Math

Measure induced on [0, 1] by infinite tosses of biased coin

It is well-known that one can get the Lebesgue measure on by tossing a fair coin infinitely (countably) many times and mapping each sequence to a real number written out in binary. I was trying to explore what happens if you follow the same procedure with a biased coin. I ....

Jacob Jacob: 7 hours ago

Math

Examples of odd-dimensional manifolds that do not admit contact structure

I'm having an hard time trying to figuring out a concrete example of an odd-dimensional closed manifold that do not admit any contact structure. Can someone provide me with some examples? ....

Greyson Greyson: 7 hours ago

Math

Any hints on how to prove that the function $\lvert\alpha\;\sin(A)+\sin(A+B)\rvert - \lvert\sin(B)\rvert$ is negative over the half of the total area?

I have this inequality with $0<A,B<\pi$ and a real $\lvert\alpha\rvert<1$: $$ f(A,B):=\bigl|\alpha\;\sin(A)+\sin(A+B)\bigr| - \bigl| \sin(B)\bigr| < 0$$ Numerically, I see that regardless of the value of $\alpha$, the area in which $f(A,B)<0$ is always half of the total area $\pi^2$. I appreciate any hints and comments on how I can ....

Aiden Aiden: 7 hours ago

Math

Sequences over finite fields

Let's we have finite field $F_q$ for some prime $q=2^M-1$. I am looking for special sequence {$a_{i}$, $i \in {1,..,q-1}$}, ($\{a_{1},...,a_{q-1}\}=F_q/\{0\}$) with the following properties: $r_{1}=a_1$ = 2; Let's ${r_{s}} = \sum_{k=1}^s{a_{k}}$ then: $\{r_{1},...,r_{q}\} =F_q$. Thanks. ....

Natalia Natalia: 8 hours ago

Math

positive expression

Let $$a_{n,k}=\sum_{s_i \geq 1 \atop \sum_{i=1}^{n-k} s_i \leq n} \frac{2^{n}}{(2(n-\sum_{i=1}^{n-k} s_i)+1)!\prod_{i=1}^{n-k} (2s_i)! }$$ for $0 \leq k \leq n-1$. Prove for $1 \leq k \leq n-1$ that $$b_{n,k}=\sum_{l=1}^k (-1)^{k-l} \sum_{s_i \geq 1 \atop \sum_{i=1}^l s_i =k} \prod_{i=1}^l a_{n,s_i}>0.$$ Motivation and alternative formulation can be found here (https://mathoverflow.net/questions/128835?sort=newest#sort-top) ....

Logan Logan: 9 hours ago

Math

Divergence form Elliptic PDE Removable Singularity/Regularity Question

Idea Given a $W^{1,2}$ solution to a linear divergence form uniformly elliptic pde with bounded coefficients, standard De Giorgi-Nash-Moser theory tells us that the solution is infact (Holder) continuous. If you have better regularity away from one isolated point, say you are $C^1$ on the puncutered ball, can the solution ....

Jade Jade: 10 hours ago

Math

How to prove monotonicity of such function?

Let $0<a \le 1, \alpha<0$ and $\beta>0$. How to prove that the function: $$f(x)=\frac{(\Gamma(a)-\Gamma(a,\alpha \ln(\beta x))) (\alpha\ln(x))^a}{(\alpha\ln(\beta x))^a (\Gamma(a)-\Gamma(a,\alpha \ln(x)))},$$ is decreasing for $\beta <1$ and increasing for $\beta>1$. By drawing the graph for some values with mathematica we can expect that the result is true. Also the sign of ....

Aubrey Aubrey: 10 hours ago

Math

Cholesky decomposition of a positive semi-definite

We know that a positive definite matrix has a Cholesky decomposition,but I want to know how a Cholesky decomposition can be done for positive semi-definite matrices?The following sentences come from a paper. "There are two assumptions on the specified correlation matrix R. The first is a general assumption that R ....

John John: 10 hours ago

Math

The monoid of lists of morphisms in a category subject to commuting diagrams

Start with a category $C$. Form a monoid $M$ whose elements are lists of morphisms in the category $C$ subject to commuting diagrams in $C$. Is there a name for this construction or a better way to categorially understand this? ....

Paisley Paisley: 11 hours ago

Math

Calculating the Ext-algebra with a computer

Given a finite dimensional quiver algebra $A$ over an arbitrary field and a module $M$ of finite injective dimension or finite projective dimension. Let $B$ be the Ext algebra of $M$, that is $B:=\bigoplus_{k=0}^{\infty}{Ext_A^i(M,M)}$. Note that $B$ is finite dimensional because of the assumptions on $M$. Questions: 1.Is there a ....

Anna Anna: 11 hours ago