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We study matrix forms of quaternionic versions of the Fourier Transform and Convolution operations. Quaternions offer a powerful representation unit, however they are related to difficulties in their use that stem foremost from non-commutativity of quaternion multiplication, and due to that $\mu^2 = -1$ possesses infinite solutions in the quaternion domain. Handling of quaternionic matrices is consequently complicated in several aspects (definition of eigenstructure, determinant, etc.). Our research findings clarify the relation of the Quaternion Fourier Transform matrix to the standard (complex) Discrete Fourier Transform matrix, and the extend on which well-known complex-domain theorems extend to quaternions. We focus especially on the relation of Quaternion Fourier Transform matrices to Quaternion Circulant matrices (representing quaternionic convolution), and the eigenstructure of the latter. A proof-of-concept application that makes direct use of our theoretical results is presented, where we present a method to bound the Lipschitz constant of a Quaternionic Convolutional Neural Network. Code is publicly available at: \url{https://github.com/sfikas/quaternion-fourier-convolution-matrix}., Comment: 25 pages, 4 figures

We consider expansions of vectors by a general class of multidimensional continued fraction algorithms. If the expansion is eventually periodic, then we describe the possible structure of a matrix corresponding to the repetend, and use it to prove that a number of vectors has an eventually periodic expansion in the Algebraic Jacobi--Perron Algorithm. Further, we give criteria for vectors to have purely periodic expansions; in particular, the vector cannot be totally positive., Comment: 31 pages

The increasing use of power electronics converters to integrate renewable energy sources has been subject of concern due to the resonance oscillatory phenomena caused by their interaction with poorly damped AC networks. Early studies are focused to assess the controller influence of a single converter connected to simple networks, and they are no longer representative for existing systems. Lately, studies of multi-infeed grid-connected converters are of particular interest, and their main aim is to apply traditional criteria and identify their difficulties in the stability assessment. An extension of traditional criteria is commonly proposed as a result of these analysis, but they can be burdensome for large and complex power systems. The present work addresses this issue by proposing a simple criterion to assess the stability of large power systems with high-penetration of power converters. The criterion has its origin in the mode analysis and positive-net damping stability criteria, and it addresses the stability in the frequency domain by studying the eigenvalues magnitude and real component of dynamic models in the admittance matrix form. Its effectiveness is tested in two case studies developed in Matlab/Simulink which compare it with traditionally criteria, proving its simplicity.

Quantization of the noncommutative geometric spectral action has so far been performed on the final component form of the action where all traces over the Dirac matrices and symmetry algebra are carried out. In this work, in order to preserve the noncommutative geometric structure of the formalism, we derive the quantization rules for propagators and vertices in matrix form. We show that the results in the case of a product of a four-dimensional Euclidean manifold by a finite space, could be cast in the form of that of a Yang-Mills theory. We illustrate the procedure for the toy electroweak model., Comment: 16 pages, 2 figures, changed equation numbering

Evolutionary Game Theory (EGT) simulations are used to model populations undergoing biological and cultural evolution in a range of fields, from biology to economics to linguistics. In this paper we present DyPy, an open source Python package that can perform evolutionary simulations for any matrix form game for three common evolutionary dynamics: Moran, Wright-Fisher and Replicator. We discuss the basic components of this package and illustrate how it can be used to run a variety of simulations. Our package allows a user to run such simulations fairly easily without much prior Python knowledge. We hope that this will be a great asset to researchers in a number of different fields.

In tabular case, when the reward and environment dynamics are known, policy evaluation can be written as $\bm{V}_{\bm{\pi}} = (I - \gamma P_{\bm{\pi}})^{-1} \bm{r}_{\bm{\pi}}$, where $P_{\bm{\pi}}$ is the state transition matrix given policy ${\bm{\pi}}$ and $\bm{r}_{\bm{\pi}}$ is the reward signal given ${\bm{\pi}}$. What annoys us is that $P_{\bm{\pi}}$ and $\bm{r}_{\bm{\pi}}$ are both mixed with ${\bm{\pi}}$, which means every time when we update ${\bm{\pi}}$, they will change together. In this paper, we leverage the notation from \cite{wang2007dual} to disentangle ${\bm{\pi}}$ and environment dynamics which makes optimization over policy more straightforward. We show that policy gradient theorem \cite{sutton2018reinforcement} and TRPO \cite{schulman2015trust} can be put into a more general framework and such notation has good potential to be extended to model-based reinforcement learning.

In this paper we focus on the linear algebra theory behind feedforward (FNN) and recurrent (RNN) neural networks. We review backward propagation, including backward propagation through time (BPTT). Also, we obtain a new exact expression for Hessian, which represents second order effects. We show that for $t$ time steps the weight gradient can be expressed as a rank-$t$ matrix, while the weight Hessian is as a sum of $t^{2}$ Kronecker products of rank-$1$ and $W^{T}AW$ matrices, for some matrix $A$ and weight matrix $W$. Also, we show that for a mini-batch of size $r$, the weight update can be expressed as a rank-$rt$ matrix. Finally, we briefly comment on the eigenvalues of the Hessian matrix., Comment: 23 pages, 4 figures

Results of research of possibility of transformation of a difference equation into a system of the first-order difference equation are presented. In contrast to the method used previously, an unknown grid function is split into two new auxiliary functions, which have definite properties. Several examples show that proposed approach can be useful in solving different physical problems., Comment: 19 pages, 14 figures

We consider the generalized pure state density matrix which depends on different time moments. The evolution equation for this density matrix is obtained in case where the density matrix corresponds to the solutions of Gross-Pitaevskii equation.

In a group trellis, the sequence of branches that split from the identity path and merge to the identity path form two normal chains. The Schreier refinement theorem can be applied to these two normal chains. The refinement of the two normal chains can be written in the form of a matrix, called the Schreier matrix form, with rows and columns determined by the two normal chains. Based on the Schreier matrix form, we give an encoder structure for a group code which is an estimator. The encoder uses the important idea of shortest length generator sequences previously explained by Forney and Trott. In this encoder the generator sequences are shown to have an additional property: the components of the generators are coset representatives in a chain coset decomposition of the branch group B of the code. Therefore this encoder appears to be a natural form for a group code encoder. The encoder has a register implementation which is somewhat different from the classical shift register structure. This form of the encoder can be extended. We find a composition chain of the branch group B and give an encoder which uses coset representatives in the composition chain of B. When B is solvable, the generators are constructed using coset representatives taken from prime cyclic groups.

The Riemann -- Silberstein -- Majorana -- Oppenheimer approach to the Maxwell electrodynamics in vacuum is investigated within the matrix formalism. The matrix form of electrodynamics includes three real 4 \times 4 matrices. Within the squaring procedure we construct four formal solutions of the Maxwell equations on the base of scalar Klein -- Fock -- Gordon solutions. The problem of separating physical electromagnetic waves in the linear space \lambda_{0}\Psi^{0}+\lambda_{1}\Psi^{1}+\lambda_{2}\Psi^{2}+ lambda_{3}\Psi^{3} is investigated, several particular cases, plane waves and cylindrical waves, are considered in detail., Comment: 26 pages 16 International Seminar NCPC, May 19-22, 2009, Minsk, Belarus

A neutrino mass matrix model M_\nu with M_\nu^T =M_\nu and a model with its inverse matrix form \widetilde{M}_\nu = m_0^2 (M_\nu^*)^{-1} can be diagonalized by the same mixing matrix U_\nu. It is investigated whether a scenario which provides a matrix model M_\nu with normal mass hierarchy can also give a model with an inverted mass hierarchy by considering a model with an inverse form of M_\nu., Comment: 6 pages, 4 figures

We show that the r-by-r minors of a d-by-n matrix of variables form a tropical basis of the ideal they generate if and only if r<4, or r=min(d,n), or else r=4 and min(d,n)<7. This answers a question asked by M. Chan, A. Jensen, and E. Rubei.

In this paper we prove theorems related to the Ramanujan-type series for $1/\pi$ (type $_3F_2$) and to the Ramanujan-like series, discovered by the author, for $1/\pi^2$ (type $_5F_4$). Our developments for the cases $_3 F_2$ and $_5 F_4$ connect with the theory of modular functions and with the theory of Calabi-Yau differential equations, respectively.

Quantum Hamiltonians that are fine-tuned to their so-called Rokhsar-Kivelson (RK) points, first presented in the context of quantum dimer models, are defined by their representations in preferred bases in which their ground state wave functions are intimately related to the partition functions of combinatorial problems of classical statistical physics. We show that all the known examples of quantum Hamiltonians, when fine-tuned to their RK points, belong to a larger class of real, symmetric, and irreducible matrices that admit what we dub a Stochastic Matrix Form (SMF) decomposition. Matrices that are SMF decomposable are shown to be in one-to-one correspondence with stochastic classical systems described by a Master equation of the matrix type, hence their name. It then follows that the equilibrium partition function of the stochastic classical system partly controls the zero-temperature quantum phase diagram, while the relaxation rates of the stochastic classical system coincide with the excitation spectrum of the quantum problem. Given a generic quantum Hamiltonian construed as an abstract operator defined on some Hilbert space, we prove that there exists a continuous manifold of bases in which the representation of the quantum Hamiltonian is SMF decomposable, i.e., there is a (continuous) manifold of distinct stochastic classical systems related to the same quantum problem. Finally, we illustrate with three examples of Hamiltonians fine-tuned to their RK points, the triangular quantum dimer model, the quantum eight-vertex model, and the quantum three-coloring model on the honeycomb lattice, how they can be understood within our framework, and how this allows for immediate generalizations, e.g., by adding non-trivial interactions to these models., Comment: 32 pages, 4 figures - accepted for publication in Annals of Physics, Elsevier

The charged lepton mass formula can be explained when the masses are propotinal to the squared vacuum expectation values (VEVs) of scalar fields. We introduce U(3) flavor symmetry and its nonet scalar field $\Phi$, whose VEV structure plays an essential role for generating the fermion mass spectrum. We can naturally obtain bilinear form of the Yukawa coupling $Y_{ij} \propto \sum_k <\Phi_{ik}> <\Phi_{kj}>$ without the non-renormalizable interactions, when the flavor symmetry is broken only through the Yukawa coupling and tadpole terms. We also speculate the possible VEV structure of $<\Phi>$., Comment: 8 pages, no figure, presentation changed

In the present paper it is shown that the Maxwell theory can be finely represented in the matrix form of Dirac's equation, if the Dirac wave function is identified with the electromagnetic wave by defined way. It seems to us, that such representation allows us to see new possibilities in the connection of the classical and quantum electrodynamics., Comment: 14 pages, submitted to Found.of Phys

The braided approach to q-deformation (due to the author and collaborators) gives natural algebras $R_{21}u_1Ru_2=u_2R_{21}u_1R$ and $R_{21}x_1x_2=x_2x_1R$ for q-Minkowski and q-Euclidean spaces respectively. These algebras are covariant under a corresponding background `rotation' quantum group. Semidirect product by this according to the bosonisation procedure (also due to the author) gives the corresponding Poincar\'e quantum groups. We review the construction and collect the resulting R-matrix formulae for both Euclidean and Minkowski cases in both enveloping algebra and function algebra form, and the duality between them. Axioms for the Poincar\'e quantum group $*$-structure and the dilaton problem are discussed., Comment: LATEX 33 pages. Mostly review. To appear in some version in a conference proceedings. Proposition~8.1 about $*$ structures was extended.