Miracle-dz/UniFAR-FTUs huggingface dataset:
On the isoperimetric problem with perimeter density r^p

In this paper the author studies the isoperimetric problem in $\re^n$ with perimeter density $|x|^p$ and volume density $1. $ We settle completely the case $n=2,$ completing a previous work by the author: we characterize the case of equality if $0\leq p\leq 1$ and deal with the case $-\infty<p<-1$ (with the additional assumption $0\in\Omega$). In the case $n\geq 3$ we deal mainly with the case $-\infty<p<0,$ showing among others that the results in $2$ dimensions do not generalize for the range $-n+1<p<0. $
Well-posedness and accuracy of the ensemble Kalman filter in discrete and continuous time

The ensemble Kalman filter (EnKF) is a method for combining a dynamical model with data in a sequential fashion. Despite its widespread use, there has been little analysis of its theoretical properties. Many of the algorithmic innovations associated with the filter, which are required to make a useable algorithm in practice, are derived in an ad hoc fashion. The aim of this paper is to initiate the development of a systematic analysis of the EnKF, in particular to do so for small ensemble size. The perspective is to view the method as a state estimator, and not as an algorithm which approximates the true filtering distribution. The perturbed observation version of the algorithm is studied, without and with variance inflation. Without variance inflation well-posedness of the filter is established; with variance inflation accuracy of the filter, with respect to the true signal underlying the data, is established. The algorithm is considered in discrete time, and also for a continuous time limit arising when observations are frequent and subject to large noise. The underlying dynamical model, and assumptions about it, is sufficiently general to include the Lorenz '63 and '96 models, together with the incompressible Navier–Stokes equation on a two-dimensional torus. The analysis is limited to the case of complete observation of the signal with additive white noise. Numerical results are presented for the Navier–Stokes equation on a two-dimensional torus for both complete and partial observations of the signal with additive white noise.
Statistically-Consistent Identification of Switched Linear Systems

This paper studies an application at the intersection of learning and control. Suppose we have a set of linear autonomous systems with bounded process noise, but the dynamics of each system are unknown. The goal of the application is to design a policy that stabilizes the system. The underlying question is how to estimate the dynamics of each system given that measurements of each system will be nonsequential. Though seemingly straightforward, existing proof techniques for proving statistical consistency of system identification procedures fail when measurements are nonsequential. Here, we provide an estimator that leverages boundedness of the process noise, and we prove its statistical consistency holds even when measurements are nonsequential. We illustrate the strong consistency (i.e., almost sure convergence) of our estimator by using it to construct a stabilizing policy for the motivating learning-based switched control application.
A type theory for synthetic $\infty$-categories

We propose foundations for a synthetic theory of $(\infty,1)$-categories within homotopy type theory. We axiomatize a directed interval type, then define higher simplices from it and use them to probe the internal categorical structures of arbitrary types. We define Segal types, in which binary composites exist uniquely up to homotopy; this automatically ensures composition is coherently associative and unital at all dimensions. We define Rezk types, in which the categorical isomorphisms are additionally equivalent to the type-theoretic identities - a "local univalence" condition. And we define covariant fibrations, which are type families varying functorially over a Segal type, and prove a "dependent Yoneda lemma" that can be viewed as a directed form of the usual elimination rule for identity types. We conclude by studying homotopically correct adjunctions between Segal types, and showing that for a functor between Rezk types to have an adjoint is a mere proposition. To make the bookkeeping in such proofs manageable, we use a three-layered type theory with shapes, whose contexts are extended by polytopes within directed cubes, which can be abstracted over using "extension types" that generalize the path-types of cubical type theory. In an appendix, we describe the motivating semantics in the Reedy model structure on bisimplicial sets, in which our Segal and Rezk types correspond to Segal spaces and complete Segal spaces.
The 1-Yamabe equation on graphs

We study the following [Formula: see text]-Yamabe equation on connected finite graphs [Formula: see text] where [Formula: see text] is the discrete [Formula: see text]-Laplacian, [Formula: see text] and [Formula: see text] are known. We show that the above [Formula: see text]-Yamabe equation always has a nontrivial solution [Formula: see text], [Formula: see text].
Partial flag incidence algebras

The $n^{th}$ partial flag incidence algebra of a poset $P$ is the set of functions from $P^n$ to some ring which are zero on non-partial flag vectors. These partial flag incidence algebras for $n>2$ are not commutative, not unitary, and not associative. However, partial flag incidence algebras contain generalized zeta, delta, and M\"obius functions which contain delicate information of the poset. Using these functions we define multi-indexed Whitney numbers, of both kinds, and develop relationships between them. Then we apply these results to recursively construct a closed formula for the coefficients of the Kazhdan-Lusztig polynomial for a matroid. We also study some generalized characteristic polynomials of posets which are not evaluations of Tutte polynomials.
Diffusion limit for the partner model at the critical value

The partner model is an SIS epidemic in a population with random formation and dissolution of partnerships, and with disease transmission only occuring within partnerships. Foxall, Edwards, and van den Driessche found the critical value and studied the subcritical and supercritical regimes. Recently Foxall has shown that (if there are enough initial infecteds $I_0$) the extinction time in the critical model is of order $\sqrt{N}$. Here we improve that result by proving the convergence of $i_N(t)=I(\sqrt{N}t)/\sqrt{N}$ to a limiting diffusion. We do this by showing that within a short time, this four dimensional process collapses to two dimensions: the number of $SI$ and $II$ partnerships are constant multiples of the the number of infected singles. The other variable, the total number of singles, fluctuates around its equilibrium like an Ornstein-Uhlenbeck process of magnitude $\sqrt{N}$ on the original time scale and averages out of the limit theorem for $i_N(t)$. As a by-product of our proof we show that if $\tau_N$ is the extinction time of $i_N(t)$ (on the $\sqrt{N}$ time scale) then $\tau_N$ has a limit.
A classification of the irreducible admissible genuine mod p representations of the metaplectic cover of p-adic SL(2)

We classify the irreducible, admissible, smooth, genuine mod p representations of the metaplectic double cover of SL(2,F), where F is a p-adic field and p is odd. We show, using a generalized Satake transform, that each such representation is isomorphic to a certain explicit quotient of a compact induction from a maximal compact subgroup by an action of a spherical Hecke operator, and we define a parameter for the representation in terms of this data. We show that our parameters distinguish genuine nonsupercuspidal representations from genuine supercuspidals, and that every irreducible genuine nonsupercuspidal representation is in fact an irreducible principal series representation. In particular, the metaplectic double cover of p-adic SL(2) has no genuine special mod p representations.
Circulating Inflammatory, Mitochondrial Dysfunction, and Senescence-Related Markers in Older Adults with Physical Frailty and Sarcopenia: A BIOSPHERE Exploratory Study

Multisystem derangements encompassing musculoskeletal, stress, and metabolic response have been described in older adults with physical frailty and sarcopenia (PF&S). Whether PF&S is also associated with markers of cellular senescence has yet to be explored. To address this research question, we quantified the serum levels of selected inflammatory, mitochondrial, and senescence-associated secretory phenotype (SASP)-related factors in 22 older adults with PF&S (mean age 75.5 ± 4.7 years; 81.8% women) and 27 nonPF&S controls (mean age 75.0 ± 4.4 years; 62.9% women) and evaluated their association with PF&S. Markers of inflammation (interleukin (IL)1-β, IL6, and tumor necrosis factor α (TNF-α)), matrix remodeling (Serpin E1, intercellular adhesion molecule 1 (ICAM-1), and tissue inhibitor of metalloproteinases 1 (TIMP-1)), mitochondrial dysfunction (growth/differentiation factor 15 (GDF15) and fibroblast growth factor 21 (FGF21)), Activin A, and glial fibrillary acidic protein (GFAP) were assayed. Serum levels of TNF-α and those of the SASP-related factors ICAM-1 and TIMP-1 were found to be higher, while IL1-β and IL6 were lower in PF&S participants compared with controls. Partial least squares discriminant analysis allowed discrimination of PF&S from nonPF&S participants with 74.0 ± 3.4% accuracy. Markers that significantly contributed to the classification were ICAM-1, TIMP-1, TNF-α, GFAP, and IL6. Future studies are warranted to establish whether inflammatory and SASP-related pathways are causally linked to the development and progression of PF&S, and may represent new targets for interventions.
Splittings of free groups from arcs and curves

We show that the arc graph of $S_g^1$ is a coarse Lipschitz retract of the free splitting complex of $F_{2g}$. We also show that the arc and curve graph of $S_g^1$ is a coarse Lipschitz retract of both the cyclic splitting graph of $F_{2g}$ and the maximally cyclic splitting graph of $F_{2g}$.
A curve of positive solutions for an indefinite sublinear Dirichlet problem

We investigate the existence of a curve $q\mapsto u_{q}$, with $q\in(0,1)$, of positive solutions for the problem $(P_{a,q})$: $-\Delta u=a(x)u^{q}$ in $\Omega$, $u=0$ on $\partial\Omega$, where $\Omega$ is a bounded and smooth domain of $\mathbb{R}^{N}$ and $a:\Omega\rightarrow\mathbb{R}$ is a sign-changing function (in which case the strong maximum principle does not hold). In addition, we analyze the asymptotic behavior of $u_{q}$ as $q\rightarrow0^{+}$ and $q\rightarrow1^{-}$. We also show that in some cases $u_{q}$ is the ground state solution of $(P_{a,q})$. As a byproduct, we obtain existence results for a singular and indefinite Dirichlet problem. Our results are mainly based on bifurcation and sub-supersolutions methods.
Exact large-scale correlations in integrable systems out of equilibrium

Using the theory of generalized hydrodynamics (GHD), we derive exact Euler-scale dynamical two-point correlation functions of conserved densities and currents in inhomogeneous, non-stationary states of many-body integrable systems with weak space-time variations. This extends previous works to inhomogeneous and non-stationary situations. Using GHD projection operators, we further derive formulae for Euler-scale two-point functions of arbitrary local fields, purely from the data of their homogeneous one-point functions. These are new also in homogeneous generalized Gibbs ensembles. The technique is based on combining a fluctuation-dissipation principle along with the exact solution by characteristics of GHD, and gives a recursive procedure able to generate nn-point correlation functions. Owing to the universality of GHD, the results are expected to apply to quantum and classical integrable field theory such as the sinh-Gordon model and the Lieb-Liniger model, spin chains such as the XXZ and Hubbard models, and solvable classical gases such as the hard rod gas and soliton gases. In particular, we find Leclair-Mussardo-type infinite form-factor series in integrable quantum field theory, and exact Euler-scale two-point functions of exponential fields in the sinh-Gordon model and of powers of the density field in the Lieb-Liniger model. We also analyse correlations in the partitioning protocol, extract large-time asymptotics, and, in free models, derive all Euler-scale nn-point functions.
Disability, Social Class and Stigma: An Intersectional Analysis of Disabled Young People’s School Experiences

Recent decades have witnessed a renewed interest in stigma and its effects on life-course trajectories of disabled people. However, sociological narratives largely adopt monolithic understandings of disability, neglecting contextual meanings of different impairments and conditions and their intersections with other ascriptive inequalities, which may be consequential for exposure to stigma. Our article provides an intersectional analysis of disabled young people’s lived experiences of stigma in mainstream school settings. Drawing on semi-structured interviews with 35 autistic, dyslexic and/or physically disabled students, we show that stigmatisation is contingent on social class background, which affects students’ location within the school. We also find substantial variation in experiences of stigma between and within sub-categories of conditions/impairments, as a consequence of the perceived distance from normative ideals of skills and behaviour attached to individuals in school settings. Our findings highlight the importance of intersectional analyses of stigma, challenging universalised views about stigmatised disabled people.
Command Filter-Based Adaptive Practical Prescribed-Time Asymptotic Tracking Control of Autonomous Underwater Vehicles With Limited Communication Angles

This work proposes a command filter-based adaptive practical prescribed-time (PPT) asymptotic tracking control scheme for autonomous underwater vehicles (AUVs) under dynamic uncertainties. Considering that the followers have limited communication angles in the leader–follower formation structure, where a novel controller is proposed to ensure communication angles to achieve preassignable precision within a predefined time. First, we present a PPT control for handling the limited communication angles and, thus, they are allowed to be limited, turnable, and controllable. Then, to achieve the asymptotic convergence of output tracking errors, the proposed adaptive controllers can effectively insure the trajectory tracking errors of AUVs asymptotically converge to zero under the influence of dynamic uncertainties. Furthermore, the derived asymptotically tracking command filter technique for AUVs not only deduce an acceleration-free strategy for followers but also achieve the asymptotic convergence of output tracking errors for the command filters themselves. In the end, the effectiveness of the proposed control strategy is demonstrated through stability analysis and simulation test.
Nonexistence of Optimal Transport Maps for the Multimarginal Repulsive Harmonic Cost

We give an example of an absolutely continuous measure $\mu$ on $\mathbb R^d$, for any $d \ge 1$, such that no minimizer of the $3$-marginal harmonic repulsive cost with all marginals equal to $\mu$ is supported on a graph over the first variable.
A symmetric 2-tensor canonically associated to Q-curvature and its applications

In this article, we define a symmetric 2-tensor canonically associated to Q-curvature called J-tensor on any Riemannian manifold with dimension at least three. The relation between J-tensor and Q-curvature is precisely like Ricci tensor and scalar curvature. Thus it can be interpreted as a higher-order analogue of Ricci tensor. This tensor can also be used to understand Chang-Gursky-Yang's theorem on 4-dimensional Q-singular metrics. Moreover, we show an Almost-Schur Lemma holds for Q-curvature, which gives an estimate of Q-curvature on closed manifolds.
Stabilization of Kelvin–Voigt viscoelastic fluid flow model

ABSTRACT In this article, stabilization result for the viscoelastic fluid flow problem is governed by Kelvin–Voigt model, that is, convergence of the unsteady solution to a steady state solution is proved under the assumption that linearized self-adjoint steady state eigenvalue problem has a minimal positive eigenvalue. Both power and exponential convergence results are derived under various conditions on the forcing function. It is shown that results are valid uniformly in the time relaxation or some times called regularization parameter as , which in turn, establishes results for the Navier–Stokes system.
Structural Compactness and Stability of Pseudo-Monotone Flows

Fitzpatrick's variational representation of maximal monotone operators is here extended to a class of pseudo-monotone operators in Banach spaces. On this basis, the initial-value problem associated with the first-order flow of such an operator is here reformulated as a minimization principle, extending a method that was pioneered by Brezis, Ekeland and Nayroles for gradient flows. This formulation is used to prove that the problem is stable w.r. t. \ arbitrary perturbations not only of data but also of operators. This is achieved by using the notion of evolutionary $\Gamma$-convergence w.r. t. \ a nonlinear topology of weak type. These results are applied to the Cauchy problem for quasilinear parabolic PDEs. This provides the structural compactness and stability of the model of several physical phenomena: nonlinear diffusion, incompressible viscous flow, phase transitions, and so on.
Yaglom limit for stable processes in cones

We give the asymptotics of the tail of the distribution of the first exit time of the isotropic $\alpha$-stable L\'evy process from the Lipschitz cone in $\mathbb{R}^d$. We obtain the Yaglom limit for the killed stable process for the cone. We construct and estimate entrance laws for the process from the vertex into the cone. For the symmetric Cauchy process and the positive half-line we give a spectral representation of the Yaglom limit. Our approach relies on the scalings of the stable process and the cone, which allow to express the temporal asymptotics of the distribution of the process at infinity by means of the spatial asymptotics of harmonic functions of the process at the vertex; on the representation of the probability of survival of the process in the cone as a Green potential; and on the approximate factorization of the heat kernel of the cone, which secures compactness and yields a limiting (Yaglom) measure by means of Prokhorov's theorem.
A counterexample to the DeMarco‐Kahn upper tail conjecture

Given a fixed graph H, what is the (exponentially small) probability that the number XH of copies of H in the binomial random graph Gn,p is at least twice its mean? Studied intensively since the mid 1990s, this so‐called infamous upper tail problem remains a challenging testbed for concentration inequalities. In 2011 DeMarco and Kahn formulated an intriguing conjecture about the exponential rate of decay of P(XH⩾(1+ε)EXH) for fixed ε > 0. We show that this upper tail conjecture is false, by exhibiting an infinite family of graphs violating the conjectured bound.
Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves

<p>We prove the first known nontrivial bounds on the sizes of the 2-torsion subgroups of the class groups of cubic and higher degree number fields <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula> (the trivial bound being <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O Subscript epsilon comma n Baseline left-parenthesis StartAbsoluteValue normal upper D normal i normal s normal c left-parenthesis upper K right-parenthesis EndAbsoluteValue Superscript 1 slash 2 plus epsilon Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>O</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>ϵ<! -- ϵ --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">D</mml:mi> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">c</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>K</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo>+</mml:mo> <mml:mi>ϵ<! -- ϵ --></mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O_{\epsilon ,n}(|\mathrm {Disc}(K)|^{1/2+\epsilon })</mml:annotation> </mml:semantics> </mml:math> </inline-formula> coming from the bound on the entire class group). This yields corresponding improvements to: (1) bounds of Brumer and Kramer on the sizes of 2-Selmer groups and ranks of elliptic curves, (2) bounds of Helfgott and Venkatesh on the number of integral points on elliptic curves, (3) bounds on the sizes of 2-Selmer groups and ranks of Jacobians of hyperelliptic curves, and (4) bounds of Baily and Wong on the number of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3. org/1998/Math/MathML" alttext="upper A 4"> <mml:semantics> <mml:msub> <mml:mi>A</mml:mi> <mml:mn>4</mml:mn> </mml:msub> <mml:annotation encoding="application/x-tex">A_4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-quartic fields of bounded discriminant. </p>
On Lusztig's map for spherical unipotent conjugacy classes

We provide an alternative description of the restriction to spherical unipotent conjugacy classes, of Lusztig's map Ψ from the set of unipotent conjugacy classes in a connected reductive algebraic group to the set of conjugacy classes of its Weyl group. For irreducible root systems, we analyze the image of this restricted map and we prove that a conjugacy class in a finite Weyl group has a unique maximal length element if and only if it has a maximum.
Scalar curvature and singular metrics

Let $M^n$, $n\ge3$, be a compact differentiable manifold with nonpositive Yamabe invariant $\sigma(M)$. Suppose $g_0$ is a continuous metric with $V(M, g_0)=1$, smooth outside a compact set $\Sigma$, and is in $W^{1,p}_{loc}$ for some $p>n$. Suppose the scalar curvature of $g_0$ is at least $\sigma(M)$ outside $\Sigma$. We prove that $g_0$ is Einstein outside $\Sigma$ if the codimension of $\Sigma$ is at least $2$. If in addition, $g_0$ is Lipschitz then $g_0$ is smooth and Einstein after a change the smooth structure. If $\Sigma$ is a compact embedded hypersurface, and $g_0$ is smooth up to $\Sigma$ from two sides of $\Sigma$, and if the difference of the mean curvatures along $\Sigma$ at two sides of $\Sigma$ has a fixed appropriate sign. Then $g_0$ is also Einstein outside $\Sigma$. For manifolds with dimension between $3$ and $7$ without spin assumption, we obtain a positive mass theorem on an asymptotically flat manifold for metrics with a compact singular set of codimension at least $2$.