International Journal of Mathematics

The 3-loop invariant (or, the 3-loop polynomial) of a knot is a rational form (or, a polynomial) presenting the 3-loop part of the Kontsevich invariant of knots. In this paper, we calculate the 3-loop polynomial of knots obtained by plumbing the doubles of two knots; this class of knots includes untwisted Whitehead doubles. We construct the 3-loop invariant by calculating the rational version of the Aarhus integral of a surgery presentation. As a consequence, we obtain an explicit presentation of the 3-loop polynomial for the knots.

In this paper, we prove strong convergence of $q$-Gaussians with respect to a parameter $q$, which implies the spectrum of any self-adjoint non-commutative polynomial in $q$-Gaussians is continuously deformed with respect to $q$. With Bożejko’s Haagerup-type inequality, we follow Brannan’s approach to proving the asymptotic strong convergence of the free orthogonal quantum group.

Let G be a unipotent algebraic group defined over a $p$-adic field of characteristic zero. We denote by $G$ the set of rational points of G. It is a $p$-adic Lie group with Lie algebra denoted by $𝔤$. Let $\pi$ be an irreducible unitary representation of $G$ in a Hilbert space ${{\mathscr{H}}}_{\pi }$, $f$ be a linear form on $𝔤$ and $𝔥$ be a polarization at $f$. We denote by ${\chi }_f$ a character of $H=exp(𝔥)$ related to $f$. The aim of this study is to give a precise description of the space of semi-invariant distribution vectors $({{\mathscr{H}}}_{\pi }^{-\infty }{)}^{H,{\chi }_f}$ of $\pi$.

Finite distributive lattices whose join-meet ideals are of König type will be classified. Furthermore, a class of polyominoes whose polyomino ideals are of König type will be studied.

For a sequence ${\{M_n\}}_{n=1}^{\infty }$ of expanding matrices with $M_n\in M_d(\mathbb{Z})$ and a sequence ${\{D_n\}}_{n=1}^{\infty }$ of finite digit sets with $D_n\subset {\mathbb{Z}}^d$, the Moran measure ${\mu }_{\{M_n\},\{D_n\}}$ is defined by the infinite convolution

In this work, a general formulation, which is based on steady boundary layer problems for the Boltzmann equation, of a half-space problem is considered. The number of conditions on the indata at the interface needed to obtain well-posedness is investigated. The solutions will converge exponentially fast “far away” from the interface. For linearized kinetic half-space problems similar to the one of evaporation and condensation in kinetic theory, slowly varying modes might occur near regime transitions where the number of conditions needed to obtain well-posedness changes (corresponding to transition between evaporation and condensation, or subsonic and supersonic evaporation/condensation), preventing uniform exponential speed of convergence. However, those modes might be eliminated by imposing extra conditions on the indata at the interface. Flow velocities at the far end for which regime transitions occur are presented for Boltzmann equations: for monatomic and polyatomic single species and mixtures; as well as bosons and fermions.

We investigate when discrete, amenable groups have $C^{\ast }$-algebras of real rank zero. While it is known that this happens when the group is locally finite, the converse is an open problem. We show that if $C^{\ast }(G)$ has real rank zero, then all normal subgroups of $G$ that are elementary amenable and have finite Hirsch length must be locally finite.

We study the structure of the supersingular locus of the Rapoport–Zink integral model of the Shimura variety for $\mathrm{\text{GU}}(2,2)$ over a ramified odd prime with the special maximal parahoric level. We prove that the supersingular locus equals the disjoint union of two basic loci, one of which is contained in the flat locus, and the other is not. We also describe explicitly the structure of the basic loci. More precisely, the former one is purely $2$-dimensional, and each irreducible component is birational to the Fermat surface. On the other hand, the latter one is purely $1$-dimensional, and each irreducible component is birational to the projective line.