Related papers: A Spectral Gap Theorem in $SU(d)$
A. Iosevich and K. Taylor showed that compact subsets of $\mathbb R^d$ with Hausdorff dimension greater than $(d+1)/2$ contain trees with gaps in an open interval. Under the same dimensional threshold, we prove the analogous result where…
The spectrum-generating algebra (SGA) for charged superstrings placed in constant background magnetic fields is constructed. Contrary to the neutral string this algebra is characteristic of including the cyclotron frequency. The Regge…
The algebra dual to Woronowicz's deformation of the 2-\-di\-men\-sion\-al Euclidean group is constructed. The same algebra is obtained from $SU_{q}(2)$ via contraction on both the group and algebra levels.
We study dynamical gauge symmetry breaking via compactified space in the framework of SU(N) gauge theory in M^{d-1}\times S^1 (d=4,5,6) space-time. In particular, we study in detail the gauge symmetry breaking in SU(2) and SU(3) gauge…
Let $\Gamma_2\subseteq \Gamma_1$ be finitely generated subgroups of ${\rm GL}_{n_0}(\mathbb{Z}[1/q_0])$. For $i=1$ or $2$, let $\mathbb{G}_i$ be the Zariski-closure of $\Gamma_i$ in $({\rm GL}_{n_0})_{\mathbb{Q}}$, $\mathbb{G}_i^{\circ}$ be…
Spectral density in the pseudoscalar and vector channels is extracted from the SU(2) lattice quenched data. It is shown to consist of three sharp poles within the energy range accessible on the lattice.
We present an approach based on density-functional theory for the calculation of fundamental gaps of both finite and periodic two-dimensional (2D) electronic systems. The computational cost of our approach is comparable to that of total…
For $n>2$, given $\phi_1,...,\phi_n$ randomly chosen isometries of $S^2$, it is well-known that the group $\G$ generated by $\phi_1,...,\phi_n$ acts ergodically on $S^2$. It is conjectured in \cite{GJS} that for almost every choice of…
SU(N) Lie algebras possess discrete symmetries which can lead naturally to stable vector dark matter (DM). In this work, we consider the possibility that the dark SU(N) sector couples to the visible sector through the Higgs portal. We find…
An extra $SU(2)_D$ gauge factor is added to the well-known left-right extension of the standard model (SM) of quarks and leptons. Under $SU(2)_L \times SU(2)_R \times SU(2)_D$, two fermion bidoublets $(2,1,2)$ and $(1,2,2)$ are assumed. The…
Assuming projective determinacy, we extend Spector's strong version of the Spector-Gandy Theorem to all odd levels of the projective hierarchy: Theorem. For every space $X$ which is a finite product of the natural numbers $N$ and Baire…
We consider, over both the integers and finite fields, Szemer\'{e}di's theorem on $k$-term arithmetic progressions where the set $S$ of allowed common differences in those progressions is restricted and random. Fleshing out a line of…
We answer a question posed by Y. Elias et al. in [8] about possible spectral distortions of algebraic numbers. We provide a closed form for the spectral distortion of certain classes of cyclotomic polynomials. Moreover, we present a bound…
It is well known from the Perron-Frobenius theory that the spectral gap of a positive square matrix is positive. In this paper, we give a more quantitative characterization of the spectral gap. More specifically, using a complex extension…
The proof of Higgs mechanism in a weakly coupled lattice gauge theory in $d \geqslant 2$ is revisited. A new power series cluster expansion is applied and the mass gap is shown to exist for the observable $F_{\mu\nu}$.
Dark matter is postulated to transform under the non-Abelian $SU(2)_D$ gauge symmetry. Its connection to Dirac neutrino masses is explored.
Let $\mathcal A$ be the adjacency matrix of a random $d$-regular graph on $N$ vertices, and we denote its eigenvalues by $\lambda_1\geq \lambda_2\cdots \geq \lambda_{N}$. For $N^{2/3}\ll d\leq N/2$, we prove optimal rigidity estimates of…
We prove that the Lie algebra $\mathfrak{sl}_n(\textbf{F}_q)$ of traceless matrices over a finite field of characteristic $p$ can be generated by $2$ elements with exceptions when $(n, p)$ is $(3, 3)$ or $(4,2)$. In the latter cases, we…
We consider a family of measures $\mu$ supported in $\br^d$ and generated in the sense of Hutchinson by a finite family of affine transformations. It is known that interesting sub-families of these measures allow for an orthogonal basis in…
I review the appearence of integrable structures in the formulation of exact nonperturbative solutions to $4d$ supersymmetric quantum gauge theories. Various examples of ${\cal N}\geq 2$ SUSY Yang-Mills nonperturbative solutions are…