The limit set of trajectory in quasi-homogeneous system in ℝ3

In this article, we study the limit set of trajectory in three-dimensional quasi-homogeneous system. For the trajectory which is close to a singular curve, we show that either it approaches a fixed point or infinity, or it oscillates. Moreover, an oscillating example is given. The behaviour of trajectory which is near a closed orbit is also studied. At the end, we classify the integral manifolds of the system with isolated singularities.     

Invariants of Ω-conjugacy of diffeomorphisms with a nongeneric homoclinic trajectory

We establish conditions for -conjugacy of two-dimensional diffeomorphisms with a nontrivial structure of the limit set in a neighborhood of a nongeneric homoclinic trajectory.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 2, pp. 153–159, February, 1990.     

Eliminating counterexamples: A Grade 8 student’s learning trajectory for contrapositive proving

This study uses a teaching experiment and retrospective analysis to develop a learning trajectory for improving a Grade 8 student’s ability to construct, critique, and validate contrapositive arguments. The study is predicated on the hypothesis that adolescents perform poorly on contrapositive reasoning tasks because they lack sufficient ways of justifying contrapositive argumentation as a viable mode of argumentation. By studying a student’s actions and comments as she develops, critiques, and validates not-the-conclusion-implies-the-conditions-are-impossible arguments for conditional claims, a promising learning trajectory for contrapositive argumentation is developed. The student’s learning trajectory demonstrates how a conception of contrapositive proving as eliminating counterexamples can be useful in developing, critiquing, and validating contrapositive arguments.     

A reassessment of the Snyman–Fatti dynamic search trajectory method for unconstrained global optimization

The aim of this paper is to present a thorough reassessment of the Snyman–Fatti (SF) Multi-start Global Minimization Algorithm with Dynamic Search Trajectories, first published twenty years ago. The reassessment is done with reference to a slightly modified version of the original method, the essentials of which are summarized here. Results of the performance of the current code on an extensive set of standard test problems commonly in use today, are presented. This allows for a fair assessment to be made of the performance of the SF algorithm relative to that of the popular Differential Evolution (DE) method, for which test results on the same standard set of test problems used here for the SF algorithm, are also given. The tests show that the SF algorithm, that requires relatively few parameter settings, is a reliably robust and competitive method compared to the DE method. The results also indicate that the SF trajectory algorithm is particularly promising to solve minimum potential energy problems to determine the structure of atomic and molecular clusters.     

Trajectory statistical solutions for the 3D Navier–Stokes equations: The trajectory attractor approach

In this article, we construct the trajectory statistical solution for the 3D incompressible Navier–Stokes equations via the natural translation semigroup and trajectory attractor. In our construction, the trajectory statistical solution is an invariant space–time probability measure which is carried by the trajectory attractor of the natural translation semigroup defined on the trajectory space, and the trajectory statistical solution possesses the invariant property under the acting of the translation semigroup.     

A trajectory in ℝ<Superscript>3</Superscript> concealed from observers

In the problem of tracking an object moving in ?³ by observers, the most concealed trajectory is characterized under the condition that the object is at any time visible to at most two observers.     

Direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using a Radau pseudospectral method

A method is presented for direct trajectory optimization and costate estimation of finite-horizon and infinite-horizon optimal control problems using global collocation at Legendre-Gauss-Radau (LGR) points. A key feature of the method is that it provides an accurate way to map the KKT multipliers of the nonlinear programming problem to the costates of the optimal control problem. More precisely, it is shown that the dual multipliers for the discrete scheme correspond to a pseudospectral approximation of the adjoint equation using polynomials one degree smaller than that used for the state equation. The relationship between the coefficients of the pseudospectral scheme for the state equation and for the adjoint equation is established. Also, it is shown that the inverse of the pseudospectral LGR differentiation matrix is precisely the matrix associated with an implicit LGR integration scheme. Hence, the method presented in this paper can be thought of as either a global implicit integration method or a pseudospectral method. Numerical results show that the use of LGR collocation as described in this paper leads to the ability to determine accurate primal and dual solutions for both finite and infinite-horizon optimal control problems.     

A learning trajectory in Kindergarten and first grade students’ thinking of variable and use of variable notation to represent indeterminate quantities

Given the importance of understanding and using indeterminate quantities in algebraic thinking, the development of learning trajectories about how Kindergarten and first grade students understand variable and use variable notation in the context of algebraic expressions is critical. Based on an empirically developed learning trajectory, we analyzed children’s responses at three different points in a classroom teaching experiment. Our purpose was to describe levels of thinking among 16 students (eight in Kindergarten and eight in first grade). Our results revealed qualitative changes in the thinking about indeterminate quantities of most student participants. As students progressed through the experiment, we found that they advanced from what we characterized as a “Pre-variable” Level to a “Letters as representing indeterminate quantities as varying unknowns; explicit operations on indeterminate quantities” Level. Learning trajectories such as that developed here hold promise for informing the design of interventions that support young children’s early algebraic thinking.