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Daniel Levin

Publications and source records attributed to Daniel Levin.

7 recordsLinked to original sources
arXiv2026

Non-Euclidean unification of isoperimetric profiles and grand Lebesgue-Sobolev scales

Let $(X,d,\mu)$ be a complete separable metric measure space satisfying a doubling condition and a $(1,1)$-Poincar\'e inequality. We develop a rigorous framework unifying two lines of analysis: the isoperimetric-profile approach of Coulhon-Grigor'yan-Levin \cite{CGL2003} and the grand/small Lebesgue-Sobolev scale introduced by Fiorenza-Formica-Gogatishvili \cite{FFG2018}. An explicit profile-to-scale transform $\PhiX$, defined via an inverse integral of $\IX$, converts geometric data into grand Lebesgue parameters. Sharp, up to universal constants, embeddings $W^{1,1}(X) \hookrightarrow \mathcal{G}_X$ with explicit constants (Theorem \ref{thmmain}). A converse: controlled grand embeddings imply explicit lower bounds on $\IX$ (Theorem \ref{thmconverse}). Concrete examples in genuinely non-Euclidean settings: the Heisenberg group $\mathbb{H}^1$, a model manifold with logarithmic volume growth, and Gaussian measure on $\R^n$ treated as a locally doubling space. All arguments are carried out on general metric measure spaces without reference to charts or a smooth structure; the gradient is the upper gradient in the sense of Heinonen-Koskela, and perimeter is the outer Minkowski content.

math.FA↗
arXiv2024

Studies of Cherenkov Photon Production in PbF$_2$ Crystals using Proton Beams at Fermilab

Future lepton colliders such as the FCC-ee, CEPC, ILC, or a muon collider will collect large data samples that allow precision physics studies with unprecedented accuracy, especially when the data is collected by innovative state-of-the-art detectors. An electromagnetic calorimeter based on scintillating crystals, designed to separately record Cherenkov and scintillation light, can achieve precision measurements of electrons and photons without sacrificing jet energy resolution, given adequate light collection efficiency and separation. This paper presents initial measurements from a program aimed at developing such a calorimeter system for future colliders. We focus on using PbF2 crystals to enhance the understanding of Cherenkov light collection, marking the first step in this endeavor.

physics.ins-det↗
arXiv2024

The hierarchies of identities and closed products for multiple complexes

We consider infinite $\Z_\Z$-index complexes $\mathcal C$ of spaces with elements depending on a number of parameters, complete with respect to a linear associative regular inseparable multilinear product. The existence of nets of vanishing ideals of orders of and powers of differentials is assumed for subspaces of $\mathcal C$-spaces. In the polynomial case of orders and powers of the differentials, we derive the hierarchies of differential identities and closed multiple products. We prove that a set of maximal orders and powers for differentials, differential conditions, together with coherence conditions on indices of a complex $\mathcal C$ elements generate families of multi-graded differential algebras.

math.FA↗
arXiv2022

The extension of cochain complexes of meromorphic functions to multiplications

Let $\mathfrak g$ be an infinite-dimensional Lie algebra and $G$ be the algebraic completion of its module. Using a geometric interpretation in terms of sewing two Riemann spheres with a number of marked points, we introduce a multiplication between elements of two spaces $\mathcal{M}^k_m(\mathfrak g, G)$ and $\mathcal{M}^n_{m'}(\mathfrak g, G)$ of meromorphic functions depending on a number of formal complex parameters $(x_1, \ldots, x_k)$ and $(y_1, \ldots, y_n)$ with specific analytic and symmetry properties, and associated to $\mathfrak g$-valued series. These spaces form a chain-cochain complex with respect to a boundary-coboundary operator. The main result of the paper shows that the multiplication is defined by an absolutely convergent series and takes values in the space $\mathcal{M}^{k+n}_{m+m'}(\mathfrak g, G)$.

math.FA↗
arXiv2013

Learning from the past, predicting the statistics for the future, learning an evolving system

We bring the theory of rough paths to the study of non-parametric statistics on streamed data. We discuss the problem of regression where the input variable is a stream of information, and the dependent response is also (potentially) a stream. A certain graded feature set of a stream, known in the rough path literature as the signature, has a universality that allows formally, linear regression to be used to characterise the functional relationship between independent explanatory variables and the conditional distribution of the dependent response. This approach, via linear regression on the signature of the stream, is almost totally general, and yet it still allows explicit computation. The grading allows truncation of the feature set and so leads to an efficient local description for streams (rough paths). In the statistical context this method offers potentially significant, even transformational dimension reduction. By way of illustration, our approach is applied to stationary time series including the familiar AR model and ARCH model. In the numerical examples we examined, our predictions achieve similar accuracy to the Gaussian Process (GP) approach with much lower computational cost especially when the sample size is large.

q-fin.ST↗
arXiv2013

Plasma panel-based radiation detectors

The plasma panel sensor (PPS) is a gaseous micropattern radiation detector under current development. It has many operational and fabrication principles common to plasma display panels. It comprises a dense matrix of small, gas plasma discharge cells within a hermetically sealed panel. As in plasma display panels, it uses nonreactive, intrinsically radiation-hard materials such as glass substrates, refractory metal electrodes, and mostly inert gas mixtures. We are developing these devices primarily as thin, low-mass detectors with gas gaps from a few hundred microns to a few millimeters. The PPS is a high gain, inherently digital device with the potential for fast response times, fine position resolution (<50-mm RMS) and low cost. In this paper, we report on prototype PPS experimental results in detecting betas, protons, and cosmic muons, and we extrapolate on the PPS potential for applications including the detection of alphas, heavy ions at low-to-medium energy, thermal neutrons, and X-rays.

physics.ins-det↗