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Let X_o be a weighted-homogeneous complete intersection germ in (R^N,o) or (C^N,o), with arbitrary singularities, possibly non-reduced. Take the foliation of the ambient space by weighted-homogeneous real arcs, \ga_s. Take a deformation of X_o by higher order terms, X_t. Does the foliation \ga_s deform compatibly with X_t? We identify the ``obstruction locus", \Sigma in X_o, outside of which such a deformation does exist, and possesses exceptionally nice properties. Using this deformed foliation we construct a contact trivialization of the family of defining equations by a homeomorphism that is real analytic (resp. Nash) off the origin, differentiable at the origin, whose presentation in weighted-polar coordinates is globally real-analytic (resp. globally Nash), and with controlled Lipschitz/C^1-properties., Comment: arXiv admin note: text overlap with arXiv:2311.13423

Nonholonomic systems are, so to speak, mechanical systems with a prescribed restriction on the velocities. A virtual nonholonomic constraint is a controlled invariant distribution associated with an affine connection mechanical control system. A Riemannian homogeneous space is, a Riemannian manifold that looks the same everywhere, as you move through it by the action of a Lie group. These Riemannian manifolds are not necessarily Lie groups themselves, but nonetheless possess certain symmetries and invariances that allow for similar results to be obtained. In this work, we introduce the notion of virtual constraint on Riemannian homogeneous spaces in a geometric framework which is a generalization of the classical controlled invariant distribution setting and we show the existence and uniqueness of a control law preserving the invariant distribution. Moreover we characterize the closed-loop dynamics obtained using the unique control law in terms of an affine connection. We illustrate the theory with new examples of nonholonomic control systems inspired by robotics applications.

We propose a general framework for deriving generalization bounds for parallel positively homogeneous neural networks--a class of neural networks whose input-output map decomposes as the sum of positively homogeneous maps. Examples of such networks include matrix factorization and sensing, single-layer multi-head attention mechanisms, tensor factorization, deep linear and ReLU networks, and more. Our general framework is based on linking the non-convex empirical risk minimization (ERM) problem to a closely related convex optimization problem over prediction functions, which provides a global, achievable lower-bound to the ERM problem. We exploit this convex lower-bound to perform generalization analysis in the convex space while controlling the discrepancy between the convex model and its non-convex counterpart. We apply our general framework to a wide variety of models ranging from low-rank matrix sensing, to structured matrix sensing, two-layer linear networks, two-layer ReLU networks, and single-layer multi-head attention mechanisms, achieving generalization bounds with a sample complexity that scales almost linearly with the network width., Comment: Under review by AISTATS 2025

We develop the concept of a homogeneity supermanifold, which is more general than most of the approaches to graded manifolds present in the literature. Instead of doing it via ringed spaces and sheaves of graded algebras, we prefer a purely differential geometric language and view gradings as provided by an additional structure on supermanifolds, namely the weight vector field. This gives us a privileged atlas with homogeneous local coordinates. The fact that the transition maps `preserve weights' is then automatic, but weights of homogeneous coordinates may vary from one chart to the other. Moreover, such local homogeneous coordinates may have arbitrary real weights, not only integers. We study homogeneity submanifolds, homogeneity Lie supergroups, tangent and cotangent lifts of homogeneity structures, homogeneous distributions and codistributions, and other related concepts. The main achievements in this framework are proofs of the homogeneous Poincar\'e Lemma and a homogeneous analog of the Darboux Theorem., Comment: 39 pages

The Fisher Information Metric (FIM) and the associated Cram\'er-Rao Bound (CRB) are fundamental tools in statistical signal processing, which inform the efficient design of experiments and algorithms for estimating the underlying parameters. In this article, we investigate these concepts for the case where the parameters lie on a homogeneous space. Unlike the existing Fisher-Rao theory for general Riemannian manifolds, our focus is to leverage the group-theoretic structure of homogeneous spaces, which is often much easier to work with than their Riemannian structure. The FIM is characterized by identifying the homogeneous space with a coset space, the group-theoretic CRB and its corollaries are presented, and its relationship to the Riemannian CRB is clarified. The application of our theory is illustrated using two examples from engineering: (i) estimation of the pose of a robot and (ii) sensor network localization. In particular, these examples demonstrate that homogeneous spaces provide a natural framework for studying statistical models that are invariant with respect to a group of symmetries., Comment: Supplementary document is available

In this paper the final stage of the Petrov classification is carried out. As it is known, the Killing vector fields specify infinitesimal transformations of the group of motions of space $V_4$. In the case when in the homogeneous space $V_4$ the group of motions $G_3$ acts simply transitive, the geometry of the non-isotropic hypersurface is determined by the geometry of the transitivity space $V_3$ of the group $G_3$. In this case, the metric tensor of the space $V_3$ can be given by a nonholonomic reper consisting of three independent vectors $\ell_{(a)}^\alpha$, which define the generators of the group $G_3$ of finite transformations in the space $V_3$. The representation of the metric tensor of $V_4$ spaces by means of vector fields $\ell_{(a)}^\alpha$ has a great physical meaning and allows to simplify substantially the equations of mathematical physics in such spaces. Therefore, the Petrov classification should be complemented by the classification of vector fields $\ell_{(a)}^\alpha$ connected to Killing vector fields. For homogeneous spaces this problem has been largely solved. A complete solution of this problem is presented in the present paper, where the Petrov classification for homogeneous spaces in which the group $G_3$, which belongs to type $VIII$ according to the Petrov classification, acts simply transitively, is refined. In addition, the complete classification of vector fields $\ell_{(a)}^\alpha$ for spaces $V_4$ in which the group $G_3$ acts simply transitivity on isotropic hypersurfaces.}, Comment: 34 pages

The rheology of dense suspensions lacks a universal description due to the involvement of a wide variety of parameters, ranging from the physical properties of solid particles to the nature of the external deformation or applied stress. While the former controls microscopic interactions, spatial variations in the latter induce heterogeneity in the flow, making it difficult to find suitable constitutive laws to describe the rheology in a unified way. For homogeneous driving with a spatially uniform strain rate, the rheology of non-Brownian dense suspensions is well described by the conventional $\mu(J)$ rheology. However, this rheology fails in the inhomogeneous case due to non-local effects, where the flow in one region is influenced by the flow in another. Here, motivated by observations from simulation data, we introduce a new dimensionless number, the suspension temperature $\Theta_s$, which contains information on local particle velocity fluctuations. We find that $\mu(J,\Theta_s)$ provides a unified description for both homogeneous and inhomogeneous flows. By employing scaling theory, we identify a set of constitutive laws for dense suspensions of frictional spherical particles and frictionless rod-shaped particles. Combining these scaling relations with the momentum balance equation for our model system, we predict the spatial variation of the relevant dimensionless numbers, the volume fraction $\phi$, the viscous number $J$, the macroscopic friction coefficient $\mu$, and $\Theta_s$ solely from the nature of the imposed external driving.

For a Banach--Lie group $G$ and an embedded Lie subgroup $K$ we consider the homogeneous Banach manifold $\mathcal M=G/K$. In this context we establish the most general conditions for a bounded operator $N$ acting on $Lie(G)$ to define a homogeneous vector bundle map $\mathcal N:T\mathcal M\to T\mathcal M$. In particular our considerations extend all previous settings on the matter and are well-suited for the case where $Lie(K)$ is not complemented in $Lie(G)$. We show that the vanishing of the Nijenhuis torsion for a homogeneous vector bundle map $\mathcal N:T\mathcal M\to T\mathcal M$ (defined by an admissible bounded operator $N$ on $Lie(G)$) is equivalent to the Nijenhuis torsion of $N$ having values in $Lie(K)$. As an application, we consider the question of integrability of an almost complex structure $\mathcal J$ on $\mathcal M$ induced by an admissible bounded operator $J$, and we give a simple characterization of integrability in terms of certain subspaces of the complexification of $Lie(G)$ (which are not eigenspaces of the complex extension of $J$)., Comment: 27 pages

Deciding on appropriate materials and designs for use in light sails, like the one proposed in the Breakthrough Starshot Initiative, is a topic that requires much care and forethought. Here, we offer a feasible option in the form of metasurfaces made of periodically arranged homogeneous and core-shell spheres. Using the re-normalized T-matrix from Mie theory, we explore the reflectance, absorptance, and acceleration time of such metasurfaces. We focus on spheres made from aluminum, silicon, silicon dioxide, and combinations thereof. Since the light sails are foreseen to be accelerated using Earth-based laser arrays to 20% of the speed of light, one needs to account for relativistic effects. As a result, a high broadband reflectance is essential for effective propulsion. We identify metasurfaces that offer such properties combined with a low absorptance to reduce heating and deformation. We highlight a promising extension to the case of a metasurface made from homogeneous silicon spheres, as already discussed in the literature, by adding a layer of silicon dioxide. The high broadband reflectance of the silicon and silicon dioxide combination is explained by the favorable interference of the multipolar contributions of the outgoing field up to quadrupolar order. We also consider the impact of an embedding material characterized by different refractive indices. Refractive indices up to 1.13 maintain over 90% reflectance without re-optimizing the light sail., Comment: 15 pages, 7 figures

Let $M=G/K$ be a compact homogeneous space and assume that $G$ and $K$ have many simple factors. We show that the topological condition of having maximal third Betti number, in the sense that $b_3(M)=s-1$ if $G$ has $s$ simple factors, so called {\it aligned}, leads to a relatively manageable algebraic structure on the isotropy representation, paving the way to the computation of Ricci curvature formulas for a large class of $G$-invariant metrics. As an application, we study the existence and classification of Einstein metrics on aligned homogeneous spaces., Comment: 37 pages

This article first introduces the concept of a general pseudo-homogeneous triangular norm. It then gives some properties of general pseudo-homogeneous triangular norms. Finally, it characterizes all general pseudo-homogeneous triangular norms completely., Comment: 18

We study the removable singularity problem for $(-1)$-homogeneous solutions of the three-dimensional incompressible stationary Navier-Stokes equations with singular rays. We prove that any local $(-1)$-homogeneous solution $u$ near a potential singular ray from the origin, which passes through a point $P$ on the unit sphere $\mathbb{S}^2$, can be smoothly extended across $P$ on $\mathbb{S}^2$, provided that $u=o(\ln \text{dist} (x, P))$ on $\mathbb{S}^2$. The result is optimal in the sense that for any $\alpha>0$, there exists a local $(-1)$-homogeneous solution near $P$ on $\mathbb{S}^2$, such that $\lim_{x\in \mathbb{S}^2, x\to P}|u(x)|/\ln |x'|=-\alpha$. Furthermore, we discuss the behavior of isolated singularities of $(-1)$-homogeneous solutions and provide examples from the literature that exhibit varying behaviors. We also present an existence result of solutions with any finite number of singular points located anywhere on $\mathbb{S}^2$.

In this paper, we consider the existence problem of Ulrich bundles on a rational homogeneous space $G/P$ of type $B$, $C$ or $D$. We show that if the Picard number of $G/P$ is greater than or equal to $2$, then there are no irreducible homogeneous Ulrich bundles on $G/P$ with respect to the minimal ample class., Comment: 30 pages

Non-homogeneous self-similar measures are generically absolute continuous in the domain of parameters for which the similarity dimension is larger than one, see \cite{[SSS]}. Using certain algebraic curves we construct here exceptional singular non-homogeneous self-similar measures in this domain.

We prove that if $G$ is a noncompact connected real reductive linear Lie group, then any discrete subgroup of $G$ acting properly discontinuously and cocompactly on some homogeneous space $G/H$ of $G$ is quasi-isometrically embedded and sharp for $G/H$, i.e. satisfies a strong, quantitative form of proper discontinuity. For noncompact reductive $H$, this was known as the Sharpness Conjecture, with applications to spectral analysis on pseudo-Riemannian locally symmetric spaces developed in arXiv:1209.4075. For $G/H$ rational of real corank one, we use sharpness to fully characterize properly discontinuous and cocompact actions on $G/H$ in terms of Anosov representations. This enables us to show that in real corank one, acting properly discontinuously and cocompactly on $G/H$ is an open property, and also to prove that a number of homogeneous spaces do not admit compact quotients, such as $\mathrm{SL}(n+1,\mathbb{K})/\mathrm{SL}(n,\mathbb{K})$ for $n>1$ and $\mathbb{K}=\mathbb{R}$, $\mathbb{C}$, or the quaternions., Comment: 45 pages

In this paper, we obtain a fractional Hardy inequality in the case $Qsp$ was extensively studied in the literature, while here we are dealing with the complementary range $Q

Homogeneous nucleation processes are important for understanding solidification and the resulting microstructure of materials. Simulating this process requires accurately describing the interactions between atoms, hich is further complicated by chemical order through cross-species interactions. The large scales needed to observe rare nucleation events are far beyond the capabilities of ab initio simulations. Machine-learning is used for overcoming these limitations in terms of both accuracy and speed, by building a high-dimensional neural network potential for binary Al-Ni alloys, which serve as a model system relevant to many industrial applications. The potential is validated against experimental diffusion, viscosity, and scattering data, and is applied to large-scale molecular dynamics simulations of homogeneous nucleation at equiatomic composition, as well as for pure Ni. Pure Ni nucleates in a single-step into an fcc crystal phase, in contrast to previous results obtained with a classical empirical potential. This highlights the sensitivity of nucleation pathways to the underlying atomic interactions. Our findings suggest that the nucleation pathway for AlNi proceeds in a single step toward a B2 structure, which is discussed in relation to the pure elements counterparts.

We present a type-independent Landau-Ginzburg (LG) model $(X_\mathrm{can}, \mathcal{W}_\mathrm{can})$ for any cominuscule homogeneous space $X=G/P$. We give a fully combinatorial construction for our superpotential $\mathcal{W}_\mathrm{can}$ as a sum of $n+1$ rational functions in the (generalized) Pl\"ucker coordinates on the "Langlands dual" minuscule homogeneous space $\mathbb{X}=P^\vee\backslash G^\vee$. Explicitly, we define the denominators $\mathcal{D}_{i_*}$ of these rational functions using the combinatorics of order ideals of the corresponding minuscule poset, which can be interpreted as (generalized) Young diagrams, by a process that can be described by "moving boxes" and hence is easily implemented. To construct the corresponding numerators, we define derivations $\delta_{i_1}$ on $\mathbb{C}[\mathbb{X}]$ that act by "adding an appropriate box if possible" and then we apply each $\delta_{i_1}$ to the corresponding $\mathcal{D}_{i_*}$. By studying certain Weyl orbits in the fundamental representations of $\widetilde{G}^\vee$ and exploiting the existence of a certain dense algebraic torus in $\mathbb{X}$, we show that the polynomials $\mathcal{D}_{i_*}$ coincide with the generalized minors $\phi_{i_*}$ appearing in the cluster structures for homogeneous spaces studied by Gei\ss-Leclerc-Schr\"oer in arXiv:math/0609138. We then define the mirror variety $X_\mathrm{can}=\mathbb{X}\setminus D_\mathrm{ac}$ to be the complement of the anticanonical divisor $D_\mathrm{ac} = \sum_{i_*}\{\mathcal{D}_{i_*}=0\}$ formed by the $\mathcal{D}_{i_*}$. Moreover, we show that the LG models $(X_\mathrm{can},\mathcal{W}_\mathrm{can})$ are isomorphic to the Lie-theoretic LG-models $(X_\mathrm{Lie},\mathcal{W}_\mathrm{Lie})$ constructed by Rietsch in arXiv:math/0511124 and our models naturally generalize the type-dependent Pl\"ucker coordinate LG-models previously studied by various authors., Comment: 72 pages

Let $R=\bigoplus_{\alpha\in\Gamma}R_{\alpha}$ be a graded integral domain. In this paper we study the space of homogeneous preserving semistar operations on $R$. We show if $\star$ is a homogeneous preserving semistar operation on $R$, then $\star_a$ is also homogeneous preserving. Let $KR(R,b)$ be the homogeneous Kronecker function ring of $R$ with respect to the $b$-operation. It is shown that the set of valuation overrings of $KR(R,b)$, endowed with the Zariski topology, is homeomorphism to $Zar_h(R)$, the set of gr-valuation overrings of $R$, endowed with the Zariski topology. We also show that the set $SStar_{f,hp}(R)$ of finite type, homogeneous preserving semistar operations on $R$, endowed with the Zariski topology, is a spectral space.

Optical flow has made great progress in clean scenes, while suffers degradation under adverse weather due to the violation of the brightness constancy and gradient continuity assumptions of optical flow. Typically, existing methods mainly adopt domain adaptation to transfer motion knowledge from clean to degraded domain through one-stage adaptation. However, this direct adaptation is ineffective, since there exists a large gap due to adverse weather and scene style between clean and real degraded domains. Moreover, even within the degraded domain itself, static weather (e.g., fog) and dynamic weather (e.g., rain) have different impacts on optical flow. To address above issues, we explore synthetic degraded domain as an intermediate bridge between clean and real degraded domains, and propose a cumulative homogeneous-heterogeneous adaptation framework for real adverse weather optical flow. Specifically, for clean-degraded transfer, our key insight is that static weather possesses the depth-association homogeneous feature which does not change the intrinsic motion of the scene, while dynamic weather additionally introduces the heterogeneous feature which results in a significant boundary discrepancy in warp errors between clean and degraded domains. For synthetic-real transfer, we figure out that cost volume correlation shares a similar statistical histogram between synthetic and real degraded domains, benefiting to holistically aligning the homogeneous correlation distribution for synthetic-real knowledge distillation. Under this unified framework, the proposed method can progressively and explicitly transfer knowledge from clean scenes to real adverse weather. In addition, we further collect a real adverse weather dataset with manually annotated optical flow labels and perform extensive experiments to verify the superiority of the proposed method.