Non-Well-Posed Problems, Unstable Manifolds, Lyapunov Functions, and Lower Bounds on Dimensions

Infinite-Dimensional Dynamical Systems in Mechanics and Physics - Applied Mathematical Sciences ◽

10.1007/978-1-4612-0645-3_8 ◽

1997 ◽

pp. 465-497

Author(s):

Roger Temam

Keyword(s):

Lower Bounds ◽

Lyapunov Functions ◽

Unstable Manifolds ◽

Well Posed

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Dichotomies with No Invariant Unstable Manifolds for Autonomous Equations

Journal of Function Spaces and Applications ◽

10.1155/2012/527647 ◽

2012 ◽

Vol 2012 ◽

pp. 1-23 ◽

Cited By ~ 3

Author(s):

Răzvan O. Moşincat ◽

Ciprian Preda ◽

Petre Preda

Keyword(s):

Differential Equation ◽

Continuous Time ◽

Orlicz Spaces ◽

Sequence Spaces ◽

Time Behavior ◽

Exponential Dichotomies ◽

Unstable Manifolds ◽

The Right ◽

Almost All ◽

Well Posed

We analyze the existence of (no past) exponential dichotomies for a well-posed autonomous differential equation (that generates aC0-semigroup ). The novelty of our approach consists in the fact that we do not assume theT(t)-invariance of the unstable manifolds. Roughly speaking, we prove that if the solution of the corresponding inhomogeneous difference equation belongs to any sequence space (on which the right shift is an isometry) for every inhomogeneity from the same class of sequence spaces, then the continuous-time solutions of the autonomous homogeneous differential equation will exhibit a (no past) exponential dichotomic behavior. This approach has many advantages among which we emphasize on the facts that the aforementioned condition is very general (since the class of sequence spaces that we use includes almost all the known sequence spaces, as the classical spaces, sequence Orlicz spaces, etc.) and that from discrete-time conditions we get information about the continuous-time behavior of the solutions.

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Mathematical Modeling and Analysis of Mathematics Anxiety Behavior on Mathematics Performance in Kenya

Journal of Advances in Mathematics and Computer Science ◽

10.9734/jamcs/2020/v35i430269 ◽

2020 ◽

pp. 46-62

Author(s):

O. M. Nathan ◽

K. O. Jackob

Keyword(s):

Mathematics Anxiety ◽

Lyapunov Functions ◽

Deterministic Model ◽

Equilibrium Points ◽

Modeling And Analysis ◽

Achievement Rate ◽

All Solutions ◽

Anxiety Behavior ◽

Sensitive Parameters ◽

Well Posed

We propose a deterministic model that describes the dynamics of students who have the capabilityWe propose a deterministic model that describes the dynamics of students who have the capabilityto perform well in mathematics examinations and engage in careers that demand its applicationand the negative inuence of individuals with mathematics anxiety on the potential students.Our model is based on SIR classical infectious model classes with Susceptible (S) and Infected (I)taken as Math anxious students (Ax) and Removed (R) adopted as achievers students (Aa) . Themodel is shown to be both epidemiologically and mathematically well posed. In particular, weprove that all solutions of the model are positive and bounded; and that every solution with initialconditions in remains in the set for all time. The existence of unique math anxious-freeand endemic equilibrium points is proved and the basic reproduction number R0 computed usingnext generation matrix approach. A global stability of anxious-free and the endemic equilibria areperformed using Lasselles invariance principle of Lyapunov functions. Sensitivity analysis showsthat achievement rate of potential achievers and achievement rate of math anxious students are the most sensitive parameters. This indicates that effort should be directed towards theseparameters, by having well trained mathematics staff and the best printed and technological resources so as to control the spread of mathematics anxiety. Furthermore, scaling up the understanding level of mathematics algorithms, lowers the mathematics anxiety level and consequently, the spread of mathematics anxiety amongst students reduces. Lastly, some numerical simulations are performed to verify the theoretical analysis result using Matlab software.

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