Consider functions from $R^n$ to a subset of $R^n$. So $f(x_1,x_2,...,x_n) = (y_1,y_2,...,y_n)$. where $x_i,y_i$ are all real. More specific consider $$f(x_1,x_2,...,x_n) = (Q_1(x_1,x_2,...,x_n),Q_2(x_1,x_2,...,x_n),...,Q_n(x_1,x_2,...,x_n))$$ where the $Q_i$ are all Quadratic forms. Even more...

Numbers that are not yet mapped to are marked prime and given their own "trees", but really they are distance $\infty$ from the other primes. Traditionally, in a connected graph, BFS forms one tree, but really this is a collection of overlapping trees. What we have on the number line is a...

$$3(a^3+b^3+c^3+a+b+c)+5(a^2+b^2+c^2)+2(a^2b+b^2c+ac^2)+4(a^2c+bc^2+ab^2+ab+bc+ac)=0$$ A table of primitives known to me $(a, b, c)$, $a \in{Z}$, $b\in \mathbb{Z}$, $c\in \mathbb{Z}^+$ $$ \boxed{\begin{array} {|r|r|r|r|}\hline №(a,b,c)& a_n & b_n & c_n \\ \hline S_1 & 0 & 0 & 0 \\ \hline S_2 & 0...

Let $$ \mathcal S=(\mathcal I,\Gamma,\Pi) $$ be a seam marked seed built from four compact oriented $2$-dimensional complex orbifold sheets. Each sheet is assumed to be a football type orbifold: its coarse underlying Riemann surface is $$ |\mathcal O_i|\cong \mathbb P^1 $$ and it has two...

Let $A\in GL(r,\mathbb{Z})$ be a positive definite matrix, $Q(x)=\dfrac{1}{2} x^t A x$ be the quadratic form associate to $A$, $Q^*(x)=\dfrac{1}{2} x^t A^{-1} x$ be the adjoint form of $Q(x)$ . Given $(c,d)=1,m\in\mathbb{Z}^r,$ we define $$G_m\left(\dfrac{d}{c}\right)=\sum_{h\,\text{mod}\,c}...

I apply the iterated forward difference operator to the sequence of primes; from $$ (p_n) = (2, 3, 5, 7, 11 \ldots) $$ I get $$ (d^1_n) = (1, 2, 2, 4 \ldots) $$ $$ (d^2_n) = (1, 0, 2 \ldots) $$ $$ (d^3_n) = (-1, 2 \ldots) $$ $$ \ldots $$ For each sequence $(x_n)$, one can reconstruct the...

Suppose that $$ x=u^{-d},\qquad y=f(u), $$ where $$ d\in \mathbb Z_{>0},\qquad f(u)\in \mathbb C((u)), \qquad y=o(x). $$ Equivalently, $\operatorname{ord}_u f(u)>-d$. Let $ (P,Q)\in \operatorname{Aut}_{\mathbb C}\mathbb C[x,y]$ be a polynomial coordinate system, and write $$ X=P(x,y),\qquad...

I am a curious outsider to mathematics and recently started reading about the Riemann Hypothesis. I am aware that many outsiders mistakenly believe they have solved the Riemann Hypothesis. I am not making such a claim. I am only trying to understand whether this perspective already exists in the...

I was solving this CSES question called Sum of Divisors and one of the solutions in the USACO Guide hints at this statement: The minimum value of $j > i$ such that $\lfloor n/j \rfloor$ < $\lfloor n/i \rfloor$ is $j = \lfloor n/q \rfloor + 1$, where $q = \lfloor n/i \rfloor$ for integers $j, i...

I am currently working through Fulton's Algebraic Curves and I have attempted the following problem: $$\text{Let P be a simple (non-singular) point on }F\text{ . Show that the tangent line to }F \text{ at } P\text{ has the equation }F_X (P )X + F_Y (P )Y + F_Z (P )Z = 0$$ My solution thus far...

In a prior post, the equation, $$x_1^5+2x_2^5 = y_1^5+2y_2^5$$ was considered. It has only one known primitive solution. This present post considers the similar, $$x_1^k+x_2^k+x_3^k = y_1^k+2y_2^k$$ valid for both $k = (1,5)$. Duncan Moore found only one primitive solution, namely, $$85333^k +...

Let $k$ be a commutative ring and let $A$ be a commutative $k$-algebra. The cotangent complex $\mathbf{L}_{A \mid k}$ can be computed using a simplicial resolution of $A$ as follows: choose a simplicial commutative $k$-algebra $P$ and an augmentation $\epsilon : P_0 \to A$ (i.e. a $k$-algebra...

Vakil, Definition 20.1.1: Let $X$ be a variety (reduced separated finite type scheme, actually I'm not sure if we need all this. I think we can get away with dropping "reduced" and "separated" and just assume $X$ is a finite type scheme) over a field $k$ (not necessarily algebraically closed)....

The trace form $(a,b)\mapsto \mathrm{tr}(ab)$ is easily motivated as a choice of bilinear form on a number field $K$ by noting that it agrees with the (standard real) dot product on $\mathbb{R}^{r_1}×\mathbb{C}^{r_2}$ as restricted to the Minkowski embedding of $K$ (the one that sends a number...

In Vaughan's paper "A variance for k-free numbers in arithmetic progressions" https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/S0024611505015352 he uses at one point a kind of dyadic decomposition for major arcs. From my understanding, I think he has $$\sum _{q\leq R}\int_{|\beta...

I am trying to work through the following problem: Let $k$ be a field, and let $X = \operatorname{Spec} k[x, y, z, w]/(xy−zw)\subseteq \mathbb{A}^4_ k.$ (a) Show that $D= V (x, z)$ is a prime (Weil) divisor on $X$ and that $\operatorname{Cl} X\simeq \mathbb{Z}$ is generated by the divisor class...

Many results in real algebraic geometry are proved by passing from a real variety to its complexification , then studying the action of complex conjugation on . For example, one often regards $X(\mathbb R)$ as the fixed-point locus of conjugation on $X(\mathbb C)$. This appears in results such...

If $f \colon X \to Y$ is a finite surjective morphism between integral Noetherian schemes, then the set $V\subseteq Y$ of points over which $f$ is flat is open. I want to show that this fails if we drop the finiteness assumption. I can think of an example given by blowing up $\mathbb A^3$ at a...