On the derivative formulas of the rotation minimizing frame in Lorentz–Minkowski 3-space

The main intention of this paper is to analyze integrability for the derivative formulas of the rotation minimizing frame in the Lorentz–Minkowski 3-space. As far as we know, no one has yet given a method to study their integrability in the Lorentz–Minkowski 3-space. So, we introduce the coordinate system in order to provide a tool for studying the integrability. As an application, the position vectors of some special curves having an important place in mathematical and physical research are obtained in the natural representation form. Finally, we support our work with examples.

SO(10) models with A4 modular symmetry

We combine SO(10) Grand Unified Theories (GUTs) with A4 modular symmetry and present a comprehensive analysis of the resulting quark and lepton mass matrices for all the simplest cases. We focus on the case where the three fermion families in the 16 dimensional spinor representation form a triplet of Γ3 ≃ A4, with a Higgs sector comprising a single Higgs multiplet H in the 10 fundamental representation and one Higgs field $$ \overline{\Delta } $$ ∆ ¯ in the $$ \overline{\mathbf{126}} $$ 126 ¯ for the minimal models, plus one Higgs field Σ in the 120 for the non-minimal models, all with specified modular weights. The neutrino masses are generated by the type-I and/or type II seesaw mechanisms and results are presented for each model following an intensive numerical analysis where we have optimized the free parameters of the models in order to match the experimental data. For the phenomenologically successful models, we present the best fit results in numerical tabular form as well as showing the most interesting graphical correlations between parameters, including leptonic CP phases and neutrinoless double beta decay, which have yet to be measured, leading to definite predictions for each of the models.

Coefficient Estimate of Toeplitz Determinant for a Certain Class of Close to Convex Functions

Let S denote the class of analytic and univalent functions in D, where D is defined as unit disk and having the Taylor representation form of S. We will determine the estimation for the Toeplitz determinants where the elements are the Taylor coefficients of the class close-to-convex functions in S.

Using Diversities to Model the Reliability of N-version Machine Learning System

N-version machine learning system (MLS) is an architectural approach to reduce error outputs from a system by redundant configuration using multiple machine learning (ML) modules. Improved system reliability achieved by N-version MLS inherently depends on how diverse ML models are employed and how diverse input data sets are given. However, neither error input spaces of individual ML models nor input data distributions are obtainable in practice, which is a fundamental barrier to understanding the reliability gain by N-version architecture. In this paper, we introduce two diversity measures quantifying the similarities of ML models’ capabilities and the interdependence of input data sets, respectively. The defined measures are used to formulate the reliability of an elemental N-version MLS called dependent double-modules double-inputs MLS. The system is assumed to fail when two ML modules output errors simultaneously for the same classification task. The reliabilities of different architecture options for this MLS are comprehensively analyzed through a compact matrix representation form of the proposed reliability model. Except for limiting cases, we observe that the architecture exploiting two diversities tends to achieve preferable reliability under reasonable assumptions. Intuitive relations between diversity parameters and architecture reliabilities are also demonstrated through numerical experiments with hypothetical settings.

Using Diversities to Model the Reliability of N-version Machine Learning System

N-version machine learning system (MLS) is an architectural approach to reduce error outputs from a system by redundant configuration using multiple machine learning (ML) modules. Improved system reliability achieved by N-version MLS inherently depends on how diverse ML models are employed and how diverse input data sets are given. However, neither error input spaces of individual ML models nor input data distributions are obtainable in practice, which is a fundamental barrier to understanding the reliability gain by N-version architecture. In this paper, we introduce two diversity measures quantifying the similarities of ML models’ capabilities and the interdependence of input data sets, respectively. The defined measures are used to formulate the reliability of an elemental N-version MLS called dependent double-modules double-inputs MLS. The system is assumed to fail when two ML modules output errors simultaneously for the same classification task. The reliabilities of different architecture options for this MLS are comprehensively analyzed through a compact matrix representation form of the proposed reliability model. Except for limiting cases, we observe that the architecture exploiting two diversities tends to achieve preferable reliability under reasonable assumptions. Intuitive relations between diversity parameters and architecture reliabilities are also demonstrated through numerical experiments with hypothetical settings.

ANALYTICAL SOLUTION OF THE SCHRÖDINGER INTEGRAL EQUATION

In this paper, the question about the use of wave dynamics for solving problems of membrane separation of helium isotopes in the gas state at cryogenic temperatures is considered. The dimensionless form of the stationary Schrödinger differential equation is obtained. Following that, the integral representation form of the wave function is written. This form, which is equivalent to the classical equation, is similar to the integral equation with a degenerate core; however, it contains a modulus of the argument with a shift along the real axis. Using the shift operator, the existing exponential function in the Schrödinger integral equation can be split into a differential operator and an exponential function of the argument module which does not contain a shift. The Fourier identity allows reducing the exponent of the modulus of the argument to a regular exponent. Next, based on the general property of a differential operator acting on an exponent, it is possible to calculate the spectral functions of the problem and write down the distribution for the wave function. This distribution is ultimately expressed through the spectra of the potential barrier. Thereafter, the structure and the spectrum of the composite barrier are considered. With the expression determining the reflection coefficient, it is found that the double-barrier system can have a resonant passage of one of the components in the sequence of distances between the layers of the membrane.

Some consequences of the rank normal form of a matrix

If A is a rectangular matrix of rank r, then A may be written as PSQ where P and Q are invertible matrices and s = \left( {\matrix{ \hfill {{{\rm{I}}_{\rm{r}}}} & \hfill {\rm{O}} \cr \hfill {\rm{O}} & \hfill {\rm{O}} \cr } } \right) . This is the rank normal form of the matrix A. The purpose of this paper is to exhibit some consequences of this representation form.