# Geometrical approach to the argumentum of bijection of one coordinate-threshold reflection

( Pp. 26-30)

More about authors

Litvinenko Vitaly Sergeevich
sotrudnik FGUP «NII «KVANT»

Federal State Unitary Enterprise Scientific Research Institute KVANT Nikonov Vladimir G. Dr. Sci. (Eng.), Professor, Member at the Presidium of the Russian Academy of Natural Sciences

Russian Academy of Natural Sciences

Moscow, Russian Federation

Federal State Unitary Enterprise Scientific Research Institute KVANT Nikonov Vladimir G. Dr. Sci. (Eng.), Professor, Member at the Presidium of the Russian Academy of Natural Sciences

Russian Academy of Natural Sciences

Moscow, Russian Federation

Abstract:

The application of threshold operations is the perspective direction of the construction of discrete information processing nodes considering the potential possibility of realization of calculating the scalar product directly in the carrier signal, for instance, perspective optical computing medium. The article analyzes the reflection of bijective binary vectors with simple implementation of both the initial and inverse transformation by means of so-called quasi-hadamard matrices A
n in threshold basis. Currently bijection of such reflection is empirically shown for n = 4, 6, 8, however there was no relevant strict proof. The first relevant proof based on the study of the geometrical properties of the reflection generated by quasi-hadamard matrice A
4 is provided in this work. During the proof it was found that it is unique and possible as proposed only for n = 4. The article highlights the interesting features of its geometrical interpretation together with the proof of important applied statement about bijection of reflection generated by quasi-hadamard matrice A
4.

How to Cite:

Litvinenko V.S., Nikonov V.G., (2015), GEOMETRICAL APPROACH TO THE ARGUMENTUM OF BIJECTION OF ONE COORDINATE-THRESHOLD REFLECTION. Computational Nanotechnology, 4 => 26-30.

Reference list:

Belevitch, V. Theorem of 2n-terminal networks with application to conference telephony. 1950. vol. 26, pp. 231-244.

Goethals, J.M., and Seidel, J.J. Orthogonal matrices with zero diagonal. Canadian Journal of Mathematics. 1967. vol. 19, pp. 1001-1010.

Glukhov M.M., Elizarov V.P., Nechaev A.A. Algebra. 2003. T. 1, 2.

Nikonov V.G., Sarantsev A.V. Metody kompaktnoy realizatsii biektivnykh otobrazheniy, zadannykh regulyarnymi sistemami odnotipnykh bulevykh funktsiy // Vestnik Rossiyskogo universiteta druzhby narodov. Seriya: Prikladnaya i komp yuternaya matematika. 2003. T. 2. № 1. S. 94-105.

Nikonov V.G., Sarantsev A.V. Postroenie i klassifikatsiya regulyarnykh sistem odnotipnykh funktsiy // Informatsionnye tekhnologii v nauke, obrazovanii, telekommunikatsii i biznese: materialy XXXI Mezhdunarodnoy konferentsii. T. 5 iz Pril. 1. - M.:.Akademiya estestvoznaniya, 2004. S. 173-174.

Nikonov V.G., Sidorov E.S. O sposobe postroeniya vzaimno odnoznachnykh otobrazheniy pri pomoshchi kvaziadamarovykh matrits // Vestnik Moskovskogo gosudarstvennogo universiteta lesa - Lesnoy vestnik. 2009. №2 (65).

Goethals, J.M., and Seidel, J.J. Orthogonal matrices with zero diagonal. Canadian Journal of Mathematics. 1967. vol. 19, pp. 1001-1010.

Glukhov M.M., Elizarov V.P., Nechaev A.A. Algebra. 2003. T. 1, 2.

Nikonov V.G., Sarantsev A.V. Metody kompaktnoy realizatsii biektivnykh otobrazheniy, zadannykh regulyarnymi sistemami odnotipnykh bulevykh funktsiy // Vestnik Rossiyskogo universiteta druzhby narodov. Seriya: Prikladnaya i komp yuternaya matematika. 2003. T. 2. № 1. S. 94-105.

Nikonov V.G., Sarantsev A.V. Postroenie i klassifikatsiya regulyarnykh sistem odnotipnykh funktsiy // Informatsionnye tekhnologii v nauke, obrazovanii, telekommunikatsii i biznese: materialy XXXI Mezhdunarodnoy konferentsii. T. 5 iz Pril. 1. - M.:.Akademiya estestvoznaniya, 2004. S. 173-174.

Nikonov V.G., Sidorov E.S. O sposobe postroeniya vzaimno odnoznachnykh otobrazheniy pri pomoshchi kvaziadamarovykh matrits // Vestnik Moskovskogo gosudarstvennogo universiteta lesa - Lesnoy vestnik. 2009. №2 (65).

Keywords:

bijective mapping, threshold function, multidimensional cones, quasidemocracy matrix.

## Related Articles

6. INFORMATION SECURITY Pages: 39-49 Issue №9439

ABOUT POSSIBILITY OF USING FRACTAL MODELS IN DATA SECURITY SYSTEM CONSTRUCTION

fractal
protection of information
function complications
threshold function

Show more
2. MATHEMATICAL MODELING, NUMERICAL METHODS AND COMPLEX PROGRAMS Pages: 132-138 Issue №7537

The usage of equalprobable functions with mutal implicantive covering of straight diameter in the problem of constructing bijective mapping ф:Vr2 → Vr2

equiprobable function
functions with full implicative coating
bijective mapping
substitution

Show more
INFORMATION SECURITY Pages: 132-139 Issue №11955

MODIFICATION OF A GEOMETRICAL ALGORITHM OF CHARACTERIZATIONk-VALUED THRESHOLD FUNCTIONS

threshold function
k-valued logic
geometric algorithm the characterization of threshold functions
the proof of convergence

Show more
Multiscale modeling for information control and processing Pages: 50-58 DOI: 10.33693/2313-223X-2021-8-3-50-58 Issue №19706

On the Complexity of Specifying a Symmetric Group of Permutations of Degree 2n in a Threshold Basis on a Promising Element Base

threshold function
symmetric group
implementation of permutations
threshold basis
complexity of implementation

Show more
3. INFORMATION TECHNOLOGY AND AUTOMATION SYSTEMS Pages: 53-59 Issue №3742

ON A DECOMPOSITION OF BOOLEAN FUNCTIONS REPRESENTED BY QUADRATIC INEQUALITIES

Boolean functions
threshold function
decomposition
quadratic inequalities

Show more
1. MATHEMATICAL MODELING, NUMERICAL METHODS AND COMPLEX PROGRAMS Pages: 6-13 Issue №6518

ABOUT BIJECTIVITY OF TRANSFORMATIONS DETERMINED BY QUASI-HADAMARD MATRIXES

bijective mapping
threshold function
quasidemocracy matrix

Show more
1. MATHEMATICAL MODELING, NUMERICAL METHODS AND COMPLEX PROGRAMS Pages: 14-23 Issue №6518

BIJECTIVE COORDINATE-FORBIDDEN k-VALUED FUNCTIONS IN A PROBLEM OF SYNTHESIS OF SUBSTITUTIONS

bijective mapping
k-valued functions with forbidden signs subfunctions

Show more
2. MATHEMATICAL MODELING, NUMERICAL METHODS AND COMPLEX PROGRAMS Pages: 31-36 Issue №5869

The constructive method for synthesis of balanced k-valued algebraic threshold functions

multivalued logic
threshold function
algebraic threshold function
balanced function

Show more
1. MATHEMATICAL MODELING, NUMERICAL METHODS AND COMPLEX PROGRAMS Pages: 7-14 Issue №9439

ABOUT THE NEW ALGORITM OF CHARACTERIZATION OF k-VALUED THRESHOLD FUNCTIONS

threshold function
k-valued logic
characterization of threshold functions
growth factors
the coefficients of increase

Show more