scholarly journals Complex Numbers Related to Semi-Antinorms, Ellipses or Matrix Homogeneous Functionals

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 340
Author(s):  
Wolf-Dieter Richter
Keyword(s):  
Algebraic Structure ◽  
Matrix Multiplication ◽  
Diagonal Matrix ◽  
Invariance Property ◽  
Complex Numbers ◽  
Quadratic Equations ◽  
Euler’S Formula ◽  
Probability Densities

We generalize the property of complex numbers to be closely related to Euclidean circles by constructing new classes of complex numbers which in an analogous sense are closely related to semi-antinorm circles, ellipses, or functionals which are homogeneous with respect to certain diagonal matrix multiplication. We also extend Euler’s formula and discuss solutions of quadratic equations for the p-norm-antinorm realization of the abstract complex algebraic structure. In addition, we prove an advanced invariance property of certain probability densities.

Model Solutions to Quadratic Equations

1986 ◽  
Vol 79 (5) ◽  
pp. 332-336
Author(s):  
Alastair McNaughton
Keyword(s):  
Three Dimensional ◽  
Quadratic Functions ◽  
Complex Numbers ◽  
Quadratic Equations ◽  
Model Solutions

Here is a method of representing quadratic functions by three-dimensional wire models. It enables one to form a simple geometric concept of the location of the imaginary zeros. I have been using this material with my students and have been delighted with the ease with which they respond to it. As a result, their confidence in dealing with complex numbers has increased, their concept of functions has shown much improvement, and they are attacking problems with real insight.

scholarly journals Translation, modulation and dilation systems in set-valued signal processing

2018 ◽  
Vol 10 (1) ◽  
pp. 143-164 ◽  
Author(s):  
H. Levent ◽  
Y. Yilmaz
Keyword(s):  
Signal Processing ◽  
Linear Space ◽  
Algebraic Structure ◽  
Compact Convex ◽  
Inner Product ◽  
Complex Numbers ◽  
Real Numbers ◽  
Aumann Integral ◽  
Definition Of ◽  
Interval Valued

In this paper, we investigate a very important function space consists of set-valued functions defined on the set of real numbers with values on the space of all compact-convex subsets of complex numbers for which the $p$th power of their norm is integrable. In general, this space is denoted by $L^{p}% (\mathbb{R},\Omega(\mathbb{C}))$ for $1\leq p<\infty$ and it has an algebraic structure named as a quasilinear space which is a generalization of a classical linear space. Further, we introduce an inner-product (set-valued inner product) on $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$ and we think it is especially important to manage interval-valued data and interval-based signal processing. This also can be used in imprecise expectations. The definition of inner-product on $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$ is based on Aumann integral which is ready for use integration of set-valued functions and we show that the space $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$ is a Hilbert quasilinear space. Finally, we give translation, modulation and dilation operators which are three fundational set-valued operators on Hilbert quasilinear space $L^{2}(\mathbb{R},\Omega(\mathbb{C}))$.

scholarly journals Partitions in the S-Box of Streebog and Kuznyechik

2019 ◽  
pp. 302-329
Author(s):  
Léo Perrin
Keyword(s):  
Finite Field ◽  
Design Process ◽  
Algebraic Structure ◽  
Matrix Multiplication ◽  
Visual Pattern ◽  
Cryptographic Primitives ◽  
Number Of Components ◽  
Linear Layer ◽  
The One ◽  
The Impact

Streebog and Kuznyechik are the latest symmetric cryptographic primitives standardized by the Russian GOST. They share the same S-Box, π, whose design process was not described by its authors. In previous works, Biryukov, Perrin and Udovenko recovered two completely different decompositions of this S-Box.We revisit their results and identify a third decomposition of π. It is an instance of a fairly small family of permutations operating on 2m bits which we call TKlog and which is closely related to finite field logarithms. Its simplicity and the small number of components it uses lead us to claim that it has to be the structure intentionally used by the designers of Streebog and Kuznyechik.The 2m-bit permutations of this type have a very strong algebraic structure: they map multiplicative cosets of the subfield GF(2m)* to additive cosets of GF(2m)*. Furthermore, the function relating each multiplicative coset to the corresponding additive coset is always essentially the same. To the best of our knowledge, we are the first to expose this very strong algebraic structure.We also investigate other properties of the TKlog and show in particular that it can always be decomposed in a fashion similar to the first decomposition of Biryukov et al., thus explaining the relation between the two previous decompositions. It also means that it is always possible to implement a TKlog efficiently in hardware and that it always exhibits a visual pattern in its LAT similar to the one present in π. While we could not find attacks based on these new results, we discuss the impact of our work on the security of Streebog and Kuznyechik. To this end, we provide a new simpler representation of the linear layer of Streebog as a matrix multiplication in the exact same field as the one used to define π. We deduce that this matrix interacts in a non-trivial way with the partitions preserved by π.

AN ALGEBRAIC PROBLEM ARISING IN BIOMATHEMATICS

2016 ◽  
Vol 78 (6-6) ◽  
Author(s):  
Sisilia Sylviani ◽  
Ema Carnia ◽  
A. K. Supriatna
Keyword(s):  
Population Growth ◽  
Matrix Model ◽  
Special Form ◽  
Matrix Multiplication ◽  
Diagonal Matrix ◽  
Permutation Matrix ◽  
Block Diagonal Matrix ◽  
Algebraic Problem ◽  
The Matrix ◽  
Block Diagonal

This paper discusses a matrix model that describes the dynamics of a population with m live stages and lives in n patch seen from algebra viewpoint. The matrix D describes population growth in a patch or location. The matrix D is defined as a matrix obtained from matrix multiplication of a permutation matrix with a block diagonal matrix that its diagonal blocks is matrices with non-negative entries and transpose of a permutation matrix [4]. It will be shown that the permutation matrix contained in D has a special form.

scholarly journals Algebras and differential equations

1977 ◽  
Vol 68 ◽  
pp. 59-122 ◽  
Author(s):  
Helmut Röhrl
Keyword(s):  
Differential Equations ◽  
Banach Algebra ◽  
Ordinary Differential Equations ◽  
Polynomial Ring ◽  
Matrix Multiplication ◽  
Complex Numbers ◽  
Unital Ring ◽  
Right Hand ◽  
The Right ◽  
Unital Banach Algebra

One purpose of this paper is a purely algebraic study of (systems of) ordinary differential equations of the typewhere the coefficients are taken from a fixed associative, commutative, unital ring R, such as the field R of real or C of complex numbers or a commutative, unital Banach algebra. The right hand sides of D are considered to be elements in the polynomial ring R[X1, …, Xn] of associating but non-commuting variables X1, …, Xn. An algebraic study calls for maps between such differential equations and, in fact, morphisms are defined between differential equations having the same arity m but not necessarily the same dimension n. These morphisms are rectangular matrices with entries in R which satisfy certain relations. This leads to a category RDiffm whose objects are precisely the differential equations of arity m and in which the composition of the morphisms is the usual matrix multiplication.

scholarly journals The algebraic structure of the semigroup of binary relations on a finite set

1981 ◽  
Vol 22 (1) ◽  
pp. 57-68 ◽  
Author(s):  
Ki Hang Kim ◽  
Fred William Roush
Keyword(s):  
Algebraic Structure ◽  
Matrix Multiplication ◽  
Boolean Matrix ◽  
Ideal Structure ◽  
Binary Relations ◽  
Finite Set ◽  
Boolean Matrix Multiplication ◽  
The Ideal

In this paper we study some questions proposed by B. Schein [8] regarding the semigroup of binary relationsBxfor a finite setX: what is the ideal structure ofBx, what are the congruences onBx, what are the endomorphisms ofBx? For |X| =nit is convenient to regardBxas the semigroupBnofn×n(0, l)-matrices under Boolean matrix multiplication.

scholarly journals Three-Complex Numbers and Related Algebraic Structures

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 342
Author(s):  
Wolf-Dieter Richter
Keyword(s):  
Algebraic Structure ◽  
Matrix Polynomial ◽  
Three Dimensional ◽  
Basis Vector ◽  
Complex Numbers ◽  
Vector Product ◽  
Exponential Functions ◽  
Spherical Coordinate ◽  
Euclidean Vector Space ◽  
Vector Representations

Three-complex numbers are introduced for using a geometric vector product in the three-dimensional Euclidean vector space R3 and proving its equivalence with a spherical coordinate product. Based upon the definitions of the geometric power and geometric exponential functions, some Euler-type trigonometric representations of three-complex numbers are derived. Further, a general l23−complex algebraic structure together with its matrix, polynomial and variable basis vector representations are considered. Then, the classes of lp3-complex numbers are introduced. As an application, Euler-type formulas are used to construct directional probability laws on the Euclidean unit sphere in R3.

scholarly journals An identity in combinatorial analysis

1962 ◽  
Vol 5 (4) ◽  
pp. 197-200 ◽  
Author(s):  
L. J. Mordell
Keyword(s):  
Number Theory ◽  
Theta Functions ◽  
Combinatorial Analysis ◽  
Complex Numbers ◽  
Striking Result ◽  
Euler’S Formula ◽  
Symmetrical Form ◽  
Image Position ◽  
Do So

In a very recent paper [1], Basil Gordon discusses generalizations of Jacobi's identitywhere x and z are complex numbers and |x| <1. He notes that some of its consequences, inter alia Euler's formulaare of interest in number theory and combinatory analysis. He proves the apparently new and striking resultwhere |s|<1, and also considers the possibility of generalizations. His methods are algebraic and quite simple, but perhaps do not make obvious what underlies such formulae. It may be worth while to do so, especially since the details become simpler and the presentation more perspicuous. The method given here assumes no more knowledge than his does, although the new proof is expressed in terms of theta-functions, in simple properties of which, formulae such as (3) have their origin. Further, (3) appears in a slightly more symmetrical form.

Imaginaries

1954 ◽  
Vol 47 (1) ◽  
pp. 11-12
Author(s):  
Fred Gruenberger
Keyword(s):  
Complex Numbers ◽  
Quadratic Equations

“Imaginary” numbers are usually presented in algebra classes merely as a necessary (and awkward) evil arising from the solution of quadratic equations. Indeed, one may go quite far in mathematics before discovering some of the ingenious practical uses of complex numbers.

Sign in / Sign up

Export Citation Format

Share Document