The study of the hypercyclicity of an operator is an old problem in mathematics; it goes back to a paper of Birkhoff in 1929 proving the hypercyclicity of the translation operators in the space of all entire functions with the topology of uniform convergence on compact subsets. This article studies the hypercyclicity of translation operators in some general reproducing kernel Hilbert spaces of entire functions. These spaces are obtained by duality in a complex separable Hilbert space $\mathcal{H}$ by means of an analytic $\mathcal{H}$-valued kernel. A link with the theory of de Branges spaces is also established. An illustrative example taken from the Hamburger moment problem theory is included.

Let $\mathcal{V}$ be an integral normal complex projective variety of dimension $n \geq 3$ and denote by $\mathcal{L}$ an ample line bundle on $\mathcal{V}$. By imposing that the linear system $\lvert\mathcal{L}\rvert$ contains an element $A = A_{1} + \cdots + A_{r}$, $r \geq 1$, where all the $A_{i}$'s are distinct effective Cartier divisors with $\mathrm{Pic}(A_{i}) = \mathbb{Z}$, we show that such a $\mathcal{V}$ is as special as the components $A_{i}$ of $A \in \lvert\mathcal{L}\rvert$. After making a list of some consequences about the positivity of the components $A_{i}$, we characterize pairs $(\mathcal{V}, \mathcal{L})$ as above when either $A_{1} \cong \mathbb{P}^{n-1}$ and $\mathrm{Pic}(A_{j}) = \mathbb{Z}$ for $j = 2, \ldots, r$, or $\mathcal{V}$ is smooth and each $A_{i}$ is a variety of small degree with respect to $[H_{i}]_{A_{i}}$, where $[H_{i}]_{A_{i}}$ is the restriction to $A_{i}$ of a suitable line bundle $H_{i}$ on $\mathcal{V}$.

Strong and weak $(1, 3)$ homotopies are equivalence relations on knot projections, defined by the first flat Reidemeister move and each of two different types of the third flat Reidemeister moves. In this paper, we introduce the cross chord number that is the minimal number of double points of chords of a chord diagram. Cross chord numbers induce a strong $(1, 3)$ invariant. We show that Hanaki's trivializing number is a weak $(1, 3)$ invariant. We give a complete classification of knot projections having trivializing number two up to the first flat Reidemeister moves using cross chord numbers and the positive resolutions of double points. Two knot projections with trivializing number two are both weak $(1, 3)$ homotopy equivalent and strong $(1, 3)$ homotopy equivalent if and only if they can be related by only the first flat Reidemeister moves. Finally, we determine the strong $(1, 3)$ homotopy equivalence class containing the trivial knot projection and other classes of knot projections.

In the present paper, we discuss the relationship between the Galois extension corresponding to the kernel of the pro-$l$ outer Galois representation associated to a hyperbolic curve over a number field and $l$-moderate points of the hyperbolic curve. In particular, we prove that, for a certain hyperbolic curve, the Galois extension under consideration is generated by the coordinates of the $l$-moderate points of the hyperbolic curve. This may be regarded as an analogue of the fact that the Galois extension corresponding to the kernel of the $l$-adic Galois representation associated to an abelian variety is generated by the coordinates of the torsion points of the abelian variety of $l$-power order. Moreover, we discuss an application of the argument of the present paper to the study of the Fermat equation.

We consider the percolation on the $k$-out graph $G_{\textup{out}}(n, k)$. The critical probability of it is $p_{c}=1/(k+\sqrt{k^{2}-k})$. Similarly to the random graph $G(n, p)$, in a scaling window $p_{c}(1+O(n^{-1/3}))$, the sequence of sizes of large components rescaled by $n^{-2/3}$ converges to the excursion lengths of a Brownian motion with some drift. Also, the size of the largest component is $O(\log n)$ in the subcritical phase, and $O(n)$ in the supercritical phase. The proof is based on the analysis of the exploration process.

We study zero divisors and minimal prime ideals in semirings, notably those of characteristic one. Thereafter we find a counterexample to the most obvious version of primary decomposition, but are able to establish a weaker version. Lastly, we study Evans' condition in this context.

We investigate the arithmetic properties of the coefficients for the normalized conformal mapping of the exterior of the Multibrot set. In this paper, an estimate of the prime factor of the denominator of these coefficients is given. Bielefeld, Fisher and Haeseler [1] presented Zagier's observation for the growth of the denominator of the coefficients for the Mandelbrot set, and Yamashita [14] verified it. Our study takes into consideration both Zagier's observation and Yamashita's estimate.

Let $(S, B_{S})$ be the log pair associated with a projective completion of a smooth quasi-projective surface $V$. Under the assumption that the boundary $B_{S}$ is irreducible, we obtain an algorithm to factorize any automorphism of $V$ into a sequence of simple links. This factorization lies in the framework of the log Mori theory, with the property that all the blow-ups and contractions involved in the process occur on the boundary. When the completion $S$ is smooth, we obtain a description of the automorphisms of $V$ which is reminiscent of a presentation by generators and relations except that the ``generators'' are no longer automorphisms. They are instead isomorphisms between different models of $V$ preserving certain rational fibrations. This description enables one to define normal forms of automorphisms and leads in particular to a natural generalization of the usual notions of affine and Jonquières automorphisms of the affine plane. When $V$ is affine, we show however that except for a finite family of surfaces including the affine plane, the group generated by these affine and Jonquières automorphisms, which we call the tame group of $V$, is a proper subgroup of $\mathrm{Aut}(V)$.

In this paper we study the backward uniqueness for parabolic equations with non-Lipschitz coefficients in time and space. The result presented here improves an old uniqueness theorem due to Lions and Malgrange [7] and some more recent results of Del Santo and Prizzi [5, 6].

We extend the notion of intrinsic linking to directed graphs. We give methods of constructing intrinsically linked directed graphs, as well as complicated directed graphs that are not intrinsically linked. We prove that the double directed version of a graph $G$ is intrinsically linked if and only if $G$ is intrinsically linked. One Corollary is that $\overline{J_{6}}$, the complete symmetric directed graph on 6 vertices (with 30 directed edges), is intrinsically linked. We further extend this to show that it is possible to find a subgraph of $\overline{J_{6}}$ by deleting 6 edges that is still intrinsically linked, but that no subgraph of $\overline{J_{6}}$ obtained by deleting 7 edges is intrinsically linked. We also show that $\overline{J_{6}}$ with an arbitrary edge deleted is intrinsically linked, but if the wrong two edges are chosen, $\overline{J_{6}}$ with two edges deleted can be embedded linklessly.

In this paper, we study power similarity of operators. In particular, we show that if $T \in \mathit{PS}(H)$ (defined below) for some hyponormal operator $H$, then $T$ is subscalar. From this result, we obtain that such an operator with rich spectrum has a nontrivial invariant subspace. Moreover, we consider invariant and hyperinvariant subspaces for $T \in \mathit{PS}(H)$.

We give a formula on the rotation number of a sequence of primitive vectors, which is a generalization of the formula on the rotation number of a unimodular sequence in [2].

We study Lauricella's hypergeometric function $F_{A}$ of $m$ variables and the system $E_{A}$ of differential equations annihilating $F_{A}$, by using twisted (co)homology groups. We construct twisted cycles with respect to an integral representation of Euler type of $F_{A}$. These cycles correspond to $2^{m}$ linearly independent solutions to $E_{A}$, which are expressed by hypergeometric series $F_{A}$. Using intersection forms of twisted (co)homology groups, we obtain twisted period relations which give quadratic relations for Lauricella's $F_{A}$.

We consider the space $[\Gamma, rr_{0}, v^{r}, \varrho]$ of all vector-valued holomorphic modular forms $f\colon \mathbb{H}_{n} \to \mathcal{Z}$ of transformation type \begin{equation*} f(MZ)=v^{r}(M)\det(CZ+D)^{r_{0}r}\varrho(CZ+D)f(Z). \end{equation*} $\varrho\colon \mathrm{GL}(n, \mathbb{C}) \to \mathrm{GL}(\mathcal{Z})$ is a rational representation on a finite dimensional complex vector space $\mathcal{Z}$. These spaces can be collected in a graded $A(\Gamma)$-module \begin{equation*} \mathcal{M} = \mathcal{M}_{\Gamma}(r_{0}, v, \varrho) := \bigoplus_{r \in \mathbb{Z}}[\Gamma, rr_{0}, v^{r}, \varrho]. \end{equation*} We treat in this paper some special cases in genus 2. The first one is essentially due to Wieber. Here the starting weight is $1/2$, the starting multiplier system is the multiplier system $v_{\Theta}$ and for $\varrho$ we take the second symmetric power of standard representation. Thus we consider a variant of this case and a new example. In this final case the starting weight is $1/2$, the starting multiplier system is the theta multiplier system $v_{\vartheta}$ and for $\varrho$ we take the standard representation. In all these cases we will determine the structure of $\mathcal{M}$.