linear algebra – Why the multiplication of a covariance matrix with the inverse of its sum with the identity matrix is symmetric?

I have an empirical result (meaning it is always true by simple simulation e.g. in R) which I cannot prove to myself:

Let $A$ be a $n times n$ covariance matrix (i.e. it is symmetric PSD), let $I_n$ be the identity matrix, $theta_1$ and $theta_2$ some scalars (in my case they are always positive but it does not matter). Let:

$V = (theta_1 A + theta_2I_n)^{-1}A$

It seems that $V$ is always symmetric! Can we prove it?

E.g. in R:

A <- cov(rbind(c(1,2.1,3), c(3,4,5.3), c(3,4.2,0)))
isSymmetric(solve(2 * A + 3 * diag(3)) %*% A)
(1) TRUE

To anyone interested: it is important to me mainly because this means I have two symmetric matrices $A, B$ which multiply to a symmetric matrix $AB$, in which case its eigenvalues are in fact multiplications of the eigenvalues of $A$ and $B$ according to this, which also simplifies its trace.

Strange Summation Identity – Mathematics Stack Exchange

I was looking at the Summation wikipedia( and found this rather strange identity:
sum_{kleq jleq ileq n} a_{i,j}=sum_{i=k}^nsum_{j=k}^i a_{i,j}=sum_{j=k}^nsum_{i=j}^n a_{i,j}=sum_{j=0}^{n-k}sum_{i=k}^{n-j} a_{i+j,i}

Can someone please explain me what the term before the first equal to sign mean and provide me with a proof of above?

hash – RNG and HMAC for identity generation

I’m working on a little side project, an online casino wallet system, where one of the things I’m doing is generating a random number and handing it out to each player that joins. This random number acts as a key to an hmac to generate a hash based on, at first, the player’s identity. The resulting hash is the player’s first wallet identifier. The entire point of doing all this is to reduce the chance of someone guessing the identity of the player’s wallet. Only if they knew this random number, the hmac algorithm used, and the player’s identity, would they be able to deduce the first wallet’s identity. Each time I exchange a player’s wallet for a new wallet I can either use the random number and the old wallet identity to compute the next wallet identity, or I can generate a new random number, use a different hmac algorithm, and use the old wallet identity in the process. Another advantage is that the wallet identity generation is deterministic.

I’m not a security expert nor a mathematical genius, which makes me wonder if this approach is even sane. I’m sensing a pattern or somewhat more formal prior art, but I honestly would not know where to look first. I’d gladly take any feedback or pointers on all this.

authentication – Consume Identity enabled WebApi in an MVC Project

I have an MVC project with no authentication. The project uses webapi as a backend to retreive and send information through the controllers.

Now we need to add authentication to the webapi project as well as introcing a secondary MVC project.

Both MVC projects needs to rely on webapi for information exchange.

I am going to add Identity support to my webapi project but how can I authenticate my controllers that way?

So what I am asking is; is it possible to secure webapi and consume it in MVC project in a way that user are able to register and login within the mvc projects. If so, how and/or any resources?

mac osx – OpenSSL identity information did not stick in certificate

We were just required to update our SSL certificate after the first year’s expiration date come up. We’ve done this several times before with no issues—Google the right flags to use with openssl, plug that into Comodo, upload the certs and we’re good to go.

This time we ran into all sorts of problems; our (outdated) macOS Servers kept kicking back our certs as invalid. This could very well be a problem on the macOS side, as these are years out of date and I wouldn’t be surprised if they don’t have to code to talk updated encryption techniques. But we also had a problem with our mail server: it’s working with the cert we requested from the CSR generator at, but it lost the identity information: name of company, city state country.

Appears to be a cosmetic problem, but important cosmetics. What might have caused this? We generated the CSR using OpenSSL 2.6.5 on macOS 10.14. (Yes, also a few years old, but we have good reasons to stick with Mojave.)

nt.number theory – A novel identity connecting permanents to Bernoulli numbers

For a matrix $(a_{j,k})_{1le j,kle n}$ over a field, its permanent is defined by
$$mathrm{per}(a_{j,k})_{1le j,kle n}:=sum_{piin S_n}prod_{j=1}^n a_{j,pi(j)}.$$
In a recent preprint of mine, I investigated arithmetic properties of some permanents.

Let $n$ be a positive integer. I have proved that
$$mathrm{per}left(leftlfloorfrac{j+k}nrightrfloorright)_{1le j,kle n}=2^{n-1}+1$$
and that
$$detleft(leftlfloorfrac{j+k}nrightrfloorright)_{1le j,kle n}=(-1)^{n(n+1)/2-1}
qquadtext{if} n>1.$$

where $lfloor cdotrfloor$ is the floor function. The proofs are relatively easy.

Recall that the Bernoulli numbers $B_0,B_1,ldots$ are given by
$$frac x{e^x-1}=sum_{n=0}^infty B_nfrac{x^n}{n!} (|x|<2pi).$$
Those $G_n=2(1-2^n)B_n (n=1,2,3,ldots)$ are sometimes called Genocchi numbers.

Based on my numerical computation, here I pose the following two conjectures.

Conjecture 1. For any positive integer $n$, we have
$$mathrm{per}left(leftlfloorfrac{2j-k}nrightrfloorright)_{1le j,kle n}=2(2^{n+1}-1)B_{n+1}.tag{1}$$

Conjecture 2. For any positive integer $n$, we have
$$detleft(leftlfloorfrac{2j-k}nrightrfloorright)_{1le j,kle n}=begin{cases}(-1)^{(n^2-1)/8}&text{if} 2nmid n,\0&text{if} 2mid n.end{cases}tag{2}$$

Conjecture 2 seems easier than Conjecture 1.

QUESTION. Are the identities $(1)$ and $(2)$ correct? How to prove them?

Your comments are welcome!

ca.classical analysis and odes – How to determine if you’ve discovered a new identity for a special function

Often times, we consult resources, like Abramowitz and Stegun’s Handbook of Mathematical Functions, NIST’s database on special functions, or Mathematica to find identities which aid us with some kind of computation.

However, what if we want to know if we have found a new identity, want to systematically check against the above resources, and want to add to the library in the case the identity is new? Also, are there journals which, even today, still consider mathematical effort toward discovering identities of classical functions?

oauth – Use Managed Identity of Azure Function when calling another Azure Function

There’s a helpful Microsoft doc that describes how to configure security for a daemon client application. This works ok but it means the client app must present a client id and client secret to the /token endpoint of AAD in order to obtain the OAuth2 access token.

This makes no use of managed identity and means I need to ensure the security of the client secret of the daemon app. If the client id and secret of this app were to fall into the wrong hands, there’s nothing to prevent a bad actor obtaining an access token and calling the target service.

I realise that a certificate can be used instead of a client secret but my question is, can this be avoided through the use of the managed identity of the client daemon app?

additive combinatorics – How to prove the combinatorial identity $sum_{k=ell}^{n}binom{2n-k-1}{n-1}k2^k=2^ell nbinom{2n-ell}{n}$ for $ngeellge0$?

With the aid of the simple identity

in Item (1.79) on page 35 of the monograph

R. Sprugnoli, Riordan Array Proofs of Identities in Gould’s Book, University of Florence, Italy, 2006. (Has this monograph been formally published somewhere?)

I proved the combinatorial identity
sum_{k=1}^{n}binom{2n-k-1}{n-1}k2^k=nbinom{2n}{n}, quad ninmathbb{N}.

My question is: how to prove the more general combinatorial identity
sum_{k=ell}^{n}binom{2n-k-1}{n-1}k2^k=2^ell nbinom{2n-ell}{n}

for $ngeellge0$?