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Knots and links which are closed 3-braids are a very special class. Like 2-bridge knots and links, they are simple enough to admit a complete classification. At the same time they are rich enough to serve as a source of examples on which,…
Dual encoder architectures like Clip models map two types of inputs into a shared embedding space and predict similarities between them. Despite their wide application, it is, however, not understood how these models compare their two…
We calculate the spectrum of the diagram for each finite $3$-transposition group. Such graphs with a given minimal eigenvalue have occurred in the context of compact Griess subalgebras of vertex operator algebras.
The control of electromechanical responses within bonding regions is essential to face frontier challenges in nanotechnologies, such as molecular electronics and biotechnology. Here, we present I\b{eta}-nanocellulose as a potentially new…
We find that several layer-phase group-III monochalcogenides, including GaS, GaSe and InSe, are piezoelectric in the monolayer form. First-principles calculations reveal that the piezoelectric coefficients of monolayer GaS, GaSe and InSe…
In this article we determine a generating set of rational invariants of minimal cardinality for the action of the orthogonal group $\mathrm{O}_3$ on the space $\mathbb{R}[x,y,z]_{2d}$ of ternary forms of even degree $2d$. The construction…
We show that if $X$ is an indecomposable $PD_3$-complex and $\pi_1(X) is the fundamental group of a reduced finite graph of finite groups but is not virtually cyclic then $X$ is orientable, the underlying graph is a tree, all the edge…
A representation $\pi$ of a locally compact group $G$ is called \e{trace class}, if for every test function $f$ the induced operator $\pi(f)$ is a trace class operator. The group $G$ is called \e{trace class}, if every $\pi\in G$ is trace…
We develop a mechanism of "isotropy separation for compact objects" that explicitly describes an invertible $G$-spectrum through its collection of geometric fixed points and gluing data located in certain variants of the stable module…
Piezoelectric and ferroelectric properties in the two dimensional (2D) limit are highly desired for nanoelectronic, electromechanical, and optoelectronic applications. Here we report the first experimental evidence of out-of-plane…
The compounds exhibit piezoelectricity, which demands to break inversion symmetry, and then to be a semiconductor. For $\mathrm{Ga_2O_3}$, the orthorhombic case ($\epsilon$-$\mathrm{Ga_2O_3}$) of common five phases breaks inversion…
A subperiodic group is a group of motions of $d$-dimensional Euclidean space $\R^d$ which contains a translation lattice $\Z^r$ of rank $r < d$ as a subgroup of finite index. A classification into abstract group isomorphism classes is…
We develop a general theory of (extended) inner autoequivalences of objects of any 2-category, generalizing the theory of isotropy groups to the 2-categorical setting. We show how dense subcategories let one compute isotropy in the presence…
Epitaxially strained SrRuO3 films have been a model system for understanding the magnetic anisotropy in metallic oxides. In this paper, we investigate the anisotropy of the Ru 4d and O 2p electronic structure and magnetic properties using…
Vision-Language Models like CLIP are extensively used for inter-modal tasks which involve both visual and text modalities. However, when the individual modality encoders are applied to inherently intra-modal tasks like image-to-image…
This article is devoted to the investigation of semidirect products of groups of loops and groups of diffeomorphisms of finite and infinte dimensional real, complex and quaternion manifolds. Necessary statements about quaternion manifolds…
We study the problem of determining the isomorphism classes of the virtually cyclic subgroups of the n-string braid groups B_n(S^2) of the 2-sphere S^2. If n is odd, or if n is even and sufficiently large, we obtain the complete…
We show that the intersection of three subgroups in a free group is related to the computation of the third homotopy group $\pi_3$. This generalizes a result of Gutierrez-Ratcliffe who relate the intersection of two subgroups with the…
This paper is a survey of some of the most elementary consequences of the JSJ-decomposition and geometrization for knot and link complements in the 3-sphere. Formulated in the language of graphs, the result is the construction of a…
Let $A \leq G$ be a subgroup of a group $G$. An $A$-complement of $G$ is a subgroup $H$ of $G$ such that $G = A H$ and $A \cap H = \{1\}$. The \emph{classifying complements problem} asks for the description and classification of all…