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?

Thanks for your help.

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

For the solution of
$$
begin{cases}
lambda u^epsilon – frac{epsilon^2}{2} Delta u^epsilon = 0 &text{in } Omega \
u^epsilon=1 & text{on } partial Omega
end{cases}
$$

Varadhan proved that
$$lim_{epsilon to 0} – epsilon log u^epsilon = sqrt{2lambda} mathrm{dist} (x,partial Omega) $$

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.