scholarly journals The Eigenvalue Complexity of Sequences in the Real Domain

Entropy ◽  
2019 ◽  
Vol 21 (12) ◽  
pp. 1194
Author(s):  
Lingfeng Liu ◽  
Hongyue Xiang ◽  
Renzhi Li ◽  
Hanping Hu
Keyword(s):  
Probability Distribution ◽  
Real Number ◽  
Complexity Measures ◽  
The Real ◽  
Dynamical Characteristics ◽  
Dynamical Complexity ◽  
Cryptographic Algorithm ◽  
Real Domain ◽  
Number Sequences ◽  
Real Number Field

The eigenvalue is one of the important cryptographic complexity measures for sequences. However, the eigenvalue can only evaluate sequences with finite symbols—it is not applicable for real number sequences. Recently, chaos-based cryptography has received widespread attention for its perfect dynamical characteristics. However, dynamical complexity does not completely equate to cryptographic complexity. The security of the chaos-based cryptographic algorithm is not fully guaranteed unless it can be proven or measured by cryptographic standards. Therefore, in this paper, we extended the eigenvalue complexity measure from the finite field to the real number field to make it applicable for the complexity measurement of real number sequences. The probability distribution, expectation, and variance of the eigenvalue of real number sequences are discussed both theoretically and experimentally. With the extension of eigenvalue, we can evaluate the cryptographic complexity of real number sequences, which have a great advantage for cryptographic usage, especially for chaos-based cryptography.

scholarly journals On Real Irreducible Representations of Lie Algebras

1959 ◽  
Vol 14 ◽  
pp. 59-83 ◽  
Author(s):  
Nagayoshi Iwahori
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Number Field ◽  
Irreducible Representations ◽  
The Real ◽  
Representations Of Lie Algebras ◽  
Real Matrices ◽  
Real Number Field

Let us consider the following two problems:Problem A. Let g be a given Lie algebra over the real number field R. Then find all real, irreducible representations of g.Problem B. Let n be a given positive integer. Then find all irreducible subalgebras of the Lie algebra ôí(w, R) of all real matrices of degree n.

On some representations of the real number field

2011 ◽  
Vol 50 (2) ◽  
pp. 189-190
Author(s):  
A. S. Morozov
Keyword(s):  
Real Number ◽  
Number Field ◽  
The Real ◽  
Real Number Field

A geometrical model for the real number field

1961 ◽  
Vol 57 (4) ◽  
pp. 722-727
Author(s):  
W. Greve
Keyword(s):  
Real Number ◽  
Number Field ◽  
Near Field ◽  
Geometrical Model ◽  
Order Structure ◽  
Affine Transformations ◽  
The Real ◽  
Abstract Group ◽  
Underlying Space ◽  
Real Number Field

Recently Cunningham and Valentine gave in (3) an axiomatic description of the one-dimensional real affine space in terms of its order structure and the (abstract) group of affine transformations It is the purpose of the present note to show that the system of axioms in (3) (cf. (L. 1)–(L. 5) of this note) leads in a natural way to a model of the real number field. Our method is suggested by a result of Hall ((4), p. 382), namely, that an infinite doubly transitive Frobenius group is isomorphic to the group of affine transformations in a near-field, provided that there is at most one transformation displacing all points and taking a given point a into a given point b. The salient point of our investigation is the redundancy of the latter condition in the case where the underlying space is endowed with a certain linear order structure which is invariant under the transformations of the given group.

scholarly journals Typical ranks ofm×n× (m− 1)ntensors with 3 ≤m≤nover the real number field

2014 ◽  
Vol 63 (5) ◽  
pp. 940-955 ◽  
Author(s):  
Toshio Sumi ◽  
Mitsuhiro Miyazaki ◽  
Toshio Sakata
Keyword(s):  
Real Number ◽  
Number Field ◽  
The Real ◽  
Real Number Field

Solvability by Real Radicals and Fermat Primes

2004 ◽  
Vol 47 (2) ◽  
pp. 229-236
Author(s):  
C. U. Jensen
Keyword(s):  
Real Number ◽  
Number Field ◽  
The Real ◽  
Real Roots ◽  
Fermat Primes ◽  
Real Number Field

AbstractWe give a survey of old and new results concerning the expressibility of the real roots of a solvable polynomial over a real number field by real radicals. A characterization of Fermat primes is obtained in terms of solvability by real radicals for certain ploynomials.

Some presentations of the real number field

2012 ◽  
Vol 51 (1) ◽  
pp. 66-88 ◽  
Author(s):  
A. S. Morozov
Keyword(s):  
Real Number ◽  
Number Field ◽  
The Real ◽  
Real Number Field

scholarly journals On the Compacity of the Orthogonal Groups

1954 ◽  
Vol 7 ◽  
pp. 111-114 ◽  
Author(s):  
Takashi Ono
Keyword(s):  
Quadratic Form ◽  
Real Number ◽  
Number Field ◽  
Orthogonal Group ◽  
Orthogonal Groups ◽  
General Statement ◽  
Valued Fields ◽  
The Real ◽  
Special Result ◽  
Real Number Field

It is a well known fact on Lorenz groups that a quadratic form f is definite if and only if the corresponding orthogonal group On(R∞, f) where R∞ is the real number field, is compact. In this note, we shall show that the analogue of this holds for the case of the p-adic orthogonal group On(Rp, f), where Rp is the rational p-adic number field, as a special result of the more general statement on the completely valued fields.

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