## Seeking a combinatorial proof \$binom{n}{2k-1}=binom{n+2}{2k+1}-2timesbinom{n+1}{2k+1}+binom{n}{2k+1}\$

I would appreciate if somebody could help me with the following problem

Q: Seeking a combinatorial proof that for all $$n,kin mathbb{N}$$, following holds

$$binom{n}{2k-1}=binom{n+2}{2k+1}-2timesbinom{n+1}{2k+1}+binom{n}{2k+1}$$

## Kolmogorovs complexity proof

Prove that there is a constant c ∈ N such that, for all n ∈ N,
|C(sn) − C(sn+1)| ≤ c.

So what I know so far is the following:
We can define 2 functions f and g such that
for f:
C(sn) − C(sn+1) ≤ c.

and for g:
C(sn+1) − C(sn) ≤ c.

We also know that C(f(x))<=C(x)+c.
So cross comparing, we can use sn=f(sn+1)for the first part and sn+1=g(sn) for part 2, but Im having a hard time defining the 2 functions. Any guidance or tips?

## Proof that a ring homomorphism between group algebras over a field has an eigenvalue

Let $$G$$ be a finite group, $$k$$ an algebraically closed field and $$kG$$ the group algebra of $$G$$ over $$k$$. Let $$M$$ be a module over $$kG$$. Let $$V$$ be an irreducible/simple $$kG$$-modules.

In the proofs of one version of schur’s lemma (for example, on Page 8 of this), it is often used that if $$phi: V to V$$ is a $$kG$$-homomorphism, then since the base field of $$V$$ is algebraically closed, as a linear mapping on $$V, phi$$ has an eigenvalue $$a in k$$.

I haven’t seen a proof of this, and I was wondering how to prove it. Also, what is meant when they say: “$$phi$$ is a $$kG$$-homomorphism”, do they just mean $$phi$$ is a module homomorphism between modules over the ring $$kG$$?

I looked at the wikipaedia page, and the proof for a normal linear map from $$k^n$$ to $$k^n$$ uses the fact that endomorphisms from vector spaces of finite dimension can be represented by a matrix on any basis and then you can use the characteristic polynomial.

I am trying to make a similar argument using the more abstract analogues of all these things. Obviously, the module homomorphism is the analogue of a linear map. Do I then show that $$kG$$ modules are finite dimension vector spaces over $$k$$, and hence $$phi$$ could be represented by some matrix with values in $$k$$, so it’s characteristic polynomial has a root?

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## reference request – Simple proof of formula related to asymptotics for eigenvalue problem for Laplacian

For the solution of
$$begin{cases} lambda u^epsilon – frac{epsilon^2}{2} Delta u^epsilon = 0 &text{in } Omega \ u^epsilon=1 & text{on } partial Omega end{cases}$$
$$lim_{epsilon to 0} – epsilon log u^epsilon = sqrt{2lambda} mathrm{dist} (x,partial Omega)$$

Is it possible to give a simple and straightforward proof of this result? Maybe relying (only or mostly) on tools like the maximum principle or the Green function of the Laplacian?

## Solve for pi; showing a proof for logistic regression

I need help solving for π and am extremely confused! I know I have to use the base e function but confused how to get there!

ln(π/1-π)=B0+B1x1
Solve the equation for π to show π=exp⁡{B0+B1x1}/(1+exp⁡{B0+B1x1})

## verification of “Concise Proof of the Riemann Hypothesis Based on Hadamard Product”

There is a circulating preprint:
Concise Proof of the Riemann Hypothesis Based on Hadamard Product.

Although, it’s short I was not able to follow the paper’s line of argument nor disprove their attempt.

Any insight?

## complexity theory – Proof that a relation is in FP

How we can prove that the relation: $$R= left{0,1right}^*times left{0,1right}^* in FP$$
I understand that we need to find a polytime algorithm to decide whether $$(x,y) in R$$ since $$(x,y)in R= left{0,1right}^*times left{0,1right}^*$$
How can we find this? And this is enough to prove that $$R in FP$$?

## bitcoin core – I can not find a clear mathematical proof method with details and example for “near to zero chance of generating the same pair key wallet”

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## real analysis – Elementary proof that an open subset of \$Bbb{R}^n\$ does not have measure zero?

There is an elementary theory of subsets of $$Bbb{R}^n$$ of measure zero, namely one defines the volume of a cube in the obvious way and one says that a subset $$A$$ has measure zero if given any $$epsilon>0$$ there exists a countable number of cubes that cover $$A$$ and such that the sum of the volumes of the cubes is $$leq epsilon$$.
One can show, with modest effort, that this notion is invariant under diffeomorphisms and thus leads to the notion of subsets of measure zero on a smooth manifold. This notion shows up in Sard’s Theorem which says that the set of critical values has measure zero.

Is there an elementary argument why non-empty open subsets do not have measure zero? Evidently it follows from standard measure theory, but for my topology course I would appreciate it if there was an elementary argument, but I can’t think of one and I can’t find one.

This is stated as an exercise in Lee’s book on smooth manifolds, but it’s not obvious to me. Note that even $$n=1$$ seems tricky.