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AppliedMath, Volume 2, Issue 4 (December 2022) – 15 articles

Cover Story (view full-size image): Extracting information about the shape or size of non-spherical aerosol particles from limited lidar data is an unstable inverse problem. In this work, we aim to find a robust regularization method including an appropriate parameter choice rule to address this. Filter functions motivated by the singular value decomposition set the conceptual basis to investigate a new iterative family of generalized Runge–Kutta regularizers. We propose a quasi-2D spheroidal model to describe the mechanism through which a non-spherical particle ensemble produces the backscatter, extinction and depolarization data collected by a lidar telescope after probing the atmosphere with a laser beam. Experiments with several synthetic atmospheric scenarios and different regularization methods favor a method of the generalized family combined with the L-curve method. View this paper
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20 pages, 7075 KiB  
Article
A Computational Fluid Dynamics Study of Flared Gas for Enhanced Oil Recovery Using a Micromodel
by Stephanie Were, Somtochukwu Godfrey Nnabuife and Boyu Kuang
AppliedMath 2022, 2(4), 738-757; https://doi.org/10.3390/appliedmath2040044 - 19 Dec 2022
Cited by 1 | Viewed by 2011
Abstract
The current handling of gas associated with oil production poses an environmental risk. This gas is being flared off due to the technical and economic attractiveness of this option. As flared gases are mainly composed of methane, they have harmful greenhouse effects when [...] Read more.
The current handling of gas associated with oil production poses an environmental risk. This gas is being flared off due to the technical and economic attractiveness of this option. As flared gases are mainly composed of methane, they have harmful greenhouse effects when released into the atmosphere. This work discusses the effectiveness of using this gas for enhanced oil recovery (EOR) purposes as an alternative to flaring. In this study, a micromodel was designed with properties similar to a sandstone rock with a porosity of 0.4, and computational fluid dynamics (CFD) techniques were applied to design an EOR system. Temperature effects were not considered in the study, and the simulation was run at atmospheric pressure. Five case studies were carried out with different interfacial tensions between the oil and gas (0.005 N/m, 0.017 N/m, and 0.034 N/m) and different injection rates for the gas (1 × 10−3 m/s, 1 × 10−4 m/s, and 1 × 10−6 m/s). The model was compared with a laboratory experiment measuring immiscible gas flooding. Factors affecting oil recoveries, such as the interfacial tension between oil and gas, the viscosity, and the pressure, were studied in detail. The results showed that the surface tension between the oil and gas interphase was a limiting factor for maximum oil recovery. The lower surface tension recovered 33% of the original oil in place. The capillary pressure was higher than the pressure in the micromodel, which lowered the amount of oil that was displaced. The study showed the importance of pressure maintenance to increase oil recovery for immiscible gas floods. It is recommended that a wider set of interfacial tensions between oil and gas be tested to obtain a range at which oil recovery is maximum for EOR with flared gas. Full article
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17 pages, 3537 KiB  
Article
A New Hybrid Multi-Objective Scheduling Model for Hierarchical Hub and Flexible Flow Shop Problems
by Sina Aghakhani and Mohammad Sadra Rajabi
AppliedMath 2022, 2(4), 721-737; https://doi.org/10.3390/appliedmath2040043 - 17 Dec 2022
Cited by 9 | Viewed by 1846
Abstract
In general, customers are looking to receive their orders in the fastest time possible and to make purchases at a reasonable price. Consequently, the importance of having an optimal delivery time is increasingly evident these days. One of the structures that can meet [...] Read more.
In general, customers are looking to receive their orders in the fastest time possible and to make purchases at a reasonable price. Consequently, the importance of having an optimal delivery time is increasingly evident these days. One of the structures that can meet the demand for large supply chains with numerous orders is the hierarchical integrated hub structure. Such a structure improves efficiency and reduces chain costs. To make logistics more cost-effective, hub-and-spoke networks are necessary as a means to achieve economies of scale. Many hub network design models only consider hub type but do not take into account the hub scale measured using freight volume. This paper proposes a multi-objective scheduling model for hierarchical hub structures (HHS), which is layered from top to bottom. In the third layer, the central hub takes factory products from decentralized hubs and sends them to other decentralized hubs to which customers are connected. In the second layer, non-central hubs are responsible for receiving products from the factory and transferring them to central hubs. These hubs are also responsible for receiving products from central hubs and sending them to customers. Lastly, the first layer contains factories responsible for producing products and providing for their customers. The factory uses the flexible flow-shop platform and structure to produce its products. The model’s objective is to minimize transportation and production costs as well as product arrival times. To validate and evaluate the model, small instances have been solved and analyzed in detail with the weighted sum and ε-constraint method. Consequently, based on the mean ideal distance (MID) metric, two methods were compared for the designed instances. Full article
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10 pages, 637 KiB  
Article
Computational Optimized Monitoring Methodology to Avoid Crowd Crush Accidents with Scattered Data
by Dongyung Kim
AppliedMath 2022, 2(4), 711-720; https://doi.org/10.3390/appliedmath2040042 - 14 Dec 2022
Cited by 2 | Viewed by 1425
Abstract
Recently, a crowd crush accident occurred in Seoul. Mathematics and data science can contribute to understanding this incident and to avoiding future accidents. In this paper, I suggest an optimized monitoring methodology to avoid crowd crush accidents with scattered data by searching the [...] Read more.
Recently, a crowd crush accident occurred in Seoul. Mathematics and data science can contribute to understanding this incident and to avoiding future accidents. In this paper, I suggest an optimized monitoring methodology to avoid crowd crush accidents with scattered data by searching the global minimum of the minimax data or minsum data. These scattered data are the position data of cell phones with time t. Mathematically, I find an exact solution of the optimized monitoring region with the suggested methodology by using the minimal constraints. The methodology is verified and validated along with the efficiency. Full article
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11 pages, 289 KiB  
Article
Posteriors in Limited Time
by Ayan Bhattacharya
AppliedMath 2022, 2(4), 700-710; https://doi.org/10.3390/appliedmath2040041 - 12 Dec 2022
Viewed by 1015
Abstract
This paper obtains a measure-theoretic restriction that must be satisfied by a prior probability measure for posteriors to be computed in limited time. Specifically, it is shown that the prior must be factorizable. Factorizability is a set of independence conditions for events in [...] Read more.
This paper obtains a measure-theoretic restriction that must be satisfied by a prior probability measure for posteriors to be computed in limited time. Specifically, it is shown that the prior must be factorizable. Factorizability is a set of independence conditions for events in the sample space that allows agents to calculate posteriors using only a subset of the dataset. The result has important implications for models in mathematical economics and finance that rely on a common prior. If one introduces the limited time restriction to Aumann’s famous Agreeing to Disagree setup, one sees that checking for factorizability requires agents to have access to every event in the measure space, thus severely limiting the scope of the agreement result. Full article
13 pages, 264 KiB  
Article
Generalizations of the Eneström–Kakeya Theorem Involving Weakened Hypotheses
by Robert Gardner and Matthew Gladin
AppliedMath 2022, 2(4), 687-699; https://doi.org/10.3390/appliedmath2040040 - 7 Dec 2022
Cited by 1 | Viewed by 1575
Abstract
The well-known Eneström–Kakeya Theorem states that, for P(z)==0naz, a polynomial of degree n with real coefficients satisfying 0a0a1an [...] Read more.
The well-known Eneström–Kakeya Theorem states that, for P(z)==0naz, a polynomial of degree n with real coefficients satisfying 0a0a1an, then all the zeros of P lie in |z|1 in the complex plane. Motivated by recent results concerning an Eneström–Kakeya “type” condition on real coefficients, we give similar results with hypotheses concerning the real and imaginary parts of the coefficients and concerning the moduli of the coefficients. In this way, our results generalize the other recent results. Full article
12 pages, 686 KiB  
Article
Signatures of Duschinsky Rotation in Femtosecond Coherence Spectra
by Paul C. Arpin, Mihail Popa and Daniel B. Turner
AppliedMath 2022, 2(4), 675-686; https://doi.org/10.3390/appliedmath2040039 - 1 Dec 2022
Cited by 2 | Viewed by 1781
Abstract
The motions of nuclei in a molecule can be mathematically described by using normal modes of vibration, which form a complete orthonormal basis. Each normal mode describes oscillatory motion at a frequency determined by the momentum of the nuclei. Near equilibrium, it is [...] Read more.
The motions of nuclei in a molecule can be mathematically described by using normal modes of vibration, which form a complete orthonormal basis. Each normal mode describes oscillatory motion at a frequency determined by the momentum of the nuclei. Near equilibrium, it is common to apply the quantum harmonic-oscillator model, whose eigenfunctions intimately involve combinatorics. Each electronic state has distinct force constants; therefore, each normal-mode basis is distinct. Duschinsky proposed a linearized approximation to the transformation between the normal-mode bases of two electronic states using a rotation matrix. The rotation angles are typically obtained by using quantum-chemical computations or via gas-phase spectroscopy measurements. Quantifying the rotation angles in the condensed phase remains a challenge. Here, we apply a two-dimensional harmonic model that includes a Duschinsky rotation to condensed-phase femtosecond coherence spectra (FCS), which are created in transient–absorption spectroscopy measurements through impulsive excitation of coherent vibrational wavepackets. Using the 2D model, we simulate spectra to identify the signatures of Duschinsky rotation. The results suggest that peak multiplicities and asymmetries may be used to quantify the rotation angle, which is a key advance in condensed-phase molecular spectroscopy. Full article
(This article belongs to the Special Issue Applications of Number Theory to the Sciences and Mathematics)
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16 pages, 7384 KiB  
Article
A Cytopathologist Eye Assistant for Cell Screening
by Débora N. Diniz, Breno N. S. Keller, Mariana T. Rezende, Andrea G. C. Bianchi, Claudia M. Carneiro, Renata R. e R. Oliveira, Eduardo J. S. Luz, Daniela M. Ushizima, Fátima N. S. de Medeiros and Marcone J. F. Souza
AppliedMath 2022, 2(4), 659-674; https://doi.org/10.3390/appliedmath2040038 - 30 Nov 2022
Viewed by 1942
Abstract
Screening of Pap smear images continues to depend upon cytopathologists’ manual scrutiny, and the results are highly influenced by professional experience, leading to varying degrees of cell classification inaccuracies. In order to improve the quality of the Pap smear results, several efforts have [...] Read more.
Screening of Pap smear images continues to depend upon cytopathologists’ manual scrutiny, and the results are highly influenced by professional experience, leading to varying degrees of cell classification inaccuracies. In order to improve the quality of the Pap smear results, several efforts have been made to create software to automate and standardize the processing of medical images. In this work, we developed the CEA (Cytopathologist Eye Assistant), an easy-to-use tool to aid cytopathologists in performing their daily activities. In addition, the tool was tested by a group of cytopathologists, whose feedback indicates that CEA could be a valuable tool to be integrated into Pap smear image analysis routines. For the construction of the tool, we evaluate different YOLO configurations and classification approaches. The best combination of algorithms uses YOLOv5s as a detection algorithm and an ensemble of EfficientNets as a classification algorithm. This configuration achieved 0.726 precision, 0.906 recall, and 0.805 F1-score when considering individual cells. We also made an analysis to classify the image as a whole, in which case, the best configuration was the YOLOv5s to perform the detection and classification tasks, and it achieved 0.975 precision, 0.992 recall, 0.970 accuracy, and 0.983 F1-score. Full article
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21 pages, 5384 KiB  
Article
Correlations of ESG Ratings: A Signed Weighted Network Analysis
by Evangelos Ioannidis, Dimitrios Tsoumaris, Dimitrios Ntemkas and Iordanis Sarikeisoglou
AppliedMath 2022, 2(4), 638-658; https://doi.org/10.3390/appliedmath2040037 - 21 Nov 2022
Viewed by 3150
Abstract
ESG ratings are data-driven indices, focused on three key pillars (Environmental, Social, and Governance), which are used by investors in order to evaluate companies and countries, in terms of Sustainability. A reasonable question which arises is how these ratings are associated to each [...] Read more.
ESG ratings are data-driven indices, focused on three key pillars (Environmental, Social, and Governance), which are used by investors in order to evaluate companies and countries, in terms of Sustainability. A reasonable question which arises is how these ratings are associated to each other. The research purpose of this work is to provide the first analysis of correlation networks, constructed from ESG ratings of selected economies. The networks are constructed based on Pearson correlation and analyzed in terms of some well-known tools from Network Science, namely: degree centrality of the nodes, degree centralization of the network, network density and network balance. We found that the Prevalence of Overweight and Life Expectancy are the most central ESG ratings, while unexpectedly, two of the most commonly used economic indicators, namely the GDP growth and Unemployment, are at the bottom of the list. China’s ESG network has remarkably high positive and high negative centralization, which has strong implications on network’s vulnerability and targeted controllability. Interestingly, if the sign of correlations is omitted, the above result cannot be captured. This is a clear example of why signed network analysis is needed. The most striking result of our analysis is that the ESG networks are extremely balanced, i.e. they are split into two anti-correlated groups of ESG ratings (nodes). It is impressive that USA’s network achieves 97.9% balance, i.e. almost perfect structural split into two anti-correlated groups of nodes. This split of network structure may have strong implications on hedging risk, if we see ESG ratings as underlying assets for portfolio selection. Investing into anti-correlated assets, called as "hedge assets", can be useful to offset potential losses. Our future direction is to apply and extend the proposed signed network analysis to ESG ratings of corporate organizations, aiming to design optimal portfolios with desired balance between risk and return. Full article
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17 pages, 673 KiB  
Article
On Critical Unicyclic Graphs with Cutwidth Four
by Zhenkun Zhang and Hongjian Lai
AppliedMath 2022, 2(4), 621-637; https://doi.org/10.3390/appliedmath2040036 - 17 Nov 2022
Cited by 1 | Viewed by 1024
Abstract
The cutwidth minimization problem consists of finding an arrangement of the vertices of a graph G on a line Pn with n=|V(G)| vertices in such a way that the maximum number of overlapping edges (i.e., [...] Read more.
The cutwidth minimization problem consists of finding an arrangement of the vertices of a graph G on a line Pn with n=|V(G)| vertices in such a way that the maximum number of overlapping edges (i.e., the congestion) is minimized. A graph G with a cutwidth of k is k-cutwidth critical if every proper subgraph of G has a cutwidth less than k and G is homeomorphically minimal. In this paper, we first verified some structural properties of k-cutwidth critical unicyclic graphs with k>1. We then mainly investigated the critical unicyclic graph set T with a cutwidth of four that contains fifty elements, and obtained a forbidden subgraph characterization of 3-cutwidth unicyclic graphs. Full article
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12 pages, 280 KiB  
Article
Some Properties of a Concircular Curvature Tensor on Generalized Sasakian-Space-Forms
by Vasant Chavan
AppliedMath 2022, 2(4), 609-620; https://doi.org/10.3390/appliedmath2040035 - 3 Nov 2022
Viewed by 1261
Abstract
The aim of the present paper is to study and investigate the geometrical properties of a concircular curvature tensor on generalized Sasakian-space-forms. In this manner, we obtained results for ϕ-concircularly flat, ϕ-semisymmetric, locally concircularly symmetric and locally concircularly ϕ-symmetric generalized [...] Read more.
The aim of the present paper is to study and investigate the geometrical properties of a concircular curvature tensor on generalized Sasakian-space-forms. In this manner, we obtained results for ϕ-concircularly flat, ϕ-semisymmetric, locally concircularly symmetric and locally concircularly ϕ-symmetric generalized Sasakian-space-forms. Finally, we construct examples of the generalized Sasakian-space-forms to verify some results. Full article
24 pages, 460 KiB  
Article
Approximate Nonlocal Symmetries for a Perturbed Schrödinger Equation with a Weak Infinite Power-Law Memory
by Stanislav Yu. Lukashchuk
AppliedMath 2022, 2(4), 585-608; https://doi.org/10.3390/appliedmath2040034 - 17 Oct 2022
Viewed by 1254
Abstract
A nonlocally perturbed linear Schrödinger equation with a small parameter was derived under the assumption of low-level fractionality by using one of the known general nonlocal wave equations with an infinite power-law memory. The problem of finding approximate symmetries for the equation is [...] Read more.
A nonlocally perturbed linear Schrödinger equation with a small parameter was derived under the assumption of low-level fractionality by using one of the known general nonlocal wave equations with an infinite power-law memory. The problem of finding approximate symmetries for the equation is studied here. It has been shown that the perturbed Schrödinger equation inherits all symmetries of the classical linear equation. It has also been proven that approximate symmetries corresponding to Galilean transformations and projective transformations of the unperturbed equation are nonlocal. In addition, a special class of nonlinear, nonlocally perturbed Schrödinger equations that admits an approximate nonlocal extension of the Galilei group is derived. An example of constructing an approximately invariant solution for the linear equation using approximate scaling symmetry is presented. Full article
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11 pages, 330 KiB  
Article
On the Universally Optimal Activation Function for a Class of Residual Neural Networks
by Feng Zhao and Shao-Lun Huang
AppliedMath 2022, 2(4), 574-584; https://doi.org/10.3390/appliedmath2040033 - 16 Oct 2022
Viewed by 1527
Abstract
While non-linear activation functions play vital roles in artificial neural networks, it is generally unclear how the non-linearity can improve the quality of function approximations. In this paper, we present a theoretical framework to rigorously analyze the performance gain of using non-linear activation [...] Read more.
While non-linear activation functions play vital roles in artificial neural networks, it is generally unclear how the non-linearity can improve the quality of function approximations. In this paper, we present a theoretical framework to rigorously analyze the performance gain of using non-linear activation functions for a class of residual neural networks (ResNets). In particular, we show that when the input features for the ResNet are uniformly chosen and orthogonal to each other, using non-linear activation functions to generate the ResNet output averagely outperforms using linear activation functions, and the performance gain can be explicitly computed. Moreover, we show that when the activation functions are chosen as polynomials with the degree much less than the dimension of the input features, the optimal activation functions can be precisely expressed in the form of Hermite polynomials. This demonstrates the role of Hermite polynomials in function approximations of ResNets. Full article
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27 pages, 1919 KiB  
Article
Modeling a Spheroidal Particle Ensemble and Inversion by Generalized Runge–Kutta Regularizers from Limited Data
by Stefanos Samaras, Christine Böckmann and Christoph Ritter
AppliedMath 2022, 2(4), 547-573; https://doi.org/10.3390/appliedmath2040032 - 10 Oct 2022
Cited by 2 | Viewed by 1361
Abstract
Extracting information about the shape or size of non-spherical aerosol particles from limited optical radar data is a well-known inverse ill-posed problem. The purpose of the study is to figure out a robust and stable regularization method including an appropriate parameter choice rule [...] Read more.
Extracting information about the shape or size of non-spherical aerosol particles from limited optical radar data is a well-known inverse ill-posed problem. The purpose of the study is to figure out a robust and stable regularization method including an appropriate parameter choice rule to address the latter problem. First, we briefly review common regularization methods and investigate a new iterative family of generalized Runge–Kutta filter regularizers. Next, we model a spheroidal particle ensemble and test with it different regularization methods experimenting with artificial data pertaining to several atmospheric scenarios. We found that one method of the newly introduced generalized family combined with the L-curve method performs better compared to traditional methods. Full article
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29 pages, 401 KiB  
Article
Using Hazard and Surrogate Functions for Understanding Memory and Forgetting
by Richard A. Chechile
AppliedMath 2022, 2(4), 518-546; https://doi.org/10.3390/appliedmath2040031 - 4 Oct 2022
Cited by 1 | Viewed by 1318
Abstract
The retention of human memory is a process that can be understood from a hazard-function perspective. Hazard is the conditional probability of a state change at time t given that the state change did not yet occur. After reviewing the underlying mathematical results [...] Read more.
The retention of human memory is a process that can be understood from a hazard-function perspective. Hazard is the conditional probability of a state change at time t given that the state change did not yet occur. After reviewing the underlying mathematical results of hazard functions in general, there is an analysis of the hazard properties associated with nine theories of memory that emerged from psychological science. Five theories predict strictly monotonically decreasing hazard whereas the other four theories predict a peaked-shaped hazard function that rises initially to a peak and then decreases for longer time periods. Thus, the behavior of hazard shortly after the initial encoding is the critical difference among the theories. Several theorems provide a basis to explore hazard for the initial time period after encoding in terms of a more practical surrogate function that is linked to the behavior of the hazard function. Evidence for a peak-shaped hazard function is provided and a case is made for one particular psychological theory of memory that posits that memory encoding produces two redundant representations that have different hazard properties. One memory representation has increasing hazard while the other representation has decreasing hazard. Full article
6 pages, 350 KiB  
Technical Note
On One Problem of the Nonlinear Convex Optimization
by Robert Vrabel
AppliedMath 2022, 2(4), 512-517; https://doi.org/10.3390/appliedmath2040030 - 21 Sep 2022
Viewed by 1552
Abstract
In this short paper, we study the problem of traversing a crossbar through a bent channel, which has been formulated as a nonlinear convex optimization problem. The result is a MATLAB code that we can use to compute the maximum length of the [...] Read more.
In this short paper, we study the problem of traversing a crossbar through a bent channel, which has been formulated as a nonlinear convex optimization problem. The result is a MATLAB code that we can use to compute the maximum length of the crossbar as a function of the width of the channel (its two parts) and the angle between them. In case they are perpendicular to each other, the result is expressed analytically and is closely related to the astroid curve (a hypocycloid with four cusps). Full article
(This article belongs to the Special Issue Applications of Number Theory to the Sciences and Mathematics)
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