. This sharpens the previous lower bound $q^{L(\frac{1}{2}+\frac{1}{g})}$ of Ram Murty. Our result is a function field analogue which is similar to a result of Soundararajan for number fields.     

Curvature estimates for submanifolds with prescribed Gauss image and mean curvature

We study that the n-graph defined by a smooth map ${f:\Omega\subset\mathbb R^{n}\to \mathbb R^{m}, m\ge 2,}$ in ${\mathbb R^{m+n}}$ of the prescribed mean curvature and the Gauss image. Under the condition $$\Delta_f=\left[\text{det}\left(\delta_{ij}+\sum_\alpha{\frac {\partial {f^\alpha}}{\partial {x^i}}}{\frac {\partial {f^\alpha}}{\partial {x^j}}}\right)\right]^{\frac{1}{2}} < 2,$$ we derive the interior curvature estimates $$\sup_{D_R(x)}|B|^2\le{\frac{C}{R^2}}$$ when 2 ≤ n ≤ 5 with constant C depending on the given geometric data. If there is no dimension limitation we obtain $$\sup_{D_R(x)}|B|^2\le CR^{-a}\sup_{D_{2R}(x)}(2-\Delta_f)^{-\left({\frac{3}{2}}+{\frac{1}{s}}\right)},\quad s=\min(m, n)$$ with a < 1. If the image under the Gauss map is contained in a geodesic ball of the radius ${{\frac{\sqrt{2}}{4}}\pi}$ in G _n,m we also derive corresponding estimates.     

Infinitely many solutions for an elliptic problem involving critical Sobolev and Hardy–Sobolev exponents

We consider the following problem $$\left\{ \begin{array}{ll}-\Delta u = \mu |u|^\frac{4}{N-2}u + \frac{|u|^\frac{4-2s}{N-2}u}{|x|^{s}} + a(x)u, & x \in \Omega,\\ u=0, & {\rm on}\; \partial \Omega \end{array}\right.$$ where ${ \mu \ge 0, 0 < s < 2, 0 \in \partial \Omega}$ and Ω is a bounded domain in R ^N . We prove that if ${N \ge 7, a(0) > 0}$ and all the principle curvatures of ?Ω at 0 are negative, then the above problem has infinitely many solutions.     

The second pinching theorem for hypersurfaces with constant mean curvature in a sphere

We generalize the second pinching theorem for minimal hypersurfaces in a sphere due to Peng–Terng, Wei–Xu, Zhang, and Ding–Xin to the case of hypersurfaces with small constant mean curvature. Let $M^n$ be a compact hypersurface with constant mean curvature $H$ in $S^{n+1}$ . Denote by $S$ the squared norm of the second fundamental form of $M$ . We prove that there exist two positive constants $\gamma (n)$ and $\delta (n)$ depending only on $n$ such that if $|H|\le \gamma (n)$ and $\beta (n,H)\le S\le \beta (n,H)+\delta (n)$ , then $S\equiv \beta (n,H)$ and $M$ is one of the following cases: (i) $S^{k}\Big (\sqrt{\frac{k}{n}}\Big )\times S^{n-k}\Big (\sqrt{\frac{n-k}{n}}\Big )$ , $\,1\le k\le n-1$ ; (ii) $S^{1}\Big (\frac{1}{\sqrt{1+\mu ^2}}\Big )\times S^{n-1}\Big (\frac{\mu }{\sqrt{1+\mu ^2}}\Big )$ . Here $\beta (n,H)=n+\frac{n^3}{2(n-1)}H^2+\frac{n(n-2)}{2(n-1)} \sqrt{n^2H^4+4(n-1)H^2}$ and $\mu =\frac{n|H|+\sqrt{n^2H^2+ 4(n-1)}}{2}$ .     

Multiple positive solutions of singular and critical elliptic problem in {\mathbb{R}^2} with discontinuous nonlinearities

Let Ω be a bounded domain in ${\mathbb{R}^2}$ with smooth boundary. We consider the following singular and critical elliptic problem with discontinuous nonlinearity: $$(P_\lambda)\left \{\begin{array}{ll} - \Delta u = \lambda \left(\frac{m(x, u) e^{\alpha{u}^2}}{|x|^{\beta}} + u^{q}g(u - a)\right),\quad{u} > 0 \quad {\rm in} \quad \Omega\\u \quad \quad = 0\quad {\rm on} \quad \partial \Omega \end{array}\right.$$ where ${0\leq q < 1 ,0< \alpha\leq4\pi}$ and ${\beta \in [0, 2)}$ such that ${\frac{\beta}{2} + \frac{\alpha}{4\pi} \leq 1}$ and ${{g(t - a) = \left\{\begin{array}{ll}1, t \leq a\\ 0, t > a.\end{array}\right.}}$ Under the suitable assumptions on m(x, t) we show the existence and multiplicity of solutions for maximal interval for λ.     

Density of primes in l-th power residues

Given a prime number l, a finite set of integers S?=?{a ₁, ...,a _m } and m many l-th roots of unity $\zeta_l^{r_i}, i=1, \ldots ,m$ we study the distribution of primes p in ?(ζ _l ) such that the l-th residue symbol of a _i with respect to p is $\zeta_l^{r_i}, \mbox{ for all } i$ . We find out that this is related to the degree of the extension $\mathbb{Q}(a_1^{\frac{1}{l}}, \ldots ,a_m^{\frac{1}{l}})/\mathbb{Q}$ . We give an algorithm to compute this degree. Also we relate this degree to rank of a matrix obtained from S?=?{a ₁, ...,a _m }. This latter argument enables one to describe the degree $\mathbb{Q}(a_1^{\frac{1}{l}}, \ldots ,a_m^{\frac{1}{l}})/\mathbb{Q}$ in much simpler terms.     

Erratum to: On the Crank�CNicolson scheme once again

Let ${\left(\tau_j\right)_{j\in\mathbb{N}}}$ be a sequence of strictly positive real numbers, and let A be the generator of a bounded analytic semigroup in a Banach space X. Put ${A_n=\prod_{j=1}^n\left(I+\frac{1}{2}\tau_jA\right)\left(I-\frac{1}{2}\tau_jA\right)^{-1}}$ , and let ${x\in X}$ . Define the sequence ${\left(x_n\right)_{n\in\mathbb{N}}\subset X}$ by the Crank?CNicolson scheme: x _n ?=?A _n x. In this erratum, it is proved that the Crank?CNicolson scheme is stable in the sense that ${\sup_{n\in\mathbb{N}}\left\Vert A_nx\right\Vert < \infty}$ provided that inequality (0.9) below holds.     

Boundedness of Generalized Riesz Transforms on Orlicz–Hardy Spaces Associated to Operators

Let ${\Phi}$ be a continuous, strictly increasing and concave function on (0, ∞) of critical lower type index ${p_\Phi^- \in(0,\,1]}$ . Let L be an injective operator of type ω having a bounded H _∞ functional calculus and satisfying the k-Davies–Gaffney estimates with ${k \in {\mathbb Z}_+}$ . In this paper, the authors first introduce an Orlicz–Hardy space ${H^{\Phi}_{L}(\mathbb{R}^n)}$ in terms of the non-tangential L-adapted square function and then establish its molecular characterization. As applications, the authors prove that the generalized Riesz transform ${D_{\gamma}L^{-\delta/(2k)}}$ is bounded from the Orlicz–Hardy space ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the Orlicz space ${L^{\widetilde{\Phi}}(\mathbb{R}^n)}$ when ${p_\Phi^- \in (0, \frac{n}{n+ \delta - \gamma}]}$ , ${0 < \gamma \le \delta < \infty}$ and ${\delta- \gamma < n (\frac{1}{p_-(L)}-\frac{1}{p_+(L)})}$ , or from ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the Orlicz–Hardy space ${H^{\widetilde \Phi}(\mathbb{R}^n)}$ when ${p_\Phi^-\in (\frac{n}{n + \delta+ \lfloor \gamma \rfloor- \gamma},\,\frac{n}{n+ \delta- \gamma}]}$ , ${1\le \gamma \le \delta < \infty}$ and ${\delta- \gamma < n (\frac{1}{p_-(L)}-\frac{1}{p_+(L)})}$ , or from ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the weak Orlicz–Hardy space ${WH^\Phi(\mathbb{R}^n)}$ when ${\gamma = \delta}$ and ${p_\Phi=n/(n + \lfloor \gamma \rfloor)}$ or ${p_\Phi^-=n/(n + \lfloor \gamma \rfloor)}$ with ${p_\Phi^-}$ attainable, where ${\widetilde{\Phi}}$ is an Orlicz function whose inverse function ${\widetilde{\Phi}^{-1}}$ is defined by ${\widetilde{\Phi}^{-1}(t):=\Phi^{-1}(t)t^{\frac{1}{n}(\gamma- \delta)}}$ for all ${t \in (0,\,\infty)}$ , ${p_\Phi}$ denotes the strictly critical lower type index of ${\Phi}$ , ${\lfloor \gamma \rfloor}$ the maximal integer not more than ${\gamma}$ and ${(p_-(L),\,p_+(L))}$ the range of exponents ${p \in[1,\, \infty]}$ for which the semigroup ${\{e^{-tL}\}_{t >0 }}$ is bounded on ${L^p(\mathbb{R}^n)}$ .     

Planar Point Sets With Large Minimum Convex Decompositions

We show the existence of sets with $n$ points ( $n\ge 4$ ) for which every convex decomposition contains more than $\frac{35}{32}n-\frac{3}{2}$ polygons, which refutes the conjecture that for every set of $n$ points there is a convex decomposition with at most $n+C$ polygons. For sets having exactly three extreme points we show that more than $n+\sqrt{2(n-3)}-4$ polygons may be necessary to form a convex decomposition.     

The first factor of the class number of the p-th cyclotomic field

Kummer’s conjecture states that the relative class number of the p-th cyclotomic field follows a strict asymptotic law. Granville has shown it unlikely to be true—it cannot be true if we assume the truth of two other widely believed conjectures. We establish a new bound for the error term in Kummer’s conjecture, and more precisely we prove that ${\log(h_p^-)=\frac{p+3}{4} \log p +\frac{p}{2}\log(2\pi)+\log(1-\beta)+O(\log_2 p)}$ , where β is a possible Siegel zero of an ${L(s,\chi), \chi}$ odd.