Triangular Matrix Research Articles

Darboux transformation of symmetric Jacobi matrices and Toda lattices

Let J be a symmetric Jacobi matrix associated with some Toda lattice. We find conditions for Jacobi matrix J to admit factorization J = LU (or J = 𝔘𝔏) with L (or 𝔏) and U (or 𝔘) being lower and upper triangular two-diagonal matrices, respectively. In this case, the Darboux transformation of J is the symmetric Jacobi matrix J(p) = UL (or J(d) = 𝔏𝔘), which is associated with another Toda lattice. In addition, we found explicit transformation formulas for orthogonal polynomials, m-functions and Toda lattices associated with the Jacobi matrices and their Darboux transformations.

Software Implementation of the Epps-Pulley Criterion in Matlab Modeling Environment

Purpose. Modeling systems and programming platforms provide ample opportunities for the use of statistical tools in research activities. Since the normal distribution is one of the most common distribution laws, the criterion for checking the sample for normality is in high demand among statistical assessment tools, among which the Epps-Pulley test has the status as one of the most powerful tests to check the deviation of the distribution from the normal one. There are a number of implementations of this test in the R and Python languages. However, this test is not implemented in one of the most popular Matlab modeling software. Thus, the purpose of this study is to develop a software implementation of the Epps-Pulley criterion in the Matlab environment and verify the correctness of the performed calculations.Materials and Methods. We implemented the calculation of Epps-Pulley statistics by two methods – classical, using cycles, and matrix-vector, using linear algebra operations. The classical method requires calculating the intermediate values necessary to obtain the criterion statistics using two independent cycles, the second cycle being a double one, in which one cycle is nested into the other. The matrix-vector method requires fewer code by performing calculations using linear algebra operations on matrices and vectors. We obtained critical statistical values for the sample size from 8 to 1000 observations with two-dimensional linear interpolation of tabular values. We used an approximation by a beta function of the third kind for a sample of over 1000 elements.Results. An assessment of the computational efficiency of the methods showed that the cyclic approach is about three times higher than the matrix-vector approach in terms of consumed time, which is presumably due to the processing of insignificant elements in triangular matrices when performing component-by-component operations. The correctness of the software implementation of the Epps-Pulley criterion was tested on several examples, which confirmed the compliance of the calculated values of the criterion statistics, as well as the critical values of statistics, with known data. We carried out a criterion statistical evaluation based on the empirical values of the error of the first kind. We obtained the error values correspondence to the specified significance levels. We performed comparative estimates of the Epps-Pulley test with the Anders-Darling and Shapiro-Wilk tests in terms of the criterion empirical power and tabulated the evaluation results. We published the software implementation of the Epps-Pulley test on the MATLAB Central Internet resource and for free use.Conclusion. We developed software implementation of the Epps-Pulley criterion as a new research tool that was previously unavailable in the Matlab modeling environment. We used the time spent on calculations to make a reasonable choice of the calculation algorithm for the criterion statistics. We confirmed correctness of the calculation algorithms by a set of selective checks and statistical estimates that showed the compliance with well-known theoretical provisions.

Upper Triangular Operator Matrices and Stability of Their Various Spectra

Upper Triangular Operator Matrices and Stability of Their Various Spectra

3D Indoor Position Estimation Based on a UDU Factorization Extended Kalman Filter Structure Using Beacon Distance and Inertial Measurement Unit Data

The development of the GPS (Global Positioning System) and related advances have made it possible to conceive of an outdoor positioning system with great accuracy; however, for indoor positioning, more efficient, reliable, and cost-effective technology is required. There are a variety of techniques utilized for indoor positioning, such as those that are Wi-Fi, Bluetooth, infrared, ultrasound, magnetic, and visual-marker-based. This work aims to design an accurate position estimation algorithm by combining raw distance data from ultrasonic sensors (Marvelmind Beacon) and acceleration data from an inertial measurement unit (IMU), utilizing the extended Kalman filter (EKF) with UDU factorization (expressed as the product of a triangular, a diagonal, and the transpose of the triangular matrix) approach. Initially, a position estimate is calculated through the use of a recursive least squares (RLS) method with a trilateration algorithm, utilizing raw distance data. This solution is then combined with acceleration data collected from the Marvelmind sensor, resulting in a position solution akin to that of the GPS. The data were initially collected via the ROS (Robot Operating System) platform and then via the Pixhawk development card, with tests conducted using a combination of four fixed and one moving Marvelmind sensors, as well as three fixed and one moving sensors. The designed algorithm is found to produce accurate results for position estimation, and is subsequently implemented on an embedded development card (Pixhawk). The tests showed that the designed algorithm gives accurate results with centimeter precision. Furthermore, test results have shown that the UDU-EKF structure integrated into the embedded system is faster than the classical EKF.

Identification of maximal C*-covers of some operator algebras

We use results on inclusions of free products and extensions of completely positive maps to determine the maximal C ∗ ^* -envelope for upper triangular 3 × 3 3 \times 3 matrices. We consider these same results in the context of larger upper triangular matrices and graph algebras associated to cycle graphs.

Modular relations involving generalized digamma functions

Generalized digamma functions ψk(x), studied by Ramanujan, Deninger, Dilcher, Kanemitsu, Ishibashi etc., appear as the Laurent series coefficients of the Hurwitz zeta function. In this paper, a modular relation of the form Fk(α)=Fk(1/α) containing infinite series of ψk(x), or, equivalently, between the generalized Stieltjes constants γk(x), is obtained for any k∈N. When k=0, it reduces to a famous transformation given on page 220 of Ramanujan's Lost Notebook. For k=1, an integral containing Riemann's Ξ-function, and corresponding to the aforementioned modular relation, is also obtained along with its asymptotic expansions as α→0 and α→∞. Carlitz-type and Guinand-type finite modular relations involving ψj(m)(x),0≤j≤k,m∈N∪{0}, are also derived, thereby extending previous results on the digamma function ψ(x). The extension of Guinand's result for ψj(m)(x),m≥2, involves an interesting combinatorial sum h(r) over integer partitions of 2r into exactly r parts. This sum plays a crucial role in an inversion formula needed for this extension. This formula has connection with the inversion formula for the inverse of a triangular Toeplitz matrix. The modular relation for ψj′(x) is subtle and requires delicate analysis.

Nonlinear optimal control for robotic exoskeletons with electropneumatic actuators

Nonlinear optimal control for robotic exoskeletons with electropneumatic actuators

Runge-Kutta Method Based on the Linear Combination of Arithmetic Mean, Geometric Mean, and Centroidal Mean: A Numerical Solution for the Hybrid Fuzzy Differential Equation

Objectives: To find the absolute error of the Hybrid fuzzy differential equation with triangular fuzzy number since its membership function has a triangular form. Methods: The third order runge-kuttta method based on the linear combination of Arithmetic mean, Geometric mean and Centroidal mean is introduced to solve Hybrid fuzzy differential equation. Seikkala’s derivatives are considered, and Numerical example is given to demonstrate the effectiveness of the proposed method. Findings: The comparative study have been done between the proposed method and the existing third order runge-kutta method based on Arithmetic mean. The proposed method gives better error tolerance than the third order runge-kutta method based on Arithmetic mean. Novelty: In this study a new formula has been developed by combining three means Arithmetic Mean, Geometric Mean and Centroidal Mean using Khattri’s formula and it is solved by the third order runge-kutta method for the first order hybrid fuzzy differential equation. The error tolerance has been calculated for the proposed method and the Arithmetic mean and it is seen that the error tolerance is better for the proposed method. Keywords: Hybrid fuzzy differential equations, Triangular fuzzy number, Seikkala’s derivative, Third order runge-kutta method, Initial value problem

Quantum geometrodynamics revived II. Hilbert space of positive definite metrics

Abstract This paper represents the second in a series of works aimed at reinvigorating the quantum geometrodynamics program. Our approach introduces a lattice regularization of the hypersurface deformation algebra, such that each lattice site carries a set of canonical variables given by the components of the spatial metric and the corresponding conjugate momenta. In order to quantize this theory, we describe a representation of the canonical commutation relations that enforces the positivity of the operators q_{ab} s^a s^b for all choices of s. Moreover, symmetry of q_{ab} and p^{ab} is ensured. This reflects the physical requirement that the spatial metric should be a positive definite, symmetric tensor. To achieve this end, we resort to the Cholesky decomposition of the spatial metric into upper triangular matrices with positive diagonal entries. Moreover, our Hilbert space also carries a representation of the vielbein fields and naturally separates the physical and gauge degrees of freedom. Finally, we introduce a generalization of the Weyl quantization for our representation. We want to emphasize that our proposed methodology is amenable to applications in other fields of physics, particularly
in scenarios where the configuration space is restricted by complicated relationships among the degrees of freedom.

Isotropy groups of the action of orthogonal *congruence on Hermitian matrices

We present a procedure which enables the computation and the description of structures of isotropy subgroups of the group of complex orthogonal matrices with respect to the action of *congruence on Hermitian matrices. A key ingredient in our proof is an algorithm giving solutions of a certain rectangular block (complex-alternating) upper triangular Toeplitz matrix equation.

On the motivic Higman conjecture

The G-representation variety RG(Σg) parametrizes the representations of the fundamental group π1(Σg) of a closed surface into an algebraic group G. For G the groups of n×n upper triangular matrices or unipotent upper triangular matrices, we compare two methods for computing algebraic invariants of RG(Σg). Using the geometric method initiated by González-Prieto, Logares and Muñoz, based on a Topological Quantum Field Theory (TQFT), we compute the virtual classes of RG(Σg) in the Grothendieck ring of varieties for n=1,…,5. Introducing the notion of algebraic representatives we are able to efficiently compute the TQFT. Using the arithmetic method initiated by Hausel and Rodriguez-Villegas, we compute the E-polynomials of RG(Σg) for n=1,…,10. For both methods, we describe how the computations can be performed algorithmically. Furthermore, we discuss how the representation varieties for the groups of unipotent upper triangular matrices are related to Higman's conjecture. The computations of this paper can be seen as positive evidence towards a generalized motivic version of the conjecture.

Performance improvement of the triangular matrix product in commodity clusters

AbstractThere are many works devoted to improving the matrix product computation, as it is used in a wide variety of scientific applications arising from many different fields. In this work, we propose alternative data distribution policies and communication patterns to reduce the elapsed time when computing triangular matrix products in distributed memory environments. In particular, we focus on commodity clusters, where the number of nodes is limited, proposing alternatives to traditional approaches in order to improve this operation’s performance. Our proposal overcomes the performance results associated with the state-of-the-art libraries, such as ScaLAPACK and SLATE, offering execution times that are up to 30% faster.

Pseudo-model-based iterative learning control for nonlinear multi-phase batch processes

In this article, a pseudo-model-based iterative learning control (ILC) is exploited for multi-phase batch processes which can be described as a nonlinear switched system with unknown functions and identical states in different phases. The nonlinear switched system is converted into a linear model whose system parameter matrix is approximated by minimizing the discrepancy from the real system output increment to the approximated system output increment. A data-driven ILC is constructed in an interactive form with system parameter matrix approximate algorithm. Meanwhile, the signs of the diagonal elements of system lower triangular parameter matrix are introduced into the construction of control input law. Theoretical analysis shows that the pseudo-model-based ILC (PM-ILC) concept can be extended to multi-phase batch processes with non-identical states in different phases. Furthermore, the approximation error of the system parameters matrix is bounded and the proposed PM-ILC is robust if the parameter is appropriately chosen. Simulation results illustrate the effectiveness and practicability of the proposed PM-ILC.

On superalgebras with pseudoautomorphism of polynomial codimension growth

Let A be an associative superalgebra endowed with a pseudoautomorphism p. In this paper we generalize the Wedderburn-Malcev Theorem in this setting and we prove that the sequence of p-codimensions of A is polynomially bounded if and only if the variety generated by A does not contain the group algebra of Z 2 and the algebra of 2 × 2 upper-triangular matrices with suitable pseudoautomorphisms.

Commuting additive maps on upper triangular and strictly upper triangular infinite matrices

Let ${\mathbb F}$ be a field, let $N_{\infty}({\mathbb F})$ be the ring of all ${\mathbb N}\times {\mathbb N}$ strictly upper triangular matrices over ${\mathbb F,}$ and let $T_{\infty}({\mathbb F})$ be the ring of all ${\mathbb N}\times {\mathbb N}$ upper triangular matrices over ${\mathbb F}$. In this paper, we completely characterize additive maps $f:N_{\infty}({\mathbb F})\to T_{\infty}({\mathbb F})$ satisfying $[f(x),x]=0$ for all $x\in N_{\infty}({\mathbb F})$. As applications, we obtain the finite fields versions of the two main results recently obtained by Slowik and Ahmed [Electron. J. Linear Algebra 37:247-255, 2021].

Kirillov Polynomials for the Exceptional Lie Algebra $$\mathfrak g_{2}$$

AbstractAs part of the development of the orbit method, Kirillov has counted the number of strictly upper triangular matrices with coefficients in a finite field of q elements and fixed Jordan type. One obtains polynomials with respect to q with many interesting properties and close relation to type A representation theory. In the present work, we develop the corresponding theory for the exceptional Lie algebra $$\mathfrak g_2$$ g 2 . In particular, we show that the leading coefficient can be expressed in terms of the Springer correspondence.

Tris-Azo Triangular Paraphenylenes: Synthesis and Reversible Interconversion into Radial π-Conjugated Macrocycles.

We report the synthesis of cycloparaphenylene derivatives featuring tris-azo groups. The smaller derivative, [3]cycloazobenzene, adopts a triangular all-cis form and exhibits thermally and photochemically stable characteristics due to significant ring strain as well as symmetric Kagome-patterned crystal packing. In contrast, the as-synthesized [3]cycloazobenzene with three biphenylene bridges adopts a triangular all-cis form, which undergoes photoinduced isomerization, leading to a photostationary state. Interestingly, the addition of an excess of acid selectively leads to the formation of an all-trans form. DFT calculations reveal that the interconversion from a triangular to a circular shape correlates with an increase in HOMO and a decrease in LUMO, characteristics intrinsic to radial π-conjugated systems.

A Novel Lie Hypergraph Based Lifetime Enhancement Routing Protocol for Environmental Monitoring in Wireless Sensor Networks

Wireless sensor network (WSN) is a rapid surging technology promising many fruitful innovations for the user community. The use of WSNs for continuous monitoring of environmental factors in severe situations, such as detection of volcanoes, forest fires, floods, and so on. Despite WSN’s various potential applications, energy and deployment are two major concerns influencing the network life span. The primary requirement is to monitor node energy consumption in order to improve network performance. Many approaches are proposed in recent times for energy efficiency, this article introduces an associated hypergraph with Lie algebra of upper triangular matrices for energy efficient clustering and routing. Initially, the sensor nodes are formulated as a hypergraph, and each hyperedge is treated as a cluster, from which cluster-head (CH) selection is exploited by minimum hypergraph transversal. The routing decision is employed by constructing the upper triangular matrix (UTM), and Lie commutators of the UTM Lie algebra identifies the best relay nodes for data forwarding. The performance of the proposed work is evaluated using simulation results which shows the effectiveness of the scheme over compared protocols in terms of, number of alive nodes, energy consumption and number of packets received in BS.

Fredholm complements of upper triangular operator matrices

Fredholm complements of upper triangular operator matrices

VDR decomposition of Chebyshev-Vandermonde matrices with the Arnoldi Process

This paper introduces the VDR decomposition of Chebyshev-Vandermonde matrices, where V represents an ordinary Vandermonde matrix, D is diagonal, and R is upper triangular. Our motivation for this work stems from the study by Brubeck et al. [Vandermonde with Arnoldi. SIAM Rev. 2021;63(2):405–415]. We explore the VDR decomposition and combine it with a QR decomposition of V for Chebyshev-Vandermonde matrices. Furthermore, we demonstrate that the upper triangular factor in the QR decomposition can be recursively obtained from the upper Hessenberg matrix in the Arnoldi process. To compare our approach with the results obtained using the default QR code in Matlab, we present experimental results. Finally, we provide an algorithm for solving linear systems with Chebyshev-Vandermonde matrices as coefficient matrices, and experimental results for an upper bound on the relative error by applying an a posteriori error analysis for the computed solutions of the systems.