Tootfinder

This https://arxiv.org/abs/2111.00486 has been replaced.
link: https://scholar.google.com/scholar?q=a

Fine-Grained Cryptanalysis: Tight Conditional Bounds for Dense k-SUM and k-XOR
An average-case variant of the $k$-SUM conjecture asserts that finding $k$ numbers that sum to 0 in a list of $r$ random numbers, each of the order $r^k$, cannot be done in much less than $r^{\lceil k/2 \rceil}$ time. On the other hand, in the dense regime of parameters, where the list contains more numbers and many solutions exist, the complexity of finding one of them can be significantly improved by Wagner's $k$-tree algorithm. Such algorithms for $k$-SUM in the dense regime have many applic…

This https://arxiv.org/abs/2111.08949 has been replaced.
link: https://scholar.google.com/scholar?q=a

Max-norm Ramsey Theory
Given a metric space $\mathcal{M}$ that contains at least two points, the chromatic number $χ\left(\mathbb{R}^n_{\infty}, \mathcal{M} \right)$ is defined as the minimum number of colours needed to colour all points of an $n$-dimensional space $\mathbb{R}^n_{\infty}$ with the max-norm such that no isometric copy of $\mathcal{M}$ is monochromatic. The last two authors have recently shown that the value $χ\left(\mathbb{R}^n_{\infty}, \mathcal{M} \right)$ grows exponentially for all finite $\math…

This https://arxiv.org/abs/2203.13966 has been replaced.
link: https://scholar.google.com/scholar?q=a

Stability of Linear Set-Membership Filters With Respect to Initial Conditions: An Observation-Information Perspective
The issue of filter stability with respect to (w.r.t.) the initial condition refers to the unreliable filtering process caused by improper prior information of the initial state. This paper focuses on analyzing and resolving the stability issue w.r.t. the initial condition of the classical Set-Membership Filters (SMFs) for linear time-invariance systems, which has not yet been well understood in the literature. To this end, we propose a new concept -- the Observation-Information Tower (OIT), wh…

This https://arxiv.org/abs/1606.00963 has been replaced.
link: https://scholar.google.com/scholar?q=a

Optimal quantization for a probability measure on a nonuniform stretched Sierpiński triangle
Quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with support containing a finite number of elements. In this paper, we have considered a Borel probability measure $P$ on $\mathbb R^2$, which has support a nonuniform stretched Sierpiński triangle generated by a set of three contractive similarity mappings on $\mathbb R^2$. For this probability measure, we investigate the optimal sets of $n$-means and the $n$th quantizat…

This https://arxiv.org/abs/2002.00502 has been replaced.
link: https://scholar.google.com/scholar?q=a

On the Erdős distance problem
In this paper, using the method of compression, we recover the lower bound for the Erdős unit distance problem and provide an alternative proof to the distinct distance conjecture. In particular, we show that for sets of points $\mathbb{E}\subset \mathbb{R}^k$ concentrated around the origin with $\# \mathbb{E}\cap \mathbb{R}^k=\frac{n}{2}$, we have \begin{align}\# \bigg\{||\vec{x_j}-\vec{x_t}||:\vec{x_j}\in \mathbb{E}\subset \mathbb{R}^k,~||\vec{x_j}-\vec{x_t}…

This https://arxiv.org/abs/2212.02593 has been replaced.
link: https://scholar.google.com/scholar?q=a

Characterizing the Nonequilibrium Response of FeRh Thin Films using Time-Domain Thermoreflectance (TDTR)
Time-Domain Thermoreflectance (TDTR) characterization of FeRh throughout its first-order antiferromagnetic (AF) to ferromagnetic (FM) transition shows that the transient reflectance, $Δ$R(t)/R, strongly depends on the magnetic order of the sample. Using TDTR, which uses optical pulses to induce small temperature excursions, we have found that the $Δ$R(t)/R of the AF phase exhibits a large negative response, while the response of the FM phase is positive. This magnetic phase sensitivity has al…

This https://arxiv.org/abs/2203.13373 has been replaced.
link: https://scholar.google.com/scholar?q=a

Pickl's Proof of the Quantum Mean-Field Limit and Quantum Klimontovich Solutions
This paper discusses the mean-field limit for the quantum dynamics of $N$ identical bosons in $\mathbf R^3$ interacting via a binary potential with Coulomb type singularity. Our approach is based on the theory of quantum Klimontovich solutions defined in [F. Golse, T. Paul, Commun. Math. Phys. 369 (2019), 1021-1053]. Our first main result is a definition of the interaction nonlinearity in the equation governing the dynamics of quantum Klimontovich solutions for a class of interaction potentials…

This https://arxiv.org/abs/2401.17719 has been replaced.
link: https://scholar.google.com/scholar?q=a

On the saddle point of a zero-sum stopper vs. singular-controller game
We construct a saddle point in a class of zero-sum games between a stopper and a singular-controller. The underlying dynamics is a one-dimensional, time-homogeneous, singularly controlled diffusion taking values either on $\mathbb{R}$ or on $[0,\infty)$. The games are set on a finite-time horizon, thus leading to analytical problems in the form of parabolic variational inequalities with gradient and obstacle constraints. The saddle point is characterised in terms of two moving boundaries: an op…

This https://arxiv.org/abs/2301.01399 has been replaced.
link: https://scholar.google.com/scholar?q=a

Global well-posedness of weak solutions to the incompressible Euler equations with helical symmetry in $\mathbb{R}^3$
We consider the three-dimensional incompressible Euler equation \begin{equation*}\left\{\begin{aligned} &\partial_t Ω+U \cdot \nabla Ω+Ω\cdot \nabla U=0 \\ &Ω(x,0)=Ω_0(x) \end{aligned}\right. \end{equation*} in the whole space $\mathbb{R}^3$. Under the assumption that the initial velocity is helical and without swirl, we prove the global well-posedness of weak solutions in $L^1_1 \bigcap L^{\infty}_1(\mathbb{R}^3)$. The vortex transport formula is also obtained in our article.

This https://arxiv.org/abs/2111.05030 has been replaced.
link: https://scholar.google.com/scholar?q=a

A central limit theorem for the variation of the sum of digits
We prove a Central Limit Theorem for probability measures defined via the variation of the sum-of-digits function, in base $b\ge 2$. For $r\ge 0$ and $d \in \mathbb{Z}$, we consider $μ^{(r)}(d)$ as the density of integers $n\in \mathbb{N}$ for which the sum of digits increases by $d$ when we add $r$ to $n$. We give a probabilistic interpretation of $μ^{(r)}$ on the probability space given by the group of $b$-adic integers equipped with the normalized Haar measure. We split the base-$b$ expans…

The Texas Legislature’s Assault on Church-State Separation in Schools | Baker Institute https://www.bakerinstitute.org/research/texas-legislatures-assault-church-state-separation-schools

This https://arxiv.org/abs/2211.15118 has been replaced.
link: https://scholar.google.com/scholar?q=a

A Faster $k$-means++ Algorithm
$k$-means++ is an important algorithm for choosing initial cluster centers for the $k$-means clustering algorithm. In this work, we present a new algorithm that can solve the $k$-means++ problem with nearly optimal running time. Given $n$ data points in $\mathbb{R}^d$, the current state-of-the-art algorithm runs in $\widetilde{O}(k )$ iterations, and each iteration takes $\widetilde{O}(nd k)$ time. The overall running time is thus $\widetilde{O}(n d k^2)$. We propose a new algorithm \textsc{Fas…

This https://arxiv.org/abs/2212.12851 has been replaced.
link: https://scholar.google.com/scholar?q=a

Constraints on real scalar inflation from preheating of LATTICEEASY
In this paper, we undertake a detailed study of real scalar inflation by employing LATTICEEASY simulations to investigate preheating phenomena. Generally, the scalar inflation potential with non-minimal coupling can be approximated by a quartic potential. We observe that the evolutionary behavior of this potential remains unaffected by the coupling coefficient. Furthermore, the theoretical predictions for the scalar spectral index ($n_s$) and tensor-to-scalar power ratio (r) are also independen…

This https://arxiv.org/abs/2206.14668 has been replaced.
link: https://scholar.google.com/scholar?q=a

Score Matching for Truncated Density Estimation on a Manifold
When observations are truncated, we are limited to an incomplete picture of our dataset. Recent methods propose to use score matching for truncated density estimation, where the access to the intractable normalising constant is not required. We present a novel extension of truncated score matching to a Riemannian manifold with boundary. Applications are presented for the von Mises-Fisher and Kent distributions on a two dimensional sphere in $\mathbb{R}^3$, as well as a real-world application of…

This https://arxiv.org/abs/1910.05655 has been replaced.
link: https://scholar.google.com/scholar?q=a

The supermoduli space of genus zero SUSY curves with Ramond punctures
We give an explicit quotient construction of the supermoduli space $\mathfrak{M}_{0, n_R}$ of genus zero super Riemann surfaces with $n_R \ge 4$ Ramond punctures and prove it is a Deligne-Mumford superstack. We also make an explicit quotient construction of the moduli space of genus zero supercurves without a SUSY structure and thereby prove it is an algebraic superstack.

This https://arxiv.org/abs/2401.08042 has been replaced.
link: https://scholar.google.com/scholar?q=a

Exponential bases for parallelepipeds with frequencies lying in a prescribed lattice
The existence of a Fourier basis with frequencies in $\mathbb{R}^d$ for the space of square integrable functions supported on a given parallelepiped in $\mathbb{R}^d$, has been well understood since the 1950s. In a companion paper, we derived necessary and sufficient conditions for a parallelepiped in $\mathbb{R}^d$ to permit an orthogonal basis of exponentials with frequencies constrained to be a subset of a prescribed lattice in $\mathbb{R}^d$, a restriction relevant in many applications. In …

This https://arxiv.org/abs/2111.06710 has been replaced.
link: https://scholar.google.com/scholar?q=a

Pósa-type results for Berge-hypergraphs
A Berge cycle of length $k$ in a hypergraph $\mathcal H$ is a sequence of distinct vertices and hyperedges $v_1,h_1,v_2,h_2,\dots,v_{k},h_k$ such that $v_{i},v_{i+1}\in h_i$ for all $i\in[k]$, indices taken modulo $k$. Füredi, Kostochka and Luo recently gave sharp Dirac-type minimum degree conditions that force non-uniform hypergraphs to have Hamiltonian Berge cycles. We give a sharp Pósa-type lower bound for $r$-uniform and non-uniform hypergraphs that force Hamiltonian Berge cycles.

This https://arxiv.org/abs/2212.00071 has been replaced.
link: https://scholar.google.com/scholar?q=a

An inequality of multiple integral of $s$ norm of vectors in $\mathbb{R}^n$
In this note, we prove some new inequalities. To facilitate this proof, we introduce the notion of the local product on a sheet and associated space.

This https://arxiv.org/abs/2402.03696 has been replaced.
link: https://scholar.google.com/scholar?q=a

Inference of multi-channel r-process element enrichment in the Milky Way using binary neutron star merger observations
Observations of GW170817 strongly suggest that binary neutron star (BNS) mergers can produce rapid neutron-capture nucleosynthesis (r-process) elements. However, it remains an open question whether BNS mergers can account for all the r-process element enrichment in the Milky Way's history. Here we demonstrate that a BNS population model informed by multimessenger neutron star observations predicts a merger rate and per-event r-process element yield consistent with geophysical and astrophysical …

This https://arxiv.org/abs/2402.01336 has been replaced.
link: https://scholar.google.com/scholar?q=a

Measurements of the branching fraction ratio $\cal{B}(ϕ\to μ^+μ^-)/\cal{B}(ϕ\to e^+e^-)$ with charm meson decays
Measurements of the branching fraction ratio ${\cal{B}(ϕ\to μ^+ μ^-)/\cal{B}(ϕ\to e^+e^-)}$ with ${D_{s}^{+} \to π^{+} ϕ}$ and ${D^{+} \to π^{+} ϕ}$ decays, denoted $R^{s}_{ϕπ}$ and $R^{d}_{ϕπ}$, are presented. The analysis is performed using a dataset corresponding to an integrated luminosity of 5.4$\,\rm{fb}^{-1}$ of $pp$ collision data collected with the LHCb experiment. The branching fractions are normalised with respect to the ${B^{+} \to K^{+} J/ψ(\to e^+e^-)}$ and ${B^{+} \t…

This https://arxiv.org/abs/2209.07848 has been replaced.
link: https://scholar.google.com/scholar?q=a

Boundedness of bundle diffeomorphism groups over a circle
In this paper we study boundedness of bundle diffeomorphism groups over a circle. For a fiber bundle $π: M \to S^1$ with fiber $N$ and structure group $Γ$ and $r \in {\Bbb Z}_{\geq 0} \cup \{ \infty \}$ we distinguish an integer $k = k(π, r) \in {\Bbb Z}_{\geq 0}$ and construct a function $\widehatν : {\rm Diff}_π(M)_0 \to {\Bbb R}_k$. When $k \geq 1$, it is shown that the bundle diffeomorphism group ${\rm Diff}_π(M)_0$ is uniformly perfect and $clb_π\,{\rm Diff}^r_π(M)_0 \leq k+3$, if …

This https://arxiv.org/abs/2109.14256 has been replaced.
link: https://scholar.google.com/scholar?q=a

Lang--Trotter Conjecture for CM Elliptic Curves
Given an elliptic curve $E$ over $\mathbb{Q}$ and non-zero integer $r$, the Lang--Trotter conjecture predicts a striking asymptotic formula for the number of good primes $p\leqslant x$, denoted by $π_{E,r}(x)$, such that the Frobenius trace of $E$ at $p$ is equal to the given integer $r$. We focus on the CM case in this memoir, and show how to realize the following two goals: (1) to give an unconditional estimate for $π_{E,r}(x)$, which confirms the upper bound part of the conjecture up to …

This https://arxiv.org/abs/2402.16812 has been replaced.
link: https://scholar.google.com/scholar?q=a

Asymptotically non-negative Ricci curvature, elliptic Kato constant and isoperimetric inequalities
The ABP method for proving isoperimetric inequalities has been first employed by Cabré in $\mathbb{R}^n$, then developed by Brendle, notably in the context of non-compact Riemannian manifolds of non-negative Ricci curvature and positive asymptotic volume ratio. In this paper, we expand upon their approach and prove isoperimetric inequalities (sharp in the limit) in the presence of a small amount of negative curvature. First, we consider smallness of the negative part $\mathrm{Ric}_-$ of the Ri…

This https://arxiv.org/abs/2206.14668 has been replaced.
link: https://scholar.google.com/scholar?q=a

Score Matching for Truncated Density Estimation on a Manifold
When observations are truncated, we are limited to an incomplete picture of our dataset. Recent methods propose to use score matching for truncated density estimation, where the access to the intractable normalising constant is not required. We present a novel extension of truncated score matching to a Riemannian manifold with boundary. Applications are presented for the von Mises-Fisher and Kent distributions on a two dimensional sphere in $\mathbb{R}^3$, as well as a real-world application of…

This https://arxiv.org/abs/2208.14713 has been replaced.
link: https://scholar.google.com/scholar?q=a

Models of Bounded Arithmetic and variants of Pigeonhole Principle
We give elementary proof that theory $T^1_2(R)$ augmented by the weak pigeonhole principle for all $Δ^b_1(R)$-definable relations does not prove the bijective pigeonhole principle for $R$. This can be derived from known more general results but our proof yields a model of $T^1_2(R)$ in which $ontoPHP^{n+1}_n(R)$ fails for some nonstandard element $n$ while $PHP^{m+1}_m$ holds for all $Δ^b_1(R)$-definable relations and all $m \leq n^{1-ε}$, where $ε> 0$ is a fixed standard rational parameter…

This https://arxiv.org/abs/2004.07259 has been replaced.
link: https://scholar.google.com/scholar?q=a

On ${\cal M}$-Theory Dual of Large-$N$ Thermal QCD-Like Theories up to ${\cal O}(R^4)$ and $G$-Structure Classification of Underlying Non-Supersymmetric Geometries
Construction of a top-down holographic dual of thermal QCD-like theories (equivalence class of theories which are UV-conformal, IR-confining and have fundamental quarks) {\it at intermediate 't Hooft coupling} and the $G$-structure (torsion classes) classification of the underlying geometries (in the Infra Red (IR)/non-conformal sector in particular) of the {\it non-supersymmetric} string/${\cal M}$-theory duals, have been missing in the literature. We take the first important steps in this dir…

This https://arxiv.org/abs/2310.03996 has been replaced.
link: https://scholar.google.com/scholar?q=a

Tightness of exponential metrics for log-correlated Gaussian fields in arbitrary dimension
We prove the tightness of a natural approximation scheme for an analog of the Liouville quantum gravity metric on $\mathbb R^d$ for arbitrary $d\geq 2$. More precisely, let $\{h_n\}_{n\geq 1}$ be a suitable sequence of Gaussian random functions which approximates a log-correlated Gaussian field on $\mathbb R^d$. Consider the family of random metrics on $\mathbb R^d$ obtained by weighting the lengths of paths by $e^{ξh_n}$, where $ξ> 0$ is a parameter. We prove that if $ξ$ belongs to the subc…

This https://arxiv.org/abs/2306.11134 has been replaced.
link: https://scholar.google.com/scholar?q=a

OpenP5: An Open-Source Platform for Developing, Training, and Evaluating LLM-based Recommender Systems
In recent years, the integration of Large Language Models (LLMs) into recommender systems has garnered interest among both practitioners and researchers. Despite this interest, the field is still emerging, and the lack of open-source R&D platforms may impede the exploration of LLM-based recommendations. This paper introduces OpenP5, an open-source platform designed as a resource to facilitate the development, training, and evaluation of LLM-based generative recommender systems for research purp…

This https://arxiv.org/abs/2212.07727 has been replaced.
link: https://scholar.google.com/scholar?q=a

End-time regularity theorem for Brakke flows
For a general $k$-dimensional Brakke flow in $\mathbb{R}^n$ locally close to a $k$-dimensional plane in the sense of measure, it is proved that the flow is represented locally as a smooth graph over the plane with estimates on all the derivatives up to the end-time. Moreover, at any point in space-time where the Gaussian density is close to $1$, the flow can be extended smoothly as a mean curvature flow up to that time in a neighborhood: this extends White's local regularity theorem to general …

This https://arxiv.org/abs/2209.00169 has been replaced.
link: https://scholar.google.com/scholar?q=a

Some Algebraic Questions about the Reed-Muller Code
Let $R_q(r,n)$ denote the $r$th order Reed-Muller code of length $q^n$ over $\Bbb F_q$. We consider two algebraic questions about the Reed-Muller code. Let $H_q(r,n)=R_q(r,n)/R_q(r-1,n)$. (1) When $q=2$, it is known that there is a "duality" between the actions of $\text{GL}(n,\Bbb F_2)$ on $H_2(r,n)$ and on $H_2(r',n)$, where $r+r'=n$. The result is false for a general $q$. However, we find that a slightly modified duality statement still holds when $q$ is a prime or $r<\text{char}\,\Bbb F_q$.…

This https://arxiv.org/abs/1908.08395 has been replaced.
link: https://scholar.google.com/scholar?q=a

A tale of two shuffle algebras
As a quantum affinization, the quantum toroidal algebra is defined in terms of its "left" and "right" halves, which both admit shuffle algebra presentations. In the present paper, we take an orthogonal viewpoint, and give shuffle algebra presentations for the "top" and "bottom" halves instead, starting from the evaluation representation of the quantum affine group and its usual R-matrix. An upshot of this construction is a new topological coproduct on the quantum toroidal algebra which extends …

This https://arxiv.org/abs/2403.01103 has been replaced.
link: https://scholar.google.com/scholar?q=a

Asymptotic order of the quantization error for a class of self-similar measures with overlaps
Let $\{f_i\}_{i=1}^N$ be a set of equi-contractive similitudes on $\mathbb{R}^1$ satisfying the finite-type condition. We study the asymptotic quantization error for self-similar measures $μ$ associated with $\{f_i\}_{i=1}^N$ and a positive probability vector. With a verifiable assumption, we prove that the upper and lower quantization coefficient for $μ$ are both bounded away from zero and infinity. This can be regarded as an extension of Graf and Luschgy's result on self-similar measures wi…

This https://arxiv.org/abs/2210.10751 has been replaced.
link: https://scholar.google.com/scholar?q=a

Global fit to $b \to cτν$ anomaly 2024 Spring breeze
Recently, several new experimental results of the test of lepton flavor universality (LFU) in $B\to D^{(\ast)}$ semi-leptonic decays were announced: the first result of $R_{D}$ from the LHCb Run 1 data, the first results of $R_{D}$ and $R_{D^\ast}$ from the LHCb Run 2 data, and the first result of $R_{D^\ast}$ from the Belle II collaboration. Including these new data, a global analysis still prefers the violation of the LFU between the tau and light leptons. A new world average of the data from…

This https://arxiv.org/abs/2211.03859 has been replaced.
link: https://scholar.google.com/scholar?q=a

From approximate to exact integer programming
Approximate integer programming is the following: For a convex body $K \subseteq \mathbb{R}^n$, either determine whether $K \cap \mathbb{Z}^n$ is empty, or find an integer point in the convex body scaled by $2$ from its center of gravity $c$. Approximate integer programming can be solved in time $2^{O(n)}$ while the fastest known methods for exact integer programming run in time $2^{O(n)} \cdot n^n$. So far, there are no efficient methods for integer programming known that are based on approxim…

This https://arxiv.org/abs/2203.16523 has been replaced.
link: https://scholar.google.com/scholar?q=a

Shifted Witten classes and topological recursion
The Witten $r$-spin class defines a non-semisimple cohomological field theory. Pandharipande, Pixton and Zvonkine studied two special shifts of the Witten class along two semisimple directions of the associated Dubrovin--Frobenius manifold using the Givental--Teleman reconstruction theorem. We show that the $R$-matrix and the translation of these two specific shifts can be constructed from the solutions of two differential equations that generalise the classical Airy differential equation. Usin…

This https://arxiv.org/abs/2403.01416 has been replaced.
link: https://scholar.google.com/scholar?q=a

An analytical approximation of the evolution of the primordial curvature perturbation in the ultraslow-roll inflation
Cosmic inflation can enter an ultraslow-roll (USR) stage, if there is a plateau on the inflaton potential. During this stage, the primordial curvature perturbation ${\mathcal R}_k$ and its power spectrum ${\mathcal P}_{\mathcal R}$ can be remarkably enhanced on small scales. In this work, an analytical approximation is provided to systematically study the evolution of ${\mathcal R}_k$ in the USR inflation. We first discuss the asymptotic solutions of the moduli and arguments of ${\mathcal R}_k$…

This https://arxiv.org/abs/2401.18027 has been replaced.
link: https://scholar.google.com/scholar?q=a

On affine Toda field theories related to ${\bf D}_r$ algebras and their real Hamiltonian forms
The paper deals with affine 2-dimensional Toda field theories related to simple Lie algebras of the classical series ${\bf D}_r$. We demonstrate that the complexification procedure followed by a restriction to a specified real Hamiltonian form commutes with the external automorphisms of $\mathfrak{g}$. This is illustrated on the examples ${\bf D}_{r+1}^{(1)} \to {\bf B}_r^{(1)}$ and ${\bf D}_4^{(1)} \to {\bf G}_2^{(1)}$ using external automorphisms of the corresponding extended Dynkin diagrams.

This https://arxiv.org/abs/2110.15065 has been replaced.
link: https://scholar.google.com/scholar?q=a

On a continuous Sárközy type problem
We prove that there exists a constant $\varepsilon > 0$ with the following property: if $K \subset \mathbb{R}^{2}$ is a compact set which contains no pair of the form $\{x, x + (z, z^{2})\}$ for $z \neq 0$, then $\mathrm{dim}_\mathrm{H} K \leq 2 - \varepsilon$.

This https://arxiv.org/abs/2209.13719 has been replaced.
link: https://scholar.google.com/scholar?q=a

Existence and regularity of steady-state solutions of the Navier-Stokes equations arising from irregular data
We analyze the forced incompressible stationary Navier-Stokes flow in $\mathbb{R}^n_+$, $n>2$. Existence of a unique solution satisfying a global integrabilty property measured in a scale of tent spaces is established for small data in homogenous Sobolev space with $s=-\frac{1}{2}$ degree of smoothness. Moreover, the velocity field is shown to be locally Hölder continuous while the pressure belongs to $L^p_{loc}$ for any $p\in (1,\infty)$. Our approach is based on the analysis of the inhomogen…

This https://arxiv.org/abs/2209.07462 has been replaced.
link: https://scholar.google.com/scholar?q=a

Quantum Mechanical Comparison between Lithiated and Sodiated Silicon Nanowires
This computational research study will compare the specific charge capacity (SCC) between lithium ions inserted into crystallize silicon (c-Si) nanowires versus sodium ions inserted into amorphous silicon (a-Si) nanowires. It will be demonstrated that the potential energy V(r) within the lithium-silicon nanowire supports a coherent energy state model with discrete electron particles while the sodium-silicon nanowire potential energy will be discovered to be essentially zero and thus the electro…

This https://arxiv.org/abs/2112.06518 has been replaced.
link: https://scholar.google.com/scholar?q=a

Volumes of subset Minkowski sums and the Lyusternik region
We begin a systematic study of the region of possible values of the volumes of Minkowski subset sums of a collection of $M$ compact sets in $\mathbb{R}^d$, which we call the Lyusternik region, and make some first steps towards describing it. Our main result is that a fractional generalization of the Brunn-Minkowski-Lyusternik inequality conjectured by Bobkov et al. (2011) holds in dimension 1. Even though Fradelizi et al. (2016) showed that it fails in general dimension, we show that a variant …

This https://arxiv.org/abs/2201.13231 has been replaced.
link: https://scholar.google.com/scholar?q=a

Solving particle-antiparticle and cosmological constant problems
We argue that fundamental objects in particle theory are not elementary particles and antiparticles but objects described by irreducible representations (IRs) of the de Sitter (dS) algebra. One might ask why, then, experimental data give the impression that particles and antiparticles are fundamental and there are conserved additive quantum numbers (electric charge, baryon quantum number and others). The matter is that, at the present stage of the universe, the contraction parameter $R$ from th…

This https://arxiv.org/abs/2112.09849 has been replaced.
link: https://scholar.google.com/scholar?q=a

Strongly Lech-independent ideals and Lech's conjecture
We introduce the notion of strongly Lech-independent ideals as a generalization of Lech-independent ideals defined by Lech and Hanes, and use this notion to derive inequalities on multiplicities of ideals. In particular we prove that if $(R,\mathfrak{m}) \to (S,\mathfrak{n})$ is a flat local extension of local rings with $\dim R = \dim S$, the completion of $S$ is the completion of a standard graded ring over a field $k$ with respect to the homogeneous maximal ideal, and the completion of $\mat…

This https://arxiv.org/abs/2206.05479 has been replaced.
link: https://scholar.google.com/scholar?q=a

Ornstein-Uhlenbeck Type Processes on Wasserstein Space
Let $\mathcal P_2$ be the space of probability measures on $\R^d$ having finite second moment, and consider the Riemannian structure on $\mathcal P_2$ induced by the intrinsic derivative on the $L^2$-tangent space. By using stochastic analysis on the tangent space, we construct an Ornstein$-$Uhlenbeck (OU) type Dirichlet form on $\mathcal P_2$ whose generator is formally given by the intrinsic Laplacian with a drift. The log-Sobolev inequality holds and the associated Markov semigroup is $L^2$-…

Don de cheveux
45 cm de cheveux coupés pour faire un don de cheveux. C'est facile Š faire : un simple envoi par lettre, possible Š partir de 10 cm de cheveux !
@… explique tout ici :
https://questionsanimalistes.com/donner-ses-cheveux/

This https://arxiv.org/abs/2202.07344 has been replaced.
link: https://scholar.google.com/scholar?q=a

A strong Borel--Cantelli lemma for recurrence
Consider a mixing dynamical systems $([0,1], T, μ)$, for instance a piecewise expanding interval map with a Gibbs measure $μ$. Given a non-summable sequence $(m_k)$ of non-negative numbers, one may define $r_k (x)$ such that $μ(B(x, r_k(x)) = m_k$. It is proved that for almost all $x$, the number of $k \leq n$ such that $T^k (x) \in B_k (x)$ is approximately equal to $m_1 + \ldots + m_n$. This is a sort of strong Borel--Cantelli lemma for recurrence. A consequence is that \[ \lim_{r \to …

This https://arxiv.org/abs/2203.16523 has been replaced.
link: https://scholar.google.com/scholar?q=a

Shifted Witten classes and topological recursion
The Witten $r$-spin class defines a non-semisimple cohomological field theory. Pandharipande, Pixton and Zvonkine studied two special shifts of the Witten class along two semisimple directions of the associated Dubrovin--Frobenius manifold using the Givental--Teleman reconstruction theorem. We show that the $R$-matrix and the translation of these two specific shifts can be constructed from the solutions of two differential equations that generalise the classical Airy differential equation. Usin…

This https://arxiv.org/abs/2211.13093 has been replaced.
link: https://scholar.google.com/scholar?q=a

Autonomous Marker-less Rapid Aerial Grasping
In a future with autonomous robots, visual and spatial perception is of utmost importance for robotic systems. Particularly for aerial robotics, there are many applications where utilizing visual perception is necessary for any real-world scenarios. Robotic aerial grasping using drones promises fast pick-and-place solutions with a large increase in mobility over other robotic solutions. Utilizing Mask R-CNN scene segmentation (detectron2), we propose a vision-based system for autonomous rapid a…

This https://arxiv.org/abs/2404.00450 has been replaced.
link: https://scholar.google.com/scholar?q=a

Planning and Editing What You Retrieve for Enhanced Tool Learning
Recent advancements in integrating external tools with Large Language Models (LLMs) have opened new frontiers, with applications in mathematical reasoning, code generators, and smart assistants. However, existing methods, relying on simple one-time retrieval strategies, fall short on effectively and accurately shortlisting relevant tools. This paper introduces a novel PLUTO (Planning, Learning, and Understanding for TOols) approach, encompassing `Plan-and-Retrieve (P&R)` and `Edit-and-Ground (E…

This https://arxiv.org/abs/2011.11816 has been replaced.
link: https://scholar.google.com/scholar?q=a

Chain Conditions for Etale Groupoid Algebras
Let $R$ be a unital commutative ring with unit and $\mathscr{G}$ an ample groupoid. Using the topology of the groupoid $\mathscr{G}$, Steinberg defined an etale groupoid algebra $R\mathscr{G}$. These etale groupoid algebras generalize various algebras including group algebras, commutative algebras over a field generated by idempotents, traditional groupoid algebras, Leavitt path algebras, higher-rank graph algebras, and inverse semigroup algebras. Steinberg later characterized the classical cha…

This https://arxiv.org/abs/2212.00147 has been replaced.
link: https://scholar.google.com/scholar?q=a

Homotopical models for metric spaces and completeness
Categories enriched in the opposite poset of non-negative reals can be viewed as generalizations of metric spaces, known as Lawvere metric spaces. In this article, we develop model structures on the categories $\mathbb{R}\text-\mathbf{Cat}$ and $\mathbb{R}\text-\mathbf{Cat}^\mathrm{sym}$ of Lawvere metric spaces and symmetric Lawvere metric spaces, each of which capture different features pertinent to the study of metric spaces. More precisely, in the three model structures we construct, the fi…

This https://arxiv.org/abs/1607.06444 has been replaced.
link: https://scholar.google.com/scholar?q=a

The Complexity of Drawing Graphs on Few Lines and Few Planes
It is well known that any graph admits a crossing-free straight-line drawing in $\mathbb{R}^3$ and that any planar graph admits the same even in $\mathbb{R}^2$. For a graph $G$ and $d \in \{2,3\}$, let $ρ^1_d(G)$ denote the smallest number of lines in $\mathbb{R}^d$ whose union contains a crossing-free straight-line drawing of $G$. For $d=2$, $G$ must be planar. Similarly, let $ρ^2_3(G)$ denote the smallest number of planes in $\mathbb{R}^3$ whose union contains a crossing-free straight-line …

This https://arxiv.org/abs/2210.03203 has been replaced.
link: https://scholar.google.com/scholar?q=a

The Aemulus Project VI: Emulation of beyond-standard galaxy clustering statistics to improve cosmological constraints
There is untapped cosmological information in galaxy redshift surveys in the non-linear regime. In this work, we use the AEMULUS suite of cosmological $N$-body simulations to construct Gaussian process emulators of galaxy clustering statistics at small scales ($0.1-50 \: h^{-1}\,\mathrm{Mpc}$) in order to constrain cosmological and galaxy bias parameters. In addition to standard statistics -- the projected correlation function $w_\mathrm{p}(r_\mathrm{p})$, the redshift-space monopole of the cor…

This https://arxiv.org/abs/2105.03805 has been replaced.
link: https://scholar.google.com/scholar?q=a

A new proof for the existence of rotationally symmetric gradient Ricci solitons
We give a new proof for the existence of rotationally symmetric steady and expanding gradient Ricci solitons in dimension $n+1$, $2\le n\le 4$, with metric $g=\frac{da^2}{h(a^2)}+a^2d\,σ$ for some function $h$ where $dσ$ is the standard metric on the unit sphere $S^n$ in $\mathbb{R}^n$. More precisely for any $λ\ge 0$, $2\le n\le 4$ and $μ_1\in\mathbb{R}$, we prove the existence of unique solution $h\in C^2((0,\infty))\cap C^1([0,\infty))$ for the equation $2r^2h(r)h_{rr}(r)=(n-1)h(r)(h(r)-…

This https://arxiv.org/abs/2209.03684 has been replaced.
link: https://scholar.google.com/scholar?q=a

Packing $K_r$s in bounded degree graphs
We study the problem of finding a maximum-cardinality set of $r$-cliques in an undirected graph of fixed maximum degree $Δ$, subject to the cliques in that set being either vertex-disjoint or edge-disjoint. It is known for $r=3$ that the vertex-disjoint (edge-disjoint) problem is solvable in linear time if $Δ=3$ ($Δ=4$) but APX-hard if $Δ\geq 4$ ($Δ\geq 5$). We generalise these results to an arbitrary but fixed $r \geq 3$, and provide a complete complexity classification for both the ver…

This https://arxiv.org/abs/2211.08929 has been replaced.
link: https://scholar.google.com/scholar?q=a

The Liouville theorem for a class of Fourier multipliers and its connection to coupling
The classical Liouville property says that all bounded harmonic functions in $\mathbb{R}^n$, i.e.\ all bounded functions satisfying $Δf = 0$, are constant. In this paper we obtain necessary and sufficient conditions on the symbol of a Fourier multiplier operator $m(D)$, such that the solutions $f$ to $m(D)f=0$ are Lebesgue a.e.\ constant (if $f$ is bounded) or coincide Lebesgue a.e.\ with a polynomial (if $f$ grows like a polynomial). The class of Fourier multipliers includes the (in general n…

This https://arxiv.org/abs/2210.10751 has been replaced.
link: https://scholar.google.com/scholar?q=a

Global fit to $b \to cτν$ anomaly 2024 Spring breeze
Recently, several new experimental results of the test of lepton flavor universality (LFU) in $B\to D^{(\ast)}$ semi-leptonic decays were announced: the first result of $R_{D}$ from the LHCb Run 1 data, the first results of $R_{D}$ and $R_{D^\ast}$ from the LHCb Run 2 data, and the first result of $R_{D^\ast}$ from the Belle II collaboration. Including these new data, a global analysis still prefers the violation of the LFU between the tau and light leptons. A new world average of the data from…

This https://arxiv.org/abs/2304.00762 has been replaced.
link: https://scholar.google.com/scholar?q=a

The (r, δ)-Locality of Repeated-Root Cyclic Codes with Prime Power Lengths
Locally repairable codes (LRCs) are designed for distributed storage systems to reduce the repair bandwidth and disk I/O complexity during the storage node repair process. A code with $(r,δ)$-locality (also called an $(r,δ)$-LRC) can simultaneously repair up to $δ-1$ symbols in a codeword by accessing at most $r$ other symbols in the codeword. In this paper, we propose a new method to calculate the $(r,δ)$-locality of cyclic codes. Initially, we give a description of the algebraic structure…

This https://arxiv.org/abs/2404.15482 has been replaced.
link: https://scholar.google.com/scholar?q=a

What drives the corpulence of galaxies? I. The formation of central compact dwarf galaxies in TNG50
Nearby dwarf galaxies display a variety of effective radii (sizes) at a given stellar mass, suggesting different evolution scenarios according to their final "stellar" size. The TNG hydrodynamical simulations present a bimodality in the z = 0 size - mass relation (SMRz0) of dwarf galaxies, at $r_{1/2,\star}$ ~ 450 pc. Using the TNG50 simulation, we explored the evolution of the most massive progenitors of dwarf galaxies (z=0 $\log( M_\star / \mathrm{M}_\odot)$ between 8.4 and 9.2) that end up a…

This https://arxiv.org/abs/2208.13489 has been replaced.
link: https://scholar.google.com/scholar?q=a

The maximal running time of hypergraph bootstrap percolation
We show that for every $r\ge 3$, the maximal running time of the $K^{r}_{r+1}$-bootstrap percolation in the complete $r$-uniform hypergraph on $n$ vertices $K_n^r$ is $Θ(n^r)$. This answers a recent question of Noel and Ranganathan in the affirmative, and disproves a conjecture of theirs. Moreover, we show that the prefactor is of the form $r^{-r} \mathrm{e}^{O(r)}$ as $r\to\infty$.

This https://arxiv.org/abs/2207.10389 has been replaced.
link: https://scholar.google.com/scholar?q=a

Geometric quadratic Chabauty and $p$-adic heights
Let $X$ be a curve of genus $g>1$ over $\mathbb{Q}$ whose Jacobian $J$ has Mordell--Weil rank $r$ and Néron--Severi rank $ρ$. When $r < g+ ρ- 1$, the geometric quadratic Chabauty method determines a finite set of $p$-adic points containing the rational points of $X$. We describe algorithms for geometric quadratic Chabauty that translate the geometric quadratic Chabauty method into the language of $p$-adic heights and $p$-adic (Coleman) integrals. This translation also allows us to give a com…

This https://arxiv.org/abs/2401.07831 has been replaced.
link: https://scholar.google.com/scholar?q=a

Reduced polygons in the hyperbolic plane
For a hyperplane $H$ supporting a convex body $C$ in the hyperbolic space $\mathbb{H}^d$ we define the width of $C$ determined by $H$ as the distance between $H$ and a most distant ultraparallel hyperplane supporting $C$. The minimum width of $C$ over all supporting $H$ is called the thickness $Δ(C)$ of $C$. A convex body $R \subset \mathbb{H}^d$ is said to be reduced if $Δ(Z) < Δ(R)$ for every convex body $Z$ properly contained in $R$. We describe a class of reduced polygons in $\mathbb{H}^…

This https://arxiv.org/abs/2208.03277 has been replaced.
link: https://scholar.google.com/scholar?q=a

V-filtrations and minimal exponents for locally complete intersection singularities
We define and study a notion of minimal exponent for a locally complete intersection subscheme $Z$ of a smooth complex algebraic variety $X$, extending the invariant defined by Saito in the case of hypersurfaces. Our definition is in terms of the Kashiwara-Malgrange $V$-filtration associated to $Z$. We show that the minimal exponent describes how far the Hodge filtration and order filtration agree on the local cohomology $H^r_Z({\mathcal O}_X)$, where $r$ is the codimension of $Z$ in $X$. We al…

This https://arxiv.org/abs/2106.09984 has been replaced.
link: https://scholar.google.com/scholar?q=a

Some factorization properties of idealization in commutative rings with zero divisors
We study some factorization properties of the idealization $R \mathop{(\! + \! )} M$ of a module $M$ in a commutative ring $R$ which is not necessarily a domain. We show that $R \mathop{(\! + \! )} M$ is ACCP if and only if $R$ is ACCP and $M$ satisfies ACC on its cyclic submodules. We give an example to show that the BF property is not necessarily preserved in idealization, and give some conditions under which $R \mathop{(\! + \! )} M$ is a BFR. We also characterize the idealization rings whic…

This https://arxiv.org/abs/2212.06274 has been replaced.
link: https://scholar.google.com/scholar?q=a

The one-sided cycle shuffles in the symmetric group algebra
We study a family of shuffling operators on the symmetric group $S_n$, which includes the top-to-random shuffle. The general shuffling scheme consists of removing one card at a time from the deck (according to some probability distribution) and re-inserting it at a (uniformly) random position further below. Rewritten in terms of the group algebra $\mathbb{R}[S_n]$, our shuffle corresponds to right multiplication by a linear combination of the elements \[t_i:=\text{cyc}_{i}+\text{cyc}_{i,i+1}+\t…

This https://arxiv.org/abs/2212.01765 has been replaced.
link: https://scholar.google.com/scholar?q=a

The Degasperis-Procesi equation on the line: Soliton resolution, asymptotic stability of $N$-solitons, Painlevé and Airy asymptotics
The Degasperis-Procesi (DP) equation is a completely integrable system and admits a $3\times3$ matrix Lax pair. In this paper, we address a complete result on the long-time asymptotics for the DP equation in the whole upper half-plane $(x,t)\in \mathbb{R}\times \mathbb{R}^+$. Using the $\bar \partial$-generalization of the Deift-Zhou nonlinear steepest descent method and a double limit technique, we derive the leading order approximation to the solution $u(x,t)$ of the DP equation for large tim…

This https://arxiv.org/abs/2108.12696 has been replaced.
link: https://scholar.google.com/scholar?q=a

Linked orbits of homeomorphisms of the plane and Gambaudo-Kolev Theorem
Let $h : \mathbb{R}^2 \to \mathbb{R}^2$ be an orientation preserving homeomorphism of the plane. For any bounded orbit $\mathcal{O}(x)=\{h^n(x):n\in\mathbb{Z}\}$ there exists a fixed point $x'\in\mathbb{R}^2$ of $h$ linked to $\mathcal{O}(x)$ in the sense of Gambaudo: one cannot find a Jordan curve $C\subseteq\mathbb{R}^2$ around $\mathcal{O}(x)$, separating it from $x'$, that is isotopic to $h(C)$ in $\mathbb{R}^2\setminus\left(\mathcal{O}(x)\cup\{x'\}\right)$.

This https://arxiv.org/abs/2202.09894 has been replaced.
link: https://scholar.google.com/scholar?q=a

Third-order affine-invariant (systems of) PDEs in two independent variables as vanishing of the Fubini-Pick invariant
If one imposes that a $3^{\mathrm{rd}}$ order PDE $\mathcal{E}:=\{F(x,y,u,u_x,\dots,u_{xyy},u_{yyy})=0\}$ in two independent variables $x,y$ and one unknown function $u$ be invariant with respect to the group of affine transformation $\mathrm{Aff}(3)$ of $\mathbb{R}^3=\{(x,y,u)\}$, then it turns out that there is (up to a nowhere zero factor) only one choice of $F$. In this paper we study more in depth the geometry behind the occurrence of such a PDE: after proving the relationship of $F$ with …

This https://arxiv.org/abs/2210.10053 has been replaced.
link: https://scholar.google.com/scholar?q=a

Nozzle Shocks, Disk Tearing and Streamers Drive Rapid Accretion in 3D GRMHD Simulations of Warped Thin Disks
The angular momentum of gas feeding a black hole (BH) is typically misaligned with respect to the BH spin, resulting in a tilted accretion disk. Rotation of the BH drags the surrounding space-time, manifesting as Lense-Thirring torques that lead to disk precession and warping. We study these processes by simulating a thin ($H/r=0.02$), highly tilted ($\mathcal{T}=65^\circ$) accretion disk around a rapidly rotating ($a=0.9375$) BH at extremely high resolutions, which we performed using the gener…

This https://arxiv.org/abs/2103.13214 has been replaced.
link: https://scholar.google.com/scholar?q=a

Finite time/Infinite time blow-up behaviors for the inhomogeneous nonlinear Schrödinger equation
In this work, we consider the following focusing inhomogeneous nonlinear Schrödinger equation \begin{align*} i\partial_t u+Δu +|x|^{-b}|u|^p u=0,\quad (t, x)\in\mathbb{R}\times\mathbb{R}^N \end{align*} with $0

This https://arxiv.org/abs/2308.12796 has been replaced.
link: https://scholar.google.com/scholar?q=a

The category of necklaces is Reedy monoidal
In the first part of this note we further the study of the interactions between Reedy and monoidal structures on a small category, building upon the work of Barwick. We define a Reedy monoidal category as a Reedy category $\mathcal{R}$ which is monoidal such that for all symmetric monoidal model categories $\textbf{A}$, the category $\mathrm{Fun}\left(\mathcal{R}^{\mathrm{op}}, \textbf{A}\right)_{\mathrm{Reedy}}$ is model monoidal when equipped with the Day convolution. In the second part, we s…

This https://arxiv.org/abs/2111.04610 has been replaced.
link: https://scholar.google.com/scholar?q=a

Sum-of-squares hierarchies for polynomial optimization and the Christoffel-Darboux kernel
Consider the problem of minimizing a polynomial $f$ over a compact semialgebraic set ${\mathbf{X} \subseteq \mathbb{R}^n}$. Lasserre introduces hierarchies of semidefinite programs to approximate this hard optimization problem, based on classical sum-of-squares certificates of positivity of polynomials due to Putinar and Schmüdgen. When $\mathbf{X}$ is the unit ball or the standard simplex, we show that the hierarchies based on the Schmüdgen-type certificates converge to the global minimum of…

This https://arxiv.org/abs/2202.10314 has been replaced.
link: https://scholar.google.com/scholar?q=a

Time complexity of the Analyst's Traveling Salesman algorithm
The Analyst's Traveling Salesman Problem asks for conditions under which a (finite or infinite) subset of $\mathbb{R}^N$ is contained on a curve of finite length. We show that for finite sets, the algorithm constructed by Schul (2007)and Badger-Naples-Vellis (2019) that solves the Analyst's Traveling Salesman Problem has polynomial time complexity and we determine the sharp exponent.

This https://arxiv.org/abs/2108.12696 has been replaced.
link: https://scholar.google.com/scholar?q=a

Linked orbits of homeomorphisms of the plane and Gambaudo-Kolev Theorem
Let $h : \mathbb{R}^2 \to \mathbb{R}^2$ be an orientation preserving homeomorphism of the plane. For any bounded orbit $\mathcal{O}(x)=\{h^n(x):n\in\mathbb{Z}\}$ there exists a fixed point $x'\in\mathbb{R}^2$ of $h$ linked to $\mathcal{O}(x)$ in the sense of Gambaudo: one cannot find a Jordan curve $C\subseteq\mathbb{R}^2$ around $\mathcal{O}(x)$, separating it from $x'$, that is isotopic to $h(C)$ in $\mathbb{R}^2\setminus\left(\mathcal{O}(x)\cup\{x'\}\right)$.

This https://arxiv.org/abs/2001.11901 has been replaced.
link: https://scholar.google.com/scholar?q=a

Solvability for non-smooth Schrödinger equations with singular potentials and square integrable data
We develop a holomorphic functional calculus for first-order operators $DB$ to solve boundary value problems for Schrödinger equations $-\mathrm{div}\, A \nabla u + a V u = 0$ in the upper half-space $\mathbb{R}^{n+1}_+$ with $n\in\mathbb{N}$. This relies on quadratic estimates for $DB$, which are proved for coefficients $A,a,V$ that are independent of the transversal direction to the boundary, and comprised of a complex-elliptic pair $(A,a)$ that are bounded and measurable, and a singular pot…

This https://arxiv.org/abs/1704.08286 has been replaced.
link: https://scholar.google.com/scholar?q=a

The First Syzygy of Hibi Rings Associated with Planar distributive lattices
Let $\mathcal{L}$ be a finite distributive lattice and $S=K[x_α: α\in \mathcal{L}]$ be a polynomial ring over a field $K$ and $I=\langle x_αx_β- x_{α\vee β} x_{α\wedgeβ} : α\nsim β,α,β\in {\mathcal{L}} \rangle$ an ideal of $S$. In this article we describe the first syzygy of the Hibi ring $R[\mathcal{L}]=S/I$, for a planar distributive lattice $\mathcal{L}$. We also derive an exact formula for the first Betti number of a planar distributive lattice. We give a characterization of pla…

This https://arxiv.org/abs/1912.12057 has been replaced.
link: https://scholar.google.com/scholar?q=a

Existence of Schrodinger Evolution with Absorbing Boundary Condition
Consider a non-relativistic quantum particle with wave function inside a region $Ω\subset \mathbb{R}^3$, and suppose that detectors are placed along the boundary $\partial Ω$. The question how to compute the probability distribution of the time at which the detector surface registers the particle boils down to finding a reasonable mathematical definition of an ideal detecting surface; a particularly convincing definition, called the absorbing boundary rule, involves a time evolution for the p…

This https://arxiv.org/abs/2404.00206 has been replaced.
link: https://scholar.google.com/scholar?q=a

Random Reed-Solomon Codes are List Recoverable with Optimal List Size
We prove that Reed-Solomon (RS) codes with random evaluation points are list recoverable up to capacity with optimal output list size, for any input list size. Namely, given an input list size $\ell$, a designated rate $R$, and any $\varepsilon > 0$, we show that a random RS code is list recoverable from $1-R-\varepsilon$ fraction of errors with output list size $L = O(\ell/\varepsilon)$, for field size $q=\exp(\ell,1/\varepsilon) \cdot n^2$. In particular, this shows that random RS codes are…

This https://arxiv.org/abs/2301.00206 has been replaced.
link: https://scholar.google.com/scholar?q=a

Melnikov's persistence for completely degenerate Hamiltonian systems with high co-dimension
In this paper, we study the Melnikov's persistence about lower-dimensional invariant tori for completely degenerate Hamiltonian systems with the following Hamiltonian \begin{equation*} H(x,y,u,v)=h(y)+g(u,v)+\varepsilon P(x,y,u,v),~~~(x,y,u,v)\in \mathbb{T}^n\times{G}\times \mathbb{R}^d\times \mathbb{R}^d, \end{equation*} where $n\geq2$ and $d\geq1$ are positive integers, $g=o(|u|^2+|v|^2)$ admits complete degeneracy and certain transversality, and $\varepsilon P$ is the sma…

This https://arxiv.org/abs/2208.04590 has been replaced.
link: https://scholar.google.com/scholar?q=a

New bounds for the number of connected components of fewnomial hypersurfaces
We prove that the number of connected components of a smooth hypersurface in the positive orthant of $\mathbb{R}^n$ defined by a real polynomial with $d + k + 1$ monomials, where $d$ is the dimension of the affine span of the exponent vectors, is smaller than or equal to $8(d+1)^{k-1} 2^{k-1 \choose 2}$, improving the previously known bounds. We refine this bound for $k = 2$ by showing that a smooth hypersurface defined by a real polynomial with $d+3$ monomials in $n$ variables has at most $\lf…

This https://arxiv.org/abs/2209.06674 has been replaced.
link: https://scholar.google.com/scholar?q=a

Analytic aspects of $q,r$-analogue of poly-Stirling numbers of both kinds
The Stirling numbers of type $B$ of the second kind count signed set partitions. In this paper we provide new combinatorial and analytical identities regarding these numbers as well as Broder's $r$-version of these numbers. Among these identities one can find recursions, explicit formulas based on the inclusion-exclusion principle, and also exponential generating functions. These Stirling numbers can be considered as members of a wider family of triangles of numbers that are characterized usi…

This https://arxiv.org/abs/2108.06708 has been replaced.
link: https://scholar.google.com/scholar?q=a

Manifolds for which Huber's Theorem holds
Extensions of Huber's Theorem to higher dimensions with $L^\frac{n}{2}$ bounded scalar curvature have been extensively studied over the years. In this paper, we delve into the properties of conformal metrics on a punctured ball with $\|R\|_{L^\frac{n}{2}}<+\infty$, aiming to identify necessary geometric constraints for Huber's theorem to be applicable. Unexpectedly, such metrics are more rigid than we initially anticipated. For instance, we found that the volume density at infinity is…

This https://arxiv.org/abs/2105.04525 has been replaced.
link: https://scholar.google.com/scholar?q=a

$2$-Modular Matrices
A rank-$r$ integer matrix $A$ is $Δ$-modular if the determinant of each $r \times r$ submatrix has absolute value at most $Δ$. The class of $1$-modular, or unimodular, matrices is of fundamental significance in both integer programming theory and matroid theory. A 1957 result of Heller shows that the maximum number of nonzero, pairwise non-parallel rows of a rank-$r$ unimodular matrix is ${r + 1 \choose 2}$. We prove that, for each sufficiently large integer $r$, the maximum number of nonzero…

This https://arxiv.org/abs/2210.11225 has been replaced.
link: https://scholar.google.com/scholar?q=a

Dirichlet Heat kernel estimates for a large class of anisotropic Markov processes
Let $Z=(Z^{1}, \ldots, Z^{d})$ be the d-dimensional Lévy {process} where {$Z^i$'s} are independent 1-dimensional Lévy {processes} with identical jumping kernel $ ν^1(r) =r^{-1}ϕ(r)^{-1}$. Here $ϕ$ is {an} increasing function with weakly scaling condition of order $\underline α, \overline α\in (0, 2)$. We consider a symmetric function $J(x,y)$ comparable to \begin{align*} \begin{cases} ν^1(|x^i - y^i|)\qquad&\text{ if $x^i \ne y^i$ for some $i$ and $x^j = y^j$ for all $j \ne …

This https://arxiv.org/abs/2208.01945 has been replaced.
link: https://scholar.google.com/scholar?q=a

Well-posedness of a parametrically forced nonlinear Schrödinger equation driven by translation-invariant noise
We prove well-posedness in $H^σ(\mathbb{R})$ for any $σ\in [0,\infty)$ of a parametrically forced nonlinear Schrödinger equation (PFNLS) in one dimension driven by multiplicative Stratonovich noise which has spatially homogeneous statistics. The noise is white in time and correlated in space. We first construct local mild solutions via a fixed-point argument. We then formulate a blow-up criterion by showing that the equation has persistence of integrability and regularity as long as the $L^2…

This https://arxiv.org/abs/2205.07492 has been replaced.
link: https://scholar.google.com/scholar?q=a

Moduli spaces of $\mathbb{Z}/k\mathbb{Z}$-constellations over $\mathbb{A}^2$
Let $ρ:\mathbb{Z}/k \mathbb{Z}\rightarrow \text{SL}(2,\mathbb{C})$ be a representation of a finite abelian group and let $Θ^{\text{gen}}\subset \text{Hom}_\mathbb{Z}(R(\mathbb{Z}/k\mathbb{Z}),\mathbb{Q})$ be the space of generic stability conditions on the set of $G$-constellations. We provide a combinatorial description of all the chambers $C\subsetΘ^{\text{gen}}$ and prove that there are $k!$ of them. Moreover, we introduce the notion of simple chamber and we show that, in order to know al…

This https://arxiv.org/abs/2202.09858 has been replaced.
link: https://scholar.google.com/scholar?q=a

The real equiangular tight frames obtained from rank $3$ graphs
We present all nontrivial real equiangular tight frames $\{φ_m\}_{m=1}^M$ in $\mathbb{R}^N$ obtained as spherical embeddings of primitive rank $3$ graphs on $M$ vertices, and those such that one of their associated $M$ strongly regular graphs on $M-1$ vertices is a primitive rank $3$ graph.

This https://arxiv.org/abs/2212.03758 has been replaced.
link: https://scholar.google.com/scholar?q=a

Finite-time blow-up to hyperbolic Keller-Segel system of consumption type with logarithmic sensitivity
This paper deals with a hyperbolic Keller-Segel system of consumption type with the logarithmic sensitivity \begin{equation*} \partial_{t} ρ= - χ\nabla \cdot \left (ρ\nabla \log c\right),\quad \partial_{t} c = - μcρ\quad (χ,\,μ>0) \end{equation*} in $\mathbb{R}^d\; (d \ge1)$ for nonvanishing initial data. This system is closely related to tumor angiogenesis, an important example of chemotaxis. We firstly show the local existence of smooth solutions corresponding to nonvanishing smooth …

This https://arxiv.org/abs/2206.10723 has been replaced.
link: https://scholar.google.com/scholar?q=a

Percolation of strongly correlated Gaussian fields I. Decay of subcritical connection probabilities
We study the decay of connectivity of the subcritical excursion sets of a class of strongly correlated Gaussian fields. Our main result shows that, for smooth isotropic Gaussian fields whose covariance kernel $K(x)$ is regularly varying at infinity with index $α\in [0, 1)$, the probability that $\{f \le \ell\}$, $\ell < \ell_c$, connects the origin to distance $R$ decays sub-exponentially in $R$ at log-asymptotic rate $c_α(\ell_c-\ell)^2 / K(R)$ for an explicit $c_α> 0$. If $α= 1$ and $\int…

This https://arxiv.org/abs/2102.12246 has been replaced.
link: https://scholar.google.com/scholar?q=a

Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces
Let $\mathcal{L}=(L,[\cdot\,,\cdot],δ)$ be an algebraic Lie algebroid over a smooth projective curve $X$ of genus $g\geq 2$ such that $L$ is a line bundle whose degree is less than $2-2g$. Let $r$ and $d$ be coprime numbers. We prove that the motivic class of the moduli space of $\mathcal{L}$-connections of rank $r$ and degree $d$ over $X$ does not depend on the Lie algebroid structure $[\cdot\,,\cdot]$ and $δ$ of $\mathcal{L}$ and neither on the line bundle $L$ itself, but only on the degree…

This https://arxiv.org/abs/2403.01338 has been replaced.
link: https://scholar.google.com/scholar?q=a

Normalized solutions of quasilinear Schrödinger equations with a general nonlinearity
We are concerned with solutions of the following quasilinear Schrödinger equations \begin{eqnarray*} -{\mathrm{div}}\left(φ^{2}(u) \nabla u\right)+φ(u) φ^{\prime}(u)|\nabla u|^{2}+λu=f(u), \quad x \in \mathbb{R}^{N} \end{eqnarray*} with prescribed mass $$ \int_{\mathbb{R}^{N}} u^{2} \mathrm{d}x=c, $$ where $N\ge 3, c>0$, $λ\in \mathbb{R}$ appears as the Lagrange multiplier and $φ\in C ^{1}(\mathbb{R} ,\mathbb{R}^{+})$. The nonlinearity $f \in C\left ( \mathbb{R}, \, \mathbb{R} …

This https://arxiv.org/abs/2205.12826 has been replaced.
link: https://scholar.google.com/scholar?q=a

Two Ramsey problems in blowups of graphs
Given graphs $G$ and $H$, we say $G \stackrel{r}{\to} H$ if every $r$-colouring of the edges of $G$ contains a monochromatic copy of $H$. Let $H[t]$ denote the $t$-blowup of $H$. The blowup Ramsey number $B(G \stackrel{r}{\to} H;t)$ is the minimum $n$ such that $G[n] \stackrel{r}{\to} H[t]$. Fox, Luo and Wigderson refined an upper bound of Souza, showing that, given $G$, $H$ and $r$ such that $G \stackrel{r}{\to} H$, there exist constants $a=a(G,H,r)$ and $b=b(H,r)$ such that for all $t \in \ma…

This https://arxiv.org/abs/2109.01746 has been replaced.
link: https://scholar.google.com/scholar?q=a

The Muirhead-Rado inequality, 1 Vector majorization and the permutohedron
Let $\mathbf{a}$ and $\mathbf{b}$ be vectors in $\mathbf{R}^n$ with nonnegative coordinates. Permuting the coordinates, we can assume that $a_1 \geq \cdots \geq a_n$ and $b_1 \geq \cdots \geq b_n$. The vector $\mathbf{a}$ majorizes the vector $\mathbf{b}$, denoted $\mathbf{b} \preceq \mathbf{a}$, if $\sum_{i=1}^n b_i = \sum_{i=1}^n a_i$ and $\sum_{i=1}^k b_i \leq \sum_{i=1}^k a_i$ for all $k \in \{1,\ldots,n-1\}$. This paper proves theorems of Hardy-Littlewood-Pólya and Rado that $\mathbf{b} \…

This https://arxiv.org/abs/2108.13413 has been replaced.
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$\mathrm{SU}(r)$ Vafa-Witten invariants, Ramanujan's continued fractions, and cosmic strings
We conjecture a structure formula for the $\mathrm{SU}(r)$ Vafa-Witten partition function for surfaces with holomorphic 2-form. The conjecture is based on $S$-duality and a structure formula for the vertical contribution previously derived by the third-named author using Gholampour-Thomas's theory of virtual degeneracy loci. For ranks $r=2,3$, conjectural expressions for the partition function in terms of the theta functions of $A_{r-1}, A_{r-1}^{\vee}$ and Seiberg-Witten invariants were know…

This https://arxiv.org/abs/2208.01842 has been replaced.
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Remarks on the determination of the Lorentzian metric by the length of geodesics
We consider a Lorentzian metric in $\mathbb{R}\times\mathbb{R}^n$. We show that if we know the lengths of space-time geodesics starting at $(0,y,η)$ when $t=0$, then we can recover the metric. We also prove the rigidity of Lorentzian metrics.

This https://arxiv.org/abs/2107.06974 has been replaced.
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Multiple twins in permutations
By an $r$-tuplet in a permutation we mean a family of $r$ pairwise disjoint subsequences with the same relative order. The length of an $r$-tuplet is defined as the length of any single subsequence in the family. Let $t^{(r)}(n)$ denote the largest $k$ such that every permutation of length $n$ contains an $r$-tuplet of length $k$. We prove that $t^{(r)}(n)=O\left(n^{\frac r{2r-1}}\right)$ and $t^{(r)}(n)=Ω\left( n^{\frac{R}{2R-1}} \right)$, where $R=\binom{2r-1}r$. We conjecture that the upper…

This https://arxiv.org/abs/2205.04397 has been replaced.
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Norm-resolvent convergence for Neumann Laplacians on manifolds thinning to graphs
Norm-resolvent convergence with order-sharp error estimate is established for Neumann Laplacians on thin domains in $\mathbb{R}^d,$ $d\ge2,$ converging to metric graphs in the limit of vanishing thickness parameter in the resonant case. The vertex matching conditions of the limiting quantum graph are revealed as being closely related to $δ'$ type.

This https://arxiv.org/abs/2204.02487 has been replaced.
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Convex polytopes in restricted point sets in $\mathbb{R}^d$
For a finite point set $P \subset \mathbb{R}^d$, denote by $\text{diam}(P)$ the ratio of the largest to the smallest distances between pairs of points in $P$. Let $c_{d, α}(n)$ be the largest integer $c$ such that any $n$-point set $P \subset \mathbb{R}^d$ in general position, satisfying $\text{diam}(P) < α\sqrt[d]{n}$, contains an $c$-point convex independent subset. We determine the asymptotics of $c_{d, α}(n)$ as $n \to \infty$ by showing the existence of positive constants $β= β(d, α)…

This https://arxiv.org/abs/2403.01220 has been replaced.
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Stability of graph pairs involving cycles
A graph pair $(Γ, Σ)$ is called stable if $\aut(Γ)\times\aut(Σ)$ is isomorphic to $\aut(Γ\timesΣ)$ and unstable otherwise, where $Γ\timesΣ$ is the direct product of $Γ$ and $Σ$. A graph is called $R$-thin if distinct vertices have different neighbourhoods. $Γ$ and $Σ$ are said to be coprime if there is no nontrivial graph $Δ$ such that $Γ\cong Γ_1 \times Δ$ and $Σ\cong Σ_1 \times Δ$ for some graphs $Γ_1$ and $Σ_1$. An unstable graph pair $(Γ, Σ)$ is called nontrivially un…

This https://arxiv.org/abs/2111.06983 has been replaced.
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Positroids are 3-colorable
We show that every positroid of rank $r \geq 3$ has a good coline. Using the definition of the chromatic number of oriented matroid introduced by J.\ Nešetřil, R.\ Nickel, and W.~Hochstättler, this shows that every orientation of a positroid is 3-colorable.