Orthonormal Basis Research Articles

An $ {\varepsilon} $-approximate solution of BVPs based on improved multiscale orthonormal basis

<abstract><p>In the present paper, we construct a set of multiscale orthonormal basis based on Legendre polynomials. Using this orthonormal basis, a new algorithm is designed for solving the second-order boundary value problems. This algorithm is to find numerical solution by seeking $ {\varepsilon} $-approximate solution. Moreover, we prove that the order of convergence depends on the boundedness of $ u^{(m)}(x) $. In addition, third numerical examples are provided to validate the efciency and accuracy of the proposed method. Numerical results reveal that the present method yields extremely accurate approximation to the exact solution. Meanwhile, compared with the other algorithms, the results obtained demonstrate that our algorithm is remarkably effective and convenient.</p></abstract>

An $ {\varepsilon} $-approximation solution of time-fractional diffusion equations based on Legendre polynomials

An $ {\varepsilon} $-approximation solution of time-fractional diffusion equations based on Legendre polynomials

Bayes synthesis of multidimensional stochastic system with high availability by wavelet canonical expansions

Paper in devoted to methodological and instrumental software synthesis support for multidimensional stochastic systems with high availability (StSHA). For modeling of nonstationary stochastic processes (StP) it is supposed to implement wavelet canonical expansions (WLCE), based on coefficients of canonical expansions of covariance matrix using two dimensional orthonormal wavelet basis with compact carrier. Vector StSHA input and output are nonlinear on vector of random parameters with known probability density and Gaussian (normal) noise. Noises do not depend upon random parameters. Functional Bayes criterion (FBC) is presented by linear functional. Optimal system is synthesized using conditional risk criterion. General StSHA synthesis method and algorithm contains 4 steps and 7 positions. FBC optimal estimate is calculated by WLCE method. Special software is developed for scalar nonlinear StSHA. Results of computer experiments are given and discussed. His shown that high efficiency of method at 8 WLCE items. Directions of future investigations are presented.

New spectrally null constrained mutually orthogonal complementary sets and Z-complementary code sets

New spectrally null constrained mutually orthogonal complementary sets and Z-complementary code sets

Fermatean Fuzzy TOPSIS Method Based on Prospect Theory and Its Application in Credit Assessment

Intuitionistic fuzzy sets, Pythagorean fuzzy sets and generalized orthogonal fuzzy sets are widely used in some multi-attribute decision-making problems. The Simple Fermatean fuzzy TOPSIS method addresses that decision makers are completely rational while making decisions, and applies the closeness index to the measurement to rank the alternatives. Prospect theory considers the irrational behavior of the decision maker in the actual decision-making process, decision makers facing the loss are risk-preferring (introducing the coefficient θ), and are risk averse when facing the gain. In this paper, prospect theory and Fermatean fuzzy sets are combined to explore the multi-attribute decision-making problem, and the Fermatean fuzzy TOPSIS method based on prospect theory is proposed. The comprehensive gain/loss ratio is calculated to rank the alternatives. In this paper, eight leading enterprises in the concept of emerging industries are used as examples to illustrate the effectiveness of the method. The results show that the optimal choice is Ganneng Co in the practical example, and the discriminant mean of the absolute value of φ+(Si)/φ-(Si) obtained by pooling expert fuzzy evaluations using the Fermatean fuzzy TOPSIS method based on prospect theory is greater than the discriminant mean of the closeness index obtained by using the Fermatean fuzzy TOPSIS method, which can be interpreted as the former being a better synthesis of the expert fuzzy evaluations. In the future, the Fermatean fuzzy TOPSIS method based on prospect theory has a wide range of application value in several fields, which can be used to solve complex decision-making problems and improve the accuracy and effectiveness of decision-making.

H-Toeplitz operators on the Dirichlet type space

<abstract><p>In this paper, we conducted a study of H-Toeplitz operators on the Dirichlet type space $ \mathfrak{D}_{t} $, which included several aspects. To begin, we established the matrix representation of the H-Toeplitz operator $ S_{\varphi} $ with respect to the orthonormal basis of $ \mathfrak{D}_t $. Subsequently, we characterized the compactness of $ S_{\varphi} $ in terms of the symbol $ \varphi $. Furthermore, we developed a new method to investigate the algebraic properties of H-Toeplitz operators, including self-adjointness, diagonality, co-isometry, partial isometry as well as commutativity.</p></abstract>

A New Method to Compute Orthonormal Basis in Three-Dimensional Euclidean Spaces

A New Method to Compute Orthonormal Basis in Three-Dimensional Euclidean Spaces

ПОСТРОЕНИЕ УПРУГОГО СОСТОЯНИЯ ТЕЛА В УСЛОВИЯХ ВТОРОЙ ОСНОВНОЙ ЗАДАЧИ ТЕОРИИ УПРУГОСТИ

The paper presents a mathematical model for constructing the stress-strain state of an anisotropic cylindrical body of a finite length with kinematic conditions specified on the body surface and the conditions being non-axisymmetric. The cylinder material is characterized by rectilinear transversal isotropy. The model is constructed on the basis of the boundary state method. The bases of the internal and boundary state spaces are formed and orthogonalized after which the desired elastic state is expanded in a Fourier series according to the elements of the orthonormal basis, the problem being to calculate the coefficients of the constructed series. A solution to the second fundamental problem of elasticity for a circular cylinder made of a hypothetical anisotropic large dark gray siltstone is presented. The accuracy of the solution is assessed and a graphical visualization of the result is provided.

Regularized Finite Impulse Response Models versus Laguerre Models: A Comparison

While in recent years, the estimation of finite impulse response (FIR) models has been improved by introducing new regularization schemes, also other orthonormal basis function (OBF) models are now becoming more prominent again. Although Laguerre models show very similar properties to the regularized FIR models, they are only rarely used for system identification. Therefore, in this paper, regularized FIR models and Laguerre models will be compared. This work focuses on the model structure and its similarities and differences, as well as the hyperparameter optimization utilizing the generalized cross-validation (GCV) error. Finally, the two model types are investigated using three different processes and the model performance is evaluated. Both model types show significant improvements compared to standard (unregularized) FIR models.

Generalized performance criteria for identified models

It is shown that some usual criteria evaluating the performances of an identified model with respect to experimental data, like the FIT criterion, can be not well-suited to fast sampled systems. This leads us to propose some generalized criteria where the signals are filtered by transfer functions belonging to an orthonormal basis. An interpretation of this filtering in the frequency domain is proposed. The basis poles selection is equivalent to making a specification about the criteria in function of the expected use of the identified model.

Vehicle Exhausts Emission Pattern Decisions for Logistic Services and Packing Industries with Orthogonal Array-Based Rough Set Theory

Precise monitoring of vehicle emissions in green logistics, focusing on the contributions of vehicles from packing industries, is crucial for many issues. It helps to understand the total emissions and gain insights into the mechanism of vehicle-associated environmental concerns. Notwithstanding, a key issue when monitoring vehicle emissions is the effective discrimination problem for different patterns generated from the parameters. Data from the packing industry are available from distribution networks but its pattern cannot be discriminated. Given this background, this article presents a new method of the orthogonal array-based rough set to discern patterns of the parametric behaviors to monitor emissions from vehicle exhausts in the packing industry. The proposed method is based on an Indian logistics network and delivery system data, which was obtained from previous work in the literature. By setting controls on the parameters of the packing industry which includes revenue obtained, packing units sold, growth rate, carbon-dioxide equivalent, materials utilized, and quantity consumed, the method was able to discern the patterns of the parametric behavior. The orthogonal arrays, which are developed, form factors (parameters) and levels to ascertain a balanced and uniform analysis of the various groups of options. Indiscernibility and approximation concepts of fuzzy sets are then applied to arrive at the outcome. Unlike previous studies, this study eliminates the need for tracking data, assumptions, and external information to establish the set membership. However, it utilizes the available information within the data. The rough set analysis indicates that there are no discernable patterns or rules that distinguish between "Yes" and "No" decisions. The method of rough set illustrated in this work shows the feasibility of the approach in the Indian packing industry. The method is useful for the logistics manager and government agencies responsible for the control of vehicle-generated greenhouse emissions.

Tree structure of spectra of spectral Moran measures with consecutive digits

Abstract Let $\{b_n\}_{n=1}^{\infty }$ be a sequence of integers larger than 1. We will study the harmonic analysis of the equal-weighted Moran measures $\mu _{\{b_n\},\{{\mathcal D}_n\}}$ with ${\mathcal D}_n=\{0,1,2,\ldots ,q_n-1\}$ , where $q_n$ divides $b_n$ for all $n\geq 1.$ In this paper, we first characterize all the maximal orthogonal sets of $L^2(\mu _{\{b_n\},\{{\mathcal D}_n\}})$ via a tree mapping. By this characterization, we give some sufficient conditions for the maximal orthogonal set to be an orthonormal basis.

On bounded coordinates in bimodules

We provide a comprehensive study on uniformly left [Formula: see text]-bounded (respectively, [Formula: see text]-bounded) orthonormal bases in infinite-dimensional cyclic bimodules associated with c.p. maps between two von Neumann algebras [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are faithful normal states on [Formula: see text] and [Formula: see text], respectively. Separate investigations on cyclic bimodules associated with Markov maps and arbitrary c.p. maps are also provided since the results differ. This generalizes the results of the authors’ previous work on Kadison’s unitary basis problem.

Data Driven Approach to Determine Linear Stability of Delay Differential Equations Using Orthonormal History Functions

Abstract Delay differential equations (DDEs) appear in many applications, and determining their stability is a challenging task that has received considerable attention. Numerous methods for stability determination of a given DDE exist in the literature. However, in practical scenarios it may be beneficial to be able to determine the stability of a delayed system based solely on its response to given inputs, without the need to consider the underlying governing DDE. In this work, we propose such a data-driven method, assuming only three things about the underlying DDE: (i) it is linear, (ii) its coefficients are either constant or time-periodic with a known fundamental period, and (iii) the largest delay is known. Our approach involves giving the first few functions of an orthonormal polynomial basis as input, and measuring/computing the corresponding responses to generate a state transition matrix M, whose largest eigenvalue determines the stability. We demonstrate the correctness, efficacy and convergence of our method by studying four candidate DDEs with differing features. We show that our approach is robust to noise, thereby establishing its suitability for practical applications, wherein measurement errors are unavoidable.

Modeling Infectious Disease Trend using Sobolev Polynomials

Trend analysis plays an important role in infectious disease control. An analysis of the underlying trend in the number of cases or the mortality of a particular disease allows one to characterize its growth. Trend analysis may also be used to evaluate the effectiveness of an intervention to control the spread of an infectious disease. However, trends are often not readily observable because of noise in data that is commonly caused by random factors, short-term repeated patterns, or measurement error. In this paper, a smoothing technique that generalizes the Whittaker-Henderson method to infinite dimension and whose solution is represented by a polynomial is applied to extract the underlying trend in infectious disease data. The solution is obtained by projecting the problem to a finite-dimensional space using an orthonormal Sobolev polynomial basis obtained from Gram-Schmidt orthogonalization procedure and a smoothing parameter computed using the Philippine Eagle Optimization Algorithm, which is more efficient and consistent than a hybrid model used in earlier work. Because the trend is represented by the polynomial solution, extreme points, concavity, and periods when infectious disease cases are increasing or decreasing can be easily determined. Moreover, one can easily generate forecast of cases using the polynomial solution. This approach is applied in the analysis of trends, and in forecasting cases of different infectious diseases.

Operator growth and Krylov complexity in Bose-Hubbard model

We study Krylov complexity of a one-dimensional Bosonic system, the celebrated Bose-Hubbard Model. The Bose-Hubbard Hamiltonian consists of interacting bosons on a lattice, describing ultra-cold atoms. Apart from showing superfluid-Mott insulator phase transition, the model also exhibits both chaotic and integrable (mixed) dynamics depending on the value of the interaction parameter. We focus on the three-site Bose Hubbard Model (with different particle numbers), which is known to be highly mixed. We use the Lanczos algorithm to find the Lanczos coefficients and the Krylov basis. The orthonormal Krylov basis captures the operator growth for a system with a given Hamiltonian. However, the Lanczos algorithm needs to be modified for our case due to the instabilities instilled by the piling up of computational errors. Next, we compute the Krylov complexity and its early and late-time behaviour. Our results capture the chaotic and integrable nature of the system. Our paper takes the first step to use the Lanczos algorithm non-perturbatively for a discrete quartic bosonic Hamiltonian without depending on the auto-correlation method.

Closed-Loop Continuous-Time Subspace Identification with Prior Information

This paper presents a closed-loop continuous-time subspace identification method using prior information. Based on a rational inner function, a generalized orthonormal basis can be constructed, and the transformed noises have ergodicity features. The continuous-time stochastic system is converted into a discrete-time stochastic system by using generalized orthogonal basis functions. As is known to all, incorporating prior information into identification strategies can increase the precision of the identified model. To enhance the precision of the identification method, prior information is integrated through the use of constrained least squares, and principal component analysis is adopted to achieve the reliable estimate of the system. Moreover, the identification of open-loop models is the primary intent of the continuous-time system identification approaches. For closed-loop systems, the open-loop subspace identification methods may produce biased results. Principal component analysis, which reliably estimates closed-loop systems, provides a solution to this problem. The restricted least-squares method with an equality constraint is used to incorporate prior information into the impulse response following the principal component analysis. The input–output algebraic equation yielded an optimal multi-step-ahead predictor, and the equality constraints describe the prior information. The effectiveness of the proposed method is provided by numerical simulations.

Spectrality of random convolutions generated by finitely many Hadamard triples

Let {(Nj,Bj,Lj):1⩽j⩽m} be finitely many Hadamard triples in R . Given a sequence of positive integers {nk}k=1∞ and ω=(ωk)k=1∞∈{1,2,…,m}N , let μω,{nk} be the infinite convolution given by μω,nk=δNω1−n1Bω1∗δNω1−n1Nω2−n2Bω2∗⋯∗δNω1−n1Nω2−n2⋯Nωk−nkBωk∗⋯. In order to study the spectrality of μω,{nk} , we first show the spectrality of general infinite convolutions generated by Hadamard triples under the equi-positivity condition. Then by using the integral periodic zero set of Fourier transform we show that if gcd(Bj−Bj)=1 for 1⩽j⩽m , then all infinite convolutions μω,{nk} are spectral measures. This implies that we may find a subset Λω,{nk}⊆R such that {eλ(x)=e2πiλx:λ∈Λω,{nk}} forms an orthonormal basis for L2(μω,{nk}) .

Formation of Brahmani Natural Arch, Eastern India: A proposed geoheritage site of India

Natural arch is one of the most fascinating geomorphological features formed by erosional process and many of them are formed within gently dipping sandstone. A natural arch (Location: 21°54′32.20″ N, 83°43′46.20″ E, RL: ∼370 m) of such type is present in the rocks of Upper Kamthi Formation of Middle Jurassic age (160–184 Ma), Upper Gondwana Group, Ib River Coalfield within dense Chhengapahar Reserve Forest near Brahmani Village, Sundargarh District, Odisha. This oval shaped natural arch with a maximum length and height of ∼30 m at the base and ∼12 m respectively. The alcove of the natural arch has maximum height and width of ∼7 m and ∼15 m respectively, is opened along NE-SW direction on the top of the hill. Flesh red colored, ripple topped, trough to planar cross stratified, coarse to gritty, texturally and mineralogically immature, ferruginous sandstone of Upper Kamthi Formation forms the hill. Kaolinisation of feldspathic members makes the sandstone loose and porous. Two orthogonal set of faults-joints (130°/60°W and 40°/60°E) initiated the formation of natural arch while the porosity of the immature, arkosic sandstone accelerate the process through subaerial weathering where surface/rain water percolation–temperature fluctuation played effective roles. Therefore, the formation of natural arch is a result of structurally triggered weathering and erosion. Efforts have already been taken to preserve this place as national geo-heritage site by Geological Survey of India, Government of India.

Unsupervised Machine Learning Identify Cis‐Regulatory Network Involved in Neurodegeneration

AbstractBackgroundCis‐Regulatory Modules (CRMs) are crucial to ensure the precise spatiotemporal gene expression, but we still lack a complete catalog of mammalian CRMs.MethodsWe developed the Mammalian Regulatory Module Detector (MrMOD), that accurately predicts a comprehensive set of high‐resolution CRMs in mouse and human genomes including enhancers, promoters, silencers, locus control regions, and microRNA regulatory modules not limited to any tissue, cell type, developmental stage, or stimuli. We validated our predictions using 19 orthogonal experimental data sets, including thousands of experimental defined CRMs and millions of putative regulatory elements from multiple databases. Next, we used an unsupervised machine learning method to annotate CRM functions. We scanned the mouse genome for transcription factor binding sites (TFBS) using position weight matrices of known transcription factors (TF) from the CIS‐BP database and obtained TFBS abundance in each CRM for every TF. Unsupervised clustering was performed using Seurat on CRMs from chromosome 17. Each CRM cluster represents a set of CRMs with similar TFBS compositions. Cluster marker genes represent TFs whose binding sites are enriched in the CRM cluster. Genes associated with the CRMs in the cluster represent the putative target genes. Functional enrichment analyses of the TFs and CRM associated genes were performed using ClusterProfiler and disgenet2r.ResultsSeurat identified a total of 43 clusters. In particular, CRM cluster #17 is linked to neurodegeneration based on five supporting evidence: 1) TF marker genes of cluster #17 include TFs known to be involved in neurodegeneration such as Spi1, Yy1, Nr4a2, Nr4a3, and Tfeb. 2) TF marker genes of cluster #17 were significantly enriched for Alzheimer’s Disease, Parkinson’s Disease, Multiple Sclerosis, among others. 3) Genes associated with cluster #17 CRMs were significantly enriched for neurodegeneration: Multiple Sclerosis and neurodegenerative disorders. Gene Ontology Cellular Components terms significantly enriched for synapse, presynapse, axon, and synaptic membrane. 4) Genes associated with cluster #17 CRMs significantly overlap the known TF target genes in the ChIP‐Atlas database defined by human ChIP‐seq experiments. 5) Top TF marker genes of cluster #17 known to activate transcription in neurons cooperatively.ConclusionsWe developed a novel approach to elucidate mechanisms involved in neurodegeneration.