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The pseudo core inverse of a lower triangular matrix

REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS 2019年第2期113卷 423-434页

作者： Gao, Yuefeng Chen, Jianlong Southeast Univ Sch Math Nanjing 210096 Jiangsu Peoples R China

The notion of core inverse was introduced by Baksalary and Trenkler for a complex matrix of index one in 2010, and then it was generalized to an arbitrary -ring case by Raki, Dini and Djordjevi in 2014. Later, the notion of pseudo core inverse extended the notion of core inverse to an element of an arbitrary index in -rings;meanwhile, it generalized the notion of core-EP inverse, introduced by Manjunatha Prasad and Mohana for complex matrices in 2014, to the case of -ring. In this paper, we characterize the pseudo core invertibility of a matrix in product form under some prescribed conditions. As applications, the existence and expression for the pseudo core inverse of a lower triangular matrix are considered.

关键词： Pseudo core inverse Core inverse Core-EP inverse lower triangular matrix 15A09 16W10 16S50

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Homogeneous Sylvester equations of lower triangular matrices

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LINEAR ALGEBRA AND ITS APPLICATIONS 2022年 653卷 183-192页

作者： Lim, Yongdo Sungkyunkwan Univ Dept Math Suwon 440746 South Korea

We study the homogeneous Sylvester equation AX = XB of lower triangular matrices with 1's along the diagonal. A necessary and sufficient condition for existence of solution is presented in terms of computable line... 详细信息

关键词： Homogeneous Sylvester equation lower triangular matrix Loos symmetric space Staircase matrix

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Optimizing smart grid flexibility with a hybrid MINLP framework for renewable integration in urban energy systems

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Energy Reports 2025年 14卷 508-523页

作者： Sameer Algburi Omer Al-Dulaimi Hassan Falah Fakhruldeen Shakhlo Isametdinova I.B. Sapaev Saiful Islam Quadri Noorulhasan Naveed Ayodele Lasisi Israa alhani Qusay Hassan Doaa H. Khalaf Michael Ssebunya Feryal Ibrahim Jabbar College of Engineering Al-Kitab University Kirkuk 36015 Iraq Electrical Technical College Al-Farahidi University Baghdad Iraq Computer Techniques Engineering Department Faculty of Information Technology Imam Ja’afar Al-Sadiq University Baghdad 10011 Iraq Associate professor Kimyo International University in Tashkent Shota Rustaveli street 156 Тashkent 100121 Uzbekistan Head of the department Physics and Chemistry Tashkent Institute of Irrigation and Agricultural Mechanization Engineers” National Research University Tashkent Uzbekistan Civil Engineering Department College of Engineering King Khalid University Abha 61421 Saudi Arabia Department of Computer Science College of Computer Science King Khalid University Abha Saudi Arabia Mazaya University College Iraq College of Engineering University of Deyala Diyala Iraq Design Department Al-Turath University College Baghdad Iraq Medical Physics Department College of Sciences Al-Mustaqbal University Babil 51001 Iraq

Urban grids face the challenge of expanding renewable deployment while curbing emissions and minimizing the capital burden of network reinforcements, all of which depend on effective flexibility integration. A hybrid optimization framework is introduced, combining Mixed-Integer Nonlinear Programming with a reformulated MILP structure to jointly size photovoltaic systems, battery storage, staged network upgrades, and the dynamic participation of electric vehicles as both load and distributed storage. Thousands of EV constraints are consolidated through a polytope-based approach, and reinforcement costs are captured using a piece-wise linear model tailored to feeder capacity increments. Application to the Tuwaiq Smart City network, covering 3780 households, employs one-minute resolution data for 2024 to benchmark five operational schemes: No Flexibility, Demand Response, Smart Charging, Vehicle-to-Grid, and Integrated Decentralised Energy Management (IDEM). Compared with the baseline, IDEM achieves a 43.8 % reduction in annualised system cost, 46 % decrease in peak imports, and capacity cuts of 75 % and 82 % for PV and storage respectively, alongside a 65 % drop in grid integration expenses. A Monte Carlo test of 150 runs confirms cost stability within ±6 %, validating the robustness of layered flexibility under stochastic solar and mobility profiles. Solving across a full-year span is achieved within minutes on standard hardware, confirming the framework’s practical value for strategic energy planning

关键词： BESS Battery Energy Storage System DER Distributed Energy Resources DR Demand Response EV Electric Vehicle IDEM Integrated Decentralized Energy Management MILP Mixed-Integer Linear Programming NF No Flexibility PV Photovoltaic SC Smart Charging V2G Vehicle-to-Grid E_ i , e_ i+1 Interval of demand A _PV , A_B, Factors for PV system and BESS B _BESS Factor for BESS B _op Factor for operational costs B _PV Factor for PV System C matrix of coefficients for linear inequalities C _active Available energy for use C _B BESS capacity C _cap Maximum capacity of the EV battery C _depart Required energy at time of departure C _EV Electric vehicle energy c _i , d_ i Coefficients in cost function C _init Initial stored energy c _PV , c_B Cost Coefficients for PV System and BESS C b0 Initial battery capacity at the start of the time period C b Current battery capacity C bmax Maximum capacity of the battery D Set of all time periods d Right-Hand side of linear inequalities D _peakt Peak demand D Set of all time periods considered E _eff Efficiency of power electronics E _S Energy Capacity of BESS Ein EVt Energy charged by the EV Eout BESSt Energy discharged by the BESS Eout EVt Energy discharged by the EV F _op Operational expenses FSPs Service providers g D_peak Cost function for peak demand g _sum Sum of costs not depending on sizing variables G gridt Power exchanged with the grid G pvt Power generated by the PV system I t Solar Irradiance K _derate Derate factor for PV system L t Load or the energy consumption l′ Unit price of PV equipment M Identity matrix MINLP Mixed-Integer Non-Linear Programming NOCT Modified nominal operating cell temperature Q _comb Combined polytope Q _gridt Grid Power Q _i Individual polytopes Q _max Maximum Grid Power Rt Charging/Discharging rate R_ min , R_ max Min/Max charging/discharging rates R c t , R dt Rates of charging and discharging Rct Rate of charging at time t R cdL Minimum charging rate R cdU Maximum charging/discharging rate R dt Rate of discharging at time t S _efft Efficiency ratio o

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RIEMANNIAN GEOMETRY OF SYMMETRIC POSITIVE DEFINITE MATRICES VIA CHOLESKY DECOMPOSITION

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SIAM JOURNAL ON matrix ANALYSIS AND APPLICATIONS 2019年第4期40卷 1353-1370页

作者： Lin, Zhenhua Natl Univ Singapore Singapore 117546 Singapore

We present a new Riemannian metric, termed Log-Cholesky metric, on the manifold of symmetric positive definite (SPD) matrices via Cholesky decomposition. We first construct a Lie group structure and a bi-invariant metric on Cholesky space, the collection of lower triangular matrices whose diagonal elements are all positive. Such group structure and metric are then pushed forward to the space of SPD matrices via the inverse of Cholesky decomposition that is a bijective map between Cholesky space and SPD matrix space. This new Riemannian metric and Lie group structure fully circumvent swelling effect, in the sense that the determinant of the Frechet average of a set of SPD matrices under the presented metric, called Log-Cholesky average, is between the minimum and the maximum of the determinants of the original SPD matrices. Comparing to existing metrics such as the affine-invariant metric and Log-Euclidean metric, the presented metric is simpler, more computationally efficient, and numerically stabler. In particular, parallel transport along geodesics under Log-Cholesky metric is given in a closed and easy-to-compute form.

关键词： Frechet mean symmetric positive definite matrix Lie group bi-invariant metric parallel transport Cholesky decomposition lower triangular matrix

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Resource Tuned Optimal Random Network Coding for Single Hop Multicast future 5G Networks

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INTERNATIONAL JOURNAL OF ELECTRONICS AND TELECOMMUNICATIONS 2019年第3期65卷 463-469页

作者： Dong, Dhawa Sang Pokhrel, Yagnya Murti Gachhadar, Anand Maharjan, Ram Krishna Qamar, Faizan Amiri, Iraj Sadegh Kathmandu Univ Sch Engn Dept Elect & Elect Engn Dhulikhel Nepal Tribhuvan Univ Inst Engn Dept Elect & Comp Engn Pulchowk Campus Lalitpur Nepal Univ Malaya Fac Engn Dept Elect Engn Kuala Lumpur 50603 Malaysia Ton Duc Thang Univ Adv Inst Mat Sci Computat Opt Res Grp Ho Chi Minh City Vietnam Ton Duc Thang Univ Adv Inst Mat Sci Fac Appl Sci Ho Chi Minh City Vietnam

Optimal random network coding is reduced complexity in computation of coding coefficients, computation of encoded packets and coefficients are such that minimal transmission bandwidth is enough to transmit coding coefficient to the destinations and decoding process can be carried out as soon as encoded packets are started being received at the destination and decoding process has lower computational complexity. But in traditional random network coding, decoding process is possible only after receiving all encoded packets at receiving nodes. Optimal random network coding also reduces the cost of computation. In this research work, coding coefficient matrix size is determined by the size of layers which defines the number of symbols or packets being involved in coding process. Coding coefficient matrix elements are defined such that it has minimal operations of addition and multiplication during coding and decoding process reducing computational complexity by introducing sparseness in coding coefficients and partial decoding is also possible with the given coding coefficient matrix with systematic sparseness in coding coefficients resulting lower triangular coding coefficients matrix. For the optimal utility of computational resources, depending upon the computational resources unoccupied such as memory available resources budget tuned windowing size is used to define the size of the coefficient matrix.

关键词： Coding coefficients computational complexity lower triangular matrix random network coding sparse coding coefficients

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A new algorithm of blind color image watermarking based on LU decomposition

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MULTIDIMENSIONAL SYSTEMS AND SIGNAL PROCESSING 2018年第3期29卷 1055-1074页

作者： Su, Qingtang Wang, Gang Zhang, Xiaofeng Lv, Gaohuan Chen, Beijing Ludong Univ Sch Informat Sci & Engn Yantai 264025 Shandong Peoples R China Nanjing Univ Informat Sci & Technol Sch Comp & Software Nanjing 210044 Jiangsu Peoples R China

This paper proposes a new algorithm of blind color image watermarking based on LU decomposition. Because of the first column the second row element and the first column the third row element of the lower triangular matrix, which obtained by LU decomposition, have higher similarity, and the color image watermark can be embedded into these elements by slightly modifying rules, which can enhance the watermark invisibility. In addition, Arnold transform is applied to improve the watermark security and the Hash pseudo-random number algorithm based on MD5 is used to improve the watermark robustness. In extraction process, the watermark embedding strength and private key are needed to extract watermark information from the attacked host image with blind manner. Experiment results reveal that the proposed method outperforms other related methods in the aspects of the invisibility, robustness, embedding payload and computational complexity.

关键词： Color image watermark Blind watermarking LU decomposition lower triangular matrix

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A generalization of zero divisor graphs associated to commutative rings

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PROCEEDINGS OF THE INDIAN ACADEMY OF SCIENCES-MATHEMATICAL SCIENCES 2018年第1期128卷 1-11页

作者： Afkhami, M. Erfanian, A. Khashyarmanesh, K. Moosavi, N. Vaez Univ Neyshabur Dept Math POB 91136-899 Neyshabur Iran Ferdowsi Univ Mashhad Dept Pure Math Int CampusPOB 1159-91775 Mashhad Iran

Let R be a commutative ring with a nonzero identity element. For a natural number n, we associate a simple graph, denoted by Gamma(n)(R), with R-n\{0} as the vertex set and two distinct vertices X and Y in R-n being a... 详细信息

Let R be a commutative ring with a nonzero identity element. For a natural number n, we associate a simple graph, denoted by Gamma(n)(R), with R-n\lower triangular matrix as the vertex set and two distinct vertices X and Y in R-n being adjacent if and only if there exists an n x n lower triangular matrix A over R whose entries on the main diagonal are nonzero and one of the entries on the main diagonal is regular such that X-T AY = 0 or Y-T AX = 0, where, for a matrix B, B-T is the matrix transpose of B. If n = 1, then Gamma(n)(R) is isomorphic to the zero divisor graph Gamma(R), and so Gamma(n)(R) is a generalization of Gamma(R) which is called a generalized zero divisor graph of R. In this paper, we study some basic properties of Gamma(n)(R). We also determine all isomorphic classes of finite commutative rings whose generalized zero divisor graphs have genus at most three.

关键词： Zero divisor graph lower triangular matrix genus complete graph

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Bounds for the norm of lower triangular matrices on the Cesaro weighted sequence space

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JOURNAL OF INEQUALITIES AND APPLICATIONS 2017年第1期2017卷 1-11页

作者： Foroutannia, Davoud Roopaei, Hadi Vali E Asr Univ Rafsanjan Dept Math Rafsanjan Iran

This paper is concerned with the problem of finding bounds for the norm of lower triangular matrix operators from l(p)(w) into c(p)(w), where c(p)(w) is the Cesaro weighted sequence space and (w(n)) is a non-negative sequence. Also this problem is considered for lower triangular matrix operators from c(p)(w) into l(p)(w), and the norms of certain matrix operators such as Cesaro, Norlund and weighted mean are computed.

关键词： norm lower triangular matrix Norlund matrix weighted mean matrix weighted sequence space

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GPU maps for the space of computation in triangular domain problems 16

GPU maps for the space of computation in triangular domain p...

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16th IEEE Int Conf on High Performance Computing and Communications/11th IEEE Int Conf on Embedded Software and Systems\6th Int Symposium on Cyberspace Safety and Security

作者： Navarro, Cristobal A. Hitschfeld, Nancy Univ Chile Dept Comp Sci Santiago Chile Ctr Estudios Cientif CECs Valdivia Chile

ISBN: (纸本)9781479961238

There is a stage in the GPU computing pipeline where a grid of thread-blocks, or space of computation, is mapped to the problem domain. Normally, the space of computation is a k-dimensional bounding box (BB) that covers a k-dimensional problem. Threads that fall inside the problem domain perform computations and threads that fall outside are discarded, all happening at runtime. For problems with non-square geometry, this approach makes the space of computation larger than what is necessary, wasting many threads. Our case of interest are the twodimensional triangular domain problems, alias td-problems, where almost half of the space of computation is unnecessary when using the BB approach. Problems such as the Euclidean distance map or collision detection are td-problems and they appear frequently as part of a larger computational problem. In this work, we study several mapping functions and their contribution to a better space of computation by reducing the number of unnecessary threads. We compare the performance of four existing mapping strategies;the bounding box (BB), the upper-triangular mapping (UTM), the rectangular box (RB) and the recursive partition (REC). In addition, we propose a map g(.), that maps any. block to a unique location (i, j) in the triangular domain. The mapping is based on the properties of the lower triangular matrix and works in block space. The theoretical improvement I obtained from using g(.) is upper bounded as I < 2 and the number of unnecessary blocks is reduced from O(n 2) to O(n). Experimental results using different Nvidia Kepler GPUs show that for computing the Euclidean distance matrix g(.) achieves an improvement of up to 18% over the basic bounding box (BB) strategy, runs faster than UTM and REC strategies and it is almost as fast as RB. Performance results on shared memory 3D collision detection show that g(.) is the fastest map of all, and the only one capable of surpassing the brute force (BB) approach by a margin of up to

关键词： GPU map function space of computation triangular domain problems mapping functions GRAPPER PICK UP Graphics Processing Unit THREADS MAPPING(MATHEMATICAL) arithmetic maps bounding boxes lower triangular matrix Domain triangular Space

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A Compact Discriminative Representation for Efficient Image-set Classification with Application to Biometric Recognition

A Compact Discriminative Representation for Efficient Image-...

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IAPR International Conference on Biometrics

作者： Muhammad Uzair Arif Mahmood Ajmal Mian Chris McDonald Computer Science & Software Engineering The University of Western Australia

ISBN: (纸本)9781479903115

We present a simple yet compact and discriminative representation for image sets which can efficiently be used for image-set based object classification. For each image-set we compute a global covariance matrix which captures correlated variations in all image-set dimensions. Without loss of information, we compact the covariance matrix into a lower triangular matrix by using Cholesky decomposition. While preserving discrimination capability of the representation, we obtain further compression by applying Multiple Discriminant Analysis. As a result, we are able to represent image sets containing N samples each of dimensionality d by a single vector whose dimensionality is << Nd. We apply the proposed representation to various biometric applications such as image-set based face recognition and person identification using image-sets of periocular regions. To show that our representation is generic, we also report results for image-set based object categorization. We observe improved accuracy and significant speedup over the current state-of-the-art techniques on standard datasets.

关键词： variance covariance matrix lower triangular matrix Biometric recognition image set Compact Compact Pattern discriminant analysis Covariance matrix Facial Recognition Neodymium

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