arXiv subjects

E. Wyman

Publications and source records attributed to E. Wyman.

9 recordsLinked to original sources

arXiv2026

High-Dimensional Signal Compression: Lattice Point Bounds and Metric Entropy

We study worst-case signal compression under an $\ell^2$ energy constraint, with coordinate-dependent quantization precisions. The compression problem is reduced to counting lattice points in a diagonal ellipsoid. Under balanced precision profiles, we obtain explicit, dimension-dependent upper bounds on the logarithmic codebook size. The analysis refines Landau's classical lattice point estimates using uniform Bessel bounds due to Olenko and explicit Abel summation.

cs.IT↗

arXiv2026

Spectral synthesis on Riemannian manifolds

We study spectral synthesis for measures supported on thin subsets of compact Riemannian manifolds. We prove that under natural non-concentration conditions, such measures admit quantitative spectral synthesis, with explicit stability bounds. We show that this phenomenon depends strongly on the underlying geometry. On the torus, synthesis holds under broad assumptions, while on the sphere we establish rigidity results demonstrating that synthesis can fail in a sharp sense. As consequences, we obtain quantitative approximation results and uncertainty principles for functions with thin spectral support. These results provide a unified framework connecting spectral synthesis, geometric structure, and stability on compact manifolds.

math.CA↗

arXiv2026

A Refinement of Vapnik--Chervonenkis' Theorem

Vapnik--Chervonenkis' theorem is a seminal result in machine learning. It establishes sufficient conditions for empirical probabilities to converge to theoretical probabilities, uniformly over families of events. It also provides an estimate for the rate of such uniform convergence. We revisit the probabilistic component of the classical argument. Instead of applying Hoeffding's inequality at the final step, we use a normal approximation with explicit Berry--Esseen error control. This yields a moderate-deviation sharpening of the usual VC estimate, with an additional factor of order $(\varepsilon\sqrt{n})^{-1}$ in the leading exponential term when $\varepsilon\sqrt{n}$ is large.

cs.LG↗

arXiv2025

Calibration Plan for the SBC 10-kg Liquid Argon Detector with 100 eV Target Threshold

The Scintillating Bubble Chamber (SBC) Collaboration is designing a new generation of low background, noble liquid bubble chamber experiments with sub-keV nuclear recoil threshold. These experiments combine the electronic recoil blindness of a bubble chamber with the energy resolution of noble liquid scintillation, and maintain electron recoil discrimination at higher degrees of superheat (lower nuclear recoil thresholds) than Freon-based bubble chambers. A 10-kg liquid argon bubble chamber has the potential to set world leading limits on the dark matter nucleon cross-section for $O(\mathrm{GeV}/c^{2})$ masses, and to perform a high statistics coherent elastic neutrino nuclear scattering measurement with reactor neutrinos. This work presents a detailed calibration plan to measure the detector response of these experiments, combining photoneutron scattering with two new techniques to induce sub-keV nuclear recoils: nuclear Thomson scattering and thermal neutron capture.

physics.ins-det↗

arXiv2024

Batch VUV4 Characterization for the SBC-LAr10 scintillating bubble chamber

The Scintillating Bubble Chamber (SBC) collaboration purchased 32 Hamamatsu VUV4 silicon photomultipliers (SiPMs) for use in SBC-LAr10, a bubble chamber containing 10~kg of liquid argon. A dark-count characterization technique, which avoids the use of a single-photon source, was used at two temperatures to measure the VUV4 SiPMs breakdown voltage ($V_{\text{BD}}$), the SiPM gain ($g_{\text{SiPM}}$), the rate of change of $g_{\text{SiPM}}$ with respect to voltage ($m$), the dark count rate (DCR), and the probability of a correlated avalanche (P$_{\text{CA}}$) as well as the temperature coefficients of these parameters. A Peltier-based chilled vacuum chamber was developed at Queen's University to cool down the Quads to $233.15\pm0.2$~K and $255.15\pm0.2$~K with average stability of $\pm20$~mK. An analysis framework was developed to estimate $V_{\text{BD}}$ to tens of mV precision and DCR close to Poissonian error. The temperature dependence of $V_{\text{BD}}$ was found to be $56\pm2$~mV~K$^{-1}$, and $m$ on average across all Quads was found to be $(459\pm3(\rm{stat.})\pm23(\rm{sys.}))\times 10^{3}~e^-$~PE$^{-1}$~V$^{-1}$. The average DCR temperature coefficient was estimated to be $0.099\pm0.008$~K$^{-1}$ corresponding to a reduction factor of 7 for every 20~K drop in temperature. The average temperature dependence of P$_{\text{CA}}$ was estimated to be $4000\pm1000$~ppm~K$^{-1}$. P$_{\text{CA}}$ estimated from the average across all SiPMs is a better estimator than the P$_{\text{CA}}$ calculated from individual SiPMs, for all of the other parameters, the opposite is true. All the estimated parameters were measured to the precision required for SBC-LAr10, and the Quads will be used in conditions to optimize the signal-to-noise ratio.

physics.ins-det↗

arXiv2024

Multi-linear forms, structure of graphs and Lebesgue spaces

Consider the operator $$T_Kf(x)=\int_{{\mathbb R}^d} K(x,y) f(y) dy,$$ where $K$ is a locally integrable function or a measure. The purpose of this paper is to study the multi-linear form $$ Λ^K_G(f_1, \dots, f_n)=\int \dots \int \prod_{ \{(i,j): 1 \leq i<j \leq n; E(i,j)=1 \} } K(x^i,x^j) \prod_{i=1}^n f_i(x^i) dx^i, $$ where $G$ is a connected graph on $n$ vertices, $E$ is the edge map on $G$, i.e $E(i,j)=1$ if and only if the $i$'th and $j$'th vertices are connected by an edge, $K$ is the aforementioned kernel, and $f_i: {\mathbb R}^d \to {\mathbb R}$, measurable. This paper establishes multi-linear inequalities of the form $$ Λ^K_G(f_1,f_2, \dots,f_n) \leq C {||f_1||}_{L^{p_1}({\mathbb R}^d)} {||f_2||}_{L^{p_2}({\mathbb R}^d)} \dots {||f_n||}_{L^{p_n}({\mathbb R}^d)}$$ and determines how the exponents depend on the structure of the kernel $K$ and the graph $G$.

math.CA↗

arXiv2022

Fractal dimension, approximation and data sets

The purpose of this paper is to study the fractal phenomena in large data sets and the associated questions of dimension reduction. We examine situations where the classical Principal Component Analysis is not effective in identifying the salient underlying fractal features of the data set. Instead, we employ the discrete energy, a technique borrowed from geometric measure theory, to limit the number of points of a given data set that lie near a $k$-dimensional hyperplane, or, more generally, near a set of a given upper Minkowski dimension. Concrete motivations stemming from naturally arising data sets are described and future directions outlined.

math.ST↗

arXiv2022

Uniform distribution and geometric incidence theory

A celebrated unit distance conjecture due to Erd\H os says that that the unit distances cannot arise more than $C_εn^{1+ε}$ times (for any $ε>0$) among $n$ points in the Euclidean plane (see e.g. \cite{SST84} and the references contained therein). In three dimensions, the conjectured bound is $Cn^{\frac{4}{3}}$ (see e.g. \cite{KMSS12} and \cite{Z19}). In dimensions four and higher, this problem, in its general formulation, loses meaning because the Lens example shows that one can construct a set of $n$ points in dimension $4$ and higher where the unit distance arises $\approx n^2$ times (see e.g. \cite{B97}). However, the Lens example is one-dimension in nature, which raises the possibility that the unit distance conjecture is still quite interesting in higher dimensions under additional structural assumptions on the point set. This point of view was explored in \cite{I19}, \cite{IS16}, \cite{IMT12}, \cite{IRU14}, \cite{OO15} and has led to some interesting connections between the unit distance problem and its continuous counterparts, especially the Falconer distance conjecture (\cite{Falc85}). In this paper, we study the unit distance problem and its variants under the assumption that the underlying family of point sets is uniformly distributed. We prove several incidence bounds in this setting and clarify some key properties of uniformly distributed sequences in the context of incidence problems in combinatorial geometry.

math.CO↗

arXiv2021

The VC-dimension and point configurations in ${\Bbb F}_q^2$

Let $X$ be a set and ${\mathcal H}$ a collection of functions from $X$ to $\{0,1\}$. We say that ${\mathcal H}$ shatters a finite set $C \subset X$ if the restriction of ${\mathcal H}$ yields every possible function from $C$ to $\{0,1\}$. The VC-dimension of ${\mathcal H}$ is the largest number $d$ such that there exists a set of size $d$ shattered by ${\mathcal H}$, and no set of size $d+1$ is shattered by ${\mathcal H}$. Vapnik and Chervonenkis introduced this idea in the early 70s in the context of learning theory, and this idea has also had a significant impact on other areas of mathematics. In this paper we study the VC-dimension of a class of functions ${\mathcal H}$ defined on ${\Bbb F}_q^d$, the $d$-dimensional vector space over the finite field with $q$ elements. Define $$ {\mathcal H}^d_t=\{h_y(x): y \in {\Bbb F}_q^d \},$$ where for $x \in {\Bbb F}_q^d$, $h_y(x)=1$ if $||x-y||=t$, and $0$ otherwise, where here, and throughout, $||x||=x_1^2+x_2^2+\dots+x_d^2$. Here $t \in {\Bbb F}_q$, $t \not=0$. Define ${\mathcal H}_t^d(E)$ the same way with respect to $E \subset {\Bbb F}_q^d$. The learning task here is to find a sphere of radius $t$ centered at some point $y \in E$ unknown to the learner. The learning process consists of taking random samples of elements of $E$ of sufficiently large size. We are going to prove that when $d=2$, and $|E| \ge Cq^{\frac{15}{8}}$, the VC-dimension of ${\mathcal H}^2_t(E)$ is equal to $3$. This leads to an intricate configuration problem which is interesting in its own right and requires a new approach.

math.CO↗