Tootfinder

Mittwoch: Handy-Neuheit mit alten Funktionen, Google-Interesse an SpaceX-Plänen
Sony mit Klinkenbuchse & MicroSD-Fach Google-Server mit SpaceX im All München wieder mit Open Source Gigabit-Anschlüsse holen auf Ankers neue Solarbank

Mittwoch: Handy-Neuheit mit alten Funktionen, Google-Interesse an SpaceX-Plänen
Sony mit Klinkenbuchse & MicroSD-Fach + Google-Server mit SpaceX im All + München wieder mit Open Source + Gigabit-Anschlüsse holen auf + Ankers neue Solarbank

On meromorphic pencils, cusp singularities and holomorphic foliations in the complex plane
Bruno Scardua
https://arxiv.org/abs/2607.18526 https://arxiv.org/pdf/2607.18526 https://arxiv.org/html/2607.18526
arXiv:2607.18526v1 Announce Type: new
Abstract: We study polynomial holomorphic $1$-forms in $\mathbb{C}^2$ that are homologically trivial along the fibers of meromorphic pencils of the form $
\phi = \frac{f^p}{g^q}, $
where $f,g$ are holomorphic functions (possibly polynomials) in general position and $(p,q)=1$. We first establish a homological characterization of relative exactness: if a polynomial $1$-form $\Omega$ has vanishing periods along every closed path contained in the fibers $\phi_c$, then $\Omega$ decomposes as $\Omega = a\omega_0 dh,$ where $\omega_0 = p gdf - q f dg,$ for suitable polynomials $a$ and $h$. In the homogeneous case, degree constraints force $a$ to be constant. We then apply this integration principle to foliations leaving invariant plane curve singularities of cusp type \[ f^p g^q = 0. \] Under a natural genericity (Morse type) condition, we prove a globalization theorem showing that homological triviality along the associated pencil implies that $\Omega$ is a polynomial cusp basic form, \[ \Omega = d(f^p g^q) \lambda (p gdf - q fdg), \qquad \lambda \in \mathbb{C}. \] In particular, such foliations admit Liouvillian first integrals of hypergeometric type.
Our results provide a bridge between relative cohomology, the geometry of rational pencils, and the analytic structure of cusp foliations, yielding explicit normal forms and first integrals under homological hypotheses.
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Sidequest incoming

Can't remember if I've already posted this, and it's worth posting again: Microsoft admits (in effect) that their only path to viability with 'AI' is to make people entirely dependent on it.... like unable to function without it. https://www.

The ancient Middle-East had a solution for surviving the heat without chillers: windcatchers (and the related solar chimneys). Such things should be employed for first-tier cooling where possible to reduce energy demand, leaving chillers to take up the slack (until we figure out how to get infinity gigawatts of free power into every house, at which point we should just capture all the excess heat from running chillers and use the Seebeck effect to convert it into electricity to power a giant…

New arXiv preprint!  With my #PurdueFortWayne colleague Yifei Pan.  🐘 Here's a thread (1/n)
https://arxiv.org/abs/2606.01447

Constructing Discontinuous but Locally Bounded Rational Functions using Łojasiewicz Inequalities
For real multivariate polynomials $P$ and $Q$ both vanishing at a point, if the zero set of $Q$ is contained in the zero set of $P$, then there exists a rational function of the form $P^{p}/Q^{q}$ which is locally bounded and such that its extension that vanishes on the zero set of $Q$ is discontinuous. The proof uses inequalities of Lojasiewicz.

Wie funktioniert die #Knotenpunktwegweisung in #Brandenburg? 🗺️🚴‍♀️ Mit nummerierten #Knotenpunkten an Kreuzungen kann man in diesem Bundesland eine Route leicht planen und auch …

Didn't get a lot of sleep last night and everything kinda hurts so I've finished working earlier than usual. I am planning on a long relaxing bath before getting ready to go out for dinner with husband. And perhaps a pre and postprandial stroll will help iron out all the remaining kinks. It's still nicely Autumn in Canberra currently.
#Ageing

A structural reduction for the symmetric hit problem in four variables
Dang Vo Phuc
https://arxiv.org/abs/2606.02626 https://arxiv.org/pdf/2606.02626 https://arxiv.org/html/2606.02626
arXiv:2606.02626v1 Announce Type: new
Abstract: Let $\mathcal{A}$ be the mod $2$ Steenrod algebra, and $P(n) = \mathbb{F}_2[x_1, \dots, x_n]$ be the polynomial algebra viewed as an unstable module over $\mathcal{A}$. The symmetric hit conjecture asks whether the symmetrization of a hit monomial in $P(n)$ is always hit in the symmetric invariant subalgebra $B(n) = P(n)^{\Sigma_n}$. While resolved for $n \leq 3$, the case $n=4$ presents significant obstructions due to combinatorial complexity, orbit cancellations intrinsically tied to $\Sigma_4$-stabilizers, and the emergence of strongly spike-free survivor modules. This paper introduces a conditional structural reduction to overcome these obstructions in the domain where the numerical weight satisfies $\mu(d) \leq 4$. By integrating Walker-Wood duality with a new $\Sigma_4$-stabilizer parity analysis, we reduce the global conjecture to localized algebraic conditions: a symmetric lower-spike reduction and a strengthened four-row digital-engineering hypothesis. Assuming these inputs, the conjecture follows by lexicographic induction on the column-sum and row-sum sequences of the binary exponent matrices.
Our approach isolates the four-variable repeated-row anomaly into exact local identities, utilizing global Steenrod-kernel functionals lifted from local spike-free quotients to detect potential survivor elements. Finally, we provide explicit monomial-level computations in degrees $8$, $12$, and $14$, explicitly illustrating the stabilizer mechanism in practice and framing the precise algebraic identities required for a future unconditional proof.
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On the fundamental solutions of two nonlocal parabolic equations related to logarithmic Laplacians
Bart Rosenzweig, Jonathan Stanfill
https://arxiv.org/abs/2606.04225 https://arxiv.org/pdf/2606.04225 https://arxiv.org/html/2606.04225
arXiv:2606.04225v1 Announce Type: new
Abstract: We answer in the affirmative a question posed by V. Maz'ya of whether one can continue as a meromorphic function of $t$ the series representation of the fundamental solution of a certain nonlocal parabolic equation associated to a logarithmic Laplacian on the circle, which arises in the study of boundary value problems associated to the ordinary Laplacian on domains with thin cavities. The $a\ln(n) O(1)$ growth of the eigenvalues of the integral operator, together with explicit formulas for the eigenfunctions and the subleading asymptotic behavior of the eigenvalues, allows us to show that the fundamental solution is reminiscent of a sum of shifted Riemann zeta functions or polylogarithms, depending on the spatial variable. We show an analogous result for an operator related to a different logarithmic Laplacian on the interval, whose structure is similar. Along the way we are led to prove and to conjecture a number of curious identities involving Bell polynomials and Bernoulli numbers related to the exponential of the digamma function which are of independent interest.
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