Can someone help me with this limit? $$ \lim_{(x,y)\to (0,0)} \frac{\sin(x-y)}{x+y} $$ Over the domain $x>0$, $y>0$. I think the result is (zero) but I can’t conclude any estimation due to $|x+y|$ at denominator.

# Category: Mathematics

## Limit of a sum of squared sines

Following up on this question Nice Limit $\lim_{n\to\infty}\sum_{k=1}^{n} \sin^2\left(\frac{\pi}{n+k}\right)$ . Fix a real $a$ and consider the sum: $S(n)=\sum_{k=1}^n sin^2\left(\frac{n^a\pi}{n+k}\right)$ In the linked question it is shown that for $a=0$ the sum converges to zero for large $n$. It is also easy to show that for $a=1$ the sum diverges, recognizing a Riemann sum. The […]

Let $X$ be a discrete random variable. We can write its expectation as $$ EX = \sum_{n=0}^\infty P(X > n).$$ Now, let $p\in (0,1)$ and then, $$ E(X^p) = \sum_{n=0}^\infty n^p P(X =n).$$ My question is, whether there exists a similar formula like the first one in this case, i.e. $$ E (X^p) = \sum_{n=0}^\infty […]

Find $$\lim_{n\rightarrow\infty}\sum\limits_{k=1}^n\frac{\sin(\frac{a}{3^k})}{3^k\sin(\frac{a}{3^{k-1}})}$$ I am not quite sure how to approach this problem, i was looking to see some ways of solving it. Any hints will be appreciated !

For a given matrix $Z_{n\times p},\;p\gt n$ and symmetric positive semidefinite matrix $B_{n\times n}$, how can I find a symmetric positive semidefinite matrix $X_{p\times p}$ that satisfies the equation : $$ZXZ^T=B$$ question: find the matrix $X$?

In Riley’s Math Methods book, there is a discussion on quadratic forms (see attached). However, I’m generally more lost about the assertion that “In any basis we can write…” the inner product as below. I am wondering why this is true. To be clear, we defined the (standard) inner product where orthogonal vectors $\mathbf a,\mathbf […]

Let $(M^3, g)$ be a (closed) Riemannian manifold and let $u: M \to S$ be a harmonic function, where $S$ is a closed orientable surface. If $\omega$ is a $2$-form on $S$, what are sufficient conditions on $\omega$ in order $u^* \omega$ to be a harmonic $2$-form on $M$? The concrete case I am analyzing […]

If $G$ is a (say) compact group and $V=\bigoplus_{i\in I}V_i$ the isotypic (a.k.a. primary) decomposition of a $G$-module, then any $G$-invariant subspace $W\subset V$ writes $W=\bigoplus_{i\in I}(W\cap V_i)$. While this isn’t hard to prove (similar to Hoffman-Kunze 1971, §7.5 for a single operator), it seems silly to redo it in a paper. Unfortunately, the only […]

Given $n$ sides of any arbitrary length, give the necessary and sufficient conditions such that they form a $n$ sided polygon. For a triangle, if the sides are given by $a\le b\le c$ then the necessary and sufficient condition will be $a+b > c$, that is sum of two smaller sides should be greater than […]

Given $f(x,y) \in C^2([0,1]^2)$ (by which I mean $C^2$ in some open neighborhood), with $f_x, f_y, f_{xy} \in L^1([0,1]^2, dxdy)$ (which is sure case since they are continuous), does the following hold? $$ \sup |f| \le \iint |f|+|f_x|+|f_y|+|f_{xy}|\, dxdy $$ I think it is safe to discretise this function under the assumptions, divide $f$ into […]