arXiv subjects

Robert Schneider

Publications and source records attributed to Robert Schneider.

At least 19 recordsLinked to original sources

arXiv2025

Partition-theoretic model of prime distribution, II

In recent work by Botkin, Dawsey, Hemmer, Just and the present author, a deterministic model of prime number distribution is developed based on properties of integer partitions that gives almost exact estimates for $π(n)$, the number of primes less than or equal to positive integer $n$, up to $n=10{,}000$. In this follow-up paper, the author summarizes the ideas behind this partition-theoretic model of primes and formulates a computational model that is practically exact in its estimates of $π(n)$ up to $n=100,000$.

math.NT↗

arXiv2025

Partition-theoretic model of prime distribution

We make an application of ideas from partition theory to a problem in multiplicative number theory. We propose a deterministic model of prime number distribution, from first principles related to properties of integer partitions, that naturally predicts the prime number theorem as well as the twin prime conjecture. The model posits that, for $n\geq 2$, $$p_{n}\ =\ 1\ +\ 2\sum_{j=1}^{n-1}\left\lceil \frac{d(j)}{2}\right\rceil\ +\ \varepsilon(n),$$ where $p_k$ is the $k$th prime number, $d(k)$ is the divisor function, and $\varepsilon(k)$ is an explicit error term that is negligible asymptotically; both the main term and error term represent enumerative functions in our conceptual model. We refine the error term to give numerical estimates of $π(n)$ similar to those provided by the logarithmic integral, and much more accurate than $\operatorname{li}(n)$ up to $n=10{,}000$ where the estimates are {\it almost exact}. We then perform computational tests of unusual predictions of the model, finding limited evidence of predictable variations in prime gaps.

math.NT↗

arXiv2025

Algebra of overpartitions

In a 2022 paper, Dawsey, Just and the present author prove that the set of integer partitions, taken as a monoid under a partition multiplication operation I defined in my Ph.D. work, is isomorphic to the positive integers as a monoid under integer multiplication. In this note, I extend partition multiplication to the set of overpartitions, which are of much interest in partition theory. I prove the overpartitions form an Abelian group under partition multiplication. Moreover, the overpartitions and the positive rational numbers are isomorphic as multiplicative groups. I then prove further overpartition isomorphisms and discuss approaches to a ring theory of overpartitions.

math.NT↗

arXiv2025

Multimodal sequences and their generating functions

We define integer multimodal sequences, which are generalizations of unimodal sequences having multiple local peaks of equal size. The generating functions for multimodal sequences represent novel types of $q$-series that combine generating functions for both integer partitions and integer compositions. We prove a bijection between multimodal sequences of equal size (sum), and show that multimodal generating functions become finite series at roots of unity like the ``strange'' function of Kontsevich, quantum modular forms, and other examples of this phenomenon in the $q$-series literature.

math.NT↗

arXiv2025

On the $q$-factorization of power series

Any power series with unit constant term can be factored into an infinite product of the form $\prod_{n\geq 1} (1-q^n)^{-a_n}$. We give direct formulas for the exponents $a_n$ in terms of the coefficients of the power series, and vice versa, as sums over partitions. As examples, we prove identities for certain partition enumeration functions. Finally, we note $q$-analogues of our enumeration formulas.

math.CO↗

arXiv2025

Infinite series identities involving $π$ and $\ln 2$

We prove identities for six infinite series whose values involve linear combinations of $π$ and $\operatorname{ln} 2$, that do not appear in standard infinite series references.

math.GM↗

arXiv2024

Mathematics of the MML functional quantizer modules for VCV Rack software synthesizer

We detail the mathematical formulation of the line of "functional quantizer" modules developed by the Mathematics and Music Lab (MML) at Michigan Technological University, for the VCV Rack software modular synthesizer platform, which allow synthesizer players to tune oscillators to new musical scales based on mathematical functions. For example, we describe the recently-released MML Logarithmic Quantizer (LOG QNT) module that tunes synthesizer oscillators to the non-Pythagorean musical scale introduced by indie band The Apples in Stereo.

cs.SD↗

arXiv2024

Partition-theoretic Frobenius-type limit formulas

Using partition generating function techniques, we prove $q$-series analogues of a formula of Frobenius generalizing Abel's convergence theorem for complex power series. Frobenius' result states that for $|q|<1$, $\lim_{q\to 1}(1-q)\sum_{n\geq 1} f(n) q^n $ is equal to the average value $\lim_{N\to \infty}$ $\frac{1}{N}\sum_{k=1}^{N}f(k)$ of the sequence $\{f(n)\}$ as $n\to \infty$, if the average value exists.

math.NT↗

arXiv2023

Engineering 2D material exciton lineshape with graphene/h-BN encapsulation

Control over the optical properties of atomically thin two-dimensional (2D) layers, including those of transition metal dichalcogenides (TMDs), is needed for future optoelectronic applications. Remarkable advances have been achieved through alloying, chemical and electrical doping, and applied strain. However, the integration of TMDs with other 2D materials in van der Waals heterostructures (vdWHs) to tailor novel functionalities remains largely unexplored. Here, the near-field coupling between TMDs and graphene/graphite is used to engineer the exciton lineshape and charge state. Fano-like asymmetric spectral features are produced in WS$_{2}$, MoSe$_{2}$ and WSe$_{2}$ vdWHs combined with graphene, graphite, or jointly with hexagonal boron nitride (h-BN) as supporting or encapsulating layers. Furthermore, trion emission is suppressed in h-BN encapsulated WSe$_{2}$/graphene with a neutral exciton redshift (44 meV) and binding energy reduction (30 meV). The response of these systems to electron-beam and light probes is well-described in terms of 2D optical conductivities of the involved materials. Beyond fundamental insights into the interaction of TMD excitons with structured environments, this study opens an unexplored avenue toward shaping the spectral profile of narrow optical modes for application in nanophotonic devices.

cond-mat.mes-hall↗

arXiv2022

Nanoimprint strain-engineering of 2D semiconductors

Mechanical strain is a powerful tool to tune the optical and optoelectronic properties of atomically thin semiconductors. Inhomogeneous strain plays an important role in exciton funneling and the activation of single-photon emitters in 2D materials. Here, we create an inhomogeneous strain profile in a 2D semiconductor on a micrometer scale by a nanoimprint process. We present a nanoimprint setup, where a mold is used to apply pressure in a controlled way to a WS2 monolayer on a heated polymer layer. After printing, the strain created in the 2D semiconductor is verified by hyperspectral optical imaging. The developed nanoimprint technique is scalable and could be transferred to commercial nanoimprint machines.

physics.app-ph↗

arXiv2022

Excitonic Absorption Signatures of Twisted Bilayer WSe$_{2}$ by Electron Energy-Loss Spectroscopy

Moiré twist angle underpins the interlayer interaction of excitons in twisted van der Waals hetero- and homo-structures. The influence of twist angle on the excitonic absorption of twisted bilayer tungsten diselenide (WSe$_{2}$) has been investigated using electron energy-loss spectroscopy. Atomic-resolution imaging by scanning transmission electron microscopy was used to determine key structural parameters, including the nanoscale measurement of the relative twist angle and stacking order. Detailed spectral analysis revealed a pronounced blueshift in the high-energy excitonic peak C with increasing twist angle, up to 200 meV when compared to the AA$^{\prime}$ stacking. The experimental findings have been discussed relative to first-principle calculations of the dielectric response of the AA$^{\prime}$ stacked bilayer WSe$_{2}$ as compared to monolayer WSe$_{2}$ by employing the \textit{GW} plus Bethe-Salpeter equation (BSE) approaches, resolving the origin of higher energy spectral features from ensembles of excitonic transitions, and thus any discrepancies between previous calculations. Furthermore, the electronic structure of moiré supercells spanning twist angles of $\sim$9.5-46.5$^{\circ}$ calculated by density functional theory (DFT) were unfolded, showing an uplifting of the conduction band minimum near the $Q$ point and minimal change in the upper valence band concurrently. The combined experiment/theory investigation provides valuable insight into the physical origins of high-energy absorption resonances in twisted bilayers, which enables to track the evolution of interlayer coupling from tuning of the exciton C transitions by absorption spectroscopy.

cond-mat.mes-hall↗

arXiv2022

Computational study of non-unitary partitions

Following Cayley, MacMahon, and Sylvester, define a non-unitary partition to be an integer partition with no part equal to one, and let $ν(n)$ denote the number of non-unitary partitions of size $n$. In a 2021 paper, the sixth author proved a formula to compute $p(n)$ by enumerating only non-unitary partitions of size $n$, and recorded a number of conjectures regarding the growth of $ν(n)$ as $n\to \infty$. Here we refine and prove some of these conjectures. For example, we prove $p(n) \sim ν(n)\sqrt{n/ζ(2)}$ as $n\to \infty$, and give Ramanujan-like congruences between $p(n)$ and $ν(n)$ such as $p(5n)\equiv ν(5n)\ (\operatorname{mod} 5)$.

math.CO↗

arXiv2022

Composition-theoretic series in partition theory

We use sums over integer compositions analogous to generating functions in partition theory, to express certain partition enumeration functions as sums over compositions into parts that are $k$-gonal numbers; our proofs employ Ramanujan's theta functions. We explore applications to lacunary $q$-series, and to a new class of composition-theoretic Dirichlet series.

math.NT↗

arXiv2022

Combinatorial formulas for arithmetic density

Let $d_S$ denote the arithmetic density of a subset $S \subseteq \mathbb N$. We derive a power series in $q\in \mathbb C$, $|q|<1$, with coëfficients related to integer partitions and integer compositions, that yields $1/d_S$ in the limit as $q\to 1$ radially.

math.NT↗

arXiv2022

High-performance broadband Faraday rotation spectroscopy of 2D materials and thin magnetic films

We present a Faraday rotation spectroscopy (FRS) technique for measurements on the micron scale. Spectral acquisition speeds of many orders of magnitude faster than state-of-the-art modulation spectroscopy setups are demonstrated. The experimental method is based on charge-coupled-device detection, avoiding speed-limiting components, such as polarization modulators with lock-in amplifiers. At the same time, FRS spectra are obtained with a sensitivity of 20 $μ$rad (0.001$^\circ$) over a broad spectral range (525 nm - 800 nm), which is on par with state-of-the-art polarization-modulation techniques. The new measurement technique also automatically cancels unwanted Faraday rotation backgrounds. Using the setup, we perform Faraday rotation spectroscopy of excitons in a hBN-encapsulated atomically thin semiconductor WS$_2$ under magnetic fields of up to 1.4 T at room temperature and liquid helium temperature. We determine the A exciton g-factor of -4.4 $\pm$ 0.3 at room temperature, and -4.2 $\pm$ 0.2 at liquid helium temperature. In addition, we perform FRS and hysteresis loop measurements on a 20 nm thick film of an amorphous magnetic Tb$_{0.2}$Fe$_{0.8}$ alloy.

cond-mat.mes-hall↗

arXiv2022

Infinite series for $π/3$ and other identities

Using techniques from calculus, we combine classical identities for $π$, $\operatorname{ln}2$, and harmonic numbers, to arrive at a nice infinite series formula for $π/3$ that does not appear to be well known. In addition, we give twenty-seven related identities involving $π$ and other irrational numbers. Posted in celebration of Pi Day 2022.

math.HO↗

arXiv2022

Substrate influence on transition metal dichalcogenide monolayer exciton absorption linewidth broadening

The excitonic states of transition metal dichacolgenide (TMD) monolayers are heavily influenced by their external dielectric environment based on the substrate used. In this work, various wide bandgap dielectric materials, namely hexagonal boron nitride (\textit{h}-BN) and amorphous silicon nitride (Si$_3$N$_4$), under different configurations as support or encapsulation material for WS$_2$ monolayers are investigated to disentangle the factors contributing to inhomogeneous broadening of exciton absorption lines in TMDs using electron energy loss spectroscopy (EELS) in a scanning transmission electron microscope (STEM). In addition, monolayer roughness in each configuration was determined from tilt series of electron diffraction patterns by assessing the broadening of diffraction spots by comparison with simulations. From our experiments, the main factors that play a role in linewidth broadening can be classified in increasing order of importance by: monolayer roughness, surface cleanliness, and substrate-induced charge trapping. Furthermore, because high-energy electrons are used as a probe, electron beam-induced damage on bare TMD monolayer is also revealed to be responsible for irreversible linewidth increases. \textit{h}-BN not only provides clean surfaces of TMD monolayer, and minimal charge disorder, but can also protect the TMD from irradiation damage. This work provides a better understanding of the mechanisms by which \textit{h}-BN remains, to date, the most compatible material for 2D material encapsulation, facilitating the realization of intrinsic material properties to their full potential.

cond-mat.mes-hall↗

arXiv2021

Partition Eisenstein series and semi-modular forms

We identify a class of "semi-modular" forms invariant on special subgroups of $GL_2(\mathbb Z)$, which includes classical modular forms together with complementary classes of functions that are also nice in a specific sense. We define an Eisenstein-like series summed over integer partitions, and use it to construct families of semi-modular forms.

math.NT↗