Ski equipment producer SWIX has recently presented a new pair of ski poles,called SWIX Triac, which di¤ers from conventional (round) ski poles by having atriangular cross section. SWIX claims that the main objective for this design is thatit has superior sti¤ness to weight ratio compared to common ski poles. We provein this paper that this claim in general is not true. More speci…c, we show thatfor thin walled cross sections, a hollow circular cross section has up to 36% bettersti¤ness to weight ratio than a corresponding triangular cross section.

We bridge mathematical number theory with optimal control and show that a generalised Fibonacci sequence enters the control function of finite-horizon dynamic optimisation problems with one state and one control variable. In particular, we show that the recursive expression describing the first-order approximation of the control function can be written in terms of a generalised Fibonacci sequence when restricting the final state to equal the steady-state of the system. Further, by deriving the solution to this sequence, we are able to write the first-order approximation of optimal control explicitly. Our procedure is illustrated in an example often referred to as the Brock-Mirman economic growth model

We bridge mathematical number theory with that of optimal control and show that a generalised Fibonacci sequence enters the control function of finite horizon dynamic optimisation problems with one state and one control variable. In particular, we show that the recursive expression describing the first-order approximation of the control function can be written in terms of a generalised Fibonacci sequence when restricting the final state to equal the steady state of the system. Further, by deriving the solution to this sequence, we are able to write the first-order approximation of optimal control explicitly. Our procedure is illustrated in an example often referred to as the Brock-Mirman economic growth model.

In this article we develop a method of solving general one-dimensional Linear Quadratic Regulator (LQR) problems in optimal control theory, using a generalized form of Fibonacci numbers. We find the solution R (k) of the corresponding discrete-time Riccati equation in terms of ratios of generalized Fibonacci numbers. An explicit Binet type formula for R (k) is also found, removing the need for recursively finding the solution at a given timestep. Moreover, we show that it is also possible to express the feedback gain, the penalty functional and the controller state in terms of these ratios. A generalized golden ratio appears in the corresponding in finite horizon problem. Finally, we show the use of the method in a few examples.

This article deals with the optimal design of long and slender compressive struts. The main objective is to minimize the mass of the struts under certain non-failure constraints and thus find the optimal material. We show that the main failure mode of the struts is Euler buckling. The results clearly show that the struts should be constructed from unidirectional carbon fiber composites. A Monte-Carlo model for random microstructure homogenization of unidirectional composites is developed. We finish by performing a numerical computation of the effective properties of the chosen carbon fiber/epoxy composite using COMSOL MULTIPHYSICS software.

Most papers on stochastic homogenization either deal with theoretical aspects or with questions regarding computational issues. Since the theoretical analysis involves the solution of a problem which is stated in a abstract probability space, it is not clear how the two areas are connected. In previous works this problem has not been considered. However, recently Bourgeat and Piatnitski investigated this connection in the scalar case for second order operators of divergence form. They proved that in the limit, the method of periodic approximation gives the same effective properties as in stochastic homogenization. In this paper we prove similar results for the vector valued case, which appears in e.g. the theory of elasticity. Moreover, we provide a numerical analysis of the results.

In this paper we investigate a set of structure conditions used in the existence theory of differential equations. More specific, we find best constants for the corresponding inequalities in the special case when the differential operator is the p-Laplace operator.

This note makes the link between theoretical results on stochastic homogenization and effective computation of averaged coefficients for diffusion operators in random media. Examples of how to construct relevant random media and numerical results on the effective coefficients are given.