Smoothness Conditions Research Articles

On the Minkowski dimension of some invariant sets of dynamical systems

Abstract. It is well known that a fractal set is not a submanifold of the ambient space. However, fractals arise as invariant subsets even in infinitely smooth conditions and the Minkowski dimension serves in this case as a characteristic of complexity of this scale. For example, when the equilibrium state during the Andronov-Hopf bifurcation losses its stability, the closure of the non-singular trajectory is a parametrically defined curve of the fractal type. In this work the fractal dimension of such curves is calculated. In addition, special two-parameter family of functions is studied such that Minkowski dimension of their graphs varies from 1 to 2. The obtained result allows us to implement a regular dynamic system with an isolated hyperbolic point such that the closure of two-dimensional stable manifold of this point may have Minkowski dimension greater than 2. To calculate the graph dimension, the segment of the argument defining the graph is split into two parts. The dimension of the first part of the graph can be estimated from above by direct calculation of the corresponding curve’s length. The upper estimation of the other part’s dimension is provided by means of the area of rectangle containing this curve. The lower estimation of the Minkowski dimension is based on calculating the cardinality of ε -distinguishable set of graph points.

Coupled Inversion of Amplitudes and Traveltimes of Primaries and Multiples for Monochannel Seismic Surveys

Engineers need to know properties of shallow marine sediments to build piers, pipelines and even offshore windfarms. We present a method for estimating the density, P velocity and thickness of these sediments. The traveltime inversion of primary and multiple reflections enables their semiquantitative estimation in marine surveys when using a minimal acquisition system such as a monochannel Boomer. Picking errors, ambient noise and interfering events lead to significant errors in the estimates. Similar, albeit milder, instabilities occur when inverting the signal amplitudes to determine the reflectivity of the layer interfaces. In this paper, we introduce a coupling between the separate inversion of amplitudes and traveltimes to obtain a better Earth model. The P velocity shows up in two stable terms provided by the separate inversions: the acoustic impedance of shallow sediments (through the amplitudes) and the transit time across the sediment layer (through the traveltimes). We couple the two inversion engines by imposing a smoothness condition on velocity and density and thickness of the layer while keeping the impedance and traveltime constant. We thus exploit the ambiguity of the solution to introduce geological criteria and reduce the noise contribution. We validated the proposed method with synthetic and real data.

Some Remarks on Smooth Mappings of Hilbert and Banach Spaces and Their Local Convexity Property

We analyze smooth nonlinear mappings for Hilbert and Banach spaces that carry small balls to convex sets, provided that the radii of the balls are small enough. We focus on the study of new and mildly sufficient conditions for the nonlinear mapping of Hilbert and Banach spaces to be locally convex, and address a suitably reformulated local convexity problem analyzed within the Leray–Schauder homotopy method approach for Hilbert spaces, and within the Lipschitz smoothness condition for both Hilbert and Banach spaces. Some of the results presented in this work prove to be interesting and novel, even for finite-dimensional problems. Open problems related to the local convexity property for nonlinear mappings of Banach spaces are also formulated.

Trust Region Methods for Nonconvex Stochastic Optimization beyond Lipschitz Smoothness

In many important machine learning applications, the standard assumption of having a globally Lipschitz continuous gradient may fail to hold. This paper delves into a more general (L0, L1)-smoothness setting, which gains particular significance within the realms of deep neural networks and distributionally robust optimization (DRO). We demonstrate the significant advantage of trust region methods for stochastic nonconvex optimization under such generalized smoothness assumption. We show that first-order trust region methods can recover the normalized and clipped stochastic gradient as special cases and then provide a unified analysis to show their convergence to first-order stationary conditions. Motivated by the important application of DRO, we propose a generalized high-order smoothness condition, under which second-order trust region methods can achieve a complexity of O(epsilon(-3.5)) for convergence to second-order stationary points. By incorporating variance reduction, the second-order trust region method obtains an even better complexity of O(epsilon(-3)), matching the optimal bound for standard smooth optimization. To our best knowledge, this is the first work to show convergence beyond the first-order stationary condition for generalized smooth optimization. Preliminary experiments show that our proposed algorithms perform favorably compared with existing methods.

On regularized polynomial functional regression

This article offers a comprehensive treatment of polynomial functional regression, culminating in the establishment of a novel finite sample bound. This bound encompasses various aspects, including general smoothness conditions, capacity conditions, and regularization techniques. In doing so, it extends and generalizes several findings from the context of linear functional regression as well. We also provide numerical evidence that using higher order polynomial terms can lead to an improved performance.

ECMWF and UK Met Office Offshore Blowing Winds: Impact of Horizontal Resolution and Coastal Orography

AbstractWe analyze the performance of the European Centre for Medium‐Range Weather Forecasts (ECMWF) and UK Met Office (UKMO) meteorological models in predicting offshore blowing wind in coastal areas. Our attention is mainly on the Mediterranean coast, up to 200 km distance from the shore. We compare forecast neutral winds with Advanced Scatterometer measurements. The results indicate that the ECMWF forecasts systematically underestimate wind speed with respect to scatterometer data, while the UKMO model tends to overestimate. A cross‐analysis suggests that, better than fetch, model biases are a function of the model horizontal discretization, hence of the number of grid steps the wind runs over the sea. The steepness and roughness of the land orography before entering the sea, together with the related drag parameters, appear to have a strong role in determining the coastal offshore wind values. Surface drag over land tends to reduce the related wind speed, and it takes a few grid points to adjust to the smooth sea surface conditions. This is supported by a detailed study with a high resolution grid, but different orography resolutions. In these simulations, practically identical coastal wind speed distributions are found when the subgrid orography schemes are switched off. Vertical cross‐sections of potential temperature and wind in bora and mistral conditions illustrate the strong role of gravity waves, wind channeling and turbulent diffusion in coastal numerical weather prediction. Besides local wave modeling, this may be relevant for offshore wind farms, typically situated within 5–50 km from the coast.

Multiple localized waves to the (2+1)-dimensional shallow water waveequation on non-flat constant backgrounds and their applications

In this paper, a new general bilinear Bäcklund transformation and Lax pair for the (2+1)-dimensional shallow water wave equation are given in terms of the binary Bell polynomials. Based on this transformation along with introducing an arbitrary function, the multi-kink soliton, line breather, and multi-line rogue wave solutions on a non-flat constant background plane are derived. Further, we found that the dynamic pattern of line breather on the background of periodic line waves are similar to the two-periodic wave solutions obtained through a multi-dimensional Riemann theta function. Also, the generation mechanism and smooth conditions of the line rogue waves on the periodic line wave background are presented with long-wave limit method. Additionally, a family of new rational solutions, consisting of line rogue waves and line solitons, are derived, which have never been reported before. Furthermore, the present work can be directly applied to other nonlinear equations.

A detection problem with a monotone observation rate

We study a quickest detection problem where the observation rate of the underlying process can be increased at any time for higher precision, but at an observation cost that grows linearly in the observation rate. This leads to a problem of combined control-and-stopping with incomplete information, with a two-dimensional sufficient statistic comprised of the current observation rate together with the conditional probability that disorder has already happened. The problem is shown to have a semi-explicit solution, where for some parameter values it is too costly to exert control at all, whereas for other parameter values the optimal strategy is to increase the observation rate in such a way that the sufficient statistic reflects at a certain boundary until the optimal stopping time. In both cases we fully characterise the optimal strategy with the help of appropriate smooth fit conditions.

Stable gravastar with large surface redshift in Einstein's gravity with two scalar fields

We propose a class of models, in which stable gravastar with large surface redshift becomes a solution. In recent decades, gravastars have become a plausible substitute for black holes. Researchers have explored stable gravastar models in various alternative gravity theories, in addition to the conventional framework of general relativity. In this paper, we present a stellar model within the framework of Einstein's gravity with two scalar fields, in accordance with the conjecture proposed by Mazur and Mottola [Proc. Nat. Acad. Sci. 101 (2004), 9545-9550]. In the model, the two scalar fields do not propagate by imposing constraints in order to avoid ghosts. The gravastar comprises two distinct regions, namely: (a) the interior region and (b) the exterior region. We assume the interior region consists of the de Sitter spacetime, and the exterior region is the Schwarzschild one. The two regions are connected with each other by the shell region. On the shell, we assume that the metric is given by a polynomial function of the radial coordinate r. The function has six constants. These constants are fixed by the smooth junction conditions, i.e., the interior region with the interior layer of the shell and the exterior region with the exterior layer of the shell. From these boundary conditions, we are able to write the coefficients of the scalar fields in terms of the interior radius and exterior radius. To clarify the philosophy of this study, we also give two examples of spacetimes that asymptote as the de Sitter spacetime for small r and as the Schwarzschild spacetime for large r. Exploration is focused on the physical attribute of the shell region, specifically, its proper length. The gravastar model's stability has frequently been examined by analyzing the relationship between surface redshift and shell thickness, a comparison we also undertake with previous models. Especially, we show that there exists a stable gravastar with a large surface redshift prohibited by the instability in the previous works. Furthermore, by checking the effective equation of state parameters, we show that the gravastar geometry realized in this paper by using two scalar fields could be difficult to generate with ordinary fluid.

Notes on Liouville‐type theorems for the 3D compressible nematic liquid crystal equations

In this note, we investigate steady compressible nematic liquid crystal flows in . We establish Liouville‐type theorems for smooth solutions that fulfill specific conditions within Lorentz spaces.

How Many Digits are Needed?

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Construction of a new class of quadrilateral spline elements based on the scaled boundary coordinates

The scaled boundary finite element method (SBFEM) is a semi-analytical numerical approach based on the scaled boundary coordinates. Recently, the method which uses interpolations in both the circumferential and radial directions has also been developed based on the scaled boundary representation. However, the stress fields are always discontinuous within the S-elements by both the original SBFEM and the latter method. In this paper, a new class of quadrilateral elements with interior smoothness is constructed by extending the B-net method based on the area coordinates into the S-net method based on the scaled boundary coordinates. In the S-net method, the Bernstein bases are separately applied to discretize both circumferential and radial directions in each sub-triangle of an S-element (or quadrilateral element), and the smoothness conditions between adjacent triangles can be simply represented in the form of linear equations. Then, by using the S-net method, we obtain a new class of quadrilateral elements with interior smoothness and high-order completeness. The shape functions or interpolation bases can be obtained explicitly and determined by the same interpolation nodes of the S-element as the original SBFEM. Some numerical tests also show that the proposed elements can obtain good results.

Influence of randomly distributed wall roughness on the flow behaviors of liquid-liquid slug flow in a co-flowing microchannel

In present study, the influence of randomly distributed wall roughness on the flow behaviors of liquid-liquid slug flow in a co-flowing microchannel has been numerically investigated, as the manufacturing tolerance of microfluidic device in engineering application is unavoidable. A circular axisymmetric microchannel model is adopted to study the effect of roughness height and correlation length on the wall roughness generation and internal laminar flow characteristics in the microchannel. The wall roughness is modeled by well-known Gaussian distribution method. The prediction of slug velocity and liquid film thickness from numerical simulations has been well validated with the empirical and analytical correlations and experimental data in the literature. The obtained results show that the rough wall condition significantly affected the internal flow behaviors, such as slug moving velocity and slug breaking-off. According to the quantitative comparison of normalized surface areas between high (larger than 0.4 m/s) and low (smaller than 0.69 m/s) velocity region, the surface area of low velocity region under rough wall condition (wall roughness: D/l=3.333 and σ/D = 0.018) is 46 % larger than smooth wall condition, which means effective flow area of microchannel is reduced. The surface area of high velocity region under rough wall condition increases around 66 % compared with smooth wall condition. This study could be used as a promising reference and provide more understanding for roughness control in industrial microfluidic device.

Torus-like solutions for the Landau-de Gennes model. Part II: Topology of [formula omitted]-equivariant minimizers

We study energy minimization of a continuum Landau-de Gennes energy functional for nematic liquid crystals, in a three-dimensional axisymmetric domain and in a restricted class of S1-equivariant (i.e., axially symmetric) configurations. We assume smooth and nonvanishing S1-equivariant (e.g. homeotropic) Dirichlet boundary condition and a physically relevant norm constraint (the Lyuksyutov constraint) in the interior. Relying on our previous results in the nonsymmetric setting [16], we prove partial regularity of minimizers away from a possible finite set of interior singularities located on the symmetry axis. For a suitable class of domains and boundary data, we show that for smooth minimizers (torus solutions) the level sets of the signed biaxiality are generically finite unions of tori of revolution. Concerning nonsmooth minimizers (split solutions), we characterize their asymptotic behavior around any singular point in terms of explicit S1-equivariant harmonic maps into S4, whence the generic level sets of the signed biaxiality contains invariant topological spheres. Finally, in the model case of a nematic droplet, we provide existence of torus solutions, at least when the boundary data are suitable uniaxial deformations of the radial anchoring, and existence of split solutions for boundary data which are suitable linearly full harmonic spheres.

Analysis of frictionless contact between a spherical indenter and a poroviscoelastic medium with a constant contact radius

The research aims to understand the force relaxation that occurs when a spherical indenter is pressed into poroviscoelastic materials, which exhibit viscoelastic properties of solid network and drainage behavior of infiltrated fluid. To achieve this, an analytical approach is proposed to model spherical indentation on a poroviscoelastic medium under the condition of a constant contact radius. This analysis is conducted in the Laplace-transformed domain, and provides closed-form expressions in the transformed domain for both the normal approach and the contact force of the indenter while maintaining a fixed contact radius. These expressions are then numerically inverse-transformed to the time domain for practical analysis. To account for different scenarios, we consider various combinations of drainage conditions at the top and smoothness conditions at the bottom of the medium. We examine three categories of viscoelastic solid behavior, as understanding how they respond under indentation is crucial for characterizing their mechanical behavior. Additionally, we conduct finite element simulations of spherical indentation on poroviscoelastic media, serving as a comparison to the semi-analytic results obtained. Notably, the thickness of the medium relative to the contact radius and the Poisson's ratio play significant roles in the evolution of both the normal approach and contact force of the spherical indenter over time.

Piecewise approximate analytical solutions of high-order reaction-diffusion singular perturbation problems with boundary and interior layers

<abstract> <p>This work aims to present a reliable algorithm that can effectively generate accurate piecewise approximate analytical solutions for third- and fourth-order reaction-diffusion singular perturbation problems. These problems involve a discontinuous source term and exhibit both interior and boundary layers. The original problem was transformed into a system of coupled differential equations that are weakly interconnected. A zero-order asymptotic approximate solution was then provided, with known asymptotic analytical solutions for the boundary and interior layers, while the outer region solution was obtained analytically using an enhanced residual power series approach. This approach combined the standard residual power series method with the Padé approximation to yield a piecewise approximate analytical solution. It satisfies the continuity and smoothness conditions and offers higher accuracy than the standard residual power series method and other numerical methods like finite difference, finite element, hybrid difference scheme, and Schwarz method. The algorithm also provides error estimates, and numerical examples are included to demonstrate the high accuracy, low computational cost, and effectiveness of the method within a new asymptotic semi-analytical numerical framewor.</p> </abstract>

Generalized composition operators on weighted Fock spaces

The generalized composition operators $J_g^{\varPhi}$ and $C_g^{\varPhi}$, induced by analytic functions $g$ and $\varPhi$ on the complex plane $\mathbb{C}$, are defined by \begin{align*} J_g^{\varPhi}(f)(z)=\int\limits_0^zf'(\varPhi(\omega))g(\omega)d\omega and C_g^{\varPhi}(f)(z)=\int\limits_0^{\varPhi(z)}f'(\omega)g(\omega)d\omega. \end{align*} In this paper, we consider these operators on weighted Fock spaces $\mathcal{F}_p^{\varPsi}$, consisting of entire functions, which are $\mathcal{L}^p(\mathbb{C})$-integrable with respect to the measure $d\lambda(z)=e^{-\varPsi (z)}d\Lambda(z)$, where $d\Lambda$ is the usual Lebesgue area measure in $\mathbb{C}$. We assume that the weight function $\varPsi$ in the spaces satisfies certain smoothness conditions, in particular, this weight function grows faster than the Gaussian weight $\frac{|z|^2}{2}$ defining the classical Fock spaces. We first consider bounded and compact properties of $J_{g}^ {\varPhi}$ and $C_{g}^{\varPhi}$, and characterize these properties in terms function theory of inducing functions $g$ and $\varPhi$, given by \begin{align*} \mathcal{M}_g^{\varPhi}(z):=\frac{|g(z)|\varPsi'(\varPhi(z))}{1+\varPsi'(z)}e^{\varPsi(\varPhi(z))-\varPsi(z)}. \end{align*} Our characterization is simpler to use than the Berezin type integral transform characterization. In some cases, our result shows that these operators experience poorer boundedness and compactness structures when acting between such spaces than the classical Fock spaces. For instance, for $\varPhi(z)=z$, there is no nontrivial bounded $J_g^{\varPhi}$ and $C_g^{\varPhi}$ on weighted Fock spaces. In the case of classical Fock spaces, they are bounded if and only if $g$ is constant. In the second part of this paper, we apply our simpler characterization of boundedness and compactness to further study the Schatten-class membership of these operators. In particular, we express the Schatten $S_p(\mathcal{F}_2^{\varPsi})$ class membership property in terms of $\mathcal{L}^p(\mathbb{C}, \Delta \varPsi d\Lambda)$-integrability of $\mathcal{M}_g^{\varPhi}$.

АППРОКСИМАЦИЯ СОПРЯЖЕНИЯ КРИВЫХ БЕЗЬЕ С СОХРАНЕНИЕМ ПОРЯДКА ГЛАДКОСТИ И ДОПОЛНИТЕЛЬНЫМИ ОГРАНИЧЕНИЯМИ

Bezier curves are a mandatory component of the geometric core of modern computer-aided design (CAD). The article proposes a mathematical approach that makes it possible to approximate the conjugation (connection) of Bezier curves of arbitrary degree, so that at the conjugation point the conditions of smoothness (continuity) are satisfied up to the order of an equal degree of the given Bezier curves. This approach helps to represent conjugate curves of a single Bezier curve with the degree equal to the degrees of the given curves. Additional restrictions can be imposed on the conjugate curves and the approximating curve in the form of a complete coincidence with one of the given curves, or the passage of the approximating curve through a given point and the equality of the derivatives to the given values at this point. To solve these problems, we introduced two difference metrics between the given curves and the approximating curve, and formulated optimization problems with constraints in the form of equalities. We applied the method of Lagrange multipliers which solves the corresponding system of linear algebraic equations. To represent Bezier curves, it is proposed to use the basic functions of B-splines, which allows you to use the software functions included in the geometric core of modern CAD systems. This greatly simplifies the derivation of all degree derivatives for curves and, without significant changes in the future, will make it possible to extend the results to conjugation problems of B-splines. The paper provides examples of approximations using various metrics and their limitations.

A survey on unsharp orthomodular lattices: A unifying framework

In this survey we consider the variety of unsharp orthomodular lattices. We will see that there are pretty smooth conditions that neatly generalize a great deal of the theory of orthomodular lattices. In particular, we will characterize the concept of block in this framework towards an appropriate notion of commutativity. Then, we capitalize on this fact and describe a categorical equivalence between orthomodular lattices, with fixed p-filters, and the variety of unsharp orthomodular lattices.

Quantized Zeroth-Order Gradient Tracking Algorithm for Distributed Nonconvex Optimization Under Polyak-Łojasiewicz Condition.

This article focuses on distributed nonconvex optimization by exchanging information between agents to minimize the average of local nonconvex cost functions. The communication channel between agents is normally constrained by limited bandwidth, and the gradient information is typically unavailable. To overcome these limitations, we propose a quantized distributed zeroth-order algorithm, which integrates the deterministic gradient estimator, the standard uniform quantizer, and the distributed gradient tracking algorithm. We establish linear convergence to a global optimal point for the proposed algorithm by assuming Polyak-Łojasiewicz condition for the global cost function and smoothness condition for the local cost functions. Moreover, the proposed algorithm maintains linear convergence at low-data rates with a proper selection of algorithm parameters. Numerical simulations validate the theoretical results.