Minimal Matrices Research Articles

Some characterizations of minimal matrices with operator norm

Some characterizations of minimal matrices with operator norm

Methods for solving ill-conditioned systems of linear algebraic equations that improve the conditionality

The problem of solving systems of linear algebraic equations (SLAE) with an ill-conditioned or degenerate exact matrix and an approximate right-hand side is considered. A scheme for solving such a problem is proposed and justified, which makes it possible to improve the conditionality of the SLAE matrix. As a result, an approximate solution that is stable to perturbations of the right-hand side is obtained with a higher accuracy than when using some other methods. The scheme is implemented by an algorithm that uses minimal pseudoinverse matrices. The results of numerical experiments are presented, confirming the theoretical provisions of the article.

Minimal Conditioned Stiffness Matrices with Frequency-Dependent Path Following for Arbitrary Elastic Layers over Half-Spaces

This paper introduces an efficient computational procedure for analyzing the propagation of harmonic waves in layered elastic media. This offers several advantages, including the ability to handle arbitrary frequencies, depths, and the number of layers above an elastic half-space, and efforts to follow dispersion curves and flag up possible singularities are investigated. While there are inherent limitations in terms of computational accuracy and capacity, this methodology is straightforward to implement for studying free or forced vibrations and obtaining relevant response data. We present computations of wavenumber dispersion diagrams, phase velocity plots, and response data in both the frequency and time domains. These computational results are provided for two example cases: plane strain and axisymmetry. Our methodology is grounded in a well-conditioned dynamic stiffness approach specifically tailored for deep-layered strata analysis. We introduce an innovative method for efficiently computing wavenumber dispersion curves. By tracking the slope of these curves, users can effectively manage continuation parameters. We illustrate this technique through numerical evidence of a layer resonance in a real-life case study characterized by a fold in the dispersion curves. Furthermore, this framework is particularly advantageous for engineers addressing problems related to ground-borne vibrations. It enables the analysis of phenomena such as zero group velocity (ZGV), where a singularity occurs, both in the frequency and time domains, shedding light on the unique characteristics of such cases. Given the reduced dimension of the problem, this formulation can considerably aid geophysicists and engineers in areas such as MASW or SASW techniques.

Methods of Correcting Errors in Messages Encoded by Fibonacci Matrices

The main problems of detection and available methods of correcting errors in encoded messages with Fibonacci matrices, which make it possible to find and correct one, two and three errors in the same or different lines of the code word, are analyzed. It has been found that even in the last decade, many scientists have published a significant number of various publications, each of which to one degree or another substantiates the expediency of using Fibonacci matrices for (de)coding data. It has been established that the elements of a codeword obtained by multiplying a message block by a Fibonacci matrix have many useful properties, which are the basis for the method for detecting and correcting errors in them. The statement is given, according to which the ratio of the corresponding elements of the code word is close to the golden ratio, which is important for the existing methods of correcting potential errors. This property of the elements makes it possible to identify the presence of double and triple false elements by checking whether their ratios belong to a fixed interval. It is found that the false affiliation indicates that there are two errors in different lines of the codeword, which require solving the corresponding Diophantine equations, the suitability of the solution of which must satisfy certain conditions for error correction. It was found that in order to correct two errors in one line of the code word, a condition was introduced according to which the set of blocks of the input message should contain only minimal matrices, which makes it possible to take the smallest solutions of the Diophantine equation, the suitability of which is specified by test ratios. It was found that in order to correct three errors in a codeword, it is necessary to check whether the relations of its corresponding elements belong to a fixed interval and to solve a nonlinear Diophantine equation, the implementation of which is extremely difficult. The proposed approach boils down to trial and error, according to which you first need to find the exact location of the erroneous elements, and only then correct them according to the appropriate methods.

Constructions of minimal Hermitian matrices related to a C*-subalgebra of 𝑀_{𝑛}(ℂ)

This paper provides a constructive method using unitary diagonalizable elements to obtain all hermitian matrices A A in M n ( C ) M_n(\Bbb C) such that ‖ A ‖ = min B ∈ B ‖ A + B ‖ , \begin{equation*} \|A\|=\min _{B\in \mathcal {B}}\|A+B\|, \end{equation*} where B \mathcal {B} is a C*-subalgebra of M n ( C ) M_n(\Bbb C) , ‖ ⋅ ‖ \|\cdot \| denotes the operator norm. Such an A A is called B \mathcal {B} -minimal. Moreover, for a C*-subalgebra B \mathcal {B} determined by a conditional expectation from M n ( C ) M_n(\Bbb C) onto it, this paper constructs ⨁ i = 1 k B \bigoplus _{i=1}^k\mathcal {B} -minimal hermitian matrices in M k n ( C ) M_{kn}(\Bbb C) through B \mathcal {B} -minimal hermitian matrices in M n ( C ) M_n(\Bbb C) , and gets a dominated condition that the matrix A ^ = diag ⁡ ( A 1 , A 2 , ⋯ , A k ) \hat {A}\!=\!\operatorname {diag}(A_1,A_2,\cdots , A_k) is ⨁ i = 1 k B \bigoplus _{i=1}^k\mathcal {B} -minimal if and only if ‖ A ^ ‖ ≤ ‖ A s ‖ \|\hat {A}\|\leq \|A_s\| for some s ∈ { 1 , 2 , ⋯ , k } s\in \{1,2,\cdots ,k\} and A s A_s is B \mathcal {B} -minimal, where A i ( 1 ≤ i ≤ k ) A_i(1\leq i\leq k) are hermitian matrices in M n ( C ) M_n(\Bbb C) .

Moment of a subspace and joint numerical range

For a subspace S of and a fixed basis, we study the compact and convex set that we call the moment of S, where . This set is relevant in the determination of minimal hermitian matrices ( such that for every diagonal D and the spectral norm ). We describe extremal points and certain curves of in terms of principal vectors that minimize the angle between S and the coordinate axes of the fixed basis. We also relate to the joint numerical range W of n rank one hermitian matrices constructed with orthogonal projection and the fixed basis used. This connection provides a new approach to the description of and to minimal matrices. As a consequence, the intersection of two of these joint numerical ranges corresponding to orthogonal subspaces allows the construction or detection of a minimal matrix.

Financial performance drivers in BRICS healthcare companies: Locally estimated scatterplot smoothing partial utility functions

AbstractThe Healthcare sector is increasing in importance and relative size in BRICS countries (Brazil, Russia, India, China, South Africa). Despite BRICS relevance, the financial performance of their healthcare companies has been scarcely studied. This research fills this literature gap not only by focusing on the impacts of such diverse business environments on the financial performance of healthcare providers but also by proposing a novel approach to estimate an overall financial performance index based on weighted additive utility functions given a set of financial performance criteria. Precisely, bootstrapped Singular Value Decomposition is the cornerstone for identifying an orthogonal base of rotated financial performance criteria, upon which partial utility functions (PUFs) are estimated using locally estimated scatterplot smoothing (LOESS) polynomial regression. A compromise weighting scheme between singular values and quadratic programming results for minimal covariance and joint entropy matrices of residuals was used for summing up the PUFs. Results indicate that the values of financial performance range between 0.7 and 0.85. We further find that current assets, level of debt and liability, the company's Tobin Q are related to the financial performance. Besides, business freedom, government integrity, tax burden, monetary freedom and government spending are also the determinants of financial performance.

Extremal matrices for the Bruhat-graph order

We consider the class of symmetric -matrices with zero trace and constant row sums k which can be identified with the class of the adjacency matrices of k-regular undirected graphs. In a previous paper, two partial orders, the Bruhat and the Bruhat-graph order, have been introduced in this class. In fact, when k = 1 or k = 2, it was shown that the two orders coincide, while for the two orders are distinct. In this paper we give general properties of minimal and maximal matrices for these orders on and study the minimal and maximal matrices when k = 1, 2 or 3.

Minimal texture of quark mass matrices and precision CKM measurements

Abstract We have carried out an extensive analysis of all possible minimal texture quark mass matrices implying 169 texture-6 zero combinations. One finds that all these combinations are ruled out: a good number of these analytically, the other possibilities being excluded by the present quark mixing data. Interestingly, even if there are future changes in the ranges of the light quark masses, these conclusions remain valid.

Supports for minimal hermitian matrices

We study certain pairs of subspaces V and W of Cn we call supports that consist of eigenspaces of the eigenvalues ±‖M‖ of a minimal hermitian matrix M (‖M‖≤‖M+D‖ for all real diagonals D).For any pair of orthogonal subspaces we define a non negative invariant δ called the adequacy to measure how close they are to form a support and to detect one. This function δ is the minimum of another map F defined in a product of spheres of hermitian matrices. We study the gradient, Hessian and critical points of F in order to approximate δ. These results allow us to prove that the set of supports has interior points in the space of flag manifolds.

The absolutely minimal realizations for first-degree nD SISO systems

Using partial fraction decomposition, limit methods and algebraic techniques, finding the absolutely minimal realization matrices for first-degree (multi-linear) nD SISO systems is transformed into solving a system of equations consisting of the all principal minors of an unknown matrix. Without using the symbolic approach by Grobner basis, the necessary and sufficient conditions and construction of the absolutely minimal realizations for the degenerate and non-degenerate of transfer functions are derived. Five examples for first-degree 3D, 4D, and 5D SISO systems are presented to illustrate the basic ideas as well as the effectiveness of the proposed procedure, and computer programs written in Mathematica are also given.

Maximal and Minimal Triangular Matrices

We call a nonscalar matrix maximal (or minimal) if its centralizer is maximal (respectively minimal) in the poset of all centralizers of matrices. We discuss the form of maximal and minimal matrices in the algebra of upper triangular matrices.

Verified iptables Firewall Analysis and Verification

This article summarizes our efforts around the formally verified static analysis of iptables rulesets using Isabelle/HOL. We build our work around a formal semantics of the behavior of iptables firewalls. This semantics is tailored to the specifics of the filter table and supports arbitrary match expressions, even new ones that may be added in the future. Around that, we organize a set of simplification procedures and their correctness proofs: we include procedures that can unfold calls to user-defined chains, simplify match expressions, and construct approximations removing unknown or unwanted match expressions. For analysis purposes, we describe a simplified model of firewalls that only supports a single list of rules with limited expressiveness. We provide and verify procedures that translate from the complex iptables language into this simple model. Based on that, we implement the verified generation of IP space partitions and minimal service matrices. An evaluation of our work on a large set of real-world firewall rulesets shows that our framework provides interesting results in many situations, and can both help and out-compete other static analysis frameworks found in related work.

Towards the minimal seesaw model via CP violation of neutrinos

We study the minimal seesaw model, where two right-handed Majorana neutrinos are introduced, focusing on the CP violating phase. In addition, we take the trimaximal mixing pattern for the neutrino flavor where the charged lepton mass matrix is diagonal. Owing to this symmetric framework, the 3 × 2 Dirac neutrino mass matrix is given in terms of a few parameters. It is found that the observation of the CP violating phase determines the flavor structure of the Dirac neutrino mass matrix in the minimal seesaw model. New minimal Dirac neutrino mass matrices are presented in the case of TM1, which is given by the additional 2-3 family mixing to the tri-bimaximal mixing basis in the normal hierarchy of neutrino masses. Our model includes the Littlest seesaw model by King et al. as one of the specific cases. Furthermore, it is remarked that our 3 × 2 Dirac neutrino mass matrix is reproduced by introducing gauge singlet flavons with the specific alignments of the VEV’s. These alignments are derived from the residual symmetry of S4 group.

Minimal matrices in the Bruhat order for symmetric (0,1)-matrices

In this paper we study the minimal matrices for the Bruhat order on the class of symmetric (0,1)-matrices with given row sum vector. We will show that, when restricted to the symmetric matrices, new minimal matrices may appear besides the symmetric matrices for the nonrestricted Bruhat order. We modify the algorithm presented by Brualdi and Hwang (2004), which gives a minimal matrix for the Bruhat order on the class of (0,1)-matrices with given row and column sum vectors, in order to obtain a minimal matrix for the Bruhat order on the class of symmetric (0,1)-matrices with given row sum vector. We identify other minimal matrices in some of these classes. Namely, we determine all the minimal matrices when the row sums are constant and equal to 3. We then describe a family of symmetric matrices that are minimal for the Bruhat order on the class of 2k-by-2k(0,1)-matrices with constant row sums equal to k+1 and identify, in terms of the term rank of a matrix, a class of symmetric matrices that are related in the Bruhat order with one of these minimal matrices.

Approximate answering of queries involving polyline–polyline topological relationships

In geographic information systems, pictorial query languages are visual languages which make easier the user to express queries by free-hand drawing. In this perspective, this article proposes an approach to provide approximate answers to pictorial queries that do not match with the content of the database, that is, the results are null. It addresses the polyline–polyline topological relationships and is based on an algorithm, called Approximate Answer Computation algorithm, which exploits the notions of Operator Conceptual Neighborhood graph and 16-intersection matrix. The operator conceptual neighborhood graph represents the conceptual topological neighborhood between Symbolic Graphical Objects and is used for relaxing constraints of queries. The nodes of the operator conceptual neighborhood graph are labeled with geo-operators whose semantics has been formalized. The 16-intersection matrix provides enriched query details with respect to the well-known Dimensionally Extended 9-Intersection Model proposed in the literature. A set of minimal 16-intersection matrices associated with each node of the operator conceptual neighborhood graph, upon the external space connectivity condition, is defined and the proof of its minimality is provided. The main idea behind each introduced notion is illustrated using a running example throughout this article.

Hybridizing Cuckoo Search with Levenberg-Marquardt Algorithms in Optimization and Training Of ANNs for Mass Appraisal of Properties

Various algorithms, including particle swarm optimization (PSO), genetic algorithm (GA), ant colony algorithm (AC), cuckoo search (CS) algorithm, and firefly algorithm (FA) have been introduced to help optimize artificial neural networks (ANNs), speed up convergence and iteration rates, and escape from trapping into local optimum. However, despite the capabilities of these algorithms, it is only GA that has been utilized in the mass appraisal of properties. Therefore, in order to deal with problems of inconsistencies in appraisal/valuation estimates that sometimes occur during predictions, CS, a meta-heuristic algorithm is introduced into the mass appraisal industry. The proposed algorithm is combined with Levenberg-Marquardt (LM) and back propagation (BP) algorithms to test their effectiveness in the prediction of property values. We analyzed a dataset of 3,494 property transactions from the city of Cape Town, South Africa. The results indicate that CSLM and CSBP outperformed standalone the conventional BP algorithm in optimizing and training of ANN for mass appraisal of properties. This is reflected in the minimal error matrices predicted by both CSLM and CSBP algorithms.

Minimal matrix centralizers over the field ℤ2

A matrix over a field is minimal if for every matrix that commutes with , the centralizer of is a subset of the centralizer of . In this paper, we study the minimal matrices over the field with two elements. We characterize the minimal matrices with their minimal polynomial of the form , where is an irreducible polynomial and . We also characterize all minimal matrices with spectrum in .

Magnetic Mixed Hemimicelles Solid-Phase Extraction of Three Food Colorants from Real Samples

Introducing a specific clean extraction procedure with a minimal matrices effect for food colorant determination is still a challengeable topic and highly recommended. Mixed hemimicelle solid-phase extraction method, based on cetyltrimethylammonium bromide-coated Fe3O4/SiO2 nanoparticles as a novel, simple, and fast preconcentration method, was applied for preconcentration and fast isolation of three synthetic food colorants in foodstuff matrices prior to HPLC-UV-vis determination. The influence of different parameters on extraction efficiency such as surfactant amount, sample pH, time of extraction, desorption condition, and nanoparticles concentration was optimized. Under the optimized conditions, a preconcentration factor of 100 was achieved by extracting 10 mL sample. The limit of detection for the three synthetic food colorants including Tartrazine, Sunset yellow FCF, and Quinoline yellow is 2.50, 1.25, and 2.12 μg/L, respectively. The proposed method was applicable for extraction and preconcentration of three food colorants in various food samples with the food dye contents in the range of 13–105 μg/L.

The normal defect of some classes of matrices

An n×n matrix A has a normal defect of k if there exists an (n+k)×(n+k) normal matrix Aext with A as a leading principal submatrix and k minimal. In this paper we compute the normal defect of a special class of 4×4 matrices, namely matrices whose only nonzero entries lie on the superdiagonal, and we provide details for constructing minimal normal completion matrices Aext. We also prove a result for a related class of n×n matrices. Finally, we present an example of a 6×6 block diagonal matrix having the property that its normal defect is strictly less than the sum of the normal defects of each of its blocks, and we provide sufficient conditions for when the normal defect of a block diagonal matrix is equal to the sum of the normal defects of each of its blocks.