Norm Of Vector Research Articles

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.

Decomposition in direct sum of seminormed vector spaces and Mazur–Ulam theorem

Abstract It was proved by S. Mazur and S. Ulam in 1932 that every isometric surjection between normed real vector spaces is affine. We generalize the Mazur–Ulam theorem and find necessary and sufficient conditions under which distance-preserving mappings between seminormed real vector spaces are linear.

Thermodynamics with conformal Killing vector in the charged Vaidya metric

We investigate the charged Vaidya spacetime with conformal symmetry by classifying the horizons and finding its connection to Hawking temperature. We find a conformal Killing vector whose existence requires the mass and electric charge functions to be proportional, as well as linear in time. Solving the Killing equations for the conformally transformed metric from the linear charged Vaidya metric yields the required form of the conformal factor. From the vanishing of the norm of the conformal Killing vector, we find three conformal Killing horizons which, under the transformation, are mapped to the Killing horizons of the associated static spacetime, if the spherical symmetry is maintained. We find that the conformal factor is not uniquely determined, but can take any function of the ratio of the radial coordinate to the dynamical mass. As an example, we illustrate a static spacetime with our choice of the conformal factor and explicitly show that the surface gravity of the conformal Killing horizons, which is conformally invariant, yield the expected Hawking temperature in the static spacetime. This static black hole spacetime contains a cosmological horizon, but it is not asymptotically de Sitter. We also investigate the case when the mass parameter is equal to the constant electric charge. While in this case the standard pair of horizons, the loci of the time component of the metric, degenerate, the conformal Killing horizons do not degenerate. This therefore leads to a non-zero Hawking temperature in the associated static spacetime.

Iterative Pseudo-Sparse Partial Least Square and Its Higher Order Variant: Application to Inference From High-Dimensional Biosignals

Partial Least Square (PLS) regression and its (L1 or L2 norm) regularized versions have been proposed to handle the high-dimensionality aspects of the problem at hand and select relevant features. Addressing these issues improves the generalizability of decoding the unseen data, with the severe challenge of high computational complexity. In order to avoid directly solving the L1 norm optimization problem or performing matrix inversion, this paper proposes two PLS-based algorithms, Pseudo-Sparse PLS (PS-PLS) and iterative Pseudo-Sparse Higher-Order PLS (iPS-HOPLS). In these proposed methods, we add the Pseudo-Sparsity term to reduce the L1 norm of the regression coefficient vector in a selective scheme for better importance interpretation while keeping the algorithm as simple as possible. Regarding the evaluation of the proposed methods, we investigate three critical high-dimensionality regression problems of 1) the prediction of 3D trajectory from Electrocorticography (ECoG) recordings, 2) decoding continuous fluctuation of the Electromyography (EMG) powers from recorded Magnetoencephalography (MEG) signals, and 3) continuous decoding of the finger forces from the High-Density surface Electromyogram (HD-sEMG) signals. As well as providing cognitive-relevant interpretations, the experimental results show significant improvements over the generic methods and competitive performance compared to the state-of-the-art regularized PLS approaches.

Analyzing Cattle Activity Patterns with Ear Tag Accelerometer Data.

In this study, we equip two breeds of cattle located in tropical and temperate climates with smart ear tags containing triaxial accelerometers to measure their activity levels across different time periods. We produce activity profiles when measured by each of four statistical features, the mean, median, standard deviation, and median absolute deviation of the Euclidean norm of either unfiltered or high-pass-filtered accelerometer readings over five-minute windows. We then aggregate the values from the 5 min windows into hourly or daily (24 h) totals to produce activity profiles for animals kept in each of the test environments. To gain a better understanding of the variation between the peak and nadir activity levels within a 24 h period, we divide each day into multiple equal-length intervals, which can range from 2 to 96 intervals. We then calculate a statistical measure, called daily differential activity (DDA), by computing the differences in feature values for each interval pair. Our findings demonstrate that patterns within the activity profile are more clearly visualised from readings that have been subject to high-pass filtering and that the median of the acceleration vector norm is the most reliable feature for characterising activity and calculating the DDA measure. The underlying causes for these differences remain elusive and is likely attributable to environmental factors, cattle breeds, or management practices. Activity profiles produced from the standard deviation (a feature routinely applied to the quantification of activity level) showed less uniformity between animals and larger variation in values overall. Assessing activity using ear tag accelerometers holds promise for monitoring animal health and welfare. However, optimal results may only be attainable when true diurnal patterns are detected and accounted for.

Selecting Indispensable Edge Patterns With Adaptive Sampling and Double Local Analysis for Data Description

Support vector data description (SVDD) inspires us in data analysis, adversarial training, and machine unlearning. However, collecting support vectors requires pricey computation, while the alternative boundary selection with O(N2) is still a challenge. The authors propose an indispensable edge pattern selection method (IEPS) for data description with direct SVDD model building. IEPS suggests a double local analysis to select the global edge patterns. Edge patterns belong to a subset of the target problem of SVDD and its variants, and neighbor analysis becomes pivotal. While an excessive number of participating data result in redundant computations, an insufficient number may impede data separability or compromise the model's quality. Consequently, a data-adaptive sampling strategy has been devised to ascertain an optimal ratio of retained data for edge pattern selection. Extensive experiments indicate that IEPS keeps indispensable edge patterns for data description while reducing the interference in the norm vector generation to guarantee the effectiveness for clustering analysis.

Algebraic method for multisensor data fusion.

In this contribution, we use Gaussian posterior probability densities to characterize local estimates from distributed sensors, and assume that they all belong to the Riemannian manifold of Gaussian distributions. Our starting point is to introduce a proper Lie algebraic structure for the Gaussian submanifold with a fixed mean vector, and then the average dissimilarity between the fused density and local posterior densities can be measured by the norm of a Lie algebraic vector. Under Gaussian assumptions, a geodesic projection based algebraic fusion method is proposed to achieve the fused density by taking the norm as the loss. It provides a robust fixed point iterative algorithm for the mean fusion with theoretical convergence, and gives an analytical form for the fused covariance matrix. The effectiveness of the proposed fusion method is illustrated by numerical examples.

A Framework with Improved Heuristics to Optimize Low-Latency Implementations of Linear Layers

In recent years, lightweight cryptography has been a hot field in symmetric cryptography. One of the most crucial problems is to find low-latency implementations of linear layers. The current main heuristic search methods include the Boyar-Peralta (BP) algorithm with depth limit and the backward search. In this paper we firstly propose two improved BP algorithms with depth limit mainly by minimizing the Euclidean norm of the new distance vector instead of maximizing it in the tie-breaking process of the BP algorithm. They can significantly increase the potential for finding better results. Furthermore, we give a new framework that combines forward search with backward search to expand the search space of implementations, where the forward search is one of the two improved BP algorithms. In the new framework, we make a minor adjustment of the priority of rules in the backward search process to enable the exploration of a significantly larger search space. As results, we find better results for the most of matrices studied in previous works. For example, we find an implementation of AES MixColumns of depth 3 with 99 XOR gates, which represents a substantial reduction of 3 XOR gates compared to the existing record of 102 XOR gates.

Conic cancellation laws and some applications in set optimization

We discuss, on finite and infinite dimensional normed vector spaces, some versions of Rådström cancellation law (or lemma) that are suited for applications to set optimization problems. In this sense, we call our results ‘conic’ variants of the celebrated result of Rådström, since they involve the presence of an ordering cone on the underlying space. Several adaptations to this context of some topological properties of sets are studied and some applications to subdifferential calculus associated to set-valued maps and to necessary optimality conditions for constrained set optimization problems are given. Finally, a stability problem is considered.

Further results on synchronization of chaotic Lur’e systems based on aperiodic time-triggered intermittent control

Aperiodic time-triggered intermittent control (ATIC) is presented to investigate the exponential synchronization issue of chaotic Lur’e systems (CLSs). In our work, for CLSs, the mathematical expression of ATIC is put forward initially. Different from time-triggered intermittent control (TIC), ATIC allows aperiodic sampling. The aperiodic sampling characteristic of ATIC makes it more flexible and adaptable in complex environments, which greatly increases its applications. Then, for the deduced ATIC error system, a key lemma is established, which demonstrates that, on control or free time intervals, the bound of the error vector norm at any instance can be described by the multiple of its norm at initial instance. Further, based on the lemma and a mixed time-dependent Lyapunov functional (MTLF), the exponential stability theorem is given for the deduced ATIC error system. Consequently, the exponential synchronization criterion with less conservativeness is obtained for CLSs via ATIC. With its aid, the design method of ATIC is proposed. Finally, the validity and superiority of ATIC are verified by a simulation example.

Spacecraft Segment Damage Identification Method Based on Fiber Optic Strain Difference Field Reconstruction and Norm Calculation

Real-time online identification of spacecraft segment damage is of great significance for realizing spacecraft structural health monitoring and life prediction. In this paper, a damage response characteristic field inversion algorithm based on the differential reconstruction of strain response is proposed to solve the problem of not being able to recognize the small damages of spacecraft structure directly by the strain response alone. Four crack damage location identification methods based on vector norm computation are proposed, which realize online identification and precise location of structural damage events without external excitation by means of spacecraft structural working loads only. A spacecraft segment structural damage monitoring system based on fiber optic grating sensors was constructed, and the average error of damage localization based on the curvature vector 2 norm calculation was 2.58 mm, and the root-mean-square error was 1.98 mm. The results show that the method has superior engineering applicability for on-orbit service environments.

An efficient equalizer for the impulsive noise environment

In some communication systems, the transmitted signal is contaminated by impulsive noise with a non-Gaussian distribution, which usually causes significant performance degradation to communication receivers. To deal with the impulsive noise, this paper proposes an efficient equalizer based on new fractional lower-order statistic (FLOS). Firstly, a nonlinear function-based new FLOS associated with the linear equalizer is designed to decrease the influence of impulsive noise. Secondly, referring to minimum mean square error (MMSE) criterion, an unconstrained optimization problem is formulated, and it is solved by the gradient descent method. Thirdly, to further improve the robustness of the proposed algorithm, the gradient is normalized by the l∞-norm of the equalizer input vector. Finally, simulation results demonstrate that the proposed algorithm achieves superior performance under Gaussian noise, even in the case of strong impulsive noise. Also, the simulation results prove the rationality of the new statistics under impulsive noise environments.

On Fragments on Lattice Normed Vector Lattices

On Fragments on Lattice Normed Vector Lattices

Miraculous hypercomplex numbers

<p>The development of the number system has been a long and difficult process, and many landmark concepts and theorems have been put forward. By briefly reviewing the development of hypercomplex systems, the rules for constructing the unit elements are discussed. As a vector space defining multiplication, division, and norm of vectors, hypercomplex numbers synthesize the advantages of mathematical tools such as algebra, geometry, and analysis, faithfully describe the intrinsic properties of space-time and physical systems, and provide a unified language and a powerful tool for basic theories and engineering technology. In the application of hypercomplex numbers, the group-like properties of the basis vectors are the most important, and the zero factor has little influence on the algebraic operation. The multiplication table of the basis vectors fully describes the intrinsic properties of the hypercomplex system, and the matrix <strong>A</strong> constructed from the multiplication table satisfies the structure equation <strong>A</strong><sup>2</sup> = <em>n</em><strong>A</strong> and thus obtains a set of faithful matrix representations of the basic elements. This paper also uses typical examples to show the simple and clear concepts and wide application of hypercomplex numbers. Therefore, hypercomplex numbers are worth learning in basic education and applying in scientific research and engineering technology.</p>

Verification and calibration of a commercial anisotropic magnetoresistive magnetometer by multivariate non-linear regression

Abstract. Commercially available anisotropic magnetoresistive (AMR) magnetometers exhibit on the order of 1 nanotesla (nT) sensitivity in small size, weight, and power (SWaP) packages. However, AMR magnetometer accuracy is diminished by properties such as static offsets, gain uncertainty, off-axis coupling, and temperature effects. This work presents a measurement of the magnitude of these effects for a Honeywell HMC1053 magnetometer and evaluates a method for calibrating the observed effects by multivariate non-linear regression using a 24-parameter measurement equation. The presented calibration method has reduced the vector norm of the root mean square error from 4300 to 72 nT for the data acquired in this experiment. This calibration method has been developed for use on the AERO (Auroral Emissions Radio Observer) and VISTA (Vector Interferometry Space Technology using AERO) CubeSat missions, but the methods and results may be applicable to other resource-constrained magnetometers whose accuracies are limited by the offset, gain, off-axis, and thermal effects that are similar to the HMC1053 AMR magnetometer.

Shallow kinetics induced by a metronome (SKIM): A novel contactless respiratory motion management.

As an alternative to conventional compression amidst the COVID-19 pandemic, we developed a contactless motion management strategy. By increasing the patient's breathing rate to induce shallow breathing with the aid of a metronome, our hypothesis is that the motion magnitude of the target may be minimized without physical contact or compression. Fourteen lung stereotactic body radiation therapy (SBRT) patients treated under fast shallow-breathing (FSB) were selected for inclusion in this retrospective study. Our proposed method is called shallow kinetics induced by a metronome (SKIM). We induce FSB by setting the beats-per-minute (BPM) high (typically in the range of 50-60). This corresponded to a patient breathing rate of 25-30 (breathing) cycles per minute. The magnitude of target motion in 3D under SKIM was evaluated using 4DCT datasets. Comparison with free breathing (FB) 4DCT was also made for a subset for which FB data available. The overall effectiveness of SKIM was evaluated with 18 targets (14 patients). Direct comparison with FB was performed with 12 targets (10patients). The vector norm mean±SD value of motion magnitude under SKIM for 18 targets was 8.2±4.1mm. The mean±SD metronome BPM was 54.9±4.0 in this group. The vector norm means±SD values of target motion for FB and SKIM in the 12 target sub-group were 14.6±8.5mm and 9.3±3.7mm, respectively. The mean±SD metronome BPM for this sub-group was 56.3±2.5. Compared with FB, SKIM can significantly reduce respiratory motion magnitude of thoracic targets. The difference in maximum motion reduction in the overall vector norm, S-I, and A-P directions was significant (p=0.033, 0.042, 0.011). Our proposed method can be an excellent practical alternative to conventional compression due to its flexibility and ease of implementation.

Nonlinear mode decomposition via physics-assimilated convolutional autoencoder for unsteady flows over an airfoil

I have developed a physics-assimilated convolutional autoencoder (CAE) neural network, namely, PhyAENet, to carry out nonlinear mode decomposition of the unsteady flow field around a National Advisory Committee for Aeronautics 4412 airfoil near stall conditions. The flow field snapshots are mapped into latent modes by the encoder part of the well-trained CAE, which are used for dynamic mode decomposition (DMD) analysis. The computed DMD modes are split into modes covering different frequency ranges. These high and low-frequency DMD modes are used to form reconstructed encoded sequences, which are then mapped back to generate the nonlinear decomposed spatiotemporal modes by the decoder of the CAE. As such, physics is assimilated into the neural network by incorporating the frequencies of the DMD modes into the latent modes in the latent space. The proposed PhyAENet is capable of extracting the dominant features of the flow fields, accounting for the nonlinearity of the underlying dynamics. Furthermore, the extracted nonlinear modes are evolving with time and physically interpretable. It is revealed that the nonlinear modes can be well represented when using more DMD modes for reconstruction of the encoded sequences. The energy spectrum of the nonlinear modes are obtained by ranking the Frobenius norm of the mode vector.

Enhanced-Rate Iterative Beamformers for Active IRS-Assisted Wireless Communications

Compared to passive intelligent reflecting surface (IRS), active IRS is viewed as a more efficient promising technique to combat the double-fading impact in IRS-aided wireless network. In this paper, in order to boost the achievable rate of user in such a wireless network, three enhanced-rate iterative beamforming methods are proposed by designing the amplifying factors and the corresponding phases at active IRS. The first method, maximizing the simplified signal-to-noise ratio (Max-SSNR) is designed by omitting the cross-term in the definition of rate. Using the Rayleigh-Ritz (RR) theorem, Max-SSNR-RR is proposed to iteratively optimize the norm of beamforming vector and its associated normalized vector. In addition, generalized maximum ratio reflection (GMRR) is presented with a closed-form expression, which is motivated by the maximum ratio combining. To further improve rate, maximizing SNR (Max-SNR) is designed by fractional programming (FP), which is called Max-SNR-FP. Simulation results show that the proposed three methods make an obvious rate enhancement over Max-reflecting signal-to-noise ratio (Max-RSNR), maximum ratio reflection (MRR), selective ratio reflecting (SRR), equal gain reflection (EGR) and passive IRS, and are in increasing order of rate performance as follows: Max-SSNR-RR, GMRR, and Max-SNR-FP.

Global Dissipativity and Quasi-Mittag-Leffler Synchronization of Fractional-Order Discontinuous Complex-Valued Neural Networks.

This article is concerned with fractional-order discontinuous complex-valued neural networks (FODCNNs). Based on a new fractional-order inequality, such system is analyzed as a compact entirety without any decomposition in the complex domain which is different from a common method in almost all literature. First, the existence of global Filippov solution is given in the complex domain on the basis of the theories of vector norm and fractional calculus. Successively, by virtue of the nonsmooth analysis and differential inclusion theory, some sufficient conditions are developed to guarantee the global dissipativity and quasi-Mittag-Leffler synchronization of FODCNNs. Furthermore, the error bounds of quasi-Mittag-Leffler synchronization are estimated without reference to the initial values. Especially, our results include some existing integer-order and fractional-order ones as special cases. Finally, numerical examples are given to show the effectiveness of the obtained theories.

New Results of Normed Approach Space Via β-Approach Structure

Researchers have identified and defined β- approach normed space if some conditions are satisfied. In this work, we show that every approach normed space is a normed space.However, the converse is not necessarily true by giving an example. In addition, we define β – normed Banach space, and some examples are given. We also solve some problems. We discuss a finite β-dimensional app-normed space is β-complete and consequent Banach app- space. We explain that every approach normed space is a metric space, but the converse is not true by giving an example. We define β-complete and give some examples and propositions. If we have two normed vector spaces, then we get two properties that are equivalent. We also explain that β-normed app- spaces are norm bounded with a condition. We show that functions of β-normed Banach spaces are β- contraction, with some results and properties. The sequentially β-contraction is also explained and the relation between metric β- app- space and Hausdorff space is studied.