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Mathematics, Volume 5, Issue 4 (December 2017)

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Research

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Open AccessArticle New Analytical Technique for Solving a System of Nonlinear Fractional Partial Differential Equations
Mathematics 2017, 5(4), 47; doi:10.3390/math5040047
Received: 24 August 2017 / Revised: 19 September 2017 / Accepted: 20 September 2017 / Published: 25 September 2017
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Abstract
This paper introduces a new analytical technique (NAT) for solving a system of nonlinear fractional partial differential equations (NFPDEs) in full general set. Moreover, the convergence and error analysis of the proposed technique is shown. The approximate solutions for a system of NFPDEs
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This paper introduces a new analytical technique (NAT) for solving a system of nonlinear fractional partial differential equations (NFPDEs) in full general set. Moreover, the convergence and error analysis of the proposed technique is shown. The approximate solutions for a system of NFPDEs are easily obtained by means of Caputo fractional partial derivatives based on the properties of fractional calculus. However, analytical and numerical traveling wave solutions for some systems of nonlinear wave equations are successfully obtained to confirm the accuracy and efficiency of the proposed technique. Several numerical results are presented in the format of tables and graphs to make a comparison with results previously obtained by other well-known methods. Full article
(This article belongs to the Special Issue Operators of Fractional Calculus and Their Applications)
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Open AccessArticle Least-Squares Solution of Linear Differential Equations
Mathematics 2017, 5(4), 48; doi:10.3390/math5040048
Received: 30 July 2017 / Revised: 11 September 2017 / Accepted: 29 September 2017 / Published: 8 October 2017
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Abstract
This study shows how to obtain least-squares solutions to initial value problems (IVPs), boundary value problems (BVPs), and multi-value problems (MVPs) for nonhomogeneous linear differential equations (DEs) with nonconstant coefficients of any order. However, without loss of generality, the approach has been applied
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This study shows how to obtain least-squares solutions to initial value problems (IVPs), boundary value problems (BVPs), and multi-value problems (MVPs) for nonhomogeneous linear differential equations (DEs) with nonconstant coefficients of any order. However, without loss of generality, the approach has been applied to second-order DEs. The proposed method has two steps. The first step consists of writing a constrained expression, that has the DE constraints embedded. These kind of expressions are given in terms of a new unknown function, g ( t ) , and they satisfy the constraints, no matter what g ( t ) is. The second step consists of expressing g ( t ) as a linear combination of m independent known basis functions. Specifically, orthogonal polynomials are adopted for the basis functions. This choice requires rewriting the DE and the constraints in terms of a new independent variable, x [ 1 , + 1 ] . The procedure leads to a set of linear equations in terms of the unknown coefficients of the basis functions that are then computed by least-squares. Numerical examples are provided to quantify the solutions’ accuracy for IVPs, BVPs and MVPs. In all the examples provided, the least-squares solution is obtained with machine error accuracy. Full article
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Open AccessArticle An Optimal Control Approach for the Treatment of Solid Tumors with Angiogenesis Inhibitors
Mathematics 2017, 5(4), 49; doi:10.3390/math5040049
Received: 24 June 2017 / Revised: 14 September 2017 / Accepted: 29 September 2017 / Published: 10 October 2017
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Abstract
Cancer is a disease of unregulated cell growth that is estimated to kill over 600,000 people in the United States in 2017 according to the National Institute of Health. While there are several therapies to treat cancer, tumor resistance to these therapies is
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Cancer is a disease of unregulated cell growth that is estimated to kill over 600,000 people in the United States in 2017 according to the National Institute of Health. While there are several therapies to treat cancer, tumor resistance to these therapies is a concern. Drug therapies have been developed that attack proliferating endothelial cells instead of the tumor in an attempt to create a therapy that is resistant to resistance in contrast to other forms of treatment such as chemotherapy and radiation therapy. In this study, a two-compartment model in terms of differential equations is presented in order to determine the optimal protocol for the delivery of anti-angiogenesis therapy. Optimal control theory is applied to the model with a range of anti-angiogenesis doses to determine optimal doses to minimize tumor volume at the end of a two week treatment and minimize drug toxicity to the patient. Applying a continuous optimal control protocol to our model of angiogenesis and tumor cell growth shows promising results for tumor control while minimizing the toxicity to the patients. By investigating a variety of doses, we determine that the optimal angiogenesis inhibitor dose is in the range of 10–20 mg/kg. In this clinically useful range of doses, good tumor control is achieved for a two week treatment period. This work shows that varying the toxicity of the treatment to the patient will change the optimal dosing scheme but tumor control can still be achieved. Full article
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Open AccessArticle The Stability of Parabolic Problems with Nonstandard p(x, t)-Growth
Mathematics 2017, 5(4), 50; doi:10.3390/math5040050
Received: 17 September 2017 / Revised: 5 October 2017 / Accepted: 8 October 2017 / Published: 12 October 2017
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Abstract
In this paper, we study weak solutions to the following nonlinear parabolic partial differential equation tudiva(x,t,u)+λ(|u|p(x,t)2
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In this paper, we study weak solutions to the following nonlinear parabolic partial differential equation t u div a ( x , t , u ) + λ ( | u | p ( x , t ) 2 u ) = 0 in Ω T , where λ 0 and t u denote the partial derivative of u with respect to the time variable t, while u denotes the one with respect to the space variable x. Moreover, the vector-field a ( x , t , · ) satisfies certain nonstandard p ( x , t ) -growth and monotonicity conditions. In this manuscript, we establish the existence of a unique weak solution to the corresponding Dirichlet problem. Furthermore, we prove the stability of this solution, i.e., we show that two weak solutions with different initial values are controlled by these initial values. Full article
Open AccessFeature PaperArticle Solutions Modulo p of Gauss–Manin Differential Equations for Multidimensional Hypergeometric Integrals and Associated Bethe Ansatz
Mathematics 2017, 5(4), 52; doi:10.3390/math5040052
Received: 22 September 2017 / Revised: 13 October 2017 / Accepted: 13 October 2017 / Published: 17 October 2017
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Abstract
We consider the Gauss–Manin differential equations for hypergeometric integrals associated with a family of weighted arrangements of hyperplanes moving parallel to themselves. We reduce these equations modulo a prime integer p and construct polynomial solutions of the new differential equations as p-analogs of
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We consider the Gauss–Manin differential equations for hypergeometric integrals associated with a family of weighted arrangements of hyperplanes moving parallel to themselves. We reduce these equations modulo a prime integer p and construct polynomial solutions of the new differential equations as p-analogs of the initial hypergeometric integrals. In some cases, we interpret the p-analogs of the hypergeometric integrals as sums over points of hypersurfaces defined over the finite field Fp. This interpretation is similar to the classical interpretation by Yu. I. Manin of the number of points on an elliptic curve depending on a parameter as a solution of a Gauss hypergeometric differential equation. We discuss the associated Bethe ansatz. Full article
Open AccessArticle Stability of a Monomial Functional Equation on a Restricted Domain
Mathematics 2017, 5(4), 53; doi:10.3390/math5040053
Received: 24 August 2017 / Revised: 2 October 2017 / Accepted: 8 October 2017 / Published: 18 October 2017
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Abstract
In this paper, we prove the stability of the following functional equation i=0nnCi(1)nif(ix+y)n!f(x)=0
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In this paper, we prove the stability of the following functional equation i = 0 n n C i ( 1 ) n i f ( i x + y ) n ! f ( x ) = 0 on a restricted domain by employing the direct method in the sense of Hyers. Full article
(This article belongs to the Special Issue Operators of Fractional Calculus and Their Applications)
Open AccessArticle An Investigation of Radial Basis Function-Finite Difference (RBF-FD) Method for Numerical Solution of Elliptic Partial Differential Equations
Mathematics 2017, 5(4), 54; doi:10.3390/math5040054
Received: 2 September 2017 / Revised: 13 October 2017 / Accepted: 17 October 2017 / Published: 23 October 2017
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Abstract
The Radial Basis Function (RBF) method has been considered an important meshfree tool for numerical solutions of Partial Differential Equations (PDEs). For various situations, RBF with infinitely differentiable functions can provide accurate results and more flexibility in the geometry of computation domains than
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The Radial Basis Function (RBF) method has been considered an important meshfree tool for numerical solutions of Partial Differential Equations (PDEs). For various situations, RBF with infinitely differentiable functions can provide accurate results and more flexibility in the geometry of computation domains than traditional methods such as finite difference and finite element methods. However, RBF does not suit large scale problems, and, therefore, a combination of RBF and the finite difference (RBF-FD) method was proposed because of its own strengths not only on feasibility and computational cost, but also on solution accuracy. In this study, we try the RBF-FD method on elliptic PDEs and study the effect of it on such equations with different shape parameters. Most importantly, we study the solution accuracy after additional ghost node strategy, preconditioning strategy, regularization strategy, and floating point arithmetic strategy. We have found more satisfactory accurate solutions in most situations than those from global RBF, except in the preconditioning and regularization strategies. Full article
(This article belongs to the Special Issue Operators of Fractional Calculus and Their Applications)
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Open AccessArticle On the Achievable Stabilization Delay Margin for Linear Plants with Time-Varying Delays
Mathematics 2017, 5(4), 55; doi:10.3390/math5040055
Received: 24 September 2017 / Revised: 19 October 2017 / Accepted: 20 October 2017 / Published: 25 October 2017
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Abstract
The paper contributes to stabilization problems of linear systems subject to time-varying delays. Drawing upon small gain criteria and robust analysis techniques, upper and lower bounds on the largest allowable time-varying delay are developed by using bilinear transformation and rational approximates. The results
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The paper contributes to stabilization problems of linear systems subject to time-varying delays. Drawing upon small gain criteria and robust analysis techniques, upper and lower bounds on the largest allowable time-varying delay are developed by using bilinear transformation and rational approximates. The results achieved are not only computationally efficient but also conceptually appealing. Furthermore, analytical expressions of the upper and lower bounds are derived for specific situations that demonstrate the dependence of those bounds on the unstable poles and nonminumum phase zeros of systems. Full article
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Open AccessFeature PaperArticle A Constructive Method for Standard Borel Fixed Submodules with Given Extremal Betti Numbers
Mathematics 2017, 5(4), 56; doi:10.3390/math5040056
Received: 29 May 2017 / Revised: 9 October 2017 / Accepted: 17 October 2017 / Published: 1 November 2017
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Abstract
Let S be a polynomial ring in n variables over a field K of any characteristic. Given a strongly stable submodule M of a finitely generated graded free S-module F, we propose a method for constructing a standard Borel-fixed submodule M
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Let S be a polynomial ring in n variables over a field K of any characteristic. Given a strongly stable submodule M of a finitely generated graded free S-module F, we propose a method for constructing a standard Borel-fixed submodule M ˜ of F so that the extremal Betti numbers of M, values as well as positions, are preserved by passing from M to M ˜ . As a result, we obtain a numerical characterization of all possible extremal Betti numbers of any standard Borel-fixed submodule of a finitely generated graded free S-module F. Full article
Open AccessArticle The Theory of Connections: Connecting Points
Mathematics 2017, 5(4), 57; doi:10.3390/math5040057
Received: 30 July 2017 / Revised: 17 September 2017 / Accepted: 24 October 2017 / Published: 1 November 2017
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Abstract
This study introduces a procedure to obtain all interpolating functions, y=f(x), subject to linear constraints on the function and its derivatives defined at specified values. The paper first shows how to express these interpolating functions passing through
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This study introduces a procedure to obtain all interpolating functions, y = f ( x ) , subject to linear constraints on the function and its derivatives defined at specified values. The paper first shows how to express these interpolating functions passing through a single point in three distinct ways: linear, additive, and rational. Then, using the additive formalism, interpolating functions with linear constraints on one, two, and n points are introduced as well as those satisfying relative constraints. In particular, for expressions passing through n points, a generalization of the Waring’s interpolation form is introduced. An alternative approach to derive additive constraint interpolating expressions is introduced requiring the inversion of a matrix with dimensions equally the number of constraints. Finally, continuous and discontinuous interpolating periodic functions passing through a set of points with specified periods are provided. This theory has already been applied to obtain least-squares solutions of initial and boundary value problems applied to nonhomogeneous linear differential equations with nonconstant coefficients. Full article
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Open AccessArticle Dynamics of Amoebiasis Transmission: Stability and Sensitivity Analysis
Mathematics 2017, 5(4), 58; doi:10.3390/math5040058
Received: 8 August 2017 / Revised: 19 September 2017 / Accepted: 24 September 2017 / Published: 1 November 2017
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Abstract
Compartmental epidemic models are intriguing in the sense that the generic model may explain different kinds of infectious diseases with minor modifications. However, there may exist some ailments that may not fit the generic capsule. Amoebiasis is one such example where transmission through
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Compartmental epidemic models are intriguing in the sense that the generic model may explain different kinds of infectious diseases with minor modifications. However, there may exist some ailments that may not fit the generic capsule. Amoebiasis is one such example where transmission through the population demands a more detailed and sophisticated approach, both mathematical and numerical. The manuscript engages in a deep analytical study of the compartmental epidemic model; susceptible-exposed-infectious-carrier-recovered-susceptible (SEICRS), formulated for Amoebiasis. We have shown that the model allows the single disease-free equilibrium (DFE) state if R 0 , the basic reproduction number, is less than unity and the unique endemic equilibrium (EE) state if R 0 is greater than unity. Furthermore, the basic reproduction number depends uniquely on the input parameters and constitutes a key threshold indicator to portray the general trends of the dynamics of Amoebiasis transmission. We have also shown that R 0 is highly sensitive to the changes in values of the direct transmission rate in contrast to the change in values of the rate of transfer from latent infection to the infectious state. Using the Routh–Hurwitz criterion and Lyapunov direct method, we have proven the conditions for the disease-free equilibrium and the endemic equilibrium states to be locally and globally asymptotically stable. In other words, the conditions for Amoebiasis “die-out” and “infection propagation” are presented. Full article
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Open AccessArticle Invariant Solutions for a Class of Perturbed Nonlinear Wave Equations
Mathematics 2017, 5(4), 59; doi:10.3390/math5040059
Received: 22 September 2017 / Revised: 20 October 2017 / Accepted: 23 October 2017 / Published: 1 November 2017
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Abstract
Approximate symmetries of a class of perturbed nonlinear wave equations are computed using two newly-developed methods. Invariant solutions associated with the approximate symmetries are constructed for both methods. Symmetries and solutions are compared through discussing the advantages and disadvantages of each method. Full article
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Open AccessArticle Graph Structures in Bipolar Neutrosophic Environment
Mathematics 2017, 5(4), 60; doi:10.3390/math5040060
Received: 14 September 2017 / Revised: 18 October 2017 / Accepted: 27 October 2017 / Published: 6 November 2017
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Abstract
A bipolar single-valued neutrosophic (BSVN) graph structure is a generalization of a bipolar fuzzy graph. In this research paper, we present certain concepts of BSVN graph structures. We describe some operations on BSVN graph structures and elaborate on these with examples. Moreover, we
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A bipolar single-valued neutrosophic (BSVN) graph structure is a generalization of a bipolar fuzzy graph. In this research paper, we present certain concepts of BSVN graph structures. We describe some operations on BSVN graph structures and elaborate on these with examples. Moreover, we investigate some related properties of these operations. Full article
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Open AccessFeature PaperArticle Mixed Order Fractional Differential Equations
Mathematics 2017, 5(4), 61; doi:10.3390/math5040061
Received: 8 September 2017 / Revised: 30 October 2017 / Accepted: 31 October 2017 / Published: 7 November 2017
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Abstract
This paper studies fractional differential equations (FDEs) with mixed fractional derivatives. Existence, uniqueness, stability, and asymptotic results are derived. Full article
(This article belongs to the Special Issue Operators of Fractional Calculus and Their Applications)
Open AccessArticle Solution of Inhomogeneous Differential Equations with Polynomial Coefficients in Terms of the Green’s Function
Mathematics 2017, 5(4), 62; doi:10.3390/math5040062
Received: 30 September 2017 / Revised: 31 October 2017 / Accepted: 6 November 2017 / Published: 10 November 2017
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Abstract
The particular solutions of inhomogeneous differential equations with polynomial coefficients in terms of the Green’s function are obtained in the framework of distribution theory. In particular, discussions are given on Kummer’s and the hypergeometric differential equation. Related discussions are given on the particular
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The particular solutions of inhomogeneous differential equations with polynomial coefficients in terms of the Green’s function are obtained in the framework of distribution theory. In particular, discussions are given on Kummer’s and the hypergeometric differential equation. Related discussions are given on the particular solution of differential equations with constant coefficients, by the Laplace transform. Full article
(This article belongs to the Special Issue Operators of Fractional Calculus and Their Applications)
Open AccessArticle Krylov Implicit Integration Factor Methods for Semilinear Fourth-Order Equations
Mathematics 2017, 5(4), 63; doi:10.3390/math5040063
Received: 26 September 2017 / Revised: 7 November 2017 / Accepted: 8 November 2017 / Published: 16 November 2017
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Abstract
Implicit integration factor (IIF) methods were developed for solving time-dependent stiff partial differential equations (PDEs) in literature. In [Jiang and Zhang, Journal of Computational Physics, 253 (2013) 368–388], IIF methods are designed to efficiently solve stiff nonlinear advection–diffusion–reaction (ADR) equations. The methods can
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Implicit integration factor (IIF) methods were developed for solving time-dependent stiff partial differential equations (PDEs) in literature. In [Jiang and Zhang, Journal of Computational Physics, 253 (2013) 368–388], IIF methods are designed to efficiently solve stiff nonlinear advection–diffusion–reaction (ADR) equations. The methods can be designed for an arbitrary order of accuracy. The stiffness of the system is resolved well, and large-time-step-size computations are achieved. To efficiently calculate large matrix exponentials, a Krylov subspace approximation is directly applied to the IIF methods. In this paper, we develop Krylov IIF methods for solving semilinear fourth-order PDEs. As a result of the stiff fourth-order spatial derivative operators, the fourth-order PDEs have much stricter constraints in time-step sizes than the second-order ADR equations. We analyze the truncation errors of the fully discretized schemes. Numerical examples of both scalar equations and systems in one and higher spatial dimensions are shown to demonstrate the accuracy, efficiency and stability of the methods. Large time-step sizes that are of the same order as the spatial grid sizes have been achieved in the simulations of the fourth-order PDEs. Full article
(This article belongs to the Special Issue Numerical Methods for Partial Differential Equations)
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Open AccessArticle On the Inception of Financial Representative Bubbles
Mathematics 2017, 5(4), 64; doi:10.3390/math5040064 (registering DOI)
Received: 9 October 2017 / Revised: 9 November 2017 / Accepted: 14 November 2017 / Published: 17 November 2017
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Abstract
In this work, we aim to formalize the inception of representative bubbles giving the condition under which they may arise. We will find that representative bubbles may start at any time, depending on the definition of a behavioral component. This result is at
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In this work, we aim to formalize the inception of representative bubbles giving the condition under which they may arise. We will find that representative bubbles may start at any time, depending on the definition of a behavioral component. This result is at odds with the theory of classic rational bubbles, which are those models that rely on the fulfillment of the transversality condition by which a bubble in a financial asset can arise just at its first trade. This means that a classic rational bubble (differently from our model) cannot follow a cycle since if a bubble exists, it will burst by definition and never arise again. Full article
(This article belongs to the Special Issue Financial Mathematics)

Review

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Open AccessReview Euclidean Submanifolds via Tangential Components of Their Position Vector Fields
Mathematics 2017, 5(4), 51; doi:10.3390/math5040051
Received: 6 September 2017 / Revised: 6 September 2017 / Accepted: 10 October 2017 / Published: 16 October 2017
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Abstract
The position vector field is the most elementary and natural geometric object on a Euclidean submanifold. The position vector field plays important roles in physics, in particular in mechanics. For instance, in any equation of motion, the position vector x (t)
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The position vector field is the most elementary and natural geometric object on a Euclidean submanifold. The position vector field plays important roles in physics, in particular in mechanics. For instance, in any equation of motion, the position vector x (t) is usually the most sought-after quantity because the position vector field defines the motion of a particle (i.e., a point mass): its location relative to a given coordinate system at some time variable t. This article is a survey article. The purpose of this article is to survey recent results of Euclidean submanifolds associated with the tangential components of their position vector fields. In the last section, we present some interactions between torqued vector fields and Ricci solitons. Full article
(This article belongs to the Special Issue Differential Geometry)
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