Nonlinear oscillations in a constant gravitational field

Exploring non-linear oscillations is a challenging task since the related differential equations cannot be directly solved in terms of elementary functions. Thus, complicated mathematical or numerical methods are usually employed to find accurate or approximate expressions that describe the behavior of the system with respect to time. In this paper, the vertical oscillations of an object under the influence of its weight and an opposite force with magnitude F=cyn, where n>0 are being explored. Accurate and approximate simple solutions regarding the object’s position with respect to time are presented and the dependence of the oscillation’s period from the oscillation’s range of displacements and the exponent n is revealed. In addition, the special case in which n=3/2 (which describes the oscillation of a rigid sphere on an elastic half space) is also highlighted. Lastly, it is shown that similar cases (such as the case of a force with magnitude F=kx+λx2) can be also treated using the same approach.

Highlighting non-parabolic bands in semiconductors

The parabolic approximation to the dispersion relation is a simplification that has often been adopted for the electronic band structure of most semiconductors near the edges of the fundamental bandgap. A non-parabolic approximation can be justified which will better describe the properties of semiconductors of narrow bandgaps for which a reduction to a quadratic form is not accurate enough, nor always warranted. It also stands for a better approximation in III–V compounds and for more complex thermoelectric materials. Some of the consequences of adopting non-parabolic bands will be highlighted, as well as approximate expressions for statistical properties. It is emphasized that many properties of semiconductors are not difficult to calculate with non-parabolic bands, which may have a wider range of applications in actual materials. These calculations can then be introduced in solid state physics and statistical physics courses through projects and homework problem sets. Specific examples are discussed designed to clarify basic physics concepts in semiconductors.

The effects of weak selection on neutral diversity at linked sites

The effects of selection on variability at linked sites have an important influence on levels and patterns of within-population variation across the genome. Most theoretical models of these effects have assumed that selection is sufficiently strong that allele frequency changes at the loci concerned are largely deterministic. These models have led to the conclusion that directional selection for new selectively favorable mutations, or against recurrent deleterious mutations, reduces nucleotide site diversity at linked neutral sites. Recent work has shown, however, that fixations of weakly selected mutations, accompanied by significant stochastic changes in allele frequencies, can sometimes cause higher diversity at linked sites when compared with the effects of fixations of neutral mutations. The present paper extends this work by deriving approximate expressions for the mean times to loss and fixation of mutations subject to selection, and analysing the conditions under which selection increases rather than reduces these times. Simulations are used to examine the relations between diversity at a neutral site and the fixation and loss times of mutations at a linked site subject to selection. It is shown that the long-term level of neutral diversity can be increased over the equilibrium expectation in the absence of selection by recurrent fixations and losses of linked, weakly selected dominant or partially dominant favorable mutations, and by linked recessive or partially recessive deleterious mutations. The results are used to examine the conditions under which associative overdominance, as opposed to background selection, is likely to operate.

Approximation of Elliptic Functions by Means of Trigonometric Functions with Applications

In this work, we give approximate expressions for Jacobian and elliptic Weierstrass functions and their inverses by means of the elementary trigonometric functions, sine and cosine. Results are reasonably accurate. We show the way the obtained results may be applied to solve nonlinear ODEs and other problems arising in nonlinear physics. The importance of the results in this work consists on giving easy and accurate way to evaluate the main elliptic functions cn, sn, and dn, as well as the Weierstrass elliptic function and their inverses. A general principle for solving some nonlinear problems through elementary functions is stated. No similar approach has been found in the existing literature.

The characteristics of average abundance function with mutation of multi-player threshold public goods evolutionary game model under redistribution mechanism

Background In recent years, the average abundance function has attracted much attention as it reflects the degree of cooperation in the population. Then it is significant to analyse how average abundance functions can be increased to promote the proliferation of cooperative behaviour. However, further theoretical analysis for average abundance function with mutation under redistribution mechanism is still lacking. Furthermore, the theoretical basis for the corresponding numerical simulation is not sufficiently understood. Results We have deduced the approximate expressions of average abundance function with mutation under redistribution mechanism on the basis of different levels of selection intensity $$\omega$$ ω (sufficiently small and large enough). In addition, we have analysed the influence of the size of group d, multiplication factor r, cost c, aspiration level $$\alpha$$ α on average abundance function from both quantitative and qualitative aspects. Conclusions (1) The approximate expression will become the linear equation related to selection intensity when $$\omega$$ ω is sufficiently small. (2) On one hand, approximation expression when $$\omega$$ ω is large enough is not available when r is small and m is large. On the other hand, this approximation expression will become more reliable when $$\omega$$ ω is larger. (3) On the basis of the expected payoff function $$\pi \left( \centerdot \right)$$ π ⋅ and function $$h(i,\omega )$$ h ( i , ω ) , the corresponding results for the effects of parameters (d,r,c,$$\alpha$$ α ) on average abundance function $$X_{A}(\omega )$$ X A ( ω ) have been explained.

Turing Instability and Hopf Bifurcation in Cellular Neural Networks

In this paper, we explore the dynamical behaviors of the 1D two-grid coupled cellular neural networks. Assuming the boundary conditions of zero-flux type, the stability of the zero equilibrium is discussed by analyzing the relevant eigenvalue problem with the aid of the decoupling method, and the conditions for the occurrence of Turing instability and Hopf bifurcation at the zero equilibrium are derived. Furthermore, the approximate expressions of the bifurcating periodic solutions are also obtained by using the Hopf bifurcation theorem. Finally, numerical simulations are provided to demonstrate the theoretical results.

Statistics of eigenvalue dispersion indices: quantifying the magnitude of phenotypic integration

Quantification of the magnitude of covariation plays a major role in the studies of phenotypic integration, for which statistics based on dispersion of eigenvalues of a covariance or correlation matrix—eigenvalue dispersion indices—are commonly used. However, their use has been hindered by a lack of clear understandings on their statistical meaning and sampling properties such as the magnitude of sampling bias and error. This study remedies these issues by investigating properties of these statistics with both analytic and simulation-based approaches. The relative eigenvalue variance of a covariance matrix is known in the statistical literature as a test statistic for sphericity, thus is an appropriate measure of eccentricity of variation. The same of a correlation matrix is exactly equal to the average squared correlation, thus is a clear measure of overall integration. Exact and approximate expressions for the mean and variance of these statistics are analytically derived for the null and arbitrary conditions under multivariate normality, clarifying the effects of sample size N, number of variables p, and parameters on the sampling bias and error. Accuracy of the approximate expressions are evaluated with simulations, confirming that most of them work reasonably well with a moderate sample size (N ≥ 16–64). Importantly, sampling properties of these indices are not adversely affected by high p:N ratio, promising their utility in high-dimensional phenotypic analyses. These statistics can potentially be applied to shape variables and phylogenetically structured data, for which necessary assumptions and modifications are presented.

Dynamic response of Mathieu–Duffing oscillator with Caputo derivative

In this paper, the harmonic balance method and its variants are used to analyze the response of Mathieu–Duffing oscillator with Caputo derivative. First, the exact and approximate expressions of the Caputo derivatives of trigonometric function and composite function are derived. Next, using the approximate expression of the Caputo derivative of the composite function, the resonance of Duffing oscillator with Caputo derivative is analyzed by the harmonic balance method. Finally, Mathieu–Duffing oscillator with Caputo derivative is approximated by three kinds of methods, i.e., the harmonic balance method, the residue harmonic balance method and the improved harmonic balance method. The corresponding numerical simulations are given to illustrate the performance of these methods as well. The results show that the residue harmonic balance method is more precise than the harmonic balance method and the improved harmonic balance method in analyzing the dynamic response of Mathieu–Duffing oscillator with Caputo derivative.

Interference Cancellation for MIMO Systems Employing the Generalized Receiver with High Spectral Efficiency

In this paper, we investigate the performan-ce in terms of symbol error probability (SEP) of multipleinput multiple-output (MIMO) systems employing the ge-neralized receiver with high spectral efficiency. In particular, we consider the coherent detection of M-PSK signals in a flat Rayleigh fading environment. We focus on spectrally efficient MIMO systems where after serial-to-parallel con-version, several sub-streams of symbols are simultaneously transmitted by using an antenna array, thereby increasing the spectral efficiency. The reception is based on linear mi-nimum mean-square-error (MMSE) combining, eventually followed by successive interference cancellation. Exact and approximate expressions are derived for an arbitrary nu-mber of transmitting and receiving antenna elements. Sim-ulation results confirm the validity of our analytical meth-odology.

Perturbation Expansion to the Solution of Differential Equations

The main purpose of this chapter is to describe the application of perturbation expansion techniques to the solution of differential equations. Approximate expressions are generated in the form of asymptotic series. These may not and often do not converge but in a truncated form of only two or three terms, provide a useful approximation to the original problem. These analytical techniques provide an alternative to the direct computer solution. Before attempting to solve these problems numerically, one should have an awareness of the perturbation approach.