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Author(s):  
C. Saranya

Abstract: The Ternary cubic Diophantine Equation represented by૟(࢞ ࢟ + ૛ ࢠૡૡ = ૛࢟࢞૚૚ − (૛ ૜ is analyzed for its infinite number of non-zero integral solutions. A few interesting among the solutions are also discussed. Keywords: Diophantine equation, Integral solutions, cubic equation with three unknowns, Ternary equation.


2022 ◽  
pp. 1-22
Author(s):  
François Baccelli ◽  
Michel Davydov ◽  
Thibaud Taillefumier

Abstract Network dynamics with point-process-based interactions are of paramount modeling interest. Unfortunately, most relevant dynamics involve complex graphs of interactions for which an exact computational treatment is impossible. To circumvent this difficulty, the replica-mean-field approach focuses on randomly interacting replicas of the networks of interest. In the limit of an infinite number of replicas, these networks become analytically tractable under the so-called ‘Poisson hypothesis’. However, in most applications this hypothesis is only conjectured. In this paper we establish the Poisson hypothesis for a general class of discrete-time, point-process-based dynamics that we propose to call fragmentation-interaction-aggregation processes, and which are introduced here. These processes feature a network of nodes, each endowed with a state governing their random activation. Each activation triggers the fragmentation of the activated node state and the transmission of interaction signals to downstream nodes. In turn, the signals received by nodes are aggregated to their state. Our main contribution is a proof of the Poisson hypothesis for the replica-mean-field version of any network in this class. The proof is obtained by establishing the propagation of asymptotic independence for state variables in the limit of an infinite number of replicas. Discrete-time Galves–Löcherbach neural networks are used as a basic instance and illustration of our analysis.


Complexity ◽  
2022 ◽  
Vol 2022 ◽  
pp. 1-16
Author(s):  
Maryam Zolfaghari-Nejad ◽  
Mostafa Charmi ◽  
Hossein Hassanpoor

In this work, we introduce a new non-Shilnikov chaotic system with an infinite number of nonhyperbolic equilibrium points. The proposed system does not have any linear term, and it is worth noting that the new system has one equilibrium point with triple zero eigenvalues at the origin. Also, the novel system has an infinite number of equilibrium points with double zero eigenvalues that are located on the z -axis. Numerical analysis of the system reveals many strong dynamics. The new system exhibits multistability and antimonotonicity. Multistability implies the coexistence of many periodic, limit cycle, and chaotic attractors under different initial values. Also, bifurcation analysis of the system shows interesting phenomena such as periodic window, period-doubling route to chaos, and inverse period-doubling bifurcations. Moreover, the complexity of the system is analyzed by computing spectral entropy. The spectral entropy distribution under different initial values is very scattered and shows that the new system has numerous multiple attractors. Finally, chaos-based encoding/decoding algorithms for secure data transmission are developed by designing a state chain diagram, which indicates the applicability of the new chaotic system.


Author(s):  
Fangcheng Fan

In this paper, we investigate a four-component Toda lattice (TL), which may be used to model the wave propagation in lattices just like the famous TL. By means of the Lax pair and gauge transformation, we construct the [Formula: see text]-fold Darboux transformation (DT), which enables us to obtain multi-soliton or multi-solitary wave solution without complex iterative process. Through the obtained DT, [Formula: see text]-fold explicit exact solutions of the system and their figures with proper parameters are presented from which we find the [Formula: see text]-fold solution shows two-solitary wave structure, the amplitude and shape of the wave change with time. Finally, we derive an infinite number of conservation laws formulaically to illustrate the integrability of the system.


2021 ◽  
Vol 3 (4) ◽  
pp. 270-278
Author(s):  
Andrei Moldavanov

Stages of natural evolution such as biogenesis and abiogenesis are the well-recognized terms to characterize the very different phases of life development. Traditionally, an abiogenesis is believed as the early stage of evolution that is mainly the chemistry phase dealing with intercoupling between the complex polymer chains when manifestations of life assumes substantial participation of cooperative effects. It its turn, a biogenesis as the subsequent stage of evolution is the period for prevalence of Darwin’s laws showing, in particular, in battle among separate species in the way of variability-heredity contest. In this article, we discuss possible nature of the transition between above stages as a normal result of progress in an evolutionary system simulated by mathematical model of open system with infinite number of conserved links with system surroundings. It is shown that the biosystem, in transition point experiences the deep reconstruction in existing pattern of energy exchange which leads to emergence of the more complicated and advanced stage of evolution. Our study showed that the found transition point can be considered as a singularity point in system evolution. In its turn, the evolution stages with the dissimilar meaning are the physical placeholders for stage of abiogenesis and biogenesis in natural evolution, correspondingly.


Entropy ◽  
2021 ◽  
Vol 24 (1) ◽  
pp. 60
Author(s):  
Ernesto P. Borges ◽  
Takeshi Kodama ◽  
Constantino Tsallis

The rich history of prime numbers includes great names such as Euclid, who first analytically studied the prime numbers and proved that there is an infinite number of them, Euler, who introduced the function ζ(s)≡∑n=1∞n−s=∏pprime11−p−s, Gauss, who estimated the rate at which prime numbers increase, and Riemann, who extended ζ(s) to the complex plane z and conjectured that all nontrivial zeros are in the R(z)=1/2 axis. The nonadditive entropy Sq=k∑ipilnq(1/pi)(q∈R;S1=SBG≡−k∑ipilnpi, where BG stands for Boltzmann-Gibbs) on which nonextensive statistical mechanics is based, involves the function lnqz≡z1−q−11−q(ln1z=lnz). It is already known that this function paves the way for the emergence of a q-generalized algebra, using q-numbers defined as ⟨x⟩q≡elnqx, which recover the number x for q=1. The q-prime numbers are then defined as the q-natural numbers ⟨n⟩q≡elnqn(n=1,2,3,⋯), where n is a prime number p=2,3,5,7,⋯ We show that, for any value of q, infinitely many q-prime numbers exist; for q≤1 they diverge for increasing prime number, whereas they converge for q>1; the standard prime numbers are recovered for q=1. For q≤1, we generalize the ζ(s) function as follows: ζq(s)≡⟨ζ(s)⟩q (s∈R). We show that this function appears to diverge at s=1+0, ∀q. Also, we alternatively define, for q≤1, ζqΣ(s)≡∑n=1∞1⟨n⟩qs=1+1⟨2⟩qs+⋯ and ζqΠ(s)≡∏pprime11−⟨p⟩q−s=11−⟨2⟩q−s11−⟨3⟩q−s11−⟨5⟩q−s⋯, which, for q<1, generically satisfy ζqΣ(s)<ζqΠ(s), in variance with the q=1 case, where of course ζ1Σ(s)=ζ1Π(s).


2021 ◽  
Author(s):  
G G
Keyword(s):  

Keto Complete Australia: There are an infinite number of supplements on the market that will trick you into believing that they can give you a slim body in no time. Many of these products are often just giddy and filled with lots of harmful additives.


Author(s):  
Neli Ilieva Stoilova ◽  
Joris Van der Jeugt

Abstract The parastatistics Fock spaces of order p corresponding to an infinite number of parafermions and parabosons with relative paraboson relations are constructed. The Fock spaces are lowest weight representations of the ℤ2 × ℤ2-graded Lie superalgebra pso(∞|∞), with a basis consisting of row-stable Gelfand-Zetlin patterns.


2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Ioseph Buchbinder ◽  
Evgeny Ivanov ◽  
Nikita Zaigraev

Abstract We present, for the first time, the complete off-shell 4D,$$ \mathcal{N} $$ N = 2 superfield actions for any free massless integer spin s ≥ 2 fields, using the $$ \mathcal{N} $$ N = 2 harmonic super-space approach. The relevant gauge supermultiplet is accommodated by two real analytic bosonic superfields $$ {h}_{\alpha \left(s-1\right)\dot{\alpha}\left(s-1\right)}^{++} $$ h α s − 1 α ̇ s − 1 + + , $$ {h}_{\alpha \left(s-2\right)\dot{\alpha}\left(s-2\right)}^{++} $$ h α s − 2 α ̇ s − 2 + + and two conjugated complex analytic spinor superfields $$ {h}_{\alpha \left(s-1\right)\dot{\alpha}\left(s-1\right)}^{+3} $$ h α s − 1 α ̇ s − 1 + 3 , $$ {h}_{\alpha \left(s-2\right)\dot{\alpha}\left(s-1\right)}^{+3} $$ h α s − 2 α ̇ s − 1 + 3 , where α(s) := (α1. . . αs),$$ \dot{\alpha} $$ α ̇ (s) := ($$ \dot{\alpha} $$ α ̇ 1. . .$$ \dot{\alpha} $$ α ̇ s). Like in the harmonic superspace formulations of $$ \mathcal{N} $$ N = 2 Maxwell and supergravity theories, an infinite number of original off-shell degrees of freedom is reduced to the finite set (in WZ-type gauge) due to an infinite number of the component gauge parameters in the analytic superfield parameters. On shell, the standard spin content (s,s−1/2,s−1/2,s−1) is restored. For s = 2 the action describes the linearized version of “minimal” $$ \mathcal{N} $$ N = 2 Einstein supergravity.


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