22 papers · ranked by Valyu relevance
Chris Heunen, Dominic Horsman
Matrix multiplication is the fundamental operation of linear algebra, modelling composition of linear maps in terms of coordinates. This article characterises matrix multiplication simply in terms of orthogonality and trace, in the following way. Suppose there were another way to combine two matrices x, y ∈ Mn into a…
Dominik Schulz, Reiner S. Thomä
Linear algebra is usually defined over a field such as the reals or complex numbers. It is possible to extend this to skew fields such as the quaternions. However, to the authors' knowledge there is no commonly accepted notation of linear algebra over skew fields. To this end, we discuss ways of notation that account…
Fabienne Chouraqui
Given two linear transformations, with representing matrices A and B with respect to some bases, it is not clear, in general, whether the Tracy-Singh product of the matrices A and B corresponds to a particular operation on the linear transformations. Nevertheless, it is not hard to show that in the particular case that…
Anna A. Taranenko
In this paper we consider four basic multidimensional matrix operations (outer product, Kronecker product, contraction, and projection) and two derivative operations (dot and circle products). We start with the interrelations between these operations and deduce some of their algebraic properties. Next, we study their…
Ayesha Gauhar, Adnan Rashid, Osman Hasan, João Bispo + 2 more
MATLAB is a software based analysis environment that supports a high-level programing language and is widely used to model and analyze systems in various domains of engineering and sciences. Traditionally, the analysis of MATLAB models is done using simulation and debugging/testing frameworks. These methods provide…
Ron Ofir, Michael Margaliot
Compound matrices play an important role in many fields of mathematics and have recently found new applications in systems and control theory. However, the explicit formulas for these compounds are non-trivial and not always easy to use. Here, we derive new formulas for the multiplicative and additive compounds of a…
Yongge Tian
Reverse order laws for generalized inverses of matrix products are a classic object of study in the theory of generalized inverses. One of the well-known reverse order laws for a matrix product AB is $(AB)(i,…,j)=B(s_{2},…,t_{2})A(s_{1},…,t_{1})$, where $(\cdot)(i,…,j)$ denotes a ${i,…,j}$-generalized inverse of…
Shanghua Zheng, Li Guo, Markus Rosenkranz
In this paper, we determine all the Rota-Baxter operators of weight zero on semigroup algebras of order two and three with the help of computer algebra. We determine the matrices for these Rota-Baxter operators by directly solving the defining equations of the operators. We also produce a Mathematica procedure to…
Stevo Stević, Ajay K. Sharma
Recently we have introduced a product-type operator and studied it on some spaces of analytic functions on the unit disc. Here we start investigating the operator on the space of analytic functions on the upper half-plane. We characterize the boundedness and compactness of the operator between Hardy and α-Bloch spaces…
Martin Schlather
Since the calculation of a genomic relationship matrix needs a large number of arithmetic operations, fast implementations are of interest. Our fastest algorithm is more accurate and 25× faster than a AVX double precision floating-point implementation.
Douglas Farenick
Using works of T. Ando and L. Gurvits, the well-known theorem of P.R. Halmos concerning the existence of unitary dilations for contractive linear operators acting on Hilbert spaces is recast as a result for d-tuples of contractive Hilbert space operators satisfying a certain matrix-positivity condition. Such operator…
Jozef Fecenko
The article presents a matrix differential operator and a pseudoinverse matrix differential operator for finding a particular solution to nonhomogeneous linear ordinary differential equations (ODE) with constant coeficients with special types of the right-hand side. Calculation requires the determination of an inverse…
Bartosz Tyrcha, Filip Brzęk, Piotr Zuchowski
This paper presents a general second-quantized form of a permutation operator interchanging $n$ pairs of electrons between interacting subsystems in the framework of the symmetry-adapted perturbation theory (SAPT). We detail the procedure for constructing this operator through the consecutive multiplication of…
Qingyun Zou, Guoqiu Wang, Qian Cao
A concept of the sub-band operator of multi-band wavelets is introduced, the theory of d-circular matrices is developed and the upper bound and the lower bound of the norm of the sub-band operator are obtained. Examples are provided to illustrate the results proposed in this paper.
Kangbien Park, Yonghee Bae
Drift, selection, and mutation are integral evolutionary factors. In this article, operator model is newly suggested to intuitively represent those evolutionary factors into mathematical operators, and to ultimately offer unconventional methodology for understanding evolutionary dynamics. To be specific, each of the…
Authors not listed
This article is the second in a two-part tutorial review on electronic spin-dependent dynamics. In Part I, we presented the fundamental theory within the adiabatic Born– Huang framework that describes the interaction between nuclear motion and the elec- tronic (spin and spatial) degrees of freedom. In particular, we…
Bingjie Wu, Ramon Grima, Chen Jia
A survey of the literature reveals notable discrepancies among the purported exact results for the spectra of stochastic gene expression models. For self-repressing gene circuits, previous studies ([Phys. Rev. Lett. 99, 108103 (2007)], [Phys. Rev. E 83,062902 (2011)], [J. Chem. Phys. 160, 074105 (2024)], and [bioRxiv…
Carl A. Whitfield, Peter Latimer, Alex Horsley, Jim M. Wild + 2 more
This paper introduces a linear operator for the purposes of quantifying the spectral properties of transport within resistive trees, such as airflow in lung airway networks. The operator, which we call the Maury matrix, acts only on the terminal nodes of the tree and is equivalent to the adjacency matrix of a complete…
Divya Shenoy
The concept of secondary range symmetric matrices is introduced here. Some characterizations as well as the equivalent conditions for a range symmetric matrix to be secondary range symmetric matrix is given. The idea of range symmetric matrices, range symmetric matrices over Minkowski space and secondary range…
Julio Candanedo
In this work we construct a detailed understanding of the distribution of electrons as described by the Full-Configuration-Interaction (FCI) and for a generalized active-space. These results are presented at an introductory level for beginning practitioners or non-experts. Suitable background are a knowledge of…
Alessandro Soncini, MATTEO PICCARDO
We present a non-orthogonal fragment ab initio methodology for the calculation of crystal field energy levels and magnetic properties in lanthanide complexes, implementing a systematic description of non-covalent contributions to metal-ligand bonding. The approach has two steps. In the first step, appropriate ab initio…
Authors not listed
We present a graph-based proof of the size-extensivity of the first-order exchange energy in symmetry-adapted perturbation theory (SAPT). The connectedness of the exchange energy expression can be established via derivatives of chromatic polynomials $\chi'_G(x)$ evaluated at $x=0$. The proof holds to all orders of the…