Non-metric Spaces Research Articles

Well-posedness for the stochastic Landau–Lifshitz–Gilbert equation with helicity driven by jump noise

We consider the stochastic Landau–Lifshitz–Gilbert equation driven by pure jump noise. We assume non-zero contribution from the helicity term to the total energy. Using finite dimensional approximation followed by a generalization of the Jakubowski’s version of the Skorohod Theorem for non-metric spaces, we show that the considered problem admits a weak martingale solution. Restricting the problem to dimension 1, we show that the obtained solution is pathwise unique, thereby concluding the existence of a strong solution.

Martingale solution of the stochastic Camassa–Holm equation with pure jump noise

We study the stochastic Camassa–Holm equation with pure jump noise. We establish the existence of the global martingale solution by the regularization method, the tightness criterion, the generalization of the Skorokhod theorem for nonmetric spaces and the stochastic renormalized formulations.

Martingale solutions of the stochastic Hall-magnetohydrodynamics equations on [formula omitted

We prove the existence of a global martingale solution of stochastic Hall-magnetohydrodynamics equations on R3 with multiplicative noise. Using the Fourier analysis we construct a sequence of approximate solutions. The existence of a solution is proved via the stochastic compactness method and the Jakubowski generalization of the Skorokhod theorem for nonmetric spaces, in particular, the spaces with weak topologies. The main difficulty is caused by the Hall term which makes the equations strongly nonlinear.

Etherington duality breaking: gravitational lensing in non-metric space–times versus intrinsic alignments

ABSTRACT The Etherington distance duality relation is well-established for metric theories of gravity, and confirms the duality between the luminosity distance and the angular diameter distance through the conservation of surface brightness. A violation of the Etherington distance duality due to lensing in a non-metric space–time would lead to fluctuations in surface brightness of galaxies. Likewise, fluctuations of the surface brightness can arise in classical astrophysics as a consequence of intrinsic tidal interaction of galaxies with their environment. Therefore, we study these in two cases in detail: First, for intrinsic size fluctuations and the resulting changes in surface brightness, and secondly, for an area-metric space–time as an example of a non-metric space–time, where the distance duality relation itself acquires modifications. The aim of this work is to quantify whether a surface brightness fluctuation effect due to area-metric gravity would be resolvable compared to the similar effect caused by intrinsic alignment. We thus compare the auto- and cross-correlations of the angular spectra in these two cases and show that the fluctuations in intrinsic brightness can potentially be measured with a cumulative signal-to-noise ratio Σ(ℓ) ≥ 3 in a Euclid-like survey. The measurement in area-metric space–times, however, depends on the specific parameter choices, which also determine the shape and amplitude of the spectra. While lensing surveys do have sensitivity to lensing-induced surface brightness fluctuations in area-metric space–times, the measurement does not seem to be possible for natural values of the Etherington-breaking parameters.

Stochastic magneto-hydrodynamic equations (MHD): Invariant measures in 2D Poincaré domains

We consider stochastic magneto-hydrodynamic equations with a multiplicative noise in 2D and 3D possibly unbounded domains. The noise terms depend both on the velocity and the magnetic field and its spatial derivatives. We prove the existence of a martingale solution and its continuous dependence on the initial state and deterministic external forces. The analysis is based on the stochastic compactness method and the Jakubowski generalization of the Skorokhod Theorem for nonmetric spaces. Moreover, in the case of 2D domains we prove the existence and uniqueness of a strong solution. Using the Maslowski-Seidler result we prove the existence of an invariant measure for the semigroup corresponding to the strong solutions of the MHD equations in 2-dimensional Poincaré domains. The results concerning the continuous dependence play an important role in proving sequentially weak Feller property of this semigroup. The aim of the paper is to extend the results obtained in [4] and [5] for the Navier-Stokes equations to the MHD equations.

New Necessary Conditions for a Fixed-Point of Maps in Non-Metric Spaces

Our purpose is to introduce new necessary conditions for a fixed point of maps on non-metric spaces. We use a contraction map on a metric topological space and a lately published definition of limit of a function between the metric topological space and the non-metric topological space. Then we show that we can create a function h on the non-metric space Y, h :Y →Y and present necessary conditions for a fixed point of this map on this map on Y. Therefore, this gives an opportunity to take a best conclusion in some sense, when non-metrizable matter is under consideration.

Martingale Solutions of the Stochastic 2D Primitive Equations with Anisotropic Viscosity

The stochastic 2D primitive equations with anisotropic viscosity are studied in this paper. The existence of the martingale solutions and pathwise uniqueness of the solutions are obtained. The proof is based on anisotropic estimates, the compactness method, tightness criteria and the Jakubowski version of the Skorokhod theorem for nonmetric spaces.

Well-posedness and large deviations for 2D stochastic constrained Navier-Stokes equations driven by Lévy noise in the Marcus canonical form

We consider stochastic two-dimensional constrained Navier-Stokes equations driven by Lévy noise in the Marcus canonical form. The aim of this work is two-fold. At first, we prove the existence of a martingale solution based on the construction relying on classical Faedo-Galerkin approximations, compactness method and the Jakubowski's version of Skorokhod representation theorem for non-metric spaces. We further prove that the martingale solution is pathwise unique and deduces the existence of a strong solution. In the second part of the paper, we establish a Wentzell-Freidlin type large deviations principle for the small noise asymptotic of solutions using weak convergence method.

K-Means+++: Outliers-Resistant Clustering

The k-means problem is to compute a set of k centers (points) that minimizes the sum of squared distances to a given set of n points in a metric space. Arguably, the most common algorithm to solve it is k-means++ which is easy to implement and provides a provably small approximation error in time that is linear in n. We generalize k-means++ to support outliers in two sense (simultaneously): (i) nonmetric spaces, e.g., M-estimators, where the distance dist(p,x) between a point p and a center x is replaced by mindist(p,x),c for an appropriate constant c that may depend on the scale of the input. (ii) k-means clustering with m≥1 outliers, i.e., where the m farthest points from any given k centers are excluded from the total sum of distances. This is by using a simple reduction to the (k+m)-means clustering (with no outliers).

Weak solutions and invariant measures of stochastic Oldroyd-B type model driven by jump noise

In this work, we consider sub-critical and critical models for viscoelastic flows driven by pure jump Lévy noise. Due to the elastic property, the noise in the equation for the stress tensor is considered in the Marcus canonical form. We investigate existence of a weak martingale solution for stochastic Oldroyd-B models, with full dissipation in whole of Rd,d=2,3. The key ingredients of the proof are classical Faedo-Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces. Pathwise uniqueness, existence of a strong solution and uniqueness in law for the two-dimensional model are also shown. We also prove, in a Poincaré domain in two-dimensions, existence of an invariant measure using bw-Feller property of the associated Markov semigroup.

The stochastic nonlinear Schrödinger equation in unbounded domains and non-compact manifolds

In this article, we construct a global martingale solution to a general nonlinear Schrödinger equation with linear multiplicative noise in the Stratonovich form. Our framework includes many examples of spatial domains like $${{\mathbb {R}}^d}$$ , non-compact Riemannian manifolds, and unbounded domains in $${{\mathbb {R}}^d}$$ with different boundary conditions. The initial value belongs to the energy space $$H^1$$ and we treat subcritical focusing and defocusing power nonlinearities. The proof is based on an approximation technique which makes use of spectral theoretic methods and an abstract Littlewood–Paley-decomposition. In the limit procedure, we employ tightness of the approximated solutions and Jakubowski’s extension of the Skorohod theorem to nonmetric spaces.

Stochastic optimal control of a evolutionary p-Laplace equation with multiplicative Lévy noise

In this article, we are interested in an initial value optimal control problem for a evolutionary p-Laplace equation driven by multiplicative Lévy noise. We first present wellposedness of a weak solution by using an implicit time discretization of the problem, along with the Jakubowski version of the Skorokhod theorem for a non-metric space. We then formulate associated control problem, and establish existence of an optimal solution by using variational method and exploiting the convexity property of the cost functional.

Topological Signal Processing Over Simplicial Complexes

The goal of this paper is to establish the fundamental tools to analyze signals defined over a topological space, i.e. a set of points along with a set of neighborhood relations. This setup does not require the definition of a metric and then it is especially useful to deal with signals defined over non-metric spaces. We focus on signals defined over simplicial complexes. Graph Signal Processing (GSP) represents a special case of Topological Signal Processing (TSP), referring to the situation where the signals are associated only with the vertices of a graph. Even though the theory can be applied to signals of any order, we focus on signals defined over the edges of a graph and show how building a simplicial complex of order two, i.e. including triangles, yields benefits in the analysis of edge signals. After reviewing the basic principles of algebraic topology, we derive a sampling theory for signals of any order and emphasize the interplay between signals of different order. Then we propose a method to infer the topology of a simplicial complex from data. We conclude with applications to real edge signals and to the analysis of discrete vector fields to illustrate the benefits of the proposed methodologies.

Martingale solutions to stochastic nonlocal Cahn–Hilliard–Navier–Stokes equations with multiplicative noise of jump type

Abstract In this paper, we are interested in proving the existence of a weak martingale solution of the stochastic nonlocal Cahn–Hilliard–Navier–Stokes system driven by a pure jump noise in both 2D and 3D bounded domains. Our goal is achieved by using the classical Faedo–Galerkin approximation, a compactness method and a version of the Skorokhod embedding theorem for nonmetric spaces. In the 2D case, we prove the pathwise uniqueness of the solution and use the Yamada–Watanabe classical result to derive the existence of a strong solution.

Weak martingale solutions for the stochastic nonlinear Schrödinger equation driven by pure jump noise

We construct a martingale solution of the stochastic nonlinear Schrodinger equation (NLS) with a multiplicative noise of jump type in the Marcus canonical form. The problem is formulated in a general framework that covers the subcritical focusing and defocusing stochastic NLS in $$H^1$$ on compact manifolds and on bounded domains with various boundary conditions. The proof is based on a variant of the Faedo-Galerkin method. In the formulation of the approximated equations, finite dimensional operators derived from the Littlewood–Paley decomposition complement the classical orthogonal projections to guarantee uniform estimates. Further ingredients of the construction are tightness criteria in certain spaces of cadlag functions and Jakubowski’s generalization of the Skorohod-Theorem to nonmetric spaces.

Binary classification in unstructured space with hypergraph case-based reasoning

Binary classification is one of the most common problem in machine learning. It consists in predicting whether a given element belongs to a particular class. In this paper, a new algorithm for binary classification is proposed using a hypergraph representation. The method is agnostic to data representation, can work with multiple data sources or in non-metric spaces, and accommodates with missing values. As a result, it drastically reduces the need for data preprocessing or feature engineering. Each element to be classified is partitioned according to its interactions with the training set. For each class, a seminorm over the training set partition is learnt to represent the distribution of evidence supporting this class.Empirical validation demonstrates its high potential on a wide range of well-known datasets and the results are compared to the state-of-the-art. The time complexity is given and empirically validated. Its robustness with regard to hyperparameter sensitivity is studied and compared to standard classification methods. Finally, the limitation of the model space is discussed, and some potential solutions proposed.

Weak Solutions of a Stochastic Landau–Lifshitz–Gilbert Equation Driven by Pure Jump Noise

In this work we study a stochastic three-dimensional Landau–Lifshitz–Gilbert equation perturbed by pure jump noise in the Marcus canonical form. We show the existence of a weak martingale solution taking values in a two-dimensional sphere $${\mathbb{S}^2}$$ and discuss certain regularity results. The construction of a solution is based on the classical Faedo–Galerkin approximation, the compactness methods and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces.

Influence of Terminalia arjuna on the riparian landscapes of the River Cauvery of South India

ABSTRACTMature and large isolated trees act as keystone structures in various ecosystems by supporting the populations of dependent organisms. Terminalia arjuna is one such species that is dominant in the riparian landscapes along the River Cauvery in Cauvery Wildlife Sanctuary. We tested the hypothesis that T. arjuna influences plant community composition in riparian landscapes and supports a distinct local riparian community. We showed plant species richness, diversity, similarity in community composition, and soil quality varied between riparian landscapes under canopy of T. arjuna and areas devoid of T. arjuna. Higher species richness, diversity, dominance, and evenness were noted in areas under canopy of T. arjuna. The non-metric multi-dimensional scaling ordination space analysis identifies the native riparian species assemblages in areas under canopy of T. arjuna with improved soil nutrient conditions. Overall, the study concludes T. arjuna has potential positive effects on riparian landscapes by showing the remarkable differences in biological diversity and it can be considered as keystone species in riparian ecosystems of South India.

Optimal relaxed control of stochastic hereditary evolution equations with Lévy noise

Existence theory of optimal relaxed control problem for a class of stochastic hereditary evolution equations driven by Lévy noise has been studied. We formulate the problem in the martingale sense of Stroock and Varadhan to establish existence of optimal controls. The construction of the solution is based on the classical Faedo–Galerkin approximation, the compactness method and the Jakubowski version of the Skorokhod theorem for nonmetric spaces, and certain compactness properties of the class of Young measures on Suslin metrizable control sets. As application of the abstract theory, Oldroyd and Jeffreys fluids have been studied and existence of optimal relaxed control is established. Existence and uniqueness of a strong solution and uniqueness in law for the two-dimensional Oldroyd and Jeffreys fluids are also shown.

Martingale solutions of nematic liquid crystals driven by pure jump noise in the Marcus canonical form

In this work we consider a stochastic evolution equation which describes the system governing the nematic liquid crystals driven by a pure jump noise in the Marcus canonical form. The existence of a martingale solution is proved for both 2D and 3D cases. The construction of the solution relies on a modified Faedo–Galerkin method based on the Littlewood–Paley-decomposition, compactness method and the Jakubowski version of the Skorokhod representation theorem for non-metric spaces. We prove that in the 2-D case the martingale solution is pathwise unique and hence deduce the existence of a strong solution.