Special least squares solutions of the reduced biquaternion matrix equation with applications

This paper presents an efficient method for obtaining the least squares Hermitian solutions of the reduced biquaternion matrix equation $(AXB, CXD) = (E, F )$. The method leverages the real representation of reduced biquaternion matrices. Furthermore, we establish the necessary and sufficient conditions for the existence and uniqueness of the Hermitian solution, along with a general expression for it. Notably, this approach differs from the one previously developed by Yuan et al. $(2020)$, which relied on the complex representation of reduced biquaternion matrices. In contrast, our method exclusively employs real matrices and utilizes real arithmetic operations, resulting in enhanced efficiency. We also apply our developed framework to find the Hermitian solutions for the complex matrix equation $(AXB, CXD) = (E, F )$, expanding its utility in addressing inverse problems. Specifically, we investigate its effectiveness in addressing partially described inverse eigenvalue problems. Finally, we provide numerical examples to demonstrate the effectiveness of our method and its superiority over the existing approach.Comment: 25 pages, 3 figure

L-structure least squares solutions of reduced biquaternion matrix equations with applications

This paper presents a framework for computing the structure-constrained least squares solutions to the generalized reduced biquaternion matrix equations (RBMEs). The investigation focuses on three different matrix equations: a linear matrix equation with multiple unknown L-structures, a linear matrix equation with one unknown L-structure, and the general coupled linear matrix equations with one unknown L-structure. Our approach leverages the complex representation of reduced biquaternion matrices. To showcase the versatility of the developed framework, we utilize it to find structure-constrained solutions for complex and real matrix equations, broadening its applicability to various inverse problems. Specifically, we explore its utility in addressing partially described inverse eigenvalue problems (PDIEPs) and generalized PDIEPs. Our study concludes with numerical examples.Comment: 30 page

(R, S) conjugate solution to coupled Sylvester complex matrix equations with conjugate of two unknowns

In this work, we are concerned with (R, S) – conjugate solutions to coupled Sylvester complex matrix equations with conjugate of two unknowns. When the considered two matrix equations are consistent, it is demonstrated that the solutions can be obtained by utilizing this iterative algorithm for any initial arbitrary (R,S) – conjugate matrices V1,W1. A necessary and sufficient condition is established to guarantee that the proposed method converges to the (R,S) – conjugate solutions. Finally, two numerical examples are provided to demonstrate the efficiency of the described iterative technique

Matris denklemleri ile ilişkili bazı özel tipli matrisler için matris yakınlık problemi

06.03.2018 tarihli ve 30352 sayılı Resmi Gazetede yayımlanan “Yükseköğretim Kanunu İle Bazı Kanun Ve Kanun Hükmünde Kararnamelerde Değişiklik Yapılması Hakkında Kanun” ile 18.06.2018 tarihli “Lisansüstü Tezlerin Elektronik Ortamda Toplanması, Düzenlenmesi ve Erişime Açılmasına İlişkin Yönerge” gereğince tam metin erişime açılmıştır.Anahtar Kelimeler: minimum kalan problemi, matris yakınlık problemi, en iyi yaklaşık çözüm, Moore-Penrose ters. İlk bölümde lineer matris denklem problemleri ile ilgili literatür bilgisine yer verilmiş ve çalışmanın içeriğini oluşturan problemler tanıtılmıştır. İkinci bölümde çalışmada kullanılan bazı tanımlar ve temel teoremlerden bahsedilmiştir. Üçüncü bölümün ilk kısmında $\left( {{A_1}X{B_1},{A_2}X{B_2}, \ldots ,{A_k}X{B_k}} \right) = \left( {{C_1},{C_2}, \ldots ,{C_k}} \right)$ matris denkleminin simetrik ve ters-simetrik matrisler için genel çözümlerinin kümesi ve en küçük kareler çözümlerinin kümesi, Moore-Penrose ters ve Kronecker çarpım kullanılarak incelenmiştir. Bu matris denkleminin en iyi yaklaşık simetrik çözümü ve en iyi yaklaşık ters-simetrik çözümü ortaya konulmuştur. İkinci kısmında AXB=C matris denkleminin (P,Q)-ortogonal simetrik ve (P,Q)-ortogonal ters-simetrik matrisler için genel çözümlerinin kümesi ve en küçük kareler çözümlerinin kümesi Moore-Penrose ters ve spektral ayrışım kullanılarak incelenmiştir. Daha sonra, en iyi yaklaşık (P,Q)-ortogonal simetrik çözümü ve (P,Q)-ortogonal ters-simetrik çözümü elde edilmiştir. Son olarak, her iki kısmın sonunda ele alınan problemlerin çözümünü elde etmek için kullanılan bir algoritma, iki örnek ve literatürden seçilmiş örnekler için karşılaştırmalı bir tablo verilmiştir. Dördüncü bölümde (AXB,CXD)=(E,F) kuaterniyon matris denkleminin merkezi-hermityen ve ters-merkezi-hermityen matrisler üzerinde minimum kalan problemi Moore-Penrose ters, Kronecker çarpım ve vec operatörü kullanılarak incelenmiştir. Daha sonra ise (AXB,CXD)=(E,F) kuaterniyon matris denkleminin en iyi yaklaşık merkezi-hermityen çözümü ve ters-merkezi-hermityen çözümü verilmiştir. Son olarak, bölüm sonunda ele alınan problemlerin çözümünü elde etmek için kullanılan bir algoritma ve iki sayısal örnek verilmiştir. Son bölüm ise sonuçların kısa bir tartışmasına ayrılmıştır.Yeni kaotik sistemin FPGA tabalı tasarım

Data-driven h2 model reduction for linear discrete-time systems

We present a new framework of $h^{2}$ optimal model reduction for linear discrete-time systems. Our main contribution is to create optimal reduced order models in the $h^{2}$-norm sense directly from the measurement data alone, without using the information of the original system. In particular, we focus on the fact that the gradient of the $h^{2}$ model reduction problem is expressed using the discrete-time Lyapunov equation and the discrete-time Sylvester equation, and derive the data-driven gradient. In the proposed algorithm, the initial point is chosen as the output of the existing data-driven methods. Numerical experiments are conducted to show that the proposed method produce better reduced order models in the $h^{2}$-norm sense than other data-driven model order reduction approaches.Comment: 8 pages, 3 figure

Method and system for detecting a failure or performance degradation in a dynamic system such as a flight vehicle

A method and system for detecting a failure or performance degradation in a dynamic system having sensors for measuring state variables and providing corresponding output signals in response to one or more system input signals are provided. The method includes calculating estimated gains of a filter and selecting an appropriate linear model for processing the output signals based on the input signals. The step of calculating utilizes one or more models of the dynamic system to obtain estimated signals. The method further includes calculating output error residuals based on the output signals and the estimated signals. The method also includes detecting one or more hypothesized failures or performance degradations of a component or subsystem of the dynamic system based on the error residuals. The step of calculating the estimated values is performed optimally with respect to one or more of: noise, uncertainty of parameters of the models and un-modeled dynamics of the dynamic system which may be a flight vehicle or financial market or modeled financial system

Monitoring and managing genetic diversity in Sitka spruce (Picea sitchensis (Bong.) Carr.)

The British breeding programme (Programme) for Sitka spruce (Picea sitchensis (Bong.) Carr.) began in 1963 by selecting individual plus trees with superior characteristics for construction-grade timber. With the establishment of seed orchards and vegetative propagation programmes, the production of improved forest reproductive material from the Programme intensified. Currently, the vast majority (>90%) of Sitka plants sold in Great Britain come from the Programme. Since the beginnings of genetic improvement of Sitka spruce in GB predates the readily available molecular genotyping methods, little is known about the genetic diversity inside the Programme. The thesis explores genetic diversity inside the different stages of the Programme. Using microsatellite (SSR) markers, chapter Chapter 1 assesses the stepwise reduction of diversity from natural populations to vegetative propagation programmes, differentiation between different seed years/provenances, and considers the origin of plus trees. Chapter 2 focuses on simple, nonmolecular methods for assessing unequal parentage contributions in a seed orchard that lead to the loss of genetic diversity. Chapter 3 utilises SSR markers to assign parents to the seed crop, quantifies and spatially analyses gene flow inside a seed orchard, and provides G-statistics of the seed crop from the orchard. Chapter 4 explores the appropriateness of SNP markers for internal control and tracking of forest reproductive material from clonal archives through seed nurseries and forest plantations. Results presented in Chapter 1 suggest that the breeding population of Sitka spruce in Britain exhibits high levels of genetic diversity. The levels are comparable and sometimes exceed those of unimproved natural populations. The highest average and effective number of alleles (12.58 and 6.27, respectively) were found in the 60 elite plus trees subsample. Similar levels are also found in the natural regeneration cores sampled under unimproved Sitka stands, which is promising for any continuous cover forest management. Genetic transfer from parents inside the studied seed orchard in Chapter 2 was incomplete and varied substantially between years. Effective population sizes based on seed viability of collected crops ranged from 7.6 in 2020 to 29.5 in 2019 (total population size is 35). The general health status of each ramet, male and female cone abundance and frost effect were all significant factors for yearly variations in genetic diversity. A molecular approach for studying the extent of genetic transfer inside the tree orchard in Chapter 3 confirms incomplete transfer and yearly variations. Low contamination rates from beyond the orchard were also detected (5.12%). A strong correlation between male flowering abundance and father contributions in successful pollinations (R2=0.41) suggests that forestry practitioners can use simple non-molecular methods for basic genetic monitoring of orchards. Finally, Chapter 4 reveals worrying levels of genetic diversity in highly improved vegetatively propagated full-sib families (VP FRM). Most samples (98%) inside the VP FRM plantations were assigned to the wrong pedigree due to mislabelled maps, clerical errors, replanting after high mortality and contaminations at the tree nurseries. Discrepancies between clonal identities inside the clonal archives were also found (27% error), suggesting that systematic testing of the breeding population would benefit further improvement efforts. Molecular markers proved to be a helpful tool for internal quality control. Comparison between SSR and SNP markers suggests that SSR markers outperform the SNP Sequenom array in the current state, and further efforts would be needed to optimize the SNP method for internal control of the breeding programme. In each of the chapters, practical advice for forestry practitioners is given. In conclusion, genetic diversity inside the Sitka spruce British breeding programme stores plenty of genetic diversity ready to be deployed. More efforts should be put into optimizing the yearly variation of seed quality from seed orchards by mixing different seed years and maintaining the general health of ramets. Additionally, levels of genetic diversity at the most improved stages of the improvement programme need to be lowered by controlling contaminations and human errors as they undermine breeding efforts to achieve desired high genetic gains

Study on Parametric Optimization of Fused Deposition Modelling (FDM) Process

Rapid prototyping (RP) is a generic term for a number of technologies that enable fabrication of physical objects directly from CAD data sources. In contrast to classical methods of manufacturing such as milling and forging which are based on subtractive and formative principles espectively, these processes are based on additive principle for part fabrication. The biggest advantage of RP processes is that an entire 3-D (three-dimensional) consolidated assembly can be fabricated in a single setup without any tooling or human intervention; further, the part fabrication methodology is independent of the mplexity of the part geometry. Due to several advantages, RP has attracted the considerable attention of manufacturing industries to meet the customer demands for incorporating continuous and rapid changes in manufacturing in shortest possible time and gain edge over competitors. Out of all commercially available RP processes, fused deposition modelling (FDM) uses heated thermoplastic filament which are extruded from the tip of nozzle in a prescribed manner in a temperature controlled environment for building the part through a layer by layer deposition method. Simplicity of operation together with the ability to fabricate parts with locally controlled properties resulted in its wide spread application not only for prototyping but also for making functional parts. However, FDM process has its own demerits related with accuracy, surface finish, strength etc. Hence, it is absolutely necessary to understand the shortcomings of the process and identify the controllable factors for improvement of part quality. In this direction, present study focuses on the improvement of part build methodology by properly controlling the process parameters. The thesis deals with various part quality measures such as improvement in dimensional accuracy, minimization of surface roughness, and improvement in mechanical properties measured in terms of tensile, compressive, flexural, impact strength and sliding wear. The understanding generated in this work not only explain the complex build mechanism but also present in detail the influence of processing parameters such as layer thickness, orientation, raster angle, raster width and air gap on studied responses with the help of statistically validated models, microphotographs and non-traditional optimization methods. For improving dimensional accuracy of the part, Taguchi‟s experimental design is adopted and it is found that measured dimension is oversized along the thickness direction and undersized along the length, width and diameter of the hole. It is observed that different factors and interactions control the part dimensions along different directions. Shrinkage of semi molten material extruding out from deposition nozzle is the major cause of part dimension reduction. The oversized dimension is attributed to uneven layer surfaces generation and slicing constraints. For recommending optimal factor setting for improving overall dimension of the part, grey Taguchi method is used. Prediction models based on artificial neural network and fuzzy inference principle are also proposed and compared with Taguchi predictive model. The model based on fuzzy inference system shows better prediction capability in comparison to artificial neural network model. In order to minimize the surface roughness, a process improvement strategy through effective control of process parameters based on central composite design (CCD) is employed. Empirical models relating response and process parameters are developed. The validity of the models is established using analysis of variance (ANOVA) and residual analysis. Experimental results indicate that process parameters and their interactions are different for minimization of roughness in different surfaces. The surface roughness responses along three surfaces are combined into a single response known as multi-response performance index (MPI) using principal component analysis. Bacterial foraging optimisation algorithm (BFOA), a latest evolutionary approach, has been adopted to find out best process parameter setting which maximizes MPI. Assessment of process parameters on mechanical properties viz. tensile, flexural, impact and compressive strength of part fabricated using FDM technology is done using CCD. The effect of each process parameter on mechanical property is analyzed. The major reason for weak strength is attributed to distortion within or between the layers. In actual practice, the parts are subjected to various types of loadings and it is necessary that the fabricated part must withhold more than one type of loading simultaneously.To address this issue, all the studied strengths are combined into a single response known as composite desirability and then optimum parameter setting which will maximize composite desirability is determined using quantum behaved particle swarm optimization (QPSO). Resistance to wear is an important consideration for enhancing service life of functional parts. Hence, present work also focuses on extensive study to understand the effect of process parameters on the sliding wear of test specimen. The study not only provides insight into complex dependency of wear on process parameters but also develop a statistically validated predictive equation. The equation can be used by the process planner for accurate wear prediction in practice. Finally, comparative evaluation of two swarm based optimization methods such as QPSO and BFOA are also presented. It is shown that BFOA, because of its biologically motivated structure, has better exploration and exploitation ability but require more time for convergence as compared to QPSO. The methodology adopted in this study is quite general and can be used for other related or allied processes, especially in multi input, multi output systems. The proposed study can be used by industries like aerospace, automobile and medical for identifying the process capability and further improvement in FDM process or developing new processes based on similar principle