MAT 544 Real Analysis
نویسنده
چکیده
3. The Implicit Function Theorem and its special case, the Inverse Function Theorem. Relation to the Contraction Mapping Fixed Point Theorem. Computation of several iterates of the approximating sequence to the implicit function. Continuity and differentiability of the implicit function. Application to differentiable variation of the fixed point in the Contraction Mapping Fixed Point Theorem. 4. The Picard-Lindelöf Theorem on existence and uniqueness of solutions of a first-order ODE whose defining equation is locally Lipschitz. Relation to the Contraction Mapping Fixed Point Theorem. Maximal extensions. 5. Diagonalization of diagonalizable matrices; Jordan form for matrices of small size. The characteristic polynomial. Eigenvalues. Eigenvectors, eigenspaces and generalized eigenspaces. The semisimple-nilpotent decomposition of a matrix. Algorithm for 2×2, 3×3 and 4×4 matrices. 6. Matrix exponentials. Definition via convergent power series. Compatibility with M 7→ UMU−1; compatibility for commuting matrices. Computation given the Jordan form. 7. Solutions of constant coefficient linear systems of ODEs, both homogeneous and inhomogeneous. Reduction of order. Solution of first-order systems by matrix exponentials. “Green’s functions” / “fundamental solutions”; solution of the inhomogeneous equation. 8. Algebras and σ-algebras in the power set of a set. The Borel σ-algebra. Product σ-algebras. 9. Measures, outer measures and premeasures. Finite and countable additivity. Subadditivity. Monotonicity. Continuity from above and below. Caratheordory’s Theorem constructing a measure from an outer measure. Compatibility with the premeasure used to construct an outer measure.
منابع مشابه
MAT 544 Real
whenever ‖~v‖V < δ. In this case the bounded linear operator dF~v0 is called the (Frechet) derivative of F at ~v0. Prove that if ‖ • ‖V is a norm on V which is equivalent to ‖ • ‖V , then F is differentiable at ~v9 with respect to ‖ • ‖V if and only if it is differentiable at ~v0 with respect to ‖ • ‖V , and in this case the derivatives are equal. Similarly, if ‖ • ‖W is a norm on W which is eq...
متن کاملReal Time Force Sensing Mat for Human Gait Analysis
This paper presents a real time force sensing instrument that is designed for human gait analysis purposes. This instrument mainly consists of three main elements: the force sensing mat, signal conditioning and switching circuit and data acquisition device. In order to control and to process the incoming signals from the force sensing mat, Force-Logger and Force-Reloader program are developed u...
متن کاملReal Time Multi-Sensory Force Sensing Mat for Sports Biomechanics and Human Gait Analysis
This paper presents a real time force sensing instrument that is designed for human gait analysis purposes. It is capable of recording and monitoring ground reaction forces exerted by human foot during various activities such as walking, running and jumping in real time. In overall, force sensing mat mainly consists of three elements: the force sensing mat, signal conditioning circuit and data ...
متن کاملApplication of Numerical Methods in the Analysis of Mat Foundations and Limitations of Classical Methods
متن کامل
Application of Numerical Methods in the Analysis of Mat Foundations and Limitations of Classical Methods
متن کامل
Changes of Matrilin Forms during Endochondral Ossification
To understand the molecular properties of matrilin-3, a newly discovered member of the novel extracellular matrix protein family, we cloned a MAT-3 cDNA from developing chicken sterna. Real time quantitative reverse-transcription polymerase chain reaction indicates that MAT-3 mRNA is mainly expressed in the proliferation zone of a growth plate. It is also expressed in the maturation zone, overl...
متن کامل