## How can I prove triangle ABC is congruent to triangle MaMbMc February 11

Ma is a point set halfway on AB, Mb halfway on BC, Mc halfway on CA.

## The projection of density 1 point on a rectifiable set. February 11

This post has also been posted here. Please see the comment on the linked page, useful information!Let $\Gamma\subset \mathbb R^N$ be $\mathcal H^{N-1}$-rectifiable. Then we know that $\mathcal H^{N-1}$ a.e. $x\in \Gamma$ has density $1$. In particul

## Hausdorff measures and densities February 11

I've been stuck on this one for a while now. It's problem 2.4 from Falconer's "The geometry of fractals"Given an $\mathcal{H}^{s}$ measurable subset $E\subset \mathbb{R}^n$ with $0<\mathcal{H}^{s}(E)<\infty$, we let $\overline{D}^{s}(E,x)$

## n-1 dimensionnal Hausdorff measure and codimension 1 measure February 11

I've been told that on a n-dimensionnal Riemannian manifold, the Hausdorff measure of dimension n-1 and the codimension 1 measure $v_{-1}$ (defined below) are mutually absolutely continuous. I've tried to prove it, but it is not obvious for me. I thi

I am working on finding the determinant of the following block matrix $$\begin{pmatrix} C & D \\ D^* & C \\ \end{pmatrix},$$ where C and D are 4x4 matrices with complex entries. I've found a theorem that states $$det\begin{pmatrix} A & B \\ ## What the curvature 2-form really represents February 11 1 Let (E,\pi,B) be a principal bundle with structure group G. The connection 1-form can be thought of as a projection on the vertical part. It allows us to characterize the horizontal subspaces as H_p E = \ker \omega_p then.Apart from that, the ## Geometric Interpretation of Complexified Tangent Vectors on a Real Manifold February 11 1 What is a good geometric way of thinking of complex tangent vectors on a manifold? I can convince myself that I understand tangent vectors by thinking of them as paths on the manifold. Is there a nice way to visualize or think of complex vectors on a ## Geometrical interpretation for curvatures February 11 What is the geometric interpretation for Ricci and Holomorphic Bisectional curvatures in the two dimensional space,like an open ball in the real plane??Any intuitive idea or source will be helpful. ## Standard deviation of function of two RVs February 11 1 I've stumbled upon a problem that basically reduces to having two random variables$$X \sim N(\mu_X,\sigma_X)Y \sim N(\mu_Y,\sigma_Y)$$and defining the third as$$Z = \sqrt{X^2 + Y^2}$$Although it would be convenient to have the exact expressi ## Variance of Signum Function of Two Random Variables February 11 Let  X  and Y be two random variables with means \mu_X and \mu_Y respectively, as well as variances \sigma_X and \sigma_Y (all of which exist). I am interested in computing the following variance:$$ Var[sgn(X-Y)]$$where, of course, sgn de ## Given M, can we find 2 primes a,b so that for all naturals x,y, a^x-b^yM February 11 For example, if M = 2, one can show that 3,17 satisfy the above:For any naturals x,y, |3^x-17^y|>2 (this pair even works for M=9). ## Distance between two symmetric equations February 11 I have been requested to solve this problem:Compute the distance between the lines:L_{1}:\frac{x-2}{3}=\frac{y-5}{2}=\frac{z-1}{-1} and L_{2}:\frac{x-4}{-4}=\frac{y-5}{4}=\frac{z+2}{1}This is my solution:I specify one point for each line:P_{L_{1 ## Division by x-a in a polynomial ring Rx February 11 1 Let R be a commutative ring with identity. Consider the polynomial ring R[x]. Suppose f \in R[x] and a \in R are such that f(a) = 0. Is it true that f(x) = (x - a)g(x) for some g \in R[x]?If f(x)=\sum_{k=0}^n a_k x^k, we can write$$f

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