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# Mathematics, Volume 8, Issue 4 (April 2020) – 192 articles

Cover Story (view full-size image): In January 2018, a tropical depression hit Nampula province, Mozambique, affecting more than 80,000 people and killing 34. Additionally, damage to the road network hampered the response operations. A mathematical model is proposed here to support pre-disaster operations and help mitigate the impact of similar disasters in developing countries. View this paper.
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16 pages, 298 KiB
Article
Robustness of Interval Monge Matrices in Fuzzy Algebra
Mathematics 2020, 8(4), 652; https://doi.org/10.3390/math8040652 - 24 Apr 2020
Viewed by 1770
Abstract
Max–min algebra (called also fuzzy algebra) is an extremal algebra with operations maximum and minimum. In this paper, we study the robustness of Monge matrices with inexact data over max–min algebra. A matrix with inexact data (also called interval matrix) is a set [...] Read more.
Max–min algebra (called also fuzzy algebra) is an extremal algebra with operations maximum and minimum. In this paper, we study the robustness of Monge matrices with inexact data over max–min algebra. A matrix with inexact data (also called interval matrix) is a set of matrices given by a lower bound matrix and an upper bound matrix. An interval Monge matrix is the set of all Monge matrices from an interval matrix with Monge lower and upper bound matrices. There are two possibilities to define the robustness of an interval matrix. First, the possible robustness, if there is at least one robust matrix. Second, universal robustness, if all matrices are robust in the considered set of matrices. We found necessary and sufficient conditions for universal robustness in cases when the lower bound matrix is trivial. Moreover, we proved necessary conditions for possible robustness and equivalent conditions for universal robustness in cases where the lower bound matrix is non-trivial. Full article
(This article belongs to the Section Fuzzy Sets, Systems and Decision Making)
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13 pages, 291 KiB
Article
On the Exponents of Exponential Dichotomies
Mathematics 2020, 8(4), 651; https://doi.org/10.3390/math8040651 - 23 Apr 2020
Cited by 1 | Viewed by 2194
Abstract
An exponential dichotomy is studied for linear differential equations. A constructive method is presented to derive a roughness result for perturbations giving exponents of the dichotomy as well as an estimate of the norm of the difference between the corresponding two dichotomy projections. [...] Read more.
An exponential dichotomy is studied for linear differential equations. A constructive method is presented to derive a roughness result for perturbations giving exponents of the dichotomy as well as an estimate of the norm of the difference between the corresponding two dichotomy projections. This roughness result is crucial in developing a Melnikov bifurcation method for either discontinuous or implicit perturbed nonlinear differential equations. Full article
22 pages, 1391 KiB
Article
Using FQFD and FGRA to Enhance the Advertising Effectiveness of Cross-Regional E-Commerce Platforms
Mathematics 2020, 8(4), 650; https://doi.org/10.3390/math8040650 - 23 Apr 2020
Cited by 8 | Viewed by 2788
Abstract
The thriving development of cross-regional e-commerce has gradually increased online marketing activities and consumers’ intention to shop online. The objective of this paper to find solutions for improving the advertising effectiveness has therefore become very important. In this article, an integrating method of [...] Read more.
The thriving development of cross-regional e-commerce has gradually increased online marketing activities and consumers’ intention to shop online. The objective of this paper to find solutions for improving the advertising effectiveness has therefore become very important. In this article, an integrating method of fuzzy quality function deployment (FQFD) and fuzzy grey relational analysis (FGRA) was proposed to identify solutions for improving the advertising effectiveness. Based on this method, the house of quality (HoQ) to facilitate investigation of the 17 advertising effectiveness needs and 10 feasible technical improvements were presented. Through the questionnaire survey of platform users, the importance and satisfaction of the attributes of advertising needs were obtained. After that, a fuzzy relationship matrix was constructed to link technical improvements and advertising effectiveness needs in a fuzzy decision environment. Finally, the priority of technical improvements regarding improving advertising effectiveness were obtained. The results show that the proposed method can help decision makers of cross-regional e-commerce to improve advertising effectiveness effectively, so that they can effectively use resources to design advertisements that meet user needs. Full article
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10 pages, 237 KiB
Article
On Wong Type Contractions
Mathematics 2020, 8(4), 649; https://doi.org/10.3390/math8040649 - 23 Apr 2020
Cited by 2 | Viewed by 1825
Abstract
In this paper, by using admissible mapping, Wong type contraction mappings are extended and investigated in the framework of quasi-metric spaces to guarantee the existence of fixed points. We consider examples to illustrate the main results. We also demonstrate that the main results [...] Read more.
In this paper, by using admissible mapping, Wong type contraction mappings are extended and investigated in the framework of quasi-metric spaces to guarantee the existence of fixed points. We consider examples to illustrate the main results. We also demonstrate that the main results of the paper cover several existing results in the literature. Full article
(This article belongs to the Special Issue Fixed Point Theory and Dynamical Systems with Applications)
20 pages, 21942 KiB
Article
Supported Evacuation for Disaster Relief through Lexicographic Goal Programming
Mathematics 2020, 8(4), 648; https://doi.org/10.3390/math8040648 - 22 Apr 2020
Cited by 9 | Viewed by 3086
Abstract
Disasters have been striking human-beings from the beginning of history and their management is a global concern of the international community. Minimizing the impact and consequences of these disasters, both natural and human-made, involves many decision and logistic processes that should be optimized. [...] Read more.
Disasters have been striking human-beings from the beginning of history and their management is a global concern of the international community. Minimizing the impact and consequences of these disasters, both natural and human-made, involves many decision and logistic processes that should be optimized. A crucial logistic problem is the evacuation of the affected population, and the focus of this paper is the planning of supported evacuation of vulnerable people to safe places when necessary. A lexicographic goal programming model for supported evacuation is proposed, whose main novelties are the classification of potential evacuees according to their health condition, so that they can be treated accordingly; the introduction of dynamism regarding the arrival of potential evacuees to the pickup points, according to their own susceptibility about the disaster and the joint consideration of objectives such us number of evacuated people, operation time and cost, among which no trade-off is possible. The performance of the proposed model is evaluated through a realistic case study regarding the earthquake and tsunami that hit Palu (Indonesia) in September 2018. Full article
(This article belongs to the Special Issue Optimization for Decision Making II)
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12 pages, 780 KiB
Article
A Fixed-Point Approach to the Hyers–Ulam Stability of Caputo–Fabrizio Fractional Differential Equations
Mathematics 2020, 8(4), 647; https://doi.org/10.3390/math8040647 - 22 Apr 2020
Cited by 12 | Viewed by 2478
Abstract
In this paper, we study Hyers–Ulam and Hyers–Ulam–Rassias stability of nonlinear Caputo–Fabrizio fractional differential equations on a noncompact interval. We extend the corresponding uniqueness and stability results on a compact interval. Two examples are given to illustrate our main results. Full article
(This article belongs to the Special Issue Topological Methods in Nonlinear Analysis)
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21 pages, 6220 KiB
Article
An AHPSort II Based Analysis of the Inequality Reduction within European Union
Mathematics 2020, 8(4), 646; https://doi.org/10.3390/math8040646 - 22 Apr 2020
Cited by 16 | Viewed by 2240
Abstract
Nowadays, sustainability is an omnipresent concept in our society, which encompasses several challenges related to poverty, inequality, climate change and so on. The United Nations adopted the Agenda 2030, a plan of action formed of universal Sustainable Developments Goals (SDGs) and targets, which [...] Read more.
Nowadays, sustainability is an omnipresent concept in our society, which encompasses several challenges related to poverty, inequality, climate change and so on. The United Nations adopted the Agenda 2030, a plan of action formed of universal Sustainable Developments Goals (SDGs) and targets, which countries have to face in order to shift the world toward a sustainable future. One of the most relevant SDGs since the onset of the financial crisis in 2007 has been the so-called reduced inequalities, which consists of dealing with the inequality of opportunities and wealth between and within countries. However, reducing inequalities depends on many heterogeneous aspects, making it difficult to make a proper analysis that evaluates the European Union (EU) countries performance of this goal. In this study, we introduce a novel approach to evaluate the inequalities in EU countries based on a sorting a multi-criteria decision-making method called AHPSort II. This approach allows to obtain a classification of the EU countries according to their achievements in reducing inequalities to subsequently carry out a deep performance analysis with the aim of drawing conclusions as to the evolution of inequality in them along the years. The results are consistent with the main international organizations’ reports and academic literature, as shown in the Discussion Section. Full article
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13 pages, 1704 KiB
Article
The Existence of a Convex Polyhedron with Respect to the Constrained Vertex Norms
Mathematics 2020, 8(4), 645; https://doi.org/10.3390/math8040645 - 22 Apr 2020
Cited by 2 | Viewed by 2936
Abstract
Given a set of constrained vertex norms, we proved the existence of a convex configuration with respect to the set of distinct constrained vertex norms in the two-dimensional case when the constrained vertex norms are distinct or repeated for, at most, four points. [...] Read more.
Given a set of constrained vertex norms, we proved the existence of a convex configuration with respect to the set of distinct constrained vertex norms in the two-dimensional case when the constrained vertex norms are distinct or repeated for, at most, four points. However, we proved that there always exists a convex configuration in the three-dimensional case. In the application, we can imply the existence of the non-empty spherical Laguerre Voronoi diagram. Full article
(This article belongs to the Special Issue Modern Geometric Modeling: Theory and Applications)
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16 pages, 5164 KiB
Article
Near-Duplicate Image Detection System Using Coarse-to-Fine Matching Scheme Based on Global and Local CNN Features
Mathematics 2020, 8(4), 644; https://doi.org/10.3390/math8040644 - 22 Apr 2020
Cited by 12 | Viewed by 3293
Abstract
Due to the great success of convolutional neural networks (CNNs) in the area of computer vision, the existing methods tend to match the global or local CNN features between images for near-duplicate image detection. However, global CNN features are not robust enough to [...] Read more.
Due to the great success of convolutional neural networks (CNNs) in the area of computer vision, the existing methods tend to match the global or local CNN features between images for near-duplicate image detection. However, global CNN features are not robust enough to combat background clutter and partial occlusion, while local CNN features lead to high computational complexity in the step of feature matching. To achieve high efficiency while maintaining good accuracy, we propose a coarse-to-fine feature matching scheme using both global and local CNN features for real-time near-duplicate image detection. In the coarse matching stage, we implement the sum-pooling operation on convolutional feature maps (CFMs) to generate the global CNN features, and match these global CNN features between a given query image and database images to efficiently filter most of irrelevant images of the query. In the fine matching stage, the local CNN features are extracted by using maximum values of the CFMs and the saliency map generated by the graph-based visual saliency detection (GBVS) algorithm. These local CNN features are then matched between images to detect the near-duplicate versions of the query. Experimental results demonstrate that our proposed method not only achieves a real-time detection, but also provides higher accuracy than the state-of-the-art methods. Full article
(This article belongs to the Special Issue Computing Methods in Steganography and Multimedia Security)
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16 pages, 245 KiB
Article
A Parametric Kind of Fubini Polynomials of a Complex Variable
Mathematics 2020, 8(4), 643; https://doi.org/10.3390/math8040643 - 22 Apr 2020
Cited by 5 | Viewed by 1648
Abstract
In this paper, we propose a parametric kind of Fubini polynomials by defining the two specific generating functions. We also investigate some analytical properties (for example, summation formulae, differential formulae and relationships with other well-known polynomials and numbers) for our introduced polynomials in [...] Read more.
In this paper, we propose a parametric kind of Fubini polynomials by defining the two specific generating functions. We also investigate some analytical properties (for example, summation formulae, differential formulae and relationships with other well-known polynomials and numbers) for our introduced polynomials in a systematic way. Furthermore, we consider some relationships for parametric kind of Fubini polynomials associated with Bernoulli, Euler, and Genocchi polynomials and Stirling numbers of the second kind. Full article
12 pages, 285 KiB
Article
A Characterization of Ruled Real Hypersurfaces in Non-Flat Complex Space Forms
Mathematics 2020, 8(4), 642; https://doi.org/10.3390/math8040642 - 21 Apr 2020
Viewed by 1655
Abstract
The Levi-Civita connection and the k-th generalized Tanaka-Webster connection are defined on a real hypersurface M in a non-flat complex space form. For any nonnull constant k and any vector field X tangent to M the k-th Cho operator [...] Read more.
The Levi-Civita connection and the k-th generalized Tanaka-Webster connection are defined on a real hypersurface M in a non-flat complex space form. For any nonnull constant k and any vector field X tangent to M the k-th Cho operator $F X ( k )$ is defined and is related to both connections. If X belongs to the maximal holomorphic distribution $D$ on M, the corresponding operator does not depend on k and is denoted by $F X$ and called Cho operator. In this paper, real hypersurfaces in non-flat space forms such that $F X S = S F X$ , where S denotes the Ricci tensor of M and a further condition is satisfied, are classified. Full article
(This article belongs to the Special Issue Differential Geometry: Theory and Applications)
9 pages, 413 KiB
Review
On the Advent of Fractional Calculus in Econophysics via Continuous-Time Random Walk
Mathematics 2020, 8(4), 641; https://doi.org/10.3390/math8040641 - 21 Apr 2020
Cited by 15 | Viewed by 2094
Abstract
In this survey article, at first, the author describes how he was involved in the late 1990s on Econophysics, considered in those times an emerging science. Inside a group of colleagues the methods of the Fractional Calculus were developed to deal with the [...] Read more.
In this survey article, at first, the author describes how he was involved in the late 1990s on Econophysics, considered in those times an emerging science. Inside a group of colleagues the methods of the Fractional Calculus were developed to deal with the continuous-time random walks adopted to model the tick-by-tick dynamics of financial markets Then, the analytical results of this approach are presented pointing out the relevance of the Mittag-Leffler function. The consistence of the theoretical analysis is validated with fitting the survival probability for certain futures (BUND and BTP) traded in 1997 at LIFFE, London. Most of the theoretical and numerical results (including figures) reported in this paper were presented by the author at the first Nikkei symposium on Econophysics, held in Tokyo on November 2000 under the title “Empirical Science of Financial Fluctuations” on behalf of his colleagues and published by Springer. The author acknowledges Springer for the license permission of re-using this material. Full article
(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)
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19 pages, 695 KiB
Article
Characterization of the Functionally Graded Shear Modulus of a Half-Space
Mathematics 2020, 8(4), 640; https://doi.org/10.3390/math8040640 - 21 Apr 2020
Cited by 11 | Viewed by 2429
Abstract
In this article, a method is proposed for determining parameters of the exponentialy varying shear modulus of a functionally graded half-space. The method is based on the analytical solution of the problem of pure shear of an elastic functionally graded half-space by a [...] Read more.
In this article, a method is proposed for determining parameters of the exponentialy varying shear modulus of a functionally graded half-space. The method is based on the analytical solution of the problem of pure shear of an elastic functionally graded half-space by a strip punch. The half-space has the depth-wise exponential variation of its shear modulus, whose parameters are to be determined. The problem is reduced to an integral equation that is then solved by asymptotic methods. The analytical relations for contact stress under the punch, displacement of the free surface outside the contact area and other characteristics of the problem are studied with respect to the shear modulus parameters. The parameters of the functionally graded half-space shear modulus are determined (a) from the coincidence of theoretical and experimental values of contact stresses under the punch and from the coincidence of forces acting on the punch, or (b) from the coincidence of theoretical and experimental values of displacement of the free surface of the half-space outside the contact and coincidence of forces acting on the punch, or (c) from other conditions. The transcendental equations for determination of the shear modulus parameters in cases (a) and (b) are given. By adjusting the parameters of the shear modulus variation, the regions of “approximate-homogeneous” state in the functionally graded half-space are developed. Full article
(This article belongs to the Special Issue Applied Mathematical Methods in Mechanical Engineering)
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7 pages, 249 KiB
Article
Fibonacci Numbers with a Prescribed Block of Digits
Mathematics 2020, 8(4), 639; https://doi.org/10.3390/math8040639 - 21 Apr 2020
Cited by 17 | Viewed by 2904
Abstract
In this paper, we prove that $F 22 = 17711$ is the largest Fibonacci number whose decimal expansion is of the form $a b … b c … c$ . The proof uses lower bounds for linear forms in three logarithms of algebraic [...] Read more.
In this paper, we prove that $F 22 = 17711$ is the largest Fibonacci number whose decimal expansion is of the form $a b … b c … c$ . The proof uses lower bounds for linear forms in three logarithms of algebraic numbers and some tools from Diophantine approximation. Full article
13 pages, 322 KiB
Article
Inertial Krasnoselskii–Mann Method in Banach Spaces
Mathematics 2020, 8(4), 638; https://doi.org/10.3390/math8040638 - 21 Apr 2020
Cited by 6 | Viewed by 2294
Abstract
In this paper, we give a general inertial Krasnoselskii–Mann algorithm for solving inclusion problems in Banach Spaces. First, we establish a weak convergence in real uniformly convex and q-uniformly smooth Banach spaces for finding fixed points of nonexpansive mappings. Then, a strong [...] Read more.
In this paper, we give a general inertial Krasnoselskii–Mann algorithm for solving inclusion problems in Banach Spaces. First, we establish a weak convergence in real uniformly convex and q-uniformly smooth Banach spaces for finding fixed points of nonexpansive mappings. Then, a strong convergence is obtained for the inertial generalized forward-backward splitting method for the inclusion. Our results extend many recent and related results obtained in real Hilbert spaces. Full article
(This article belongs to the Special Issue Applications of Inequalities and Functional Analysis)
13 pages, 309 KiB
Article
Theoretical Bounds on Performance in Threshold Group Testing Schemes
Mathematics 2020, 8(4), 637; https://doi.org/10.3390/math8040637 - 21 Apr 2020
Cited by 4 | Viewed by 2196
Abstract
A threshold group testing (TGT) scheme with lower and upper thresholds is a general model of group testing (GT) which identifies a small set of defective samples. In this paper, we consider the TGT scheme that require the minimum number of tests. We [...] Read more.
A threshold group testing (TGT) scheme with lower and upper thresholds is a general model of group testing (GT) which identifies a small set of defective samples. In this paper, we consider the TGT scheme that require the minimum number of tests. We aim to find lower and upper bounds for finding a set of defective samples in a large population. The decoding for the TGT scheme is exploited by minimization of the Hamming weight in channel coding theory and the probability of error is also defined. Then, we derive a new upper bound on the probability of error and extend a lower bound from conventional one to the TGT scheme. We show that the upper and lower bounds well match with each other at the optimal density ratio of the group matrix. In addition, we conclude that when the gaps between the two thresholds in the TGT framework increase, the group matrix with a high density should be used to achieve optimal performance. Full article
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11 pages, 787 KiB
Article
Statistical Deferred Nörlund Summability and Korovkin-Type Approximation Theorem
Mathematics 2020, 8(4), 636; https://doi.org/10.3390/math8040636 - 21 Apr 2020
Cited by 27 | Viewed by 2456
Abstract
The concept of the deferred Nörlund equi-statistical convergence was introduced and studied by Srivastava et al. [Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. (RACSAM) 112 (2018), 1487–1501]. In the present paper, we have studied the notion of the deferred Nörlund [...] Read more.
The concept of the deferred Nörlund equi-statistical convergence was introduced and studied by Srivastava et al. [Rev. Real Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. (RACSAM) 112 (2018), 1487–1501]. In the present paper, we have studied the notion of the deferred Nörlund statistical convergence and the statistical deferred Nörlund summability for sequences of real numbers defined over a Banach space. We have also established a theorem presenting a connection between these two interesting notions. Moreover, based upon our proposed methods, we have proved a new Korovkin-type approximation theorem with algebraic test functions for a sequence of real numbers on a Banach space and demonstrated that our theorem effectively extends and improves most of the earlier existing results (in classical and statistical versions). Finally, we have presented an example involving the generalized Meyer–König and Zeller operators of a real sequence demonstrating that our theorem is a stronger approach than its classical and statistical versions. Full article
(This article belongs to the Special Issue New Frontiers in Applied Mathematics and Statistics)
9 pages, 237 KiB
Article
A Rational Approximation for the Complete Elliptic Integral of the First Kind
Mathematics 2020, 8(4), 635; https://doi.org/10.3390/math8040635 - 21 Apr 2020
Cited by 12 | Viewed by 2582
Abstract
Let $K ( r )$ be the complete elliptic integral of the first kind. We present an accurate rational lower approximation for $K ( r )$ . More precisely, we establish the inequality [...] Read more.
Let $K ( r )$ be the complete elliptic integral of the first kind. We present an accurate rational lower approximation for $K ( r )$ . More precisely, we establish the inequality $2 π K ( r ) > 5 ( r ′ ) 2 + 126 r ′ + 61 61 ( r ′ ) 2 + 110 r ′ + 21$ for $r ∈ ( 0 , 1 )$ , where $r ′ = 1 − r 2$ . The lower bound is sharp. Full article
17 pages, 256 KiB
Article
On Convergence Rates of Some Limits
Mathematics 2020, 8(4), 634; https://doi.org/10.3390/math8040634 - 21 Apr 2020
Cited by 1 | Viewed by 1605
Abstract
In 2019 Seneta has provided a characterization of slowly varying functions L in the Zygmund sense by using the condition, for each $y > 0$ , [...] Read more.
In 2019 Seneta has provided a characterization of slowly varying functions L in the Zygmund sense by using the condition, for each $y > 0$ , . Very recently, we have extended this result by considering a wider class of functions U related to the following more general condition. For each $y > 0$ , , for some functions r and g. In this paper, we examine this last result by considering a much more general convergence condition. A wider class related to this new condition is presented. Further, a representation theorem for this wider class is provided. Full article
(This article belongs to the Special Issue Stability Problems for Stochastic Models: Theory and Applications)
18 pages, 3507 KiB
Article
Deep Assessment Methodology Using Fractional Calculus on Mathematical Modeling and Prediction of Gross Domestic Product per Capita of Countries
Mathematics 2020, 8(4), 633; https://doi.org/10.3390/math8040633 - 20 Apr 2020
Cited by 8 | Viewed by 3134
Abstract
In this study, a new approach for time series modeling and prediction, “deep assessment methodology,” is proposed and the performance is reported on modeling and prediction for upcoming years of Gross Domestic Product (GDP) per capita. The proposed methodology expresses a function with [...] Read more.
In this study, a new approach for time series modeling and prediction, “deep assessment methodology,” is proposed and the performance is reported on modeling and prediction for upcoming years of Gross Domestic Product (GDP) per capita. The proposed methodology expresses a function with the finite summation of its previous values and derivatives combining fractional calculus and the Least Square Method to find unknown coefficients. The dataset of GDP per capita used in this study includes nine countries (Brazil, China, India, Italy, Japan, the UK, the USA, Spain and Turkey) and the European Union. The modeling performance of the proposed model is compared with the Polynomial model and the Fractional model and prediction performance is compared to a special type of neural network, Long Short-Term Memory (LSTM), that used for time series. Results show that using Deep Assessment Methodology yields promising modeling and prediction results for GDP per capita. The proposed method is outperforming Polynomial model and Fractional model by 1.538% and by 1.899% average error rates, respectively. We also show that Deep Assessment Method (DAM) is superior to plain LSTM on prediction for upcoming GDP per capita values by 1.21% average error. Full article
(This article belongs to the Special Issue Mathematical Economics: Application of Fractional Calculus)
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17 pages, 1179 KiB
Article
Some Identities Involving Two-Variable Partially Degenerate Hermite Polynomials Induced from Differential Equations and Structure of Their Roots
Mathematics 2020, 8(4), 632; https://doi.org/10.3390/math8040632 - 20 Apr 2020
Cited by 7 | Viewed by 1827
Abstract
In this paper, we introduce two-variable partially degenerate Hermite polynomials and get some new symmetric identities for two-variable partially degenerate Hermite polynomials. We study differential equations induced from the generating functions of two-variable partially degenerate Hermite polynomials to give identities for two-variable partially [...] Read more.
In this paper, we introduce two-variable partially degenerate Hermite polynomials and get some new symmetric identities for two-variable partially degenerate Hermite polynomials. We study differential equations induced from the generating functions of two-variable partially degenerate Hermite polynomials to give identities for two-variable partially degenerate Hermite polynomials. Finally, we study the symmetric properties of the structure of the roots of the two-variable partially degenerate Hermite equations. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications)
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15 pages, 1735 KiB
Article
A Revisit of the Boundary Value Problem for Föppl–Hencky Membranes: Improvement of Geometric Equations
Mathematics 2020, 8(4), 631; https://doi.org/10.3390/math8040631 - 20 Apr 2020
Cited by 16 | Viewed by 1865
Abstract
In this paper, the well-known Föppl–Hencky membrane problem—that is, the problem of axisymmetric deformation of a transversely uniformly loaded and peripherally fixed circular membrane—was resolved, and a more refined closed-form solution of the problem was presented, where the so-called small rotation angle assumption [...] Read more.
In this paper, the well-known Föppl–Hencky membrane problem—that is, the problem of axisymmetric deformation of a transversely uniformly loaded and peripherally fixed circular membrane—was resolved, and a more refined closed-form solution of the problem was presented, where the so-called small rotation angle assumption of the membrane was given up. In particular, a more effective geometric equation was, for the first time, established to replace the classic one, and finally the resulting new boundary value problem due to the improvement of geometric equation was successfully solved by the power series method. The conducted numerical example indicates that the closed-form solution presented in this study has higher computational accuracy in comparison with the existing solutions of the well-known Föppl–Hencky membrane problem. In addition, some important issues were discussed, such as the difference between membrane problems and thin plate problems, reasonable approximation or assumption during establishing geometric equations, and the contribution of reducing approximations or relaxing assumptions to the improvement of the computational accuracy and applicability of a solution. Finally, some opinions on the follow-up work for the well-known Föppl–Hencky membrane were presented. Full article
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16 pages, 2106 KiB
Article
On the Consecutive k1 and k2-out-of-n Reliability Systems
Mathematics 2020, 8(4), 630; https://doi.org/10.3390/math8040630 - 19 Apr 2020
Cited by 9 | Viewed by 2275
Abstract
In this paper we carry out a reliability study of the consecutive-k1 and k2-out-of-n systems with independent and identically distributed components ordered in a line. More precisely, we obtain the generating function of the structure’s reliability, while recurrence [...] Read more.
In this paper we carry out a reliability study of the consecutive-k1 and k2-out-of-n systems with independent and identically distributed components ordered in a line. More precisely, we obtain the generating function of the structure’s reliability, while recurrence relations for determining its signature vector and reliability function are also provided. For illustration purposes, some numerical results and figures are presented and several concluding remarks are deduced. Full article
(This article belongs to the Special Issue Probability, Statistics and Their Applications)
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16 pages, 311 KiB
Article
Some Janowski Type Harmonic q-Starlike Functions Associated with Symmetrical Points
Mathematics 2020, 8(4), 629; https://doi.org/10.3390/math8040629 - 19 Apr 2020
Cited by 38 | Viewed by 2504
Abstract
The motive behind this article is to apply the notions of q-derivative by introducing some new families of harmonic functions associated with the symmetric circular region. We develop a new criterion for sense preserving and hence the univalency in terms of q [...] Read more.
The motive behind this article is to apply the notions of q-derivative by introducing some new families of harmonic functions associated with the symmetric circular region. We develop a new criterion for sense preserving and hence the univalency in terms of q-differential operator. The necessary and sufficient conditions are established for univalency for this newly defined class. We also discuss some other interesting properties such as distortion limits, convolution preserving, and convexity conditions. Further, by using sufficient inequality, we establish sharp bounds of the real parts of the ratios of harmonic functions to its sequences of partial sums. Some known consequences of the main results are also obtained by varying the parameters. Full article
(This article belongs to the Special Issue New Frontiers in Applied Mathematics and Statistics)
31 pages, 4841 KiB
Article
Spectrally Sparse Tensor Reconstruction in Optical Coherence Tomography Using Nuclear Norm Penalisation
Mathematics 2020, 8(4), 628; https://doi.org/10.3390/math8040628 - 18 Apr 2020
Viewed by 2438
Abstract
Reconstruction of 3D objects in various tomographic measurements is an important problem which can be naturally addressed within the mathematical framework of 3D tensors. In Optical Coherence Tomography, the reconstruction problem can be recast as a tensor completion problem. Following the seminal work [...] Read more.
Reconstruction of 3D objects in various tomographic measurements is an important problem which can be naturally addressed within the mathematical framework of 3D tensors. In Optical Coherence Tomography, the reconstruction problem can be recast as a tensor completion problem. Following the seminal work of Candès et al., the approach followed in the present work is based on the assumption that the rank of the object to be reconstructed is naturally small, and we leverage this property by using a nuclear norm-type penalisation. In this paper, a detailed study of nuclear norm penalised reconstruction using the tubal Singular Value Decomposition of Kilmer et al. is proposed. In particular, we introduce a new, efficiently computable definition of the nuclear norm in the Kilmer et al. framework. We then present a theoretical analysis, which extends previous results by Koltchinskii Lounici and Tsybakov. Finally, this nuclear norm penalised reconstruction method is applied to real data reconstruction experiments in Optical Coherence Tomography (OCT). In particular, our numerical experiments illustrate the importance of penalisation for OCT reconstruction. Full article
(This article belongs to the Special Issue New Trends in Machine Learning: Theory and Practice)
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14 pages, 289 KiB
Article
P-Tensor Product for Group C*-Algebras
Mathematics 2020, 8(4), 627; https://doi.org/10.3390/math8040627 - 18 Apr 2020
Viewed by 1494
Abstract
In this paper, we introduce new tensor products $⊗ p ( 1 ≤ p ≤ + ∞ )$ on $C ℓ p * ( Γ ) ⊗ C ℓ p * ( Γ )$ and $⊗ c 0$ on [...] Read more.
In this paper, we introduce new tensor products $⊗ p ( 1 ≤ p ≤ + ∞ )$ on $C ℓ p * ( Γ ) ⊗ C ℓ p * ( Γ )$ and $⊗ c 0$ on $C c 0 * ( Γ ) ⊗ C c 0 * ( Γ )$ for any discrete group $Γ$ . We obtain that for $1 ≤ p < + ∞$ $C ℓ p * ( Γ ) ⊗ m a x C ℓ p * ( Γ ) = C ℓ p * ( Γ ) ⊗ p C ℓ p * ( Γ )$ if and only if $Γ$ is amenable; $C c 0 * ( Γ ) ⊗ m a x C c 0 * ( Γ ) = C c 0 * ( Γ ) ⊗ c 0 C c 0 * ( Γ )$ if and only if $Γ$ has Haagerup property. In particular, for the free group with two generators $F 2$ we show that $C ℓ p * ( F 2 ) ⊗ p C ℓ p * ( F 2 ) ≇ C ℓ q * ( F 2 ) ⊗ q C ℓ q * ( F 2 )$ for $2 ≤ q < p ≤ + ∞$ . Full article
17 pages, 288 KiB
Article
Labelled Natural Deduction for Public Announcement Logic with Common Knowledge
Mathematics 2020, 8(4), 626; https://doi.org/10.3390/math8040626 - 18 Apr 2020
Viewed by 2568
Abstract
Public announcement logic is a logic that studies epistemic updates. In this paper, we propose a sound and complete labelled natural deduction system for public announcement logic with the common knowledge operator (PAC). The completeness of the proposed system is proved indirectly through [...] Read more.
Public announcement logic is a logic that studies epistemic updates. In this paper, we propose a sound and complete labelled natural deduction system for public announcement logic with the common knowledge operator (PAC). The completeness of the proposed system is proved indirectly through a Hilbert calculus for PAC known to be complete and sound. We conclude with several discussions regarding the system including some problems of the system in attaining normalisation and subformula property. Full article
29 pages, 450 KiB
Article
Asymptotic Results in Broken Stick Models: The Approach via Lorenz Curves
Mathematics 2020, 8(4), 625; https://doi.org/10.3390/math8040625 - 18 Apr 2020
Cited by 1 | Viewed by 1851
Abstract
A stick of length 1 is broken at random into n smaller sticks. How much inequality does this procedure produce? What happens if, instead of breaking a stick, we break a square? What happens asymptotically? Which is the most egalitarian distribution of the [...] Read more.
A stick of length 1 is broken at random into n smaller sticks. How much inequality does this procedure produce? What happens if, instead of breaking a stick, we break a square? What happens asymptotically? Which is the most egalitarian distribution of the smaller sticks (or rectangles)? Usually, when studying inequality, one uses a Lorenz curve. The more egalitarian a distribution, the closer the Lorenz curve is to the first diagonal of $[ 0 , 1 ] 2$. This is why in the first section we study the space of Lorenz curves. What is the limit of a convergent sequence of Lorenz curves? We try to answer these questions, firstly, in the deterministic case and based on the results obtained there in the stochastic one. Full article
(This article belongs to the Special Issue Probability, Statistics and Their Applications)
25 pages, 14195 KiB
Article
Gas–Liquid Two-Phase Flow Investigation of Side Channel Pump: An Application of MUSIG Model
Mathematics 2020, 8(4), 624; https://doi.org/10.3390/math8040624 - 18 Apr 2020
Cited by 5 | Viewed by 2913
Abstract
This paper introduces a novel application of a multiphase flow model called the Multi-Size-Group model (MUSIG) to solve 3D complex flow equations in a side channel pump, in order to analyze the flow dynamics of the gas phase distribution and migration under different [...] Read more.
This paper introduces a novel application of a multiphase flow model called the Multi-Size-Group model (MUSIG) to solve 3D complex flow equations in a side channel pump, in order to analyze the flow dynamics of the gas phase distribution and migration under different inlet gas volume fractions (IGVFs). Under different IGVF, the suction side is more likely to concentrate bubbles, especially near the inner radius of the impeller, while there is very little or no gas at the outer radius of the impeller. The diameter of bubbles in the impeller are similar and small for most regions even at IGVF = 6% due to the strong shear turbulence flow which eliminates large bubbles. Additionally, this method also can capture the coalescence and breakage evolution of bubbles. Once a mixture of fluid goes into the impeller from the inlet pipe, the large bubbles immediately break, which accounts for the reason why nearly all side channel pumps have the capacity to deliver gas–liquid two-phase flow. The results in this study provide a foundation and theoretical value for the optimal design of side channel pumps under gas–liquid two-phase conditions to increase their application. Full article
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14 pages, 318 KiB
Article
Bohr Radius Problems for Some Classes of Analytic Functions Using Quantum Calculus Approach
Mathematics 2020, 8(4), 623; https://doi.org/10.3390/math8040623 - 18 Apr 2020
Cited by 5 | Viewed by 2368
Abstract
The main purpose of this investigation is to use quantum calculus approach and obtain the Bohr radius for the class of q-starlike (q-convex) functions of order $α$ . The Bohr radius is also determined for a generalized class of q [...] Read more.
The main purpose of this investigation is to use quantum calculus approach and obtain the Bohr radius for the class of q-starlike (q-convex) functions of order $α$ . The Bohr radius is also determined for a generalized class of q-Janowski starlike and q-Janowski convex functions with negative coefficients. Full article
(This article belongs to the Special Issue Complex Analysis and Its Applications)
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