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This paper studies the Bondi-Metzner-Sachs group in homogeneous projective coordinates, because it is then possible to write all transformations of such a group in a manifestly linear way. The 2-sphere metric, Bondi-Metzner-Sachs metric, asymptotic Killing vectors, generators of supertranslations, as well as boosts and rotations of Minkowski spacetime, are all re-expressed in homogeneous projective coordinates. Last, the integral curves of vector fields which generate supertranslations are evaluated in detail. This work prepares the ground for more advanced applications of the differential geometry of asymptotically flat spacetimes in projective coordinates., Comment: 26 pages, Latex, 9 figures. Misprints corrected and presentation improved

We present a numerical method for the velocity-space, spatially homogeneous, collisional Boltzmann equation for electron transport in low-temperature plasma (LTP) conditions. Modeling LTP plasmas is useful in many applications, including advanced manufacturing, material processing, semiconductor processing, and hypersonics, to name a few. Most state-of-the-art methods for electron kinetics are based on Monte-Carlo sampling for collisions combined with Lagrangian particle-in-cell methods. We discuss an Eulerian solver that approximates the electron velocity distribution function using spherical harmonics (angular components) and B-splines (energy component). Our solver supports electron-heavy elastic and inelastic binary collisions, electron-electron Coulomb interactions, steady-state and transient dynamics, and an arbitrary nmber of angular terms in the electron distribution function. We report convergence results and compare our solver to two other codes: an in-house particle Monte-Carlo ethod; and Bolsig+, a state-of-the-art Eulerian solver for electron transport in LTPs. Furthermore, we use our solver to study the relaxation time scales of the higher-order anisotropic correction terms. Our code is open-source and provides an interface that allows coupling to multiphysics simulations of low-temperature plasmas.

We show that closures of homogeneous braids are visually prime, addressing a question of Cromwell. The key technical tool for the proof is the following criterion concerning primeness of open books, which we consider to be of independent interest. For open books of 3-manifolds the property of having no fixed essential arcs is preserved under essential Murasugi sums with a strictly right-veering open book, if the plumbing region of the original open book veers to the left. We also provide examples of open books in S^3 demonstrating that primeness is not necessarily preserved under essential Murasugi sum, in fact not even under stabilizations a.k.a. Hopf plumbings. Furthermore, we find that trefoil plumbings need not preserve primeness. In contrast, we establish that figure-eight knot plumbings do preserve primeness., Comment: 25 pages, 14 figures. Comments welcome!

In this paper, we introduce the notion of a multiplicative unimodularity for a coisotropic Poisson homogeneous space. Then, we discuss the unimodularity and the multiplicative unimodularity for these spaces and the existence of an invariant volume form for explicit Hamiltonian systems on such spaces. Several interesting examples illustrating the theoretical results are also presented., Comment: 36 pages

We construct a unique global solution to the Cauchy problem of the 3D Boltzmann equation for initial data around the Maxwellian in the spatially critical homogeneous Besov space $\widetilde{L}^2_{\xi}(\dot{B}_{2,1}^{1/2}\cap\dot{B}_{2,1}^{3/2})$. In addition, under the condition that the low-frequency part of initial perturbation is bounded in $\widetilde{L}^2_{\xi}(\dot{B}_{2,\infty}^{\sigma_{0}})$ with $-3/2\leq\sigma_{0}<1/2$, it is shown that the solution converges to its equilibrium in large times with the optimal rate of $\mathcal{O}(t^{-(\sigma-\sigma_{0})/2})$ in $\widetilde{L}^2_{\xi}(\dot{B}_{2,1}^{\sigma})$ with some $\sigma>\sigma_0$, and the microscopic part decays at an enhanced rate of $\mathcal{O}(t^{-(\sigma-\sigma_{0})/2-1/2})$. In contrast to [19], the usual $L^2$ estimates are not necessary in our approach, which provides a new understanding of hypocoercivity theory for the Boltzmann equation allowing to construct the Lyapunov functional with different dissipation rates at low and high frequencies. Furthermore, a time-weighted Lyapunov energy argument can be developed to deduce the optimal time-decay estimates., Comment: 35 pages

In this article, we study the homogeneous Dirichlet problem for a degenerate parabolic-hyperbolic PDE perturbed by Levy noise. In particular, we develop the well-posedness theory of entropy solution based on the Kru\v{z}kov's semi-entropy formulation. In comparison to the pioneered work by Bauzet et al. (J. Funct. Anal. 266, (2014), 2503-2545), concerning the existence and uniqueness of entropy solution for the Dirichlet problem for conservation laws driven by Brownian noise, our present analysis involves a simpler approach to obtain the global Kato's inequality.

This paper is devoted to the study of the $m$-point homogeneity property for the vertex sets of polytopes in Euclidean spaces. In particular, we present the classifications of $2$-point and $3$-point homogeneous polyhedra in $\mathbb{R}^3$., Comment: 34 pages, 22 figures, some arguments simplified, typos corrected, comments are welcome!

Starting from a fairly explicit homogeneous realization of the toroidal Lie algebra $\mathcal{L}^{\rm tor}_{r+1}(\mathfrak{sl}_\ell)$ via lattice vertex algebra, we derive an integrable hierarchy of Hirota bilinear equations. Moreover, we represent this hierarchy in the form of Lax equations, and show that it is an extension of a certain reduction of the $\ell$-component KP hierarchy.

Based on the rapid development of dyadic analysis and the theory of variable weighted function spaces over the spaces of homogeneous type $(X,d,\mu)$ in recent years, we systematically consider the quantitative variable weighted characterizations for fractional maximal operators. On the one hand, a new class of variable multiple weight $A_{\vec{p}(\cdot),q(\cdot)}(X)$ is established, which enables us to prove the strong and weak type variable multiple weighted estimates for multilinear fractional maximal operators ${{{\mathscr M}_{\eta }}}$. More precisely, \[ {\left[ {\vec \omega } \right]_{{A_{\vec p( \cdot ),q( \cdot )}}(X)}} \lesssim {\left\| \mathscr{M}_\eta \right\|_{\prod\limits_{i = 1}^m {{L^{p_i( \cdot )}}({X,\omega _i})} \to {L^{q( \cdot )}}(X,\omega )({WL^{q( \cdot )}}(X,\omega ))}} \le {C_{\vec \omega ,\eta ,m,\mu ,X,\vec p( \cdot )}}. \] On the other hand, on account of the classical Sawyer's condition $S_{p,q}(\mathbb{R}^n)$, a new variable testing condition $C_{{p}(\cdot),q(\cdot)}(X)$ also appears in here, which allows us to obtain quantitative two-weighted estimates for fractional maximal operators ${{{M}_{\eta }}}$. To be exact, \begin{align*} \|M_{\eta}\|_{L^{p(\cdot)}(X,\omega)\rightarrow L^{q(\cdot)}(X,v)} \lesssim \sum\limits_{\theta = \frac{1}{{{p_{\rm{ - }}}}},\frac{1}{{{p_{\rm{ + }}}}}} {{{\left( {{{[\omega ,v]}_{C_{p( \cdot ),q( \cdot )}^2(X)}} + {{[\omega ]}_{C_{p( \cdot ),q( \cdot )}^1(X)}}{{[\omega ,v]}_{C_{p( \cdot ),q( \cdot )}^2(X)}}} \right)}^\theta }}. \end{align*} The implicit constants mentioned above are independent on the weights., Comment: Some fallacies about the second main result have been corrected. arXiv admin note: substantial text overlap with arXiv:2404.15550

Let M\"ob be the biholomorphic automorphism group of the unit disc of the complex plane, $\mathcal{H}$ be a complex separable Hilbert space and $\mathcal{U}(\mathcal{H})$ be the group of all unitary operators. Suppose $\mathcal{H}$ is a reproducing kernel Hilbert space consisting of holomorphic functions over the poly-disc $\mathbb D^n$ and contains all the polynomials. If $\pi : \mbox{M\"ob} \to \mathcal{U}(\mathcal{H})$ is a multiplier representation, then we prove that there exist $\lambda_1, \lambda_2, \ldots, \lambda_n > 0$ such that $\pi$ is unitarily equivalent to $(\otimes_{i=1}^{n} D_{\lambda_i}^+)|_{\mbox{M\"ob}}$, where each $D_{\lambda_i}^+$ is a holomorphic discrete series representation of M\"ob. As an application, we prove that if $(T_1, T_2)$ is a M\"ob - homogeneous pair in the Cowen - Douglas class of rank $1$ over the bi-disc, then each $T_i$ posses an upper triangular form with respect to a decomposition of the Hilbert space. In this upper triangular form of each $T_i$, the diagonal operators are identified. We also prove that if $\mathcal{H}$ consists of symmetric (resp. anti-symmetric) holomorphic functions over $\mathbb D^2$ and contains all the symmetric (resp. anti-symmetric) polynomials, then there exists $\lambda > 0$ such that $\pi \cong \oplus_{m = 0}^\infty D^+_{\lambda + 4m}$ (resp. $\pi \cong \oplus_{m=0}^\infty D^+_{\lambda + 4m + 2}$).

Let U be a homogeneous variety over rational numbers of some linear algebraic group. Fix an integral model and assume the existence of infinitely many integral points. Then one would like to give an asymptotic count with the help of some height function. In many cases, with the help of measure rigidity of unipotent flows, we reduce this problem to one on equivariant birational geometry. For instance, we show that if G and H are both connected, semisimple, simply connected and without compact factors, then U=G/H is strongly Hardy-Littlewood with respect to certain height function. We also show that when H is "large" in H and both G and H are connected, reductive and without non-trivial Q-characters, then for every equivariant height, the asymptotic of integral points is the same as the volume asymptotic up to a constant. Three concrete examples with explicit heights are also provided to illustrate our approach., Comment: 81 pages

This work is a thorough and detailed study on the geometry of the unit sphere of certain Banach spaces of homogeneous polynomials in ${\mathbb{R}}^2$. Specifically, we provide a complete description of the unit spheres, identify the extreme points of the unit balls, derive explicit formulas for the corresponding polynomial norms, and describe the techniques required to tackle these questions. To enhance the comprehensiveness of this work, we complement the results and their proofs with suitable diagrams and figures. The new results presented here settle some open questions posed in the past. For the sake of completeness of this work, we briefly discuss previous known results and provide directions of research and applications of our results., Comment: 46 pages, 18 figures

We give explicit parametrizations for all the homogeneous contact Riemannian structures on $3$-dimensional Sasakian space forms.

We derive the transport equations from the Vlasov-Fokker-Planck equation when the velocity space is spherically symmetric. The Shkarofsky's form of Fokker-Planck-Rosenbluth collision operator is employed in the Vlasov-Fokker-Planck equation. A closed-form relaxation model for homogeneous plasmas could be presented in terms of Gauss hypergeometric2F1 functions. This has been accomplished based on the Maxwellian mixture model. Furthermore, we demonstrate that classic models such as two-temperature thermal equilibrium model and thermodynamic equilibrium model are special cases of our relaxation model and the zeroth-order Braginskii heat transfer model can also be derived. The present relaxation model is a nonequilibrium model based on the hypothesis that the plasmas system possesses finitely distinguishable independent features, without relying on the conventional near-equilibrium assumption., Comment: 15 pages, 1 figure

The purpose of this paper is to characterize the homogeneous Besov space in the Dunkl setting. We utilize a new discrete reproducing formula, that is, the building blocks are differences of the Dunkl-Poisson kernel which involves both the Euclidean metric and the Dunkl metric. To introduce the Besov spaces in the Dunkl setting, new test functions and distributions are introduced, and a new decomposition is established.

The equivalence of spectral convergence and Benjamini-Schramm convergence is extended from homogeneous spaces to spaces which are compact modulo isometry group. The equivalence is proven under the condition of a uniform discreteness property. It is open, which implications hold without this condition.

In this paper, we give the Assouad dimension formula and the upper bound of the lower dimension for homogeneous Moran sets under the condition $\sup_{k\ge 1}\{n_{k}\}<+\infty$. We also give the Assouad spectrum and the lower spectrum formulas for Cantor-like sets.

We consider the dynamics of a population spatially structured in colonies that are vulnerable to catastrophic events occurring at random times, which randomly reduce their population size and compel survivors to disperse to neighboring areas. The dispersion behavior of survivors is critically significant for the survival of the entire species. In this paper, we consider an uniform dispersion scheme, where all possible survivor groupings are equally probable. The aim of the survivors is to establish new colonies, with individuals who settle in empty sites potentially initiating a new colony by themselves. However, all other individuals succumb to the catastrophe. We consider the number of dispersal options for surviving individuals in the aftermath of a catastrophe to be a fixed value $d$ within the neighborhood. In this context, we conceptualize the evolution of population dynamics occurring over a homogeneous tree. We investigate the conditions necessary for these populations to survive, presenting pertinent bounds for survival probability, the number of colonized vertices, the extent of dispersion within the population, and the mean time to extinction for the entire population., Comment: 22 pages

We propose a constrained generalized method of moments estimator (GMM) incorporating theoretical conditions for the unique existence of equilibrium prices for estimating conduct parameters in a log-linear model with homogeneous goods markets. First, we derive such conditions. Second, Monte Carlo simulations confirm that in a log-linear model, incorporating the conditions resolves the problems of implausibly low or negative values of conduct parameters., Comment: 7 pages, 12 page appendix

We study weighted Sobolev inequalities on open convex cones endowed with $\alpha$-homogeneous weights satisfying a certain concavity condition. We establish a so-called reduction principle for these inequalities and characterize optimal rearrangement-invariant function spaces for these weighted Sobolev inequalities. Both optimal target and optimal domain spaces are characterized. Abstract results are accompanied by general yet concrete examples of optimal function spaces. For these examples, the class of so-called Lorentz--Karamata spaces, which contains in particular Lebesgue spaces, Lorentz spaces, and some Orlicz spaces, is used.

In this paper we consider a rotating black hole with electric and magnetic monopole charges that moves in a static homogeneous electromagnetic field. We assume that both the charges and the fields are weak, so that they have no effect on the spacetime geometry, which is described by the Kerr metric. We present exact solutions to Maxwell's equations describing the field of a charged rotating black hole moving in an external field background. We use these solutions to calculate the energy, momentum, and angular momentum fluxes of the electromagnetic field into the black hole. Using these results, we obtain expressions for the torque and force acting on the moving charged rotating black hole arising as a result of its interaction with the external electromagnetic field. We calculate these quantities both in the frame comoving with the black hole and in the frame of the external background field. We use this result to derive the equations that govern the change in the black hole's mass and spin, as well as the motion of the black hole. We provide exact solutions for specific cases which illustrate the role of charge and spin on the motion of the black hole., Comment: 12 pages

The goal of this paper is to establish a new and efficient characterization of quasi-homogeneous isolated singularities of free curves and nearly free curves $C$ in $\mathbb{P}^2$. The criterion will be in terms of the first syzygy matrix associated with the Jacobian ideal of $f$ where $f=0$ is the equation of the plane curve $C$., Comment: 20 pages, 4 figures

The application of data-intensive automatic speech recognition (ASR) technologies to dysarthric and elderly adult speech is confronted by their mismatch against healthy and nonaged voices, data scarcity and large speaker-level variability. To this end, this paper proposes two novel data-efficient methods to learn homogeneous dysarthric and elderly speaker-level features for rapid, on-the-fly test-time adaptation of DNN/TDNN and Conformer ASR models. These include: 1) speaker-level variance-regularized spectral basis embedding (VR-SBE) features that exploit a special regularization term to enforce homogeneity of speaker features in adaptation; and 2) feature-based learning hidden unit contributions (f-LHUC) transforms that are conditioned on VR-SBE features. Experiments are conducted on four tasks across two languages: the English UASpeech and TORGO dysarthric speech datasets, the English DementiaBank Pitt and Cantonese JCCOCC MoCA elderly speech corpora. The proposed on-the-fly speaker adaptation techniques consistently outperform baseline iVector and xVector adaptation by statistically significant word or character error rate reductions up to 5.32% absolute (18.57% relative) and batch-mode LHUC speaker adaptation by 2.24% absolute (9.20% relative), while operating with real-time factors speeding up to 33.6 times against xVectors during adaptation. The efficacy of the proposed adaptation techniques is demonstrated in a comparison against current ASR technologies including SSL pre-trained systems on UASpeech, where our best system produces a state-of-the-art WER of 23.33%. Analyses show VR-SBE features and f-LHUC transforms are insensitive to speaker-level data quantity in testtime adaptation. T-SNE visualization reveals they have stronger speaker-level homogeneity than baseline iVectors, xVectors and batch-mode LHUC transforms., Comment: In submission to IEEE/ACM Transactions on Audio, Speech, and Language Processing