This is a problem from my exam on functional analysis. I did only trivial case and now I'm just curious about another cases. Let $T : H \rightarrow H$ is a linear continuous unit …

For probably twenty years, category theorists have known of some objects in the Platonic universe called "(weak) $\infty$-categories", in which there are $k$-morphisms for all $k\i …

Hello ! If I'm not mistaken, an atomic topos decompose as a disjoint sum of connected atomic topos, and Connected Atomic topos with a point corresponds to classifying topos of loc …

Hello all! I'm just wondering if there is an online encyclopedia of Diophantine equations (OEDE), analogous to the OEIS for sequences. While trying to solve one Diophantine equat …

Let $\cal D$ be the set of all $D=1,2,3,\dots$ such that $D\equiv 0,1 \mod(4)$ and $D$ is not a square. For $D\in\cal D$ let $\varepsilon(D)$ denote the smallest number $\varepsilo …

A measurable space $(X,\mathcal X)$ consists of a set $X$ equipped with a $\sigma$-algebra of subsets $\mathcal X$. I would like to write computer programs involving measurable spa …

Assume that $\gamma$ is a symmetric convex curve w.r.t. two orthogonal lines (such a curve is the ellipse). Is the following statement true. There exists a $(l,L)$-bi-Lipschitz map …

Let $x_1,\ldots,x_N \in \{0,1\}^D$ be $N$ binary vectors, assumed affinely independent. Is there an efficient algorithm for determining whether a new binary vector $x_{N+1}$ is in …

Let $A=\{a_i\in\mathbb{Q}:a_i> 1\land 1\le i \le b-2\}$, (cardinality of $A$ is $b-2$) and $\displaystyle B=\bigcup_{c=0}^{\infty} \left[\dfrac{bc+1}{d},\dfrac{b(c+1)-1}{d}\righ …

For a real number $\alpha$, let the irrationality measure $\mu(\alpha) \in \mathbb{R}\cup {\infty}$ be defined as the supremum of all real numbers $\mu$ such that $$ \left| \alpha …

Have the 2D matrix operations been generalized to n-dimensional matrices? Are there any books that define various operations on multidimensional matrix? I'd like to see operations …

Can someone give me an example of a Banach Algebra which does not have an isometric representation in a Hilbert Space ? (if possible, can you add a proof or a reference ? ) Thank …

Let $G$ be a countable, Gromov-hyperbolic group. We say that $H$ is hyperbolically embedded in $G$ if $G$ is relatively hyperbolic to {$H$} (in the strong sense). This definition …

Let M be a Riemannian manifold, S a closed immersed submanifold of M, and consider the space of smooth perturbations, i.e. functions X : S → NS (the normal bundle of S in M), p → …

Given a distribution $F$ defined as a linear combination of Gaussian distributions: $F = \sum_{i=1}^n C_i*N(\mu_i,\sigma_i)$ with $\sum_{i=1}^n C_i = 1$ I want to find a Gaussian …

Hello, I'd like to know the square root of the following $n$ by $n$ matrix, for $n > 2$ and $r>0$: $R_{ii}=r+1$ for $i < n$ $R_{ij}=r$ otherwise The $2$ by $2$ case is given …

I'm trying to prove that $S^1 \times S^n$ is parallelizable for any $n \geq 1$. When $n$ is odd I can construct the $n+1$ linearly independent vector fields explicitly, but I can't …

Let $\mathbb{P}^2_\mathbb{K}$ be the projective space, with $\mathbb{K}=\bar{\mathbb{K}}$. Let $n\geq 7$ be odd and $f_1, \dotsc, f_n$ be $n$ general forms of degree $\frac{n-1}{2} …

I am trying to understand Henselian Weierstrass Theorem in Hironaka's Idealistic exponents of singularity, page 76 - 77. At some point he has $R$, Noetherian, Henselian, and loca …

Assume Goldbach's conjecture. Then for every $n\ge 2$ there exists at least one non-negative integer $r\le n-2$ such that both $n+r$ and $n-r$ are primes. Let's write $r_{0}(n):=\i …

I'd like to ask for references on the status of modularity results for elliptic curves with CM which are not necessarily defined over $\mathbb Q$. In the case of an elliptic curve …

Because I have heard the phrase "totally ordered abelian group", I imagine there should be non-abelian ones. By this I mean a group with a total ordering (not to be confused with …

There is a vast literature on embeddings of graphs into surfaces. I am interested in embeddings of graphs that belong to the given homotopy class. Here is the precise formulation …

Suppose I have a sequence of stochastic processes $X_{N}(t)$, $N=1,2,3,\ldots$ with mean zero and that I know for every fixed $t$, the random variable $X_{N}(t)$ converges in law t …

By a variety over a field $k$, I mean a scheme that is separated and of finite type over $k$. I indicate changes of the base ring by subscripts. Does there exist a smooth and pro …

As a graded $\mathbb{Z}$-module, the structure of the group cohomology $H^{*}(\mathbb{Z}/n\mathbb{Z};\mathbb{Z})$ is extremely well-known. Yet, I am having difficulty finding a ref …

In the equivariant Atiyah-Singer index theorem, when $G$ is a compact group acting on a manifold $M$ and $R(G)$ is the representation ring of $G$. We have the analytic index $$ \te …

We can map from set of coverings over $X$ to symmetric group $\mathfrak{S}_n$ via monodromy (if we fix a loop at the basepoint). Also we can consider braid group $Br_n(Y)$, allow s …

For $\Re s = 1/2$ numerical evidence suggest: $$ \Re \zeta'(s)/\zeta(s) = 1/2 \log(\pi) - 1/2 \Re \psi(s/2) \qquad (1) $$ How this was found. Consider the symmetrized zeta functio …

Here are the two versions of Arnold's conjecture on Hamiltonian orbits: Let $(M,\omega)$ be a closed symplectic manifold, and let $H: \mathbb{R/Z} \times M \to \mathbb{R}$ be a …