Smooth Spaces Research Articles

Local linear independence of bilinear (and higher degree) B-splines on hierarchical T-meshes

We generate hierarchical T-meshes by repeatedly inserting new line segments, in order to adapt both the size and the shape of the cells to the specific requirements of the underlying application. The associated spaces of bilinear spline functions are spanned by the locally refined (LR) B-splines of Dokken et al. (2013), which are products of univariate B-splines defined over local knot vectors. Our method guarantees that these functions are linearly independent for the obtained class of hierarchical T-meshes. Furthermore, they even possess the property of local linear independence, since exactly 4 functions take non-zero values on each cell of the mesh, and they form a partition of unity without the need to perform additional scaling or truncation.The correctness of our refinement algorithm is verified by enumerating the (newly introduced) local configurations, which represent the possible topologies of the mesh in the vicinity of a cell. It is then shown that the method works well in all the resulting situations. The fairly large number (385) of cases sheds some light on the challenges associated with linear independence in the context of LR B-spline refinement.Additionally we apply our method to meshes that possess the additional property local semi-regularity (first studied by Weller and Hagen, 1995), where only 49 topologically different situations may arise. However, we observe that the resulting class of hierarchical T-meshes is less flexible, which leads to a substantial amount of excess refinement in applications.Finally we note that the same construction can be applied to generate basis functions for Cs–smooth spline spaces of degree p=2s+1, again with the property of local linear independence.

Contrastive blind denoising autoencoder for real time denoising of industrial IoT sensor data

In an industrial IoT setting, ensuring the quality of sensor data is a must when data-driven algorithms operate on the upper layers of the control system. Unfortunately, the common place in industrial facilities is to find sensor time series heavily corrupted by noise and outliers. In this work, a purely data-driven self-supervised learning-based approach based on a blind denoising autoencoder is proposed for real time denoising of industrial sensor data. The term \textit{blind} stresses that no prior knowledge about the noise is required for denoising, in contrast to typical denoising autoencoders where prior knowledge about the noise is required. Blind denoising is achieved by using a noise contrastive estimation (NCE) regularization on the latent space of the autoencoder, which not only helps to denoise but also induces a meaningful and smooth latent space. Experimental evaluation in both a simulated system and a real industrial process shows that the proposed technique outperforms classical denoising methods.

Sobolev inequality and isoperimetric inequality for submanifolds in a smooth metric measure space

Brendle recently proved a sharp Sobolev inequality and logarithmic Sobolev inequality for submanifolds in Euclidean space. From the sharp Sobolev inequality, he achieved a breakthrough in the conjecture of isoperimetric inequality for minimal submanifolds. In this paper, we extend Brendle’s results to submanifolds in a smooth metric measure space. As an application, we prove some new isoperimetric-type inequalities in some smooth metric measure spaces. For example, we obtain a new isoperimetric-type inequality for self-expander.

Smoothing and global attractors for the Hirota-Satsuma system on the torus

We consider the Hirota-Satsuma system, a coupled KdV-type system, with periodic boundary conditions. The first part of the paper concerns with the smoothing estimates for the system. More precisely, it is shown that, for initial data in a Sobolev space, the difference of the nonlinear and linear evolutions lies in a smoother space. The smoothing gain we obtain depends very much on the arithmetic nature of the coupling parameter a which determines the structure of the resonant sets in the estimates. In the second part, we address the forced and damped Hirota-Satsuma system and obtain counterpart smoothing estimates. As a consequence of these estimates, we prove the existence and smoothness of a global attractor in the energy space.

Morrey smoothness spaces: A new approach

In the recent years, the so-called Morrey smoothness spaces attracted a lot of interest. They can (also) be understood as generalisations of the classical spaces A(skp,q/s}(ℝn) with A ∈{B,F} in ℝn, where the parameters satisfy s ∈ ℝ (smoothness), 0 < p ⩽ ∞ (integrability) and 0 < q ⩽ ∞ (summability). In the case of Morrey smoothness spaces, additional parameters are involved. In our opinion, among the various approaches at least two scales enjoy special attention, also in view of applications: the scales $$\cal{A}_{u,p,q}^{s}(\mathbb{R}^{n})$$ with $$\cal{A}\in\{\mathcal{N},\cal{E}\}$$ and u ⩾ p, and A (ℝn) with A ∈ {B, F} and τ ⩾ 0. We reorganise these two prominent types of Morrey smoothness spaces by adding to (s,p, q) the so-called slope parameter ϱ, preferably (but not exclusively) with −n ⩽ ϱ < 0. It comes out that ∣ϱ∣ replaces n, and min(∣ϱ∣, 1) replaces 1 in slopes of (broken) lines in the ( $$1\over p$$ , s)-diagram characterising distinguished properties of the spaces A (ℝn) and their Morrey counterparts. Special attention will be paid to low-slope spaces with −1 < ϱ < 0, where the corresponding properties are quite often independent of n ∈ ℕ. Our aim is two-fold. On the one hand, we reformulate some assertions already available in the literature (many of which are quite recent). On the other hand, we establish on this basis new properties, a few of which become visible only in the context of the offered new approach, governed, now, by the four parameters (s, p, q, ϱ).

Gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces with evolving metrics and potentials

This article presents new parabolic and elliptic type gradient estimates for positive smooth solutions to nonlinear parabolic equations involving the Witten Laplacian in the context of smooth metric measure spaces. The metric and potential here are time dependent and evolve under a super Perelman–Ricci flow. The estimates are derived under natural lower bounds on the associated generalised Bakry–Émery Ricci curvature tensors and are utilised in establishing fairly general local and global bounds, Harnack-type inequalities and Liouville-type global constancy theorems to mention a few. Other implications and consequences of the results are also discussed.

Theorising the memescape: The spatial politics of Internet memes

AbstractDespite the increasing centrality of Internet memes for everyday political circulations and practices, their emergent implications as low-cultural artefacts of global politics have received little theoretical attention. In this article, I develop a critical theory of memes to provide a conceptual apparatus to understand the global political implications and possibilities of this pop-cultural phenomenon. I argue that, in order to attend to the emergent implications of memes and consider their differentiations from other pop-cultural phenomena, we need to unpack the spatial logic through which memes emerge and circulate. Analysing this spatial logic through the concept of the ‘memescape’ and deploying Gilles Deleuze and Felix Guattari’s notions of striated and smooth spaces, this article articulates the spatial logic of the memescape as comprising rhizomatic, decentralised circulations of digital content; nomadic, playful, and humorous disruptions of once-stable signs; and affective congregations of a multiplicity of subjects. Through two examples exploring how these smooth spatial tendencies produce divergent political potentials in the resistant memes of Indigenous digital communities and reactionary memes of the Alt-Right, I conclude that the global politics of the memescape is open-ended and undetermined which requires careful and nuanced political and ethical attention to actualise its futures for emancipatory horizons.

Gradient estimates for nonlinear elliptic equations involving the Witten Laplacian on smooth metric measure spaces and implications

AbstractThis article presents new local and global gradient estimates of Li-Yau type for positive solutions to a class of nonlinear elliptic equations on smooth metric measure spaces involving the Witten Laplacian. The estimates are derived under natural lower bounds on the associated Bakry-Émery Ricci curvature tensor and find utility in proving fairly general Harnack inequalities and Liouville-type theorems to name a few. The results here unify, extend and improve various existing results in the literature for special nonlinearities already of huge interest and applications. Some consequences are presented and discussed.

Non-compact Einstein manifolds with symmetry

For Einstein manifolds with negative scalar curvature admitting an isometric action of a Lie group G \mathsf {G} with compact, smooth orbit space, we show that the nilradical N \mathsf {N} of G \mathsf {G} acts polarly and that the N \mathsf {N} -orbits can be extended to minimal Einstein submanifolds. As an application, we prove the Alekseevskii conjecture: Any homogeneous Einstein manifold with negative scalar curvature is diffeomorphic to a Euclidean space.

Gradient estimates for a nonlinear elliptic equation on a smooth metric measure space

Let (M, g, e −f dv) be a smooth metric measure space. We consider local gradient estimates for positive solutions to the following elliptic equation ∆ f u + au log u + bu = 0 where a, b are two real constants and f be a smooth function defined on M. As an application, we obtain a Liouville type result for such equation in the case a &lt; 0 under the m-dimensions Bakry-Émery Ricci curvature.

Forecasting daily admissions to an emergency department considering single and multiple seasonal patterns

When dealing with several years of daily data, such as the number of daily admissions to a hospital’s emergency department (ED), how complex does it get to forecast into the future? With that in mind, this study has two main goals: to explore the differences between several methodologies, considering both single and multiple-seasonal patterns; and to select the most suitable model for the administration of a Portuguese hospital to use while managing their ED. To that end, we first considered the data as a time series with a single weekly seasonal pattern. We then modelled the data using time series regression, linear regression with autoregressive integrated moving average (ARIMA) errors, seasonal ARIMA and exponential smoothing techniques. Second, the data was set to be a time series with weekly and annual seasonal patterns. Then, using Fourier terms, we applied time series regression, linear regression with ARIMA errors and trigonometric exponential smoothing state space models with Box–Cox transformation, ARMA errors, Trend and Seasonal components (TBATS) for the analysis. After selecting the best-fitting models using the Akaike Information Criteria (AIC) values, we forecasted into the future and compared the results using both training and test datasets’ root mean square error (RMSE), mean absolute error (MAE) and mean absolute percentage error (MAPE) values. The time series regression model based on seasonal variables and a weekly seasonal pattern gives the best results. However, we decided to use linear regression with ARIMA errors, seasonal variables, and both weekly and annual seasonal patterns. This produces similar results but allows for the annual seasonality to be considered, which is useful when more data is added.

Deleuze and AlphaGo

It is time to update Deleuze and Guattari's contrast between Chess and Go in the ‘Nomadology’ Plateau with a discussion of AlphaGo, the artificial intelligence that revolutionised Go in 2016. I focus less on the political issues in Go nomadology, central as they are, and more on smooth space and time. I explain and speculate on some details in Go strategy, as well as some processes of machine learning. The relations between human Go, computer Go, and smooth-time nomadology remain unsettled, in my view, but many resources for nomadological analysis open up for readers who learn to play, and program, Go.

Nonabelian Hodge theory for stacks and a stacky P=W conjecture

We introduce a version of the P=W conjecture relating the Borel–Moore homology of the stack of representations of the fundamental group of a genus g Riemann surface with the Borel–Moore homology of the stack of degree zero semistable Higgs bundles on a smooth projective complex curve of genus g. In order to state the conjecture we propose a construction of a canonical isomorphism between these Borel–Moore homology groups. We relate the stacky P=W conjecture with the original P=W conjecture concerning the cohomology of smooth moduli spaces, and the PI=WI conjecture concerning the intersection cohomology groups of singular moduli spaces. In genus zero and one, we prove the conjectures that we introduce in this paper.

Exponential and sub-exponential stability times for the derivative wave equation

We prove a Nekhoroshev type theorem for the derivative wave equation$ \begin{equation*} u_{tt} = u_{xx}-mu-(D^{\alpha}u)^3, \quad \ D = \sqrt{-\partial_{xx}+m} \end{equation*} $under Dirichlet boundary conditions and $ 0\leq \alpha\leq 1 $. More precisely, we prove the sub-exponential long time stability result in a smooth function space and the exponential long time stability result in a modified Sobolev space around the origin for the above equation, where using Birkhoff normal form technique for infinite dimensional Hamiltonian systems and the so-called $ {tame} $ property of the non-linearity. This result is inspired by Bambusi-Grébert (2006, Duke) and a recent work by Feola-Massetti (arXiv: 2207.09986v1).

Density of quantized approximations

The paper contains a review of known results and proofs of new results on conditions on a set $M$ in a Banach space $X$ that are necessary or sufficient for the additive semigroup $R(M)=\{x_1+…+x_n\colon x_k\in M, n\in {\mathbb N}\}$ to be dense in $X$. We prove, in particular, that if $M$ is a rectifiable curve in a uniformly smooth real space $X$, and $M$ does not lie entirely in any closed half-space, then $R(M)$ is dense in $X$. We present known and new results on the approximation by simple partial fractions (logarithmic derivatives of polynomials) in various spaces of functions of a complex variable. Meanwhile, some well-known theorems, in particular, Korevaar's theorem, are derived from new general results on the density of a semigroup. We also study approximation by sums of shifts of one function, which are a natural generalization of simple partial fractions. Bibliography: 79 titles.

Multiuser Physical-Layer Authentication Based on Latent Perturbed Neural Networks for Industrial Internet of Things

Recently, learning (DL)-based physical-layer authentication (PLA) has attracted much attention since artificial neural networks (ANNs) can be built to extract useful features from complex wireless environments, thus achieving high authentication performance and lightweight deployment in mobile edge computing (MEC)-Industrial Internet of Things (IIoT) scenario. However, the low latency characteristic of MEC makes it impossible to have much time to obtain sufficient signals for training the authentication system, which will cause over-fitting issues and deteriorate the authentication performance. Data augmentation is an effective method to address this problem. However, existing PLA with data augmentation can not generate representative and high-quality samples, consequently lacking generality in the actual identity authentication. To tackle this problem, a novel channel impulse response (CIR)-based multiuser authentication named latent perturbed neural networks (LPNNs) is proposed in this article, aiming at achieving high authentication performance even when trained a few data. Instead of relying on the generation of synthetic samples, the proposed LPNN adds Gaussian noise in the smooth latent space to avoid underdetermined and poor generalization, which has better interpretability. Specifically, to obtain a better understanding than a black box that connects input CIRs to authentication results, we defined Fingerprint Library and provided post-hoc explanations to answer the following question: which library examples explain the authentication results issued for a given CIR sample? Moreover, the simulations under the static and dynamic IIoT scenarios verify the superiority in authentication accuracy of the proposed LPNN over vanilla deep neural network (DNN) and convolutional neural network (CNN).

MATLAB-Based Visualization of Helical Structures and Their Applications

A helix is a type of smooth space curve, i.e. a curve in three-dimensional space. This study aims to present a MATLAB-based approach to visualize helical structures and to demonstrate their applications in various fields. As a qualitative study, primary and secondary sources of data have been consulted. Primarily, observing the real-life activities of humans and secondly, the analysis of archival documents such as related research articles, analytical books, and related reports. In conclusion, most of the architectural work in the world is possible by different types of geometrical design through the helix. It also found that we can visualize the different types of helices by using MATLAB in mathematics teaching. The implication of helix is everywhere in different fields and areas such as constructing a basket, screw, etc. Finally, we used helical concepts related to daily life activities in the field of instruction in the mathematics classroom.

Towards mathematics curriculum recontextualisation: developing a rhizocurrere with roma students

Students out of mainstream like Roma students often face problems following the mathematics curriculum which is designed without taking into consideration them and this fact contributes to the reproduction of inequalities. This paper suggests indicatively due to the limited layout an activity in the direction of reconceptualization of the Mathematics curriculum through rhizomatic thinking. The aim of the rhizomatic inquiry is to avoid representationalism often inherent in qualitative research and to challenge the dichotomies apparent in Mathematics education. Roma students’ ideas on parallel and intersecting straight line sections become a smooth space for both knowledge consolidation and application of mathematical knowledge in contexts that allow the restoration of social justice and honor movements against marginalization softening, thus, power relations.

The deep interface of the effectuated voluminous territories: gates, smooth and striated spaces, and the royal science in the Air Silk Road

ABSTRACT While the scholarship on volumetric territory is now gaining momentum, our knowledge of the in-betweens of spheres is still limited. This paper studies the voluminous territory through a new lens: the deep interface as a destabilizing ontology that invites, enables and disciplines movements across the different spheres. The conceptualization of the deep interface unfolds in three layers: (1) an effective territory that treats territory as formed by the mobility of disruptive object–spaces; (2) a deep interface constituted by a series of gates distributed in various spheres for the succession of a relay along particular paths; and (3) the striation of smooth space by imposing the ‘royal science’ on the moving things, translating voluminous space to be a vector of political–rational calculations. Tracing the moving aeroplane as the disruptive object–space, we demonstrate how the three gates of the Air Silk Road scheme, Dara Sakor Airport and the mangrove forests constitute a deep interface after the construction of their interdependence by instantly allied forces.

Some Inequalities of Hardy Type Related to Witten–Laplace Operator on Smooth Metric Measure Spaces

A complete Riemannian manifold equipped with some potential function and an invariant conformal measure is referred to as a complete smooth metric measure space. This paper generalizes some integral inequalities of the Hardy type to the setting of a complete non-compact smooth metric measure space without any geometric constraint on the potential function. The adopted approach highlights some criteria for a smooth metric measure space to admit Hardy inequalities related to Witten and Witten p-Laplace operators. The results in this paper complement in several aspect to those obtained recently in the non-compact setting.