We extend the Genocchi polynomials and investigate their Fourier expansions and integral representations. We obtain their formulas at rational arguments in terms of Hurwitz zeta function and show an explicit relationship with Gaussian hypergeometric functions. Some known results for the classical Genocchi polynomials are also deduced.

Existence of weak (martingale) solutions and pathwise uniqueness are established for stochastic evolution equations driven by Lévy processes.

Let $G$ be a finite $p$-solvable group, where $p$ is an odd prime. Suppose that $\chi \in \operatorname{Irr}(G)$ lifts an irreducible $p$-Brauer character. If $G/N$ is a $p$-group, then we prove that the irreducible constituents of $\chi_{N}$ lift irreducible Brauer characters of $N$. This result was proven for $|G|$ odd by J.P. Cossey.

Motivated by the study of ribbon knots we explore symmetric unions, a beautiful construction introduced by Kinoshita and Terasaka in 1957. For symmetric diagrams $D$ we develop a two-variable refinement $W_{D}(s,t)$ of the Jones polynomial that is invariant under symmetric Reidemeister moves. Here the two variables $s$ and $t$ are associated to the two types of crossings, respectively on and off the symmetry axis. From sample calculations we deduce that a ribbon knot can have essentially distinct symmetric union presentations even if the partial knots are the same. If $D$ is a symmetric union diagram representing a ribbon knot $K$, then the polynomial $W_{D}(s,t)$ nicely reflects the geometric properties of $K$. In particular it elucidates the connection between the Jones polynomials of $K$ and its partial knots $K_{\pm}$: we obtain $W_{D}(t,t) = V_{K}(t)$ and $W_{D}(-1,t) = V_{K_{-}}(t) \cdot V_{K_{+}}(t)$, which has the form of a symmetric product $f(t) \cdot f(t^{-1})$ reminiscent of the Alexander polynomial of ribbon knots.

We consider the operation of Whitehead double on a component of a link and study the behavior of Milnor invariants under this operation. We show that this operation turns a link whose Milnor invariants of length $\leq k$ are all zero into a link with vanishing Milnor invariants of length $\leq 2k+1$, and we provide formulae for the first non-vanishing ones. As a consequence, we obtain statements relating the notions of link-homotopy and self $\Delta$-equivalence via the Whitehead double operation. By using our result, we show that a Brunnian link $L$ is link-homotopic to the unlink if and only if the link $L$ with a single component Whitehead doubled is self $\Delta$-equivalent to the unlink.

In this paper, we develop a moduli theory of transverse structures given by calibrations on foliated manifolds, including transverse Calabi--Yau structures. We show that the moduli space of the transverse structures is a smooth manifold of finite dimension under a cohomological assumption. We also prove a local Torelli type theorem. If the foliation is taut, we can construct a Riemannian metric on the set of transverse Riemannian structures. This metric induces a distance on the moduli space of the transverse structures given by a calibration. As an application, we show the moduli space of transverse Calabi--Yau structures is a Hausdorff and smooth manifold of finite dimension.

Let $\mathbf{F}$ be a smooth flow on a smooth manifold $M$ and $\mathcal{D}(\mathbf{F})$ be the group of diffeomorphisms of $M$ preserving orbits of $\mathbf{F}$. We study the homotopy type of the identity components $\mathcal{D}_{\id}(\mathbf{F})^{r}$ of $\mathcal{D}(\mathbf{F})$ with respect to distinct Whitney topologies $\mathsf{W}^{r}$, ($0 \leq r \leq \infty$). The main result presents a class of flows $\mathbf{F}$ for which $\mathcal{D}_{\id}(\mathbf{F})^{r}$ coincide for all $r$ and are either contractible or homotopy equivalent to the circle. The group $\mathcal{D}_{\id}(\mathbf{F})^{0}$ was studied in the author's paper [13]. Unfortunately that article contains a gap in estimations of continuity of local inverses of the so-called shift map. The present paper also repairs these estimations and shows that they hold under additional assumptions on the behavior of regular points of $\mathbf{F}$.

The well-posedness of hyperbolic initial boundary value problems is linked to the occurrence of zeros of the so-called Lopatinskiĭ determinant. For an important class of problems, the Lopatinskiĭ determinant vanishes in the hyperbolic region of the frequency domain and nowhere else. In this paper, we give a criterion that ensures that the hyperbolic region coincides with the projection of the forward cone. We give some examples of strictly hyperbolic operators that show that our criterion is sharp.

We study the boundary of the $3$-dimensional Rauzy fractal $\mathcal{E} \subset \mathbb{R} \times \mathbb{C}$ generated by the polynomial $P(x) = x^{4}-x^{3}-x^{2}-x-1$. The finite automaton characterizing the boundary of $\mathcal{E}$ is given explicitly. As a consequence we prove that the set $\mathcal{E}$ has $18$ neighboors where $6$ of them intersect the central tile $\mathcal{E}$ in a point. Our construction shows that the boundary is generated by an iterated function system starting with $2$ compact sets.

By definition, Kähler--Ricci solitons are defined on Fano manifolds. In this note, we shall generalize the notion of Kähler--Ricci solitons to the case of general polarized manifolds from the view point of K-energy, which are called ``generalized Kähler--Ricci solitons''. Moreover, ``generalized Kähler--Ricci solitons'' are also one of generalizations of constant scalar curvature Kähler metrics. Furthermore, we shall give a non-trivial example of a ``generalized Kähler--Ricci soliton''.

For a finite group $G$ and its subgroup $H$ which does not contain any normal subgroup of $G$ except the identity, criterions for the existence or non-existence of Hurwitz families of special type of $(G,H)$-coverings of the Riemann sphere are given.

We illustrate homology 3-spheres which never yield any lens spaces by integral Dehn surgery by using Ozsváth and Szabó's contact invariant.

A moment-angle complex $\mathcal{Z}_{K}$ is a compact topological space associated with a finite simplicial complex $K$. It is realized as a subspace of a polydisk $(D^{2})^{m}$, where $m$ is the number of vertices in $K$ and $D^{2}$ is the unit disk of the complex numbers $\mathbb{C}$, and the natural action of a torus $(S^{1})^{m}$ on $(D^{2})^{m}$ leaves $\mathcal{Z}_{K}$ invariant. The Buchstaber invariant $s(K)$ of $K$ is the largest integer for which there is a subtorus of rank $s(K)$ acting on $\mathcal{Z}_{K}$ freely. The story above goes over the real numbers $\mathbb{R}$ in place of $\mathbb{C}$ and a real analogue of the Buchstaber invariant, denoted $s_{\mathbb{R}}(K)$, can be defined for $K$ and $s(K)\leqq s_{\mathbb{R}}(K)$. In this paper we will make some computations of $s_{\mathbb{R}}(K)$ when $K$ is a skeleton of a simplex. We take two approaches to find $s_{\mathbb{R}}(K)$ and the latter one turns out to be a problem of integer linear programming and of independent interest.

In the previous paper we determined $\widetilde{\Aut}(X)$ of each Hopf surface $X = W/G$ with $W=\mathbf{C}^{2} - (0,0)$ so that its holomorphic automorphism group is given by $\Aut(X) = \widetilde{\Aut}(X)/G$. We calculate the group of connected components $\pi_{0}(\Aut(X))$ by reviewing the classification.