21 papers · ranked by Valyu relevance
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…
Daizhan Cheng, Zhengping Ji
A cubic hypermatrix of order d can be considered as a structure matrix of a tensor with covariant order r and contravariant order s = d − r. Corresponding to this matrix expression of the hypermatrix, an eigenvector x with respect to an eigenvalue λ is proposed, called the universal eigenvector and eigenvalue of the…
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…
Takuya Okuyama, André Röhm, Takatomo Mihana, Makoto Naruse + 1 more
Matrix multiplication is important in various information-processing applications, including the computation of eigenvalues and eigenvectors, and in combinatorial optimization algorithms. Therefore, reducing the computation time of matrix products is essential to speed up scientific and practical calculations. Several…
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…
Jaydeep Bhattacharjee, Jaydeb Sarkar
| 1. | Introduction | 1 | | --- | --- | --- | | 2. | Crimmins' perspective | 4 | | 3. | Nagy-Foias and Langer's perspective | 6 | | 4. | von Neumann's perspectives | 9 | | 5. | Inner projections | 11 | | 6. | Model projections | 16 | | 7. | Blaschke products | 18 | | References | | 21 |
Trieu Le, Tomas Miguel Rodriguez, Sönmez Şahutoğlu
We study compactness of product of Toeplitz operators with symbols continuous on the closure of the polydisc in terms of behavior of the symbols on the boundary. For certain classes of symbols f and g, we show that $T_fT_g$ is compact if and only if fg vanishes on the boundary. We provide examples to show that for more…
Chengshen Xu, Lin Li, Xiao Huang, Rong Liu + 1 more
In textbooks and historical literature, the cross product has been defined only in 2-dimensional and 3-dimensional Euclidean spaces and the cross product of only two vectors has been defined only in the high dimensional Euclidean space whose metric matrix is the unit matrix. Nobody has given a universal definition for…
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…
Maksim Kosmakov, Vitaly Tarasov
We give new combinatorial formulae for vector-valued weight functions (off-shell nested Bethe vectors) for the evaluation modules over the Yangian $Y\mathfrak{gl}_n$.
Günter Hempel
Analytical expressions for the description of the time evolution of spin systems beyond product-operator formalism (POF) can be obtained if a low-dimensional subspace of the Liouville space has been found in which the time evolution of the spin system takes place completely. This can be achieved using a procedure that…
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…
Enrico De Santis, Alessio Martino, Antonello Rizzi, Muhammad Aleem
In many real-world applications concerning pattern recognition techniques, it is of utmost importance the automatic learning of the most appropriate dissimilarity measure to be used in object comparison. Real-world objects are often complex entities and need a specific representation grounded on a composition of…
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…
M. L. Kringelbach, G. Deco
Brain dynamics can be described in three different convenient mathematical languages, namely connectome harmonics, turbulence and complex harmonics (CHARM). Here we demonstrate that these theoretical frameworks can be rigorously unified, under the functional calculus, as one self-adjoint operator and its single…
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…
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…
Yonatan Kleerekoper, Mohammad Kurtam, Yitzhak Schiller, Hadas Benisty
Understanding the temporal dynamics of high-dimensional, time-varying systems remains a fundamental challenge across scientific disciplines. In many real-world systems, the interactions between sensors are also time-varying, which presents an additional level of complexity in extracting a meaningful representation and…
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…