I am trying to determine whether the following, $ n $ by $ n $ matrix: $$ A = \begin{pmatrix} \Delta_{11} - 1 & \Delta_{12} & ... & \Delta_{1n} \\ \Delta_{21} & \Delta_{22} - 1 & ... & \Delta_{2n} \\ ... \\ \Delta_{n1} & \Delta_{n2} & ... & \Delta_{nn} - 1 \end{pmatrix} $$ Subject to $ \Delta_{i...

In stochastic calculus, it is often standard to write a DE in differential form, such as $\mathrm dY = H \, \mathrm dX$ for the stochastic integral $$\displaystyle\int\limits_0^t H \, \mathrm d X := \displaystyle\int\limits_0^t H_s \, \mathrm d X_s.$$ The most common sense-interpretation of an ex...

Examine the convergence of the series of functions $$\displaystyle\mathop{\sum}\limits_{n=1}^{+\infty}\Big({\frac{x}{1+x^n}}\Big)^n$$ a) pointwise in $[0,1]$, b) uniformly in $[0,1]$. My attempt for pointwise convergence: For all $x\in[0,1)$ exists $n_0(x)\in{\mathbb{N}}$ such that for al...

Consider a mass $m$ is placed on a horizontal level surface and attached to a spring, whose other end is attached to a vertical wall. The mass moves in a viscous medium, where the resistance force acting on it is proportional to the velocity: $\vec{F}_p=-\gamma\vec{\nu}$, where $\gamma$ is a kno...

The above picture comes from Fulton's "Introduction to Intersection Theory in Algebraic Geometry". The variety $I$ is the partial flag variety $F(0,d;n)$, also known as the incidence variety of points on $d$-planes in $\mathbf P^n$. In other words, points in $I$ are pairs $(p,P)$ where $P$ is a ...

This question stems from [The kinetic limit of a system of coagulating Brownian particles] (https://arxiv.org/abs/math/0408395), specifically the last step in the proof of Lemma 4.2. The setting is as follows. We fix some finite time $T<\infty$, some constant $K\in\mathbb R$, a positive mollifier...

So, I am trying to see if something would be visible to someone standing on the surface of a planet or the top of mountain on it. So, imagine a perfect sphere for the planet, then imagine a moon which orbits perpendicular to the line from the observer to the centre of the sphere. Let's give a orb...

The vertices of a hexagon are uniformly random points on a unit circle; $a,b,c$ are the lengths of three distinct random sides. A simulation with $10^7$ such random hexagons yielded a proportion of $0.60008$ of them satisfying $ab<c$. Is the following conjecture true: $P(ab<c)=\dfrac35$. Probab...

Let $f(n)$ be the number of groups of order $n$ up to isomorphism. We want to prove that: $$f(a) \cdot f(b) \leq f(a \cdot b)$$ for all nonnegative integers $a$ and $b$. Our progress: If $a \cdot b \leq 2048$, then our conjecture holds. If $a + b \leq 94$, then our conjecture holds. If $\gcd(a...

This is an attempt to clarify an issue from Section 6 in the proof of Theorem 2.2. Bass, R. F., Uniqueness in law for pure jump Markov processes, Probab. Theory Relat. Fields 79, No. 2, 271-287 (1988). ZBL0664.60080. Let $D([0,\infty))$ be the set of all $f : [0,\infty) \to \mathbb R^d$ which are...

Consider these sets $$ A\equiv \bigcap_{\delta>0} \liminf_{n\rightarrow \infty} \{x \in X: d(p_n, [\ell(x), u(x)])\leq \delta\} $$ $$ A_n\equiv \{x \in X: d(p_n, [\ell(x), u(x)])=0\} $$ where: $A$ is non-empty. $(p_n)_n$ is a sequence of reals taking values in $[0,1]$. $\ell(\cdot)$ and $u(\c...

Furthermore, let $(E,\mathcal E,\lambda)$ be a $\sigma$-finite measure space; $\mu$ be a probability measure on $(E,\mathcal E)$; $\zeta$ be a Markov kernel on $(E,\mathcal E)$; $\pi$ be a probability measure on $(E,\mathcal E)$ with density $p$ with respect to $\lambda$; $c>0$ and $\ell\in[0,1]...

I solved this problem, but I want to know if there's an easier or more magical solution for it. problem statement Let integer $n\ge 3,$ $\tbinom n2$ nonnegative real numbers $a_{i,j}$ satisfy $ a_{i,j}+a_{j,k}\le a_{i,k}$ holds for all $1\le i <j<k\le n$. Proof $$\left\lfloor\frac{n^2}4\right\rf...

I already asked this question for general $B$ and it was answered negatively here, so if the statement is true, one has to use the assumption on $B$ as well. In the original question I already discussed possibilities for tackling this problem. In short, one can write $A$ (and $B$) as either the s...

We are given a circle with radius $1$, its center point and an inscribed isosceles triangle with $AB=AC$ and its height (as shown in the picture below). Can we express the area $(ABC)$ as a function of $θ$ where $θ=B \hat{A} C$? How I tried: I put the diagram in a Cartesian coordinate system. A...