Let $A$ be a non-empty finite subset of ${\bf R}^n$, $n\geq 2$.Prove or disprove the following:There exists a norm $\|\cdot\|$ on ${\bf R}^n$ and a map $T:{\bf R}^n\to {\bf R}^n$ with the following ...

I am trying to prove that the number of non-isomorphic finite abelien groups of order $n = p_1^{a_1} \cdots p_n^{a_n} = P(a_1) \cdots P(a_n)$, where $P$ gives the number of partitions of an integer....

I have been studying about heat kernel bounds for Markov chains from Woess' book. In Section 14, there are some heat kernel bounds when the underlying Markov chain $(X, P)$ satisfies an isoperimetric ...

The Onsager-Machlup function is a generalization of probability densities to infinite dimensional spaces. It is defined for probability measure $\mu$ on Banach space $(\mathcal B,\|\cdot\|)$ by$$\...

How can divergent series of the form$\frac{9 + 90 + 900 + 9000 + \dots}{1 + 10 + 100 + 1000 + \dots} = \frac{999\ldots}{111\ldots} = \frac{9}{1}$or$\frac{2 + 20 + 200 + 2000 + \dots}{3 + 30 + 300 + ...

For $x$ in the interval $(-1,0)$, my intention is to write, if $\gamma$ is euler's constant, and $\zeta(s)$ the zeta function, the following:$$\tag{1}\int_0^1\frac{1-t^{x-1}}{1-t}\,dt-\gamma(x-1)-\...

I just started studying the paper by Christ, Dyckerhoff, and Walde. I am a bit confused about the first categorification rule table on page 4. As far as I understand the word "Cone" in rule ...

If we fix a function on sets $f : X \to Y$, and a set $Z$ and consider the precomposition map$$ f^* : \text{Hom}(Y, Z) \to \text{Hom}(X, Z), \quad \phi \mapsto \phi \circ f. $$Then $f^*$ injective $\...

I have the following questions while reading Cisinski's paper Descente par éclatements en K-théorie invariante par homotopie. The paper is in French so hopefully I am interpreting everything correctly ...

I am trying to understand the proof of Proposition 4.25 in David Harari's book "Galois Cohomology and Class Field Theory".Proposition 4.25: Let $G$ be a profinite group. Let $H$ be a closed ...

Motivation. Suppose the ordinal $\omega+1$ is given the order topology of $\omega+1$ (that is, $A\subseteq (\omega+1)$ is open if $A\subseteq \omega$, or $(\omega+1)\setminus A$ is finite). It has ...

Let $T$ be a $0$-truncated algebraic theory, and consider $A : T[\text{Anima}]$. By definition, $A$ is a colimit $F_n$, meaning (rewriting the colimit) that it is the colimit of a simplicial diagram ...

I am looking for an example of a discrete probability measure $ \nu $ and a diffuse (or at least not discrete) probability measure $ \mu $ such that the spaces $ L^1(\nu) $ and $L^1(\mu)$ are ...

I am a master’s student in mathematics with a focus on probability theory, and I find myself overwhelmed by the vastness of the field. While I am eager to explore current research directions, I ...

Can one give me an explicit proof that if the Fourier transform of $f$ is compactly supported in a region, then $f$ is essentially constant on the dual region,i.e., $f \sim 1 $ on the dual region,i.e. ...