Tootfinder

Winners of the 2025 Close-Up Photographer of the Year #nature #photography

I feel like—from all sides—this case is The Big Moment for SCOTUS.
Have they completely abdicated or is there any semblance of separation of powers and rule of law?
If they side with Trump it feels like a death knell of sorts.
... If we aren’t there already.
-- Kendyl Hanks
https://

Chris Geidner (@chrisgeidner.bsky.social)
NEW: The birthright citizenship arguments will show what Donald Trump has done to the United States. Trump's executive order to end birthright citizenship is at the Supreme Court on Wednesday. The case will test his anti-immigrant authoritarian project — and the court. Today, at Law Dork: https://www.lawdork.com/p/birthright-citizenship-arguments-preview

Polynomials in $c$-free random variables with applications to free denoising
Adrian Celestino, Franz Lehner, Kamil Szpojankowski
https://arxiv.org/abs/2603.21372 https://arxiv.org/pdf/2603.21372 https://arxiv.org/html/2603.21372
arXiv:2603.21372v1 Announce Type: new
Abstract: We study distributions of polynomials in conditionally free (c-free) random variables, a notion of independence for two-state noncommutative probability spaces introduced by Bozejko, Leinert and Speicher. To this end we establish recursive relations between the joint Boolean cumulants of c-free random variables, analogous to previously found recursions for Boolean cumulants of free random variables. The algebraic reformulation of these recursions on the free associative algebra provides an effective formal machinery for the computation of the moment generating functions and thus the distributions of arbitrary self-adjoint polynomials in c-free random variables. As an application of a recent observation, our approach can be used to determine conditional expectations of the form $E[a|P(a,b)]$, where $P(a,b)$ is a self-adjoint polynomial in free (in the sense of Voiculescu) random variables $a,b$. We illustrate this with an example where $P(a,b)=i[a,b]$. Finally we define orthogonal projections that formally play the role of conditional expectations in the framework of c-freeness and share some properties with the conditional expectations of free variables. In particular they can be used to re-derive by purely algebraic methods the formula of Popa and Wang for the $\Sigma$-transform for the c-free multiplicative convolution.
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