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Axioms, Volume 8, Issue 1 (March 2019)

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Open AccessArticle General Relativity with a Positive Cosmological Constant Λ as a Gauge Theory
Received: 1 January 1970 / Revised: 30 January 2019 / Accepted: 4 February 2019 / Published: 21 February 2019
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Abstract
In this paper, we show that the general relativity action (and Lagrangian) in recent Einstein–Palatini formulation is equivalent in four dimensions to the action (and Langrangian) of a gauge field. First, we briefly showcase the Einstein–Palatini (EP) action, and then we present how [...] Read more.
In this paper, we show that the general relativity action (and Lagrangian) in recent Einstein–Palatini formulation is equivalent in four dimensions to the action (and Langrangian) of a gauge field. First, we briefly showcase the Einstein–Palatini (EP) action, and then we present how Einstein fields equations can be derived from it. In the next section, we study Einstein–Palatini action integral for general relativity with a positive cosmological constant Λ in terms of the corrected curvature Ω c o r . We see that in terms of Ω c o r this action takes the form typical for a gauge field. Finally, we give a geometrical interpretation of the corrected curvature Ω c o r . Full article
(This article belongs to the Special Issue Applications of Differential Geometry)
Open AccessArticle First Order Coupled Systems With Functional and Periodic Boundary Conditions: Existence Results and Application to an SIRS Model
Received: 6 January 2019 / Revised: 13 February 2019 / Accepted: 13 February 2019 / Published: 16 February 2019
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Abstract
The results presented in this paper deal with the existence of solutions of a first order fully coupled system of three equations, and they are split in two parts: 1. Case with coupled functional boundary conditions, and 2. Case with periodic boundary conditions. [...] Read more.
The results presented in this paper deal with the existence of solutions of a first order fully coupled system of three equations, and they are split in two parts: 1. Case with coupled functional boundary conditions, and 2. Case with periodic boundary conditions. Functional boundary conditions, which are becoming increasingly popular in the literature, as they generalize most of the classical cases and in addition can be used to tackle global conditions, such as maximum or minimum conditions. The arguments used are based on the Arzèla Ascoli theorem and Schauder’s fixed point theorem. The existence results are directly applied to an epidemic SIRS (Susceptible-Infectious-Recovered-Susceptible) model, with global boundary conditions. Full article
(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)
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Open AccessArticle Heteroclinic Solutions for Classical and Singular ϕ-Laplacian Non-Autonomous Differential Equations
Received: 28 December 2018 / Revised: 28 January 2019 / Accepted: 11 February 2019 / Published: 15 February 2019
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Abstract
In this paper, we consider the second order discontinuous differential equation in the real line, at,uϕu=ft,u,u,a.e.tR,u( [...] Read more.
In this paper, we consider the second order discontinuous differential equation in the real line, a t , u ϕ u = f t , u , u , a . e . t R , u ( ) = ν , u ( + ) = ν + , with ϕ an increasing homeomorphism such that ϕ ( 0 ) = 0 and ϕ ( R ) = R , a C ( R 2 , R ) with a ( t , x ) > 0 for ( t , x ) R 2 , f : R 3 R a L 1 -Carathéodory function and ν , ν + R such that ν < ν + . The existence and localization of heteroclinic connections is obtained assuming a Nagumo-type condition on the real line and without asymptotic conditions on the nonlinearities ϕ and f . To the best of our knowledge, this result is even new when ϕ ( y ) = y , that is for equation a t , u ( t ) u ( t ) = f t , u ( t ) , u ( t ) , a . e . t R . Moreover, these results can be applied to classical and singular ϕ -Laplacian equations and to the mean curvature operator. Full article
(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)
Open AccessArticle 3-Lie Superalgebras Induced by Lie Superalgebras
Received: 21 November 2018 / Revised: 30 January 2019 / Accepted: 31 January 2019 / Published: 6 February 2019
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Abstract
We show that given a Lie superalgebra and an element of its dual space, one can construct the 3-Lie superalgebra. We apply this approach to Lie superalgebra of (m,n)-block matrices taking a supertrace of a matrix as the [...] Read more.
We show that given a Lie superalgebra and an element of its dual space, one can construct the 3-Lie superalgebra. We apply this approach to Lie superalgebra of ( m , n ) -block matrices taking a supertrace of a matrix as the element of dual space. Then we also apply this approach to commutative superalgebra and the Lie superalgebra of its derivations to construct 3-Lie superalgebra. The graded Lie brackets are constructed by means of a derivation and involution of commutative superalgebra, and we use them to construct 3-Lie superalgebras. Full article
Open AccessArticle Solution Estimates for the Discrete Lyapunov Equation in a Hilbert Space and Applications to Difference Equations
Received: 15 January 2019 / Revised: 1 February 2019 / Accepted: 2 February 2019 / Published: 6 February 2019
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Abstract
The paper is devoted to the discrete Lyapunov equation X-A*XA=C, where A and C are given operators in a Hilbert space H and X should be found. We derive norm estimates for solutions of that [...] Read more.
The paper is devoted to the discrete Lyapunov equation X - A * X A = C , where A and C are given operators in a Hilbert space H and X should be found. We derive norm estimates for solutions of that equation in the case of unstable operator A, as well as refine the previously-published estimates for the equation with a stable operator. By the point estimates, we establish explicit conditions, under which a linear nonautonomous difference equation in H is dichotomic. In addition, we suggest a stability test for a class of nonlinear nonautonomous difference equations in H . Our results are based on the norm estimates for powers and resolvents of non-self-adjoint operators. Full article
(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)
Open AccessArticle Note on Limit-Periodic Solutions of the Difference Equation xt1-[h(xt)λ]xt=rt, λ>1++
Received: 7 January 2019 / Revised: 31 January 2019 / Accepted: 1 February 2019 / Published: 5 February 2019
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Abstract
As a nontrivial application of the abstract theorem developed in our recent paper titled “Limit-periodic solutions of difference and differential systems without global Lipschitzianity restricitions”, the existence of limit-periodic solutions of the difference equation from the title is proved, both in the scalar [...] Read more.
As a nontrivial application of the abstract theorem developed in our recent paper titled “Limit-periodic solutions of difference and differential systems without global Lipschitzianity restricitions”, the existence of limit-periodic solutions of the difference equation from the title is proved, both in the scalar as well as vector cases. The nonlinearity h is not necessarily globally Lipschitzian. Several simple illustrative examples are supplied. Full article
(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)
Open AccessArticle Complex Soliton Solutions to the Gilson–Pickering Model
Received: 23 December 2018 / Revised: 30 January 2019 / Accepted: 31 January 2019 / Published: 1 February 2019
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Abstract
In this paper, an analytical method based on the Bernoulli differential equation for extracting new complex soliton solutions to the Gilson–Pickering model is applied. A set of new complex soliton solutions to the Gilson–Pickering model are successfully constructed. In addition, 2D and 3D [...] Read more.
In this paper, an analytical method based on the Bernoulli differential equation for extracting new complex soliton solutions to the Gilson–Pickering model is applied. A set of new complex soliton solutions to the Gilson–Pickering model are successfully constructed. In addition, 2D and 3D graphs and contour simulations to the complex soliton solutions are plotted with the help of computational programs. Finally, at the end of the manuscript a conclusion about new complex soliton solutions is given. Full article
Open AccessArticle Best Proximity Point Theorems on Rectangular Metric Spaces Endowed with a Graph
Received: 3 January 2019 / Revised: 23 January 2019 / Accepted: 28 January 2019 / Published: 1 February 2019
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Abstract
In this paper, we ensure the existence and uniqueness of a best proximity point in rectangular metric spaces endowed with a graph structure. Full article
(This article belongs to the Special Issue Fixed Point Theory and Related Topics)
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Open AccessEditorial Advanced Numerical Methods in Applied Sciences
Received: 25 January 2019 / Accepted: 29 January 2019 / Published: 31 January 2019
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Abstract
The use of scientific computing tools is, nowadays, customary for solving problems in Applied Sciences at several levels of complexity. The great need for reliable software in the scientific community conveys a continuous stimulus to develop new and more performing numerical methods which [...] Read more.
The use of scientific computing tools is, nowadays, customary for solving problems in Applied Sciences at several levels of complexity. The great need for reliable software in the scientific community conveys a continuous stimulus to develop new and more performing numerical methods which are able to grasp the particular features of the problem at hand. This has been the case for many different settings of numerical analysis, and this Special Issue aims at covering some important developments in various areas of application. Full article
(This article belongs to the Special Issue Advanced Numerical Methods in Applied Sciences)
Open AccessArticle Interval Methods with Fifth Order of Convergence for Solving Nonlinear Scalar Equations
Received: 25 December 2018 / Revised: 26 January 2019 / Accepted: 28 January 2019 / Published: 31 January 2019
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Abstract
In this paper, based on Kou’s classical iterative methods with fifth-order of convergence, we propose new interval iterative methods for computing a real root of the nonlinear scalar equations. Some numerical experiments have executed with the program INTLAB in order to confirm the [...] Read more.
In this paper, based on Kou’s classical iterative methods with fifth-order of convergence, we propose new interval iterative methods for computing a real root of the nonlinear scalar equations. Some numerical experiments have executed with the program INTLAB in order to confirm the theoretical results. The computational results have described and compared with Newton’s interval method, Ostrowski’s interval method and Ostrowski’s modified interval method. We conclude that the proposed interval schemes are effective and they are comparable to the classical interval methods. Full article
Open AccessArticle PSO with Dynamic Adaptation of Parameters for Optimization in Neural Networks with Interval Type-2 Fuzzy Numbers Weights
Received: 4 December 2018 / Revised: 9 January 2019 / Accepted: 9 January 2019 / Published: 24 January 2019
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Abstract
A dynamic adjustment of parameters for the particle swarm optimization (PSO) utilizing an interval type-2 fuzzy inference system is proposed in this work. A fuzzy neural network with interval type-2 fuzzy number weights using S-norm and T-norm is optimized with the proposed method. [...] Read more.
A dynamic adjustment of parameters for the particle swarm optimization (PSO) utilizing an interval type-2 fuzzy inference system is proposed in this work. A fuzzy neural network with interval type-2 fuzzy number weights using S-norm and T-norm is optimized with the proposed method. A dynamic adjustment of the PSO allows the algorithm to behave better in the search for optimal results because the dynamic adjustment provides good synchrony between the exploration and exploitation of the algorithm. Results of experiments and a comparison between traditional neural networks and the fuzzy neural networks with interval type-2 fuzzy numbers weights using T-norms and S-norms are given to prove the performance of the proposed approach. For testing the performance of the proposed approach, some cases of time series prediction are applied, including the stock exchanges of Germany, Mexican, Dow-Jones, London, Nasdaq, Shanghai, and Taiwan. Full article
(This article belongs to the Special Issue Type-2 Fuzzy Logic: Theory, Algorithms and Applications)
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Open AccessArticle Fixed Point Results in Partial Symmetric Spaces with an Application
Received: 8 December 2018 / Revised: 5 January 2019 / Accepted: 15 January 2019 / Published: 22 January 2019
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Abstract
In this paper, we first introduce the class of partial symmetric spaces and then prove some fixed point theorems in such spaces. We use one of the our main results to examine the existence and uniqueness of a solution for a system of [...] Read more.
In this paper, we first introduce the class of partial symmetric spaces and then prove some fixed point theorems in such spaces. We use one of the our main results to examine the existence and uniqueness of a solution for a system of Fredholm integral equations. Furthermore, we introduce an analogue of the Hausdorff metric in the context of partial symmetric spaces and utilize the same to prove an analogue of the Nadler contraction principle in such spaces. Our results extend and improve many results in the existing literature. We also give some examples exhibiting the utility of our newly established results. Full article
(This article belongs to the Special Issue Mathematical Analysis and Applications II)
Open AccessArticle A New Identity for Generalized Hypergeometric Functions and Applications
Received: 19 November 2018 / Revised: 28 December 2018 / Accepted: 14 January 2019 / Published: 18 January 2019
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Abstract
We establish a new identity for generalized hypergeometric functions and apply it for first- and second-kind Gauss summation formulas to obtain some new summation formulas. The presented identity indeed extends some results of the recent published paper (Some summation theorems for generalized [...] Read more.
We establish a new identity for generalized hypergeometric functions and apply it for first- and second-kind Gauss summation formulas to obtain some new summation formulas. The presented identity indeed extends some results of the recent published paper (Some summation theorems for generalized hypergeometric functions, Axioms, 7 (2018), Article 38). Full article
(This article belongs to the Special Issue Mathematical Analysis and Applications II)
Open AccessArticle Relations between Shadowing and Inverse Shadowing in Dynamical Systems
Received: 15 December 2018 / Revised: 13 January 2019 / Accepted: 15 January 2019 / Published: 17 January 2019
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Abstract
In this paper, we study relations between shadowing and inverse shadowing for homeomorphisms of a compact space. We present an example of a smooth diffeomorphism of a compact three-dimensional manifold that has the shadowing property and does not have the inverse shadowing property. [...] Read more.
In this paper, we study relations between shadowing and inverse shadowing for homeomorphisms of a compact space. We present an example of a smooth diffeomorphism of a compact three-dimensional manifold that has the shadowing property and does not have the inverse shadowing property. For some classes of inverse shadowing, we construct examples of homeomorphisms that have the inverse shadowing property but do not have the shadowing property. Full article
(This article belongs to the Special Issue Shadowing in Dynamical Systems)
Open AccessArticle Some Features of Rank One Real Solvable Cohomologically Rigid Lie Algebras with a Nilradical Contracting onto the Model Filiform Lie Algebra Qn
Received: 25 November 2018 / Revised: 11 January 2019 / Accepted: 14 January 2019 / Published: 16 January 2019
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Abstract
The generic structure and some peculiarities of real rank one solvable Lie algebras possessing a maximal torus of derivations with the eigenvalue spectrum spec(t)=1,k,k+1,,n+k3 [...] Read more.
The generic structure and some peculiarities of real rank one solvable Lie algebras possessing a maximal torus of derivations with the eigenvalue spectrum spec ( t ) = 1 , k , k + 1 , , n + k 3 , n + 2 k 3 for k 2 are analyzed, with special emphasis on the resulting Lie algebras for which the second Chevalley cohomology space vanishes. From the detailed inspection of the values k 5 , some series of cohomologically rigid algebras for arbitrary values of k are determined. Full article
Open AccessArticle Harrod–Domar Growth Model with Memory and Distributed Lag
Received: 6 December 2018 / Revised: 9 January 2019 / Accepted: 11 January 2019 / Published: 15 January 2019
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Abstract
In this paper, we propose a macroeconomic growth model, in which we take into account memory with power-law fading and gamma distributed lag. This model is a generalization of the standard Harrod–Domar growth model. Fractional differential equations of this generalized model with memory [...] Read more.
In this paper, we propose a macroeconomic growth model, in which we take into account memory with power-law fading and gamma distributed lag. This model is a generalization of the standard Harrod–Domar growth model. Fractional differential equations of this generalized model with memory and lag are suggested. For these equations, we obtain solutions, which describe the macroeconomic growth of national income with fading memory and distributed time-delay. The asymptotic behavior of these solutions is described. Full article
(This article belongs to the Special Issue Fractional Differential Equations)
Open AccessArticle Optimal Genetic Design of Type-1 and Interval Type-2 Fuzzy Systems for Blood Pressure Level Classification
Received: 1 December 2018 / Revised: 4 January 2019 / Accepted: 10 January 2019 / Published: 15 January 2019
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Abstract
The use of artificial intelligence techniques such as fuzzy logic, neural networks and evolutionary computation is currently very important in medicine to be able to provide an effective and timely diagnosis. The use of fuzzy logic allows to design fuzzy classifiers, which have [...] Read more.
The use of artificial intelligence techniques such as fuzzy logic, neural networks and evolutionary computation is currently very important in medicine to be able to provide an effective and timely diagnosis. The use of fuzzy logic allows to design fuzzy classifiers, which have fuzzy rules and membership functions, which are designed based on the experience of an expert. In this particular case a fuzzy classifier of Mamdani type was built, with 21 rules, with two inputs and one output and the objective of this classifier is to perform blood pressure level classification based on knowledge of an expert which is represented in the fuzzy rules. Subsequently different architectures were made in type-1 and type-2 fuzzy systems for classification, where the parameters of the membership functions used in the design of each architecture were adjusted, which can be triangular, trapezoidal and Gaussian, as well as how the fuzzy rules are optimized based on the ranges established by an expert. The main contribution of this work is the design of the optimized interval type-2 fuzzy system with triangular membership functions. The final type-2 system has a better classification rate of 99.408% than the type-1 classifier developed previously in “Design of an optimized fuzzy classifier for the diagnosis of blood pressure with a new computational method for expert rule optimization” with 98%. In addition, we also obtained a better classification rate than the other architectures proposed in this work. Full article
(This article belongs to the Special Issue Type-2 Fuzzy Logic: Theory, Algorithms and Applications)
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Open AccessArticle The Laplacian Flow of Locally Conformal Calibrated G2-Structures
Received: 8 November 2018 / Revised: 31 December 2018 / Accepted: 3 January 2019 / Published: 11 January 2019
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Abstract
We consider the Laplacian flow of locally conformal calibrated G2-structures as a natural extension to these structures of the well-known Laplacian flow of calibrated G2-structures. We study the Laplacian flow for two explicit examples of locally conformal calibrated G [...] Read more.
We consider the Laplacian flow of locally conformal calibrated G 2 -structures as a natural extension to these structures of the well-known Laplacian flow of calibrated G 2 -structures. We study the Laplacian flow for two explicit examples of locally conformal calibrated G 2 manifolds and, in both cases, we obtain a flow of locally conformal calibrated G 2 -structures, which are ancient solutions, that is they are defined on a time interval of the form ( , T ) , where T > 0 is a real number. Moreover, for each of these examples, we prove that the underlying metrics g ( t ) of the solution converge smoothly, up to pull-back by time-dependent diffeomorphisms, to a flat metric as t goes to , and they blow-up at a finite-time singularity. Full article
(This article belongs to the Special Issue Applications of Differential Geometry)
Open AccessReview Contact Semi-Riemannian Structures in CR Geometry: Some Aspects
Received: 26 September 2018 / Revised: 20 December 2018 / Accepted: 2 January 2019 / Published: 9 January 2019
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Abstract
There is one-to-one correspondence between contact semi-Riemannian structures (η,ξ,φ,g) and non-degenerate almost CR structures (H,ϑ,J). In general, a non-degenerate almost CR structure is not a CR structure, that [...] Read more.
There is one-to-one correspondence between contact semi-Riemannian structures ( η , ξ , φ , g ) and non-degenerate almost CR structures ( H , ϑ , J ) . In general, a non-degenerate almost CR structure is not a CR structure, that is, in general the integrability condition for H 1 , 0 : = X i J X , X H is not satisfied. In this paper we give a survey on some known results, with the addition of some new results, on the geometry of contact semi-Riemannian manifolds, also in the context of the geometry of Levi non-degenerate almost CR manifolds of hypersurface type, emphasizing similarities and differences with respect to the Riemannian case. Full article
(This article belongs to the Special Issue Applications of Differential Geometry)
Open AccessEditorial Acknowledgement to Reviewers of Axioms in 2018
Published: 8 January 2019
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Abstract
Rigorous peer-review is the corner-stone of high-quality academic publishing [...] Full article
Open AccessArticle Lipschitz Stability for Non-Instantaneous Impulsive Caputo Fractional Differential Equations with State Dependent Delays
Received: 21 November 2018 / Revised: 24 December 2018 / Accepted: 25 December 2018 / Published: 29 December 2018
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Abstract
In this paper, we study Lipschitz stability of Caputo fractional differential equations with non-instantaneous impulses and state dependent delays. The study is based on Lyapunov functions and the Razumikhin technique. Our equations in particular include constant delays, time variable delay, distributed delay, etc. [...] Read more.
In this paper, we study Lipschitz stability of Caputo fractional differential equations with non-instantaneous impulses and state dependent delays. The study is based on Lyapunov functions and the Razumikhin technique. Our equations in particular include constant delays, time variable delay, distributed delay, etc. We consider the case of impulses that start abruptly at some points and their actions continue on given finite intervals. The study of Lipschitz stability by Lyapunov functions requires appropriate derivatives among fractional differential equations. A brief overview of different types of derivative known in the literature is given. Some sufficient conditions for uniform Lipschitz stability and uniform global Lipschitz stability are obtained by an application of several types of derivatives of Lyapunov functions. Examples are given to illustrate the results. Full article
(This article belongs to the Special Issue New Trends in Differential and Difference Equations and Applications)
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Open AccessArticle Separability of Topological Groups: A Survey with Open Problems
Received: 25 November 2018 / Revised: 23 December 2018 / Accepted: 25 December 2018 / Published: 29 December 2018
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Abstract
Separability is one of the basic topological properties. Most classical topological groups and Banach spaces are separable; as examples we mention compact metric groups, matrix groups, connected (finite-dimensional) Lie groups; and the Banach spaces C(K) for metrizable compact spaces K [...] Read more.
Separability is one of the basic topological properties. Most classical topological groups and Banach spaces are separable; as examples we mention compact metric groups, matrix groups, connected (finite-dimensional) Lie groups; and the Banach spaces C ( K ) for metrizable compact spaces K; and p , for p 1 . This survey focuses on the wealth of results that have appeared in recent years about separable topological groups. In this paper, the property of separability of topological groups is examined in the context of taking subgroups, finite or infinite products, and quotient homomorphisms. The open problem of Banach and Mazur, known as the Separable Quotient Problem for Banach spaces, asks whether every Banach space has a quotient space which is a separable Banach space. This paper records substantial results on the analogous problem for topological groups. Twenty open problems are included in the survey. Full article
(This article belongs to the collection Topological Groups)
Open AccessArticle Type I Almost-Homogeneous Manifolds of Cohomogeneity One—IV
Received: 16 November 2018 / Revised: 17 December 2018 / Accepted: 19 December 2018 / Published: 25 December 2018
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Abstract
This paper is one of a series in which we generalize our earlier results on the equivalence of existence of Calabi extremal metrics to the geodesic stability for any type I compact complex almost homogeneous manifolds of cohomogeneity one. In this paper, we [...] Read more.
This paper is one of a series in which we generalize our earlier results on the equivalence of existence of Calabi extremal metrics to the geodesic stability for any type I compact complex almost homogeneous manifolds of cohomogeneity one. In this paper, we actually carry all the earlier results to the type I cases. In Part II, we obtained a substantial amount of new Kähler–Einstein manifolds as well as Fano manifolds without Kähler–Einstein metrics. In particular, by applying Theorem 15 therein, we obtained complete results in the Theorems 3 and 4 in that paper. However, we only have partial results in Theorem 5. In this note, we provide a report of recent progress on the Fano manifolds N n , m when n > 15 and N n , m when n > 4 . We provide two pictures for these two classes of manifolds. See Theorems 1 and 2 in the last section. Moreover, we present two conjectures. Once we solve these two conjectures, the question for these two classes of manifolds will be completely solved. By applying our results to the canonical circle bundles, we also obtain Sasakian manifolds with or without Sasakian–Einstein metrics. These also provide open Calabi–Yau manifolds. Full article
(This article belongs to the Special Issue Applications of Differential Geometry)
Open AccessArticle (L)-Semigroup Sums
Received: 12 November 2018 / Revised: 14 December 2018 / Accepted: 17 December 2018 / Published: 22 December 2018
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Abstract
An (L)-semigroup S is a compact n-manifold with connected boundary B together with a monoid structure on S such that B is a subsemigroup of S. The sum S+T of two (L)-semigroups S and T having boundary B is [...] Read more.
An (L)-semigroup S is a compact n-manifold with connected boundary B together with a monoid structure on S such that B is a subsemigroup of S. The sum S + T of two (L)-semigroups S and T having boundary B is the quotient space obtained from the union of S × { 0 } and T × { 1 } by identifying the point ( x , 0 ) in S × { 0 } with ( x , 1 ) in T × { 1 } for each x in B. It is shown that no (L)-semigroup sum of dimension less than or equal to five admits an H-space structure, nor does any (L)-semigroup sum obtained from (L)-semigroups having an Abelian boundary. In particular, such sums cannot be a retract of a topological group. Full article
(This article belongs to the collection Topological Groups)
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