How can I transform the following proposition that is gotten in $real$ space into the corresponding one used in the $complex$ space,i.e.,$A\in C^{n\times n},x=(x_1,...,x_n)\in C^n …

In ZF we have the two relations $A \leq B$ and $A \leq^\ast B$ which relate the size of sets: the first says there is an injection from $A$ to $B$, the second that there is a surje …

Given the vector space $\mathbb{R}^n$, endowed with the standard inner (dot) product $\langle\cdot,\cdot\rangle:\mathbb{R}^n\times \mathbb{R}^n\to\mathbb{R}$, the problem of almost …

For $k\geq 3$, and $k$ arbitrary points $S=( z_1,\cdots,z_k ) \in \mathbb{P}^1$, we can write $$ P^1 \setminus S \cong \mathbb{H}/G $$ where $\mathbb{H}$ is the upper-half plane …

Let $$ F: \mathcal{C} \leftrightarrows \mathcal{D} :G $$ be a Quillen adjunction between model categories. Consider the corresponding adjunction of total derived functors $$ \mathb …

For a given set $S$ of complex square matrices $M1,M2\cdots,Mn$, one can obtain a matrix group $G$ generated by matrx multiplication. For any $i$, we can define a matrix space $Gi$ …

Let $f(x,y)=0$ and $g(x,y)=0$ be bivariate polynomial equations where the polynomials have the same degree, say, $N\geq 3$. Furthermore, both of them have the same terms but differ …

Let $W$ be a weighted adjacency/affinity matrix for some connected graph. $W$ is symmetric and non-negative. If $W_{ij}$ is large, then vertex $i$ and vertex $j$ have high affinity …

I know the construction of a 3D topological quantum field theory (TQFT) from a modular tensor category. I heard that we can even (mathematically) construct 4D TQFT from a modular …

By "a triangulation of $X$", I mean a simplicial complex whose geometric realization is homeomorphic to $X$. Tutte showed that the number of combinatorially distinct triangulations …

Is there a general notion/description of fields $K$ such that the image of any embedding $K \hookrightarrow \mathbb{C}$ is invariant under complex conjugation, thus inducing an inv …

A unitary fusion category is a fusion category with a $C^*$-tensor structure. Hence, in principle, a fusion categoriy could have more than one unitary structure. Does exist a fusio …

Hi, I was wondering about the following question: if you have a faithful action of a group G on the real line R by orientation-preserving homeomorphisms, it is easy to construct …

Suppose that C is a fusion C*-cateogry and that A is an irreducible Frobenius algebra object in C, is there always a Frobenius algebra A' isomorphic to A such that A' is a Longo Q- …

Iwaniec and Kowalski, in their famous book Analytic Number Theory states a strong form of the effective Chebotarev density theorem page 143, and prove it assuming both GRH for Arti …