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Let W be an infinite Coxeter group. We initiate the study of the set E of limit points of "normalized" positive roots (representing the directions of the roots) of W. We show that E is contained in the isotropic cone of the bilinear form B…

Let $0\le \alpha \le \beta\le 1$. For any finite set $B\subset\mathbb{N}$, we show that there exists a set $A\subset\mathbb{N}$ such that $\underline{d}(A+B) = \alpha$ and $\bar{d}(A+B) = \beta$, where $\underline{d}(A+ B)$ and…

Nuclear density functional theory (DFT) is one of the main theoretical tools used to study the properties of heavy and superheavy elements, or to describe the structure of nuclei far from stability. While on-going efforts seek to better…

We explore a variety of unsolved problems in density functional theory, where mathematicians might prove useful. We give the background and context of the different problems, and why progress toward resolving them would help those doing…

Let $f: B^n \rightarrow {\mathbb R}$ be a $d+1$ times continuously differentiable function on the unit ball $B^n$, with $\max_{z\in B^n} \Vert f(z) \Vert=1$. A well-known fact is that if $f$ vanishes on a set $Z\subset B^n$ with a non-empty…

This paper concerns algorithms that give correct answers with (asymptotic) density $1$. A dense description of a function $g : \omega \to \omega$ is a partial function $f$ on $\omega$ such that $\left\{n : f(n) = g(n)\right\}$ has density…

For any real numbers $B \ge 1$ and $\delta \in (0, 1)$ and function $f: [0, B] \rightarrow \mathbb{R}$, let $d_{B; \delta} (f) \in \mathbb{Z}_{> 0}$ denote the minimum degree of a polynomial $p(x)$ satisfying $\sup_{x \in [0, B]} \big| p(x)…

For a real quadratic field $\mathbb{Q}(\sqrt{d})$, we study the norm-form energy $N = S_\zeta^2 - d \cdot S_L^2$, where $S_\zeta$ and $S_L$ are Lorentzian-weighted zero sums with $w(\rho) = 2/(\tfrac{1}{4} + \gamma^2)$. We prove three main…

Let $f: \mathbb{N} \to \mathbb{C}$ be a multiplicative function for which $$ \sum_{p : \, |f(p)| \neq 1} \frac{1}{p} = \infty. $$ We show under this condition alone that for any integer $h \neq 0$ the set $$ \{n \in \mathbb{N} : f(n) =…

Classical density functional theory (DFT) is a statistical mechanical theory for calculating the density profiles of the molecules in a liquid. It is widely used, for example. to calculate the density distribution of the molecules in the…

Density functional theory (DFT) is a powerful theoretical tool widely used in such diverse fields as computational condensed matter physics, atomic physics, and quantum chemistry. DFT establishes that a system of $N$ interacting electrons…

Given any 1-random set $X$ and any $r\in(0,1)$, we construct a set of intrinsic density $r$ which is computable from $r\oplus X$. For almost all $r$, this set will be the first known example of an intrinsic density $r$ set which cannot…

We characterize the phenomenon of "crowding" near the largest eigenvalue $\lambda_{\max}$ of random $N \times N$ matrices belonging to the Gaussian $\beta$-ensemble of random matrix theory, including in particular the Gaussian orthogonal…

We provide conditions which yield a strong law of large numbers for expressions of the form $1/N\sum_{n=1}^{N}F\big(X(q_1(n)),..., X(q_\ell(n))\big)$ where $X(n),n\geq 0$'s is a sufficiently fast mixing vector process with some moment…

A density functional theory (DFT) framework is presented that links functional derivatives of free-energy functionals to non-linear static density response functions in quantum many-body systems. Within this framework, explicit expressions…

Hecke studies the distribution of fractional parts of quadratic irrationals with Fourier expansion of Dirichlet series. This method is generalized by Behnke and Ash-Friedberg, to study the distribution of the number of totally positive…

Modern density functional approximations achieve moderate accuracy at low computational cost for many electronic structure calculations. Some background is given relating the gradient expansion of density functional theory to the WKB…

Quadric hypersurfaces are well-known to satisfy the Hasse principle. However, this is no longer true in the case of the Hasse principle for integral points, where counter-examples are known to exist in dimension 1 and 2. This work explores…

Let $f(x)$ be a polynomial of degree $n \ge 1$ with real coefficients and let $X \ge 2$ and $\delta \ge 0$ be real numbers. Let $\|\cdot\|$ be the distance to the nearest integer. We obtain upper bounds for the number of solutions to the…

Given $\rho\in(0, 1/3]$, let $\mu$ be the Cantor measure satisfying $\mu=\frac{1}{2}\mu f_0^{-1}+\frac{1}{2}\mu f_1^{-1}$, where $f_i(x)=\rho x+i(1-\rho)$ for $i=0, 1$. The support of $\mu$ is a Cantor set $C$ generated by the iterated…