Differences between Robin and Neumann eigenvalues on metric graphs

We consider the Laplacian on a metric graph, equipped with Robin ($\delta$-type) vertex condition at some of the graph vertices and Neumann-Kirchhoff condition at all others. The corresponding eigenvalues are called Robin eigenvalues, whereas they are called Neumann eigenvalues if the Neumann-Kirchhoff condition is imposed at all vertices. The sequence of differences between these pairs of eigenvalues is called the Robin-Neumann gap. We prove that the limiting mean value of this sequence exists and equals a geometric quantity, analogous to the one obtained for planar domains. Moreover, we show that the sequence is uniformly bounded and provide explicit upper and lower bounds. We also study the possible accumulation points of the sequence and relate those to the associated probability distribution of the gaps. To prove our main results, we prove a local Weyl law, as well as explicit expressions for the second moments of the eigenfunction scattering amplitudes.Comment: 30 pages, 10 figures. arXiv admin note: text overlap with arXiv:2212.0914

Parallel random number generation

We present a library of 19 pseudo-random number generators, implemented for graphical processing units. The library is implemented in the OpenCL framework and empirically evaluated using the TestU01 library. Most of the presented generators pass the tests. The generators' performance is evaluated on five different devices. The Tyche-i generator is the best choice overall, while on some specific devices other generators are better

Problem of random number generation and solutions

The general information about random number generation is presented in the paper. The reason why it is a problem is explained and the methods used for the solution are examined. The advantages and disadvantages of each method are provided

Three-dimensional representation of the photofission process on heavy nuclei, induced by real photons with energy between 0.1 and 0.6 GeV

La fisión de núcleos pesados inducida por fotones reales (fotofisión), en el rango de energías entre 0,1 GeV y el umbral de fotoproducción de un pión (0,6 GeV), se representa usando un motor de gráficos en 3D creado por el autor. Para modelar los núcleos pesados se usan los modelos nucleares de gota líquida y gas de Fermi. El proceso de fotofisión se lleva a cabo en dos etapas: la etapa de cascada intranuclear y la etapa de cascada evaporativa. En la primera etapa, el haz de fotones incidente da inicio a una cascada de reacciones nucleares en las que algunas de las partículas involucradas pueden escapar del núcleo. Como resultado, el núcleo queda en un estado excitado (núcleo compuesto), dando paso a la segunda etapa de fotofisión. En esta etapa, el núcleo compuesto es desexcitado por medio de los mecanismos de evaporación de partículas y/o de fisión. Así, contando la tasa de fisión, se calcula la probabilidad de fotofisión. Las etapas antes mencionadas son representadas usando ilustraciones tridimensionales a través de una aplicación creada en C/C++ y vinculada con OpenGL.The fission of heavy nuclei induced by real photons (photofission) in the energy range between 0.1 GeV and the one-pion-photoproduction threshold (0.6 GeV) is represented using a 3D graphics engine created by the author. Nuclear models: liquid drop and Fermi gas are used to represent the heavy nuclei. The photofission process is divided into two stages: the intranuclear cascade and the evaporative cascade. In the first stage, the incident photon beam initiates an intranuclear cascade in which some of the particles involved can escape from the nucleus. The nucleus is then left in an excited state (compound nucleus), triggering the evaporative cascade in which the compound nucleus is de-excited via particle evaporation and/or fission mechanisms. The photofission probability is computed by counting the fission rate. The aforementioned stages were depicted using three-dimensional illustrations through an application written in C/C++ and linked to OpenGL

Finding Fibonacci : an interdisciplinary liberal arts course based on mathematical patterns

The purpose of this study was to design, teach, and evaluate an undergraduate interdisciplinary mathematics course based on certain patterns, primarily the Fibonacci sequence. Rationale for the course includes the benefits of connected learning and the scarcity of liberal arts courses based on mathematics. The course is intended to emphasize pattern exploration in mathematics as well as in other disciplines. It is hoped that students in the course will find connections between mathematics and history, art, architecture, music, literature, nature, and economics. Course design includes a syllabus, student textbook, and sample lesson plans. The student textbook explores mathematical connections with the Fibonacci sequence such as the golden ratio, Pascal\u27s triangle, Pythagorean triples, combinatorics, and fractal geometry. Historical background of Leonardo Fibonacci\u27s life and times in the High Middle Ages is used to introduce the course. Applications of Fibonacci numbers in art, architecture, music, literature, nature, and economics are discussed. Students are asked to assess the meaning of these connections in light of their liberal arts experience. Evaluation of the course, primarily qualitative in nature, gives evidence that the pilot offering of the course enabled students to see relationships between various fields of study in a new way