First we introduce the notion of Killing structure Jacobi operator for real hypersurfaces in the complex hyperbolic quadric ${Q^m}^*=SO^0_{2,m}/SO_2 SO_m$ . Next we give a complete classification of real hypersurfaces in ${Q^m}^*=SO^0_{2,m}/SO_2 SO_m$ with Killing structure Jacobi operator.

It was shown by Usher that any fiber sum of Lefschetz fibrations over $S^2$ is minimal, which was conjectured by Stipsicz. We prove that the converse does not hold by showing that there exists a genus-2 indecomposable minimal Lefschetz fibration (IMLF for short).

In this short note, we give a lower bound on the Loewy length of the center of a block of a group algebra in terms of a defect group.

In [4], we introduced special types of fusions, so called simple-ribbon fusions on links. A knot obtained from the trivial knot by a finite sequence of simple-ribbon fusions is called a simple-ribbon knot. Every ribbon knot with $łeq 9$ crossings is a simple-ribbon knot. In this paper, we give a formula for the Alexander polynomials of simple-ribbon knots. Using the formula, we determine if a knot with $10$ crossings is a simple-ribbon knot. Every simple-ribbon fusion can be realized by ``elementary" simple-ribbon fusions. We call a knot an $m$-simple-ribbon knot if the knot is obtained from the trivial knot by a finite sequence of elementary $m$-simple-ribbon fusions for a fixed positive integer $m$. We provide a condition for a simple-ribbon knot to be both of an $m$-simple-ribbon knot and an $n$-simple-ribbon knot for positive integers $m$ and $n$.

The time-global existence of unique smooth positive solutions to the reaction diffusion equations of the Keener-Tyson model for the Belousov-Zhabotinsky reaction in the whole space is established with bounded non-negative initial data. Deriving estimates of semigroups and time evolution operators, and applying the maximum principle, the unique existence and the positivity of solutions are ensured by construction of time-local solutions from certain successive approximation. Invariant regions and large time behavior of solutions are also discussed.

We introduce and study a class of field extensions that we call $pre$-$Galois$; viz. extensions that become Galois after some linearly disjoint Galois base change $L/k$. Among them are $geometrically Galois$ extensions of $\kappa(T)$, with $\kappa$ a field: extensions that become Galois and remain of the same degree over $\overline \kappa(T)$. We develop a pre-Galois theory that includes a Galois correspondence, and investigate the corresponding variants of the inverse Galois problem. We provide answers in situations where the classical analogs are not known. In particular, for every finite simple group $G$, some power $G^n$ is a geometric Galois group over $k$, and is a pre-Galois group over $k$ if $k$ is Hilbertian. For every finite group $G$, the same conclusion holds for $G$ itself ($n=1$) if $k=\mathbb{Q}^{\rm ab}$ and $G$ has a weakly rigid tuple of conjugacy classes; and then $G$ is a regular Galois group over an extension of $\mathbb{Q}^{\rm ab}$ of degree dividing the order of Out$(G)$. We also show that the inverse problem for pre-Galois extensions over a field $k$ (that every finite group is a pre-Galois group over $k$) is equivalent to the $a$ $priori$ stronger inverse Galois problem over $k$, and similarly for the geometric vs. regular variants.

In this paper, we shall establish the Mackey formulas in the following two setups: (i) on the tensor induction and restriction functors on the modules over cyclotomic Hecke algebras (Ariki-Koike algebras) and their standard subalgebras of parabolic subgroups; (ii) on the Bezrukavnikov-Etingof induction and restriction functors [3] among categories $\mathcal{O}$ [11] of rational Cherednik algebras for the complex reflection group of type $G(r,1,n)$ and their parabolic subgroups.

In this paper we are concerned with weighted boundary limits of monotone Sobolev functions in Orlicz spaces on bounded $(\eta,\psi)$-John domains in a metric space. We also deal with Lindelöf type theorems for monotone Sobolev functions on uniform domains in a metric space.

In this note we show how the main conjecture of the Iwasawa theory over $\mathbb Q$ has a natural place in the context of the Galois representation of $Gal (\bar{\mathbb Q}/\mathbb Q)$ on the etale pro-$p$ fundamental group of the projective line minus three points. However we still need to assume the Vandiver conjecture to get a proof of the main conjecture in this context.

We introduce a topological combinatorial game called the Region Smoothing Swap Game. The game is played on a game board derived from the connected shadow of a link diagram on a (possibly non-orientable) surface by smoothing at crossings. Moves in the game are performed on regions of the diagram and can switch the direction of certain crossings' smoothings. The players' goals relate to the connectedness of the diagram produced by game play.

We introduce the notion of a $nested$ $open$ $book$, a submanifold equipped with an open book structure compatible with an ambient open book, and describe in detail the special case of a $push$-$off$ of the binding of an open book. This enables us to explicitly describe a natural open book decomposition of a fibre connected sum of two open books along their bindings, provided they are diffeomorphic and admit an open book structure themselves. Furthermore, we apply the results to contact open books, showing that the natural open book structure of a contact fibre connected sum of two adapted open books along their contactomorphic bindings is again adapted to the resulting contact structure.

Given a quaternionic manifold $M$ with a certain $\mathrm{U}(1)$-symmetry, we construct a hypercomplex manifold $M'$ of the same dimension. This construction generalizes the quaternionic Kähler/ hyper-Kähler-correspondence. As an example of this construction, we obtain a compact homogeneous hypercomplex manifold which does not admit any hyper-Kähler structure. Therefore our construction is a proper generalization of the quaternionic Kähler/hyper-Kähler-correspondence.