Mathematical Background Research Articles

Distributivity of a segmentation lattice

Closure spaces, namely the finite ones, with closed singletons are studied on the level of segmentations — partitions of the space into closed subsets. Segmentations form a lattice and we study spaces for which this lattice is distributive. Studying these spaces may help understanding mathematical background for segmentation of a digital image. A crucial notion is that of connectively irreducible sets which can be defined in any finite closure space. The paper provides several equivalent conditions for segmentational distributivity in terms of triples of closed sets, connected systems of closed sets, property of induced closure operator on down-sets of connectively irreducible sets, and finally by restriction (or disability) of existence of certain sublattices.

Data Labeling for Machine Learning Engineers: Project-Based Curriculum and Data-Centric Competitions

The process of training and evaluating machine learning (ML) models relies on high-quality and timely annotated datasets. While a significant portion of academic and industrial research is focused on creating new ML methods, these communities rely on open datasets and benchmarks. However, practitioners often face issues with unlabeled and unavailable data specific to their domain. We believe that building scalable and sustainable processes for collecting data of high quality for ML is a complex skill that needs focused development. To fill the need for this competency, we created a semester course on Data Collection and Labeling for Machine Learning, integrated into a bachelor program that trains data analysts and ML engineers. The course design and delivery illustrate how to overcome the challenge of putting university students with a theoretical background in mathematics, computer science, and physics through a program that is substantially different from their educational habits. Our goal was to motivate students to focus on practicing and mastering a skill that was considered unnecessary to their work. We created a system of inverse ML competitions that showed the students how high-quality and relevant data affect their work with ML models, and their mindset changed completely in the end. Project-based learning with increasing complexity of conditions at each stage helped to raise the satisfaction index of students accustomed to difficult challenges. During the course, our invited industry practitioners drew on their first-hand experience with data, which helped us avoid overtheorizing and made the course highly applicable to the students’ future career paths.

An Example of the Use of Photogrammetry for Archiving the Shapes of Natural Materials and Creations

We can find cultural monuments everywhere. Smaller ones are often made from available and cheap natural materials - stones. However, due to the weather, due to their smaller dimensions, are often subject to vandalism, damage caused by careless handling or accidental damage. Sometimes they are just pieces of stone, but they have a deeper meaning in connection with cultural value, and for example by studying them in office spaces in the form of 3D models. In our contribution, we focus on the protection, study and preservation of such cultural and artistic monuments made of natural materials or generally stone natural formations or even large stone or brick surfaces. In our article, we want to focus on the practical feasibility rather than the mathematical background, as we believe that this information is much more useful in practice. We point out that it is advisable to use drones to check the condition of materials.

Factors Affecting First Year Students’ Performance in Fundamental Accounting Course: Case Study Kampala International University in Tanzania (KIUT)

Purpose: The purpose of this study was to determine factors that potentially influence students’ academic performance in Fundamental Accounting course at Kampala International University in Tanzania (KIUT).
 Methodology: The study adopted a quantitative research design. The sample of the study involved 134 first year students majoring in Bachelor of Business Administration at KIUT. Data was analyzed using the chi-square test at p=.05.
 Findings: The results revealed that students’ Mathematics Background, Accounting background, Combination, interest and study positioning had significant association with students’ performance in fundamental accounting course while students’ gender, English background, Weekly study hours, financial support, financial support rate, usage of hardcopies, school type, understanding level, had no significant with students’ performance in fundamental accounting course. While several variables to examine factors affecting students’ performance have been done; this study included students’ positioning and students study style (by hard copies or soft copies) which have not been exhausted in the past.
 Unique Contribution to Theory, Practice and Policy: On the basis of the findings this study recommended that KIUT should adjust its selection criteria by including previous academic performance particularly Mathematics and Accounting related subjects as among the metrics to be used in admitting students for programmers having fundamental accounting as one of its core course. However, to maintain stability, students with poor Mathematics and Accounting backgrounds could be provided with remedial programmes to cover up for their weakness.

Three‐Dimensional, Molecule‐Sensitive Mapping of Structured Falling Liquid Films Using Raman Scanning

AbstractA method for molecule‐sensitive, non‐contact and marker‐free 3D mapping of vertical falling liquid films using Raman spectroscopy is presented. Technical liquid films are common in many applications in the chemical industry such as columns, evaporators, dryers and absorbers/desorbers. Measurements of an aqueous sodium sulfate film on a plain surface with a small pyramid attached to it, as well as a pillow plate are presented using a 3D surface plot. Suppressing surface gloss as well as other optical reflections from the metallic backplates is realized by using only inelastic light scattering and mathematical background correction. The results show that liquid mapping with Raman spectroscopy is possible and 3D maps can be created from different structured surfaces.

Bridging the Gap Between Artificial Neural Networks and Kernel Regressions for Vector-Valued Problems in Microwave Applications

Thanks to their convex formulation, kernel regressions have shown an improved accuracy with respect to artificial neural network (ANN) structures in regression problems where a reduced set of training samples are available. However, despite the above interesting features, kernel regressions are inherently less flexible than ANN structures since their implementations are usually limited to scalar-output regression problems. This article presents a vector-valued (multioutput) formulation of the kernel ridge regression (KRR) aimed at bridging the gap between multioutput ANN structures and scalar kernel-based approaches. The proposed vector-valued KRR relies on a generalized definition of the reproducing kernel Hilbert space (RKHS) and on a new multioutput kernel structure. The mathematical background of the proposed vector-valued formulation is extensively discussed together with different matrix kernel functions and training schemes. Moreover, a compression strategy based on the Nystrom approximation is presented to reduce the computational complexity of the model training. The effectiveness and the performance of the proposed vector-valued KRR are discussed on an illustrative example consisting of a high-speed link and on the optimization of a Doherty amplifier.

Multi‐Spectrum Method for the Determination of the Spectral Responsivity and the Short‐Circuit Current of Photovoltaic Devices

Herein, a method for the determination of the spectral responsivity (SR) and the short‐circuit current under standard test conditions of photovoltaic devices (e.g., solar cells) is presented. This multi‐spectrum SR method requires a spectrally tunable broadband light source irradiating the photovoltaic device with a large number of different spectra. For each spectrum, the light response of the device and the spectral irradiance in the measuring plane are measured. The spectral irradiances are integrated within predefined wavelength intervals and are incorporated together with the measured light response into an equation system which relates them to the (unknown) SR of the photovoltaic device. By solving the equation system, mathematically using regression algorithms, the SR is determined. Due to the usage of a broadband light source, the device operates at realistic injection conditions during measurements. The mathematical background of the multi‐spectrum SR method is described and its applicability is demonstrated on three world‐photovoltaic‐scale‐type solar cells and one large‐area reference cell. Short‐circuit currents from all SR curves are calculated using the tabulated AM1.5 G spectrum. In comparison to the SR reference data, the short‐circuit currents from the multi‐spectrum SR method deviate by less than 0.68%.

Generation of Escher‐like spiral drawings in a modified hyperbolic space

Dutch graphic artist M.C. Escher created many famous drawings with a deep mathematical background based on wallpaper symmetry, hyperbolic geometry, spirals, and regular polyhedra. However, he did not attempt any spiral drawings in hyperbolic space. In this paper, we consider a modified hyperbolic geometry by removing the condition that a geodesic is orthogonal to the unit circle in the Poincaré model. We show that spiral symmetry and the similarity property exist in this modified geometry so that the creation of uncommon hyperbolic spiral drawings is possible. To this end, we first establish the theoretical foundation for the proposed method by deriving a contraction mapping and a rotation for constructing modified hyperbolic spiral tilings (MHSTs) and introduce symmetry groups to analyze the structure of MHSTs. Then, to embed a pre‐designed wallpaper template into the tiles, we derive a one‐to‐one mapping between a tile of MHST and a rectangle. Finally, we specify some technical implementation details and give a gallery of the resulting MHST drawings. Using existing wallpaper templates, the proposed method is able to generate a great variety of exotic Escher‐like drawings.

Writing statistical methods for ecologists

AbstractTheMethodssection is a key component of any ecology research publication containing detailed information on how the data were collected and analyzed. However, descriptions of which statistical methods were used and how they were applied can substantially vary and may not provide enough information for the analyses to be reproducible. Computational and statistical programming advances have allowed ecological researchers without a strong statistical or mathematical background to access and use increasingly complex statistical methods. Thus, statistical methods are written by and need to be accessible to researchers across a range of quantitative expertise. Poorly writtenMethodssections can incorrectly inflate the strength of or call into question the results of an analysis. Although there are resources available, we have not found one that is specific to writing statistical methods, includes all the elements we discuss, and is targeted for ecologists. Here we provide guidelines for ecological researchers when writing statistical methods and review frequent errors made inStatistical Methodssections. We highlight some common dos and don'ts when writingStatistical Methodssections and present a simple checklist to help guide authors with their writing to ensure reproducibility. We illustrate the use of this guidance with two examples.

Mathematics Behind the Heisenberg Uncertainty Principle

Most physics books do not reveal clearly how the Heisenberg uncertainty principle was derived. This uncertainty comes from the consequence of the wave-particle duality of matter giving statement that position and momentum cannot be measured in the same time. This article tries to reveal mathematics background behind the expression the Heisenberg uncertainty using supported mathematics background such as Fourier transform, Fourier transform integral, the probability of Gaussian distribution and it ends up with the expression of wave function which describe the localized particle giving relation Heisenberg uncertainty principle.

Identification of Nonlinear Systems Using the Hammerstein-Wiener Model with Improved Orthogonal Functions

Hammerstein-Wiener systems present a structure consisting of three serial cascade blocks. Two are static nonlinearities, which can be described with nonlinear functions. The third block represents a linear dynamic component placed between the first two blocks. Some of the common linear model structures include a rational-type transfer function, orthogonal rational functions (ORF), finite impulse response (FIR), autoregressive with extra input (ARX), autoregressive moving average with exogenous inputs model (ARMAX), and output-error (O-E) model structure. This paper presents a new structure, and a new improvement is proposed, which is consisted of the basic structure of Hammerstein-Wiener models with an improved orthogonal function of Müntz-Legendre type. We present an extension of generalised Malmquist polynomials that represent Müntz polynomials. Also, a detailed mathematical background for performing improved almost orthogonal polynomials, in combination with Hammerstein-Wiener models, is proposed. The proposed approach is used to identify the strongly nonlinear hydraulic system via the transfer function. To compare the results obtained, well-known orthogonal functions of the Legendre, Chebyshev, and Laguerre types are exploited.

On Modeling Antennas Using MoM-Based Algorithms: Wire-Grid versus Surface Triangulation

This paper focuses on antenna modeling using wire-grid and surface triangulation as two of the most commonly used MoM-based approaches in this field. A comprehensive overview is provided for each of them, including their history, applications, and limitations. The mathematical background of these approaches is briefly presented. Two working algorithms were developed and described in detail, along with their implementations using acceleration techniques. The wire-grid-based algorithm enables modeling of arbitrary antenna solid structures using their equivalent grid of wires according to a specific modeling recommendation proposed in earlier work. On the other hand, the surface triangulation-based algorithm enables calculation of antenna characteristics using a novel excitation source model. Additionally, a new mesh generator based on the combined use of the considered algorithms is developed. These algorithms were used to estimate the characteristics of several antenna types with different levels of complexity. The algorithms computational complexities were also obtained. The results obtained using these algorithms were compared with those obtained using the finite difference time domain numerical method, as well as those calculated analytically and measured. The analysis and comparisons were performed on the example of a rectangle spiral, a spiral, rounded bow-tie planar antennas, biconical, and horn antennas. Furthermore, the validity of the proposed algorithms is verified using the Monte Carlo methodology.

Testing Distributions in Banking Sector Loans with Different Computer Programs: An Experimental Analysis for Turkey

Within the fields of risk management and banking, the normality condition is one of the basic assumptions to apply value at risk, capital asset pricing or linear regression models on credit risk assessment. However, banking sector data related to loans may not be normally distributed. Hence, it needs to be put through scientific tests. For this purpose, firstly, Anderson-Darling, Jarque-Bera, Kolmogorov-Smirnov, Shapiro-Wilk, and Shapiro-Francia tests are applied to ninety-two banking sector loan variables and it is demonstrated that most of the variables are not normally distributed. Additionally, the parameters of Normal, Birnbaum-Saunders, Exponential, Extreme Value, Gamma, Generalized Extreme Value, Inverse Gaussian, Log-Logistic, Logistic, Lognormal, Nakagami, Negative Binomial, Non-Parametric, Poisson, Rayleigh, Rician, t- Location-Scale, and Weibull distributions are estimated for loan variables. Thirdly, when the data are not normally distributed, it is necessary to examine the other test results. Therefore, Kolmogorov-Smirnov, Anderson Darling, and Chi-square test results are employed for sixty-one distributions related to the variables and best fitted distribution per variable is aimed at. The results indicate that different computer codes and programs may give different outcomes in connection with the normality and best fitting distribution. Therefore, the use of different strategies may also be adopted in risk management courses along with the traditional ones since the normality assumption is an essential first step for the application of such techniques. Finally, pedagogically speaking, it should be noted that teaching the essence of mathematical background and computer codes could be strategically useful for students in internalizing these distribution concepts. Keywords: distribution, banking sector loans, risk management

Procrustes cross-validation of multivariate regression models

A generalization of Procrustes Cross-Validation — recently introduced novel approach for validation of chemometric models — is proposed. The generalized approach is faster than its predecessor by several orders of magnitude and can be used for validation of a wider range of models. Furthermore, it provides new tools for exploring the heterogeneity of the dataset, quality of cross-validation splits, presence of outliers, etc. The paper describes methodological aspects of the generalized approach, based on using Procrustean rules, the mathematical background, as well as presents practical results obtained using real chemical datasets.

Pre-service Mathematics Teachers’ Multicultural Mathematics Competency: A Case of a University in Northern Thailand

This study aimed to explore pre-service mathematics teachers’ perceptions of their strengths and deficiencies in dealing with students from diverse mathematics cultural backgrounds. The pre-service mathematics teachers in the final year at a university located in Northern Thailand were participants. The survey and focus group interview were manipulated to attain more profound confirmation about how the pre-service mathematics teachers performed in their diverse mathematics classrooms. The pre-service mathematics teachers’ strengths and deficiencies were represented and divided into the affective, cognitive, and psychomotor domains. Their strengths included 1) consciousness about the multicultural processes, 2) acknowledgment of students’ sources of resistance, and 3) possession of a feedback cycle for dealing with diverse mathematics classrooms. Nevertheless, it was found their deficiencies, including 1) unawareness of multicultural diversity through many perspectives, 2) limited knowledge to handle irrelevant students’ culture, and 3) lack of skills to respond to a problem existing such as time limits. The implication included that pre-service mathematics teachers should have the opportunity to 1) interact with other different cultural mathematics backgrounds, 2) study more content courses about multicultural mathematics, and 3) engage in various activities involving multiculturalism and be exposed to multicultural instruction courses.

Analysis of Students' Critical Thinking Skills Based on Prior Knowledge Mathematics

The development of critical thinking skills has emerged as a global concern in a variety of fields, including education. The purpose of this research is to shed light on the connection that exists between one's prior knowledge and the mathematical critical thinking skills that are acquired through the use of quantitative methods. Collecting data through the use of mathematical instruments, mathematical critical thinking tests, and daily test data for the prerequisite material based on the average student's performance. High school students in the eleventh grade made up the study's sample population. The findings of the research indicate that there is a positive connection between mathematical background knowledge and critical thinking abilities in that subject area. It is possible to draw the conclusion that the material that serves as initial knowledge and is therefore a prerequisite needs to be studied extensively in order to have broad insight that contributes to the success of learning the subsequent material. In spite of this, the researchers discovered that the number of students who fell into the high and medium categories of critical thinking skills was lower when compared to the number of students who fell into the second category when it came to their initial mathematical abilities. In other words, there has been a rise in the proportion of students who score poorly on tests of their ability to engage in critical thinking. These additional aspects should be sought out and incorporated as research fodder in subsequent studies.

USAGE OF MASK R-CNN FOR AUTOMATIC LICENSE PLATE RECOGNITION

The subject of study is the creation process of an artificial intelligence system for automatic license plate detection. The goal is to achieve high license plate recognition accuracy on large camera angles with character extraction. The tasks are to study existing license plate recognition technics and to create an artificial intelligence system that works on big shooting camera angles with the help of modern machine learning solution – deep learning. As part of the research, both hardware and software-based solutions were studied and developed. For testing purposes, different datasets and competing systems were used. Main research methods are experiment, literature analysis and case study for hardware systems. As a result of analysis of modern methods, Mask R-CNN algorithm was chosen due to high accuracy. Conclusions. Problem statement was declared; solution methods were listed and characterized; main algorithm was chosen and mathematical background was presented. As part of the development procedure, accurate automatic license plate system was presented and implemented in different hardware environments. Comparison of the network with existing competitive systems was made. Different object detection characteristics, such as Recall, Precision and F1-Score, were calculated. The acquired results show that developed system on Mask R-CNN algorithm process images with high accuracy on large camera shooting angles.

Two Comments on the Binomial Theorem

Summary We present a calculus and a probabilistic proofs of the binomial theorem. We believe that the material presented in this article would be most suitable to students with some background in calculus and discrete mathematics. It can be used by instructors looking for interesting applications in a calculus or a discrete mathematics/probability course.

Fascination with Fluctuation: Luria and Delbrück’s Legacy

While Luria and Delbrück’s seminal work has found its way to some college biology textbooks, it is now largely absent from those in mathematics. This is a significant omission, and we consider it a missed opportunity to present a celebrated conceptual model that provides an authentic and, in many ways, intuitive example of the quantifiable nature of stochasticity. We argue that it is an important topic that could enrich the educational literature in mathematics, from the introductory to advanced levels, opening many doors to undergraduate research. The paper has two main parts. First, we present in detail the mathematical theory behind the Luria–Delbrück model and make suggestions for further readings from the literature. We also give ideas for inclusion in various mathematics courses and for projects that can be used in regular courses, independent projects, or as starting points for student research. Second, we briefly review available hands-on activities as pedagogical ways to facilitate problem posing, problem-based learning, and investigative case-based learning and to expose students to experiments leading to Poisson distributions. These help students with even limited mathematics backgrounds understand the significance of Luria–Delbrück’s work for determining mutation rates and its impact on many fields, including cancer chemotherapy, antibiotic resistance, radiation, and environmental screening for mutagens and teratogens.

CAD-Based Design Optimization of Four-Bar Mechanisms: An Emergency Ventilator Case Study

The design optimization of mechanisms is promising as it results in more energy-efficient machines without compromising performance. However, machine builders do not apply state-of-the-art methods, as these algorithms require case-specific theoretical analysis. Moreover, the design synthesis approaches in the literature predominantly utilize heuristic optimizers, leading to suboptimal local minima. This paper introduces a widely applicable workflow, guaranteeing the global optimum. The constraints describing the feasible region of the possible designs are essential to find the global optimum. Therefore, kinematic analysis of the point-to-point planar four-bar mechanism is discussed. Within the feasible design space, objective value samples were generated through the CAD multi-body software. These motion simulations determine the required torque to fulfill the movement for a combination of design parameters. This replaces the cumbersome analytic derivation of the torque. This paper introduces sparse interpolation techniques to avoid brute force sampling of the design space. The advantage of this approach is that it is easily scalable to more design parameters, as the interpolation method minimizes the number of necessary samples. This paper explains the mathematical background of our developed interpolation approach. However, a step-by-step procedure is introduced to allow the employment of the interpolation technique by machine designers without the necessity to understand the underlying mathematics. Finally, the mathematical expression, obtained from the interpolation, enables applying global optimizers. In a case study of an emergency ventilator mechanism with three design parameters, 1870 CAD motion simulations allowed reducing the RMS torque of the mechanism by 67%.