Let $m$ and $n$ be two integers and $m \le n$. There are a matrix $A$ of $m$-by-$m$ with $A(i,j) = 1/(2n+2j-2i)!$ and a vector $r$ of $m$ entries with $r(i) = 2/(2n+2i)!$. Is there a formula for the inner product of $r$ and the first column of the inverse of $A$? Or, can it be shown that the inn...

I am a bit stuck on the following second order differential equation, using the method of undertermined coefficients. $$y'' + 3y' + 2y = xe^{-x}$$ The homogenous solution is easy to find, but I run into some issues with the particular solution. $y = Axe^{-x}$ doesn't seem to be a good enough gues...

I don't understand Spivak's comment at the end that $f$ means something different on the two sides of the equation. Don't they both refer to the same function? Also, the expression $D_1(f \circ (g, h))$ isn't clear about which variable should be first. The first var of $f$ is $u$, but the first ...

Consider these two sets: $$ A\equiv \{x\in X: \forall \xi>0 \text{ }\exists N_{\xi, x} \text{ s.t. } \forall N\geq N_{\xi, x} \text{ } d(p_N, I(x))\leq \xi)\}, $$ $$ B\equiv \bigcap_{\xi>0} \bigcup_{N=1}^\infty\bigcap_{K=N}^\infty \{x\in X: d(p_K, I(x))\leq \xi\}, $$ where $A$ and $B$ are non-em...

The double integral over the region: $$ R = \left\{ \left( x,\: y \right) : a \leqslant x \leqslant b,\: g\left( x \right) \leqslant y \leqslant h\left( x \right) \right\} $$ is expressed as $$ \iint_R f\left( x,\: y \right) \mathrm{d}A = \int_a^b \left[ \int_{g\left( x \right)}^{h\left( x \right...

I am reading Serre's Lectures on the Mordell-Weil Theorem, where he specifically talks about a Large Sieve inequality and proceeds to give an example. Theorem. (Section 12.1) Let $K$ be a number field, $\Lambda$ be a free $O_K$ -module of rank $n$. Let $\|\cdot \|$ be a norm over $\Lambda_{\math...

I am considering formulating an applied research problem as a simultaneous zero-sum game with two players. The first player's set of actions is an infinite and compact subset of $\mathbb{R}^n$, while the second player has a finite number of actions. For a specific action $s_2$ chosen by Player 2,...