Seeds of sunflowers are often modelled by $n⟼{\varphi }_{\theta }\left(n\right)=\sqrt{n}{e}^{2i\pi n\theta }$ leading to a roughly uniform repartition with seeds indexed by consecutive integers at angular distance $2\pi \theta$ for $\theta$ the golden ratio. We associate to such a map ${\varphi }_{\theta }$ a geodesic path ${\gamma }_{\theta }:{ℝ}_{\>0}⟶{\mathrm{PSL}}_2\left(\mathbb{Z}\right)\setminus ℍ$ of the modular curve and use it for local descriptions of the image ${\varphi }_{\theta }\left(\mathbb{N}\right)$ of the phyllotactic map ${\varphi }_{\theta }$.

Let $(A_0,A_1)$ be a regular interpolation couple. Under several different assumptions on a fixed $A^{\beta }$, we show that $A^{\theta }=A_{\theta }$ for every $\theta \in (0,1)$. We also deal with assumptions on ${\overline{A}}^{\beta }$, the closure of $A^{\beta }$ in the dual of ${(A_0^{*},A_1^{*})}_{\beta }$.

Let $L(\theta ,\lambda )$ be the set of limit points of the fractional parts $\left\{\lambda {\theta }^n\right\}$, $n=0,1,2,\cdots$, where $\theta$ is a Pisot number and $\lambda \in \mathbb{Q}\left(\theta \right)$. Using a description of $L(\theta ,\lambda )$, due to Dubickas, we show that there is a sequence ${\left({\lambda }_n\right)}_{n\ge 0}$ of elements of $\mathbb{Q}\left(\theta \right)$ such that $Card\left(L(\theta ,{\lambda }_n)\right)<Card\left(L(\theta ,{\lambda }_{n+1})\right)$, $\forall$ $n\ge 0$. Also, we prove that the fractional parts of Pisot numbers, with a fixed degree greater than 1, are dense in the unit interval.

The extremal functions  $f_0\left(z\right)$  realizing the maxima of some functionals (e.g. $max|a_3|$, and  $maxarg{f}^{{}^{\text{'}}}\left(z\right)$) within the so-called universal linearly invariant family $U_{\alpha }$ (in the sense of Pommerenke [10]) have such a form that $f_0^{{}^{\text{'}}}\left(z\right)$  looks similar to generating function for Meixner-Pollaczek (MP) polynomials [2], [8]. This fact gives motivation for the definition and study of the generalized Meixner-Pollaczek (GMP) polynomials $P_n^{\lambda }(x;\theta ,\psi )$ of a real variable $x$ as coefficients of $$G^{\lambda }(x;\theta ,\psi ;z)=\frac{1}{{(1-z{e}^{i\theta })}^{\lambda -ix}{(1-z{e}^{i\psi })}^{\lambda +ix}}=\sum _{n=0}^{\infty }{P}_n^{\lambda }(x;\theta ,\psi )z^n,\left|z\right|<1,$$ where the parameters $\lambda$, $\theta$, $\psi$ satisfy the conditions:...

We consider the linear system $|2{\Theta }_0|$ of second order theta functions over the Jacobian $JC$ of a non-hyperelliptic curve $C$. A result by J.Fay says that a divisor $D\in |2{\Theta }_0|$ contains the origin $𝒪\in JC$ with multiplicity $4$ if and only if $D$ contains the surface $C-C=\left\{𝒪\right(p-q)\mid p,q\in C\}\subset JC$. In this paper we generalize Fay’s result and some previous work by R.C.Gunning. More precisely, we describe the relationship between divisors containing $𝒪$ with multiplicity $6$, divisors containing the fourfold $C_2-C_2=\{𝒪(p+q-r-s)\mid p,q,r,s\in C\}$, and divisors singular along $C-C$, using...

We prove that the global geometric theta-lifting functor for the dual pair $(H,G)$ is compatible with the Whittaker functors, where $(H,G)$ is one of the pairs $({\mathrm{S}𝕆}_{2n},{𝕊p}_{2n})$, $({𝕊p}_{2n},{\mathrm{S}𝕆}_{2n+2})$ or $({𝔾\mathrm{L}}_n,{𝔾\mathrm{L}}_{n+1})$. That is, the composition of the theta-lifting functor from $H$ to $G$ with the Whittaker functor for $G$ is isomorphic to the Whittaker functor for $H$.

We fix $\mathrm{\ell }$ a prime and let $M$ be an integer such that $\mathrm{\ell }|̸M$; let $f\in S_2({\mathrm{\Gamma }}_1(M{\mathrm{\ell }}^2))$ be a newform supercuspidal of fixed type at $\mathrm{\ell }$ and special at a finite set of primes. For an indefinite quaternion algebra over $Q$, of discriminant dividing the level of $f$, there is a local quaternionic Hecke algebra $T$ associated to $f$. The algebra $T$ acts on a module $M_f$ coming from the cohomology of a Shimura curve. Applying the Taylor-Wiles criterion and a recent Savitt's theorem, $T$ is the universal deformation ring of a global...

Let $(M,\theta )$ be a compact CR manifold of dimension $2n+1$ with a contact form $\theta$, and $L=(2+2/n){\Delta }_b+R$ its associated CR conformal laplacien. The CR Yamabe conjecture states that there is a contact form $\tilde{\theta }$ on $M$ conformal to $\theta$ which has a constant Webster curvature. This problem is equivalent to the existence of a function $u$ such that $Lu=u^{1+2/n}$, $u>0$ on $M$. D. Jerison and J. M. Lee solved the CR Yamabe problem in the case where $n\ge 2$ and $(M,\theta )$ is not locally CR equivalent to the sphere $S^{2n+1}$ of ${𝐂}^n$. In a join work with R. Yacoub, the CR Yamabe...

The object of this note is to study certain 2-dimensional $\lambda$-adic representations of $Gal(\overline{Q}/Q)$; fixed $p_1,\mathrm{\dots },p_n$ distinct primes, we will consider representations $\rho :G\to G{L}_2(A)$, given by the matrix which are unramified outside $p_1,\mathrm{\dots },p_n,\mathrm{\infty }$ and the residue characteristic of $\lambda$, which are a product of $m$ representations over finite extensions of the ring of Witt vectors of the residue field and which are reducible modulo $\lambda$. In analogy with the theory of the modular representations, we will introduce the analogue of Mazur's Hecke...

Dati due elementi $f$ e $g$ in un'algebra uniforme $A$, sia $G=f(M_A/f({\partial }_A)$. Nella presente Nota si danno, fra l’altro, due nuove dimostrazioni elementari del fatto che la funzione $\lambda \to \log \max g(f^{-1}(\lambda ))$ è subarmonica su $G$ e che l’applicazione $\lambda \to g(f^{-1}(\lambda ))$ è analitica nel senso di Oka.

Let $X\subset {\mathbb{P}}^N$ be a smooth irreducible non degenerate surface over the complex numbers, $N\ge 4$. We define the projective genus of $X$, denoted by $PG(X)$, as the geometric genus of the singular curve of the projection of $X$ from a general linear subspace of codimension four. Denote by $g(X)$ the sectional genus of $X$. In this paper we conjecture that the only surfaces for which $PG(X)=g(X)-1$ are the del Pezzo surface in ${\mathbb{P}}^4$, in ${\mathbb{P}}^5$ and a conic bundle of degree 5 in ${\mathbb{P}}^4$. We prove that for $N\ge 5$ if $PG(X)=g(X)-1+\lambda$, $\lambda$ a non negative integer, then $g(X)\le \lambda +1+\alpha$ where...

In this paper we give a recursive formula for the sequence of primes $\{p_n\}$ and apply it to find a necessary and sufficient condition in order that a prime number $p_{n+1}$ is equal to $p_n+2$. Applications of previous results are given to evaluate the probability that $p_{n+1}$ is of the form $p_n+2$; moreover we prove that the limit of this probability is equal to zero as $n$ goes to $\mathrm{\infty }$. Finally, for every prime $p_n$ we construct a sequence whose terms that are in the interval $[p_n^2-2,p_{n+1}^2-2[$ are the first terms of two twin primes. This...

Let $D$ be an integral domain with quotient field $D$. Recall that $D$ is Schreier if $D$ is integrally closed and for all $x,y,z\in D\setminus \{0\}$, $x|yz$ implies that $x=r\cdot s$ where $r|y$ e $s|z$. A GCD domain is Schreier. We show that an integral domain $D$ is a GCD domain if and only if (i) for each pair $a,b\in D\setminus \{0\}$, there is a finitely generated ideal $B$ such that $aD\bigcap bD=B_v$ and (ii) every quadratic in $D[X]$ that is a product of two linear polynomials in $K[X]$ is a product of two linear polynomials in $D[X]$. We also show that $D$ is Schreier if and only if every...

This article is devoted to sequences ${(u_n)}_n$ of subharmonic functions in ${ℝ}^N$, with finite order, whose means $J_{u_n}(r)$ (over spheres centered at the origin, with radius r) satisfy such a condition as: $\forall r>0$, $\exists A_r>0$ such that $J_{u_n}(r)\le A_r$, $\forall n\in 𝐍$. The paper investigates under which conditions one may extract a pointwise or uniformly convergent subsequence.

We introduce a new class of spaces, called Hausdorff modulo $ℐ$ or $T_2$ mod $ℐ$ spaces with respect to an ideal $ℐ$ which contains the class of all Hausdorff spaces. Characterizations of these spaces are given and their properties are investigated. The concept of compactness modulo an ideal $ℐ$ was introduced by Newcomb in 1967 and studied by Hamlett and Jankovic in 1990. We study the properties of $ℐ$-compact subsets in Hausdorff modulo $ℐ$ spaces and generalize some results of Hamlett and Jankovic....

Let ${𝕊}^1$ and $𝔻$ be the unit circle and the unit disc in the plane and let us denote by $𝒜({𝕊}^1)$ the algebra of the complex-valued continuous functions on ${𝕊}^1$ which are traces of functions in the Sobolev class $W^{1,2}(𝔻)$. On $𝒜({𝕊}^1)$ we define the following norm $$\parallel u\parallel ={\parallel u\parallel }_{L^{\mathrm{\infty }}({𝕊}^1)}+{\left({\iint }_{𝔻}{|\nabla \tilde{u}|}^2\right)}^{1/2}$$ where is the harmonic extension of $u$ to $𝔻$. We prove that every isomorphism of the functional algebra $𝒜({𝕊}^1)$ is a quasitsymmetric change of variables on ${𝕊}^1$.

We study the irreducibility of the Dirichlet polynomial $P_G(s)$ when $G$ is the alternating group on $p+1$ elements with $p$ prime and we prove that $P_G(s)$ is irreducible for infinitely many choiches of $p$.

Let $X$ be a smooth projective curve over an algebraically closed field of characteristic $\&gt;2$. Consider the dual pair $H={\mathrm{SO}}_{2m},G={\mathrm{Sp}}_{2n}$ over $X$ with $H$ split. Write ${\mathrm{Bun}}_G$ and ${\mathrm{Bun}}_H$ for the stacks of $G$-torsors and $H$-torsors on $X$. The theta-kernel ${\mathrm{Aut}}_{G,H}$ on ${\mathrm{Bun}}_G\times {\mathrm{Bun}}_H$ yields theta-lifting functors $F_G:\mathrm{D}\left({\mathrm{Bun}}_H\right)\to \mathrm{D}\left({\mathrm{Bun}}_G\right)$ and $F_H:\mathrm{D}\left({\mathrm{Bun}}_G\right)\to \mathrm{D}\left({\mathrm{Bun}}_H\right)$ between the corresponding derived categories. We describe the relation of these functors with Hecke operators. In two particular cases these functors realize the geometric Langlands functoriality for the above pair (in the non...

Se $X$ ed $Y$ sono spazi topologici, una funzione $f:X\to Y$ è detta regolarmente chiusa [5] se essa trasforma ogni insieme regolarmente chiuso di $X$ in un insieme chiuso di $Y$. Si dimostra che una funzione regolarmente chiusa $f:X\to Y$ risulta chiusa se $X$ è normale.

We study the problem of correctors in the framework of the homogenization of linear parabolic equations posed in a heterogeneous medium $\mathrm{\Omega }$ made of two materials. The first one is located in a set $F_{ϵ}$ of cylindrical parallel fibers periodically distributed with a period of size $ϵ$, and the second one is located in the "matrix" $M_{ϵ}=\mathrm{\Omega }\setminus F_{ϵ}$. The ratio between the conductivity coefficients of the two materials is of order $1/{ϵ}^2$. After writing the homogenized problem, we give a corrector result and prove...

Let $f(\cdot ,t)$ be the probability density function representing the solution of Kac's Boltzmann-like equation at time $t$, with initial data $f_0$, and let $g_{\sigma }$ be the Gaussian density with zero mean and variance ${\sigma }^2$, ${\sigma }^2$ being the value of the second moment of $f_0$. Henry McKean Jr. put forward the conjecture that the total variation distance between $f(\cdot ,t)$ and $g_{\sigma }$ goes to zero, as $t\to +\mathrm{\infty }$, with an exponential rate equal to $-1/4$. This lecture aims at explaining the main efforts made to a view to validating this conjecture,...

We give a new proof using the classic Calderón-Zygmund decomposition that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space $L^{p(\cdot )}$ whenever the exponent function $p(\cdot )$ satisfies log-Hölder continuity conditions. We include the case where $p(\cdot )$ assumes the value infinity. The same proof also shows that the fractional maximal operator $M_a$, , maps $L^{p(\cdot )}$ into $L^{q(\cdot )}$, where $1/p(\cdot )-1/q(\cdot )=a/n$.

We prove that the $\mathrm{\Gamma }$-limit in $L_{\mu }^{\mathrm{\infty }}$ of a sequence of supremal functionals of the form $F_k(u)={\mu -\mathrm{ess}\sup }_{\mathrm{\Omega }}{f}_k(x,u)$ is itself a supremal functional. We show by a counterexample that, in general, the function which represents the $\mathrm{\Gamma }$-lim $F(\cdot ,B)$ of a sequence of functionals $F_k(u,B)={\mu -\mathrm{ess}\sup }_B{f}_k(x,u)$ can depend on the set $B$ and wegive a necessary and sufficient condition to represent $F$ in the supremal form$F(u,B)={\mu -\mathrm{ess}\sup }_{B}f(x,u)$. As a corollary, if $f$ represents a supremal functional, then the level convex envelope of $f$ represents its weak* lower semicontinuous envelope. ...

The following results are proved: (i) if $\alpha$, $\beta \in {ℝ}^{+}$ and $A$ is a sectorial operator, then the set $\{t^{\alpha }{A}^{\beta }(t+A);t>0\}$ is bounded; (ii) the same set of operators is R-bounded if $A$ is R-sectorial.

The aim of this work is to prove a chain rule and an $L^1$-lower semicontinuity theorems for integral functional defined on $BV(\mathrm{\Omega })$. Moreover we apply this result in order to obtain new relaxation and $\mathrm{\Gamma }$-convergence result without any coerciveness and any continuity assumption of the integrand $f(x,s,p)$ with respect to the variable $s$.

The purpose of this paper is to investigate efficient representations of the residue classes modulo $q$, by performing sum and product set operations starting from a given subset $A$ of ${\mathbb{Z}}_q$. We consider the case of very small sets $A$ and composite $q$ for which not much seemed known (nontrivial results were recently obtained when $q$ is prime or when log $\left|A\right|\sim logq$). Roughly speaking we show that all residue classes are obtained from a $k$-fold sum of an $r$-fold product set of $A$, where $r\ll logq$ and $logk\ll logq$, provided the...

The range of the bigenus $P_2$ is one of the unsolved problems concerning smooth complex projective regular threefolds of general type with $p_g=0$: The examples in the literature have $P_2\le 5$. In the present paper we present a non-singular threefold with $p_g=q_1=q_2=0$; $P_2=6$; the bicanonical map is stably birational.

We study the codimension two locus $H$ in ${𝒜}_g$ consisting of principally polarized abelian varieties whose theta divisor has a singularity that is not an ordinary double point. We compute the class $\left[H\right]\in C{H}^2\left({𝒜}_g\right)$ for every $g$. For $g=4$, this turns out to be the locus of Jacobians with a vanishing theta-null. For $g=5$, via the Prym map we show that $H\subset {𝒜}_5$ has two components, both unirational, which we describe completely. We then determine the slope of the effective cone of $\overline{{𝒜}_5}$ and show that the component $\overline{N_0^{\text{'}}}$ of the Andreotti-Mayer...

Let $X$, $X^{\prime }$, $Y$ be smooth projective complex curves with $Y$ curve of genus $\ge 1$. Let $d$ be an integer $\ge 3$, let $\underset{¯}{e}=(e_1,\mathrm{\dots },e_r)$ be a partition of $d$ and let $|e|={\sum }_{i=1}^r(e_i-1)$. Let $X\stackrel{𝜋}{\to }{X}^{\prime }\stackrel{𝑓}{\to }Y$ be a sequence of coverings where $\pi$ is a degree 2 branched covering and $f$ is a degree $d$ covering, with monodromy group $S_d$, branched in $n_2+1$ points, one of which is special point $c$ whose local monodromy has cycle type given by $\underset{¯}{e}$. Moreover the branch locus of the covering $\pi$ is contained in $f^{-1}(c)$. In this paper we prove the irreducibility...

We consider the Grassmannian $𝔾r(k,n)$ of $(k+1)$-dimensional linear subspaces of $V_n=H^0({\mathbb{P}}^1,{𝒪}_{{\mathbb{P}}_1}(n))$. We define ${𝔛}_{k,r,d}$ as the classifying space of the $k$-dimensional linear systems of degree $n$ on ${\mathbb{P}}^1$, whose bases realize a fixed number $r$ of polynomial relations of fixed degree $d$, say $r$ syzygies of degree $d$. Firstly, we compute the dimension of ${𝔛}_{k,r,d}$. In the second part we make a link between ${𝔛}_{k,r,d}$ and the Poncelet varieties. In particular, we prove that the existence of linear syzygies implies the existence of singularities on the...