Three-dimensional exact solutions of the diffusion equation

При помощи метода быстрых разложений решается задача диффузии в параллелепипеде с граничными условиями 1-го рода и внутренним источником вещества, зависящим от координат точек параллелепипеда. Получено в общем виде решение, содержащее свободные параметры, с помощью которых можно получить множество новых точных решений с различными свойствами. Показан пример построения точного решения для случая внутреннего источника переменного только по оси OZ . Приведен анализ особенностей диффузионных потоков в параллелепипеде с указанным внутреннем источником. Получено, что концентрация вещества в центре параллелепипеда равна сумме среднеарифметического значения концентраций вещества в его вершинах и амплитуды внутреннего источника умноженного на величину The authors solve the problem of diffusion in a parallelepiped-shaped body with boundary conditions of the 1st kind and an internal source of substance, depending on the parallelepiped points coordinates with the fast expansions method. The proposed exact solution in general form contains free parameters, which can be used to obtain many new exact solutions with different properties. An example of constructing an exact solution with a variable internal source depending on one coordinate z is shown in the work. An analysis of the features of diffusion flows in a parallelepiped with the indicated internal source is given. It was found that the concentration of a substance in the center of a parallelepiped is equal to the sum of the arithmetic mean of the concentration of a substance at its vertices and the amplitude of the internal source multiplied by the value

APPLICATION OF FAST EXPANSIONS TO OBTAIN EXACT SOLUTIONS TO A PROBLEM ON RECTANGULAR MEMBRANE DEFLECTION UNDER ALTERNATING LOAD

The problem of rectangular membrane deflection under alternating loads is solved in general terms by means of the method of fast expansions. The exact solution is represented by the finite expression borrowed from the theory of fast expansions as a sum of the boundary function and Fourier sine series with two Fourier coefficients taken into account. The obtained exact solution includes free parameters. Changing the values of these parameters, one can derive many new exact solutions. Obtaining of exact solutions to a problem of the rigidly fixed membrane under two types of loads (dome-shaped and sinusoidal) is shown as an example. Graphs of the dome-shaped and sinusoidal loads on the membrane and the curves of the corresponding deflections and stress components are presented in the paper. From the analysis of the exact solutions, it is obvious that only when a symmetrical alternating load is used, the membrane maximum deflection is attained in the center of the membrane, and the stresses reach the highest values in the middle of both long sides. In the case of a non-symmetrical load, the maximum stress occurs in the middle of either one of two long sides of the rectangular membrane, and the maximum deflection is found in the central region.

Mutational and Selective Processes Involved in Evolution during Bacterial Range Expansions

Bacterial populations have been shown to accumulate deleterious mutations during spatial expansions that overall decrease their fitness and ability to grow. However, it is unclear if and how they can respond to selection in face of this mutation load. We examine here if artificial selection can counteract the negative effects of range expansions. We examined the molecular evolution of 20 mutator lines selected for fast expansions (SEL) and compared them to 20 other mutator lines freely expanding without artificial selection (CONTROL). We find that the colony size of all 20 SEL lines have increased relative to the ancestral lines, unlike CONTROL lines, showing that enough beneficial mutations are produced during spatial expansions to counteract the negative effect of expansion load. Importantly, SEL and CONTROL lines have similar numbers of mutations indicating that they evolved for the same number of generations and that increased fitness is not due to a purging of deleterious mutations. We find that loss of function mutations better explain the increased colony size of SEL lines than nonsynonymous mutations or a combination of the two. Interestingly, most loss of function mutations are found in simple sequence repeats (SSRs) located in genes involved in gene regulation and gene expression. We postulate that such potentially reversible mutations could play a major role in the rapid adaptation of bacteria to changing environmental conditions by shutting down expensive genes and adjusting gene expression.