Approximation Theory Research Articles

Approximations of semiring-valued fuzzy sets with applications in new fuzzy structures

Many of the new types of fuzzy sets, such as intuitionistic, neutrosophic or fuzzy soft sets, can be transformed into one common type of fuzzy sets, called (R,R⁎)-fuzzy sets, with values in a set R that is a common underlying set of complete commutative idempotent semirings R and R⁎. For (R,R⁎)-fuzzy sets, the theory of lower and upper approximations by (R,R⁎)-relations is defined and properties of these approximations are investigated. The transformation of new types of fuzzy sets into (R,R⁎)-fuzzy sets is used to show the possibility of introducing approximation operators in new types of fuzzy sets without having to define these operators separately for each new type of fuzzy set.

Lanczos method for spatio‐temporal graph convolutional networks to forecast expressway flow

AbstractTraffic forecasting has made pronounced progress with the development of graph convolution networks and the use of the topology of road networks. However, existing works face some limitations when it comes to modelling spatial dependencies. For example, pre‐defined graphs rely on global information to establish spatial relationships, and the spatial receptive field is limited by the polynomial convolutional method. To address these limitations, the authors propose the Lanczos method for Spatio‐Temporal Graph Convolutional Networks (LSTGCN); this approach uses the low‐rank approximation theory to drive pre‐defined graphs to collect important information and eliminate spatial redundancy. Additionally, a learnable dynamic graph feature (LDGF) module generates adaptive graphs and perceives the latent invariance‐variability between graph nodes. To further improve the model's ability to capture spatial dependencies and temporal correlations, multi‐span spatial learning is employed for enlarged receptive fields, which can be well integrated into gated recurrent units. The authors conducted baselines comparison and ablation experiments on real‐world datasets, and the findings show that the LSTGCN model outperforms the baselines and improves prediction accuracy. Notably, this work is the first attempt to use graph low‐rank theory for traffic prediction.

Bi-Kappa Proton Mirror and Cyclotron Instabilities in the Solar Wind

The charged particles in the solar wind are often observed to possess a nonthermal tail in the velocity distribution function, a feature that can be fitted with the Kappa model. The anisotropic, or bi-Kappa, model of protons, electrons, and other charged particles is thus adopted in the literature for interpreting the data as well as in the context of the analysis of wave–particle interactions. The present paper develops an approximate but efficient theory of the mirror and cyclotron instabilities excited by the bi-Kappa protons in the solar wind. A velocity moment-based quasi-linear theory of these instabilities is also formulated in order to investigate the saturation behavior. Applications of the formalism are made for instabilities close to the marginally unstable state, which is typical of the solar wind near 1 au.

Strict $\mathcal{C}^p$-triangulations – a new approach to desingularization

Let R be any real closed field expanded by some o-minimal structure. Let f: A\to R^d be a definable and continuous mapping defined on a definable, closed, bounded subset A of R^n . Let \mathcal{E} be a finite family of definable subsets of R^n contained in A . Let p be any positive integer. We prove that then there exists a finite simplicial complex \mathcal{T} in R^n and a definable homeomorphism h: |\mathcal{T} |\to A , where |\mathcal{T}|:= \bigcup \mathcal{T} , such that for each simplex \varDelta\in \mathcal{T} , the restriction of h to its relative interior \mathring{\varDelta} is a \mathcal{C}^p -embedding of \mathring{\varDelta} into R^n and moreover both h and f\circ h are of class \mathcal{C}^p in the sense that they have definable \mathcal{C}^p -extensions defined on an open definable neighborhood of | \mathcal{T}| in R^n . We then call a pair ( \mathcal{T}, h) a strict \mathcal{C}^p -triangulation of A . In addition, this triangulation can be made compatible with \mathcal{E} in the sense that for each E\in \mathcal{E} , h^{-1}(E) is a union of some \mathring{\varDelta} , where \varDelta\in \mathcal{T} . We also give an application to approximation theory.

Democracy of quasi-greedy bases in -Banach spaces with applications to the efficiency of the Thresholding Greedy Algorithm in the Hardy spaces

We use new methods, specific for non-locally convex quasi-Banach spaces, to investigate when the quasi-greedy bases of a $p$-Banach space for $0< p<1$ are democratic. The novel techniques we obtain permit to show in particular that all quasi-greedy bases of the Hardy space $H_p({\mathbb {D}})$ for $0< p<1$ are democratic while, in contrast, no quasi-greedy basis of $H_p({\mathbb {D}}^d)$ for $d\ge 2$ is, solving thus a problem that was raised in [7]. Applications of our results to other spaces of interest both in functional analysis and approximation theory are also provided.

Another Torture Track for Quantum Chemistry: Reinvestigation of the Benzaldehyde Amidation by Nitrogen‐Atom Transfer from Platinum(II) and Palladium(II) Metallonitrenes

AbstractWe showcase here a dramatic failure of CCSD(T) theory that originates from the pronounced multi‐reference character of a key intermediate formed in the benzaldehyde amidation by N‐atom transfer from Pd(II) and Pt(II) metallonitrenes studied recently in combined experimental and theoretical work. For detailed analysis we devised a minimal model system, for which we established reliable reference energies based on approximate full configuration interaction theory, to assess the performance of single‐reference coupled‐cluster theory up to the CCSDTQ(P) excitation level. While RHF‐based CCSD(T) theory suffered dramatic errors, in one case exceeding 220 kcal mol−1, we show that the use of broken‐symmetry (BS) or Kohn‐Sham (KS) orbital references yields substantially improved CCSD(T) results. Further, the EOM‐SF‐CCSD(T)(a)* approach met the reference data with excellent accuracy. We applied the KS‐CCSD(T*)‐F12b variant as high‐level part of an ONIOM(KS‐CC:DFT) scheme to reinvestigate the reactivity of the full Pt(II) and Pd(II) metallonitrenes. The revised reaction pathway energetics provide a detailed mechanistic rationale for the experimental observations.

Construction of fractional granular model and bright, dark, lump, breather types soliton solutions using Hirota bilinear method

The present article designs the granular metamaterials considering the granular structures of discrete particles which are different from elastic metamaterials consisting of continuous media. In granular metamaterials, the wave propagates through contact with neighboring particles. To identify the propagating properties of wave quantities in the rough granular medium the fractional granular equation is formulated directly in a pre-compressed spherical chain adopting Hertz law and long wave approximation theory. Using phase and group velocities, Caputo fractional derivatives are used to illustrate normal and anomalous dispersion wave dependence. To demonstrate in depth the dynamical behavior of the wave profile, various types of complex solutions like multi-shock, multi-solitons, lump, and breather solutions of the one-dimensional time fractional granular equations are explored employing Hirota’s bilinear approach. Finally, the more complicated hybrid solutions such as kink with the lump, soliton with the lump, etc. are exhibited from numerical understanding. The numerical graphs and figures demonstrate the crucial role of the order of derivative (roughness parameter) in the formation of different types of soliton solutions.

The Fuzzy Width Theory in the Finite-Dimensional Space and Sobolev Space

This paper aims to fuzzify the width problem of classical approximation theory. New concepts of fuzzy Kolmogorov n-width and fuzzy linear n-width are introduced on the basis of α-fuzzy distance which is induced by the fuzzy norm. Furthermore, the relationship between the classical widths in linear normed space and the fuzzy widths in fuzzy linear normed space is discussed. Finally, the exact asymptotic orders of the fuzzy Kolmogorov n-width and fuzzy linear n-width corresponding to a given fuzzy norm in finite-dimensional space and Sobolev space are estimated.

Generalization Analysis of Pairwise Learning for Ranking with Deep Neural Networks.

Pairwise learning is widely employed in ranking, similarity and metric learning, area under the curb maximization, and many other learning tasks involving sample pairs. Pairwise learning with deep neural networks was considered for ranking, but enough theoretical understanding about this topic is lacking. In this letter, we apply symmetric deep neural networks to pairwise learning for ranking with a hinge loss ϕh and carry out generalization analysis for this algorithm. A key step in our analysis is to characterize a function that minimizes the risk. This motivates us to first find the minimizer of ϕh-risk and then design our two-part deep neural networks with shared weights, which induces the antisymmetric property of the networks. We present convergence rates of the approximation error in terms of function smoothness and a noise condition and give an excess generalization error bound by means of properties of the hypothesis space generated by deep neural networks. Our analysis is based on tools from U-statistics and approximation theory.

Towards the theory of how a constitutional supercooling layer appears ahead of the planar crystallization front

This study, the effect of constitutional supercooling appearing ahead of the crystallization front and leading to the mushy layer origination is considered. An approximate analytical theory determining the time of mushy layer initiation is constructed. Theoretical predictions are in good agreement with numerical simulations carried out in previous studies.

Connectedness in asymmetric spaces

Inverse theorems for geometric approximation theory related to the structure of approximating sets are obtained. Various types of connectedness of sets in asymmetric spaces (generally nonmetrizable) are studied. In particular, conditions are established for an asymmetric space and a subset of this space that guarantee that the intersection of an open ball with this set is path-connected.

Failure Mode and Effects Analysis on the Air System of an Aero Turbofan Engine Using the Gaussian Model and Evidence Theory.

Failure mode and effects analysis (FMEA) is a proactive risk management approach. Risk management under uncertainty with the FMEA method has attracted a lot of attention. The Dempster-Shafer (D-S) evidence theory is a popular approximate reasoning theory for addressing uncertain information and it can be adopted in FMEA for uncertain information processing because of its flexibility and superiority in coping with uncertain and subjective assessments. The assessments coming from FMEA experts may include highly conflicting evidence for information fusion in the framework of D-S evidence theory. Therefore, in this paper, we propose an improved FMEA method based on the Gaussian model and D-S evidence theory to handle the subjective assessments of FMEA experts and apply it to deal with FMEA in the air system of an aero turbofan engine. First, we define three kinds of generalized scaling by Gaussian distribution characteristics to deal with potential highly conflicting evidence in the assessments. Then, we fuse expert assessments with the Dempster combination rule. Finally, we obtain the risk priority number to rank the risk level of the FMEA items. The experimental results show that the method is effective and reasonable in dealing with risk analysis in the air system of an aero turbofan engine.

Hydrodynamic approach to many-body systems: Exact conservation laws

In this paper I present a pedagogical derivation of continuity equations manifesting exact conservation laws in an interacting electronic system based on the nonequilibrium Keldysh technique. The purpose of this exercise is to lay the groundwork for extending the hydrodynamic approach to electronic transport to strongly correlated systems where the quasiparticle approximation and Boltzmann kinetic theory fail.

Jain's operator: A new construction and applications in approximation theory

The new and more comprehensive Jain‐type operators based on a function and two sequences of functions and have been introduced. This newly defined operator has an important place in which these three functions can create both existing and new operators with special selections. In order to show their approximation properties, we present direct approximation results and Voronovkskaya‐type results for these operators. Finally, we provide a series of numerical examples to show their performance visually.

Electroosmosis oriented flow of Jeffrey viscoelastic model through scraped surface heat exchanger

Scraped surface heat exchanger (SSHE) is vibrantly utilized in various industries. The heat exchanger devices play pivotal role in the energy transfer phenomenon between the fluids, and it improves the efficient use of energy. The steady isothermal flow of a rheological aqueous ionic solution in a narrow gap scraped surface heat exchanger is explored. The mathematical model is established with the aid of lubrication approximation theory and extracted analytical solution. Exact analytical expressions are gathered for velocities, volume fluxes, pressure gradients, pressure at the edges of the blades and stream functions in the various stations of SSHE. Strength of significant flow parameters on the velocities, volume fluxes, pressure gradients, pressure at the edges of the blades and stream functions is revealed with aid of various plots. Significant enhancement in the velocity profile is observed in main domain of the flow with increasing ratio of relaxation to retardation time parameter and Helmholtz Smoluchowski velocity. While, it is observed that electro-osmotic parameter creates impediments to the flow.

Structural properties and optical modeling of thermally annealed silicon carbide thin films

Non–hydrogenated amorphous silicon carbide films were processed by electron beam-physical vapour deposition on c-Si (100) and glass substrates. The films were isochronally annealed at varying temperatures. FTIR and Raman spectroscopies as well as XRD studies were used for structural investigation. For the pristine sample, it is found that the film is Si rich in an amorphous SiC matrix and that the observed graphitization is exclusively described by a single broad D-band in Raman. During the annealing process, a start of rearrangement of the network is observed at low temperatures of annealing but the amorphous phase still dominates in the microstructure. Still at low temperatures, the graphitization band broadens and expands to the G-bands region while at higher temperatures, an improved stochiometric crystalline SiC phase is achieved and distinct graphitic D and G bands are resolved. The induced crystalline network through annealing is prone to oxidation both in Si-O, bridged oxygen Si-O-Si and/or to silicon oxycarbide SiCx -O groups. An optical model based on effective medium approximation (EMA) theory is proposed in order to extract the optical properties of the material. It is established, from the evolution of the bandgap, that the films become transparent over a wider region of the visible spectrum as the temperature of annealing is increased to high values.

Best Approximation of Unbounded Functions by Modulus of Smoothness

In this paper, we study the approximation of unbounded functions in a weighted space by modulus of smoothness using various linear operators. We establish direct theorems for such approximations and analyze the properties of the modulus of smoothness within the same space. Specifically, we investigate the behavior of the modulus of smoothness under different types of linear operators, including the Bernstein-Durrmeyer operator, the Fejer operator, and the Jackson operator. We also provide a detailed analysis of the convergence rate of these operators. Furthermore, we discuss the relationship between the modulus of smoothness and the Lipschitz constant of a function. Our findings have important implications for the field of approximation theory and may help to inform future research in this area.

A manifold two-sample test study: integral probability metric with neural networks

Abstract Two-sample tests are important areas aiming to determine whether two collections of observations follow the same distribution or not. We propose two-sample tests based on integral probability metric (IPM) for high-dimensional samples supported on a low-dimensional manifold. We characterize the properties of proposed tests with respect to the number of samples $n$ and the structure of the manifold with intrinsic dimension $d$. When an atlas is given, we propose a two-step test to identify the difference between general distributions, which achieves the type-II risk in the order of $n^{-1/\max \{d,2\}}$. When an atlas is not given, we propose Hölder IPM test that applies for data distributions with $(s,\beta )$-Hölder densities, which achieves the type-II risk in the order of $n^{-(s+\beta )/d}$. To mitigate the heavy computation burden of evaluating the Hölder IPM, we approximate the Hölder function class using neural networks. Based on the approximation theory of neural networks, we show that the neural network IPM test has the type-II risk in the order of $n^{-(s+\beta )/d}$, which is in the same order of the type-II risk as the Hölder IPM test. Our proposed tests are adaptive to low-dimensional geometric structure because their performance crucially depends on the intrinsic dimension instead of the data dimension.

Heat transfer analysis of the non-Newtonian polymer in the calendering process with slip effects

In this paper, a non-isothermal study of the calendering processes is presented using Carreau–Yasuda model along with nonlinear slip condition introduced at the upper roll surface. The flow equations for the problem are developed and converted into dimensionless form with the help of dimensionless variables and then finally simplified by a well-known lubrication approximation theory. The final equations are solved numerically using “bvp4c” to find stream function and velocity profiles, while the hybrid numerical method which is the combination of shooting and finite difference methods is used to solve the energy equation. Graphs show the impact of the concerned material parameters on various quantities of interest. The pressure distribution decreases with the increasing values of the slip parameter and Weissenberg number. The mechanical variables show an increasing trend with the increasing values of the slip parameter and Weissenberg number. The temperature distribution increases with an increase in the Brinkman number, while temperature shows declining trend near the roll surface with the increasing values of the slip parameter. The force separating the two rollers, total power input into both rolls, increase with the increasing values of the Weissenberg number and slip parameters. The results show that the Newtonian model predicts higher pressure in the nip zone than the Carreau–Yasuda model. It is interesting to note that for the case of shear thinning, the Carreau–Yasuda model predicts 30% less pressure in the nip region when compared to the Newtonian model.

Fast Linear Canonical Transform for Nonequispaced Data

The investigations of the discrete and fast linear canonical transform (LCT) are becoming one of the hottest research topics in modern signal processing and optics. Among them, the fast calculation of LCT for non-uniform data is one of key problems. Focus on this problem, a new fast algorithm of the LCT has been proposed in this paper firstly by interpolation and approximation theory. The proposed algorithms can calculate quickly the LCT of the data, whether the input or output data is uniform. Secondly, the complexity and precision of derived algorithms have been analyzed for different situations. Finally, the experimental results are presented to verify the correctness of the obtained results.