Norm Of Vector Research Articles

Ricci flow of Kaehlerian slant submanifolds in complex space forms and its applications

AbstractThe normalized Ricci flow converges to a constant curvature metric for a connected Kaehlerian slant submanifold in a complex space form if the squared norm of the second fundamental form satisfies certain upper bounds. These bounds include the constant sectional curvature, the slant angle, and the squared norm of the mean curvature vector. Additionally, we demonstrate that the submanifold is diffeomorphic to the sphere $$\mathbb {S}^{n_1}$$ S n 1 under some restriction on the mean curvature. We claim that some of our previous results are rare cases.

On the adaptive deterministic block Kaczmarz method with momentum for solving large-scale consistent linear systems

The Kaczmarz method is a traditional and widely used iterative algorithm for solving large-scale consistent linear systems, while its improved block Kaczmarz-type methods have received much attention and research in recent years due to their excellent numerical performance. Hence, in this paper, we present a deterministic block Kaczmarz method with momentum, which is based on Polyak’s heavy ball method and a row selection criterion for a set of block-controlled indices defined by the Euclidean norm of the residual vector. The proposed method does not need to compute the pseudo-inverses of a row submatrix at each iteration and it adaptively selects and updates the set of block control indices, thus this is different from the block Kaczmarz-type methods that are based on projection and pre-partitioning of row indices. The theoretical analysis of the proposed method shows that it converges linearly to the unique least-norm solutions of the consistent linear systems. Numerical experiments demonstrate that the deterministic block Kaczmarz method with momentum method is more efficient than the existing block Kaczmarz-type methods.

Some Applications of Cauchy-Schwarz Inequality on Lightlike Warped Product Submanifolds

The well-known Cauchy-Schwarz inequality is a fundamental tool of analysis in Hilbert spaces. It is an upper bound on the inner product between two vectors in an inner product space in terms of the product of the vector norms. It is the most important and widely used inequalities in mathematics. We explore that inequality in case of lightlike warped product hypersurface of lorentzian space form to reach to the existence of minimal screen conformal lightlike hypersurface.

510 Utilizing python programming for modern genetic evaluation systems

Abstract Advancements in agricultural data collection and methodologies for data analysis offer opportunities for beef cattle producers to improve herd genetics. Genetic evaluation systems (GES) can benefit from modern programming approaches to keep up with these advancements. Python, a fast-developing programming language with scientific computing libraries such as NumPy and SciPy, offers flexibility that allows for quick adoption of superior algorithms and new data sources, as well as easy maintenance of the software over the long-term. This study evaluated the use of Python (v3.12.1) for developing a GES capable of executing all the necessary functions expected of modern implementation of GES. A simulated dataset was generated with the R package AlphaSimR to mimic characteristics of the Angus beef cattle breed. The dataset included pedigree information, phenotypic data, and true breeding values (TBVs) for each animal. Phenotypic traits included birth weight (BWT), weaning gain (WG), and post-weaning gain (PWG), with a total of 976,400 animals simulated across 15 generations. The GES software was developed in house, and included components such as data import, pedigree and data preprocessing, calculating inbreeding, constructing the inverse of the numerator relationship matrix, solver initialization, estimation of breeding value, and writing results. The genetic evaluation employed pedigree-based best linear unbiased prediction (BLUP) and a multiple-trait animal model, using a preconditioned conjugate gradient iteration algorithm to solve the mixed model equations. Convergence was assessed using the residual vector norm divided by the right-hand-side norm, with a threshold to declare convergence set to 1x10-5. The software was run on a Linux server with an Intel(R) Xeon(R) Gold 6242 CPU @ 2.80GHz, 96 GB of RAM, and eight cores. Accuracy was measured using Pearson correlation coefficient between TBVs and estimated breeding values (EBVs). To benchmark computing time and solutions, the same dataset and model were input into a commonly used commercial GES, MiXBLUP (v3.0.1). The solver was the most time-consuming component of the GES, with a run time of 27 min. However, the application of multiprocessing reduced the run time by 72% to 8 min. The Pearson correlation coefficient between EBV and TBV was 0.86, 0.79, and 0.73 (P < 0.01) for BWT, WG, and PWG, respectively. In contrast, MiXBLUP completed breeding value prediction in less than 1 min. The Pearson correlation coefficient between Python and MiXBLUP predictions was 1.0 (P < 0.01). While Python is an ideal language for modern GES, further updates are necessary to optimize performance and operations. Future research will focus on refining computational algorithms, exploring parallel processing techniques, and enhancing user interfaces to ensure easy integration with industry practices.

Leader-Following Synchronization Control of Multiagent Systems Under Hybrid Cyber Attacks via Impulsive Control Based on Topology Switching.

This article investigates the leader-following synchronization problem of multiagent systems (MASs) under hybrid cyber attacks, which refers to deception attacks and multichannel independent denial-of-service (DoS) attacks in communication channels. In order to achieve the secure control of MASs under hybrid cyber attacks, a novel impulsive control method based on topology switching is proposed, and a new algorithm for determining impulsive instants is designed. In addition, the cooperative-competitive relationship between agents is also considered, which is more in line with reality. Sufficient conditions for ensuring secure control of MASs and a parametric upper bound on the error vector norm between the agents and the leader are obtained. Finally, the numerical simulation verified the effectiveness of the proposed algorithm.

Data depth functions for non-standard data by use of formal concept analysis

In this article we introduce a notion of depth functions for data types that are not given in standard statistical data formats. We focus on data that cannot be represented by one specific data structure, such as normed vector spaces. This covers a wide range of different data types, which we refer to as non-standard data. Depth functions have been studied intensively for normed vector spaces. However, a discussion of depth functions for non-standard data is lacking. In this article, we address this gap by using formal concept analysis to obtain a unified data representation. Building on this representation, we then define depth functions for non-standard data. Furthermore, we provide a systematic basis by introducing structural properties using the data representation provided by formal concept analysis. Finally, we embed the generalised Tukey depth into our concept of data depth and analyse it using the introduced structural properties. Thus, this article presents the mathematical formalisation of centrality and outlyingness for non-standard data and increases the number of spaces in which centrality can be discussed. In particular, we provide a basis for defining further depth functions and statistical inference methods for non-standard data.

A novel hybrid features grey incidence model and its application in identifying key factors influencing air pollution in Jiangsu Province

Accurately assessing the key factors influencing air pollution is crucial for effective air pollution control. To address this need, we propose a novel Hybrid Features Grey Incidence Model (HFGIM), which integrates geometric feature differences from both proximity and similarity perspectives. Firstly, we extract geometric feature difference vectors of proximity and similarity from time series data and measure the overall feature difference degree by calculating vector norms. Secondly, we calculate the relative feature differences and information contribution rates of proximity and similarity to derive the hybrid feature differences coefficient between sequences, thereby obtaining the hybrid features incidence degree. After detailing the model’s properties and modelling steps, we introduce the Cross-sectional Data Hybrid Features Grey Incidence Model (C-HFGIM) and the Panel Data Hybrid Features Grey Incidence Model (P-HFGIM) for handling cross-sectional and panel data, respectively. Applying HFGIM, we identified the key pollutants and primary pollution source indicators of air pollution in Jiangsu Province. We also compared HFGIM with other classical grey incidence models to verify the proposed model’s effectiveness. Based on the analysis results, we propose policy recommendations for air pollution control in Jiangsu Province.

Geometric Inequalities of Slant Submanifolds in Locally Metallic Product Space Forms

In this particular article, our focus revolves around the establishment of a geometric inequality, commonly referred to as Chen’s inequality. We specifically apply this inequality to assess the square norm of the mean curvature vector and the warping function of warped product slant submanifolds. Our investigation takes place within the context of locally metallic product space forms with quarter-symmetric metric connections. Additionally, we delve into the condition that determines when equality is achieved within the inequality. Furthermore, we explore a number of implications of our findings.

Robust hyperspectral unmixing using weighted group sparsity and minimum volume regularization

ABSTRACT Hyperspectral unmixing (HU) has been a vital technique in hyperspectral image processing. Over the past years, many algorithms have been developed based on the nonnegative matrix factorization (NMF) framework. However, the conventional methods based on the NMF need prior information of the number of endmembers, which is usually estimated by a separate procedure. Therefore, the precision of estimating the number of endmembers will impact the subsequent unmixing effect. Thus, collaborating between the two processes is desired to avoid the propagation of errors. This paper proposes a new NMF-based method, which can estimate the number of endmembers and extract the endmembers and the abundance maps. Firstly, we introduce the weighted group sparsity regularization to adaptively eliminate the abundances corresponding to the redundant endmembers, while the abundances of the true endmembers are preserved. Then the number of endmembers can be determined by the reconstruction error function and the l 2 -norm of the abundance vectors. Secondly, since the simplex determined by the endmembers has the minimum volume among all simplexes that circumscribe the data points, we impose the total variational regularization to force the endmembers to be close to each other to minimize the simplex volume. And because the total variational regularization avoids the use of noisy pixels, it can decrease the influence of noise and improve the robustness of the solution. For the proposed model, we design an effective algorithm to solve it. Also, the experimental results based on the synthetic data and real data demonstrate that the proposed method can identify the endmembers number well and achieve robust unmixing results.

Оценка хозяйственных рисков при реализации технологий угледобычи в современных условиях недропользования

Introduction. This paper presents research in the field of methodology of assessment and monitoring of associated under ground coalmining risks in order to level their impact on the resulting technical and economic indicators of production and economic activities in modern conditions of subsoiluse. The functioning of technological systems of coal mining enterprises at the present stage of subsurface use occurs in a highly dynamic and highly variable functional environment with a fair degree of uncertainty, which is generated by a significant variety of parameters and characteristics of the initial mining, geological and mining engineering environment, which, in turn, generates the emergence of various groups and risk factors for subsurface use. The established procedures and methods of risk forecasting currently do not give satisfactory results due to the globalization of world markets and high volatility of prices for coal products. Leveling the negative degree of influence of the risk component on the resulting indicators – indicators of technical and economic activity can be achieved through the introduction of an operational system for assessing and monitoring various groups of risk factors. Materials and methods. For the purpose of evaluation, it is proposed to use the methodological and methodological foundations of the method called “vector norms”, which involves the use of Affine space, the constituent areas of compromises and the introduction of utility estimates. A complete model representation of the assessment and monitoring of the impact of various risk factors of subsurface use, taking into account their clustering and ranking, is presented. The algorithmic support of this method is reduced to the following components-iterations: 1. Formation of criteria-based assessment matrices of various factors of subsurface use risk groups. 2. Assignment of optima to evaluation criteria. 3. In order to ensure the legality of the use and implementation of mathematical procedures and operations, the reduction of multi-dimensional evaluation criteria to a relative form, taking into account the assigned optima. 4. Formation of the objective function of the “norm of the vector” (integral indicator) and the development of a procedure for selecting the optimal option. Since the use of the methodological and methodological foundations of the method called “vector norms” allows us to form quantitative estimates of the presented factors-types of risk of subsurface use, this makes it possible to implement the clustering procedure or ranking of a selected set of coal mining enterprises in conjunction with existing risk areas. Results. Quite high values of correlation coefficients were obtained with a significant closeness of the relationship of inversely proportional dependencies of generalizing integral indicators of the degree of risk and the level of achievement of planned indicators and development of production capacity with the level of operational profitability from the conduct of production and economic activities of coal mining enterprises with a directly proportional dependence on the operating costs of mining 1t of coal. Discussion. Сomplete model representation of the assessment and monitoring of the impact of various risk factors of subsurface use, taking into account their clustering and ranking, is presented. As a practical component of the research, variable series of areas of integral degree of risk in the functional environment of Kuzbass coal mines are presented. The use of software modules for correlation and regression analysis, taking into account three-year retrospective dynamics, made it possible to establish the legality and objectivity of the proposed methodological apparatus for integrated risk assessment of the implementation of coal mining technologies in modern conditions of subsurface use. Conclusion. The results of the implementation of the theoretical components of the methodology and model representations of the developed conceptual foundations for the integrated risk assessment of the implementation of coal mining technologies in modern conditions of subsurface use, taking into account five-year retrospective dynamics, showed that in the Kuznetsk basin there was a steady tendency to reduce the cluster of coal mines, which is characterized by high risks of carrying out production and economic activities in the presence of a disproportion of resource potential (the emergent part). The accompanying processes of formation of this trend are manifested in a quantitative increase in the functional units of the rating cluster of group 2B with an average resource potential and inherent moderate risk. It is within this cluster that strategies for prioritizing investments for the development and renewal of technological systems and structures of coal mining are being formed, since only they are able to provide the fastest possible return on invested funds. Resume. The article presents the results of research in the field of methodology for assessing and monitoring the risks associated with underground coal mining in order to level out their impact on the resulting technical and economic indicators of production and economic activity in modern conditions of subsoil use. The research results can be useful in the implementation of coal mining technology in modern conditions of subsoil use.

Adaptive position control using backstepping technique for the leader‐follower multiple quadrotor unmanned aerial vehicle formation

SummaryThis article addresses the position control issue of multi‐quadrotor unmanned aerial vehicle (QUAV) formation. Concerning the translational dynamic of a multi‐QUAV system, on the one hand, it is an under‐actuation dynamic; on the other hand, it does not satisfy the matching condition. These features will cause inevitable thorny in the formation position control design. Furthermore, because of the state coupling problem, the formation control of multi‐QUAV system is more challenging and knotty than the control of single QUAV system. To achieve this control, both backstepping technique and neural network (NN) approximation strategy are combined by introducing an intermediary control, where NN is employed to compensate the system uncertainty. However, since the traditional adaptive NN control methods need to train a large number of adaptive parameters for the high approximation accuracy, it will cause the heavy computing burden if traditional adaptive method is used for the QUAV formation control. The proposed adaptive NN strategy in this paper only requires training a scalar adaptive parameter, which is generated from the norm of NN weight vector or matrix, thereby significantly reducing computational burden. Finally, according to Lyapunov stability proof and computer simulation, it is demonstrated that the control tasks can be successfully accomplished.

Smoothed Analysis with Adaptive Adversaries

We prove novel algorithmic guarantees for several online problems in the smoothed analysis model. In this model, at each time step an adversary chooses an input distribution with density function bounded above pointwise by \(\tfrac{1}{\sigma }\) times that of the uniform distribution; nature then samples an input from this distribution. Here, σ is a parameter that interpolates between the extremes of worst-case and average case analysis. Crucially, our results hold for adaptive adversaries that can base their choice of input distribution on the decisions of the algorithm and the realizations of the inputs in the previous time steps. An adaptive adversary can nontrivially correlate inputs at different time steps with each other and with the algorithm’s current state; this appears to rule out the standard proof approaches in smoothed analysis. This paper presents a general technique for proving smoothed algorithmic guarantees against adaptive adversaries, in effect reducing the setting of an adaptive adversary to the much simpler case of an oblivious adversary (i.e., an adversary that commits in advance to the entire sequence of input distributions). We apply this technique to prove strong smoothed guarantees for three different problems: (1) Online learning: We consider the online prediction problem, where instances are generated from an adaptive sequence of σ-smooth distributions and the hypothesis class has VC dimension d . We bound the regret by \(\tilde{O}(\sqrt {T d\ln (1/\sigma)} + d\ln (T/\sigma))\) and provide a near-matching lower bound. Our result shows that under smoothed analysis, learnability against adaptive adversaries is characterized by the finiteness of the VC dimension. This is as opposed to the worst-case analysis, where online learnability is characterized by Littlestone dimension (which is infinite even in the extremely restricted case of one-dimensional threshold functions). Our results fully answer an open question of Rakhlin et al. [ 64 ]. (2) Online discrepancy minimization: We consider the setting of the online Komlós problem, where the input is generated from an adaptive sequence of σ-smooth and isotropic distributions on the ℓ 2 unit ball. We bound the ℓ ∞ norm of the discrepancy vector by \(\tilde{O}(\ln ^2(\frac{nT}{\sigma }))\) . This is as opposed to the worst-case analysis, where the tight discrepancy bound is \(\Theta (\sqrt {T/n})\) . We show such \(\mathrm{polylog}(nT/\sigma)\) discrepancy guarantees are not achievable for non-isotropic σ-smooth distributions. (3) Dispersion in online optimization: We consider online optimization with piecewise Lipschitz functions where functions with ℓ discontinuities are chosen by a smoothed adaptive adversary and show that the resulting sequence is \(({\sigma }/{\sqrt {T\ell }}, \tilde{O}(\sqrt {T\ell }))\) -dispersed. That is, every ball of radius \({\sigma }/{\sqrt {T\ell }}\) is split by \(\tilde{O}(\sqrt {T\ell })\) of the partitions made by these functions. This result matches the dispersion parameters of Balcan et al. [ 13 ] for oblivious smooth adversaries, up to logarithmic factors. On the other hand, worst-case sequences are trivially (0, T )-dispersed. 1

Normalized frequency‐domain block sign algorithm using ℓ1$\ell _{1}$‐norm minimization

AbstractIn this letter, a normalized frequency‐domain block sign‐error algorithm (NFBSA) is proposed by minimizing the ‐norm of the posteriori error vector for filter weight. The NFBSA updates the filter weight vector by employing the sign value of the priori error vector, which can mitigate the impact of impulsive noise. Furthermore, a variable step‐size algorithm (VSS) suitable for application in the NFBSA is proposed, which minimizes the ‐norm of the posteriori error vector with respect to the step‐size. The system identification simulations are conducted using a first‐order autoregressive signal as an input, assuming that impulsive noise occurs. The simulation results demonstrate that the NFBSA exhibits superior misalignment performance compared to the conventional time‐domain sign algorithm, and the NFBSA with the proposed VSS outperformed other VSS‐based sign algorithms.

Analyzing cross-talk between superimposed signals: Vector norm dependent hidden Markov models and applications to ion channels

Analyzing cross-talk between superimposed signals: Vector norm dependent hidden Markov models and applications to ion channels

IMU/Magnetometer-Based Azimuth Estimation with Norm Constraint Filtering.

A typical magnetometer-based measurement-while-drilling (MWD) system determines the azimuth of the bottom hole assembly during the drilling process by employing triaxial accelerometers and magnetometers. The geomagnetic azimuth solution is susceptible to magnetic interference, especially strong magnetic interference and so a rotary norm constraint filtering (RNCF) method for azimuth estimation, designed to support a gyroscope-aided magnetometer-based MWD system, is proposed. First, a new magnetic dynamical system, one whose output is observed by the magnetometers triad, is designed based on the Coriolis equation of the desired geomagnetic vector. Second, given that the norm of the non-interfered geomagnetic vector can be approximated as a constant during a short-term drilling process, a norm constraint procedure is introduced to the Kalman filter. This is achieved by the normalization of the geomagnetic part of the state vector of the dynamical system and is undertaken in order to obtain a precise geomagnetic component. Simulation and actual drilling experiments show that the proposed RNCF method can effectively improve the azimuth measurement precision with 98.5% over the typical geomagnetic solution and 37.1% over the KF in a RMSE sense when being strong magnetic interference environment.

Iterative solution methods for high-order/hp–DGFEM approximation of the linear Boltzmann transport equation

In this article we consider the iterative solution of the linear system of equations arising from the discretisation of the poly-energetic linear Boltzmann transport equation using a high-order/hp–version discontinuous Galerkin finite element approximation in space, angle, and energy. In particular, we develop preconditioned Richardson iterations which may be understood as generalisations of source iteration in the mono-energetic setting, and derive computable a posteriori bounds for the solver error incurred due to inexact linear algebra, measured in a relevant problem-specific norm. We prove that the convergence of the resulting schemes and a posteriori solver error estimates are independent of the mesh size h and polynomial degree p. We also discuss how the poly-energetic Richardson iteration may be employed as a preconditioner for the generalised minimal residual (GMRES) method. Furthermore, we show that standard implementations of GMRES based on minimising the Euclidean norm of the residual vector can be utilized to yield computable a posteriori solver error estimates at each iteration, through judicious selections of left- and right-preconditioners for the original linear system. The effectiveness of poly-energetic source iteration and preconditioned GMRES, as well as their respective a posteriori solver error estimates, is demonstrated through numerical examples arising in the modelling of photon transport. While the convergence of poly-energetic source iteration is independent of h and p, we observe that the number of iterations required to attain convergence when employing GMRES only depends mildly on h and p. Moreover, this latter approach is highly effective in the low energy regime.

Shape Sensing of Cantilever Column Using Hybrid Frenet-Serret Homogeneous Transformation Matrix Method.

The Frenet-Serret (FS) framework stands as a pivotal tool in shape sensing for various infrastructures. However, this tool suffers from accumulative errors, particularly at inflection points where the normal vector undergoes sign changes. To minimize the error, the traditional FS framework is modified by incorporating the homogeneous matrix transformation (HMT) method for segments containing inflection points. Additionally, inclination information is also used to calculate the unit tangent vector and the unit norm vector at the start point of each segment. This novel approach, termed the FS-HMT method, aims to enhance accuracy. To validate the effectiveness of the proposed method, a simulation of a cantilever column was conducted using finite element software ANSYS 19.2. The numerical results demonstrate the capability of the proposed method to accurately predict curves with inflection points, yielding a maximum error of 1.1%. Subsequently, experimental verification was performed using a 1 m long spring steel sheet, showcasing an error of 4.9%, which is notably lower than that of the traditional FS framework. Our proposed modified FS framework exhibits improved accuracy, especially in scenarios involving inflection points. These findings underscore its potential as a valuable tool for enhanced shape sensing in practical applications.

M-fuzzifying normed vector spaces

M-fuzzifying normed vector spaces

A Robust Sidelobe Cancellation Algorithm Based on Beamforming Vector Norm Constraint

Sidelobe cancellation (SLC) is a well-established beamforming technique for mitigating interference, particularly in the context of satellite communication (SATCOM). However, traditional SLC suffers from the issue of partially canceling the desired signal at high signal-to-noise ratio (SNR), primarily due to unconstrained beamforming processing. Extensive research has been conducted to address this problem; however, existing algorithms have limitations such as dependence on knowledge of signal array vectors or number of interferers and involve high computational complexity. In this paper, we propose a robust SLC algorithm based on beamforming vector norm constraint. Our proposal offers a practical solution by only requiring knowledge of the earth station antenna gain and maximum auxiliary array gain to the desired signal, both of which are fully known. Furthermore, compared to traditional SLC, our proposed method introduces additional computational complexity that only scales linearly with the size of the auxiliary array. Simulation results demonstrate comparable performance between our proposed method and existing techniques such as diagonal loading and spatial degrees-of-freedom control-based algorithms.

Taking into account the distribution of the noise level along the wavelength during broadband control of the deposition process of optical coatings

The paper presents numerical results of simulations of the deposition process of a complex optical coating using broadband monitoring of this process in the case when the noise level in the measured transmittance coefficient has an uneven distribution along the wavelength. The influence on the accuracy of data control at the edges of the measured wavelength range, where the error in the transmittance coefficient is maximum, is investigated. It is shown that when a part of the data is removed at the edges of the range, the self-compensation coefficient of the deposition process errors almost does not change, while the average norm of the error vector of layer thicknesses significantly decreases. Thus, it is shown for the first time that taking into account the effect of self-compensation of errors in broadband control opens up the possibility of a reasonable choice of the optimal spectral range of optical control for practice.