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A Complete Characterization of Finite-Order Entire Solutions to Fermat-Type Partial Differential-Difference Systems in $\mathbb{C}^n$
Sujoy Majumder, Jhilik Banerjee, Abhijit Banerjee
https://arxiv.org/abs/2606.05240 https://arxiv.org/pdf/2606.05240 https://arxiv.org/html/2606.05240
arXiv:2606.05240v1 Announce Type: new
Abstract: The primary objective of this paper is to determine the explicit existence form and structure of finite-order entire solutions in $\mathbb{C}^n$ of the following system of Fermat-type partial differential-difference equations: \[\begin{cases} \left(\frac{\partial f_1\left(z\right)}{\partial z_1}\right)^{n_1} (f_2 \left(z c\right)-f_1(z) )^{m_1}= 1,
\medskip \left(\frac{\partial f_2\left(z\right)}{\partial z_1}\right)^{n_2} (f_1 \left(z c \right)-f_2(z) )^{m_2}= 1, \end{cases}\]
for different choices of the positive integers $n_1$, $n_2$, $m_1$, and $m_2$, where $c=(c_1,c_2,\ldots,c_n)$. We characterize the precise structure of finite-order transcendental entire solutions and extend the results of Xu et al. \cite{XLL1} from the setting of $\mathbb{C}^2$ to the more general space $\mathbb{C}^m$. In addition, several examples are presented to demonstrate the effectiveness and sharpness of the main results.
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EU Member States are deprioritising 'non-wires' solutions or non-fossil flexibility as alternatives for network expansion.
https://www.consilium.europa.eu/en/press/press-releases/20…

One click sends a command
from the Home Assistant app
on my mobile phone
via WiFi
to the WiFi Router
to a Satellite
to a ground station in Germany
to my VPN endpoint in Iceland
to London
to Amsterdam
to Frankfurt
to Zurich
to Winterthur
to my router
to my reverse proxy
to my Home Assistant server
to my access point
to the WiFi
to the WiFi-Bridge
to the door lock
*click* *motor whirring* *door…

Seattle-based Brinc Drones, which makes 911 response drones, raised $125M led by Motorola Solutions, bringing its total funding to more than $280M (Todd Bishop/GeekWire)
https://www.geekwire.com/2026/motorola-leads-125m…

Motorola leads $125M round for Brinc, fueling 911 drone expansion amid U.S. import crackdown
Seattle-based 911 drone maker Brinc raised $125 million led by Motorola Solutions, at a valuation the company says has nearly doubled from a year ago. The funding will support expanded manufacturing and a new Seattle headquarters and factory later this year.

Quebec hydropower is now flowing to New York City—1.25 GW of clean electricity displacing fossil generation.
Meanwhile, Twelve launched America's first commercial e-jet fuel plant, turning CO2 and renewable power into synthetic jet fuel.
And China brought online a massive solar-battery-hydrogen complex to power industry with renewables.
Real infrastructure. Real progress. Today.
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Following a blockbuster IPO earlier this month,
Elon Musk’s SpaceX experienced a rude awakening.
The rocket company’s shares have now been sliding for four consecutive days,
wiping out nearly all the gains the public offering had initially made.
Now the broader stock market is experiencing a similar and intensifying sell-off.
S&P 500 futures slid 1.6 percent on Tuesday,
while Nasdaq 100 futures dropped 2.8 percent, on track to wipe out over $1 trillio…

Tech Stocks Are Abruptly Collapsing
The global stock market is experiencing an intensifying sell-off, intensifying fears of a bursting AI bubble.

#goodmorning
I spent yesterday loosing myself in my IT setup and still don't really know if I'm following a good solution. But I've learned a bit about Forgejo...

“Awake! for Morning in the Bowl of Night
Has flung the Stone that puts the Stars to Flight:        
And Lo! The Hunter of the East has caught                        
The Sultan’s Turret in a Noose of Light”
(Edward FitzGerald and the Rubšiyšt of Omar Khayyšm, 1st edition 1859)
#Poetry #Translation

"Dr. Chen’s lawyers said that Ms. Herridge’s “claimed right to protect her source(s) glosses over exactly what she is shielding from Chen, the courts, and the public: the identity of federal official(s) who broke the law, abused their access to sensitive records to harm a private citizen and laundered that corrupt, unlawful conduct through a reporter to escape detection.”
Supreme Court Allows Reporter to Be Fined for Failing to Disclose Source - The New York Times
https://www.nytimes.com/2026/07/02/us/politics/supreme-court-herridge-first-amendment.html

Juvenile white sharks are already appearing along the California coastline,
fleeing warmer than usual waters in Mexico during what’s expected to be an incredibly strong El Niño.
But while shark experts say it could be the sharkiest summer in California in a decade,
they also say beachgoers shouldn’t be too alarmed as the apex predators have spent generations learning what’s on the menu.
And humans aren’t their preferred food

California braces for its sharkiest summer in a decade – but experts say don’t panic
Young great white sharks are returning in large numbers, but researchers say humans are not on the menu