Mixed-type System Research Articles

Retraction Note: Second-order impulsive differential systems of mixed type: oscillation theorems

Retraction Note: Second-order impulsive differential systems of mixed type: oscillation theorems

Incremental Learning Based on Granular Ball Rough Sets for Classification in Dynamic Mixed-Type Decision System

Granular computing, a new paradigm for solving large-scale and complex problems, has made significant progresses in knowledge discovery. Granular ball computing (GBC) is a novel granular computing method, which can rapidly generate scalable and robust information granules, that is, granular balls. However, a comprehensive index for measuring the performance of a granular ball does not exist. Furthermore, GBC lacks a mechanism to deal with dynamic decision systems. Therefore, in this study, the quality index of a granular ball is first formulated. Next, with this index, a novel granular ball rough sets model (GBRS) based on GBC is proposed. GBRS is more conducive to learning knowledge from uncertain datasets and more suited to incremental learning than the latest granular ball neighborhood rough sets model based on GBC. Subsequently, an incremental mechanism is introduced into GBRS, and two incremental learning models are developed for objects increasing in stream patterns and batch patterns, respectively. In the incremental learning process, three patterns of granular balls, that is, update, fusion, and split, were well studied when a set of objects was added to the decision system. Finally, to verify the effectiveness and efficiency, we apply GBRS and these two incremental learning models into classification tasks. Compared with four current state-of-the-art classification methods based on granular computing and four classical classifiers in machine learning, the proposed classifiers in this paper achieve higher classification accuracy as well as better efficiency on benchmark datasets.

The Shock Waves for a Mixed-Type System from Chemotaxis

The Shock Waves for a Mixed-Type System from Chemotaxis

Second-order impulsive differential systems of mixed type: oscillation theorems

In this paper, we obtain necessary and sufficient conditions for the oscillation of solutions to a second-order neutral differential equation with impulses. Two examples are provided to show the effectiveness and feasibility of the main results. Our main tool is Lebesgue’s dominated convergence theorem.

Impulsive Functional-Controllability Problem for Fractional Integro-Differential Evolution Systems of Mixed Type with the Measure of Noncompactness

Impulsive Functional-Controllability Problem for Fractional Integro-Differential Evolution Systems of Mixed Type with the Measure of Noncompactness

НЕКОТО РЫЕ ВОПРОСЫ СТАТ ИКИ ГЕОМЕТРИЧЕСКИ И ФИЗИЧЕСКИ НЕЛИНЕЙНОЙ СПЛОШНОЙ СРЕДЫ

We consider the generalized plane deformation of physically and geometrically nonlinear continuous medium. The mathematical model of medium is described by physically nonlinear relations in the form of arbitrary cross dependences between the first invariants of tensors and the second invariants of stress and strain deviators, taking into account geometrical nonlinearity (according to V.V. Novozhilov). The system of resolving partial differential equations of equilibrium, done in displacements, is quasilinear. This type is being investigated. Despite the fact that the type of the system of quasilinear partial differential equations in a certain part of space can be determined only for a specific solution, it is stated that in the general case the system of differential equations of equilibrium of generalized plane deformation is a system of mixed type. The type of the system of partial differential equations in a given area of space is completely determined by the values of its coefficients, and therefore depends both on the value of physical constants of continuous medium material and also on the value of derivatives of displacements in spatial coordinates.

Semi-analytic treatment of mixed hyperbolic–elliptic Cauchy problem modeling three-phase flow in porous media

This work provides a technical applied description of the residual power series method (RPSM) to develop a fast and accurate algorithm for mixed hyperbolic–elliptic systems of conservation laws with Riemann initial datum. The RPSM does not require discretization, reduces the system to an explicit system of algebraic equations and consequently of massive and complex computations, and provides the solution in a form of Taylor power series expansion of closed-form exact solution (if exists). Theoretically, convergence hypotheses are discussed, and error bounds of exponential rates are derived. Numerically, the convergence and stability of approximate solutions are achieved for systems of mixed type. The reported results, with application to general Cauchy problems, which rise in diverse branches of physics and engineering, reveal the reliability, efficiency, and economical implementation of the proposed algorithm for handling nonlinear partial differential equations in applied mathematics.

Mass transport in multicomponent compressible fluids: Local and global well-posedness in classes of strong solutions for general class-one models

We consider a system of partial differential equations describing mass transport in a multicomponent isothermal compressible fluid. The diffusion fluxes obey the Fick–Onsager or Maxwell–Stefan closure approach. Mechanical forces result into one single convective mixture velocity, the barycentric one, which obeys the Navier–Stokes equations. The thermodynamic pressure is defined by the Gibbs–Duhem equation. Chemical potentials and pressure are derived from a thermodynamic potential, the Helmholtz free energy, with a bulk density allowed to be a general convex function of the mass densities of the constituents.The resulting PDEs are of mixed parabolic–hyperbolic type. We prove two theoretical results concerning the well-posedness of the model in classes of strong solutions: 1. The solution always exists and is unique for short-times and 2. If the initial data are sufficiently near to an equilibrium solution, the well-posedness is valid on arbitrary large, but finite time intervals. Both results rely on a contraction principle valid for systems of mixed type that behave like the compressible Navier–Stokes equations. The linearised parabolic part of the operator possesses the self map property with respect to some closed ball in the state space, while being contractive in a lower order norm only. In this paper, we implement these ideas by means of precise a priori estimates in spaces of exact regularity.

Ill-posed boundary value problem for mixed type system equations with two degenerate lines

Ill-posed boundary value problem for mixed type system equations with two degenerate lines

Logarithmic stability of an inverse problem for Biot’s consolidation system in poro-elasticity

In this paper, we consider a coupled system of mixed hyperbolic-parabolic type, which describes Biot’s consolidation model in poro-elasticity. We study an inverse problem of determining five spatially varying coefficients in the model, i.e. two Lamé coefficients, the secondary consolidation effects and two densities, by three measurements of displacement in an arbitrary subboundary and temperature in an arbitrary neighborhood of the boundary over a time interval. By assuming that, in a neighborhood of the boundary of the spatial domain, the densities, secondary consolidation effects and the Lamé coefficients are known, we prove a logarithmic stability estimate for the inverse problem.

An analytical approach for the three-dimensional water entry problem of solids creating elliptical contact regions

The purpose of the present study is to approach analytically the water entry problem of solids creating elliptical contact regions with the free surface during impact. The governing boundary value problem has been linearized using asymptotic analysis and considering the early stages of the phenomenon. The fundamental assumption is made that the contact region between the solid and the free surface is always an ellipse. This allows to consider the underlying boundary value problem in elliptical coordinates exploiting the properties of elliptical harmonics. It is shown that for elliptical contact regions the solution sought can be detached from the angular dependence. This finding is extremely beneficial as it allows the derivation of a perturbation sequence of mathematical systems of mixed type, involving Fredholm integral equations, which can be treated using rigorous methods of solution for mixed boundary value problems in potential theory. The theory developed for elliptical contact regions during water entry situations, is accordingly applied for anticipating the details of the slamming phenomenon during the free-surface impact of a nonaxisymmetric prolate spheroid assuming that the contact line between the liquid and the solid is always an ellipse. The solid is assumed to descend steadily, without rotating, towards an initially undisturbed free surface. The perturbation technique requires that the slenderness parameter is small compared to unity, while slenderness affects significantly the hydrodynamic load exerted on the solid during water entry. Numerical results are presented and discussed using both the von Karman and the Wagner models of water impact. Finally, the study was extended to consider the 3D Wagner water impact problem of an elliptic paraboloid and to compare computations against available data reported in the past.

Ein als „Präsidialsystem“ verkleidetes (autokratisch-)parlamentarisches Regierungssystem: Die Präsidentschafts- und Parlamentswahlen in der Türkei vom 24. Juni 2018

On the 24th of June 2018, for the first time Turkish citizens voted in separate elections for both parliament and the head of government . The constitutional amendments passed a year earlier in a controversial referendum became effective at the same time and transformed the parliamentary system of government into a so-called presidential system . However, regarding the relationship between the executive and legislative branches, the constitution reveals strong similarities to an Israeli experiment (1996 to 2002), which conversely was described as a mixed type or quasi-parliamentary system . The “alla Turca” variant presents a clear asymmetry concerning the horizontal separation of powers in favor of the president and creates a concentration of powers . Parliament is rigorously constrained in its powers, while the president permanently remains capable of acting unless an arithmetic three-fifths majority is formed that could recall him prematurely for political reasons . Under the undemocratic conditions of a semi-competitive electoral system and in a state of emergency that applied at the time, the AKP won both elections . [ZParl, vol . 50 (2019), no . 4, pp . 791 - 809]

Structure, size, and statistical properties of chaotic components in a mixed-type Hamiltonian system.

We perform a detailed study of the chaotic component in mixed-type Hamiltonian systems on the example of a family of billiards [introduced by Robnik in J. Phys. A: Math. Gen. 16, 3971 (1983)JPHAC50305-447010.1088/0305-4470/16/17/014]. The phase space is divided into a grid of cells and a chaotic orbit is iterated a large number of times. The structure of the chaotic component is discerned from the cells visited by the chaotic orbit. The fractal dimension of the border of the chaotic component for various values of the billiard shape parameter is determined with the box-counting method. The cell-filling dynamics is compared to a model of uncorrelated motion, the so-called random model [Robnik et al. J.Phys. A: Math. Gen. 30, L803 (1997)JPHAC50305-447010.1088/0305-4470/30/23/003], and deviations attributed to sticky objects in the phase space are found. The statistics of the number of orbit visits to the cells is analyzed and found to be in agreement with the random model in the long run. The stickiness of the various structures in the phase space is quantified in terms of the cell recurrence times. The recurrence time distributions in a few selected cells as well as the mean and standard deviation of recurrence times for all cells are analyzed. The standard deviation of cell recurrence time is found to be a good quantifier of stickiness on a global scale. Three methods for determining the measure of the chaotic component are compared and the measure is calculated for various values of the billiard shape parameter. Lastly, the decay of correlations and the diffusion of momenta is analyzed.

Interactions of Composite Waves of the Two-Dimensional Full Euler Equations for Van Der Waals Gases

This paper studies two types of wave interactions arising from two-dimensional (2D) Riemann problems for the full Euler equations: interaction of jump-fan composite waves and interaction of fan-jump-fan composite waves, which are absent in polytropic gases, but appear in van der Waals gases. Because of the occurrence of shock waves, the flow considered here is unsteady, nonisentropic, and rotational. In order to construct the solutions to these wave interaction problems, we consider two types of discontinuous Goursat problems for a 2D self-similar Euler system. Global piecewise smooth solutions to these discontinuous Goursat problems are obtained by the method of characteristics. The 2D self-similar Euler system is a mixed-type system, and the type in each point is determined by the local pseudo-Mach number. So, one of the main difficulties for the global existence is that hyperbolicity is hard to control in the composite wave interaction zones. By constructing some invariant regions of wave characteristic angles, we prove that the system is uniformly hyperbolic in these zones. In order to obtain the a priori gradient estimates of solutions to these discontinuous Goursat problems, we derive some characteristic decompositions for the 2D self-similar full Euler system. The solutions constructed in this paper can be used as building blocks of solutions to 2D Riemann problems.

Integro-differential systems of mixed type involving higher order fractional Laplacian

ABSTRACTIn this paper, we deal with a new class of non-local operators that we term integro-differential systems of mixed type. We study the behaviour of solutions of this system when the diffusion term involves higher order fractional powers of the Laplacian. Moreover, we prove that the solution of the system decays faster than a power with an exponent given by the smallest index of the fractional power of the Laplacian.

Practical Stability of the “Cross” Scheme in the Numerical Integration of Dynamic Equations for Flexible Thin-Walled Structural Elements Obeying the Hypotheses of the Timoshenko Theory

In the von Karman approximation, we formulate the initial boundary-value problem of the dynamics of flexible isotropic and composite elastic beams-walls within the framework of two versions of the Timoshenko theory. We perform a qualitative analysis of the resolving system of equations of motion. It is shown that, in the geometrically linear statement, the dynamics of elastic beams is described by a hyperbolic system. At the same time, in the case of deformation of flexible beams, the system of resolving motion equations may change its type degenerating from a hyperbolic system into a system of mixed (composite) type. We develop finite-difference and variation-difference versions of the explicit (in time) “cross” scheme for the numerical integration of the posed initial boundary-value problems. On the basis of these numerical methods, we perform the numerical analyses of the dynamic flexural deformation of flexible metallic and composite beams under explosive-type loads. The result of these calculations demonstrate that, in almost cases, one can indicate the levels of loading of flexible beams under which the “cross” scheme becomes unstable, although the condition of stability obtained in the linear approximation is satisfied with a considerable margin. Thus, it is shown that, in the case of dynamic analysis of flexible beams, one can speak only about the practical stability of the “cross” scheme but not about its conditional stability.

R&D-based Calibrated Growth Models with Finite-Length Patents: A Novel Relaxation Algorithm for Solving an Autonomous FDE System of Mixed Type

The statutory patent length is 20 years in most countries. R&D-based growth models, however, often presume an infinite patent length. In this paper, finite-length patents are embedded in a non-scale R&D-based growth model, while allowing any patent's effective life to be terminated prematurely, subject to two idiosyncratic hazards from imitation and creative destruction. This gives rise to an autonomous system of mixed-type functional differential equations (FDEs) that had never been encountered in the growth literature. Its dynamics are driven by current, delayed and advanced states. We present a relaxation algorithm to solve these FDEs by solving a sequence of standard boundary value problems for systems of ordinary differential equations. We use this algorithm to simulate a calibrated U.S. economy's transitional dynamics by making discrete changes from the baseline 20 years patent length. We find that if transitional impacts are taken into account, the switch to the long-run optimal patent length can incur a welfare loss, albeit rather small.

THE NATIONAL INNOVATION SYSTEM IN RUSSIA: CONDITIONS OF FUNC-TIONING

The article covers the problem actuality of creating the national innovation system in Russia. Тhe national innovation system is considered as a factor in international competitiveness and a basis for effective economic development. The authors examine the genesis of the concept of the national innovation system and its structure. We consider a comprehensive approach to the creation of the national innovation system in the re-search of some Russian authors. We present statistics on the level of innovation activity in Russia and compare them with those in developed economies. We describe several approaches to establish an effective national innovation system. We describe a well-known in world practice types of innovation development: innovation and imitation. The author conclude that in Russia may create a national innovation system of mixed type.

Some New Generalized Retarded Gronwall-Like Inequalities and Their Applications in Nonlinear Systems

The Gronwall inequalities are of significance in mathematics and engineering. This paper generalizes the Gronwall-like inequalities from different perspectives. Using the proposed inequalities, the difficulties to discuss the controllability of integrodifferential systems of mixed type can be solved. Meanwhile, two examples as their applications are also given to show the effectiveness of our main results.

On the expansion of a wedge of van der Waals gas into a vacuum

We study the existence of global in time classical solution to the expansion of a wedge of van der Waals gas into a vacuum. We reduce this problem to a Goursat-type boundary value problem for 2D self-similar Euler system. The 2D self-similar Euler system is a mixed type system, the type in each point is determined by the local pseudo-Mach number. By introducing the concept of invariant square region of solution, we prove that this system is strictly hyperbolic in the flow region of the Goursat problem. A prior C1 estimate of the solution to the Goursat problem is obtained by using the method of characteristic decomposition. Due to the existence of vacuum boundary, the classical approach to extend local solution to global solution does not work here. We extend the local solution of the Goursat problem up to the interface of gas with vacuum by solving many “small” Goursat problems in each extension step.