Mathematical Community Research Articles

The integral of complex variables on three defined contours

Complex variables have aroused strong interest in the community of mathematics in the past few decades. Our aim is to obtain the integral results of a particular equationin three defined contours. To determine the value of integral, the information we need to find is the residue values ABCD on four specific points and find out what residue values each contour includes. After finding the residue values and analysing four contours, we should use Cauchy integral formula to find the integral value. Firstly, by using the partial fraction to separate the equation into four independent parts, we obtain four values on each of the four fractions are the value of integral. Thus we can find exactly the four residue values, which are -1 on (1,0), 3 on (0,i), -2 on (-1.0) and 1 on (-I,0) as shown in the following expression. and then we must analyse each contour and find out the residue values they include. By drawing a diagram of each of the three contours. We find that contour 1 contains A, contour 2 contains A and D, and contour 3 contains ABCD. The final step to obtaining the value of residue is to use Cauchy integral formulaAccording to Cauchy integral formula, the value of integral of the equation on a specific contour is obtained by multiplying the sum of the residue values included in that contour by 2πi, the value of integral of the equation can be found as we now have all the necessary information. For the first contour the value of integral is calculated by using 2πi multiply by -1. We can get the result -2πi. For the second contour the value of integral is calculated by using 2πi multiply by the sum of -1 and 1. As we can see, the sum is 0. We can get the result 0. For the third contour the value of integral is calculated by using 2πi multiply by the sum of -1,3, -2, and 1.As we can see, the sum of these residue values 1. We can get the result 2πi. Hence all of the values of integral of the equation is found.

Welcoming students to the mathematics community: obstacles to “belonging”

This paper reflects on some of the obstacles which lead some students, particularly those from non-traditional academic backgrounds, to question whether they “belong” to the mathematics community.

An Open-Access Clicker Question Bank for Numerical Analysis

ABSTRACT We present a question bank consisting of over 250 multiple-choice and true–false questions covering a broad range of material typically taught in an introductory undergraduate course in numerical analysis or computational science. The questions are ideal for polling students during lectures by means of a student response system that uses clicker remotes or smartphones running a suitable app. We describe our experiences implementing these clicker questions in a recent class and provide evidence of their effectiveness in terms of testing students' prior knowledge, gauging understanding of new material, increasing participation, and especially improving student satisfaction. Our conclusions are supported by a mid-semester student survey as well as anecdotal observations. The question bank has been released as an open-access educational resource under a Creative Commons license (CC BY-NC-SA) for free use by the mathematics community.

A Graphic Method for Detecting Multiple Roots Based on Self-Maps of the Hopf Fibration and Nullity Tolerances

The aim of this paper is to study, from a topological and geometrical point of view, the iteration map obtained by the application of iterative methods (Newton or relaxed Newton’s method) to a polynomial equation. In fact, we present a collection of algorithms that avoid the problem of overflows caused by denominators close to zero and the problem of indetermination which appears when simultaneously the numerator and denominator are equal to zero. This is solved by working with homogeneous coordinates and the iteration of self-maps of the Hopf fibration. As an application, our algorithms can be used to check the existence of multiple roots for polynomial equations as well as to give a graphical representation of the union of the basins of attraction of simple roots and the union of the basins of multiple roots. Finally, we would like to highlight that all the algorithms developed in this work have been implemented in Julia, a programming language with increasing use in the mathematical community.

Borsuk’s partition conjecture

In 1933, Borsuk proposed the following problem: Can every bounded set in \({{\mathbb{E}}^n}\) be divided into n + 1 subsets of smaller diameter? This problem has been studied by many authors, and a lot of partial results have been discovered. In particular, Kahn and Kalai’s counterexamples surprised the mathematical community in 1993. Nevertheless, the problem is still far away from being completely resolved. This paper presents a broad review on related subjects and, based on a novel reformulation, introduces a computer proof program to deal with this challenging problem.

Romania helps Uganda on its way to the International Mathematical Olympiad

This article reports on an ongoing privately-financed project started in 2019, promoted by members of the Romanian mathematical community, supporting Uganda’s participation in the International Mathematical Olympiads.

Regularization for the inversion of fibre Bragg grating spectra

Fibre Bragg Gratings have become widespread measurement devices in engineering and other fields of application. In all but a few cases, the relation between cause and effect is simplified to a proportional model. However, at its mathematical core lies a nonlinear inverse problem which appears not to have received much attention in the literature. In this paper, we present this core problem to the mathematical community and provide a first report on opportunities and limitations of a regularization approach. In particular, we show that difficulties arise from non-uniqueness and the absence of established parameter selection rules for nonlinear inverse problems with multiple regularization parameters. Nevertheless, the paper takes a first step toward extracting more information from a single FBG measurement.

The Legacy of Peter Wynn

After the death of Peter Wynn in December 2017, manuscript documents he left came to our knowledge. They concern continued fractions, rational (Padé) approximation, Thiele interpolation, orthogonal polynomials, moment problems, series, and abstract algebra. The purpose of this paper is to analyze them and to make them available to the mathematical community. Some of them are in quite good shape, almost finished, and ready to be published by anyone willing to check and complete them. Others are rough notes, and need to be reworked. Anyway, we think that these works are valuable additions to the literature on these topics and that they cannot be left unknown since they contain ideas that were never exploited. They can lead to new research and results. Two unpublished papers are also mentioned here for the first time.

John Conway and Mathematical Communities

John Conway and Mathematical Communities

A mathematician’s perspective on the Oldroyd B model: Progress and future challenges

We review progress and open challenges for a number of problems related to the Oldroyd B model which are of current interest to the mathematical community. We discuss existence results for initial value problems and steady flows, flow stability, and the problem of understanding the asymptotics of the high Weissenberg number limit. We also discuss efforts and challenges in numerical simulation, exemplified by three problems: extensional flow in a four roll mill geometry, flow past a confined cylinder, and locomotion of microorganisms in a viscoelastic fluid. • Reviews the current state of progress and open challenges. • Existence theory, flow stability, boundary layers and singularities. • Numerical simulations of problems which illustrate mathematical difficulties. • Extensional flow in four roll mill geometries, flow past a cylinder. • Locomotion of microorganisms.

Biomechanical modelling of cancer: Agent‐based force‐based models of solid tumours within the context of the tumour microenvironment

AbstractOnce cancer is initiated, with normal cells mutated into malignant ones, a solid tumour grows, develops and spreads within its microenvironment invading the local tissue; the disease progresses and the cancer cells migrate around the body leading to metastasis, the formation of distant secondary tumours. Interactions between the tumour and its microenvironment drive this cascade of events which have devastating, if not fatal, consequences for the human host/patient. Among these interactions, biomechanical interactions are a vital component. In this review paper, key biomechanical relationships are discussed through a presentation of modelling efforts by the mathematical and computational oncology community. The main focus is directed, naturally, towards lattice‐free agent‐based, force‐based models of solid tumour growth and development. In such models, interactions between pairs of cancer cells (as well as between cells and other structures of the tumour microenvironment) are governed by forces. These forces are ones of repulsion and adhesion, and are typically modelled via either an extended Hertz model of contact mechanics or using Johnson–Kendal–Roberts theory, both of which are discussed here. The role of the extracellular matrix in determining disease progression is outlined along with important cell‐vessel interactions which combined together account for a great proportion of Hanahan and Weinberg's Hallmarks of Cancer.

A Symbolic-Numeric Validation Algorithm for Linear ODEs with Newton–Picard Method

A symbolic-numeric validation algorithm is developed to compute rigorous and tight uniform error bounds for polynomial approximate solutions to linear ordinary differential equations, and in particular D-finite functions. It relies on an a posteriori validation scheme, where such an error bound is computed afterwards, independently from how the approximation was built. Contrary to Newton–Galerkin validation methods, widely used in the mathematical community of computer-assisted proofs, our algorithm does not rely on finite-dimensional truncations of differential or integral operators, but on an efficient approximation of the resolvent kernel using a Chebyshev spectral method. The result is a much better complexity of the validation process, carefully investigated throughout this article. Indeed, the approximation degree for the resolvent kernel depends linearly on the magnitude of the input equation, while the truncation order used in Newton–Galerkin may be exponential in the same quantity. Numerical experiments based on an implementation in C corroborate this complexity advantage over other a posteriori validation methods, including Newton–Galerkin.

On the Up-to-date Course of Mathematical Logic for the Future Math Teachers

All-round development of the everyday logic of students should be considered as one of the most important tasks of general secondary education on the whole and general secondary mathematics education in particular. We discuss the problem of organization in teachers' training institutions of higher education and the expedient training of the future math teachers at institutions of general secondary education. The main goal is to ensure their ability to realize all their future professional activities and the necessary participation in forming the everyday logic of their pupils. The authors think that vocational educational program of training is that the future secondary school math teachers must contain a separate course of mathematical logic including at least 90 training hours (3 credits ECTS). Although the content filling of the course cannot be irrespective of the general level of arrangement of mathematics education in the corresponding country, it ought to be a subject of discussion of the international mathematics community and managers in the sphere of higher mathematics education. Simultaneously, the role, the place, and the expedient structure of such a course in the corresponding training programs should be under discussion. The article represents the authors' point of view on the problems indicated above. The research has a qualitative characteristic as a whole. Only some of its conclusions have statistical corroboration.

Efficient kinetic thermal inverse modeling for organic material decomposition

The prevalent use of organic materials in manufacturing is a fire safety concern, and motivates the need for predictive thermal decomposition models. A critical component of predictive modeling is numerical inference of kinetic parameters from bench scale data. Currently, an active area of computational pyrolysis research focuses on identifying efficient, robust methods for optimization. This paper demonstrates that kinetic parameter calibration problems can successfully be solved using classical gradient-based optimization. We explore calibration examples that exhibit characteristics of concern: high nonlinearity, high dimensionality, complicated schemes, overlapping reactions, noisy data, and poor initial guesses. The examples demonstrate that a simple, non-invasive change to the problem formulation can simultaneously avoid local minima, avoid computation of derivative matrices, achieve a computational efficiency speedup of 10x, and make optimization robust to perturbations of parameter components. Techniques from the mathematical optimization and inverse problem communities are employed. By re-examining gradient-based algorithms, we highlight opportunities to develop kinetic parameter calibration methods that should outperform current methods.

New perspectives in bibliometric indicators: Moving from citations to citing authors

• Definition of a bibliometric indicator based on the number of citing authors: ca-index(r). • Innovative application on data coming from MathSciNet database restricted to Pure Mathematics. • Testing of the indicator at aggregated data level. • Applying the indicator to evaluate single researchers’ performances in comparison with other usual bibliometric indexes. We propose a new bibliometric indicator, called citing authors index , to assess the impact and spread of the individual scientific research. With this new index we change the paradigm by focusing the attention to the number of citing authors instead of the mere number of citations. We test this index in the mathematical community by using the MathSciNet database of the American Mathematical Society. An aggregate version of the index is also proposed for the comparison of the performances of different institutions.

A Random Dynamical Systems Perspective on Isochronicity for Stochastic Oscillations

For an attracting periodic orbit (limit cycle) of a deterministic dynamical system, one defines the isochron for each point of the orbit as the cross-section with fixed return time under the flow. Equivalently, isochrons can be characterized as stable manifolds foliating neighborhoods of the limit cycle or as level sets of an isochron map. In recent years, there has been a lively discussion in the mathematical physics community on how to define isochrons for stochastic oscillations, i.e. limit cycles or heteroclinic cycles exposed to stochastic noise. The main discussion has concerned an approach finding stochastic isochrons as sections of equal expected return times versus the idea of considering eigenfunctions of the backward Kolmogorov operator. We discuss the problem in the framework of random dynamical systems and introduce a new rigorous definition of stochastic isochrons as random stable manifolds for random periodic solutions with noise-dependent period. This allows us to establish a random version of isochron maps whose level sets coincide with the random stable manifolds. Finally, we discuss links between the random dynamical systems interpretation and the equal expected return time approach via averaged quantities.

Explicit construction of frames and pairs of dual frames on locally compact abelian groups

Locally compact abelian (LCA) groups form a natural setting for frame theory in signal processing: while classical frame theory was developed in the setting of \( L^2({\mathbb {R}}),\) concrete applications have to be implemented on \({\mathbb {R}}^d,\) two scenarios that are covered under the setup of LCA groups. Due to rapid progress over the past few years, it is known within the mathematical community that large parts of the standard frame theory for \( L^2({\mathbb {R}})\) can be carried over to the setting of LCA groups. We show that this also holds on the level of explicit constructions, which is a crucial issue for applications within signal processing; in fact we provide explicit constructions of frames and pairs of dual frames on LCA groups. Special attention will be given to the so-called elementary LCA groups, i.e., groups that are tensor products of \({\mathbb {R}},\) \({\mathbb {Z}},\) the torus group \({{\mathbb {T}}},\) and the finite group \({\mathbb {Z}}_N.\) The frames will have the form of a Fourier-like system, i.e., the Fourier transform of a generalized shift-invariant system. The constructions are based on a partition of unity; in particular we provide a general and flexible approach for constructing a partition of unity that makes it easy to control key features of the underlying functions, e.g., support size and continuity. The results apply to arbitrary LCA groups with a supply of “sufficiently fine” lattices, a condition that is satisfied for all elementary LCA groups. Concrete constructions of pairs of dual frames for several elementary LCA groups are provided as illustration. In the case of a system of functions generated by a single lattice, such as a Gabor system, the general results become particularly transparent.

Some open problems in low dimensional dynamical systems

The aim of this paper is to share with the mathematical community a list of 33 problems that I have found along the years in my research. I believe that it is worth to think about them and, hopefully, solve some of the problems or make some substantial progress. Many of them are about planar differential equations but there are also questions about other mathematical aspects: Abel differential equations, difference equations, global asymptotic stability, geometrical questions, problems involving polynomials and some recreational problems with a dynamical component.

Francesco Altomare - the remarkable mathematician and human being

We laconically describe the great contributions of Professor Francesco Altomare to mathematical research and PhD education, and his unique status in the mathematical community. In particular, we present and give examples of his innovative and great achievements related to the following areas of mathematics: Functional Analysis, Operator Theory, Potential Theory, Approximation Theory, Probability Theory, Function Spaces, Choquet's Theory, Dirichlet's Problem and Semigroup Theory. 
 Moreover, we report on and give concrete examples of his unique way to work together with PhD students, both before and sometimes also after their dissertation. Finally, we shortly describe his remarkable “class travel” from “simple” conditions with no academic traditions in his family in the small hometown Giovinazzo to finally become the broad, ingenious, and powerful mathematician he is regarded to be today.

Monitoring trucks to reveal Belgian geographical structures and dynamics: From GPS traces to spatial interactions

Despite the fact that freight transport has a huge impact on the economy and the environment, Belgian datasets have always been scarce or restricted to very small a-spatial samples. Spatial data collected in Belgium for toll-paying trucks are here examined, and geographical structures and dynamics are extracted from this massive dataset. The originality of this dataset is its exhaustivity and its real-time approach: the location of all the trucks circulating in Belgium is collected every 30 s.The paper first relates to the methodology applied when using and transforming big data generated by On Board Units GNSS (cleaning, transforming and pre-processing). Second, it maps and comments on the movements (traffic) and stops of trucks within the whole country, providing a clear picture of the Belgian situation, useful for regional planners and logistics companies. Finally, the flows of trucks observed between Belgian locations enable the country to be divided into mathematical communities of places that interact the most. Analyses are performed for sub-categories based on the country of registration, underlining the spatial specificities of freight transit in Belgium. This exploratory spatial data analysis enables to reveal not only multi-level spatial structures associated with urban hierarchies and the transport infrastructure, but also firm locations or political organizations and to consider the complexity and interconnectivity of any measure taken for a more sustainable future. With a clear methodological framework to cope with the data pre-processing, this paper opens the way to various potential applications linked with freight transportation in Belgium.