Compactification Methods Research Articles

Viscosity solutions approach to finite-horizon continuous-time Markov decision process

This paper investigates the optimal control problems for the finite-horizon continuous-time Markov decision processes with delay-dependent control policies. We develop compactification methods in decision processes and show that the existence of optimal policies. Subsequently, through the dynamic programming principle of the delay-dependent control policies, the differential-difference Hamilton-Jacobi-Bellman (HJB) equation in the setting of discrete space is established. Under certain conditions, we give the comparison principle and further prove that the value function is the unique viscosity solution to this HJB equation. Based on this, we show that among the class of delay-dependent control policies, there is an optimal one which is Markovian.

Compactification method in linear programming approach to infinite-horizon optimal control problems with a noncompact state constraint

Compactification method in linear programming approach to infinite-horizon optimal control problems with a noncompact state constraint

From the Optimal Singular Stochastic Control to the Optimal Stopping for Regime-Switching Processes

This work generalizes the connection between the optimal singular stochastic control problem and the optimal stopping problem for regime-switching processes. Via the optimal singular stochastic control, the optimal stopping time and the continuation region are characterized. Moreover, we prove the existence of optimal singular stochastic control for a finite horizon singular control problem with the cost function containing the terminal cost. We prove it directly by the compactification method, which is based on an elaborate application of the properties of probability measures over the càdlàg space. Such a problem was left open in Haussmann and Suo [SIAM J. Control Optim., 33 (1995), pp. 916–936] and [SIAM J. Control Optim., 33 (1995), pp. 937–959]). In addition, our compactification method can remove the convexity condition on the coefficients used in Dufour and Miller [SIAM J. Control Optim., 43 (2004), pp. 708–730].

On a simple model for describing convection of the rotating fluid: Integrability, bifurcations and global dynamics

<p style='text-indent:20px;'>The Glukhovsky-Dolzhansky (GD) model arises naturally from geophysical science, which describes rotating fluid convection inside the ellipsoid. This work aims to provide some new insights into the GD model. (<i>i</i>) We first show that, under some conditions there are homothetic transformations which covert the GD model into other similar quadric physical models, therefore, our results on the GD model can be naturally applied to the investigation of these models. (<i>ii</i>) We propose a complete classification of Darboux polynomials and exponent factors for the GD model, which implies that the GD model has no polynomial, rational, or Darboux first integrals. In addition, some integrable cases of the GD model are also given when the physical parameters are allowed to be non-positive. (<i>iii</i>) The existence of global attractor is proved. The stability and local bifurcations of all co-dimension one and two are investigated. Particularly, we show that the GD model undergoes two dynamical transitions as the Rayleigh number increases. (<i>iv</i>) To understand the asymptotic behavior of the orbits for the GD model, we use the Poincaré compactification method to study its dynamical behavior at infinity. More precisely, we prove that the phase portraits of the GD model at infinity consist of an infinite sequence of periodic solutions and two heteroclinic loops. Our results may help us better understand the complex and rich dynamics of rotating fluid convection.</p>

Global dynamics of a structurally orthotropic stringer shell system

In this paper we are interested in a structurally orthotropic stringer shell system, which can be transformed into a planar reversible polynomial Hamiltonian system via the Galerkin method. Numerous articles have employed numerical methods to study its dynamics, but only partial dynamical behaviors are obtained due to the limitation of numerical methods. To capture all possible dynamical behaviors of this shell system, we resort to the Poincaré compactification method and the homogeneous directional blow up technique in the qualitative theory of dynamical systems. After reducing the original system to a normal form with less parameters, we prove that this shell system exhibits exactly 14 non-equivalent global dynamics. Moreover, the parameter range corresponding to each kind of global dynamics is accurately determined, and bifurcation diagrams are drawn.

Ancient solutions of the homogeneous Ricci flow on flag manifolds

For any flag manifold M=G/K of a compact simple Lie group G we describe non-collapsing ancient invariant solutions of the homogeneous unnormalized Ricci flow. Such solutions emerge from an invariant Einstein metric on M, and by [13] they must develop a Type I singularity in their extinction finite time, and also to the past. To illustrate the situation we engage ourselves with the global study of the dynamical system induced by the unnormalized Ricci flow on any flag manifold M=G/K with second Betti number b2(M) = 1, for a generic initial invariant metric. We describe the corresponding dynamical systems and present non-collapsed ancient solutions, whose α-limit set consists of fixed points at infinity of MG. Based on the Poincaré compactification method, we show that these fixed points correspond to invariant Einstein metrics and we study their stability properties, illuminating thus the structure of the system’s phase space.

Optimal Singular Control Problem in Infinite Horizon for Stochastic Processes with Regime-Switching

We develop the compactification method to prove the existence of an optimal control for multidimensional stochastic processes with regime-switching. We consider separately the situations with and without state constraints. Three kinds of control policies are studied in this work: the classical controls on the drifts, the singular controls, and the controls on the transition rates of switching processes. Based on the existence of the optimal controls, the dynamic programming principle is also established.

Complex dynamics in a quasi-periodic plasma perturbations model

<p style='text-indent:20px;'>In this paper, the complex dynamics of a quasi-periodic plasma perturbations (QPP) model, which governs the interplay between a driver associated with pressure gradient and relaxation of instability due to magnetic field perturbations in Tokamaks, are studied. The model consists of three coupled ordinary differential equations (ODEs) and contains three parameters. This paper consists of three parts: (1) We study the stability and bifurcations of the QPP model, which gives the theoretical interpretation of various types of oscillations observed in [Phys. Plasmas, 18(2011):1-7]. In particular, assuming that there exists a finite time lag <inline-formula><tex-math id="M1">\begin{document}$ \tau $\end{document}</tex-math></inline-formula> between the plasma pressure gradient and the speed of the magnetic field, we also study the delay effect in the QPP model from the point of view of Hopf bifurcation. (2) We provide some numerical indices for identifying chaotic properties of the QPP system, which shows that the QPP model has chaotic behaviors for a wide range of parameters. Then we prove that the QPP model is not rationally integrable in an extended Liouville sense for almost all parameter values, which may help us distinguish values of parameters for which the QPP model is integrable. (3) To understand the asymptotic behavior of the orbits for the QPP model, we also provide a complete description of its dynamical behavior at infinity by the Poincaré compactification method. Our results show that the input power <inline-formula><tex-math id="M2">\begin{document}$ h $\end{document}</tex-math></inline-formula> and the relaxation of the instability <inline-formula><tex-math id="M3">\begin{document}$ \delta $\end{document}</tex-math></inline-formula> do not affect the global dynamics at infinity of the QPP model and the heat diffusion coefficient <inline-formula><tex-math id="M4">\begin{document}$ \eta $\end{document}</tex-math></inline-formula> just yield quantitative, but not qualitative changes for the global dynamics at infinity of the QPP model.

Fan-Gottesman Compactification and Scattered Spaces

Abstract Compactification is the process or result of making a topological space into a compact space. An embedding of a topological space X as a dense subset of a compact space is called a compactification of X. There are a lot of compactification methods but we study with Fan- Gottesman compactification. A topological space X is said to be scattered if every nonempty subset S of X contains at least one point which is isolated in S. Compact scattered spaces are important for analysis and topology. In this paper, we investigate the relation between the Fan-Gottesman compactification of T 3 space and scattered spaces. We show under which conditions the Fan-Gottesman compactification X * is a scattered.

Students’ Mastering of Structural Analysis of Substance as a Method to Form Future Specialists’ Scientific Thinking. Part II

Introduction. In today’s knowledge society, the amount of scientific-applied information, which university graduates have to acquire, continues to increase continuously. There is a concurrent reduction in the number of study hours to undertake educational programmes in order to increase the hours for students’ independent work. Against this background, higher school is required to increase future experts’ competencies. Therefore, the content of fundamental and special disciplines of entire period of training and independent work of students should be thoroughly coordinated by increasing students’ motivation to self-education and self-development. Classroom-based and independent learning of disciplines and sections of fundamental academic courses, especially chemistry, is impossible without formation of students’ scientific thinking. Today, it is difficult to consider the activity of most professionals without the ability to think scientifically: active expansion of science into professional sphere has a strong tendency to be increased.The aim of the present research is to show the possibilities of formation and development of scientific thinking in the students of natural-scientific and technical directions of education using the example of studying of one of the elements of programmes in chemistry (the method of nuclear magnetic resonance (NMR) analysis).Methodology and research methods. The research was carried out on the basis of competency-based, systematic and interdisciplinary approaches. The methods of analysis, synthesis, integration, differentiation and compactification of fundamental knowledge and training material were used.Results and scientific novelty. The high potential of chemical education for formation of scientific thinking, subject content (chemical), natural-scientific and holistic scientific thinking is emphasised. However, chemistry education in higher education institution is complicated by the absence of the unified structure of fundamental preparation, the preservation of extensive approach to the content of chemical disciplines, the irrational organisation of students’ independent work, which now is accounted for a half of instructional time. Overcoming these problems lies in the dialectic unity of fundamental and practice-oriented knowledge, which is provided by the compliance with the principles of continuity and interdisciplinarity. It is necessary to provide deductive structurisation of training material in order to give integrity and systemacity to the content of education, without which it is impossible to create a comprehensive natural-scientific picture of the world in students. The key initial element of vocational training stimulating the formation of reflexive skills and scientific thinking of future experts is mastering by students of a categorical-conceptual framework of science, which is consistently and comprehensively revealed throughout a high school stage of education. The authors designated phases of development of scientific thinking (formal-logical, reflexive-theoretical, hypothetico-deductive thinking), which are not clearly differentiated due to interpenetration and entanglement of their components and identity of thought processes in terms of their speed and quality. However, the allocation of these stages allows to structure and to correct the content of educational material taking into account the characteristics and the level of students’ readiness.From these standpoints, the expediency of more detailed examination of the NMR method is proved within the disciplines such as “Chemistry”, “General Chemistry”, “Inorganic Chemistry” and “Analytical Chemistry” (a part of material about the NMR method can be worked out by students independently). This method, based on one phenomenon, includes hundreds of various types of the experiments, which are intended for receiving particular information. The NMR method is widely used both in scientific research, including master’s thesis, and in the most various manufacturing spheres. Today, the spectroscopy of NMR is recognised as the most powerful informative and perspective method of structural analysis of substance. The fundamental nature, interdisciplinarity and universality of the method provide students with basic professional knowledge on physics, chemistry, medicine, biology, technology and ecology. The authors of the present research propose the option of configuration of educational information on NMR. According to the suggested version, the principle of work is the following: firstly, bachelors study the system of key concepts and terms, moving gradually from formal-logical to substantial generalisations; then, students learn to explain the phenomena scientifically and to make forecasts, and, as a result, they become the “owners” of hypothetico-deductive thinking. The acquired competencies are the key to professional literacy, which is improved in master’s degree programme, when the previously compactified scientific knowledge in a contracted form is developed in the form suitable for an optimal solution of a particular research or practical aim. The similar scheme of vocational training makes it possible to overcome traditional orientation of high school programmes of the natural-science block (i.e. retention of permanently growing amount of factual material).Practical significance. The research materials can be useful for methodologists of the higher school, for experts engaged in methodological development and the organisation of educational process, for high school teachers of chemistry and related disciplines, for post-graduate students and master’s students of chemical and chemico-technological specialties as well.

Students’ Mastering of Structural Analysis of Substance as a Method to Form Future Specialists’ Scientific Thinking. Part I

Introduction. In today’s knowledge society, the amount of scientificapplied information, which university graduates have to acquire, continues to increase continuously. There is a concurrent reduction in the number of study hours to undertake educational programmes in order to increase the hours for students’ independent work. Against this background, higher school is required to increase future experts’ competencies. Therefore, the content of fundamental and special disciplines of entire period of training and independent work of students should be thoroughly coordinated by increasing students’ motivation to self-education and self-development. Classroom-based and independent learning of disciplines and sections of fundamental academic courses, especially chemistry, is impossible without formation of students’ scientific thinking. Today, it is difficult to consider the activity of most professionals without the ability to think scientifically: active expansion of science into professional sphere has a strong tendency to be increased.The aim of the present research is to show the possibilities of formation and development of scientific thinking in the students of natural-scientific and technical directions of education using the example of studying of one of the elements of programmes in chemistry (the method of nuclear magnetic resonance (NMR) analysis).Methodology and research methods. The research was carried out on the basis of competency-based, systematic and interdisciplinary approaches. The methods of analysis, synthesis, integration, differentiation and compactification of fundamental knowledge and training material were used.Results and scientific novelty. The high potential of chemical education for formation of scientific thinking, subject content (chemical), natural-scientific and holistic scientific thinking is emphasised. However, chemistry education in higher education institution is complicated by the absence of the unified structure of fundamental preparation, the preservation of extensive approach to the content of chemical disciplines, the irrational organisation of students’ independent work, which now is accounted for a half of instructional time. Overcoming these problems lies in the dialectic unity of fundamental and practice-oriented knowledge, which is provided by the compliance with the principles of continuity and interdisciplinarity. It is necessary to provide deductive structurisation of training material in order to give integrity and systemacity to the content of education, without which it is impossible to create a comprehensive natural-scientific picture of the world in students. The key initial element of vocational training stimulating the formation of reflexive skills and scientific thinking of future experts is mastering by students of a categoricalconceptual framework of science, which is consistently and comprehensively revealed throughout a high school stage of education. The authors designated phases of development of scientific thinking (formal-logical, reflexive-theoretical, hypotheticodeductive thinking), which are not clearly differentiated due to interpenetration and entanglement of their components and identity of thought processes in terms of their speed and quality. However, the allocation of these stages allows to structure and to correct the content of educational material taking into account the characteristics and the level of students’ readiness. From these standpoints, the expediency of more detailed examination of the NMR method is proved within the disciplines such as “Chemistry”, “General Chemistry”, “Inorganic Chemistry” and “Analytical Chemistry” (a part of material about the NMR method can be worked out by students independently). This method, based on one phenomenon, includes hundreds of various types of the experiments, which are intended for receiving particular information. The NMR method is widely used both in scientific research, including master’s thesis, and in the most various manufacturing spheres. Today, the spectroscopy of NMR is recognised as the most powerful informative and perspective method of structural analysis of substance. The fundamental nature, interdisciplinarity and universality of the method provide students with basic professional knowledge on physics, chemistry, medicine, biology, technology and ecology. The authors of the present research propose the option of configuration of educational information on NMR. According to the suggested version, the principle of work is the following: firstly, bachelors study the system of key concepts and terms, moving gradually from formal-logical to substantial generalisations; then, students learn to explain the phenomena scientifically and to make forecasts, and, as a result, they become the “owners” of hypothetico-deductive thinking. The acquired competencies are the key to professional literacy, which is improved in master’s degree programme, when the previously compactified scientific knowledge in a contracted form is developed in the form suitable for an optimal solution of a particular research or practical aim. The similar scheme of vocational training makes it possible to overcome traditional orientation of high school programmes of the natural-science block (i.e. retention of permanently growing amount of factual material).Practical significance. The research materials can be useful for methodologists of the higher school, for experts engaged in methodological development and the organisation of educational process, for high school teachers of chemistry and related disciplines, for post-graduate students and master’s students of chemical and chemico-technological specialties as well.

An analytic study of the Vaidyanathan chaotic dynamics in Lorentzian metric which has no similar dynamics in Riemannian metric

In this paper we investigate behaviour at infinity of a physical 3-dimensional chaotic system via Poincare compactification method. This system has been introduced by Vaidyanathan et al. (2017a). We plot the phase portrait of the system for parameters a and b which appear in the nonlinear part of the system. We will see a set of non-isolated singular points at infinity is a hyperbolic set by considering a Lorentzian metric g on R2 and it is not a hyperbolic set in the sense of Riemannian metrics. We compute a first integral for the resulted system and we prove there is at most a generalised rational first integral when one of its parameters is equal to zero.

Cosmological dynamics of brane gravity: A global dynamical system perspective

The braneworld model of gravity is well-known for several notable cosmological features such as self-acceleration originating from a geometric and not matter source, effective dark energy behavior with phantom characteristics but not leading to a Big-Rip singularity, rough resemblance to the $\Lambda$CDM evolution, etc. The dynamical system tools usually allow us to obtain generic conclusions on the global dynamics of a system over a wide range of initial conditions. With this motivation, in order to recover the important features of the braneworld model from a more global perspective, here, we investigate the global cosmological dynamics of the braneworld model using dynamical system techniques. We first analyze the case where there is just a normal matter on the brane and then extend the analysis to the case with an extra scalar field also trapped on the brane. In the presence of a scalar field, potentials belonging to different classes are considered. The stability behavior of critical points is examined using linear stability analysis and when necessary center manifold theory as well as numerical perturbation techniques are also used. To understand the global dynamics of a dynamical system, we utilized the Poincar\'e compactification method to capture the properties of all possible critical points. Applying dynamical system analysis, we found that brane gravity is consistent with observed actions of the Universe. In particular, our analysis shows that important cosmological behaviors like the long-lasting matter-dominated era, late time acceleration as well as the avoidance of Big-Rip singularity can be realized in brane gravity for a wide range of initial conditions.

A model to study finite-size and magnetic effects on the phase transition of a fermion interacting system

We present a model to study the effects from external magnetic field, chemical potential and finite size on the phase structures of a massive four- and six-fermion interacting systems. These effects are introduced by a method of compactification of coordinates, a generalization of the standard Matsubara prescription. Through the compactification of the z-coordinate and of imaginary time, we describe a heated system with the shape of a film of thickness L, at temperature [Formula: see text] undergoing first- or second-order phase transition. We have found a strong dependence of the temperature transition on the coupling constants [Formula: see text] and [Formula: see text]. Besides inverse magnetic catalysis and symmetry breaking for both kinds of transition, we have found an inverse symmetry breaking phenomenon with respect to first-order phase transition.

The Chow Motives of Relative Fulton-Macpherson Space

Suppose that $X$ is a complex nonsingular projective variety and $D$ is a smooth divisor. Compactifications of configuration spaces of distinct and non-distinct $n$ points in $X$ away from $D$ were constructed by the author and B. Kim in "A generalization of Fulton-MacPherson configuration spaces" by using the method of wonderful compactification. In this paper, we give explicit presentations of Chow motives and Chow groups of these configuration spaces.

Divergence of solutions of polynomial finite-difference equations

A theorem is proven for kth-order polynomial finite-difference equations that guarantees the divergence of solutions. A ‘basin of divergence’ is characterized and an order of divergence is provided. The basin of divergence is shown to depend on k independent parameters. An unconventional compactification method is used. Applications include the multi-step method in the numerical integration of ordinary differential equations, quadratic equations and the Henon map.

Finite-size effects on the phase diagram of difermion condensates in two-dimensional four-fermion interaction models

We investigate finite-size effects on the phase structure of chiral and difermion condensates at finite temperature and density in the framework of the two-dimensional large-$N$ Nambu-Jona-Lasinio model. We take into account size-dependent effects by making use of $zeta$-function and compactification methods. The thermodynamic potential and the gap equations for the chiral and difermion condensed phases are then derived in the mean-field approximation. Size-dependent critical lines separating the different phases are obtained considering anti-periodic boundary conditions for the spatial coordinate.

A CHARACTERIZATION OF COMPACTIFICATIONS OF LOCAL COMPACT HAUSDORFF SPACE

It is known that different compactification methods are applied to different spaces. In this paper, it is examined some relations among spaces obtained by being applied different compactification methods (One-point compactification, Stone–Cech compactification, Wallman compactification, Fan–Gottesman compactification) to local compact Hausdorff space and these compactifications are characterized with the inequality αX ≅ wX ≅ X* ≤ ⋯ ≤ βX by the addition of lindelöf conditions to studied space. Moreover, it is found maximum and minimum element of the class of all compactifications of the space. We also give an example to illustrate how to apply mathematical result to an application of compactification in physics.

Fan-Gottesman compactification of some specific spaces is Wallman-type compactification

It is known that some compactification methods have similar properties when studied spaces are local compact hausdorff. But this compactification methods (one-point compactification, Stone-Cech compactification, Wallman compactification, Fan-Gottesman compactification, Freudenthal compactification, etc.) have very few similarity in some specific spaces. In this paper especially it is investigated relation between Fan-Gottesman and Wallman compactifications and it is shown that Fan-Gottesman compactification of some interesting and specific spaces such as normal A 2 and T 4 is Wallman-type compactification.

Global nonexistence of solutions to a semilinear wave equation in the Minkowski space

This work presents the finite-time blow-up of solutions to the equation u t t − Δ u = a − k | u | p , in the Minkowski space. We extend the previous result of Belchev, Kepka and Zhou [E. Belchev, M. Kepka, Z. Zhou, Finite-time blow-up of solutions to semilinear wave equations, J. Funct. Anal. 190 (1) (2002) 233–254] comprehensively. Due to a modification of the so-called method of conformal compactification used by Belchev, Kepka and Zhou, we show that the solutions blow up in finite time with more relaxed initial data and extended index p .