shamisen
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Are all optimization problems convex?
4 votes

Suppose that your optimization problem is $(P1) \quad \min_{x} f(x)$ such that $x \in \Omega \qquad$, where $f:\mathbb{R}^n \rightarrow \mathbb{R}$ is a continuous real-valued function and $\Omega \...

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How to show unitary decomposition is continuous
3 votes

A short proof can be found in Problems 17 and 18 from 7.2 of Horn, Roger A., and Charles R. Johnson. Matrix Analysis. Cambridge University Press, 1990. Proof. We first prove the following inequality: ...

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frechet differentiable implies uniformly continuous/ absolutely continuous?
3 votes

We have that $f: \mathbb{R} \rightarrow \mathbb{R}$ given by $f(x)=x^2$ is not absolutely continuous (not even uniformly continuous), yet $f$ is Fréchet differentiable with its Fréchet derivative ...

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What is the implication of Perron Frobenius Theorem?
2 votes

To check the stability of dynamical systems, one can try to search for a real-valued positive function named Lyapunov function. If the dynamical system is LTI, that is, of the format $$ \dot{x}(t) = ...

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Relation between Riccati Algebraic Equation and optimization problem
2 votes

Suppose that $(A,B)$ is stabilizable and $Q$ is Hermitian. Define the operator $\mathcal{R}(X)$ as $$ \mathcal{R}(X) = A^TX + XA - XBB^TX+ Q. $$ If there exists a Hermitian solution $X$ of the ...

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Extreme Value Theorem and Semicontinuity
2 votes

There is a more straightforward way by using that a function is lower semicontinuous if and only if for each $c \in \mathbb{R}$, the set $\{ x \in X : f(x) \le c \}$ is closed. For the proof of this ...

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Applications of simultaneous diagonalization of quadratic forms
Accepted answer
2 votes

This is also known as the modal decomposition in engineering applications, specially in vibrations. It is very useful for the decoupling of differential equations. A good example of its use in this ...

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Generalization of Fuchs' Theorem for Differential Equations
Accepted answer
1 votes

The higher order differential equation you proposed to analyze is known in the literature as a Fuchsian differential equation. According to pp. 75-77 of the thesis Differential equations and group ...

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Describing a Dual Cone
1 votes

1) It is defined for all kinds of cone, even for an arbitrary set (see this link for more information). 2) A figure is shown in p. 143 of Dattorro, Jon. Convex optimization & Euclidean distance ...

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Conditions that Roots of a Polynomial be Less than Unity
1 votes

Routh–Hurwitz stability criterion is a necessary and sufficient criterion for the Hurwitz stability of a polynomial, meaning that the real part of all the roots of the polynomial are negative. Note ...

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what's the difference between RDE and SDE?
0 votes

As the other answers have already stated, the concepts of stochastic differential equations (SDE) and random differential equations (RDE) are different. I would like to complement the answers with a ...

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Application of argument principle
0 votes

Suppose first that $p(z)=\sum_{k=0}^{n} a_k z^k$ is a polynomial without zeros on the imaginary axis and let $\gamma(t):= R e^{\frac{\pi}{2}jt}$, for $t\in[-1,1]$. By argument principle, $$\int_{...

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Solution of differential lyapunov equation
0 votes

Recently, (1) proposed novel formulations of solution algorithms for differential matrix equations based on an $LDL^{T}$ decomposition that keep the computations in real arithmetic. According to the ...

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formula for computing inverse of a matrix
0 votes

Could it be matrix inversion lemma (also known as Woodbury formula)?

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