Bifurcation of limit cycles in a quintic Hamiltonian system under a sixth-order perturbation

This paper intends to explore the bifurcation of limit cycles for planar polynomial systems with even number of degrees. To obtain the maximum number of limit cycles, a sixth-order polynomial perturbation is added to a quintic Hamiltonian system, and both local and global bifurcations are considered. By employing the detection function method for global bifurcations of limit cycles and the normal form theory for local degenerate Hopf bifurcations, 31 and 35 limit cycles and their configurations are obtained for different sets of controlled parameters. It is shown that: H(6)  35 = 6² − 1, where H(6) is the Hilbert number for sixth-degree polynomial systems.     

Linear estimate of the number of limit cycles for a class of non-linear systems

A dynamic system has a finite number of limit cycles. However, finding the upper bound of the number of limit cycles is an open problem for general non-linear dynamical systems. In this paper, we investigated a class of non-linear systems under perturbations. We proved that the upper bound of the number of zeros of the related elliptic integrals of the given system is 7n + 5 including multiple zeros, which also gives the upper bound of the number of limit cycles for the given system.     

Curious Characterizations of Projective and Affine Geometries

We prove the following characterization theorem: If any three of the following four matroid invariants—the number of points, the number of lines, the coefficient of λ^n − 2 in the characteristic polynomial, and the number of three-element dependent sets—of a rank-n combinatorial geometry (or simple matroid) are the same as those of a rank-n projective geometry, then it is a projective geometry (of the same order). To do this, we use a lemma which is of independent interest: If H is a geometry in which all the lines have exactly ℓ − 1 or ℓ points and G is a geometry with at least three of the four matroid invariants the same as H, then all the lines in G also have exactly ℓ − 1 or ℓ points. An analogue of the characterization theorem holds for affine geometries. Our methods also yield inequalities amongst the four matroid invariants.     

Magnetohydrodynamic stationary and oscillatory convective stability in a mushy layer during binary alloy solidification

For the case of solidification of a bottom cooled binary alloy, the magnetohydrodynamic stationary and oscillatory convective stability in the mushy layer is investigated analytically using normal mode linear stability analysis. In the limit of large Stefan number (S_t), a near–eutectic approximation with large far field temperature is considered in the present research. To ascertain the instability in the mushy layer, the strength of the superimposed magnetic field is so chosen that it corresponds to a given mush Hartmann number (Ha_m) of the problem. The results are presented for various values of mush Hartmann numbers in the range, 0 ≤ Ha_m ≤ 50. The critical Rayleigh number for stationary convection shows a linear relationship with increasing Ha_m. The magnetohydrodynamic effect imparts a stabilizing influence during stationary convection. In comparison to that of the stationary convective mode, the oscillatory mode appears to be critically susceptible at higher values of β (β = S_t/℘² ϒ², ℘ is the compositional ratio, ϒ = 1 + S_t/℘), and vice versa for lower β values. Analogous to the behavior for stationary convection, the magnetic field also offers a stabilizing effect in oscillatory convection and thus influences global stability of the mushy layer. Increasing magnetic strength shows reduction in the wavenumber and in the number of rolls formed in the mushy layer.     

Numerical simulation of magnetohydrodynamic buoyancy-induced flow in a non-isothermally heated square enclosure

Numerical simulation of magnetohydrodynamic (MHD) buoyancy-induced heat transfer and fluid flow has been analyzed in a non-isothermally heated square enclosure using finite volume method. The bottom wall of enclosure were heated and cooled with a sinusoidal function and top wall was cooled isothermally. Vertical walls of the enclosure were adiabatic. Effects of Rayleigh number (Ra = 10⁴, 10⁵ and 10⁶), Hartman number (Ha = 0, 50 and 100) and amplitude of sinusoidal function (n = 0.25, 0.5 and 1) on temperature and flow fields were analyzed. It was observed that heat transfer was decreased with increasing Hartmann number and decreasing value of amplitude of sinusoidal function.     

Integrals,Quantum Galois Extensions,and the Affineness Criterion for Quantum Yetter–Drinfel'd Modules

In this paper we shall generalize the notion of an integral on a Hopf algebra introduced by Sweedler, by defining the more general concept of an integral of a threetuple (H, A, C), where H is a Hopf algebra coacting on an algebra A and acting on a coalgebra C. We prove that there exists a total integral γ: C → Hom(C, A) of (H, A, C) if and only if any representation of (H, A, C) is injective in a functorial way, as a corepresentation of C. In particular, the quantum integrals associated to Yetter–Drinfel'd modules are defined. Let now A be an H-bicomodule algebra, ^H$\text{Y}$$\text{D}$_A the category of quantum Yetter–Drinfel'd modules, and B = {a ∈ A|∑S^− 1(a_〈1〉)a_{〈 − 1〉} ⊗ a_〈0〉 = 1_H ⊗ a}, the subalgebra of coinvariants of the Verma structure A ∈ ^H$\text{Y}$$\text{D}$_A. We shall prove the following affineness criterion: if there exists γ: H → Hom(H, A) a total quantum integral and the canonical map β: A ⊗ _B A → H ⊗ A, β(a ⊗ _B b) = ∑S^− 1(b_〈1〉)b_{〈 − 1〉} ⊗ ab_〈0〉 is surjective (i.e., A/B is a quantum homogeneous space), then the induction functor – ⊗ _B A: $\text{M}$_B → ^H$\text{Y}$$\text{D}$_A is an equivalence of categories. The affineness criteria proven by Cline, Parshall, and Scott, and independently by Oberst (for affine algebraic groups schemes) and Schneider (in the noncommutative case), are recovered as special cases.     

Fast Rectangular Matrix Multiplication and Applications

First we study asymptotically fast algorithms for rectangular matrix multiplication. We begin with new algorithms for multiplication of ann×nmatrix by ann×n²matrix in arithmetic timeO(n^ω),ω=3.333953…, which is less by 0.041 than the previous record 3.375477…. Then we present fast multiplication algorithms for matrix pairs of arbitrary dimensions, estimate the asymptotic running time as a function of the dimensions, and optimize the exponents of the complexity estimates. For a large class of input matrix pairs, we improve the known exponents. Finally we show three applications of our results:   (a) we decrease from 2.851 to 2.837 the known exponent of the work bounds for fast deterministic (NC) parallel evaluation of the determinant, the characteristic polynomial, and the inverse of ann×nmatrix, as well as for the solution to a nonsingular linear system ofnequations,   (b) we asymptotically accelerate the known sequential algorithms for the univariate polynomial composition mod xⁿ, yielding the complexity boundO(n^1.667) versus the old record ofO(n^1.688), and for the univariate polynomial factorization over a finite field, and   (c) we improve slightly the known complexity estimates for computing basic solutions to the linear programming problem withmconstraints andnvariables.     

Vacuum polarization in light two-electron atoms and ions

A general theory of the vacuum polarization in light atomic and muon-atomic systems is considered. We derive the closed analytical expression for the Uehling potential and evaluate corrections on vacuum polarization for the 1¹S-state of the two-electron ³He and ⁴He atoms and for some two-electron ions, including the Li⁺, Be²⁺, B³⁺ and C⁴⁺ ions. The correction for vacuum polarization in two-electron He atoms has been evaluated as ΔE_ueh ≈ 7.253 ± 0.0025 × 10⁻⁷ a.u. The analogous corrections in the two-electron He-like ions rapidly increase with the nuclear charge Q (ΔE_ueh ≈ 2.7061 × 10⁻⁶ a.u. for the Li⁺ ion and ΔE_ueh ≈ 2.3495 × 10⁻⁵ a.u. for the C⁴⁺ ion). The corresponding corrections have also been evaluated for the electron–nucleus and electron–electron interactions.     

Fractional dynamics of systems with long-range interaction

We consider one-dimensional chain of coupled linear and nonlinear oscillators with long-range powerwise interaction defined by a term proportional to 1/∣n − m∣^α+1. Continuous medium equation for this system can be obtained in the so-called infrared limit when the wave number tends to zero. We construct a transform operator that maps the system of large number of ordinary differential equations of motion of the particles into a partial differential equation with the Riesz fractional derivative of order α, when 0 < α < 2. Few models of coupled oscillators are considered and their synchronized states and localized structures are discussed in details. Particularly, we discuss some solutions of time-dependent fractional Ginzburg–Landau (or nonlinear Schrodinger) equation.     

Free in-plane vibration analysis of a curved beam (arch) with arbitrary various concentrated elements

In the existing literature, the information regarding the exact solutions for free in-plane vibrations of the curved beams (or arches) carrying various concentrated elements is rare, particularly for the case with multiple attachments including eccentricities and mass moments of inertias. For this reason, this paper aims at presenting an effective approach to tackle the title problem. First of all, the un-coupled equation of motion for the circumferential displacement of an arch segment is derived. Next, based on the value of the discriminate parameter for a cubic equation, the exact solutions for the three types of roots of the un-coupled equation are determined and, corresponding to each type of roots, all displacement functions for the arch segment in terms of the real numbers (instead of the complex ones) are obtained. Finally, use of the compatible equations for the displacements and slopes together with the equilibrium equations for the forces and moments at each intermediate node and two ends of the entire curved beam, a frequency equation of the form ∣H(ω)∣ = 0 is obtained. It is found that the conventional approach by using the condition “∣H(ω^t)∣ ? ε” to search for the approximate value of ω^t is difficult even if the convergence tolerance ε is greater than 10⁺³ (i.e., ε > 10⁺³) instead of less than 10^?3 (i.e., ε < 10^?3), however, the half-interval method is one of the effective tools for solving the problem if all coefficients of the determinant ∣H(ω)∣ are the real numbers. In addition to comparing with the existing literature, most of the numerical results obtained from the presented method are compared with those obtained from the conventional finite element method (FEM) and good agreement is achieved.     

Statistical properties of nucleotides in human chromosomes 21 and 22

In this paper the statistical properties of nucleotides in human chromosomes 21 and 22 are investigated. The n-tuple Zipf analysis with n = 3, 4, 5, 6, and 7 is used in our investigation. It is found that the most common n-tuples are those which consist only of adenine (A) and thymine (T), and the rarest n-tuples are those in which GC or CG pattern appears twice. With the n-tuples become more and more frequent, the double GC or CG pattern becomes a single GC or CG pattern. The percentage of four nucleotides in the rarest ten and the most common ten n-tuples are also considered in human chromosomes 21 and 22, and different behaviors are found in the percentage of four nucleotides. Frequency of appearance of n-tuple f(r) as a function of rank r is also examined. We find the n-tuple Zipf plot shows a power-law behavior for r < 4ⁿ⁻¹ and a rapid decrease for r > 4ⁿ⁻¹. In order to explore the interior statistical properties of human chromosomes 21 and 22 in detail, we divide the chromosome sequence into some moving windows and we discuss the percentage of ξη (ξ, η = A, C, G, T) pair in those moving windows. In some particular regions, there are some obvious changes in the percentage of ξη pair, and there maybe exist functional differences. The normalized number of repeats N₀(l) can be described by a power law: N₀(l) ∼ l^−μ. The distance distributions P₀(S) between two nucleotides in human chromosomes 21 and 22 are also discussed. A two-order polynomial fit exists in those distance distributions: log P₀(S) = a + bS + cS², and it is quite different from the random sequence.     

Characterizations of PG(n − 1, q)\PG(k − 1, q) by Numerical and Polynomial Invariants

We show that the simple matroid PG(n − 1, q)\PG(k − 1, q), for n ≥ 4 and 1 ≤ k ≤ n − 2, is characterized by a variety of numerical and polynomial invariants. In particular, any matroid that has the same Tutte polynomial as PG(n − 1, q)\PG(k − 1, q) is isomorphic to PG(n − 1, q)\PG(k − 1, q).     

Maximal subgroups of the Coxeter group W(H4) and quaternions

The largest finite subgroup of O(4) is the non-crystallographic Coxeter group W(H₄) of order 14,400. Its derived subgroup is the largest finite subgroup W(H₄)/Z₂ of SO(4) of order 7200. Moreover, up to conjugacy, it has five non-normal maximal subgroups of orders 144, two 240, 400 and 576. Two groups [W(H₂) × W(H₂)] ⋊ Z₄ and W(H₃) × Z₂ possess non-crystallographic structures with orders 400 and 240 respectively. The groups of orders 144, 240 and 576 are the extensions of the Weyl groups of the root systems of SU(3) × SU(3), SU(5) and SO(8) respectively. We represent the maximal subgroups of W(H₄) with sets of quaternion pairs acting on the quaternionic root systems.     

Neural networks analysis of free laminar convection heat transfer in a partitioned enclosure

The feasibility of using neural networks (NNs) to predict the complete thermal and flow variables throughout a complicated domain, due to free convection, is demonstrated. Attention is focused on steady, laminar, two-dimensional, natural convective flow within a partitioned cavity. The objective is to use NN (trained on a database generated by a CFD analysis of the problem of a partitioned enclosure) to predict new cases; thus saving effort and computation time. Three types of NN are evaluated, namely General Regression NNs, Polynomial NNs, and a versatile design of Backpropagation neural networks. An important aspect of the study was optimizing network architecture in order to achieve best performance. For each of the three different NN architectures evaluated, parametric studies were performed to determine network parameters that best predict the flow variables.A CFD simulation software was used to generate a database that covered the range of Rayleigh number Ra = 10⁴–5 × 10⁶. The software was used to calculate the temperature, the pressure, and the horizontal and vertical components of flow speed. The results of the CFD were used for training and testing the neural networks (NN). The robustness of the trained NNs was tested by applying them to a “production” data set (1500 patterns for Ra = 8 × 10⁴ and 1500 patterns for Ra = 3 × 10⁶), which the networks have never been “seen” before. The results of applying the technique on the “production” data set show excellent prediction when the NNs are properly designed. The success of the NN in accurately predicting free convection in partitioned enclosures should help reduce analysis-time and effort. Neural networks could potentially help solve some cases in which CFD fails to solve because of numerical instability.     

Calculation of a static potential created by plane fractal cluster

In this paper we demonstrate new approach that can help in calculation of electrostatic potential of a fractal (self-similar) cluster that is created by a system of charged particles. For this purpose we used the simplified model of a plane dendrite cluster [1] that is generated by a system of the concentric charged rings located in some horizontal plane (see Fig. 2). The radiuses and charges of the system of concentric rings satisfy correspondingly to relationships: r_n = r₀ξⁿ and e_n = e₀bⁿ, where n determines the number of a current ring. The self-similar structure of the system considered allows to reduce the problem to consideration of the functional equation that similar to the conventional scaling equation. Its solution represents itself the sum of power-low terms of integer order and non-integer power-law term multiplied to a log-periodic function [5], [6]. The appearance of this term was confirmed numerically for internal region of the self-similar cluster (r₀ ≪ r ≪ r_N−1), where r₀, r_N−1 determine the smallest and the largest radiuses of the limiting rings correspondingly. The results were obtained for homogeneously (b > 0) and heterogeneously (b < 0) charged rings. We expect that this approach allows to consider more complex self-similar structures with different geometries of charge distributions.     

Dynamics of landslide model with time delay and periodic parameter perturbations

In present paper, we analyze the dynamics of a single-block model on an inclined slope with Dieterich–Ruina friction law under the variation of two new introduced parameters: time delay T_d and initial shear stress μ. It is assumed that this phenomenological model qualitatively simulates the motion along the infinite creeping slope. The introduction of time delay is proposed to mimic the memory effect of the sliding surface and it is generally considered as a function of history of sliding. On the other hand, periodic perturbation of initial shear stress emulates external triggering effect of long-distant earthquakes or some non-natural vibration source. The effects of variation of a single observed parameter, T_d or μ, as well as their co-action, are estimated for three different sliding regimes: β < 1, β = 1 and β > 1, where β stands for the ratio of long-term to short-term stress changes. The results of standard local bifurcation analysis indicate the onset of complex dynamics for very low values of time delay. On the other side, numerical approach confirms an additional complexity that was not observed by local analysis, due to the possible effect of global bifurcations. The most complex dynamics is detected for β < 1, with a complete Ruelle–Takens–Newhouse route to chaos under the variation of T_d, or the co-action of both parameters T_d and μ. These results correspond well with the previous experimental observations on clay and siltstone with low clay fraction. In the same regime, the perturbation of only a single parameter, μ, renders the oscillatory motion of the block. Within the velocity-independent regime, β = 1, the inclusion and variation of T_d generates a transition to equilibrium state, whereas the small oscillations of μ induce oscillatory motion with decreasing amplitude. The co-action of both parameters, in the same regime, causes the decrease of block’s velocity. As for β > 1, highly-frequent, limit-amplitude oscillations of initial stress give rise to oscillatory motion. Also for β > 1, in case of perturbing only the initial shear stress, with smaller amplitude, velocity of the block changes exponentially fast. If the time delay is introduced, besides the stress perturbation, within the same regime, the co-action of T_d (T_d < 0.1) and small oscillations of μ induce the onset of deterministic chaos.     

Effect of variable density on hydromagnetic mixed convection flow of a non-Newtonian fluid past a moving vertical plate

The effect of temperature-dependent density on MHD mixed convection flow of power-law fluid past a moving semi-infinite vertical plate for high temperature differences between the plate and the ambient fluid is studied. The fluid density is assumed to decrease exponentially with temperature. The usual Boussinesq approximations are not considered due to the large temperature differences. The surface temperature of the moving plate was assumed to vary according to a power-law form, that is, T_w(x) = T_∞ + Ax^γ. The fluid is permeated by a uniform magnetic field imposed perpendicularly to the plate on the assumption of small magnetic Reynolds number. A numerical shooting algorithm for two unknown initial conditions with fourth-order Runge–Kutta integration scheme has been used to solve the coupled non-linear boundary value problem. The effects of various parameters on the velocity and temperature profiles as well as the local skin-friction coefficient and the local Nusselt number are presented graphically and in the tabular form. The results show that application of Boussinesq approximations in a non-Newtonian fluid subjected to high temperature differences gives a significant error in the values of the skin-friction coefficient and the application of an external magnetic field reduces this error markedly in the case of shear-thickening fluid.     

Bifurcation of limit cycles by perturbing a piecewise linear Hamiltonian system with a homoclinic loop

In this paper, we study limit cycle bifurcations for a kind of non-smooth polynomial differential systems by perturbing a piecewise linear Hamiltonian system with the center at the origin and a homoclinic loop around the origin. By using the first Melnikov function of piecewise near-Hamiltonian systems, we give lower bounds of the maximal number of limit cycles in Hopf and homoclinic bifurcations, and derive an upper bound of the number of limit cycles that bifurcate from the periodic annulus between the center and the homoclinic loop up to the first order in ε$\epsilon$. In the case when the degree of perturbing terms is low, we obtain a precise result on the number of zeros of the first Melnikov function.     

Bifurcations of travelling wave solutions for the combined kdv–mkdv equation

By using the theory of bifurcations of dynamical systems to the combined k dv–mk dv equation, the existence of solitary wave, kink and anti-kink wave solutions and uncountably infinite many smooth periodic wave solutions are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given.