Let a matrix the $A \in M_{n\times n}(\mathbb{R})$, and has set of eigenvalues, $\sigma(A)$={$\lambda_1$,$\lambda_2$........,$\lambda_k$}, that is $\forall \lambda \in \sigma(A)$ such that orthonormal base $B_\lambda \subseteq \mathcal{E}_{\lambda(A)},$ the eigen space of $A.$ My question is that...

Today we learned about partial limits of a sequence, i.e. a limit of some subsequence. We wanted to prove a theorem where a sequence has a limit if every subsequence has the same limit (can be infinity or -infinity). The teacher showed us a pretty long proof of this fact, but I came up with a muc...

A group $G$ is metacyclic if has the following exact sequence $$1 \to N \to G \to K \to 1$$ where $N$ and $K$ are cyclic groups. In the split extensions, the wikipedia says that direct and semidirect product of metacyclic groups is metacyclic. What about non split extensions? Is it true? I mean, ...

I have a semi-explicit DAE (Differential Algebraic Equation) with the following form: $$ \begin{align} \mathbf{\dot{s}} &= \mathbf{f}(\mathbf{s}, \mathbf{x}) \\ \mathbf{0} &= \mathbf{g}(\mathbf{s}, \mathbf{x}) \end{align} $$ where $\mathbf{s}\in\mathbb{R}^M$ is the states variable and $\mathbf{x}...

Maximum value of $\displaystyle f(x)=(\cos x-x)\left(x+\sqrt{x^2+\sin^2x}\right)$ What I try : I am Trying to solve it using $\displaystyle 4ab\leq (a+b)^2$: $\displaystyle 4f(x)\leq \left(\cos x+\sqrt{x^2+\sin^2x}\right)^2$ $\displaystyle 4f(x)\leq 1+x^2+2\cos x\sqrt{x^2+\sin^2 x}$ I am struck...

What is a Viterbi algorithm that exploits sparseness of transition matrix? In PYIN: A FUNDAMENTAL FREQUENCY ESTIMATOR USING PROBABILISTIC THRESHOLD DISTRIBUTIONS Matthias Mauch and Simon Dixon https://www.eecs.qmul.ac.uk/~simond/pub/2014/MauchDixon-PYIN-ICASSP2014.pdf the authors write: the HMM ...

I have the below problem from a research paper that I would like to solve $$\frac{1}{2}(y^2 - \bar{y}^2) = -\gamma\sin\theta-\gamma_{\theta}\cos\theta + a_1$$ where, $$a_1 = \gamma\sin\theta +\gamma_{\theta}\cos\theta$$ at $\theta=\bar{\theta}$. This gives me the following $$y=[2(-\gamma\sin\thet...

Let $E\to M$ be a Riemannian vector bundle over an oriented Riemannian manifold $(M,g)$ with a connection $\nabla$. Let $\Gamma(E)$ denote the vector space of sections of $E\to M$. For $\sigma \in \Gamma(E)$, its Sobolev $k$-norm $(k=1,2,\dots)$ is defined by $$ |\sigma|_k^2:=\int_M ||\sigma||^2+...

the question Consider the tetrahedron $ABCD$, with the perpendicular edge $AD$ on the $(BCD)$ plane. Let $X$ be some point of the edge $AD$ and the point $Y$ on $(AD)$ such that $m(∠Y CD) = m(∠DAB)$. The circle with the center $O_1$,which is on $CD$ passing through through $C$ and $X$ intersect $...

I consider an optimal control problem using the Bolza formulation : the state and control variable are respectively $x(t)\in X\subset\mathbb{R}^n$ $u(t)\in\mathbb{R}^m$ for $t\in[t_0,T]$ where $X$ and $U$ are assumed to be compact. The dynamics is given by $$ \dot{x}(t)=f(x(t),u(t)) $$ and the op...

In set theory, $A \cup B$ is logically defined as $\{x : x \in A \lor x \in B\}$. In set theory, the result of unionizing A with B is a bigger set, but in logic, "or" is a softening operation. In set theory, $A \cap B$ is logically defined as $\{x : x \in A \land x \in B\}$. In set theory, the re...

I have a question about Contour Integrals. Contour Integrals are defined as $$\int_{C} f(z)dz = \int_{a}^{b} f(z(t))z'(t)dt \tag{1}$$. But why define it this way? The textbook I'm using (Complex Variables and Applications by Brown and Churchill) simply gave this definition with little explanation...

Consider the exact sequence $0\rightarrow M'\rightarrow M\rightarrow M''\rightarrow0$ and its induced sequence $0\rightarrow T(M')\rightarrow T(M)\rightarrow T(M'')$, where $T$ denotes the torsion submodule. I aim to present an example where the last arrow is not surjective. Could it be that the ...

Edit: I managed to find the integral representations to solve the integral, but I am getting the wrong answer. So, I must be doing something wrong. It is probably a silly mistake. The steps are as follows: Start from the Gaussian integral and expand to get the integral in the first link \begin{...

As mentioned in the bounty, I'm actually looking for a set $E$ such that $d(D)=[0,1]$. The original question follows for context. Let $E \subset \mathbb{R}$ be Lebesgue measurable, let $D \subseteq \mathbb{R}$ the set of all points for which the Lebesgue density of $E$ exists, and let $d : D \t...

I am working with FFT using NumPy in Python, and I noticed that it's common to divide the output of the np.fft.fft function by the length of the data array. Here's a simplified example of my code: L = 1024 # Length of the data array x = np.array([3, 1, 0, ..., 2]) # Data array with 1024 element...