Set Theory and Proofs for Engineering Education

Through evaluations of learning objectives on several Engineering courses, the majority of students at some point will struggle with demonstration of proof of a principle in their homework assignments, quizzes, and exams. An early introduction of "Set Theory and Proofs" to engineering students can enrich their intuition and ability to solve comprehensive problems. As illustrated in this paper, set theory can be recognized by students as a simple and unnecessary topic. However, the understanding of principles in set theory and its derived concepts are essential to engineering students so they can improve their problem-solving skills when approaching a more complex problem using mathematics. Set Theory is a vast field of study which includes: Operations and algebra with sets, power sets, product sets, relations, functions, quantifiers, family of sets, index sets, just to name a few [1]. At The University of Texas at Tyler, the authors experienced set theory embedded in the learning objectives of Manufacturing Systems (MENG 5318) course offered by the Mechanical Engineering Department to its graduate students. In Fall 2016 and 2017, most of the students in the class failed to apply some of the principles in set theory. Overall, set theory is an important topic to engineering students where an understanding of the principal will ensure the success in completing advanced level courses.Cockrell School of Engineerin

Complexities of Bi-Colored Rubik\u27s Cubes

Which of two bi-colored cubes is the simpler puzzle? The differences in the coloring of the cubes creates different symmetries that dramatically reduce the number of states each cube can reach. Which of the symmetries is most reductive? The answer to these questions can be achieved by discovering and comparing the “God’s Number” for these cubes

Can Modern Nuclear Hamiltonians Tolerate a Bound Tetraneutron?

I show that it does not seem possible to change modern nuclear Hamiltonians to bind a tetraneutron without destroying many other successful predictions of those Hamiltonians. This means that, should a recent experimental claim of a bound tetraneutron be confirmed, our understanding of nuclear forces will have to be significantly changed. I also point out some errors in previous theoretical studies of this problem.Comment: 4 pages, 4 figures Revision corrects a pronou

Linear convergence of accelerated conditional gradient algorithms in spaces of measures

A class of generalized conditional gradient algorithms for the solution of optimization problem in spaces of Radon measures is presented. The method iteratively inserts additional Dirac-delta functions and optimizes the corresponding coefficients. Under general assumptions, a sub-linear $\mathcal{O}(1/k)$ rate in the objective functional is obtained, which is sharp in most cases. To improve efficiency, one can fully resolve the finite-dimensional subproblems occurring in each iteration of the method. We provide an analysis for the resulting procedure: under a structural assumption on the optimal solution, a linear $\mathcal{O}(\zeta^k)$ convergence rate is obtained locally.Comment: 30 pages, 7 figure

A domain-specific language and matrix-free stencil code for investigating electronic properties of Dirac and topological materials

We introduce PVSC-DTM (Parallel Vectorized Stencil Code for Dirac and Topological Materials), a library and code generator based on a domain-specific language tailored to implement the specific stencil-like algorithms that can describe Dirac and topological materials such as graphene and topological insulators in a matrix-free way. The generated hybrid-parallel (MPI+OpenMP) code is fully vectorized using Single Instruction Multiple Data (SIMD) extensions. It is significantly faster than matrix-based approaches on the node level and performs in accordance with the roofline model. We demonstrate the chip-level performance and distributed-memory scalability of basic building blocks such as sparse matrix-(multiple-) vector multiplication on modern multicore CPUs. As an application example, we use the PVSC-DTM scheme to (i) explore the scattering of a Dirac wave on an array of gate-defined quantum dots, to (ii) calculate a bunch of interior eigenvalues for strong topological insulators, and to (iii) discuss the photoemission spectra of a disordered Weyl semimetal.Comment: 16 pages, 2 tables, 11 figure