Diffusion and Superdiffusion from Hydrodynamic Projections

Hydrodynamic projections, the projection onto conserved charges representing ballistic propagation of fluid waves, give exact transport results in many-body systems, such as the exact Drude weights. Focussing one one-dimensional systems, I show that this principle can be extended beyond the Euler scale, in particular to the diffusive and superdiffusive scales. By hydrodynamic reduction, Hilbert spaces of observables are constructed that generalise the standard space of conserved densities and describe the finer scales of hydrodynamics. The Green–Kubo formula for the Onsager matrix has a natural expression within the diffusive space. This space is associated with quadratically extensive charges, and projections onto any such charge give generic lower bounds for diffusion. In particular, bilinear expressions in linearly extensive charges lead to explicit diffusion lower bounds calculable from the thermodynamics, and applicable for instance to generic momentum-conserving one-dimensional systems. Bilinear charges are interpreted as covariant derivatives on the manifold of maximal entropy states, and represent the contribution to diffusion from scattering of ballistic waves. An analysis of fractionally extensive charges, combined with clustering properties from the superdiffusion phenomenology, gives lower bounds for superdiffusion exponents. These bounds reproduce the predictions of nonlinear fluctuating hydrodynamics, including the Kardar–Parisi–Zhang exponent 2/3 for sound-like modes, the Levy-distribution exponent 3/5 for heat-like modes, and the full Fibonacci sequence.

Advanced fibonacci sequence and its sum by second order variable co-efficient difference operator

We introduce a second order difference operator with specific powers of variable co-efficient and its inverse in this study, which allows us to derive the (α1tr1, α2tr2 )-Fibonacci sequence and its summation. This series is known as the Fibonacci sequence with variable co-efficients (VCFS). On the sum of the terms of the variable co-efficient Fibonacci sequence, some theorems and intriguing findings are generated. To demonstrate our findings, appropriate instances arepresented.

On Two Problems Related to Divisibility Properties of z(n)

The order of appearance (in the Fibonacci sequence) function z:Z≥1→Z≥1 is an arithmetic function defined for a positive integer n as z(n)=min{k≥1:Fk≡0(modn)}. A topic of great interest is to study the Diophantine properties of this function. In 1992, Sun and Sun showed that Fermat’s Last Theorem is related to the solubility of the functional equation z(n)=z(n2), where n is a prime number. In addition, in 2014, Luca and Pomerance proved that z(n)=z(n+1) has infinitely many solutions. In this paper, we provide some results related to these facts. In particular, we prove that limsupn→∞(z(n+1)−z(n))/(logn)2−ϵ=∞, for all ϵ∈(0,2).

Spin wave propagation and nonreciprocity in metallic magnonic quasi-crystals

The characteristics of spin waves propagating in Fibonacci magnonic quasi-crystals (MQCs) were investigated in micromagnetic simulations. The spin waves feel 1/3rd of the characteristic Fibonacci sequence length as a period, and mini band gaps reflected by MQCs are formed. The effect of the MQC on the spin wave’s propagation becomes prominent above the first band gap frequency. The properties of spin waves in MQCs generally depend on the propagation direction, because spin waves feel different structures depending on the direction. Therefore, the nonreciprocity (NR) characteristics becomes complex. The NR characteristics change at every band gap frequency and hence across the frequency regions defined by them. In particular, some frequency regions have almost no NR, while others have enhanced NR and some have even negative NR. These characteristics provide a new way to control NR.

Leonardo’s three-dimensional relations and some identities

In this work, new results are explored in relation to the Leonardo sequence. With that, a study about this second order recursive sequence, little explored in the mathematical scope, is briefly presented, relating it to the Fibonacci sequence. Thus, its complexification process is carried out, where from its one-dimensional model, imaginary units are inserted, obtaining Leonardo’s three-dimensional numbers. In this way, the imaginary units i and j are inserted. Finally, some three-dimensional identities are presented for Leonardo’s numbers.

The Proof of a Conjecture on the Density of Sets Related to Divisibility Properties of z(n)

Let (Fn)n be the sequence of Fibonacci numbers. The order of appearance (in the Fibonacci sequence) of a positive integer n is defined as z(n)=min{k≥1:n∣Fk}. Very recently, Trojovská and Venkatachalam proved that, for any k≥1, the number z(n) is divisible by 2k, for almost all integers n≥1 (in the sense of natural density). Moreover, they posed a conjecture that implies that the same is true upon replacing 2k by any integer m≥1. In this paper, in particular, we prove this conjecture.

From the Golden Ratio to the Golden Series and Their Social Application

This paper is about the logic of golden ratio. It is about the calculation of its value and the inverse value, examination of its uniqueness, the relation with Fibonacci sequence and its spiral and the logic of development of an organism. We expand the logic of golden ratio up until the sequence of Zeno from Elea that tends to infinity. We find the differentiate logic of golden ratio coming from ancient years and its unknown relation to the golden ratio. Also, we calculate the values φ of series that follows the logic of golden ratio, reaching the golden (normal) series, as a result of its logic, with its modern applications. Finally, it is criticized the fact that we do not include golden ratio in our education and the consequences that this has, by compare it with the achievements of its era. The application of golden ratio’s logic in social sciences results in possible examples of its use and their advantages.

Simulation of Opening Angle of Archimedes Wind Turbine Design Based on the Fibonacci Series

The configuration of new type of turbine is the Archimedes wind turbine with a spiral structure whose design is inspired by logarithmic spirals. This type of wind turbine uses lift and drag to harness the kinetic energy of the wind. The eccentric design has aerodynamic characteristics that have been the focus of previous research. The design is made from the arrangement of the Fibonacci sequence (1x1, 2x2, 3x3, 5x5, 8x8) or commonly known as the golden ratio. This study aims to analyze the coefficient of lift (CL) and coefficient of drag (CD) with variations opening angle of 35°, 45°, 65°, air fluid, turbulent flow, Re 1200, pressure distribution 1 atm, wind speed 5, 5 and 15 m/s. The results is at wind speed of 5.5 m/s, an angle of 35°, the CL value is 1.07E+02, the CD value is 4.02E+04. At wind speed of 5.5 m/s, an angle of 45°, the CL value is 1.08E+04, the CD value is 1.77E+01. At wind speed of 5.5 m/s, an angle of 65°, the CL value is 1.84E+06, the CD value is 3.68E+04. At wind speed of 15 m/s, an angle of 35°, the CL value is 2.20E+03, the CD value is 9.76E+02. At wind speed of 15 m/s, an angle of 45°, the CL value is 5.51E+04, the CD value is 4.12E+02. At wind speed of 15 m/s, an angle of 65°, the CL value is 5.96E+01, the CD value is 1.33E+03. Based on this, it can be concluded that at wind speed of 5.5 m/s the higher the opening angle, the higher CL produced. At wind speed of 15 m/s the larger the opening angle the CD increases. This is because the higher the angle, the more it receives sweeps or catches the wind. While the unstable value generated in this simulation is generally a weakness in the wind turbine design.

The Proof of a Conjecture Related to Divisibility Properties of z(n)

The order of appearance of n (in the Fibonacci sequence) z(n) is defined as the smallest positive integer k for which n divides the k—the Fibonacci number Fk. Very recently, Trojovský proved that z(n) is an even number for almost all positive integers n (in the natural density sense). Moreover, he conjectured that the same is valid for the set of integers n ≥ 1 for which the integer 4 divides z(n). In this paper, among other things, we prove that for any k ≥ 1, the number z(n) is divisible by 2k for almost all positive integers n (in particular, we confirm Trojovský’s conjecture).