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Given integers $n$ and $k$, Alice is given $k$ numbers $1 \le a_1 < a_2 < \cdots < a_k \le n$. She then writes down a message $x\ (1 \le x \le m)$. Bob is given the message $x$ and one ...

first time poster so I'm sorry if any of the formatting is slightly off. I am trying to use this equation to find the inverse of a 3x3 matrix. $$\mathbf A^{-1} = \frac{1}{det(\mathbf A)} \sum_{s=0}^{n-...

I'm trying to learn the concept of nilpotent groups. On the one hand, there's this formal definition: $Z_0(G)=1, \; Z_{i+1}(G)/Z_i(G) = Z(G/Z_i(G))$. Least $i$ for which $Z_i(G) = G$ (if exists) is ...

Let $f$ be a decreasing function on $[0,1]$ and $a\in(0,1)$. Prove that $$\int_0^af(x)\mathrm dx\ge a\int_0^1f(x)\mathrm dx.$$ This would be quite obvious if $f$ were continuous. But for non-...

I was recently reading I.N.Herstein's "Topics in algebra" and stumbled across interesting proposition and it's proof: For any three sets, $A, B, C$ we have: $$A \cap (B \cup C) = (A \cap B) ...

Let $S$ be a smooth complex projective surface. We consider the double cover $\pi:X\to S$ branched along $B\equiv 2L$(suppose $B$ is smooth). Now let $C\subset X$ be an irreducible curve. If $C$ is ...

In chapter 3 of Analysis I by Terence Tao, the following definition of empty set is given: (Empty set). There exists a set $\phi$, known as the empty set, which contains no elements, i.e., for every ...

I’ve been exploring a measurement approach for NP and NP-complete problems based on average time per logical step. I define: ...

The action \begin{align}\label{action} S=\int d^4x\sqrt{-g}\bigg[\frac{1}{2\kappa}\bigg(R-\varepsilon\, B^{\mu\lambda}B^\nu\, _\lambda R_{\mu\nu}\bigg)-\frac{1}{12}H_{\lambda\mu\nu}H^{\lambda\mu\nu}-V(...

The divergence test is inconclusive for $f(x) = 1/x$. The sum of the $1/n$'s is in fact divergent, $n\in \mathbb{N}$ and $n$ in $[1, ∞)$. We can say the same for the function $g(x) = 1/x^p$, with $0 &...

Let $z = f(u, v)$, where $u = u(x, y)$ and $v = v(x, y)$ are defined on an open region $D \subseteq \mathbb{R}^2$. Suppose that: The functions $u(x, y)$ and $v(x, y)$ are differentiable at a point $(...

I am reading this post : Ideal Class Group of $ \mathbb{Q}(\sqrt[3]{3}) $ and I don't understand some one part. I hope this post isn't considered as a duplicate :) In book ' A conversational ...

Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be defined as; $$f(x) = \begin{cases} 3x & \text{if} \ x \ \leq \frac{1}{2} \\ 3-3x & \text{if} \ x \ > \frac{1}{2} \end{cases} $$ Proposition ...

On a closed oriented Riemannian surface $M$, a vector field $X$ has divergence and curl defined via the metric and Hodge star. If $\alpha = X^\flat$ is the metric dual 1-form, then $$ \operatorname{...

My advisor asked me to describe the spectral properties of the operator $T_z: L^2(\mathbb{R})\longrightarrow L^2(\mathbb{R})$ that is defined by $T_z(f)(x) = f(x+z)$. It was fairly direct to verify ...