Four-dimensional Matrix Research Articles

Characterizations of some new classes of four-dimensional matrices and dual spaces on the double spaces of Cesàro means

The main purpose in this study is to investigate some topological and algebraic properties of the absolutely double series spaces |C_{1,1}|_{k} defined by combining the first order Cesàro means with the concept of absolute summability k≥1. Beside this, we determine the α-dual of the space |C_{1,1}|₁ and the β(bp)- and γ-duals of the spaces |C_{1,1}|_{k} for k≥1. Finally, we characterize some new four-dimensional matrix classes (|C_{1,1}|_{k},υ), (|C_{1,1}|₁,υ), (|C_{1,1}|₁,L_{k}), (|C_{1,1}|_{k},L_{u}), (L_{u},|C_{1,1}|_{k}) and (L_{k},|C_{1,1}|₁), where υ∈{M_{u},C_{bp}} for 1≤k

Damage identification of steel bridge based on data augmentation and adaptive optimization neural network

With the advancement of deep learning, data-driven structural damage identification (SDI) has shown considerable development. However, collecting vibration signals related to structural damage poses certain challenges, which can undermine the accuracy of the identification results produced by data-driven SDI methods in scenarios where data is scarce. This paper introduces an innovative approach to bridge SDI in a few-shot context by integrating an adaptive simulated annealing particle swarm optimization-convolutional neural network (ASAPSO-CNN) as the foundational framework, augmented by data enhancement techniques. Firstly, three specific types of noise are introduced to augment the source signals used for training. Subsequently, the source signals and augmented signals are recombined to construct a four-dimensional matrix as the input to the CNN, while defining the damage feature vector as the output. Secondly, a CNN is constructed to establish the mapping relationship between the input and output. Then, an adaptive fitness function is proposed that simultaneously considers the accuracy of SDI, model complexity, and training efficiency. The ASAPSO is employed to adaptively optimize the hyperparameters of the CNN. The proposed method is validated on an experimental model of a three-span continuous beam. It is compared with four other data-driven methods, demonstrating good effectiveness and robustness of SDI under cases of scarce data. Finally, the effectiveness of this SDI method is validated in a real-world case of a steel truss bridge.

Investigation into the thermal noise propagation mechanism within composite materials for gravitational wave detection systems

The thermal transmission properties of polymer-based composites, commonly used in satellites, are crucial for high-precision thermal management applications. The study constructed the structures of polyimide polymers and polyimide/graphene interfaces based on molecular dynamics and fluctuation dissipation theorem. Optical parameters were determined through first-principles calculations, and the thermal transport from a microthermal source was simulated using Reverse Non-Equilibrium Molecular Dynamics (RNEMD) and the modified four-dimensional transfer matrix method (M4D-TMM). Simulations of polymers, interface structures, and interfacial thermal radiation models investigated the transmission dynamics of microthermal sources during heat transport. Findings showed that the differences in the temperature fields formed within the polymer from microthermal heat sources were less than 20 K for heat source inputs ranging from 0.002 W/m² to 20 GW/m². With a microthermal source, the heat diffusion was extensive, the heat transport was continuous, and the temperature field remained stable. At the level of the heat source up to the intermolecular energy level, the temperature field distribution showed significant instability, leading to nonlinear effects. Within the interface structure, the continuity of heat diffusion increased as the heat source intensified. The interface structure demonstrated enhanced temperature sensitivity and improved stability, with minimal fluctuations of only 0.59 %. The increase in heat flux perpendicular to the interface structure was ascribed to near-field radiative heat transfer. As the temperature difference decreased to less than 0.1 K, the influence of evanescent waves on radiative heat flux nearly disappeared. Phonon polaritons maintained temperature stability during thermal transport through micro- and nanoscale gaps. Integrating thermal conduction principles in polyimide polymers and interface structures with near-field thermal radiation provides a theoretical framework for high-precision thermal management in gravitational wave detection.

Framework of managerial wisdom: an analysis using the TOPSIS and MICMAC process

PurposeBehavioral characteristics and the existence of personal knowledge and skills that are essential in forming a capable manager with specialized insight, is one of the most important evaluation processes in appointing a manager. Wisdom as a trait of a set of personal characteristics and cognitive knowledge of a capable manager will gain competitive advantages for the company and external stakeholders. The purpose of this study is to present the framework of managerial financial wisdom through qualitative and quantitative research methodologies.Design/methodology/approachThe study uses grounded theory methodology to identify factors for managerial financial wisdom and uses a MICMAC process to evaluate the power of driving and dependence among factors. MaxQDA software was used for data coding and analysis. The MICMAC process by determining the location of each of the sub-themes in the four-dimensional matrix, seeks to determine the most effective criterion in the context of the study. The method of data collection is mixed and in terms of philosophical nature, this study is inductive/comparative.FindingsThe results of the research in the qualitative part indicate the presentation of a framework based on two categories, four components, and thirty themes. The results of the quantitative part of the study also showed that the most important factor of managerial financial wisdom is market-oriented values. Finally, it was found that the social trust caused by managerial financial wisdom is considered a factor in gaining more shares in the capital market due to the positive stimulus in this dimension.Originality/valueThe study is relevant for both practitioners and academia and has significant implications. For practitioners, the structural linkages identified will help enhance managerial wisdom effectiveness and drive the critical metrics for financial decisions and shareholders' rights protection. The study may help financial market practitioners better understand future policies. Academia and researchers can take reference from methodologies used in this study for exploring factors of interest and developing matrix linkages among them.

Policy drivers and tools influencing the development of social enterprises in Taiwan

PurposeTaiwan promotes social enterprises with the Social Enterprise Action Plan and Social Innovation Action Plan, focusing on solving social problems and achieving sustainable development goals. This study aims to clarify the impetus for the above policies and determines whether the relevant policy tools can achieve the established policy goals.Design/methodology/approachThis study adopts the in-depth interview method and uses the “social impact investment framework” constructed by OEDC and Dunn’s definition of public policy stakeholders to select 22 respondents, who were divided into three groups.FindingsTaiwan’s decision-making in social enterprise policy is mainly driven by the pressure of youth unemployment and the 318-student movement. Intrinsic motivation strategies popularized the concept of social enterprise. Various strategies are used to break the limitations and diversify the organization, trigger more diverse social investments, broaden the goals of investment and complete the social enterprise ecosystem through these. Taiwan’s social enterprise policies are consistent with the global trend of “replacing subsidies with investment.”Originality/valueThis study shows that social enterprises generate mutual benefits between investors and social enterprises, that is, achieve free matching through external mechanisms. This study fills the gaps in Oliver’s “behavioral cube” framework of policy instruments. A “behavioral four-dimensional matrix” composed of nudge, shove, budge and reciprocity is proposed to complete the framework for social enterprise policy analysis tools.

Mean-based Borda count for paradox-free comparisons of optimization algorithms

Numerical comparisons are essential for selecting an efficient optimization algorithm for specific problems arising from various human practices. Note that recent researches have shown that paradoxes may occur for comparisons of the numerical performance of optimization algorithms, particularly the cycle-ranking paradox. Paradox-free is still open for some popular data analysis methods based on hypothesis testing (HT), which motivates us to design a class of HT-based paradox-free data analysis methods. The numerical comparison of optimization algorithms is analyzed for dimensional reduction in a matrix. The data collected during the experiment is stored in a four-dimensional matrix, which is then reduced to three-dimensional, two-dimensional, and finally a one-dimensional ranking vector. Then a mean-based Borda count (MeanBordaCount/T) is proposed to eliminate the cycle-ranking paradox that arises from the HT-based data analysis methods where HTs are performed on each problem. Specifically, hypothesis testing is replaced primarily by mean comparison, which has been proven that the result of hypothesis testing is the same as the result based only on mean comparisons, except that the former contains more equal or tied results. The Borda count is adopted to eliminate cycle-ranking in the final dimensional reduction. Finally, MeanBordaCount/T is proved to be the best choice among all HT-type methods, at least in the sense that it can minimize the error of pairwise comparisons.

A new double series space derived by factorable matrix and four-dimensional matrix transformations

<p>In this study, we introduce a new double series space $ \left\vert F_{a, b}^{u, \theta }\right\vert _{k} $ using the four dimensional factorable matrix $ F $ and absolute summability method for $ k\geq 1 $. Also, examining some algebraic and topological properties of $ \left\vert F_{a, b}^{u, \theta }\right\vert _{k} $, we show that it is norm isomorphic to the well-known double sequence space $ \mathcal{L}_{k} $ for $ 1\leq k < \infty. $ Furthermore, we determine the $ \alpha $-, $ \beta \left(bp\right) $- and $ \gamma $-duals of the spaces $ \left\vert F_{a, b}^{u, \theta }\right\vert _{k} $ for $ k\geq 1. $ Additionally, we characterize some new four dimensional matrix transformation classes on double series space $ \left\vert F_{a, b}^{u, \theta }\right\vert _{k} $. Hence, we extend some important results concerned on Riesz and Cesàro matrix methods to double sequences owing to four dimensional factorable matrix.</p>

Towards just futures: a capabilitarian approach to transforming undergraduate learning outcomes

ABSTRACT Learning outcomes are predominantly framed in narrow and measurable terms, with students as decontextualised learners. As an alternative, the paper outlines a capabilitarian approach, building a four-dimensional matrix for reconceptualising learning outcomes. It is made up of a varied, multi-dimensional set of opportunities, processes and outcomes to enable students to flourish and become people who can live fulfilling and connected lives in and beyond higher education. The matrix dimensions are made up of a capability set, corresponding functionings, material resources and conversion processes, underpinned by Constitutional and human development values. The paper draws on a project on higher education outcomes in South Africa, especially the longitudinal life history data. It makes the case for a reparative futures orientation which might dismantle the colonial-apartheid past and present, and outlines key capabilitarian concepts. The social and higher education context is sketched, followed by a focus on the capability set and what data supports its value to students.

Spatial-Temporal Evolution and Risk Assessment of Land Finance: Evidence from China

Land finance is a special land financing mode in China under the nationalization of urban land since 1954. The policy authorizes local governments to collect fiscal revenue from land grant premiums and land taxes. As China is experiencing the social and economic transformation, heavily replying on land finance starts causing financial sustainable problems. Based on the spatial panel data of 30 provinces in China in the last two decades, we analyzed the spatial-temporal evolution of land finance. We found that the spatial variation of land finance declined during the period of study and decreased from east to west. The results revealed that land finance had significant positive spatial autocorrelation and robust spatial clustering characteristics. In addition, the spatial distribution of land finance was consistent with the population-based Hu Line. We also assessed land finance risks via a four-dimensional risk matrix through spatial panel regression (SPR). The spatial spillover effects suggested that there is inter-provincial imitation and collaboration but no competition. Our forecast indicates that most provinces will be at a relatively low risk level in the next decade except some southwest provinces. Based on the findings, we highlight the policy implications to mitigate risks and maintain sustainable land finance.

On the recursive equivalence to Smith form of multivariate polynomial matrices

Abstract The Smith form of an $n$-$D$ polynomial matrix plays an important role in many areas of mathematics and engineering. In this paper, we investigate the recursive equivalence problem of $n$-dimensional polynomial matrices, i.e. if diag$(1,B)$ is equivalent to diag$(1,1,C)$, is B equivalent to diag$(1,C)$? We give a negative answer to this question by explicitly constructing a four-dimensional polynomial matrix which is not equivalent to its Smith form.

Анализ регулярности матриц

The paper investigates the problem of analyzing the regularity of multidimensional matrices based on the repetition of significant (non-empty) characters in the matrix cells. Such a repetition means that when the matrix is shifted along one or more of its coordinates, some significant characters are preserved. For each shift repeated r times, the regularity number is entered as the product of rs, where s is the number of significant symbols that persist for all r repetitions of the shift. Two numerical characteristics of matrix regularity are introduced: the regularity sum and the regularity coefficient. The regularity sum is defined as the sum of the regularity numbers for all possible matrix shifts and allows you to compare the regularity of matrices of the same form, i.e. the same dimension and the same size with the same arrangement of non-empty characters. The regularity coefficient allows you to compare the regularity of arbitrary matrices and is defined as the percentage of the sum of the regularity of a matrix to the sum of the regularity of the «most regular» matrix (all significant symbols of which are the same) of the same form. Algorithms for calculating the sum and regularity coefficient of a matrix are proposed and implemented in computer programs. As an applied area, the article uses the analysis of the regular structure of the poems of the ancient Chinese «Canon of Poems» (Shih-ching). The poem is represented by a four-dimensional matrix, its coordinates are a stanza, a line in a stanza, a verse in a line, and a hieroglyph in a verse; blank characters equalize the sizes of verses, lines and stanzas. The article presents generalizing results of computer experiments with all 305 poems of Shih-ching.

Multi-dimensional modeling of radiation belt electron pitch-angle diffusion coefficients caused by plasmaspheric hiss

Plasmaspheric hiss is an important wave mode in the Earth’s radiation belts. Hiss waves can scatter energetic electrons into loss cones to precipitate into the atmosphere, and therefore become an important source of fluctuations, leading the radiation belt to lose electrons . As a function of electron energy and pitch angle, the diffusion coefficient of hiss waves for radiation belt electrons is significantly influenced by the solar wind and geomagnetic activity, and also strongly depends on the spatial position, the background magnetic field, and the plasma density distribution. In order to quickly obtain the diffusion coefficients of hiss waves on electrons in the radiation belt for modelling the global dynamics of the radiation belt, we systematically calculate the diffusion coefficients of hiss waves on electrons in the radiation belt by using the full diffusion code (FDC), and build a four-dimensional matrix database of diffusion coefficients for the spatial region <i>L</i> = 1.5–6, the cold plasma parameter <i>α</i><sup>*</sup> = 3–30, electron energy 1 keV–10 MeV, and electron throw angle 0°–90°. According to the database, we can quickly obtain diffusion coefficients with different <i>L</i> and <i>α</i><sup>*</sup> values through linear interpolations. By comparing the errors between diffusion coefficients calculated by the FDC code and those linearly interpolated from the diffusion coefficient database, the accuracies of interpolated coefficients are validated, showing that most of the errors lie in 10%. The four-dimensional database of hiss wave pitch angle diffusion coefficients for radiation belt electrons and the validated linear interpolation method established in this paper can significantly reduce the time required to obtain global information about hiss wave diffusion coefficients, thereby rapidly improving the computational efficiency of carrying out simulations of spatial and temporal changes in the radiation belts over long periods of time, which in turn is expected to provide favourable conditions for the development of dynamic forecasting models of the Earth's radiation belts.

On the domain of difference double sequence spaces of arbitrary orders

Abstract The prime objective of this paper is to define a new double difference operator with arbitrary order via which new classes of difference double sequences are introduced. Results on topological structures, dual spaces and four-dimensional matrix mappings related to the proposed difference double sequence spaces are discussed. As an application of this work, the proposed operator is being used to approximate partial derivatives of fractional orders. Some numerical examples are also given in support of the validity or the clear visualization of the results obtained.

Some Double Sequence Spaces Generated by Four-Dimensional Pascal Matrix

The purpose of this paper is to discover and examine a four-dimensional Pascal matrix domain on Pascal sequence spaces. We show that they are spaces and also establish their Schauder basis, topological properties, isomorphism and some inclusions.

Six degrees of freedom dynamic motion-including dose reconstruction in a commercial treatment planning system.

Intrafractional motion during radiotherapy delivery can deteriorate the delivered dose. Dynamic rotational motion of up to 38 degrees has been reported during prostate cancer radiotherapy, but methods to determine the dosimetric consequences of such rotations are lacking. Here, we create and experimentally validate a dose reconstruction method that accounts for dynamic rotations and translations in a commercial treatment planning system (TPS). Interplay effects are quantified by comparing dose reconstructions with dynamic and constant rotations. The dose reconstruction accumulates the dose in points of interest while the points are moved in six degrees of freedom (6DoF) in a precalculated time-resolved four-dimensional (4D) dose matrix to emulate dynamic motion in a patient. The required 4D dose matrix was generated by splitting the original treatment plan into multiple sub-beams, each representing 0.4s dose delivery, and recalculating the dose of the split plan in the TPS (Eclipse). The dose accumulation was performed via TPS scripting by querying the dose of each sub-beam in dynamically moving points, allowing dose reconstruction with any dynamic motion. The dose reconstruction was validated with film dosimetry for two prostate dual arc VMAT plans with intra-prostatic lesion boosts. The plans were delivered to a pelvis phantom with internal dynamic rotational motion of a film stack (21 films with 2.5mm separation). Each plan was delivered without motion and with three prostate motion traces. Motion-including dose reconstruction was performed for each motion experiment using the actual dynamic rotation as well as a constant rotation equal to the mean rotation during the experiment. For each experiment, the 3%/2mm γ failure rate of the TPS dose reconstruction was calculated with the film measurement being the reference. For each motion experiment, the motion-induced 3%/2mm γ failure rate was calculated using the static delivery as the reference and compared between film measurements and TPS dose reconstruction. DVH metrics for RT structures fully contained in the film volume were also compared between film and TPS. The mean γ failure rate of the TPS dose reconstructions when compared to film doses was 0.8% (two static experiments) and 1.7% (six dynamic experiments). The mean (range) of the motion-induced γ failure rate in film measurements was 35.4% (21.3-59.2%). The TPS dose reconstruction agreed with these experimental γ failure rates with root-mean-square errors of 2.1% (dynamic rotation dose reconstruction) and 17.1% (dose reconstruction assuming constant rotation). By DVH metrics, the mean (range) difference between dose reconstructions with dynamic and constant rotation was 4.3% (-0.3-10.6%) (urethra ), -0.6% (-5.6%-2.5%) (urethra ), 1.1% (-7.1-7.7%) (GTV ), -1.4% (-17.4-7.1%) (GTV ), -1.2% (-17.1-5.7%) (GTV ), and -0.1% (-3.2-7.6%) (GTV mean dose). Dose reconstructions with dynamic motion revealed large interplay effects (cold and hot spots). A method to perform dose reconstructions for dynamic 6DoF motion in a TPS was developed and experimentally validated. It revealed large differences in dose distribution between dynamic and constant rotations not identifiable through dose reconstructions with constant rotation.

AlphaTims: Indexing Trapped Ion Mobility Spectrometry–TOF Data for Fast and Easy Accession and Visualization

High-resolution MS-based proteomics generates large amounts of data, even in the standard LC–tandem MS configuration. Adding an ion mobility dimension vastly increases the acquired data volume, challenging both analytical processing pipelines and especially data exploration by scientists. This has necessitated data aggregation, effectively discarding much of the information present in these rich datasets. Taking trapped ion mobility spectrometry (TIMS) on a quadrupole TOF (Q-TOF) platform as an example, we developed an efficient indexing scheme that represents all data points as detector arrival times on scales of minutes (LC), milliseconds (TIMS), and microseconds (TOF). In our open-source AlphaTims package, data are indexed, accessed, and visualized by a combination of tools of the scientific Python ecosystem. We interpret unprocessed data as a sparse four-dimensional matrix and use just-in-time compilation to machine code with Numba, accelerating our computational procedures by several orders of magnitude while keeping to familiar indexing and slicing notations. For samples with more than six billion detector events, a modern laptop can load and index raw data in about a minute. Loading is even faster when AlphaTims has already saved indexed data in an HDF5 file, a portable scientific standard used in extremely large-scale data acquisition. Subsequently, data accession along any dimension and interactive visualization happens in milliseconds. We have found AlphaTims to be a key enabling tool to explore high-dimensional LC-TIMS-Q-TOF data and have made it freely available as an open-source Python package with a stand-alone graphical user interface at https://github.com/MannLabs/alphatims or as part of the AlphaPept ‘ecosystem’.

Eigenvectors of polarization coherency matrices.

Calculation of the eigenvectors of two- and three-dimensional coherency matrices, and the four-dimensional coherency matrix associated with a Mueller matrix, is considered, especially for algebraic cases, in the light of recently published algorithms. The preferred approach is based on a combination of an evaluation of the characteristic polynomial and an adjugate matrix. The diagonal terms of the coherency matrix are given in terms of the characteristic polynomial of reduced matrices as functions of the eigenvalues of the coherency matrix. The analogous polynomial form for the off-diagonal elements of the coherency matrix is also presented. Simple expressions are given for the pure component in the characteristic decomposition.

On the Domain of the Four-Dimensional Sequential Band Matrix in Some Double Sequence Spaces

Let E represent any of the spaces M u , C ϑ ( ϑ = { b , b p , r } ) , and L q ( 0 < q < ∞ ) of bounded, ϑ -convergent, and q-absolutely summable double sequences, respectively, and E ˜ be the domain of the four-dimensional (4D) infinite sequential band matrix B ( r ˜ , s ˜ , t ˜ , u ˜ ) in the double sequence space E, where r ˜ = ( r m ) m = 0 ∞ , s ˜ = ( s m ) m = 0 ∞ , t ˜ = ( t n ) n = 0 ∞ , and u ˜ = ( u n ) n = 0 ∞ are given sequences of real numbers in the set c ∖ c 0 . In this paper, we investigate the double sequence spaces E ˜ . First, we determine some topological properties and prove several inclusion relations under some strict conditions. Then, we examine α -, β ( ϑ ) -, and γ -duals of E ˜ . Finally, we characterize some new classes of 4D matrix mappings related to our new double sequence spaces and conclude the paper with some significant consequences.

Four-Dimensional Characteristic Matrix for Electromagnetic Coupling of Multilayer Sub-Wavelength Metasurface System

In this article, we propose a 4-D characteristic matrix theory for the electromagnetic coupling of the multilayer metasurface system. The multilayer metasurface system is generally composed of periodic sub-wavelength metallic elements printed on the substrates. In the system, not only the coupling between multilayer metallic elements but also the coupling between different polarized waves can exist. In order to analyze the couplings and determine the total transmission (reflection) characteristic, a series of 4-D characteristic matrices are established for the substrates and metallic elements to relate the total tangential fields (two electrical components and two magnetic components) at adjacent interfaces. Finally, a four-layer metasurface system is designed to verify the theory whose results agree well with the simulated ones at arbitrary incidence angle. Unlike the existing methods, the four-dimensional characteristic matrix theory is not only independent of the anisotropy of the metallic elements, but also insensitive to the layer number of the metasurface. Thus, the proposed theory holds valid for arbitrary multilayer metasurface system, regardless of the layer number and the metallic elements.

Refining the evaluation of eigen emittances measured by the dedicated four-dimensional emittance scanner ROSE

A dedicated device to fully determine the four-dimensional beam matrix, called ROSE (ROtating System for Emittance measurements) was successfully commissioned. Results obtained with 83Kr13+ at 1.4 MeV/u are reported in Xiao et al. (2016). Coupled moments were determined with an accuracy of about 10%, which is sufficiently low to reliably determine a lattice which could decouple the beam. However, the remaining uncertainty on the corresponding eigen emittances was still considerable high. The present paper reports on improvement of the evaluation procedure which lowers the inaccuracy of measured eigen emittances significantly to the percent level. The method is based on trimming directly measured data within their intrinsic measurement resolution such that the finally resulting quantity is determined with high precision.