I have a set of coupled differential equations of the form $$ \omega y_1 \frac{\partial q_1}{\partial y_1}+\omega y_2 \frac{\partial q_1}{\partial y_2}+\omega y_3 \frac{\partial q_1}{\partial y_3}+g\frac{\partial q_1}{\partial y_1}+t q_{3}e^{-g/\omega (y_{3}-y_1)}-t q_{2}e^{-g/\omega (y_{2}-y_3)}...

The idea that the product of two permutation matrices gives another permutation matrix makes sense to me, since we know that they only have one entry of 1 in each row and column (and 0s everywhere else). However, how would we show/prove this mathematically?

I am having trouble getting the transformation right. Let me show my work: We know $b^2-4ac=-4x$, so if $x>0$ the equation is eliptic. Let's make our substitution: $\epsilon= -bx+2ay=2xy $ $n= \sqrt{4ac-b^2}x=\sqrt{4x}x=2x^{\frac{3}{2}}$ Notice that $\epsilon_{x}=2y, \epsilon_{y}=2x, n_{x}=3x^{\f...

$\DeclareMathOperator{\Hom}{Hom}$ In a category C, an exponential object $X^Y$ (if it exists) is an object of C such that (by def) $$\forall Z, \Hom_C(Z,X^Y) \cong \Hom_C(Z \times Y,X)$$ This of course implies that $$\Hom_C(1,X^Y) \cong \Hom_C(Y,X)$$ i.e. the generalized elements of the exponenti...

Let it be $A=\{a_0,...,a_n\}$, $B=\{b_0,...,b_j\}$ and $C=\{c_0,...,c_k\}$ three sets of equal or distinct integers, not necessarily ordered, each set having at least two values other than $0$. I am having troubles to find "interesting" conditions on the elements of these sets for this equality t...

Suppose that $G$ is a locally compact (Hausdorff) group. It is well-known that the compactly supported functions $C_c(G)$ are dense in $L^2(G)$. In the book "Operator algebras, theory of $C^{*}$-algebras and von Neumann algebras" written by Bruce Blackadar it is claimed (without proof) that $L^...

Let $J=\left(\begin{array}{cc}1 & 1 \\ 1 & -1\end{array}\right)$ and $E=\left\{\left(\begin{array}{ll}a & b \\ c & d\end{array}\right) \in M_{2}(\mathbb{R}) : a - d=0\right\}$ We define the application $\varphi: E \times E \rightarrow \mathbb{R}$ such that for all M,N$\in E, \varphi(M,N)=Tr(M J N...

If we have a right triangle then the inradius is equal to $$r=\frac{a+b-c}2,$$ where $c$ is the hypothenuse and $a$ and $b$ are the legs. This formula is mentioned in various places and it can be useful both in geometric problems and in problems on Pythagorean triples.1 Questions: How can one der...

Consider a differential operator of the form $\sum_{j=-\infty}^\infty(z_{j+1}\partial_{z_j}+z_j\partial_{z_{j+1}}-g z_j-g\partial_{z_j})$ Here g is a constant. The operator consisting up the first two terms $\sum_j(z_{j+1}\partial_{z_j}+z_j\partial_{z_{j+1}})$ has an eigenfunction of the form $\s...

I was asked to find the inverse of matrix $A=\text{diag}(a_1,a_2,...,a_n)$ Let $B$ be the inverse of $A$, then we have to find $B$ such that $AB=I$ where $I$ is identity matrix of same order as of $A$. I began by assuming that $B$ is diagonal matrix and by using the fact that the product of two d...

Let $X$ be a topological space that has a universal cover $p:\tilde{X}\to X$. Fix a basepoint $\tilde{x}_0\in \tilde{X}$ and let $x_0=p(\tilde{x}_0)$. Then the group $G$ of deck transformations of $\tilde{X}$ is isomorphic to $\pi_1(X,x_0)$: for a loop $\gamma$, let $\tilde{\gamma}$ be the lift o...

For a prime $p$, I am trying to determine the set of all $n$ for which the numerator of $H_n$ is divisible by $p$, with $H_n$ being the $n$'th harmonic number. After going through a lot of literature, this turns out to be a difficult thing to do. The most promising paper regarding my question see...

I have set of answers obtained from different respondents to the following questions (with available answers): Q1 --> Do you think this practice is bad? A1 --> YES or NO. Q2 --> What is the severity of this practice? A2 --> 1-5 Likert scale answer. Q3 --> What is the priority of this practice? A3 --

Let $S_3$ be the symmetry group of vertices permutations of equilateral triangle in $\Bbb{R}^2$ with origin in $0$. Show that the group operation of $S_3$ could be extended to a linear map $\rho (\sigma), \sigma\in S_3$ over the plane. Show that $\rho : S_3 \to GL(\Bbb{R}^2)$ is a group represe...

For a curve $C$ on an orientable surface $Q$ I want to define (not uniquely) $r>0$ and disjoint subsets $L,R$ of $Q$ "on each side" of $C$ such that the set $ \{ B(v,r): \ \ v \in C \} $ (where $B(v,r)$ is the open ball around $v$ of radius $r$ in $Q$ ) is the union of $L$ and $R$. And for a...

Searching for materials on discrete probability I find "first course in probability" type stuff and some seemingly rather specific things. Is there a principled, step-by-step introduction to advanced discrete probabilistic topics (e.g. in a book)? Are there any advanced general theorems in this a...

The PDE: $$\frac1D C_t-Q=\frac2rC_r+C_{rr}$$ on the domain $r \in [0,\bar{R}]$ and $t \in [0,+\infty]$ and where $D$ and $Q$ are Real constants. We're looking for a function $C(r,t)$. The BC: $$C(0,t)=f(t)$$ $$C_r(\bar{R},t)=0$$ The IC: $$C(r,0)=C_0$$ If $f(t)=0$ then I know the solution. Assume:...

We all know the 2nd derivative test in its original form, if $f'(x)=0$, then if $f''(x)<0$ the point is max, and if $f''(x)>0$ the point is min. We also know the generalization for the case is inconclusive with one variable: (I) https://en.wikipedia.org/wiki/Derivative_test#Higher-order_derivativ...

I am new to differential equation, and I realized that I don't have even an intuition as to what solutions to ordinary differential equations would roughly look like. For example, given the governing equation: $$ \frac{\partial^2 y}{\partial x^2} -\frac{\partial y}{\partial x} + y = 0 $$ for the ...

The problem is as follows: The figure from below shows two triangles intersected on point $E$. Assume $AE=3\,m$ and $ED=1\,m$. Find the maximum value of $AB+EC$ The given choices in my book are as follows: $\begin{array}{ll} 1.&\sqrt{10}\,m\\ 2.&\4\,m\\ 3.&\sqrt{5}\,m\\ 4.&\4\,m\\ \end{array}$...