In the Introduction to Weibel's homological algebra book, he states that homological algebra gives nonconstructive results. He doesn't elaborate on this further, so I wanted to know where exactly the obstruction to being constructible is. For a concrete example, is Riemann-Roch 'constructible'? M...

Stumbled onto the following observation. Defining a Sierpinski matrix recursively import numpy as np def sierpinski(n): """ Generate a Sierpinski triangle matrix""" size = 2**n + 1 triangle = np.zeros((size, size), dtype=int) def fill_triangle(x, y, side_length): if side_...

There is a population that is equally likely to grow by $10\%$ or decrease by $10\%$ every day. What can we say about it in a year's time: will it be bigger, smaller, or the same as it was in the beginning? And in one year and one day? Let the population size initially be equal to $x$. Then after...

I am trying to expand the following: $$ \frac{2p + 2p^2}{2+2p+p^2} $$. Using the taylor series at $a=0$, I get: $p - p^3/2 + p^4/2 - p^5/4 + \dots$. But in the book, it is also equated to $p + p^3 + 2p^5 + p^7 + p^8 + ...$. How is this done?

Corollary 5.3.12 in Schmidt's "subgroup lattices of groups" states that if groups $A,B$ have lower semimodular subgroup lattices, then so does their direct product $A \times B$. This paper examines groups whose subgroup lattices are consistent, ie. for any two subgroups $A,B \le G$, if $A$ is joi...

the question Consider the tetrahedron $ABCD$ and $M$ a point inside the triangle $BCD$. Parallels taken from $M$ to the edges $AB$, $AC$, $AD$ intersect the faces $(ACD)$, $(ABD)$, respectively, $(ABC)$ at the points $A', B',$ respectively, $ C'$. If $(BCD) || (A' B 'C')$ , prove that $M$ is the ...

From the following estimate $$ \int_0^X |ψ(x) − x|^{2k} dx ≪ (ck^2)^k X^{k+1} \ \ \ \ (1)$$ where $c$ is an absolute constant, I want to prove the following estimate $$ µ ( \{ x \le X : |ψ(x) − x| \ge εx^{1/2} (\log x)^2 \} ) ≪ X^{1−c′ε^{1/2}} \ \ \ \ (2)$$ where $c' = 2 \exp(−c/2 − 1...

I want to show that $$\lim_{n\to \infty} (2\pi)^{-d}n^{d/2}d^{-2n}\int_{[-\pi, \pi]^d} (\cos(x_1)+\cdots +\cos(x_d))^{2n} dx_1\cdots dx_d =2(d/4\pi)^{d/2}$$ holds. How do I prove this? It seems that the central limit theorem can be used for this problem, but I don't know how to apply it. Or is th...

I am reading this thesis https://www.few.vu.nl/~trt800/theses/haroldnieuwboer.pdf and on pag 11 there is Example 2.2.3 that I am trying to figure out. Let $W_K=\{ ([a : b], [x : y : z]) \in \Bbb{CP^1} \times \Bbb{CP^2}: a^ky=b^kx \}$ This is a smooth complex hypersurface. Let $j : W_k \to \Bbb...

If a parabola is of the form $\frac{(42x-15x^2)}{8}$, what would be the maximum area of a rectangle that can be fitted inside it if the bottom (width) of the rectangle is aligned on the X axis i.e the y isn't less than zero? I tried to use 2nd derivatives to get a maxmimum width value but I keep ...

I am having a party with 20 guests and I want to find the probability to share my birthday with exactly one of the guests (depending on the result, I may buy a second birthday cake, just in case!!). I know how to calculate the probability for at least one of the guests: The probability for everyo...

Frequently my son's teacher will show him fun little math "tricks." I usually take this as a moment to show him what's really underlying the trick (e.g., why two consecutive squares will differ by a predictable odd number). However, the most recent one is eluding me. Consider the square array $...

Is there some general way of expressing the fact that the operation of taking an average is sensitive not only to the set of numbers on which it is operating but also the number of instances of those numbers? That is, for the 2-set example, (ma+nb)/(m+n) is not necessarily equal to (a+b)/2. 2....