Irrational Factor of Order k and ITS Connections With k-Free Integers

We introduce an irrational factor of order k defined by \({I_{k}(n) ={\prod_{i=1}^{l}} p_{i}^{\beta_{i}}}\) , where \({n = \prod_{i=1}^{l} p_{i}^{\alpha_{i}}}\) is the factorization of n and \({\beta_{i} = \left\{\begin{array}{ll}\alpha_i, \quad \quad {\rm if} \quad \alpha_i < k \\ \frac{1}{\alpha_i},\quad \quad {\rm if} \quad \alpha_i \geqq k \end{array}\right.}\) . It turns out that the function \({\frac{I_{k} (n)}{n}}\) well approximates the characteristic function of k-free integers. We also derive asymptotic formulas for \({\prod_{v=1}^{n} I_{k}(v)^{\frac{1}{n}}, \sum_{n \leqq x} I_{k}(n)}\) and \({\sum_{n \leqq x} (1 - \frac{n}{x}) I_{k}(n)}\) .     

Boundedness of Singular Integrals in Journé’s Class on Weighted Multiparameter Hardy Spaces

In this paper, we obtain the boundedness of singular integral operators T in Journé’s class on weighted multiparameter Hardy spaces \(H^{p}_{w}\) of arbitrary k number of parameters (k≥3) under the assumption that \(T^{\ast}_{i}(1)=0\) , i=1,…,k, and the kernel of T has a regularity of order ?>0, where \(w \in A_{r}(\Bbb{R}^{n_{1}}\times \cdots \times \Bbb{R}^{n_{k}}), r \geq 1\) and \(\max\{ \frac{r n_{1} }{n_{1}+\varepsilon}, \ldots, \frac{r n_{k} }{n_{k}+\varepsilon}\} .     

An upper bound for the Waring rank of a form

Let \({\phi(n)}\) denote the Euler-totient function. We study the error term of the general k-th Riesz mean of the arithmetical function \({\frac {n}{\phi(n)}}\) for any positive integer \({k \ge 1}\) , namely the error term \({E_k(x)}\) where $${\frac{1}{k!} \sum_{n \leq x} \frac{n}{\phi(n)} \left(1-\frac{n}{x}\right)^k = M_k(x) + E_k(x).}$$ The upper bound for \({| E_k(x)|}\) established here thus improves the earlier known upper bounds for all integers \({k\geq 1}\) .     

Generalized absolute Hausdorff summability of orthogonal series

For 1≦k≦2 and a sequence $\gamma :={\{\gamma(n)\}}_{n=1}^{\infty}$ that is quasi β-power monotone decreasing with ${\beta>1-\frac{1}{k}}$ , we prove the |A,γ| _k summability of an orthogonal series, where A is either a regular or Hausdorff matrix. For ${\beta>-\frac{3}{4}}$ , we give a necessary and sufficient condition for |A,γ| _k summability, where A is Hausdorff matrix. Our sufficient condition for ${\beta>-\frac{3}{4}}$ is weaker than that of Kantawala [1], ${\beta>-\frac{1}{k}}$ for |E,q,γ| _k summability; and of Leindler [4], β>?1 for |C,α,γ| _k , ${\alpha<\frac{1}{4}}$ . Also, our result generalizes the result of Spevakov [6] for |E,q,1|₁ summability.     

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 Note on the Largest Size of Families of Sets with a Forbidden Poset

Consider families of subsets of [n]:?=?{1,2,...,n} that do no contain a given poset P as a subposet. Let La(n, P) denote the largest size of such families and h(P) denote the height of P. The best known general upper bound for La(n, P) is $\left(\frac{1}{2}(|P|+h(P))-1\right)\left( \begin{array}{l}\,\,\,n \\ \lfloor \frac{n}{2} \rfloor\end{array}\right)$ , due to Bursi and Nagy (2012). This paper provides an improved upper bound $\frac{1}{m+1} \left(|P|+\frac{1}{2}(m^2+3m-2)(h(P)-1)-1\right) \left( \begin{array}{l} \,\,\,n \\ \lfloor \frac{n}{2} \rfloor\end{array}\right) $ , where m can be any positive integer less than $\lceil \frac{n}{2}\rceil$ .     

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 ${L^p(\mathbb{R}^n)}$ .     

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 ${\theta<\frac{\pi}{2}}$ .     

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.     

Convex hulls in the hyperbolic space

We show that the convex hull of any N points in the hyperbolic space ${\mathbb{H}^{n}}$ is of volume smaller than ${\frac{2 (2 \sqrt \pi)^n}{\Gamma(\frac n 2)} N}$ , and that for any dimension n there exists a constant C _n > 0 such that for any set ${A \subset \mathbb{H}^{n}}$ , $$Vol(Conv(A_1)) \leq C_n Vol(A_1)$$ where A ₁ is the set of points of hyperbolic distance to A smaller than 1.     

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).     

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.     

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 ${\varepsilon}$ .     

The circle method and bounds for L-functions - I

Let $f$ be a Hecke-Maass or holomorphic primitive cusp form of arbitrary level and nebentypus, and let $\chi $ be a primitive character of conductor $M$ . For the twisted $L$ -function $L(s, f\otimes \chi )$ we establish the hybrid subconvex bound $$\begin{aligned} L\left( \frac{1}{2}+it, f\otimes \chi \right) \ll (M(3+|t|))^{\frac{1}{2}-\frac{1}{18}+\varepsilon }, \end{aligned}$$ for $t\in \mathbb{R }$ . The implied constant depends only on the form $f$ and $\varepsilon $ .     

Pinned distance sets, k-simplices, Wolff’s exponent in finite fields and sum-product estimates

An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold α?>?0 such that ${|\Delta(E)| \gtrsim q}$ whenever ${|E| \gtrsim q^{\alpha}}$ , where ${E \subset {\mathbb {F}}_q^d}$ , the d-dimensional vector space over a finite field with q elements (not necessarily prime). Here ${\Delta(E)=\{{(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2: x,y \in E\}}$ . Iosevich and Rudnev (Trans Am Math Soc 359(12):6127–6142, 2007) established the threshold ${\frac{d+1}{2}}$ , and in Hart et?al. (Trans Am Math Soc 363:3255–3275, 2011) proved that this exponent is sharp in odd dimensions. In two dimensions we improve the exponent to ${\tfrac{4}{3}}$ , consistent with the corresponding exponent in Euclidean space obtained by Wolff (Int Math Res Not 10:547–567, 1999). The pinned distance set ${\Delta_y(E)=\{{(x_1-y_1)}^2+\dots+{(x_d-y_d)}^2: x\in E\}}$ for a pin ${y\in E}$ has been studied in the Euclidean setting. Peres and Schlag (Duke Math J 102:193–251, 2000) showed that if the Hausdorff dimension of a set E is greater than ${\tfrac{d+1}{2}}$ , then the Lebesgue measure of Δ _y (E) is positive for almost every pin y. In this paper, we obtain the analogous result in the finite field setting. In addition, the same result is shown to be true for the pinned dot product set ${\Pi_y(E)=\{x\cdot y: x\in E\}}$ . Under the additional assumption that the set E has Cartesian product structure we improve the pinned threshold for both distances and dot products to ${\frac{d^2}{2d-1}}$ . The pinned dot product result for Cartesian products implies the following sum-product result. Let ${A\subset \mathbb F_q}$ and ${z\in \mathbb F^*_q}$ . If ${|A|\geq q^{\frac{d}{2d-1}}}$ then there exists a subset ${E'\subset A\times \dots \times A=A^{d-1}}$ with ${|E'|\gtrsim |A|^{d-1}}$ such that for any ${(a_1,\dots, a_{d-1}) \in E'}$ , $$ |a_1A+a_2A+\dots +a_{d-1}A+zA| > \frac{q}{2}$$ where ${a_j A=\{a_ja:a \in A\},j=1,\dots,d-1}$ . A generalization of the Falconer distance problem is to determine the minimal α?>?0 such that E contains a congruent copy of a positive proportion of k-simplices whenever ${|E| \gtrsim q^{\alpha}}$ . Here the authors improve on known results (for k?>?3) using Fourier analytic methods, showing that α may be taken to be ${\frac{d+k}{2}}$ .     

Regularity of semi-stable solutions to fourth order nonlinear eigenvalue problems on general domains

We examine the fourth order problem $\Delta ^2 u = \lambda f(u) $ in $ \Omega $ with $ \Delta u = u =0 $ on $ {\partial \Omega }$ , where $ \lambda > 0$ is a parameter, $ \Omega $ is a bounded domain in $\mathbb{R }^N$ and where $f$ is one of the following nonlinearities: $ f(u)=e^u$ , $ f(u)=(1+u)^p $ or $ f(u)= \frac{1}{(1-u)^p}$ where $ p>1$ . We show the extremal solution is smooth, provided $$\begin{aligned} N < 2 + 4 \sqrt{2} + 4 \sqrt{ 2 - \sqrt{2}} \approx 10.718 \text{ when} f(u)=e^u, \end{aligned}$$ and $$\begin{aligned} N < \frac{4p}{p-1} + \frac{4(p+1)}{p-1} \left( \sqrt{ \frac{2p}{p+1}} + \sqrt{ \frac{2p}{p+1} - \sqrt{ \frac{2p}{p+1}}} - \frac{1}{2} \right) \end{aligned}$$ when $ f(u)=(u+1)^p$ . New results are also obtained in the case where $ f(u)=(1-u)^{-p}$ . These are substantial improvements to various results on critical dimensions obtained recently by various authors. To do that, we derive a new stability inequality satisfied by minimal solutions of the above equation, which is more amenable to estimates as it allows a method of proof reminiscent of the second order case.     

Two families of approximations for the gamma function

In this paper, we establish two families of approximations for the gamma function: $$ \begin{array}{lll} {\varGamma}(x+1)&=\sqrt{2\pi x}{\left({\frac{x+a}{{\mathrm{e}}}}\right)}^x {\left({\frac{x+a}{x-a}}\right)}^{-\frac{x}{2}+\frac{1}{4}} {\left({\frac{x+b}{x-b}}\right)}^{\sum\limits_{k=0}^m\frac{{\beta}_k}{x^{2k}}+O{{\left(\frac{1}{x^{2m+2}}\right)}}},\\ {\varGamma}(x+1)&=\sqrt{2\pi x}\cdot(x+a)^{\frac{x}{2}+\frac{1}{4}}(x-a)^{\frac{x}{2}-\frac{1}{4}} {\left({\frac{x-1}{x+1}}\right)}^{\frac{x^2}{2}}\\ &\quad\times {\left({\frac{x-c}{x+c}}\right)}^{\sum\limits_{k=0}^m\frac{{\gamma}_k}{x^{2k}}+O{\left({\frac{1}{x^{2m+2}}}\right)}}, \end{array}$$ where the constants ${\beta }_k$ and ${\gamma }_k$ can be determined by recurrences, and $a$ , $b$ , $c$ are parameters. Numerical comparison shows that our results are more accurate than Stieltjes, Luschny and Nemes’ formulae, which, to our knowledge, are better than other approximations in the literature.     

Weak-strong uniqueness criterion for the \beta -generalized surface quasi-geostrophic equation

We prove that weak-strong uniqueness holds for the $\beta $ -generalized surface quasi-geostrophic equation in the regular class $\nabla \theta \in L^{q}(0,T; L^{p}(\mathbb{R }^{2}))$ with $\frac{\alpha }{q}+\frac{2}{p}=\alpha +\beta -1$ , where $\alpha \in (0,1], \beta \in [1,2)$ and $\frac{2}{\alpha +\beta -1}<p<\infty $ .     

Bounding S(t) and S_1(t) on the Riemann hypothesis

Let $\pi S(t)$ denote the argument of the Riemann zeta-function, $\zeta (s)$ , at the point $s=\frac{1}{2}+it$ . Assuming the Riemann hypothesis, we present two proofs of the bound $$\begin{aligned} |S(t)| \le \left(\frac{1}{4} + o(1) \right)\frac{\log t}{\log \log t} \end{aligned}$$ for large $t$ . This improves a result of Goldston and Gonek by a factor of 2. The first method consists of bounding the auxiliary function $S_1(t) = \int _0^{t} S(u) \> \text{ d}u$ using extremal functions constructed by Carneiro, Littmann and Vaaler. We then relate the size of $S(t)$ to the size of the functions $S_1(t\pm h)-S_1(t)$ when $h\asymp 1/\log \log t$ . The alternative approach bounds $S(t)$ directly, relying on the solution of the Beurling–Selberg extremal problem for the odd function $f(x) = \arctan \left(\frac{1}{x}\right) - \frac{x}{1 + x^2}$ . This draws upon recent work by Carneiro and Littmann.