Analytical and Qualitative Study of Some Families of FODEs via Differential Transform Method

This current work is devoted to develop qualitative theory of existence of solution to some families of fractional order differential equations (FODEs). For this purposes we utilize fixed point theory due to Banach and Schauder. Further using differential transform method (DTM), we also compute analytical or semi-analytical results to the proposed problems. Also by some proper examples we demonstrate the results.

Topological Invariants of Vapor–Liquid, Vapor–Liquid–Liquid and Liquid–Liquid Phase Diagrams

The study of topological invariants of phase diagrams allows for the development of a qualitative theory of the processes being researched. Studies of the properties of objects in the same equivalence class may be carried out with the aim of predicting the properties of unexplored objects from this class, or predicting the behavior of a whole system. This paper describes a number of topological invariants in vapor–liquid, vapor–liquid–liquid and liquid–liquid equilibrium diagrams. The properties of some invariants are studied and illustrated. It is shown that the invariant of a diagram with a miscibility gap can be used to distinguish equivalence classes of phase diagrams, and that the balance equation of the singular-point indices, based on the Euler characteristic, may be used to analyze the binodal-surface structure of a quaternary system.

The acyclicity of a quadratic differential system

Дан краткий обзор некоторых основных публикаций, посвященных исследованию вопроса о предельных циклах и сепаратрисах квадратичных дифференциальных систем. Рассмотрено наличие замкнутых траекторий для определенного класса автономных квадратичных систем на плоскости. Доказательство основано на применении теории прямых изоклин, признаков Дюлака и Бендиксона качественной теории дифференциальных уравнений. Предложенное доказательство покрывает результаты известной работы Л.А. Черкаса и Л.С. Жилевич. We now give a brief overview of some of the main publications devoted to the study of the question of limit cycles and separatrices of quadratic differential systems. In this paper, we consider the existence of closed trajectories for a certain class of autonomous quadratic systems on the plane. The proof is based on the application of the theory of straight line isoclines, Dulac and Bendixon criteria of the qualitative theory of differential equations. The proposed proof covers the results of the well-known work of L.A. Cherkas and L.S. Zhilevich.

A weight-homogenous condition to the real Jacobian conjecture in

It is known that a polynomial local diffeomorphism $(f,\, g): {\mathbb {R}}^{2} \to {\mathbb {R}}^{2}$ is a global diffeomorphism provided the higher homogeneous terms of $f f_x+g g_x$ and $f f_y+g g_y$ do not have real linear factors in common. Here, we give a weight-homogeneous framework of this result. Our approach uses qualitative theory of differential equations. In our reasoning, we obtain a result on polynomial Hamiltonian vector fields in the plane, generalization of a known fact.

Global Dynamics of the Hořava–Lifshitz Cosmological Model in a Non-Flat Universe with Non-Zero Cosmological Constant

When the cosmological constant is non-zero, the dynamics of the cosmological model based on Hořava–Lifshitz gravity in a non-flat universe are characterized by using the qualitative theory of differential equations.

Qualitative Theory of Two-Dimensional Polynomial Dynamical Systems

A qualitative theory of two-dimensional quadratic-polynomial integrable dynamical systems (DSs) is constructed on the basis of a discriminant criterion elaborated in the paper. This criterion enables one to pick up a single parameter that makes it possible to identify all feasible solution classes as well as the DS critical and singular points and solutions. The integrability of the considered DS family is established. Nine specific solution classes are identified. In each class, clear types of symmetry are determined and visualized and it is discussed how transformations between the solution classes create new types of symmetries. Visualization is performed as series of phase portraits revealing all possible catastrophic scenarios that result from the transition between the solution classes.

Sharp asymptotics in a fractional Sturm-Liouville problem

The current research of fractional Sturm-Liouville boundary value problems focuses on the qualitative theory and numerical methods, and much progress has been recently achieved in both directions. The objective of this paper is to explore a different route, namely, construction of explicit asymptotic approximations for the solutions. As a study case, we consider a problem with left and right Riemann-Liouville derivatives, for which our analysis yields asymptotically sharp estimates for the sequence of eigenvalues and eigenfunctions.

A Qualitative Theory of Cognitive Attitudes and their Change

Abstarct We present a general logical framework for reasoning about agents’ cognitive attitudes of both epistemic type and motivational type. We show that it allows us to express a variety of relevant concepts for qualitative decision theory including the concepts of knowledge, belief, strong belief, conditional belief, desire, conditional desire, strong desire, and preference. We also present two extensions of the logic, one by the notion of choice and the other by dynamic operators for belief change and desire change, and we apply the former to the analysis of single-stage games under incomplete information. We provide sound and complete axiomatizations for the basic logic and for its two extensions.

Introspection, Adverbialism, and Rich Contents

The qualitative view is distinguished from sense-datum theories. Introspection is distinguished from attention. An account of sensory introspection is developed. A new theory of phenomenal concepts is outlined. The qualitative theory is compared with adverbialism and shown not to be subject to the same criticisms. The qualitative theory is shown to transform the debate about rich contents and allow a wide range of such contents.

The Metaphysics of Sensory Experience

What are the materials of conscious perceptual experience? What is going on when we are consciously aware of a visual scene, or hear sounds, or otherwise enjoy sensory experience? In The Metaphysics of Sensory Experience David Papineau exposes the flaws in contemporary answers to this central philosophical question and defends a new alternative. Contemporary theories of perceptual experience all hold that conscious experiences reach out into the world beyond the mind. According to naïve realism, experiences literally incorporate perceived facts, while representationalism holds that experiences contain ordinary properties of the kind possessed by physical objects. These ideas might seem attractive at first sight, but Papineau shows that they do not stand up to examination. Instead Papineau argues for a purely qualitative account of sensory experience. Conscious sensory experiences are intrinsic states of people with no essential connection to external circumstances or represented properties. This might run counter to initial intuition, but Papineau shows that it is the only view that fits the facts. He develops this qualitative theory in detail, showing how it can accommodate the rich structure of sensory experience. Papineau’s qualitative account has respectable antecedents in the history of philosophy, and is also probably the view adopted by most non-specialists, be they reflective high school students, practising neuroscientists, or philosophers working outside the philosophy of perception. By placing the qualitative theory on a firm footing, Papineau shows all those curious about experience that they need not be restricted to the options on the contemporary philosophical menu.