Proof of the stability results

AbstractThe morphological and compositional instabilities in the heteroepitaxial strained alloy films have attracted intense interest from both experimentalists and theorists. To understand the mechanisms and properties for the generation of instabilities, we have developed a nonequilibrium, continuum model for the dislocation-free and coherent film systems. The early evolution processes of surface pro.les for both growing and postdeposition (non-growing) thin alloy films are studied through a linear stability analysis. We consider the coupling between top surface of the film and the underlying bulk, as well as the combination and interplay of different elastic effects. These e.ects are caused by filmsubstrate lattice misfit, composition dependence of film lattice constant (compositional stress), and composition dependence of both Young's and shear elastic moduli. The interplay of these factors as well as the growth temperature and deposition rate leads to rich and complicated stability results. For both the growing.lm and non-growing alloy free surface, we determine the stability conditions and diagrams for the system. These show the joint stability or instability for film morphology and compositional pro.les, as well as the asymmetry between tensile and compressive layers. The kinetic critical thickness for the onset of instability during.lm growth is also calculated, and its scaling behavior with respect to misfit strain and deposition rate determined. Our results have implications for real alloy growth systems such as SiGe and InGaAs, which agree with qualitative trends seen in recent experimental observations.

Download Full-text

Analysis of dengue model with fractal-fractional Caputo–Fabrizio operator

Advances in Difference Equations ◽

10.1186/s13662-020-02881-w ◽

2020 ◽

Vol 2020 (1) ◽

Cited By ~ 1

Author(s):

Fatmawati ◽

Muhammad Altaf Khan ◽

Cicik Alfiniyah ◽

Ebraheem Alzahrani

Keyword(s):

Statistical Data ◽

Dengue Infection ◽

Fractional Operator ◽

Novel Approach ◽

Reduction Number ◽

Best Fitting ◽

The Stability ◽

Parameter Values ◽

Graphical Illustration ◽

Stability Results

AbstractIn this work, we study the dengue dynamics with fractal-factional Caputo–Fabrizio operator. We employ real statistical data of dengue infection cases of East Java, Indonesia, from 2018 and parameterize the dengue model. The estimated basic reduction number for this dataset is $\mathcal{R}_{0}\approx2.2020$ R 0 ≈ 2.2020 . We briefly show the stability results of the model for the case when the basic reproduction number is $\mathcal{R}_{0} <1$ R 0 < 1 . We apply the fractal-fractional operator in the framework of Caputo–Fabrizio to the model and present its numerical solution by using a novel approach. The parameter values estimated for the model are used to compare with fractal-fractional operator, and we suggest that the fractal-fractional operator provides the best fitting for real cases of dengue infection when varying the values of both operators’ orders. We suggest some more graphical illustration for the model variables with various orders of fractal and fractional.

Download Full-text

Boundary Stabilization of a Semilinear Wave Equation with Variable Coefficients under the Time-Varying and Nonlinear Feedback

Abstract and Applied Analysis ◽

10.1155/2014/728760 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6 ◽

Cited By ~ 1

Author(s):

Bei Gong ◽

Xiaopeng Zhao

Keyword(s):

Wave Equation ◽

Riemannian Geometry ◽

Nonlinear Term ◽

Variable Coefficients ◽

Time Varying ◽

Nonlinear Feedback ◽

Boundary Stabilization ◽

Semilinear Wave Equation ◽

The Stability ◽

Stability Results

We study the boundary stabilization of a semilinear wave equation with variable coefficients under the time-varying and nonlinear feedback. By the Riemannian geometry methods, we obtain the stability results of the system under suitable assumptions of the bound of the time-varying term and the nonlinearity of the nonlinear term.

Download Full-text

On the probabilistic stability of some functional equations

Carpathian Journal of Mathematics ◽

10.37193/cjm.2013.01.01 ◽

2013 ◽

Vol 29 (1) ◽

pp. 125-132

Author(s):

CLAUDIA ZAHARIA ◽

◽

DOREL MIHET ◽

Keyword(s):

Fixed Point ◽

Functional Equations ◽

Fixed Point Theory ◽

Point Theory ◽

Quadratic Functional ◽

Normed Spaces ◽

Stability Of Functional Equations ◽

The Stability ◽

Stability Results

We establish stability results concerning the additive and quadratic functional equations in complete Menger ϕ-normed spaces by using fixed point theory. As particular cases, some theorems regarding the stability of functional equations in β - normed and quasi-normed spaces are obtained.

Download Full-text

On Cauchy’s Interlacing Theorem and the Stability of a Class of Linear Discrete Aggregation Models Under Eventual Linear Output Feedback Controls

Symmetry ◽

10.3390/sym11050712 ◽

2019 ◽

Vol 11 (5) ◽

pp. 712 ◽

Cited By ~ 2

Author(s):

Manuel De la Sen

Keyword(s):

Dynamic System ◽

Output Feedback ◽

Iteration Step ◽

Stability And Convergence ◽

Feedback Controls ◽

Time Aggregation ◽

The Stability ◽

The Given ◽

Stability Results

This paper links the celebrated Cauchy’s interlacing theorem of eigenvalues for partitioned updated sequences of Hermitian matrices with stability and convergence problems and results of related sequences of matrices. The results are also applied to sequences of factorizations of semidefinite matrices with their complex conjugates ones to obtain sufficiency-type stability results for the factors in those factorizations. Some extensions are given for parallel characterizations of convergent sequences of matrices. In both cases, the updated information has a Hermitian structure, in particular, a symmetric structure occurs if the involved vector and matrices are complex. These results rely on the relation of stable matrices and convergent matrices (those ones being intuitively stable in a discrete context). An epidemic model involving a clustering structure is discussed in light of the given results. Finally, an application is given for a discrete-time aggregation dynamic system where an aggregated subsystem is incorporated into the whole system at each iteration step. The whole aggregation system and the sequence of aggregated subsystems are assumed to be controlled via linear-output feedback. The characterization of the aggregation dynamic system linked to the updating dynamics through the iteration procedure implies that such a system is, generally, time-varying.

Download Full-text

Complexity factor for dynamical spherically symmetric fluid distributions in f(R) gravity

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887819501706 ◽

2019 ◽

Vol 16 (11) ◽

pp. 1950170

Author(s):

H. Nazar ◽

G. Abbas

Keyword(s):

Physical Nature ◽

The Self ◽

Fluid Distribution ◽

Spherically Symmetric ◽

Static Case ◽

System Configuration ◽

Prior Investigation ◽

Text Model ◽

The Stability ◽

Stability Results

In this study, we analyze the complexity factor that is extended up to the dynamical spherically symmetric non-static case with anisotropic dissipative self-gravitating fluid distribution in context of [Formula: see text] theory of gravity. For this evaluation we choose the particular [Formula: see text] model that signifies the physical nature of the self-gravitating system. The proposed work discusses not only the complexity factor of the structure of the fluid distribution, but also defines the minimization rate of complexity of the pattern of evolution. Here, first we have applied similar approach for obtaining the structure scalar [Formula: see text] of the complexity factor as used for in the static case, and next we have described explicitly the dissipative and non-dissipative cases by assuming the simplest pattern of evolution (homologous condition). It has been found that the system configuration fulfills the vanishing condition of complexity factor and emerging homologously, corresponds to a energy density homogeneity, shearfree and geodesic, isotropic in pressure. Moreover, we define the stability results for the vanishing complexity factor condition. Finally, we would like to mention that these results are satisfying the prior investigation about complexity factor in General Relativity (GR) by setting [Formula: see text].

Download Full-text

Development of Microemulsions Containing Carthamus tinctorius Extract for 5α-Reductase Inhibition

Key Engineering Materials ◽

10.4028/www.scientific.net/kem.819.92 ◽

2019 ◽

Vol 819 ◽

pp. 92-97

Author(s):

Sitthiphong Soradech ◽

Khaunnapa Panapong ◽

Surada Soonthornsatitwong ◽

Somkamol Manchun ◽

Sirinan Tubtimted ◽

...

Keyword(s):

Transdermal Delivery ◽

Reductase Activity ◽

Skin Permeation ◽

Carthamus Tinctorius ◽

Permeation Rate ◽

Du145 Cells ◽

Thai Medicinal Plants ◽

The Stability ◽

Stability Results

The purpose of this study was to develop micromeulsion consisting of Carthamus tinctorius floret extract (CT) as an ingredient to inhibit 5α-reductase activity. CT was extracted using a simple maceration technique with ethanol and inhibition of 5α-reductase activities was determined. Solutions of 2% CT extract were loaded into four microemulsion (ME) formulas (F1, F2, F3 and F4) and investigated for their physical properties, skin permeation and stability. Results showed that crude CT extract had no toxic effects on DU145 cells at concentrations of 0.0001-1.0 mg/mL. For reduction of 5α-reductase activities, concentration of CT extract at 0.05 mg/mL exhibited highest 5-reductase type-1 inhibition activity on the DU-145 cell line at 89.96% of the control, higher than standard finasteride (31.39%) and dutasteride (38.58%). The results indicated that a thermodynamically stable microemulsion improved the stability and permeation rate of CT extract. Among the ME formula, F3 was most appropriate for ME formulation with highest permeation rate and good stability during 30 days of storage. Therefore, using nanotechnology for stable transdermal delivery systems of bioactive compounds from Thai medicinal plants is one approach to improve skin and hair follicle permeation.

Download Full-text

Stability of Solutions of Ordinary Differential Equations with Respect to a Closed Set

Canadian Journal of Mathematics ◽

10.4153/cjm-1968-070-3 ◽

1968 ◽

Vol 20 ◽

pp. 720-726

Author(s):

T. G. Hallam ◽

V. Komkov

Keyword(s):

Differential Equation ◽

Ordinary Differential Equation ◽

Differential Equations ◽

Ordinary Differential Equations ◽

Compact Set ◽

Stability Of Solutions ◽

Closed Set ◽

The Stability ◽

Stability Results

The stability of the solutions of an ordinary differential equation will be discussed here. The purpose of this note is to compare the stability results which are valid with respect to a compact set and the stability results valid with respect to an unbounded set. The stability of sets is a generalization of stability in the sense of Liapunov and has been discussed by LaSalle (5; 6), LaSalle and Lefschetz (7, p. 58), and Yoshizawa (8; 9; 10).

Download Full-text

ON THE STABILITY OF THE DIFFERENTIAL PROCESS GENERATED BY COMPLEX INTERPOLATION

Journal of the Institute of Mathematics of Jussieu ◽

10.1017/s1474748020000080 ◽

2020 ◽

pp. 1-32

Author(s):

Jesús M. F. Castillo ◽

Willian H. G. Corrêa ◽

Valentin Ferenczi ◽

Manuel González

Keyword(s):

Unit Circle ◽

Banach Spaces ◽

Local Stability ◽

Interpolation Method ◽

Complex Interpolation ◽

Analytic Family ◽

Köthe Spaces ◽

The Stability ◽

Stability Results ◽

Differential Process

We study the stability of the differential process of Rochberg and Weiss associated with an analytic family of Banach spaces obtained using the complex interpolation method for families. In the context of Köthe function spaces, we complete earlier results of Kalton (who showed that there is global bounded stability for pairs of Köthe spaces) by showing that there is global (bounded) stability for families of up to three Köthe spaces distributed in arcs on the unit circle while there is no (bounded) stability for families of four or more Köthe spaces. In the context of arbitrary pairs of Banach spaces, we present some local stability results and some global isometric stability results.

Download Full-text

STABILITY COMPUTATIONS FOR NILPOTENT HOPF BIFURCATIONS IN COUPLED CELL SYSTEMS

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127407018658 ◽

2007 ◽

Vol 17 (08) ◽

pp. 2595-2603 ◽

Cited By ~ 9

Author(s):

M. GOLUBITSKY ◽

M. KRUPA

Keyword(s):

Hopf Bifurcations ◽

Cell Systems ◽

Codimension One ◽

Partial Stability ◽

Branching Patterns ◽

Coupled Cell Systems ◽

The Stability ◽

Identity Transformations ◽

Partial Synchrony ◽

Stability Results

Vanderbauwhede and van Gils, Krupa, and Langford studied unfoldings of bifurcations with purely imaginary eigenvalues and a nonsemisimple linearization, which generically occurs in codimension three. In networks of identical coupled ODE these nilpotent Hopf bifurcations can occur in codimension one. Elmhirst and Golubitsky showed that these bifurcations can lead to surprising branching patterns of periodic solutions, where the type of bifurcation depends in part on the existence of an invariant subspace corresponding to partial synchrony. We study the stability of some of these bifurcating solutions. In the absence of partial synchrony the problem is similar to the generic codimension three problem. In this case we show that the bifurcating branches are generically unstable. When a synchrony subspace is present we obtain partial stability results by using only those near identity transformations that leave this subspace invariant.

Download Full-text