On Nonstrongly Regular Matrices

An algebraic description of basic discrete symmetries (space reversal P, time reversal T, and their combination PT) is studied. Discrete subgroups of orthogonal groups of multidimensional spaces over the fields of real and complex numbers are considered in terms of fundamental automorphisms of Clifford algebras. In accordance with a division ring structure, a complete classification of automorphism groups is established for the Clifford algebras over the field of real numbers. The correspondence between eight double coverings (Dąbrowski groups) of the orthogonal group and eight types of the real Clifford algebras is defined with the use of isomorphisms between the automorphism groups and finite groups. Over the field of complex numbers there is a correspondence between two nonisomorphic double coverings of the complex orthogonal group and two types of complex Clifford algebras. It is shown that these correspondences associate with a well-known Atiyah–Bott–Shapiro periodicity. Generalized Brauer–Wall groups are introduced on the extended sets of the Clifford algebras. The structure of the inequality between the two Clifford–Lipschitz groups with mutually opposite signatures is elucidated. The physically important case of the two different double coverings of the Lorentz groups is considered in details.

It is shown that for real,m x n matricesA andB the system of matrix equationsAX=B, BY=A is solvable forX andY doubly stochastic if and only ifA=BP for some permutation matrixP. This result is then used to derive other equations and to characterize the Green’s relations on the semigroup Ω n of alln x n doubly stochastic matrices. The regular matrices in Ω n are characterized in several ways by use of the Moore-Penrose generalized inverse. It is shown that a regular matrix in Ω n is orthostochastic and that it is unitarily similar to a diagnonal matrix if and only if it belongs to a subgroup of Ω n . The paper is concluded with extensions of some of these results to the convex setS n of alln x n nonnegative matrices having row and column sums at most one.

It is shown that any regular polynomial matrix over the field of complex numbers that has at most one elementary divisor of degree 3 and whose remaining elementary divisors are of degree no greater than 2 can be factored into linear regular factors.

A quantum theory representations of real (R) and complex (C) numbers is given that is based on states of single, finite strings of qukits for any base k > 1. Both unary representations and the possibility that qukits with k a prime number are elementary and the rest composite are discussed. Cauchy sequences of qukit string states are defined from the arithmetic properties. The representations of R and C, as equivalence classes of these sequences, differ from classical kit string state representations in two ways: the freedom of choice of basis states, and the fact that each quantum theory representation is part of a mathematical structure that is itself based on the real and complex numbers. These aspects enable the description of 3 dimensional frame fields labeled by different k values, different basis or gauge choices, and different iteration stages. The reference frames in the field are based on each R and C representation where each frame contains representations of all physical theories as mathematical structures based on the R and C representation. Approaches to integrating this with physics are described. It is observed that R and C values of physical quantities, matrix elements, etc. which are viewed in a frame as elementary and featureless, are seen in a parent frame as equivalence classes of Cauchy sequences of qukit string states.

For many years, vector space models have been used in information retrieval and computational linguistics to represent terms, queries, and documents, using vector addition as a simple operator to model semantic composition. Though surprisingly successful, many aspects of meaning including word order, typed relationships, and nested structures are not captured by this modelling process.

We show that any regular quadratic matrix polynomial can be reduced to an upper triangular quadratic matrix polynomial over the complex numbers preserving the finite and infinite elementary divisors. We characterize the real quadratic matrix polynomials that are triangularizable over the real numbers and show that those that are not triangularizable are quasi-triangularizable with diagonal blocks of sizes $1\times 1$ and $2 \times 2$. We also derive complex and real Schur-like theorems for linearizations of quadratic matrix polynomials with nonsingular leading coefficients. In particular, we show that for any monic linearization $\lambda I+A$ of an $n\times n$ quadratic matrix polynomial there exists a nonsingular matrix defined in terms of $n$ orthonormal vectors that transforms $A$ to a companion linearization of a (quasi-)triangular quadratic matrix polynomial. This provides the foundation for designing numerical algorithms for the reduction of quadratic matrix polynomials to upper (quasi-)triangular form.

A new, nonclassical convergence acceleration concept, called \(\mu \)-acceleration of convergence (where \(\mu \) is a positive monotonically increasing sequence), is introduced and compared with the classical convergence acceleration concept. It is shown that this concept allows to compare the speeds of convergence for a larger set of sequences than the classical convergence acceleration concept. Also, regular matrix methods that improve and accelerate the convergence of sequences and series are studied. Some problems related to the speed of convergence of sequences and series with respect to matrix methods are discussed. Several theorems on the improvement and acceleration of the convergence are proved. As an application, the results obtained are used to increase the order of approximation of Fourier expansions and Zygmund means of Fourier expansions in certain Banach spaces.

Humans make sense of mathematics by relating it to personal conceptions and experiences.This might not be a smooth process as it may involve conceptions that work in an old context but do not work in a new context.The framework proposed by Chin and Tall (2012), Chin (2013) and Chin (2014) highlights how prior experiences shape humans' conceptions can be either supportive or problematic in making sense of extended contexts of mathematics.A supportive conception is a conception that supports learning and generalisation in a new context while a problematic conception is a conception that impedes learning and generalisation in a new context.By employing this framework, this study aims to explore how a group of respondents that consists of 30 mathematics undergraduate students from a public university in Malaysia make sense of complex numbers.The system of complex numbers is an extension of real numbers thus students will learn real numbers prior to learning complex numbers.How the respondents cope with the transition from real numbers context to complex numbers context is the focus of this study.Data were collected through a questionnaire and follow up interviews.Respondents participated in the follow up interviews voluntarily and data from three respondents were reported in this study.The result shows that these respondents have several supportive conceptions, such as the concept of addition and multiplication of real variable.The data also support the existence of problematic conceptions in making sense of complex numbers, namely the concept of positive-negative numbers and inequalities.These conceptions are originated from the context of real numbers.

Networks are often made up of several layers that exhibit diverse degrees of interdependencies. An interdependent network consists of a set of graphs G that are interconnected through a weighted interconnection matrix B, where the weight of each intergraph link is a non-negative real number p. Various dynamical processes, such as synchronization, cascading failures in power grids, and diffusion processes, are described by the Laplacian matrix Q characterizing the whole system. For the case in which the multilayer graph is a multiplex, where the number of nodes in each layer is the same and the interconnection matrix B=pI, I being the identity matrix, it has been shown that there exists a structural transition at some critical coupling p^{*}. This transition is such that dynamical processes are separated into two regimes: if p>p^{*}, the network acts as a whole; whereas when p<p^{*}, the network operates as if the graphs encoding the layers were isolated. In this paper, we extend and generalize the structural transition threshold p^{*} to a regular interconnection matrix B (constant row and column sum). Specifically, we provide upper and lower bounds for the transition threshold p^{*} in interdependent networks with a regular interconnection matrix B and derive the exact transition threshold for special scenarios using the formalism of quotient graphs. Additionally, we discuss the physical meaning of the transition threshold p^{*} in terms of the minimum cut and show, through a counterexample, that the structural transition does not always exist. Our results are one step forward on the characterization of more realistic multilayer networks and might be relevant for systems that deviate from the topological constraints imposed by multiplex networks.

In this paper we discuss the important Abel's summation formula, which is a very powerful tool for analysing series of real or complex numbers. We derive from it an integral test which may be useful in cases where the classical integral test may not be applied. We also discuss how this new integral test may be used when one is dealing with alternating series of real or complex numbers. The main goal of this note is to provide a supplementary material for courses dealing with infinite series of real or complex numbers.

Although complex numbers are essential in mathematics, they are not needed to describe physical experiments, as those are expressed in terms of probabilities, hence real numbers. Physics, however, aims to explain, rather than describe, experiments through theories. Although most theories of physics are based on real numbers, quantum theory was the first to be formulated in terms of operators acting on complex Hilbert spaces1,2. This has puzzled countless physicists, including the fathers of the theory, for whom a real version of quantum theory, in terms of real operators, seemed much more natural3. In fact, previous studies have shown that such a ‘real quantum theory’ can reproduce the outcomes of any multipartite experiment, as long as the parts share arbitrary real quantum states4. Here we investigate whether complex numbers are actually needed in the quantum formalism. We show this to be case by proving that real and complex Hilbert-space formulations of quantum theory make different predictions in network scenarios comprising independent states and measurements. This allows us to devise a Bell-like experiment, the successful realization of which would disprove real quantum theory, in the same way as standard Bell experiments disproved local physics.

Abel's theorem on power series is well known: If the series += a, converges and has the sum S, then the power series 2o anX converges for I x < 1 and its sum tends to S as x tends to 1 through real values. The converse theorem is not unconditionally true: it may occur that the power series O anoX converges for I x 1 < 1 and its sum tends to a finite limit S as x tends to 1, but this series does not converge when x = 1. Nevertheless Tauber [12]' proved in 1897 that, if nan , or even if (1 /n) a, tends to zero when n tends to + so, the assumption that f(x) = + an X tends to S when x tends to 1 through real smaller values implies the convergence of the series +O a. to the sum S. This result was the origin of a long series of works which led to various converses of Abel's theorem and its extensions to Dirichlet series and Laplace integrals. We shall quote here only some of them. Let us consider a Dirichlet series EOw ake , where the ak are real or complex numbers and the Xk are real numbers which satisfy

By studying the chapter on complex numbers, one found that they are very different from real numbers. Complex numbers can solve problems that cannot be solved by real numbers, such as computation, geometry problems in the complex plane, etc. In this article one will introduce the basic operations of complex numbers, complex conjugates, algebraic and trigonometric forms of complex numbers and their transformations. And how to find the roots of complex numbers is something that one will cover in the text. When one talk about these knowledge points, one will combine the title to help one better explain and readers' understanding. Before writing this article, one read a lot of literature, which helped one learn more knowledge and made one more aware of complex numbers. Therefore, in this article, one will also refer to the literature one refer to and cite them.

We introduce a novel method for complex number representation. The proposed, redundant complex binary number system (RCBNS) is developed by combining a redundant binary number and a complex number in base (-1+j). Donald (1960) and Walter Penny (1964, 1965) represented complex numbers using base -j and (-1+j) in the classified algorithmic models. A redundant complex binary number system consists of both real and imaginary-radix number systems that form a redundant integer digit set. This system is formed by using complex radix of (-1+j) and a digit set of /spl alpha/=3, where /spl alpha/ assumes a value of -3, -2, -1, 0, 1, 2, 3. The arithmetic operations of complex numbers with this system treat the real and imaginary parts as one unit. The carry-free addition has the advantage of redundancy in number representation in the arithmetic operations. Results of the arithmetic operations are in the RCBNS form. The two methods for conversion from the RCBNS form to the standard binary number form have been presented. The RCBNS reduces the number of steps required to perform complex number arithmetic operations, thus enhancing the speed.

We investigate the concept of semicomputability of relations on abstract structures. We consider three possible definitions of this concept, which all reduce to the classical notion of recursive enumerability over the natural numbers. By working in the algebra of the reals, with and without order, we find examples of sets which distinguish between these three notions. We also find interesting examples of sets of real and complex numbers which are semicomputable but not computable.