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 $A,ϵ>0$ be arbitrary. Suppose that $x$ is a sufficiently large positive number. We prove that the number of integers $n\in (x,x+x^{\theta }]$, satisfying some natural congruence conditions, which cannot be written as the sum of three squares of primes is $\ll x^{\theta }{(\log x)}^{-A}$, provided that $\frac{7}{16}+ϵ\le \theta \le 1$.

We consider the stochastic differential equation $$dX(t)=a(X(t);\xi (t))dt+{\int }_{\mathrm{\Theta }}b(X(t);\theta ){𝒩}_p(dt;d\theta )$$ for $t\ge 0$ with the initial condition $X(0)=x_0$. We give sufficient conditions for the asymptotic stability of the semigroup ${\{P^t\}}_{t\ge 0}$ generated by the stochastic differential equation (1).

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 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...

For nonnegative integers a, b, c and positive integer n, let N(a,b,c;n) denote the number of representations of n by the form ${\sum }_{i=1}^a(x{²}_i+x_i{y}_i+y{²}_i)+2{\sum }_{j=1}^b(u{²}_j+u_j{v}_j+v{²}_j)+4{\sum }_{k=1}^c(r{²}_k+r_k{s}_k+s{²}_k)$. Explicit formulas for N(a,b,c;n) for some small values were determined by Alaca, Alaca and Williams, by Chan and Cooper, by Köklüce, and by Lomadze. We establish formulas for N(2,1,0;n), N(2,0,1;n), N(1,2,0;n), N(1,0,2;n) and N(1,1,1;n) by employing the (p, k)-parametrization of three 2-dimensional theta functions due to Alaca, Alaca and Williams.

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 $H$.

In this paper we introduce and investigate three new subclasses of $p$-valent analytic functions by using the linear operator $D_{\lambda ,p}^m(f*g)\left(z\right)$. The various results obtained here for each of these function classes include coefficient bounds, distortion inequalities and associated inclusion relations for $(n,\theta )$-neighborhoods of subclasses of analytic and multivalent functions with negative coefficients, which are defined by means of a non-homogenous differential equation.

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...