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Symmetry
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Feature Papers in Mathematics Section
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Feature Papers in Mathematics Section

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: 31 December 2024 | Viewed by 6055

Special Issue Editor

Dr. Calogero Vetro

E-Mail Website
Guest Editor
Associate Professor, Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy
Interests: difference equations; flow invariance; nonlinear regularity theory; ordinary differential equations; partial differential equations; reduction methods; symmetry operators; weak symmetries
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

This Special Issue is devoted to collecting research works and reviews in all the fields of Mathematics. We aim to receive papers highlighting the latest advances in pure mathematics and in applied mathematics, as well as papers providing applications of mathematics in real-life processes. Hence, we encourage both scientists in leadership positions and young researchers at the beginning of their careers to contribute. We hope this Special Issue will provide a suitable platform to share new interdisciplinary ideas, to support emerging topics and to disseminate consolidated theories, hence increasing the level of knowledge and understanding of mathematical research in the scientific community. Particular attention will be given to refinements of the roles of symmetries and asymmetries.

Dr. Calogero Vetro
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

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Published Papers (5 papers)

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Research

Jump to: Review

22 pages, 495 KiB  
Article
The BiCG Algorithm for Solving the Minimal Frobenius Norm Solution of Generalized Sylvester Tensor Equation over the Quaternions
by Mengyan Xie, Qing-Wen Wang and Yang Zhang
Symmetry 2024, 16(9), 1167; https://doi.org/10.3390/sym16091167 - 6 Sep 2024
Viewed by 688
Abstract
In this paper, we develop an effective iterative algorithm to solve a generalized Sylvester tensor equation over quaternions which includes several well-studied matrix/tensor equations as special cases. We discuss the convergence of this algorithm within a finite number of iterations, assuming negligible round-off [...] Read more.
In this paper, we develop an effective iterative algorithm to solve a generalized Sylvester tensor equation over quaternions which includes several well-studied matrix/tensor equations as special cases. We discuss the convergence of this algorithm within a finite number of iterations, assuming negligible round-off errors for any initial tensor. Moreover, we demonstrate the unique minimal Frobenius norm solution achievable by selecting specific types of initial tensors. Additionally, numerical examples are presented to illustrate the practicality and validity of our proposed algorithm. These examples include demonstrating the algorithm’s effectiveness in addressing three-dimensional microscopic heat transport and color video restoration problems. Full article
(This article belongs to the Special Issue Feature Papers in Mathematics Section)
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13 pages, 2079 KiB  
Article
A Genetic Algorithm for Three-Dimensional Discrete Tomography
by Elena Toscano and Cesare Valenti
Symmetry 2024, 16(7), 923; https://doi.org/10.3390/sym16070923 - 19 Jul 2024
Viewed by 511
Abstract
Discrete tomography is a specific case of computerized tomography that deals with the reconstruction of objects made of a few density values on a discrete lattice of points (integer valued coordinates). In the general case of computerized tomography, several hundreds of projections are [...] Read more.
Discrete tomography is a specific case of computerized tomography that deals with the reconstruction of objects made of a few density values on a discrete lattice of points (integer valued coordinates). In the general case of computerized tomography, several hundreds of projections are required to obtain a single high-resolution slice of the object; in the case of discrete tomography, projections of an object made by just one homogeneous material are sums along very few angles of the pixel values, which can be thought to be 0’s or 1’s without loss of generality. Genetic algorithms are global optimization techniques with an underlying random approach and, therefore, their convergence to a solution is provided in a probabilistic sense. We present here a genetic algorithm able to straightforwardly reconstruct binary objects in the three-dimensional space. To the best of our knowledge, our methodology is the first to require no model of the shape (e.g., periodicity, convexity or symmetry) to reconstruct. Experiments were carried out to test our new approach in terms of computational time and correctness of the solutions. Over the years, discrete tomography has been studied for many interesting applications to computer vision, non-destructive reverse engineering and industrial quality control, electron microscopy, X-rays crystallography, biplane angiography, data coding and compression. Full article
(This article belongs to the Special Issue Feature Papers in Mathematics Section)
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21 pages, 2209 KiB  
Article
New Parametric 2D Curves for Modeling Prostate Shape in Magnetic Resonance Images
by Rosario Corso, Albert Comelli, Giuseppe Salvaggio and Domenico Tegolo
Symmetry 2024, 16(6), 755; https://doi.org/10.3390/sym16060755 - 17 Jun 2024
Viewed by 1128
Abstract
Geometric shape models often help to extract specific contours in digital images (the segmentation process) with major precision. Motivated by this idea, we introduce two models for the representation of prostate shape in the axial plane of magnetic resonance images. In more detail, [...] Read more.
Geometric shape models often help to extract specific contours in digital images (the segmentation process) with major precision. Motivated by this idea, we introduce two models for the representation of prostate shape in the axial plane of magnetic resonance images. In more detail, the models are two parametric closed curves of the plane. The analytic study of the models includes the geometric role of the parameters describing the curves, symmetries, invariants, special cases, elliptic Fourier descriptors, conditions for simple curves and area of the enclosed surfaces. The models were validated for prostate shapes by fitting the curves to prostate contours delineated by a radiologist and measuring the errors with the mean distance, the Hausdorff distance and the Dice similarity coefficient. Validation was also conducted by comparing our models with the deformed superellipse model used in literature. Our models are equivalent in fitting metrics to the deformed superellipse model; however, they have the advantage of a more straightforward formulation and they depend on fewer parameters, implying a reduced computational time for the fitting process. Due to the validation, our models may be applied for developing innovative and performing segmentation methods or improving existing ones. Full article
(This article belongs to the Special Issue Feature Papers in Mathematics Section)
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14 pages, 294 KiB  
Article
Integral Transforms and the Hyers–Ulam Stability of Linear Differential Equations with Constant Coefficients
by Douglas R. Anderson
Symmetry 2024, 16(2), 135; https://doi.org/10.3390/sym16020135 - 23 Jan 2024
Viewed by 841
Abstract
Integral transform methods are a common tool employed to study the Hyers–Ulam stability of differential equations, including Laplace, Kamal, Tarig, Aboodh, Mahgoub, Sawi, Fourier, Shehu, and Elzaki integral transforms. This work provides improved techniques for integral transforms in relation to establishing the Hyers–Ulam [...] Read more.
Integral transform methods are a common tool employed to study the Hyers–Ulam stability of differential equations, including Laplace, Kamal, Tarig, Aboodh, Mahgoub, Sawi, Fourier, Shehu, and Elzaki integral transforms. This work provides improved techniques for integral transforms in relation to establishing the Hyers–Ulam stability of differential equations with constant coefficients, utilizing the Kamal transform, where we focus on first- and second-order linear equations. In particular, in this work, we employ the Kamal transform to determine the Hyers–Ulam stability and Hyers–Ulam stability constants for first-order complex constant coefficient differential equations and, for second-order real constant coefficient differential equations, improving previous results obtained by using the Kamal transform. In a section of examples, we compare and contrast our results favorably with those established in the literature using means other than the Kamal transform. Full article
(This article belongs to the Special Issue Feature Papers in Mathematics Section)

Review

Jump to: Research

25 pages, 2118 KiB  
Review
An Extensive Investigation into the Use of Machine Learning Tools and Deep Neural Networks for the Recognition of Skin Cancer: Challenges, Future Directions, and a Comprehensive Review
by Syed Ibrar Hussain and Elena Toscano
Symmetry 2024, 16(3), 366; https://doi.org/10.3390/sym16030366 - 18 Mar 2024
Cited by 4 | Viewed by 2289
Abstract
Skin cancer poses a serious risk to one’s health and can only be effectively treated with early detection. Early identification is critical since skin cancer has a higher fatality rate, and it expands gradually to different areas of the body. The rapid growth [...] Read more.
Skin cancer poses a serious risk to one’s health and can only be effectively treated with early detection. Early identification is critical since skin cancer has a higher fatality rate, and it expands gradually to different areas of the body. The rapid growth of automated diagnosis frameworks has led to the combination of diverse machine learning, deep learning, and computer vision algorithms for detecting clinical samples and atypical skin lesion specimens. Automated methods for recognizing skin cancer that use deep learning techniques are discussed in this article: convolutional neural networks, and, in general, artificial neural networks. The recognition of symmetries is a key point in dealing with the skin cancer image datasets; hence, in developing the appropriate architecture of neural networks, as it can improve the performance and release capacities of the network. The current study emphasizes the need for an automated method to identify skin lesions to reduce the amount of time and effort required for the diagnostic process, as well as the novel aspect of using algorithms based on deep learning for skin lesion detection. The analysis concludes with underlying research directions for the future, which will assist in better addressing the difficulties encountered in human skin cancer recognition. By highlighting the drawbacks and advantages of prior techniques, the authors hope to establish a standard for future analysis in the domain of human skin lesion diagnostics. Full article
(This article belongs to the Special Issue Feature Papers in Mathematics Section)
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