Considering Instructional Approach & Question Design with the Hardy-Weinberg Principle

The Hardy-Weinberg principle (HWP) is an application of the binomial expansion theorem that is foundational to the field of population genetics. Because of the important history of the HWP in answering how variation is preserved during evolution, and the ability of Hardy-Weinberg equilibrium (HWE) to detect natural and sexual selection acting on a trait, the HWP is a staple of the introductory biology undergraduate curriculum in the United States. Introductory courses often cover a wide range of topics in ecology and evolution, and it is important that students have enough time during the semester to grasp the foundations of population genetics. At the same time, information needs to be presented clearly to ensure that the student gains a correct understanding of the HWP. This article discusses the importance of the HWP to undergraduate education in biology and describes misconceptions from the instructor’s perspective. These misconceptions are pervasive and risk undermining a proper understanding of the HWP. We provide examples adapted from university- and AP-level standardized tests.

Sensitivity analysis of truck structure based on CA algorithm

With the continuous improvement of modern CAE technology, structural reanalysis algorithm has gradually come into people’s vision and developed rapidly. The structure reanalysis algorithm introduced in this paper is an accelerated calculation method. The core idea of this algorithm is to avoid the complete analytical calculations after the structure modification, and reduce the calculation scale, save the calculation time, improve the efficiency of CAE simulation effectively on the premise of meeting the requirements of structure accuracy. The objective of this paper that is based on the initial third-order modal information of the truck structure is to control the overall quality of the structure. And it has important guiding significance for practical production. In this paper, different design variables are set in combination with the structural reanalysis algorithm. While the parameters of design variables are modified, sensitivity information analysis and Taylor expansion theorem are used to verify the feasibility and accuracy of the structural reanalysis method in optimal calculation

Wave propagation dynamics in a fractional Zener model with stochastic excitation

Equations of motion for a Zener model describing a viscoelastic rod are investigated and conditions ensuring the existence, uniqueness and regularity properties of solutions are obtained. Restrictions on the coefficients in the constitutive equation are determined by a weak form of the dissipation inequality. Various stochastic processes related to the Karhunen-Loéve expansion theorem are presented as a model for random perturbances. Results show that displacement disturbances propagate with an infinite speed. Some corrections of already published results for a non-stochastic model are also provided.

NONSTATIONARY HEAT CONDUCTION IN MULTILAYER GLAZING SUBJECTED TO DISTRIBUTED HEAT SOURCES

A method for calculation of nonstationary thermal fields in a multilayer glazing of vehicles under the effect of impulse film heat sources is offered. The glazing is considered as a rectangular multilayer plate made up of isotropic layers with constant thickness. Film heat sources are arranged on layers' interfaces. The heat conduction equation is solved using the Laplace transformation, series expansion and the second expansion theorem. The method offered can be used for designing a safe multilayer glazing under operational and emergency thermal and force loading in vehicles.

On the complex Hermite polynomials

In this paper we use a set of partial differential equations to prove an expansion theorem for multiple complex Hermite polynomials. This expansion theorem allows us to develop a systematic and completely new approach to the complex Hermite polynomials. Using this expansion, we derive the Poisson Kernel, the Nielsen type formula, the addition formula for the complex Hermite polynomials with ease. A multilinear generating function for the complex Hermite polynomials is proved.

Calculation of Temperature Fields in Multilayer Plates and Shells with Distributed Heat Sources

The aircraft multilayer glazing is considered as arectangular multilayer plate made up of isotropic layers withconstant thickness. The temperature on the side surface of theplate is zero. Convective heat transfer occurs on outer surfaces ofthe plate; on layers' interfaces film heat sources are arranged.The heat conduction equation for an arbitrary plate layer isreduced to the functional equation. A solution of the functionalequation we search in the form of three space functions product.We get the system of ordinary differential equations. Seriesexpansion factors are determined from a system of linearalgebraic equations. A transform of the required function isfound by the second expansion theorem, and the problemsolution has the form of double trigonometrical series.The comparative analysis of the results is carried out with theresults of other method. The method offered can be used fordesigning a safe multilayer glazing under operational andemergency thermal and force loading in different vehicles.

A nuclear matter calculation with the tensor-optimized Fermi sphere method with central interaction

The tensor-optimized Fermi sphere (TOFS) theory is applied first for the study of the property of nuclear matter using the Argonne V4$^\prime$$NN$ potential. In the TOFS theory, the correlated nuclear matter wave function is taken to be a power-series type of the correlation function $F$, where $F$ can induce central, spin–isospin, tensor, etc. correlations. This expression has been ensured by a linked cluster expansion theorem established in the TOFS theory. We take into account the contributions from all the many-body terms arising from the product of the nuclear matter Hamiltonian $\mathcal{H}$ and $F$. The correlation function is optimally determined in the variation of the total energy of nuclear matter. It is found that the density dependence of the energy per particle in nuclear matter is reasonably reproduced up to the nuclear matter density $\rho \simeq 0.20$ fm$^{-3}$ in the present numerical calculation, in comparison with other methods such as the Brueckner–Hartree–Fock approach.

q-Analogs of Lidstone expansion theorem, two-point Taylor expansion theorem, and Bernoulli polynomials

In this paper, we introduce a generalization of the [Formula: see text]-Taylor expansion theorems. We expand a function in a neighborhood of two points instead of one in three different theorems. The first is a [Formula: see text]-analog of the Lidstone theorem where the two points are 0 and 1 and we expand the function in [Formula: see text]-analogs of Lidstone polynomials which are in fact [Formula: see text]-Bernoulli polynomials as in the classical case. The definitions of these [Formula: see text]-Bernoulli polynomials and numbers are introduced. We also introduce [Formula: see text]-analogs of Euler polynomials and numbers. On the other two expansion theorems, we expand an analytic function around arbitrary points [Formula: see text] and [Formula: see text] either in terms of the polynomials [Formula: see text] or in terms of the polynomials [Formula: see text]. As an application, we introduce a new series expansion for the basic hypergeometric series [Formula: see text].