On the average indices of closed geodesics on positively curved Finsler spheres

In this paper, we prove that on every Finsler n-sphere (S ⁿ , F) for n ≥  6 with reversibility λ and flag curvature K satisfying ${(\frac{\lambda}{\lambda+1})^2 \, < \, K \, \le \, 1}$ , either there exist infinitely many prime closed geodesics or there exist ${[\frac{n}{2}]-2}$ closed geodesics possessing irrational average indices. If in addition the metric is bumpy, then there exist n?3 closed geodesics possessing irrational average indices provided the number of prime closed geodesics is finite.     

The moment problem for continuous positive semidefinite linear functionals

Let τ be a locally convex topology on the countable dimensional polynomial ${\mathbb{R}}$ -algebra ${\mathbb{R} [\underline{X}] := \mathbb{R} [X_1, \ldots, X_{n}]}$ . Let K be a closed subset of ${\mathbb{R} ^{n}}$ , and let ${M := M_{\{g_1, \ldots, g_s\}}}$ be a finitely generated quadratic module in ${\mathbb{R} [\underline{X}]}$ . We investigate the following question: When is the cone Psd(K) (of polynomials nonnegative on K) included in the closure of M? We give an interpretation of this inclusion with respect to representing continuous linear functionals by measures. We discuss several examples; we compute the closure of ${M = \sum \mathbb{R} [\underline{X}]^{2}}$ with respect to weighted norm-p topologies. We show that this closure coincides with the cone Psd(K) where K is a certain convex compact polyhedron.     

A Categorical Approach to Groupoid Frobenius Algebras

In this paper, we show that $\mathcal{G}$ -Frobenius algebras (for $\mathcal{G}$ a finite groupoid) correspond to a particular class of Frobenius objects in the representation category of $D(k[\mathcal{G}])$ , where $D(k[\mathcal{G}])$ is the Drinfeld double of the quantum groupoid $k[\mathcal{G}]$ (Nikshych et al. 2000).     

Pfaffian representations of cubic surfaces

Let $\mathbb{K }$ be a field of characteristic zero. We describe an algorithm which requires a homogeneous polynomial $F$ of degree three in $\mathbb{K }[x_{0},x_1,x_{2},x_{3}]$ and a zero ${\mathbf{a }}$ of $F$ in $\mathbb{P }^{3}_{\mathbb{K }}$ and ensures a linear Pfaffian representation of $\text{ V}(F)$ with entries in $\mathbb{K }[x_{0},x_{1},x_{2},x_{3}]$ , under mild assumptions on $F$ and ${\mathbf{a }}$ . We use this result to give an explicit construction of (and to prove the existence of) a linear Pfaffian representation of $\text{ V}(F)$ , with entries in $\mathbb{K }^{\prime }[x_{0},x_{1},x_{2},x_{3}]$ , being $\mathbb{K }^{\prime }$ an algebraic extension of $\mathbb{K }$ of degree at most six. An explicit example of such a construction is given.     

The universality theorem for Hecke L-functions

We extend the universality theorem for Hecke L-functions attached to ray class characters from the previously known strip ${ \max \{\frac{1}{2}, 1-\frac{1}{d}\} < {\rm Re}\,s < 1}$ for ${d=\left[K:\mathbb{Q}\right]}$ to the maximal strip ${\frac{1}{2} < {\rm Re}\,s < 1}$ under an assumption of a weak version of the density hypothesis. As a corollary, we give a new proof of the universality theorem for the Dedekind zeta function ζ _K (s) in the case of ${K/\mathbb{Q}}$ finite abelian.     

Some applications of the double large sieve

We sharpen a procedure of Cao and Zhai (J Théorie Nombres Bordeaux,11: 407–423, 1999) to estimate the sum $$\begin{aligned} \sum _{m\sim M} \sum _{n\sim N} a_m b_n \, e\left(\frac{F m^\alpha n^\beta }{M^\alpha N^\beta }\right) \end{aligned}$$ with $|a_m|,\ |b_n| \le 1$ . We apply this to give bounds for the discrepancy (mod 1) of the sequence $\{p^c: p\le X\}$ where $p$ is a prime variable, in the range $\frac{130}{79}\le c \le \frac{11}{5}$ . An alternative strategy is used for the range $1.48 \le c \le \frac{130}{79}$ . We use further exponential sum estimates to show that for large $R>0$ , and a small constant $\eta >0$ , the inequality $$\begin{aligned} \left| p_1^c+p_2^c+p_3^c+p_4^c+p_5^c - R\right| < R^{-\eta } \end{aligned}$$ holds for many prime tuples, provided $2<c\le 2.041$ . This improves work of Cao and Zhai (Monatsh Math, 150:173–179, 2007) and a theorem claimed by Shi and Liu (Monatsh Math, published online, 2012).     

Symplectic Lefschetz fibrations of curves as gonal coverings

We show that after a finite base change every symplectic Lefschetz fibration ${f \colon X \rightarrow B}$ of genus g >  3 curves over a closed oriented surface becomes a finite covering of degree ${\frac{g}{2} + 1}$ or ${\frac{g}{2} + \frac{3}{2}}$ of a family of spheres over a Riemann surface, with a branch locus admitting complex algebraic curves as local models. In the case of fibers of genus 4, it is shown that after a 2:1 base change the family admits a trigonal covering to a symplectic ruled surface, with symplectic branch locus.     

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 ${\theta<\frac{\pi}{2}}$ .     

Liouville theorems for stable solutions of biharmonic problem

We prove some Liouville type results for stable solutions to the biharmonic problem $\Delta ^2 u= u^q, \,u>0$ in $\mathbb{R }^n$ where $1 < q < \infty $ . For example, for $n \ge 5$ , we show that there are no stable classical solution in $\mathbb{R }^n$ when $\frac{n+4}{n-4} < q \le \left(\frac{n-8}{n}\right)_+^{-1}$ .     

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 $ .     

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 ${L^p(\mathbb{R}^n)}$ .     

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 $||T_f||= \sup (1-|z|^2)|T_f(z)|$ .     

A spinorial characterization of hyperspheres

Let M be a compact orientable n-dimensional hypersurface, with nowhere vanishing mean curvature H, immersed in a Riemannian spin manifold ${\overline{M}}$ admitting a non trivial parallel spinor field. Then the first eigenvalue ${\lambda_1(D_{M}^{H})}$ (with the lowest absolute value) of the Dirac operator ${D_{M}^{H}}$ corresponding to the conformal metric ${\langle\;,\;\rangle^{H}=H^{2}\,\langle\;,\;\rangle}$ , where ${\langle\;,\;\rangle}$ is the induced metric on M, satisfies ${\left|\lambda_1(D_{M}^{H})\right|\le \frac{n}{2}}$ . By applying the Bourguignon-Gauduchon first variational formula, we obtain a necessary condition for ${\left|\lambda_1(D_{M}^{H})\right|=\frac{n}{2}}$ . As a consequence, we prove that round hyperspheres are the only hypersurfaces of the Euclidean space satisfying the equality in the Bär inequality $$\lambda_1(D_{M})^{2}\le \frac{n^{2}}{4{vol}(M)}\int_{M} H^{2}\, dV,$$ where D _M stands now for the Dirac operator of the induced metric.     

Forbidden configurations and Steiner designs

Let ${\mathcal{F}}$ be a (0, 1) matrix. A (0, 1) matrix ${\mathcal{M}}$ is said to have ${\mathcal{F}}$ as a configuration if there is a submatrix of ${\mathcal{M}}$ which is a row and column permutation of ${\mathcal{F}}$ . We say that a matrix ${\mathcal{M}}$ is simple if it has no repeated columns. For a given ${v \in \mathbb{N}}$ , we shall denote by forb ${(v, \mathcal{F})}$ the maximum number of columns in a simple (0, 1) matrix with v rows for which ${\mathcal{F}}$ does not occur as a configuration. We say that a matrix ${\mathcal{M}}$ is maximal for ${\mathcal{F}}$ if ${\mathcal{M}}$ has forb ${(v, \mathcal{F})}$ columns. In this paper we show that for certain natural choices of ${\mathcal{F}}$ , forb ${(v, \mathcal{F})\leq\frac{\binom{v}{t}}{t+1}}$ . In particular this gives an extremal characterization for Steiner t-designs as maximal (0, 1) matrices in terms of certain forbidden configurations.     

Howe correspondence and Springer correspondence for real reductive dual pairs

We consider a real reductive dual pair (G′, G) of type I, with rank ${({\rm G}^{\prime}) \leq {\rm rank(G)}}$ . Given a nilpotent coadjoint orbit ${\mathcal{O}^{\prime} \subseteq \mathfrak{g}^{{\prime}{*}}}$ , let ${\mathcal{O}^{\prime}_\mathbb{C} \subseteq \mathfrak{g}^{{\prime}{*}}_\mathbb{C}}$ denote the complex orbit containing ${\mathcal{O}^{\prime}}$ . Under some condition on the partition λ′ parametrizing ${\mathcal{O}^{\prime}}$ , we prove that, if λ is the partition obtained from λ by adding a column on the very left, and ${\mathcal{O}}$ is the nilpotent coadjoint orbit parametrized by λ, then ${\mathcal{O}_\mathbb{C}= \tau (\tau^{\prime -1}(\mathcal{O}_\mathbb{C}^{\prime}))}$ , where ${\tau, \tau^{\prime}}$ are the moment maps. Moreover, if ${chc(\hat\mu_{\mathcal{O}^{\prime}}) \neq 0}$ , where chc is the infinitesimal version of the Cauchy-Harish-Chandra integral, then the Weyl group representation attached by Wallach to ${\mu_{\mathcal{O}^{\prime}}}$ with corresponds to ${\mathcal{O}_\mathbb{C}}$ via the Springer correspondence.     

Cubic symmetric polynomials yielding variations of the Erdős–Ginzburg–Ziv theorem

Let p be a prime and let $\varphi\in\mathbb{Z}_{p}[x_{1},x_{2},\ldots, x_{p}]$ be a symmetric polynomial, where  $\mathbb {Z}_{p}$ is the field of p elements. A sequence T in  $\mathbb {Z}_{p}$ of length p is called a φ-zero sequence if φ(T)=0; a sequence in $\mathbb {Z}_{p}$ is called a φ-zero free sequence if it does not contain any φ-zero subsequence. Motivated by the EGZ theorem for the prime p, we consider symmetric polynomials $\varphi\in \mathbb {Z}_{p}[x_{1},x_{2},\ldots, x_{p}]$ , which satisfy the following two conditions: (i) every sequence in  $\mathbb {Z}_{p}$ of length 2p?1 contains a φ-zero subsequence, and (ii) the φ-zero free sequences in  $\mathbb {Z}_{p}$ of maximal length are all those containing exactly two distinct elements, where each element appears p?1 times. In this paper, we determine all symmetric polynomials in $\mathbb {Z}_{p}[x_{1},x_{2},\ldots, x_{p}]$ of degree not exceeding 3 satisfying the conditions above.     

1D Schrödinger Operators with Short Range Interactions: Two-Scale Regularization of Distributional Potentials

For real ${L_\infty(\mathbb{R})}$ -functions ${\Phi}$ and ${\Psi}$ of compact support, we prove the norm resolvent convergence, as ${\varepsilon}$ and ${\nu}$ tend to 0, of a family ${S_{\varepsilon \nu}}$ of one-dimensional Schrödinger operators on the line of the form $$S_{\varepsilon \nu} = -\frac{d^2}{dx^2} + \frac{\alpha}{\varepsilon^2} \Phi \left( \frac{x}{\varepsilon} \right) + \frac{\beta}{\nu} \Psi \left(\frac{x}{\nu} \right),$$ provided the ratio ${\nu/\varepsilon}$ has a finite or infinite limit. The limit operator S ₀ depends on the shape of ${\Phi}$ and ${\Psi}$ as well as on the limit of ratio ${\nu/\varepsilon}$ . If the potential ${\alpha\Phi}$ possesses a zero-energy resonance, then S ₀ describes a non trivial point interaction at the origin. Otherwise S ₀ is the direct sum of the Dirichlet half-line Schrödinger operators.     

Calabi Product Lagrangian Immersions in Complex Projective Space and Complex Hyperbolic Space

Starting from two Lagrangian immersions and a Legendre curve ${\tilde{\gamma}(t)}$ in ${\mathbb{S}^3(1)}$ $({\rm or\,in}\,{\mathbb{H}_1^3(-1)})$ , it is possible to construct a new Lagrangian immersion in ${\mathbb{CP}^n(4)}$ $({\rm or\,in}\,{\mathbb{CH}^n(-4)})$ , which is called a warped product Lagrangian immersion. When ${\tilde{\gamma}(t)=(r_1e^{i(\frac{r_2}{r_1}at)}, \;r_2e^{i(- \frac{r_1}{r_2}at)})}$ $({\rm or}\,{\tilde{\gamma}(t)=(r_1e^{i(\frac{r_2}{r_1}at)}, \;r_2e^{i( \frac{r_1}{r_2}at)})})$ , where r ₁, r ₂, and a are positive constants with ${r_1^2+r_2^2=1}$ $({\rm or}\,{-r_1^2+r_2^2=-1})$ , we call the new Lagrangian immersion a Calabi product Lagrangian immersion. In this paper, we study the inverse problem: how to determine from the properties of the second fundamental form whether a given Lagrangian immersion of ${\mathbb{CP}^n(4)}$ or ${\mathbb{CH}^n(-4)}$ is a Calabi product Lagrangian immersion. When the Calabi product is minimal, or is Hamiltonian minimal, or has parallel second fundamental form, we give some further characterizations.     

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.