Eisermann and Lamm introduced a notion of symmetric equivalence among symmetric union diagrams and studied it using a refined form of the Jones polynomial. We introduced invariants of symmetric equivalence via refined versions of topological spin models and provided a partial answer to a question left open by Eisermann and Lamm. In this paper we adopt a new approach to the symmetric equivalence problem and give a complete answer to the original question left open by Eisermann and Lamm.

We introduce the 2-parity colour. It is a theory of parity for a large class of virtual links, defined using the interaction between orientations of the link components and a certain type of colouring. The 2-colour parity is an extension of the Gaussian parity, to which it reduces on virtual knots. We show that the 2-colour parity descends to a parity on free links. We compare the 2-colour parity to other parity theories of virtual links, focusing on a theory due to Im and Park. The 2-colour parity yields a strictly stronger invariant than the Im-Park parity.

We introduce an invariant, the 2-colour writhe, that takes the form of a string of integers. The 2-colour writhe is a concordance invariant, and so obstructs sliceness. It is also an obstruction to ($\pm$)-amphichirality and chequerboard colourability within a concordance class.

Let $S_{\boldsymbol{\lambda} }$ be a weighted shift on a rooted directed tree with one branching vertex $\widetilde{u}$, $\eta $ branches ($2\leq \eta \lt\infty $) and positive weight sequence $\boldsymbol{\lambda }$. We define a collection of (classical) weighted shifts, the so-called ``the $i$-th branching weighted shifts'' $W^{(i)}$ for $0\leq i\leq \eta $, whose weights are derived from those of $S_{\boldsymbol{\lambda}}$. In this note we discuss the relationships between $n$-contractivity, $n$-hypercontractivity and hyponormality of $S_{\boldsymbol{\lambda} }$ and these properties for the $W^{(i)}$ $(0\leq i\leq \eta )$.

We prove that for genus $g=3,4$, the extended mapping class group $\text{Mod}^{\pm}(S_g)$ can be generated by two elements of finite orders. But for $g=1$, $\text{Mod}^{\pm}(S_1)$ cannot be generated by two elements of finite orders.

We give a characterization of projective spaces for quasi-log canonical pairs from the Mori theoretic viewpoint.

Let the compact torus $T^{n-1}$ act on a smooth compact manifold $X^{2n}$ effectively with nonempty finite set of fixed points. We pose the question: what can be said about the orbit space $X^{2n}/T^{n-1}$ if the action is cohomologically equivariantly formal (which essentially means that $H^{\rm odd}(X^{2n};\mathbb{Z})=0$)? It happens that homology of the orbit space can be arbitrary in degrees $3$ and higher. For any finite simplicial complex $L$ we construct an equivariantly formal manifold $X^{2n}$ such that $X^{2n}/T^{n-1}$ is homotopy equivalent to $\Sigma^3L$. The constructed manifold $X^{2n}$ is the total space of a projective line bundle over the permutohedral variety hence the action on $X^{2n}$ is Hamiltonian and cohomologically equivariantly formal. We introduce the notion of an action in $j$-general position and prove that, for any simplicial complex $M$, there exists an equivariantly formal action of complexity one in $j$-general position such that its orbit space is homotopy equivalent to $\Sigma^{j+2}M$.

We consider the problem of constructing weak solutions to the Itô and to the Stratonovich stochastic differential equations having critical-order singularities in the drift and critical-order discontinuities in the dispersion matrix.

The subgroup posets of finite groups are illustrated by graphs (Hasse diagrams). The properties of these graphs have been studied by many researchers — initiated by K. Brown and D. Quillen for $p$-subgroups. We consider the opposite direction, that is, a realization problem: Given a graph, when does a finite group with its Hasse diagram being the graph exist?, and if any, classify all such finite groups. The Hasse diagram of a group is not arbitrary — it has the top and the bottom vertices, and they are connected by paths of edges. We divide such graphs into two types ``branched'' and “unbranched”, where unbranched graphs are birdcage-shaped, and finite groups with their Hasse diagrams being such graphs are called birdcage groups. We completely classify the unbranched case: A birdcage group is either a cyclic group of prime power order or a semidirect product of two cyclic groups of prime orders (). In the former, the Hasse diagram is a straight line (a birdcage with a single bar) and in the latter, a birdcage with all bars being of length 2.

In this paper we show that any two birational Mori fiber spaces of $\mathbb{Q}$-factorial gklt g-pairs are connected by a finite sequence of Sarkisov links.

We investigate the existence of a holomorphic and isometric immersion in the complex projective space for the complete Ricci-flat Kähler metrics constructed by M. B. Stenzel in [15] on the cotangent bundle of a compact, rank one, globally symmetric space.

This manuscript is concerned with long-time dynamics for a laminated beam which consists of two identical layers of uniform thickness, taking into account that an adhesive of small thickness is bonding the two surfaces thereby producing an interfacial slip. Using the variable norm technique of Kato, we prove the global well-posedness of solutions. For asymptotic behavior, we apply the Energy Method. Assuming the control through a time-varying delay just on the transverse displacement of the beam, we establish the exponential decay of energy to the system by using an appropriate Lyapunov functional.

In mathematical physics, Minkowski space (or Minkowski space-time) is a combination of three-dimensional Euclidean space and time into a four-dimensional manifold.

The hyperbolic surface and de Sitter surface of a curve are defined in the spacelike hypersurface $M$ in Minkowski $4$-space and located, respectively, in hyperbolic 3-space and de Sitter 3-space. In this study, techniques from singularity theory were applied to obtain the generic shape of such surfaces and their singular value sets and the geometrical meanings of these singularities were investigated.

In this paper, we prove that the Bergman projection extends continuously to a projection from harmonic $L^1$-functions onto holomorphic $L^1$-functions and maps continuously $L^{\infty}$-functions onto the space of Bloch holomorphic functions in a certain class of infinite type, convex domains in $\mathbb{C}^2$.