updates
Find the inverse function of $f(x) = \frac{x^3+3x}{2}$
Find the inverse function of $f(x) = \frac{x^3+3x}{2}$ My thought on this are as follows: I've solved a similar problem where I had to find the inver...
Apr 21, 2026
Archive: Updates
![Gram Matrices Rank [duplicate]](https://news.conceivablytech.com/uploads/gram-matrices-rank-duplicate_720.jpg){kind=icon}
updates
Gram Matrices Rank [duplicate]
Let $A$ be an $m \times n$ matrix. Show that, even though they may be of different sizes, both Gram matrices $K = A^TA$ and $L = AA^T$ have the same rank....
Apr 21, 2026
updates
Chebyshev center of a polyhedron: nonnegativity issue
Let us have a polyhedron, defined by the inequalities of the form: $$ \mathcal{P} = \{ x \ | \ a_i^T x \leq b_i, \ i=1,\ldots,m \} $$ Here on page 19, the...
Apr 21, 2026
updates
Fibred product of schemes
I'm learning algebraic geometry, and I'm having some difficulties in developing intuition for the fibred product of schemes. I can take schemes ...
Apr 21, 2026
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Expectation of Double Dice Throw
I was wondering about finding expected value of dice throw. I know that in case of single dice throw,the expected value is 3.5 = ((1+2+3+4+5+6)/6). So for...
Apr 21, 2026
updates
Motivation for Tom Lehrer's song "Lobachevsky"?
I am trying to understand the motivation for the jingle about plagiarism written by Tom Lehrer. A YouTube version can be found here http://www.youtube.com...
Apr 21, 2026
updates
Binomial coefficients that are powers of 2
I would like a proof that $$ {{n}\choose{k}} = \frac{n!}{k!(n-k)!} = 2^m $$ for $n,k,m\in \mathbb{N}$, only if $k=1$ or $k=n-1$. It seems to me that this ...
Apr 21, 2026
updates
Determining percentage errors for inverse trig functions in conjunction with other errors
Let's say I have the following equation in which the unknown is $θ$: $$tan(θ)=\frac{a}{b}$$ $$tan(θ)=\frac{5}{3}$$ $$θ=arctan(1.667)$$ $$θ=59.036°$$ ...
Apr 21, 2026
updates
Finding the shaded area in a triangle
Here is the diagram: I only know that the middle segment is a median of the big triangle. But nothing else.
Apr 21, 2026
updates
What are the connections between the three Mertens' theorem?
In number theory the three Mertens' theorems are the following. Mertens' $1$st theorem. For all $n\geq2$ $$\left\lvert\sum_{p\leqslant n} \frac{...
Apr 21, 2026
