Real and Imaginary Parts of Complex Numbers: A Model of Design to be Generalized to All Objects?

AbstractWe analyze the mathematical structure of the classical Grover’s algorithm and put it within the framework of linear algebra over the complex numbers. We also generalize it in the sense, that we are seeking not the one ‘chosen’ element (sometimes called a ‘solution’) of the dataset, but a set of m such ‘chosen’ elements (out of $$n>m)$$ n > m ) . Besides, we do not assume that the so-called initial superposition is uniform. We assume also that we have at our disposal an oracle that ‘marks,’ by a suitable phase change $$\varphi $$ φ , all these ‘chosen’ elements. In the first part of the paper, we construct a unique unitary operator that selects all ‘chosen’ elements in one step. The constructed operator is uniquely defined by the numbers $$\varphi $$ φ and $$\alpha $$ α which is a certain function of the coefficients of the initial superposition. Moreover, it is in the form of a composition of two so-called reflections. The result is purely theoretical since the phase change required to reach this heavily depends on $$\alpha $$ α . In the second part, we construct unitary operators having a form of composition of two or more reflections (generalizing the constructed operator) given the set of orthogonal versors. We find properties of these operations, in particular, their compositions. Further, by considering a fixed, ‘convenient’ phase change $$\varphi ,$$ φ , and by sequentially applying the so-constructed operator, we find the number of steps to find these ‘chosen’ elements with great probability. We apply this knowledge to study the generalizations of Grover’s algorithm ($$m=1,\phi =\pi $$ m = 1 , ϕ = π ), which are of the form, the found previously, unitary operators.

Download Full-text

Toeplitz nonnegative realization of spectra via companion matrices

Special Matrices ◽

10.1515/spma-2019-0017 ◽

2019 ◽

Vol 7 (1) ◽

pp. 230-245

Author(s):

Macarena Collao ◽

Mario Salas ◽

Ricardo L. Soto

Keyword(s):

Eigenvalue Problem ◽

Half Plane ◽

Toeplitz Matrix ◽

Toeplitz Matrices ◽

Nonnegative Matrix ◽

Inverse Eigenvalue Problem ◽

Complex Numbers ◽

Inverse Eigenvalue ◽

Companion Matrices ◽

Solution Matrix

Abstract The nonnegative inverse eigenvalue problem (NIEP) is the problem of finding conditions for the existence of an n × n entrywise nonnegative matrix A with prescribed spectrum Λ = {λ1, . . ., λn}. If the problem has a solution, we say that Λ is realizable and that A is a realizing matrix. In this paper we consider the NIEP for a Toeplitz realizing matrix A, and as far as we know, this is the first work which addresses the Toeplitz nonnegative realization of spectra. We show that nonnegative companion matrices are similar to nonnegative Toeplitz ones. We note that, as a consequence, a realizable list Λ= {λ1, . . ., λn} of complex numbers in the left-half plane, that is, with Re λi≤ 0, i = 2, . . ., n, is in particular realizable by a Toeplitz matrix. Moreover, we show how to construct symmetric nonnegative block Toeplitz matrices with prescribed spectrum and we explore the universal realizability of lists, which are realizable by this kind of matrices. We also propose a Matlab Toeplitz routine to compute a Toeplitz solution matrix.

Download Full-text

EIGENVALUES OF FIBONACCI STOCHASTIC ADDING MACHINE

Stochastics and Dynamics ◽

10.1142/s0219493710002966 ◽

2010 ◽

Vol 10 (02) ◽

pp. 291-313 ◽

Cited By ~ 6

Author(s):

A. MESSAOUDI ◽

D. SMANIA

Keyword(s):

Julia Set ◽

Transition Operator ◽

Complex Numbers ◽

Topological Properties

In this work, we compute the eigenvalues of the transition operator associated to the Fibonacci stochastic adding machine. In particular, we show that the eigenvalues are connected to the set [Formula: see text] of complex numbers z where (z2, z) belongs to the filled Julia set of a particular endomorphism of ℂ2. We also study some topological properties of the set [Formula: see text].

Download Full-text

Winding Numbers, Unwinding Numbers, and the Lambert W Function

Computational Methods and Function Theory ◽

10.1007/s40315-021-00398-1 ◽

2021 ◽

Author(s):

A. F. Beardon

Keyword(s):

Complex Plane ◽

Complex Number ◽

Lambert W Function ◽

Complex Numbers ◽

Winding Number ◽

Line Integral ◽

Complex Functions

AbstractThe unwinding number of a complex number was introduced to process automatic computations involving complex numbers and multi-valued complex functions, and has been successfully applied to computations involving branches of the Lambert W function. In this partly expository note we discuss the unwinding number from a purely topological perspective, and link it to the classical winding number of a curve in the complex plane. We also use the unwinding number to give a representation of the branches $$W_k$$ W k of the Lambert W function as a line integral.

Download Full-text

The Superstability of the Generalized D'alembert Functional Equation

Georgian Mathematical Journal ◽

10.1515/gmj.2003.503 ◽

2003 ◽

Vol 10 (3) ◽

pp. 503-508 ◽

Cited By ~ 1

Author(s):

Elhoucien Elqorachi ◽

Mohamed Akkouchi

Keyword(s):

Functional Equation ◽

Integral Equation ◽

Compact Group ◽

Locally Compact Group ◽

Complex Numbers ◽

Locally Compact

Abstract We generalize the well-known Baker's superstability result for the d'Alembert functional equation with values in the field of complex numbers to the case of the integral equation where 𝐺 is a locally compact group, μ is a generalized Gelfand measure and σ is a continuous involution of 𝐺.

Download Full-text

Iterants, Majorana Fermions and the Majorana-Dirac Equation

Symmetry ◽

10.3390/sym13081373 ◽

2021 ◽

Vol 13 (8) ◽

pp. 1373

Author(s):

Louis H. Kauffman

Keyword(s):

Dirac Equation ◽

Schrödinger Equation ◽

Schrodinger Equation ◽

Clifford Algebras ◽

Discrete Systems ◽

Complex Numbers ◽

Majorana Fermions ◽

Matrix Algebras ◽

Dirac Equations ◽

Points Of View

This paper explains a method of constructing algebras, starting with the properties of discrimination in elementary discrete systems. We show how to use points of view about these systems to construct what we call iterant algebras and how these algebras naturally give rise to the complex numbers, Clifford algebras and matrix algebras. The paper discusses the structure of the Schrödinger equation, the Dirac equation and the Majorana Dirac equations, finding solutions via the nilpotent method initiated by Peter Rowlands.

Download Full-text

A Generalized Mandelbrot Set of Polynomials of Type 𝐸𝑑 for Bicomplex Numbers

Georgian Mathematical Journal ◽

10.1515/gmj.2008.189 ◽

2008 ◽

Vol 15 (1) ◽

pp. 189-194

Author(s):

Ahmad Zireh

Keyword(s):

Mandelbrot Set ◽

Complex Numbers ◽

Quadratic Polynomials ◽

Bicomplex Numbers

Abstract We use a commutative generalization of complex numbers called bicomplex numbers to introduce the bicomplex dynamics of polynomials of type 𝐸𝑑, 𝑓𝑐(𝑤) = 𝑤(𝑤 + 𝑐)𝑑. Rochon [Fractals 8: 355–368, 2000] proved that the Mandelbrot set of quadratic polynomials in bicomplex numbers of the form 𝑤2 + 𝑐 is connected. We prove that our generalized Mandelbrot set of polynomials of type 𝐸𝑑, 𝑓𝑐(𝑤) = 𝑤(𝑤 + 𝑐)𝑑, is connected.

Download Full-text