Arbitrary Space Research Articles

Gravitational memory and compact extra dimensions

We develop a general formalism for treating radiative degrees of freedom near ${\mathcal{I}}^{+}$ in theories with an arbitrary Ricci-flat internal space. These radiative modes are encoded in a generalized news tensor which decomposes into gravitational, electromagnetic, and scalar components. We find a preferred gauge which simplifies the asymptotic analysis of the full nonlinear Einstein equations and makes the asymptotic symmetry group transparent. This asymptotic symmetry group extends the Bondi--Metzner--Sachs (BMS) group to include angle-dependent isometries of the internal space. We apply this formalism to study memory effects, which are expected to be observed in future experiments, that arise from bursts of higher-dimensional gravitational radiation. We outline how measurements made by gravitational wave observatories might probe properties of the compact extra dimensions.

On the Nonlocal Problems in Time for Time-Fractional Subdiffusion Equations

The nonlocal boundary value problem, dtρu(t)+Au(t)=f(t) (0<ρ<1, 0<t≤T), u(ξ)=αu(0)+φ (α is a constant and 0<ξ≤T), in an arbitrary separable Hilbert space H with the strongly positive selfadjoint operator A, is considered. The operator dt on the left hand side of the equation expresses either the Caputo derivative or the Riemann–Liouville derivative; naturally, in the case of the Riemann–Liouville derivatives, the nonlocal boundary condition should be slightly changed. Existence and uniqueness theorems for solutions of the problems under consideration are proved. The influence of the constant α on the existence of a solution to problems is investigated. Inequalities of coercivity type are obtained and it is shown that these inequalities differ depending on the considered type of fractional derivatives. The inverse problems of determining the right-hand side of the equation and the function φ in the boundary conditions are investigated.

An anisotropic vector hysteresis model of ferromagnetic behavior under alternating and rotational magnetic field

Vector quantities and tensorial properties rule the magnetization mechanisms in ferromagnetic materials. Every ferromagnetic material is anisotropic to some degree in its magnetic response, so this anisotropy has to be considered for a general model. In this study, a multiscale approach based on a statistical description of the magnetic domain distribution and the knowledge of the crystallographic texture is used to predict the anhysteretic behavior along an arbitrary space direction. Combined with the vector Bergqvist dry-friction hysteresis model, qualitatively reliable simulation results are obtained under alternating and rotational magnetization. FeSi 3% grain-oriented (FeSi GO) electrical steel is chosen as study material: FeSi GO are widespread so that extensive data are available and strongly anisotropic, forcing the model toward the “worst-case” scenario from the viewpoint of anisotropy.Moreover, under high excitation of rotational magnetization, losses drop due to the disappearance of the magnetic domains. This behavior is represented correctly by the proposed simulation method. In such conditions, the magnetization behavior is led mainly by the anhysteretic behavior, strengthening the predictive ability of the proposed model. In this manuscript, comparisons between simulations and measurements under many amplitudes of alternating and rotational magnetization for different levels of imposed excitation or magnetization are provided and used to validate the simulation method.

Stable and Efficient Petrov--Galerkin Methods for a Kinetic Fokker--Planck Equation

We propose a stable Petrov-Galerkin discretization of a kinetic Fokker-Planck equation constructed in such a way that uniform inf-sup stability can be inferred directly from the variational formulation. Inspired by well-posedness results for parabolic equations, we derive a lower bound for the dual inf-sup constant of the Fokker-Planck bilinear form by means of stable pairs of trial and test functions. The trial function of such a pair is constructed by applying the kinetic transport operator and the inverse velocity Laplace-Beltrami operator to a given test function. For the Petrov-Galerkin projection we choose an arbitrary discrete test space and then define the discrete trial space using the same application of transport and inverse Laplace-Beltrami operator. As a result, the spaces replicate the stable pairs of the continuous level and we obtain a well-posed numerical method with a discrete inf-sup constant identical to the inf-sup constant of the continuous problem independently of the mesh size. We show how the specific basis functions can be efficiently computed by low-dimensional elliptic problems, and confirm the practicability and performance of the method for a numerical example.

Mobility in Unsupervised Word Embeddings for Knowledge Extraction—The Scholars’ Trajectories across Research Topics

In the knowledge discovery field of the Big Data domain the analysis of geographic positioning and mobility information plays a key role. At the same time, in the Natural Language Processing (NLP) domain pre-trained models such as BERT and word embedding algorithms such as Word2Vec enabled a rich encoding of words that allows mapping textual data into points of an arbitrary multi-dimensional space, in which the notion of proximity reflects an association among terms or topics. The main contribution of this paper is to show how analytical tools, traditionally adopted to deal with geographic data to measure the mobility of an agent in a time interval, can also be effectively applied to extract knowledge in a semantic realm, such as a semantic space of words and topics, looking for latent trajectories that can benefit the properties of neural network latent representations. As a case study, the Scopus database was queried about works of highly cited researchers in recent years. On this basis, we performed a dynamic analysis, for measuring the Radius of Gyration as an index of the mobility of researchers across scientific topics. The semantic space is built from the automatic analysis of the paper abstracts of each author. In particular, we evaluated two different methodologies to build the semantic space and we found that Word2Vec embeddings perform better than the BERT ones for this task. Finally, The scholars’ trajectories show some latent properties of this model, which also represent new scientific contributions of this work. These properties include (i) the correlation between the scientific mobility and the achievement of scientific results, measured through the H-index; (ii) differences in the behavior of researchers working in different countries and subjects; and (iii) some interesting similarities between mobility patterns in this semantic realm and those typically observed in the case of human mobility.

Extending the Reach of the Point-to-Set Principle

The point-to-set principle of J. Lutz and N. Lutz (2018) has recently enabled the theory of computing to be used to answer open questions about fractal geometry in Euclidean spaces $\mathbb{R}^n$. These are classical questions, meaning that their statements do not involve computation or related aspects of logic. In this paper we extend the reach of the point-to-set principle from Euclidean spaces to arbitrary separable metric spaces $X$. We first extend two fractal dimensions--computability-theoretic versions of classical Hausdorff and packing dimensions that assign dimensions $\dim(x)$ and $\textrm{Dim}(x)$ to individual points $x\in X$--to arbitrary separable metric spaces and to arbitrary gauge families. Our first two main results then extend the point-to-set principle to arbitrary separable metric spaces and to a large class of gauge families. We demonstrate the power of our extended point-to-set principle by using it to prove new theorems about classical fractal dimensions in hyperspaces. (For a concrete computational example, the stages $E_0, E_1, E_2, \ldots$ used to construct a self-similar fractal $E$ in the plane are elements of the hyperspace of the plane, and they converge to $E$ in the hyperspace.) Our third main result, proven via our extended point-to-set principle, states that, under a wide variety of gauge families, the classical packing dimension agrees with the classical upper Minkowski dimension on all hyperspaces of compact sets. We use this theorem to give, for all sets $E$ that are analytic, i.e., $\mathbf{\Sigma}^1_1$, a tight bound on the packing dimension of the hyperspace of $E$ in terms of the packing dimension of $E$ itself.

On the extension of surjective isometries whose domain is the unit sphere of a space of compact operators

We prove that every surjective isometry from the unit sphere of the space K(H), of all compact operators on an arbitrary complex Hilbert space H, onto the unit sphere of an arbitrary real Banach space Y can be extended to a surjective real linear isometry from K(H) onto Y. This is probably the first example of an infinite dimensional non-commutative C*-algebra containing no unitaries and satisfying the Mazur-Ulam property. We also prove that all compact C*-algebras and all weakly compact JB*-triples satisfy the Mazur-Ulam property.

Joint Optimization for RIS-Assisted Wireless Communications: From Physical and Electromagnetic Perspectives

Reconfigurable intelligent surfaces (RISs) are envisioned to be a disruptive wireless communication technique that is capable of reconfiguring the wireless propagation environment. In this paper, we study a free-space RIS-assisted multiple-input single-output (MISO) communication system in far-field operation. To maximize the received power from the physical and electromagnetic nature point of view, a comprehensive optimization, including beamforming of the transmitter, phase shifts of the RIS, orientation and position of the RIS is formulated and addressed. After exploiting the property of line-of-sight (LoS) links, we derive closed-form solutions of beamforming and phase shifts. For the non-trivial RIS position optimization problem in arbitrary three-dimensional space, a dimensional-reducing theory is proved. The simulation results show that the proposed closed-form beamforming and phase shifts approach the upper bound of the received power. The robustness of our proposed solutions in terms of the perturbation is also verified. Moreover, the RIS significantly enhances the performance of the mmWave/THz communication system.

$ L^p $-exact controllability of partial differential equations with nonlocal terms

<p style='text-indent:20px;'>The paper deals with the exact controllability of partial differential equations by linear controls. The discussion takes place in infinite dimensional state spaces since these equations are considered in their abstract formulation as semilinear equations. The linear parts are densely defined and generate strongly continuous semigroups. The nonlinear terms may also include a nonlocal part. The solutions satisfy nonlocal properties, which are possibly nonlinear. The states belong to Banach spaces with a Schauder basis and the results exploit topological methods. The novelty of this investigation is in the use of an approximation solvability method which involves a sequence of controllability problems in finite-dimensional spaces. The exact controllability of nonlocal solutions can be proved, with controls in <inline-formula><tex-math id="M2">\begin{document}$ L^p $\end{document}</tex-math></inline-formula> spaces, <inline-formula><tex-math id="M3">\begin{document}$ 1<p<\infty $\end{document}</tex-math></inline-formula>. The results apply to the study of the exact controllability for the transport equation in arbitrary Euclidean spaces and for the equation of the nonlinear wave equation.</p>

Jordan maps and zero Lie product determined algebras

Let $A$ be an algebra over a field $F$ with $(F)\ne 2$. If $A$ is generated as an algebra by $[[A,A],[A,A]]$, then for every skew-symmetric bilinear map $\Phi:A\times A\to X$, where $X$ is an arbitrary vector space over $F$, the condition that $\Phi(x^2,x)=0 $ for all $x\in A$ implies that $\Phi(xy,z) +\Phi(zx,y) + \Phi(yz,x)=0$ for all $x,y,z\in A$. This is applicable to the question of whether $A$ is zero Lie product determined and is also used in proving that a Jordan homomorphism from $A$ onto a semiprime algebra $B$ is the sum of a homomorphism and an antihomomorphism.

Nondegenerate homotopy and geometric flows

The Euler characteristic is an invariant of a topological space that in a precise sense captures its canonical notion of size, akin to the cardinality of a set. The Euler characteristic is closely related to the homology of a space, as it can be expressed as the alternating sum of its Betti numbers, whenever the sum is well-defined. Thus, one says that homology categorifies the Euler characteristic. In his work on the generalisation of cardinality-like invariants, Leinster introduced the magnitude of a metric space, a real number that counts the "effective number of points" of the space and has been shown to encode many invariants of metric spaces from integral geometry and geometric measure theory. In 2015, Hepworth and Willerton introduced a homology theory for metric graphs, called magnitude homology, which categorifies the magnitude of a finite metric graph. This work was subsequently generalised to enriched categories by Leinster and Shulman, and the homology theory that they introduced categorifies magnitude for arbitrary finite metric spaces. When studying a metric space, one is often only interested in the metric space up to a rescaling of the distance of the points by a non-negative real number. The magnitude function describes how the effective number of points changes as one scales the distance, and is completely encoded by magnitude homology. When studying a finite metric space in topological data analysis using persistent homology, one approximates the space through a nested sequence of simplicial complexes so as to recover topological information about the space by studying the homology of this sequence. Here we relate magnitude homology and persistent homology as two different ways of computing homology of filtered simplicial sets.

Testing the Sobolev property with a single test plan

We prove that on an arbitrary metric measure space the following property holds: a single test plan can be used to recover the minimal weak upper gradient of any Sobolev function. This means that, in order to identify which are the exceptional curves in t

On existence theorems for coupled systems of quadratic Hammerstein-Urysohn integral equations in Orlicz spaces

<abstract><p>We present two existence theorems for a general system of functional quadratic Hammerstein-Urysohn integral equations in arbitrary Orlicz spaces $ L_\varphi $, namely when the generating $ N $-functions fulfill $ \Delta' $ and $ \Delta_3 $-conditions. The studied system contains many integral equations as special cases such as the Chandrasekhar equations, which have significant applications in technology and different disciplines of science. Our analysis is concerned with the fixed point approach and a measure of noncompactness.</p></abstract>

On existence theorems for coupled systems of quadratic Hammerstein-Urysohn integral equations in Orlicz spaces

<abstract><p>We present two existence theorems for a general system of functional quadratic Hammerstein-Urysohn integral equations in arbitrary Orlicz spaces $ L_\varphi $, namely when the generating $ N $-functions fulfill $ \Delta' $ and $ \Delta_3 $-conditions. The studied system contains many integral equations as special cases such as the Chandrasekhar equations, which have significant applications in technology and different disciplines of science. Our analysis is concerned with the fixed point approach and a measure of noncompactness.</p></abstract>

Atomic lattices of subspaces of an arbitrary vector space and associated operator algebras

Atomic lattices of subspaces of an arbitrary vector space and associated operator algebras

О взаимоотношении движений динамических систем

In the earlier articles by the authors [A.P. Afanasiev, S.M. Dzyuba, “On new properties of recurrent motions and minimal sets of dynamical systems”, Russian Universities Reports. Mathematics, 26:133 (2021), 5–14] and [A.P. Afanasiev, S.M. Dzyuba, “New properties of recurrent motions and limit motions sets of dynamical systems”, Russian Universities Reports. Mathematics, 27:137 (2022), 5–15], there was actually established the interrelation of motions of dynamical systems in compact metric spaces. The goal of this paper is to extend these results to the case of dynamical systems in arbitrary metric spaces. Namely, let Σ, be an arbitrary metric space. In this article, first of all, a new important property is established that connects arbitrary and recurrent motions in such a space. Further, on the basis of this property, it is shown that if the positive (negative) semitrajectory of some motion f(t,p) located in Σ is relatively compact, then ω- (α-) limit set of the given motion is a compact minimal set. It follows, that in the space Σ, any nonrecurrent motion is either positively (negatively) outgoing or positively (negatively) asymptotic with respect to the corresponding minimal set.

Blang: Bayesian Declarative Modeling of General Data Structures and Inference via Algorithms Based on Distribution Continua

Consider a Bayesian inference problem where a variable of interest does not take values in a Euclidean space. These "non-standard" data structures are in reality fairly common. They are frequently used in problems involving latent discrete factor models, networks, and domain specific problems such as sequence alignments and reconstructions, pedigrees, and phylogenies. In principle, Bayesian inference should be particularly wellsuited in such scenarios, as the Bayesian paradigm provides a principled way to obtain confidence assessment for random variables of any type. However, much of the recent work on making Bayesian analysis more accessible and computationally efficient has focused on inference in Euclidean spaces. In this paper, we introduce Blang, a domain specific language and library aimed at bridging this gap. Blang allows users to perform Bayesian analysis on arbitrary data types while using a declarative syntax similar to the popular family of probabilistic programming languages, BUGS. Blang is augmented with intuitive language additions to create data types of the user's choosing. To perform inference at scale on such arbitrary state spaces, Blang leverages recent advances in sequential Monte Carlo and non-reversible Markov chain Monte Carlo methods.

Analysis and asymptotic reduction of a bulk-surface reaction-diffusion model of Gierer-Meinhardt type

<p style='text-indent:20px;'>We consider a Gierer-Meinhardt system on a surface coupled with a parabolic PDE in the bulk, the domain confined by this surface. Such a model was recently proposed and analyzed for two-dimensional bulk domains by Gomez, Ward and Wei (<i>SIAM J. Appl. Dyn. Syst. 18</i>, 2019). We prove the well-posedness of the bulk-surface system in arbitrary space dimensions and show that solutions remain uniformly bounded in parabolic Hölder spaces for all times. The cytosolic diffusion is typically much larger than the lateral diffusion on the membrane. This motivates to a corresponding asymptotic reduction, which consists of a nonlocal system on the membrane. We prove the convergence of solutions of the full system towards unique solutions of the reduction.</p>

Regularity of the inverse mapping in Banach function spaces

AbstractWe study the regularity properties of the inverse of a bilipschitz mapping f belonging to , where X is an arbitrary Banach function space. Namely, we prove that the inverse mapping is also in . Furthermore, the paper shows that the class of bilipschitz mappings in is closed with respect to composition and multiplication.

On the empirical spectral distribution for certain models related to sample covariance matrices with different correlations

Given [Formula: see text], we study two classes of large random matrices of the form [Formula: see text] where for every [Formula: see text], [Formula: see text] are iid copies of a random variable [Formula: see text], [Formula: see text], [Formula: see text] are two (not necessarily independent) sets of independent random vectors having different covariance matrices and generating well concentrated bilinear forms. We consider two main asymptotic regimes as [Formula: see text]: a standard one, where [Formula: see text], and a slightly modified one, where [Formula: see text] and [Formula: see text] while [Formula: see text] for some [Formula: see text]. Assuming that vectors [Formula: see text] and [Formula: see text] are normalized and isotropic “in average”, we prove the convergence in probability of the empirical spectral distributions of [Formula: see text] and [Formula: see text] to a version of the Marchenko–Pastur law and the so-called effective medium spectral distribution, correspondingly. In particular, choosing normalized Rademacher random variables as [Formula: see text], in the modified regime one can get a shifted semicircle and semicircle laws. We also apply our results to the certain classes of matrices having block structures, which were studied in [G. M. Cicuta, J. Krausser, R. Milkus and A. Zaccone, Unifying model for random matrix theory in arbitrary space dimensions, Phys. Rev. E 97(3) (2018) 032113, MR3789138; M. Pernici and G. M. Cicuta, Proof of a conjecture on the infinite dimension limit of a unifying model for random matrix theory, J. Stat. Phys. 175 (2) (2019) 384–401, MR3968860].