A new generalization of binomial coefficients

Let t be a fixed parameter and x some indeterminate. We give some properties of the generalized binomial coefficients $\genfrac{\langle }{\rangle}{0pt}{}{x}{k}$ inductively defined by $k/x \genfrac{\langle}{\rangle}{0pt}{}{x}{k}= t\genfrac{\langle}{\rangle}{0pt}{}{x-1}{k-1} +(1-t)\genfrac{\langle}{\rangle}{0pt}{}{x-2}{k-2}$ .     

A decomposition of the universal embedding space for the near polygon {{\mathbb H}_n}

Let ${{\mathbb H}_n, n \geq 1}$ , be the near 2n-gon defined on the 1-factors of the complete graph on 2n?+?2 vertices, and let e denote the absolutely universal embedding of ${{\mathbb H}_n}$ into PG(W), where W is a ${\frac{1}{n+2} \left(\begin{array}{c}2n+2 \\ n+1\end{array}\right)}$ -dimensional vector space over the field ${{\mathbb F}_2}$ with two elements. For every point z of ${{\mathbb H}_n}$ and every ${i \in {\mathbb N}}$ , let Δ _i (z) denote the set of points of ${{\mathbb H}_n}$ at distance i from z. We show that for every pair {x, y} of mutually opposite points of ${{\mathbb H}_n, W}$ can be written as a direct sum ${W_0 \oplus W_1 \oplus \cdots \oplus W_n}$ such that the following four properties hold for every ${i \in \{0,\ldots,n \}}$ : (1) ${\langle e(\Delta_i(x) \cap \Delta_{n-i}(y)) \rangle = {\rm PG}(W_i)}$ ; (2) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(x) \right) \right\rangle = {\rm PG}(W_0 \oplus W_1 \oplus \cdots \oplus W_i)}$ ; (3) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(y) \right) \right\rangle = {\rm PG}(W_{n-i}\oplus W_{n-i+1} \oplus \cdots \oplus W_n)}$ ; (4) ${\dim(W_i) = |\Delta_i(x) \cap \Delta_{n-i}(y)| = \left(\begin{array}{c}n \\ i\end{array}\right)^2 - \left(\begin{array}{c}n \\ i-1\end{array}\right) \cdot \left(\begin{array}{c}n \\ i+1\end{array}\right)}$ .     

On cyclic compositions of positive integers

Say that two compositions of n into k parts are related if they differ only by a cyclic shift. This defines an equivalence relation on the set of such compositions. Let ${\left\langle \begin{array}{c}n \\ k\end{array} \right\rangle}$ denote the number of distinct corresponding equivalence classes, that is, the number of cyclic compositions of n into k parts. We show that the sequence ${\left\langle\begin{array}{c}n \\ k\end{array}\right\rangle}$ is log-concave and prove some results concerning ${\left\langle \begin{array}{c}n \\ k \end{array} \right\rangle}$ modulo two.     

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.     

Eta Cocycles, Relative Pairings and the Godbillon–Vey Index Theorem

We prove a Godbillon?CVey index formula for longitudinal Dirac operators on a foliated bundle with boundary ${(X,\mathcal{F})}$ ; in particular, we define a Godbillon?CVey eta invariant on ${(\partial X,\mathcal{F}_{\partial}),}$ that is, a secondary invariant for longitudinal Dirac operators on type III foliations. Moreover, employing the Godbillon?CVey index as a pivotal example, we explain a new approach to higher index theory on geometric structures with boundary. This is heavily based on the interplay between the absolute and relative pairings of K-theory and cyclic cohomology for an exact sequence of Banach algebras, which in the present context takes the form ${0 \to \mathbf{\mathfrak{J}} \to \mathbf{\mathfrak{A}} \to \mathbf{\mathfrak{B}} \to 0}$ with ${ \mathbf{\mathfrak{J}}}$ dense and holomorphically closed in ${C^* (X,\mathcal{F})}$ and ${ \mathbf{\mathfrak{B}}}$ depending only on boundary data. Of particular importance is the definition of a relative cyclic cocycle ${(\tau_{GV}^r,\sigma_{GV})}$ for the pair ${\mathbf{\mathfrak{A}} \to \mathbf{\mathfrak{B}}}$ ; ${\tau_{GV}^r}$ is a cyclic cochain on ${\mathbf{\mathfrak{A}}}$ defined through a regularization à la Melrose of the usual Godbillon?CVey cyclic cocycle ?? _GV ; ?? _GV is a cyclic cocycle on ${\mathbf{\mathfrak{B}}}$ , obtained through a suspension procedure involving ?? _GV and a specific 1-cyclic cocycle (Roe??s 1-cocycle). We call ?? _GV the eta cocycle associated to ?? _GV . The Atiyah?CPatodi?CSinger formula is obtained by defining a relative index class ${{\rm Ind} (D,D^\partial) \in K_* (\mathbf{\mathfrak{A}}, \mathbf{\mathfrak{B}})}$ and establishing the equality ${\langle {\rm Ind} (D), [\tau_{GV}] \rangle\,=\,\langle {\rm Ind} (D,D^\partial), [(\tau^r_{GV}, \sigma_{GV})] \rangle}$ . The Godbillon?CVey eta invariant ?? _GV is obtained through the eta cocycle ?? _GV .     

Harmonic maps of foliated Riemannian manifolds

We study ${({\mathcal{F}}, {\mathcal{G}})}$ -harmonic maps between foliated Riemannian manifolds ${(M, {\mathcal{F}}, g)}$ and ${(N, {\mathcal{G}}, h)}$ i.e. smooth critical points ? : M → N of the functional ${E_T (\phi ) = \frac{1}{2} \int_M \| d_T \phi \|^2 \,d \, v_g}$ with respect to variations through foliated maps. In particular we study ${({\mathcal{F}}, {\mathcal{G}})}$ -harmonic morphisms i.e. smooth foliated maps preserving the basic Laplace equation Δ _B u =  0. We show that CR maps of compact Sasakian manifolds preserving the Reeb flows are weakly stable ${({\mathcal{F}}, {\mathcal{G}})}$ -harmonic maps. We study ${({\mathcal{F}}, {\mathcal{G}}_0 )}$ -harmonic maps into spheres and give foliated analogs to Solomon’s (cf., J Differ Geom 21:151–162, 1985) results.     

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).     

Gamma-convergence of nonlocal perimeter functionals

Given ${\Omega\subset\mathbb{R}^{n}}$ open, connected and with Lipschitz boundary, and ${s\in (0, 1)}$ , we consider the functional $$\mathcal{J}_s(E,\Omega)\,=\, \int_{E\cap \Omega}\int_{E^c\cap\Omega}\frac{dxdy}{|x-y|^{n+s}}+\int_{E\cap \Omega}\int_{E^c\cap \Omega^c}\frac{dxdy}{|x-y|^{n+s}}\,+ \int_{E\cap \Omega^c}\int_{E^c\cap \Omega}\frac{dxdy}{|x-y|^{n+s}},$$ where ${E\subset\mathbb{R}^{n}}$ is an arbitrary measurable set. We prove that the functionals ${(1-s)\mathcal{J}_s(\cdot, \Omega)}$ are equi-coercive in ${L^1_{\rm loc}(\Omega)}$ as ${s\uparrow 1}$ and that $$\Gamma-\lim_{s\uparrow 1}(1-s)\mathcal{J}_s(E,\Omega)=\omega_{n-1}P(E,\Omega),\quad \text{for every }E\subset\mathbb{R}^{n}\,{\rm measurable}$$ where P(E, ??) denotes the perimeter of E in ?? in the sense of De Giorgi. We also prove that as ${s\uparrow 1}$ limit points of local minimizers of ${(1-s)\mathcal{J}_s(\cdot,\Omega)}$ are local minimizers of P(·, ??).     

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 ${L^p(\mathbb{R}^n)}$ .     

On the Hall algebra of semigroup representations over \mathbb F _1

Let $\mathrm{A }$ be a finitely generated semigroup with 0. An $\mathrm{A }$ -module over $\mathbb F _1$ (also called an $\mathrm{A }$ -set), is a pointed set $(M,*)$ together with an action of $\mathrm{A }$ . We define and study the Hall algebra $\mathbb H _{\mathrm{A }}$ of the category $\mathcal C _{\mathrm{A }}$ of finite $\mathrm{A }$ -modules. $\mathbb H _{\mathrm{A }}$ is shown to be the universal enveloping algebra of a Lie algebra $\mathfrak n _{\mathrm{A }}$ , called the Hall Lie algebra of $\mathcal C _{\mathrm{A }}$ . In the case of $\langle t \rangle $ —the free monoid on one generator $\langle t \rangle $ , the Hall algebra (or more precisely the Hall algebra of the subcategory of nilpotent $\langle t \rangle $ -modules) is isomorphic to Kreimer’s Hopf algebra of rooted forests. This perspective allows us to define two new commutative operations on rooted forests. We also consider the examples when $\mathrm{A }$ is a quotient of $\langle t \rangle $ by a congruence, and the monoid $G \cup \{ 0\}$ for a finite group $G$ .     

The Algebra of Polynomial Integro-Differential Operators is a Holonomic Bimodule over the Subalgebra of Polynomial Differential Operators

In contrast to its subalgebra $A_n:=K\langle x_1, \ldots , x_n, \frac{\partial}{\partial x_1}, \ldots ,\frac{\partial}{\partial x_n}\rangle $ of polynomial differential operators (i.e. the n’th Weyl algebra), the algebra ${\mathbb{I}}_n:=K\langle x_1, \ldots ,$ $ x_n, \frac{\partial}{\partial x_1}, \ldots ,\frac{\partial}{\partial x_n}, \int_1, \ldots , \int_n\rangle $ of polynomial integro-differential operators is neither left nor right Noetherian algebra; moreover it contains infinite direct sums of nonzero left and right ideals. It is proved that ${\mathbb{I}}_n$ is a left (right) coherent algebra iff n?=?1; the algebra ${\mathbb{I}}_n$ is a holonomic A _n -bimodule of length 3 ⁿ and has multiplicity 3 ⁿ with respect to the filtration of Bernstein, and all 3 ⁿ simple factors of ${\mathbb{I}}_n$ are pairwise non-isomorphic A _n -bimodules. The socle length of the A _n -bimodule ${\mathbb{I}}_n$ is n?+?1, the socle filtration is found, and the m’th term of the socle filtration has length ${n\choose m}2^{n-m}$ . This fact gives a new canonical form for each polynomial integro-differential operator. It is proved that the algebra ${\mathbb{I}}_n$ is the maximal left (resp. right) order in the largest left (resp. right) quotient ring of the algebra ${\mathbb{I}}_n$ .     

Graphs, partitions and classes of groups

If ${\mathfrak X}$ is a class of groups, Delizia et?al. (Bull Austral Math Soc 75:313–320, 2007) call a group G ${\mathfrak X}$ -transitive (or an ${\mathfrak XT}$ -group) if whenever ${\langle a,b\rangle}$ and ${\langle b,c\rangle}$ are in ${\mathfrak X} \langle a,c\rangle$ is also in ${\mathfrak X}$ ( ${a,b,c\in G}$ ). The structure of ${\mathfrak XT}$ -groups has been investigated for a number of classes of groups, by Delizia, Moravec and Nicotera and others. A graph can be associated with a group in many ways. Delizia, Moravec and Nicotera introduce a graph which is a generalisation of the commuting graph of a group, but do not make use of the graph. We will use the properties of the graph to investigate further classes of groups and to obtain more detailed structural information.     

Graph invariants and the positivity of the height of the Gross-Schoen cycle for some curves

Let X be a projective curve over a global field K. Gross and Schoen defined a modified diagonal cycle Δ on X ³, and showed that the height ${\langle \Delta, \Delta \rangle}$ is defined in general. Zhang recently proved a formula which describe ${\langle \Delta, \Delta \rangle}$ in terms of the self pairing of the admissible dualizing sheaf and the invariants arising from the reduction graphs. In this note, we calculate explicitly those graph invariants for the reduction graphs of curves of genus 3 and examine the positivity of ${\langle \Delta, \Delta \rangle}$ . We also calculate them for so-called hyperelliptic graphs. As an application, we find a characterization of hyperelliptic curves of genus 3 by the configuration of the reduction graphs and the property ${\langle \Delta, \Delta \rangle = 0}$ .     

Infinitely many solutions for an elliptic problem involving critical Sobolev growth and Hardy potential

In this paper, we will prove the existence of infinitely many solutions for the following elliptic problem with critical Sobolev growth and a Hardy potential: $$-\Delta u-\frac{\mu}{|x|^2}u = |u|^{2^{\ast}-2}u+a u\quad {\rm in}\;\Omega,\quad u=0 \quad {\rm on}\; \partial\Omega,\qquad (*)$$ under the assumptions that N ≥ 7, ${\mu\in \left[0,\frac{(N-2)^2}4-4\right)}$ and a > 0, where ${2^{\ast}=\frac{2N}{N-2}}$ , and Ω is an open bounded domain in ${\mathbb{R}^N}$ which contains the origin. To achieve this goal, we consider the following perturbed problem of (*), which is of subcritical growth, $$-\Delta u-\frac{\mu}{|x|^2}u = |u|^{2^{\ast}-2-\varepsilon_n}u+au \quad {\rm in}\,\Omega, \quad u=0 \quad {\rm on}\;\partial\Omega,\qquad(\ast\ast)_n$$ where ${\varepsilon_{n} > 0}$ is small and ${\varepsilon_n \to 0}$ as n → + ∞. By the critical point theory for the even functionals, for each fixed ${\varepsilon_{n} > 0}$ small, (**) _n has a sequence of solutions ${u_{k,\varepsilon_{n}} \in H^{1}_{0}(\Omega)}$ . We obtain the existence of infinitely many solutions for (*) by showing that as n → ∞, ${u_{k,\varepsilon_{n}}}$ converges strongly in ${H^{1}_{0}(\Omega)}$ to u _k , which must be a solution of (*). Such a convergence is obtained by applying a local Pohozaev identity to exclude the possibility of the concentration of ${\{u_{k,\varepsilon_n}\}}$ .     

Nonscattering solutions and blowup at infinity for the critical wave equation

We consider the critical focusing wave equation $(-\partial _t^2+\Delta )u+u^5=0$ in ${\mathbb{R }}^{1+3}$ and prove the existence of energy class solutions which are of the form $$\begin{aligned} u(t,x)=t^\frac{\mu }{2}W(t^\mu x)+\eta (t,x) \end{aligned}$$ in the forward lightcone $\{(t,x)\in {\mathbb{R }}\times {\mathbb{R }}^3: |x|\le t, t\gg 1\}$ where $W(x)=(1+\frac{1}{3} |x|^2)^{-\frac{1}{2}}$ is the ground state soliton, $\mu $ is an arbitrary prescribed real number (positive or negative) with $|\mu |\ll 1$ , and the error $\eta $ satisfies $$\begin{aligned} \Vert \partial _t \eta (t,\cdot )\Vert _{L^2(B_t)} +\Vert \nabla \eta (t,\cdot )\Vert _{L^2(B_t)}\ll 1,\quad B_t:=\{x\in {\mathbb{R }}^3: |x|<t\} \end{aligned}$$ for all $t\gg 1$ . Furthermore, the kinetic energy of $u$ outside the cone is small. Consequently, depending on the sign of $\mu $ , we obtain two new types of solutions which either concentrate as $t\rightarrow \infty $ (with a continuum of rates) or stay bounded but do not scatter. In particular, these solutions contradict a strong version of the soliton resolution conjecture.     

{\mathcal{C}^{\alpha}}-regularity for a class of non-diagonal elliptic systems with p-growth

We consider weak solutions to nonlinear elliptic systems in a W ^1,p -setting which arise as Euler equations to certain variational problems. The solutions are assumed to be stationary in the sense that the differential of the variational integral vanishes with respect to variations of the dependent and independent variables. We impose new structure conditions on the coefficients which yield everywhere ${\mathcal{C}^{\alpha}}$ -regularity and global ${\mathcal{C}^{\alpha}}$ -estimates for the solutions. These structure conditions cover variational integrals like ${\int F(\nabla u)\; dx}$ with potential ${F(\nabla u):=\tilde F (Q_1(\nabla u),\ldots, Q_N(\nabla u))}$ and positively definite quadratic forms in ${\nabla u}$ defined as ${Q_i(\nabla u)=\sum_{\alpha \beta} a_i^{\alpha \beta} \nabla u^\alpha \cdot \nabla u^\beta}$ . A simple example consists in ${\tilde F(\xi_1,\xi_2):= |\xi_1|^{\frac{p}{2}} + |\xi_2|^{\frac{p}{2}}}$ or ${\tilde F(\xi_1,\xi_2):= |\xi_1|^{\frac{p}{4}}|\xi_2|^{\frac{p}{4}}}$ . Since the Q _i need not to be linearly dependent our result covers a class of nondiagonal, possibly nonmonotone elliptic systems. The proof uses a new weighted norm technique with singular weights in an L ^p -setting.     

On the sup-norm of Maass cusp forms of large level. III

Let $f$ be a Hecke–Maass cuspidal newform of square-free level $N$ and Laplacian eigenvalue $\lambda $ . It is shown that $\left||f \right||_\infty \ll _{\lambda ,\epsilon } N^{-\frac{1}{6}+\epsilon } \left||f \right||_2$ for any $\epsilon >0$ .     

Valuation Extensions of Algebras Defined by Monic Gr?bner Bases

Let K be a field, $\mathcal {O}_v$ a valuation ring of K associated to a valuation v: K → Γ?∪?{?∞?}, and m _v the unique maximal ideal of $\mathcal {O}_v$ . Consider an ideal $\mathcal {I}$ of the free K-algebra $K\langle X\rangle =K\langle X_1,...,X_n\rangle$ on X ₁,...,X _n . If ${\cal I}$ is generated by a subset $\mathcal {G}\subset{\cal O}_v\langle X\rangle$ which is a monic Gr?bner basis of ${\cal I}$ in $K\langle X\rangle$ , where $\mathcal {O}_v\langle X\rangle =\mathcal{O}_v\langle X_1,...,X_n\rangle$ is the free $\mathcal{O}_v$ -algebra on X ₁,...,X _n , then the valuation v induces naturally an exhaustive and separated Γ-filtration F ^v A for the K-algebra $A=K\langle X\rangle /\mathcal {I}$ , and moreover $\mathcal{I}\cap\mathcal{O}_v\langle X\rangle =\langle\mathcal{G}\rangle$ holds in $\mathcal{O}_v\langle X\rangle$ ; it follows that, if furthermore $\mathcal{G}\not\subset {\bf m}_v{O}_v\langle X\rangle$ and $k\langle X\rangle /\langle\overline{\mathcal G}\rangle$ is a domain, where $k=\mathcal{O}_v/{\bf m}_v$ is the residue field of $\mathcal{O}_v$ , $k\langle X\rangle =k\langle X_1,...,X_n\rangle$ is the free k-algebra on X ₁,...,X _n , and $\overline{\mathcal G}$ is the image of $\mathcal{G}$ under the canonical epimorphism $\mathcal{O}_v\langle X\rangle\rightarrow k\langle X\rangle$ , then F ^v A determines a valuation function A → Γ?∪?{?∞?}, and thereby v extends naturally to a valuation function on the (skew-)field Δ of fractions of A provided Δ exists.