Narrow Results

The stability in stock markets is the theme of this work. It has been demonstrated that the random walk theory alone is insufficient to explain the dynamic behavior of asset or stock prices. Deviations from the random walk theory reveal collective behaviors that produce waves and patterns. Research has shown that the Fibonacci sequence and the golden ratio emerge in such dynamic systems, representing states of minimal stability. Their sustained stability is ensured by the presence of Landau damping within the system. The complex dynamics of the stock market can be analyzed by considering the exchanged shares as fluctuating and the unexchanged shares as nonfluctuating entities. The traders exhibit a kind of group oscillations that resemble the waves in physical plasma. At steady state, the waves can be expressed by a cosine term, and at the least stable state, the dynamics involves the golden ratio, such that the cosine of $36^{\circ }$ is equal to half of the golden ratio. Using the trigonometric cosine formula, it is possible to obtain other angles which are the multiples of $9^{\circ }$. They can be expressed in terms of the golden ratio, and they stand as Fibonacci angles. The stabilization is achieved by a mechanism so-called Landau damping, and the waves thus created are called Elliott waves, and they keep the system near the instability border. It was found that these angles appear quite often in the motive and corrective Elliott waves in the weekly price change of crude oil between January 2001 and June 2023. The percent occurrence of these angles increases through oscillations in motive waves with peaks at 18°, $45^{\circ }$ and 72°. The corrective wave has a maximum peak at 36°, and the percent occurrence decreases through oscillations having smaller peak values at 54° and 72°. The highest values are such that the motive waves appear at 50% in the bull market, and the corrective waves at 27.8% in the bear market.

We continue our study of the class of Fibonacci-automatic words. These are infinite words whose nth term is defined in terms of a finite-state function of the Fibonacci representation of n. In this paper, we show how enumeration questions (such as counting the number of squares of length n in the Fibonacci word) can be decided purely mechanically, using a decision procedure. We reprove some known results, in a unified way, using our technique, and we prove some new results. We also examine abelian properties of these words. As a consequence of our results on abelian properties, we get the result that every nontrivial morphic image of the Fibonacci word is Fibonacci-automatic.

The magnetization of the magnetic multilayers with Fibonacci sequence was studied theoretically. By considering the Zeeman, magnetic anisotropy, bilinear and biquadratic exchange energies in the multiplayer system, the angles of the magnetization between all magnetic layers were calculated numerically by minimizing the magnetic energy. The magnetization curves and hysteresis loops were obtained. By considering the different strengths of biquadratic interaction between the magnetic layers some novel phenomena appears. The magnetization in magnetic sub-layers may rotates reversibly or irreversibly by magnetic field.

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 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 .

Let * denote the canonical involution of the group algebra KG induced by the map x ↦ x^-1 for x ∈ G. In case K is a real extension of ℚ, we consider Cayley unitary elements built out of skew elements k = α(x - x^-1) in KG such that 1 + k is invertible in KG, for α ∈ K and x ∈ G. The constructions involve an interesting sequence in the coefficients of (1 + k)^-1 which is the Fibonacci sequence when α = 1.

It has been known for centuries that the configurations of phylla (flowers, bracts) on plants have special properties. The difference in angular positions of successive phylla is locally constant (called the divergence angle) and is close to the golden angle; the radial positions are related by the locally constant plastichrone ratio. The phylla themselves lie on clockwise and counterclockwise families of spirals and the numbers in each family follow, for many plants, the regular Fibonacci sequence. We introduce a new invariant related to the self-similar nature of the phyllotactic pattern which essentially tells one about the shapes of the deformations from which the pattern is formed.

Since emotion is important in influencing cognition, perception of daily activities such as learning, communication and even rational decision-making, it must be considered in human-computer interaction. In this paper, we compare four different weighting functions in weighted KNN-based classifiers to recognize five emotions, including anger, happiness, sadness, neutral and boredom, from Mandarin emotional speech. The classifiers studied include weighted KNN, weighted CAP, and weighted D-KNN. We use the result of traditional KNN classifier as the line performance measure. The experimental results show that the used Fibonacci weighting function outperforms others in all weighted classifiers. The highest accuracy achieves 81.4% with weighted D-KNN classifier.

In this paper, we prove a conjecture of Sun concerning the arithmetic progression of integers which contains no integers of the form $F_n+p$. In fact, we show that $d-F_n$ has at least two distinct prime factors whenever $d\equiv r(modM)$, where $F_n$ is a Fibonacci number and $r$ and $M$ are suitable positive integers.

In this paper, we find all integers $c$ having at least two representations as a difference between a Fibonacci number and a Tribonacci number.

We derive an inequality on the discrepancy of sequences on the ring of $p$-adic integers ${\mathbb{Z}}_p$ using techniques from Fourier analysis. This is a $p$-adic analogue of the classical LeVeque inequality on the circle group $ℝ/\mathbb{Z}$. Some applications of the inequality are also given.

The reflection properties of light wave propagation in one-dimensional quasi-periodic metallic photonic crystal (PC) are comprehensively analyzed by transfer matrix method. In this work, we form a Fibonacci sequence quasi-periodic PC composed of metal and dielectric. The results demonstrate that the reflection stop band is strongly dependent on the periodic structure, metal thickness and incident angle. For this structure, the reflection stop band ranges from the visible light region to near-infrared region. Compared with the periodic metallic PC, the reflection stop bandwidth of our structure is wider. When the metal thickness increases, the reflection stop band is significantly enlarged. Furthermore, the reflection stop bandwidth slowly gets narrow and shifts to short wavelength region with the increase of incidence angle. Considering TE and TM wave at all incident angles, there is an omnidirectional reflection bandgap with width of 241nm for our investigated quasi-periodic metal PC.

The integrable hungry Lotka–Volterra (hLV) system stands for a prey–predator model in mathematical biology. The discrete-time hLV (dhLV) system is derived from a time discretization of the hLV system. The solution to the dhLV system is known to be represented by using the Casorati determinant. In this paper, we show that if the entries of the Casorati determinant become an extended Fibonacci sequence at the initial discrete time, then those are also an extended Fibonacci sequence at any discrete time. In other words, the extended Fibonacci sequence always appears in the entries of the Casorati determinant under the time evolution of the dhLV system with suitable initial setting. We also show that one of the dhLV variables converges to the ratio of two successive extended Fibonacci numbers as the discrete time goes to infinity.

In this paper, we define a function $Q$ which gives the sequence of partial sums of the Fibonacci quaternion sequence $(Q_n)$. We provide an identity about $Q^k$ which is the sequence obtained by applying $Q$ to $(Q_n)$$k$-times.

In this study, unlike other power sequences, we define new ones in the integer and integer-valued polynomial module. We call these sequences the power-generalized $k$-Horadam sequence in the integer module and the power-generalized $k$-Horadam sequence in the integer-valued polynomial module. Using these sequences, we rebuild the ElGamal cryptosystem and then obtain a new cryptographic method by combining this and symmetric systems. We provide an encryption example where our method is applied. Finally, we see the advantages of the new cryptographic method when we compare the obtained new cryptographic method with some of the asymmetric cryptographic systems such as RSA and ElGamal. And we achieve that new cryptographic method that has more advantageous.