Related papers: On the Maximum Flatness Factor over Unimodular Lat…

The Flat Wall Theorem of Robertson and Seymour states that there is some function $f$, such that for all integers $w,t>1$, every graph $G$ containing a wall of size $f(w,t)$, must contain either (i) a $K_t$-minor; or (ii) a small subset…

The Theta rank of a finite point configuration $V$ is the maximal degree necessary for a sum-of-squares representation of a non-negative linear function on $V$. This is an important invariant for polynomial optimization that is in general…

We characterize supersolvable lattices in terms of a certain modular type relation. McNamara and Thomas earlier characterized this class of lattices as those graded lattices having a maximal chain that consists of left-modular elements. Our…

We prove that the very simple lattices which consist of a largest, a smallest and $2n$ pairwise incomparable elements where $n$ is a positive integer can be realized as the lattices of intermediate subfactors of finite index and finite…

We introduce a vector-valued generalization of the Epstein zeta functions associated with the root lattices of ADE-type Lie algebras. The quadratic forms defining these lattices correspond to the Gram matrices of the simple roots. Using the…

We consider a two-dimensional analogue of Jacobi theta functions and prove that, among all lattices $\Lambda \subset \mathbb{R}^2$ with fixed density, the minimal value is maximized by the hexagonal lattice. This result can be interpreted…

The smoothing parameter $\eta_{\epsilon}(\mathcal{L})$ of a Euclidean lattice $\mathcal{L}$, introduced by Micciancio and Regev (FOCS'04; SICOMP'07), is (informally) the smallest amount of Gaussian noise that "smooths out" the discrete…

We study lattice fermions from the viewpoint of spectral graph theory (SGT). We find that a fermion defined on a certain lattice is identified as a spectral graph. SGT helps us investigate the number of zero eigenvalues of lattice Dirac…

Cellular solids and micro-lattices are a class of lightweight architected materials that have been established for their unique mechanical, thermal, and acoustic properties. It has been shown that by tuning material architecture, a…

The zeta-dimension of a set A of positive integers is the infimum s such that the sum of the reciprocals of the s-th powers of the elements of A is finite. Zeta-dimension serves as a fractal dimension on the positive integers that extends…

I review recent approaches to constructing supersymmetric lattice theories focusing in particular on the concept of topological twisting. The latter technique is shown to expose a nilpotent, scalar supersymmetry which can be implemented…

We study Zariski-like topologies on a proper class $X\varsubsetneqq L$ of a complete lattice $\mathcal{L}=(L,\wedge ,\vee ,0,1)$. We consider $X$ with the so called classical Zariski topology $(X,\tau ^{cl})$ and study its topological…

In this paper, we investigate two properties concerning the unimodality of the $\delta$-vectors of lattice polytopes, which are log-concavity and alternatingly increasingness. For lattice polytopes $\mathcal{P}$ of dimension $d$, we prove…

We prove a highly uniform stability or "almost-near" theorem for dual lattices of lattices $L \subseteq \Bbb R^n$. More precisely, we show that, for a vector $x$ from the linear span of a lattice $L \subseteq \Bbb R^n$, subject to…

Is module-lattice reduction better than unstructured lattice reduction? This question was highlighted as 'Q8' in the Kyber NIST standardization submission (Avanzi et al., 2021), as potentially affecting the concrete security of Kyber and…

We study magnetic geometries with Lifshitz and/or hyperscaling violation exponents (both with a hard wall cutoff in the IR and a smooth black brane horizon) which have a complex scalar field which couples to the magnetic field. The complex…

We study topological aspects of a compact lattice superconductor, and show that the characteristic energy splitting, $\Delta$, between almost degenerate ground states, is simply related to a novel order parameter $\tilde W$, which is…

Let $x_\alpha$ be a net in a locally solid vector lattice $(X,\tau)$; we say that $x_\alpha$ is unbounded $\tau$-convergent to a vector $x\in X$ if $\lvert x_\alpha-x \rvert\wedge w \xrightarrow{\tau} 0$ for all $w\in X_+$. In this paper,…

We study the lattice effects on the simple holographic toy model; massive U(1) gauge theory for the bulk action. The mass term is for the U(1) gauge symmetry breaking in the bulk. Without the lattice, the AC conductivity of this model shows…

This work concerns representations of a finite flat group scheme $G$, defined over a noetherian commutative ring $R$. The focus is on lattices, namely, finitely generated $G$-modules that are projective as $R$-modules, and on the full…