Triangular Matrix Research Articles

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.

Leonardo and hyper-Leonardo numbers via Riordan arrays

UDC 511 A generalization of the Leonardo numbers is defined and called the hyper-Leonardo numbers. Infinite lower triangular matrices, whose elements are Leonardo and hyper-Leonardo numbers are considered. Then the A - and Z -sequences of these matrices are obtained. Finally, the combinatorial identities between the hyper-Leonardo and Fibonacci numbers are obtained using the fundamental theorem of the Riordan arrays.

Abelian model structures on comma categories

UDC 512.64 Let A and B be bicomplete Abelian categories, which both have enough projectives and injectives and let T : A → B be a right exact functor. Under some mild conditions, we show that hereditary Abelian model structures on A and B can be amalgamated into a global hereditary Abelian model structure on the comma category ( T ↓ B ) . As an application of this result, we give an explicit description of a subcategory that consists of all trivial objects of the Gorenstein flat model structure on the category of modules over a triangular matrix ring.

A Strong Gram Classification of Non-negative Unit Forms of Dynkin Type 𝔸r

An integral quadratic form q : ℤn → ℤ is usually identified with a bilinear form 𝕓∨q:ℤn×ℤn→ℤ satisfying q(x)=𝕓∨q(x,x) for any vector x in ℤn, and such that its Gram matrix with respect to the canonical basis of ℤn is upper triangular. Two integral quadratic forms are called strongly (resp. weakly) Gram congruent if their corresponding upper triangular bilinear forms (resp. their symmetrizations) are equivalent. If q is unitary, such upper triangular bilinear form is unimodular, and one considers the associated Coxeter transformation and its characteristic polynomial, the so-called Coxeter polynomial of q with this identification. Two strongly Gram congruent quadratic unit forms are weakly Gram congruent and have the same Coxeter polynomial. Here we show that the converse of this statement holds for the connected non-negative case of Dynkin type 𝔸r (r ≥ 1) and arbitrary corank, and use this characterization to complete a combinatorial classification of such quadratic forms started in [Fundamenta Informaticae 184(1):49–82, 2021] and [Fundamenta Informaticae 185(3):221–246, 2022].

Mining frequent temporal duration-based patterns on time interval sequential database

Sequential databases have wide applications, such as market basket analysis, medical prediction, and sign language recognition. Most prior research is based on pointed-based sequential databases, which assume each item/event occurs instantaneously. However, in many real-world scenarios, events persist over intervals of varying durations, such as varying time intervals of a symptom or a gesture of sign language. Assigning the same weight to different times of events and neglecting the duration of events can hinder the recognition of interesting patterns, such as concurrent symptoms preceding a disease. To address these issues, this paper integrates duration with temporal patterns in interval-based sequential databases, introduces the concept of temporal duration-based patterns (TDPs), and designs two algorithms called FTDPMiner-EP (Frequent TDPMiner based on endpoint representation) and FTDPMiner-TM (Frequent TDPMiner based on triangular matrix representation) by using different extension methods to mine frequent TDPs. Due to the complex relationships between events, temporal pattern mining is more challenging than sequential pattern mining. Strategies are used in this paper to accelerate the algorithms' search process. Experiments are conducted on both real and synthetic databases, which show good performance of the two algorithms.

Lie triple centralizers of the algebra of dominant block upper triangular matrices

Lie triple centralizers of the algebra of dominant block upper triangular matrices

Tunable resource allocation dynamics for interpreting economic complexity

A lot of effort has been devoted to network structural interpretations of the economic complexity which can affect the level of economic growth and the activities of economic entities, although it remains unknown how network dynamical interpretations on which the economic complexity index operates affects its performance. Here, we regard export trade flows as the dynamics of resource allocation, and design a tunable resource allocation (TRA) process for interpreting economic complexity. The TRA could degrade into the classical economic complexity index (ECI) when the tunable parameter equals zero. We apply the TRA dynamics to the perfectly nested triangular matrix and the real country-product bipartite network from the world trade data to interpret economic complexity respectively. The parametric interpretations provide conceptual descriptions of the economic complexity index, shedding new light on a pattern of specialization of resource allocation, and the fluctuation of the economic complexity in countries and products. Moreover, the relationship of economic complexity between countries and products in the framework of TRA reveals that higher-ECI countries usually specialize in higher-ECI products while products not so as to countries, and the correlation strength varies with tunable parameters.

The Development of New Type GFRP Hollow Panels

Under the background of global energy transformation, pumped storage power stations are becoming more and more important. Most of their water retaining structures adopt face rockfill dams, but their concrete panels will have many problems such as durability. In this study, a new type of GFRP hollow panel was developed for face rockfill dam. The nonlinear behavior and damage process of the material were simulated by finite element analysis and UMAT subroutine. Through analysis and comparison, it is found that the triangular section form is superior to other forms in terms of stress distribution and bearing capacity. After optimization design, the optimal section takes into account the stability, safety and economic benefits of the structure. The research results provide a high-performance panel solution with high construction efficiency and low maintenance cost for the face rockfill dam, and provide an important design reference for the application of GFRP composite materials in water conservancy projects in the future.

Generalized Gorenstein modules with respect to duality pairs over triangular matrix rings

Let A and B be rings and [Formula: see text] with M an A-B-bimodule. We first construct a semi-complete duality pair [Formula: see text] over T using duality pairs over A and B, respectively. Then we characterize when a left T-module is Gorenstein projective, injective or flat with respect to the duality pair [Formula: see text]. As applications, we investigate when a left T-module is projectively coresolved Gorenstein flat, Ding injective or Gorenstein flat. In addition, we give an estimate of Gorenstein flat dimension of a left T-module with respect to [Formula: see text]. Finally, we construct model structures and recollements relative to these generalized Gorenstein modules.

Lie centralizing mappings on generalized matrix algebras through two-sided zero products

Let [Formula: see text] be a generalized matrix algebra defined by the Morita context [Formula: see text] and [Formula: see text] the center of [Formula: see text]. In this paper, under some certain conditions on [Formula: see text], we prove that if [Formula: see text] is an additive map satisfying [Formula: see text] for any [Formula: see text] with [Formula: see text], then [Formula: see text] has the form [Formula: see text] for all [Formula: see text], where [Formula: see text] and [Formula: see text] is an additive map from [Formula: see text] into [Formula: see text]. Finally as its applications, we characterize Lie centralizer maps and generalized Lie derivations on [Formula: see text]. Moreover we prove that the similar conclusions remain valid on full matrix algebras, triangular algebras, upper triangular matrix algebras.

A note on a generalized Jordan form of an infinite upper triangular matrix

In this paper, several equivalent conditions for the existence of a generalized Jordan form for matrices of locally nilpotent linear operators acting on an infinite countable dimensional vector space are presented, and an example is given showing that these conditions are not always fulfilled, which means the existence of infinite upper triangular matrices that are not similar to any generalized infinite Jordan matrix. Besides, it is shown that these conditions are not applicable in the case of an arbitrary dimensional of a vector space.

A New Fixed Coefficient Diagonally Implicit Block Backward Differentiation Formula for Solving Stiff Initial Value Problems

Stiff initial value problems in ordinary differential equations occur when solution components evolve at varying rates, posing challenges for traditional computational methods. Specialized techniques are crucial for maintaining accuracy and stability during rapid transitions, emphasizing their significance in developing reliable numerical algorithms across scientific and engineering applications. This study aims to develop a new fixed coefficient 3-point diagonally implicit block backward differentiation formula for the numerical solution of first order stiff initial value problems. The method is constructed by integrating a triangular matrix into the coefficient matrix of an existing extended 3-point super class of block BDF for solving stiff initial value problems. The selection of a fixed coefficient within the interval accompanies this integration to ensure optimal stability. The method is found to order five. Stability analysis indicates that the method is consistent, zero-stable, and almost A-stable, validating its applicability to stiff initial value problems. Implementation of the method involves Newton’s iteration, and a code in the C programming language is devised to demonstrate its effectiveness. Comparative examination of numerical outcomes with the existing 3BBDF and 3ESBBDF methods highlights the proposed method's enhanced accuracy and reduced computation time.

Integer-valued polynomials on subsets of upper triangular matrix rings

S. Frisch, showed that the integer-valued polynomials on upper triangular matrix ring Int T n ( K ) ( T n ( D ) ) : = { f ∈ T n ( K ) [ x ] | f ( T n ( D ) ) ⊆ T n ( D ) } is a ring, where D is an integral domain with field of fractions K. Let R 1 ⊆ R 2 be commutative rings with identity. In this paper, we study the set Int T n ( R 2 ) ( Ω , T n ( R 1 ) ) : = { f ∈ T n ( R 2 ) [ x ] | f ( Ω ) ⊆ T n ( R 1 ) } for some subsets Ω ⊆ T n ( R 1 ) . We generalize Frisch’s result and show that Int T n ( R 2 ) ( T n ( R 1 ) ) : = Int T n ( R 2 ) ( T n ( R 1 ) , T n ( R 1 ) ) is a ring. We state a lower bound for the Krull dimension of the integer-valued polynomials on upper triangular matrix rings. Finally, we state the concept of Skolem closure of an ideal of the integer-valued polynomials on upper triangular matrix rings and as a consequence, we obtain a classification of maximal ideals of the integer-valued polynomials on upper triangular matrix rings.

Graded identities with involution for the algebra of upper triangular matrices

Let F be a field of characteristic zero and let m≥2 be an integer. In this paper, we prove that if a group grading on UTm(F) admits a graded involution then this grading is a coarsening of a Z⌊m2⌋-grading on UTm(F) and the graded involution is equivalent to the reflection or symplectic involution on UTm(F), this grading is called the finest grading on UTm(F). Furthermore, if m≤4 the algebra UTm(F) with the finest grading satisfies no non-trivial monomial identities. For the finest grading, a finite basis for the (Z⌊m2⌋,⁎)-identities is exhibited with the reflection and symplectic involutions and the asymptotic growth of the (Z⌊m2⌋,⁎)-codimensions is determined. As a consequence, we prove that for any G-grading on UTm(F) and any graded involution the (G,⁎)-exponent is m. Finally, we study the algebra UT3(F). For this algebra, there are, up to equivalence, two non-trivial gradings that admit a graded involution: the canonical Z-grading and the Z2-grading induced by (0,1,0). We determine a basis for the (Z2,⁎)-identities and we compute the codimension sequence for the (Z2,⁎)-graded identities for UT3(F).

Transposed Poisson structures on the Lie algebra of upper triangular matrices

We describe transposed Poisson structures on the upper triangular matrix Lie algebra T_{n}(F) , n>1 , over a field F of characteristic zero. We prove that, for n>2 , any such structure is either of Poisson type or the orthogonal sum of a fixed non-Poisson structure with a structure of Poisson type, and for n=2 , there is one more class of transposed Poisson structures on T_{n}(F) . We also show that, up to isomorphism, the full matrix Lie algebra M_{n}(F) admits only one non-trivial transposed Poisson structure, and it is of Poisson type.

A Triangular-Matrix-Based Spectral Encoding Method for Broadband Filtering and Reconstruction-Based Spectral Measurement.

Broadband filtering and reconstruction-based spectral measurement represent a hot technical route for miniaturized spectral measurement; the measurement encoding scheme has a great effect on the spectral reconstruction fidelity. The existing spectral encoding schemes are usually complex and hard to implement; thus, the applications are severely limited. Considering this, here, a simple spectral encoding method based on a triangular matrix is designed. The condition number of the proposed spectral encoding system is estimated and demonstrated to be relatively low theoretically; then, verification experiments are carried out, and the results show that the proposed encoding can work well under precise or unprecise encoding and measurement conditions; therefore, the proposed scheme is demonstrated to be an effective trade-off of the spectral encoding efficiency and implementation cost.

The new notion of Bohl dichotomy for non-autonomous difference equations and its relation to exponential dichotomy

In A. Czornik et al. [Spectra based on Bohl exponents and Bohl dichotomy for non-autonomous difference equations, J. Dynam. Differ. Equ. (2023)] the concept of Bohl dichotomy is introduced which is a notion of hyperbolicity for linear non-autonomous difference equations that is weaker than the classical concept of exponential dichotomy. In the class of systems with bounded invertible coefficient matrices which have bounded inverses, we study the relation between the set BD of systems with Bohl dichotomy and the set ED of systems with exponential dichotomy. It can be easily seen from the definition of Bohl dichotomy that ED ⊆ BD . Using a counterexample we show that the closure of ED is not contained in BD . The main result of this paper is the characterization int ⁡ BD = ED . The proof uses upper triangular normal forms of systems which are dynamically equivalent and utilizes a diagonal argument to choose subsequences of perturbations each of which is constructed with the Millionshikov Rotation Method. An Appendix describes the Millionshikov Rotation Method in the context of non-autonomous difference equations as a universal tool.