Miracle-dz/UniFAR-FTUs huggingface dataset:
On semi-invariants of filtered representations of quivers and the cotangent bundle of the enhanced Grothendieck-Springer resolution

We introduce the notion of filtered representations of quivers, which is related to usual quiver representations, but is a systematic generalization of conjugacy classes of $n\times n$ matrices to (block) upper triangular matrices up to conjugation by invertible (block) upper triangular matrices. With this notion in mind, we describe the ring of invariant polynomials for interesting families of quivers, namely, finite $ADE$-Dynkin quivers and affine type $\widetilde{A}$-Dynkin quivers. We then study their relation to an important and fundamental object in representation theory called the Grothendieck-Springer resolution, and we conclude by stating several conjectures, suggesting further research.
Simple random walk on distance-regular graphs

A survey is presented of known results concerning simple random walk on the class of distance-regular graphs. One of the highlights is that electric resistance and hitting times between points can be explicitly calculated and given strong bounds for, which leads in turn to bounds on cover times, mixing times, etc. Also discussed are harmonic functions, moments of hitting and cover times, the Green's function, and the cutoff phenomenon. The main goal of the paper is to present these graphs as a natural setting in which to study simple random walk, and to stimulate further research in the field.
Phase Transitions of the Multifractal Spectrum

We consider the multifractal analysis of the pointwise dimension for Gibbs measures on countable Markov shifts. Our paper analyses the set of non-analytic points or phase transitions of the multifractal spectrum. By Sarig's thermodynamic formalism for countable Markov shifts and Iommi's expression of the multifractal spectrum, we apply analyticity arguments on the pressure function for the countable shift. Finally, we apply our results about the phase transitions of the multifractal spectrum to the Gauss map.
On the convexity of piecewise-defined functions

Functions that are piecewise defined are a common sight in mathematics while convexity is a property especially desired in optimization. Suppose now a piecewise-defined function is convex on each of its defining components - when can we conclude that the entire function is convex? In this paper we provide several convenient, verifiable conditions guaranteeing convexity (or the lack thereof). Several examples are presented to illustrate our results.
Graphical Fourier Theory and the Cost of Quantum Addition

The ZX-calculus is a convenient formalism for expressing and reasoning about quantum circuits at a low level, whereas the recently-proposed ZH-calculus yields convenient expressions of mid-level quantum gates such as Toffoli and CCZ. In this paper, we will show that the two calculi are linked by Fourier transform. In particular, we will derive new Fourier expansion rules using the ZH-calculus, and show that we can straightforwardly pass between ZH- and ZX-diagrams using them. Furthermore, we demonstrate that the graphical Fourier expansion of a ZH normal-form corresponds to the standard Fourier transform of a semi-Boolean function. As an illustration of the calculational power of this technique, we then show that several tricks for reducing the T-gate cost of Toffoli circuits, which include for instance quantum adders, can be derived using graphical Fourier theory and straightforwardly generalized to more qubits.
Horseshoes and Lyapunov exponents for Banach cocycles over non-uniformly hyperbolic systems

Abstract We extend Katok’s result on ‘the approximation of hyperbolic measures by horseshoes’ to Banach cocycles. More precisely, let f be a $C^r(r>1)$ diffeomorphism of a compact Riemannian manifold M, preserving an ergodic hyperbolic measure $\mu $ with positive entropy, and let $\mathcal {A}$ be a Hölder continuous cocycle of bounded linear operators acting on a Banach space $\mathfrak {X}$ . We prove that there is a sequence of horseshoes for f and dominated splittings for $\mathcal {A}$ on the horseshoes, such that not only the measure theoretic entropy of f but also the Lyapunov exponents of $\mathcal {A}$ with respect to $\mu $ can be approximated by the topological entropy of f and the Lyapunov exponents of $\mathcal {A}$ on the horseshoes, respectively. As an application, we show the continuity of sub-additive topological pressure for Banach cocycles.
The continuum Pólya-like random walk

The P\'olya urn scheme is a discrete-time process concerning the addition and removal of colored balls. There is a known embedding of it in continuous-time, called the P\'olya process. We introduce a generalization of this stochastic model, where the initial values and the entries of the transition matrix (corresponding to additions or removals) are not necessarily fixed integer values as in the standard P\'olya process. In one of our scenarios, we even allow the entries of the matrix to be random variables. As a result, we no longer have a combinatorial model of "balls in an urn", but a broader interpretation as a random walk, in a possibly high number of dimensions. In this paper, we study several parametric classes of these generalized continuum P\'olya-like random walks.
Identification of a Quorum Sensing System Regulating Capsule Polysaccharide Production and Biofilm Formation in Streptococcus zooepidemicus

Streptococcus zooepidemicus is an important opportunistic pathogen of several species including humans. This organism is also well-known as the main producing strain in industrial production of hyaluronic acid (HA), which is the component of its capsule polysaccharide. How its virulence and capsule polysaccharide production is regulated remains poorly understood. Intercellular chemical signaling among bacteria provides communities of microbes the opportunity to coordinate gene expression to facilitate group behavior, such as pathogenicity, capsule polysaccharide production, etc. Yet no conserved cell-to-cell signaling system has been elucidated in S. zooepidemicus. Encoded within the genome of S. zooepidemicus is one Rgg regulator encoding gene (rgg) with low similarity to both rgg2 and rgg3 from Streptococcus pyogenes. A small ORF (named as shp) encoding a novel short hydrophobic peptide (SHP) was found in the vicinity of rgg. We found that the active form of pheromone is short and hydrophobic (LLLLKLA), corresponding to the C terminal 7 amino acids of the pre-peptide Shp, which shows divergent sequence to all peptide pheromones reported in streptococci. In response to active SHP, Rgg functions as a transcriptional activator to induce the expression of shp, forming a positive feedback circuit. Bacteria social behaviors, such as capsule polysaccharide production and biofilm formation, were significantly affected when the rgg-shp pathway was inactivated. These data provide the first demonstration that Rgg/Shp signaling pathway comprises an active quorum sensing system in S. zooepidemicus.
Spatial Distribution of Leisure Agriculture in Xinjiang and Its Influencing Factors Based on Geographically Weighted Regression

As a new industry in modern agriculture, leisure agriculture has a strong correlation with rural tourism, and provides rural areas with positive prospects for sustainable development. However, leisure agriculture tends to include a number of bottlenecks. In this study, we investigated the spatial distribution of leisure agriculture in Xinjiang, and the factors that affect it. Kernel density analysis, the nearest-neighbor index, and the geographic concentration index were used to analyze the distribution characteristics of leisure agriculture. Following the conclusion of the ordinary least squares tests, geographically weighted regression (GWR) was conducted to explore the factors affecting spatial distribution. The findings were as follows: (1) The spatial distribution of leisure agriculture in Xinjiang is uneven, and is concentrated in the northern and southern parts of the Tianshan Mountains in western Xinjiang. (2) In terms of the distribution density, there are four high-concentration centers (Bosten Lake, Hami, and the east and west sides of the Ili River Valley) and one subconcentration center (spreading outward from Urumqi). (3) Population, transportation, tourism resources, urban factors, and rainfall, all had significant effects on the distribution of leisure agriculture. These factors had positive and negative effects on the distribution of leisure agriculture, forming high- and low-value areas in space. (4) The leisure agricultural sector responded in varying degrees to the different factors, with large internal variability. Rainfall and population had greater differential effects on the spatial distribution of leisure agriculture compared to transportation, tourism resources, and urban factors, and there were significant two-way effects. Transportation, urban factors, and tourism resources all had consistent, predominantly positive, effects on the distribution of leisure agriculture.
Quantum transport of Dirac electrons in graphene in the presence of a spatially modulated magnetic field

We have investigated the electrical transport properties of Dirac electrons in a monolayer graphene sheet in the presence of a perpendicular magnetic field that is modulated weakly and periodically along one direction.We find that the Landau levels broaden into bands and their width oscillates as a function of the band index and the magnetic field.We determine the $\sigma_{yy}$ component of the magnetoconductivity tensor for this system which is shown to exhibit Weiss oscillations.We also determine analytically the asymptotic expressions for $\sigma_{yy}$. We compare these results with recently obtained results for electrically modulated graphene as well as those for magnetically modulated conventional two-dimensional electron gas (2DEG) system.We find that in the magnetically modulated graphene system cosidered in this work,Weiss oscillations in $\sigma_{yy}$ have a reduced amplitude compared to the 2DEG but are less damped by temperature while they have a higher amplitude than in the electrically modulated graphene system. We also find that these oscillations are out of phase by $\pi$ with those of the electrically modulated system while they are in phase with those in the 2DEG system.
One-sided curvature estimates for H-disks

In this paper we prove an extrinsic one-sided curvature estimate for disks embedded in $\mathbb{R}^3$ with constant mean curvature which is independent of the value of the constant mean curvature. We apply this extrinsic one-sided curvature estimate in [24] to prove to prove a weak chord arc type result for these disks. In Section 4 we apply this weak chord arc result to obtain an intrinsic version of the one-sided curvature estimate for disks embedded in $\mathbb{R}^3$ with constant mean curvature. In a natural sense, these one-sided curvature estimates generalize respectively, the extrinsic and intrinsic one-sided curvature estimates for minimal disks embedded in $\mathbb{R}^3$ given by Colding and Minicozzi in Theorem 0.2 of [8] and in Corollary 0.8 of [9].
On differential modules associated to de Rham representations in the imperfect residue field case

Let $K$ be a complete discrete valuation field of mixed characteristic $(0,p)$, whose residue field may not be perfect, and $G_K$ the absolute Galois group of $K$. In the first part of this paper, we prove that Scholl's generalization of fields of norms over $K$ is compatible with Abbes-Saito's ramification theory. In the second part, we construct a functor $\mathbb{N}_{\mathrm{dR}}(V)$ associating a de Rham representation $V$ with a $(\varphi,\nabla)$-module in the sense of Kedlaya. Finally, we prove a compatibility between Kedlaya's differential Swan conductor of $\mathbb{N}_{\mathrm{dR}}(V)$ and Swan conductor of $V$, which generalizes Marmora's formula.
The quantile transform of a simple walk

We examine a new path transform on 1-dimensional simple random walks and Brownian motion, the quantile transform. This transformation relates to identities in fluctuation theory due to Wendel, Port, Dassios and others, and to discrete and Brownian versions of Tanaka’s formula. For an n-step random walk, the quantile transform reorders increments according to the value of the walk at the start of each increment. We describe the distribution of the quantile transform of a simple random walk of n steps, using a bijection to characterize the number of pre-images of each possible transformed path. We deduce, both for simple random walks and for Brownian motion, that the quantile transform has the same distribution as Vervaat’s transform. For Brownian motion, the quantile transforms of the embedded simple random walks converge to a time change of the local time profile. We characterize the distribution of the local time profile, giving rise to an identity that generalizes a variant of Jeulin’s description of the local time profile of a Brownian bridge or excursion.
Reidemeister-Schreier's Algorithm For 2-Coverings of Seifert Manilfolds

It is classical that given any Seifert structure on N, Reidemeister-Schreier's algorithm produces a presentation of all index 2 subgroups of the fundamental group of N, described as the fundamental group of some Seifert manifolds. The new result of this article is concise formulas that gather all possible cases.
NEW TREATMENT OF THE NONCOMMUTATIVE DIRAC EQUATION WITH A COULOMB POTENTIAL

Using the approach of the modified Euler–Lagrange field equation together with the corresponding Seiberg–Witten maps of the dynamical fields, a noncommutative Dirac equation with a Coulomb potential is derived. We then find the noncommutative modification to the energy levels and the possible new transitions. In the nonrelativistic limit a general form of the Hamiltonian of the hydrogen atom is obtained, and we show that the noncommutativity plays the role of spin and magnetic field which gives the hyperfine structure.
Planar and Nanowire InP Thin Solar Cells for Ultralight Space Power Applications

Indium phosphide (InP) thin film solar cells have considerable potential for low-cost space photovoltaic applications due to their efficiency, ultralight weight form factor, favorable surface recombination properties, optimal bandgap, and innately high radiation resistance compared to silicon and gallium arsenide (GaAs). However, InP cells have received less attention than their GaAs and GaInP/GaAs counterparts for space photovoltaic application. However, future ultralight space photovoltaics with specific power greater than 1kW/kg will require innovative designs for flexible radiation-hard cells without conventional cover glass radiation shielding. We investigate here designs for ultralight InP space photovoltaics, including nanowire array cells, planar thin-film cells with TiO2/ITO optimized antireflection layers, and planar thin-film cells with pyramidally-textured TiO2 antireflection layers, and assess these designs with respect to efficiency, specific power, and scalable fabrication.
A note on submanifolds of generalized K\"ahler manifolds

In this note, we consider submanifolds of a generalized Kahler manifold that are CR-submanifolds for the two associated Hermitian structures. Then, we establish the conditions for the induced, generalized F structure to be a CRFK structure. The results extend similar conditions which we obtained for hypersurfaces in an earlier paper.
On the notion of spectral decomposition in local geometric Langlands

The geometric Langlands program is distinguished in assigning spectral decompositions to all representations, not only the irreducible ones. However, it is not even clear what is meant by a spectral decomposition when one works with non-abelian reductive groups and with ramification. The present work is meant to clarify the situation, showing that the two obvious candidates for such a notion coincide. More broadly, we study the moduli space of (possibly irregular) de Rham local systems from the perspective of homological algebra. We show that, in spite of its infinite-dimensional nature, this moduli space shares many of the nice features of an Artin stack. Less broadly, these results support the belief in the existence of a version of geometric Langlands allowing arbitrary ramification. Along the way, we give some apparently new, if unsurprising, results about the algebraic geometry of the moduli space of connections, using Babbitt-Varadarajan's reduction theory for differential equations.
Ding projective complexes with respect to a semidualizing module

Let R be a commutative ring and C a semidualizing R-module. In this article, we introduce and investigate the notion of DC-projective complexes. We first prove that a complex X is DC-projective if and only if each degree of X is a DC-projective module and Hom(X;H) is exact for any C-flat complex H. As immediate consequences of this result, some properties of DC-projective complexes are given. Secondly, we investigate a kind of stability of DC-projective complexes by showing that an iteration of the procedure used to defi? ne the DC-projective complexes yields exactly the DC-projective complexes. Finally, We introduce and characterize the notion of DC-projective dimension of complexes.
Will zigzag graphene nanoribbon turn to half metal under electric field

At B3LYP level of theory, we predict that the half-metallicity in zigzag edge graphene nanoribbon (ZGNR) can be realized when an external electric field is applied across the ribbon. The critical electric field decreases with the increase of the ribbon width to induce the half-metallicity. Both the spin polarization and half-metallicity are removed when the edge state electrons fully transferred from one side to the other under very strong electric field. The electric field range under which ZGNR remains half-metallic increases with the ribbon width. Our study demonstrates a rich field-induced spin polarization behavior, which may lead to some important applications in spinstronics.
Energy-norm error estimates for finite element discretization of parabolic problems

We consider the discretization of parabolic initial boundary value problems by finite element methods in space and a Runge-Kutta time stepping scheme. Order optimal a-priori error estimates are derived in an energy-norm under natural smoothness assumptions on the solution and without artificial regularity conditions on the parameters and the domain. The key steps in our analysis are the careful treatment of time derivatives in the H(-1)-norm and the the use of an L2-projection in the error splitting instead of the Ritz projector. This allows us to restore the optimality of the estimates with respect to smoothness assumptions on the solution and to avoid artificial regularity requirements for the problem, usually needed for the analysis of the Ritz projector, which limit the applicability of previous work. The wider applicability of our results is illustrated for two irregular problems, for which previous results can either not by applied or yield highly sub-optimal estimates.
Relative entropy minimization over Hilbert spaces via Robbins-Monro

One way of getting insight into non-Gaussian measures, posed on infinite dimensional Hilbert spaces, is to first obtain best fit Gaussian approximations, which are more amenable to numerical approximation. These Gaussians can then be used to accelerate sampling algorithms. This begs the questions of how one should measure optimality and how the optimizers can be obtained. Here, we consider the problem of minimizing the distance with respect to relative entropy. We examine this minimization problem by seeking roots of the first variation of relative entropy, taken with respect to the mean of the Gaussian, leaving the covariance fixed. Adapting a convergence analysis of Robbins-Monro to the infinite dimensional setting, we can justify the application of this algorithm and highlight necessary assumptions to ensure convergence, not only in the context of relative entropy minimization, but other infinite dimensional problems as well. Numerical examples in path space, showing the robustness of this method with respect to dimension, are provided.
Energy Harvesting Management Unit for Wearable Devices

Energy harvesting materials and systems have become a popular study topic that is rapidly expanding. The harvesters will be used for a variety of applications, including distributed wireless sensor nodes for structural health monitoring, embedded and implanted sensor nodes for medical applications, recharging large system batteries, monitoring pressure in automobiles, powering unmanned vehicles, and running security systems in domestic settings. Components and devices at micro-macro sizes, spanning materials, electronics, and integration, have recently been developed. Energy harvesting has piqued public attention in recent years, with the number of study reports steadily increasing over the previous decade. In this paper an energy harvesting from body heat is presented. The harvesters will be used for a many of purposes. The sensing system is self-powered by harvesting thermal energy from human body heat via a thermoelectric generator (TEG) coupled to a low voltage, self-starting boost converter and integrated power management system.
Lifting the Carter Transform of Ogus-Vologodsky modulo p^n

Let $W$ be the ring of the Witt vectors of a perfect field of characteristic $p$, $\mathfrak{X}$ a smooth formal scheme over $W$, $\mathfrak{X}'$ the base change of $\mathfrak{X}$ by the Frobenius morphism of $W$, $\mathfrak{X}_{2}'$ the reduction modulo $p^{2}$ of $\mathfrak{X}'$ and $X$ the special fiber of $\mathfrak{X}$. We lift the Cartier transform of Ogus-Vologodsky defined by $\mathfrak{X}_{2}'$ modulo $p^{n}$. More precisely, we construct a functor from the category of $p^{n}$-torsion $\mathscr{O}_{\mathfrak{X}'}$-modules with integrable $p$-connection to the category of $p^{n}$-torsion $\mathscr{O}_{\mathfrak{X}}$-modules with integrable connection, each subject to suitable nilpotence conditions. Our construction is based on Oyama's reformulation of the Cartier transform of Ogus-Vologodsky in characteristic $p$. If there exists a lifting $F:\mathfrak{X}\to \mathfrak{X}'$ of the relative Frobenius morphism of $X$, our functor is compatible with a functor constructed by Shiho from $F$. As an application, we give a new interpretation of Faltings' relative Fontaine modules and of the computation of their cohomology.