Abstract:
We investigate the effect of turning delays on the behaviour of groups of differential wheeled robots and show that the group-level behaviour can be described by a transport equation with a suitably incorporated delay. The results of our mathematical analysis are supported by numerical simulations and experiments with e-puck robots. The experimental quantity we compare to our revised model is the mean time for robots to find the target area in an unknown environment. The transport equation with delay better predicts the mean time to find the target than the standard transport equation without delay.

Abstract:
In this paper we will look at the book Mathematical Go by Elwyn Berlekamp and David Wolfe \cite{MG}, and argue that the definitions and theories that they use are not the correct ones. We will argue that the new theory of scoring play games as developed by Fraser Stewart \cite{FS, FSP} is the proper way to analyse the game because it gives us the actual mathematical foundation for developing further theories for Go. We will also show that the reason the methods in the book Mathematical Go appear to work is coincidentally because the game Go has some very nice properties and show that these theories would not work for other scoring play games.

Abstract:
The mathematical model of the task of compiling the time-table in High-school has been carried out. It has been showed, that the task may be reduced to canonical form of extrimal combinatorial tasks with unlinear structure after identical transformations. The algorithm of the task’s decision for realizing the scheme of the directed sorting of variants is indicated. Приведена математическая модель задачи составления расписания занятий в высшем учебном заведении. Показано, что после тождественных преобразований она сводится к каноническому виду экстремальных комбинаторных задач с нелинейной структурой. Рассмотрен алгоритм решения задачи, реализующий схему направленного перебора вариантов. Наведено математичну модель задач складання розкладу занять у вищому навчальному заклад . Показано, що п сля тотожних перетворень вона зводиться до канон чного вигляду екстремальних комб наторних задач з нел н йною структурою. Розглянуто алгоритм розв’язання задач , що реал зу схему спрямованого перебору вар ант в.

Abstract:
The purpose of this paper is to present for the first time an elementary summary of a few recent results obtained through the application of the formal theory of partial differential equations and Lie pseudogroups in order to revisit the mathematical foundations of general relativity. Other engineering examples (control theory, elasticity theory, electromagnetism) will also be considered in order to illustrate the three fundamental results that we shall provide. The paper is therefore divided into three parts corresponding to the different formal methods used. 1) CARTAN VERSUS VESSIOT: The quadratic terms appearing in the " Riemann tensor " according to the " Vessiot structure equations " must not be identified with the quadratic terms appearing in the well known " Cartan structure equations " for Lie groups and a similar comment can be done for the " Weyl tensor ". In particular, " curvature+torsion" (Cartan) must not be considered as a generalization of "curvature alone" (Vessiot). Roughly, Cartan and followers have not been able to " quotient down to the base manifold ", a result only obtained by Spencer in 1970 through the "nonlinear Spencer sequence" but in a way quite different from the one followed by Vessiot in 1903 for the same purpose and still ignored. 2) JANET VERSUS SPENCER: The " Ricci tensor " only depends on the nonlinear transformations (called " elations " by Cartan in 1922) that describe the "difference " existing between the Weyl group (10 parameters of the Poincar\'e subgroup + 1 dilatation) and the conformal group of space-time (15 parameters). It can be defined by a canonical splitting, that is to say without using the indices leading to the standard contraction or trace of the Riemann tensor. Meanwhile, we shall obtain the number of components of the Riemann and Weyl tensors without any combinatoric argument on the exchange of indices. Accordingly, the Spencer sequence for the conformal Killing system and its formal adjoint fully describe the Cosserat/Maxwell/Weyl theory but General Relativity is not coherent at all with this result. 3) ALGEBRAIC ANALYSIS: Contrary to other equations of physics (Cauchy equations, Cosserat equations, Maxwell equations), the Einstein equations cannot be " parametrized ", that is the generic solution cannot be expressed by means of the derivatives of a certain number of arbitrary potential-like functions, solving therefore negatively a 1000 $ challenge proposed by J. Wheeler in 1970. Accordingly, the mathematical foundations of mathematical physics must be revisited within this formal framework,

We start
recalling with critical eyes the mathematical methods used in gauge theory and
prove that they are not coherent with continuum mechanics, in particular the
analytical mechanics of rigid bodies (despite using the same group theoretical
methods) and the well known couplings existing between elasticity and
electromagnetism (piezzo electricity, photo elasticity, streaming
birefringence). The purpose of this paper is to avoid such contradictions by
using new mathematical methods coming from the formal theory of systems of
partial differential equations and Lie pseudo groups. These results finally
allow unifying the previous independent tentatives done by the brothers E. and
F. Cosserat in 1909 for elasticity or H. Weyl in 1918 for electromagnetism by
using respectively the group of rigid motions of space or the conformal group
of space-time. Meanwhile we explain why the Poincaré duality scheme existing between geometry and physics has to do with homological algebra
and algebraic analysis. We insist on the fact that these results could not have
been obtained before 1975 as the corresponding tools were not known before.

The purpose of this paper is to present for the first time an elementary summary of a few recent results obtained through the application of the formal theory of partial differential equations and Lie pseudogroups in order to revisit the mathematical foundations of general relativity. Other engineering examples (control theory, elasticity theory, electromagnetism) will also be considered in order to illustrate the three fundamental results that we shall provide successively. 1) VESSIOT VERSUS CARTAN: The quadratic terms appearing in the “Riemann tensor” according to the “Vessiot structure equations” must not be identified with the quadratic terms appearing in the well known “Cartan structure equations” for Lie groups. In particular, “curvature+torsion” (Cartan) must not be considered as a generalization of “curvature alone” (Vessiot). 2) JANET VERSUS SPENCER: The “Ricci tensor” only depends on the nonlinear transformations (called “elations” by Cartan in 1922) that describe the “difference” existing between the Weyl group (10 parameters of the Poincaré subgroup + 1 dilatation) and the conformal group of space-time (15 parameters). It can be defined without using the indices leading to the standard contraction or trace of the Riemann tensor. Meanwhile, we shall obtain the number of components of the Riemann and Weyl tensors without any combinatoric argument on the exchange of indices. Accordingly and contrary to the “Janet sequence”, the “Spencer sequence” for the conformal Killing system and its formal adjoint fully describe the Cosserat equations, Maxwell equations and Weyl equations but General Relativity is not coherent with this result. 3) ALGEBRA VERSUS GEOMETRY: Using the powerful methods of “Algebraic Analysis”, that is a mixture of homological agebra and differential geometry, we shall prove that, contrary to other equations of physics (Cauchy

Abstract:
This paper is concerned with the verification of mathematical modeling of the container cranes under earthquake loadings with shake table test results. Comparison of the shake table tests with the theoretical studies has an important role in the estimation of the seismic behavior of the engineering structures. For this purpose, a new shake table and mathematical model were developed. Firstly, a new physical model is directly fixed on the shake table and the seismic response of the container crane model against the past earthquake ground motion was measured. Secondly, a four degrees-of-freedom mathematical model is developed to understand the dynamic behaviour of cranes under the seismic loadings. The results of the verification study indicate that the developed mathematical model reasonably represents the dynamic behaviour of the crane structure both in time and frequency domains. The mathematical model can be used in active-passive vibration control studies to decrease structural vibrations on container cranes. 1. Introduction The 1995 Kobe Earthquake exposed the destructive effects of earthquake motions on cranes. During the earthquake, many cranes were damaged and one collapsed primarily due to spreading of support girders. Recent studies indicated that modern jumbo cranes are susceptible to damage highlighting the importance of improving the seismic performance of cranes. Seismic risk to cranes is becoming better understood. In Figure 1, the collapsed crane at the 1995 Kobe Earthquake is represented. Figure 1: Collapsed crane at the Kobe Earthquake [ 21]. Some of the researches about seismic behaviour of the cranes aim to understand the behaviour of cranes under earthquake loadings while some of them aim to investigate methods to mitigate these risks. An important tool to understand the dynamic behaviour of crane structures in earthquakes is to develop a scaled physical model of the crane and perform the shake table tests. Kanayama and Kashiwazaki [1] and Kanayama et al. [2] investigated the dynamic behavior of cranes conducting a series of shaking table tests. 1/25 scaled crane model was used for the first experiment and, for the second one, a 1/8 scaled model was used. Experiments were performed under uniaxial earthquake excitations that were applied along the direction of the boom. Real earthquake records were used during the experiments and horizontal component of the records considered which component is the most destructive. Performed experiments were focused on investigating the major failure mechanisms and monitoring the rigidity of the

采用参数化–相关点法模型对折叠桌的动态变化过程进行数学描述。首先通过分析木条在桌面边缘铰接处的坐标、木条与钢筋的交点坐标以及桌脚边缘点坐标三者和桌面铰接处与x轴的夹角θ、桌面半径r之间的关系，将各坐标变量用参数表示，通过参数值的变化观测相应坐标变量的变化，进而用此参数方程对折叠桌的动态变化过程进行数学描述。而具体解决问题时，也可以直接利用桌面边缘铰接处的坐标、木条与钢筋的交点的坐标表示出桌脚边缘线上各点的坐标，根据三者的函数关系式来描述折叠桌的动态变化过程。
Parametric-related point method model is used to give a mathematical description of the dynamic table folding process. Firstly by analyzing the relationship between the coordinate of the batten on the edge of the table articulated point, the intersection point coordinates of batten and steel and the coordinates of leg edge points and the inclined angle—θ of table articulated point and the x axis, and radius of the table—r, we can conclude the equation of coordinate variables indicated by parameters and observe the change of the corresponding coordinate variables through the change of parameter values. This parametric equation of folding table is used to give a mathematical description for the process of dynamic change. When solving specific problems, we can also directly use the coordinate of the batten on the edge of the table articulated point and the coordinate of the intersection between batten and steel to calculate the coordinate of leg edge points. Using these three function formulas, the folding process is described.