An equation involving the Smarandache primitive function and triangular numbers

Introduction: The Hadamard conjecture about the existence of Hadamard matrices in all orders multiple of 4, and the Gauss problem about the number of points in a circle are among the most important turning points in the development of mathematics. They both stimulated the development of scientific schools around the world with an immense amount of works. There are substantiations that these scientific problems are deeply connected. The number of Gaussian points (Z3 lattice points) on a spheroid, cone, paraboloid or parabola, along with their location, determines the number and types of Hadamard matrices.Purpose: Specification of the upper and lower bounds for the number of Gaussian points (with odd coordinates) on a spheroid depending on the problem size, in order to specify the Gauss theorem (about the solvability of quadratic problems in triangular numbers by projections onto the Liouville plane) with estimates for the case of Hadamard matrices. Methods: The authors, in addition to their previous ideas about proving the Hadamard conjecture on the base of a one-to-one correspondence between orthogonal matrices and Gaussian points, propose one more way, using the properties of generalized circles on Z3 .Results: It is proved that for a spheroid, the lower bound of all Gaussian points with odd coordinates is equal to the equator radius R, the upper limit of the points located above the equator is equal to the length of this equator L=2πR, and the total number of points is limited to 2L. Due to the spheroid symmetry in the sector with positive coordinates (octant), this gives the values of R/8 and L/4. Thus, the number of Gaussian points with odd coordinates does not exceed the border perimeter and is no less than the relative share of the sector in the total volume of the figure.Practical significance: Hadamard matrices associated with lattice points have a direct practical significance for noise-resistant coding, compression and masking of video information.

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Modification of the Logistic Map Using Fuzzy Numbers with Application to Pseudorandom Number Generation and Image Encryption

Entropy ◽

10.3390/e22040474 ◽

2020 ◽

Vol 22 (4) ◽

pp. 474 ◽

Cited By ~ 5

Author(s):

Lazaros Moysis ◽

Christos Volos ◽

Sajad Jafari ◽

Jesus M. Munoz-Pacheco ◽

Jacques Kengne ◽

...

Keyword(s):

Lyapunov Exponent ◽

Image Encryption ◽

Fuzzy Numbers ◽

Statistical Tests ◽

Logistic Map ◽

Simple Rule ◽

Pseudorandom Number ◽

Bifurcation Diagrams ◽

Triangular Numbers ◽

Pseudorandom Number Generation

A modification of the classic logistic map is proposed, using fuzzy triangular numbers. The resulting map is analysed through its Lyapunov exponent (LE) and bifurcation diagrams. It shows higher complexity compared to the classic logistic map and showcases phenomena, like antimonotonicity and crisis. The map is then applied to the problem of pseudo random bit generation, using a simple rule to generate the bit sequence. The resulting random bit generator (RBG) successfully passes the National Institute of Standards and Technology (NIST) statistical tests, and it is then successfully applied to the problem of image encryption.

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Existence of inner addition and inner multiplication on Set of Triangular numbers and some inherent properties of Triangular numbers

Materials Today Proceedings ◽

10.1016/j.matpr.2021.05.502 ◽

2021 ◽

Author(s):

Sridevi K ◽

Srinivas Thiruchinapalli

Keyword(s):

Triangular Numbers

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Optimization of Financial Asset Neutrosophic Portfolios

Mathematics ◽

10.3390/math9111162 ◽

2021 ◽

Vol 9 (11) ◽

pp. 1162

Author(s):

Marcel-Ioan Boloș ◽

Ioana-Alexandra Bradea ◽

Camelia Delcea

Keyword(s):

Capital Market ◽

Financial Risk ◽

Financial Asset ◽

Financial Assets ◽

Portfolio Risk ◽

Financial Return ◽

Additional Information ◽

Triangular Numbers ◽

The Capital Market

The purpose of this paper was to model, with the help of neutrosophic fuzzy numbers, the optimal financial asset portfolios, offering additional information to those investing in the capital market. The optimal neutrosophic portfolios are those categories of portfolios consisting of two or more financial assets, modeled using neutrosophic triangular numbers, that allow for the determination of financial performance indicators, respectively the neutrosophic average, the neutrosophic risk, for each financial asset, and the neutrosophic covariance as well as the determination of the portfolio return, respectively of the portfolio risk. There are two essential conditions established by rational investors on the capital market to obtain an optimal financial assets portfolio, respectively by fixing the financial return at the estimated level as well as minimizing the risk of the financial assets neutrosophic portfolio. These conditions allowed us to compute the financial assets’ share in the total value of the neutrosophic portfolios, for which the financial return reaches the level set by investors and the financial risk has the minimum value. In financial terms, the financial assets’ share answers the legitimate question of rational investors in the capital market regarding the amount of money they must invest in compliance with the optimal conditions regarding the neutrosophic return and risk.

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TRANSFORMATION FORMULAS FOR THE NUMBER OF REPRESENTATIONS OF BY LINEAR COMBINATIONS OF FOUR TRIANGULAR NUMBERS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972719001400 ◽

2020 ◽

Vol 102 (1) ◽

pp. 39-49

Author(s):

ZHI-HONG SUN

Keyword(s):

Theta Function ◽

Linear Combinations ◽

Triangular Numbers ◽

Transformation Formulas ◽

Positive Integers ◽

Theta Function Identities

Let $\mathbb{Z}$ and $\mathbb{Z}^{+}$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\in \mathbb{Z}^{+}$, let $t(a,b,c,d;n)$ be the number of representations of $n$ by $\frac{1}{2}ax(x+1)+\frac{1}{2}by(y+1)+\frac{1}{2}cz(z+1)+\frac{1}{2}dw(w+1)$ with $x,y,z,w\in \mathbb{Z}$. Using theta function identities we prove 13 transformation formulas for $t(a,b,c,d;n)$ and evaluate $t(2,3,3,8;n)$, $t(1,1,6,24;n)$ and $t(1,1,6,8;n)$.

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