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With the World Cup looming, there’s still no clear replacement for sports Twitter

Jayden Nelson of Canada celebrates 2-0 during the International Friendly match between Canada v Uzbekistan at the Commonwealth Stadium on June 1, 2026, in Edmonton Canada. | Photo by Dale MacMillan/Soccrates/Getty Images

Three years ago, when the women's World Cup kicked off in Australia and New Zealand, my social feeds were in a strange place. Twitter had just transformed into X, newcomer Threads was seemingly ascendant, and places like Bluesky had yet to garner much momentum. It left me with an odd, and admittedly silly, dilemma: I had nowhere good to post dumb jokes during games. And now, as the next World Cup is set to begin across Canada, the US, and Mexico, not much has changed.

There were a lot of problems with Twitter even when it was at its most popular, but it also introduced a new kind of live experience for many. Its real-time nature and huge au …

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Three key vital signs make up the "urban pulse" of a city

People often speak metaphorically of the heartbeat or pulse of a city, but according to the authors of a new paper published in the Proceedings of the National Academy of Sciences, cities do indeed have an "urban pulse"—an indication of urban "metabolic activity" that can be measured to suss out telltale patterns. And those patterns could help inform future public policy around urban planning.

The precise definition of urbanization has shifted over the centuries. Zhe Zhu of the University of Connecticut and his fellow authors adopted a broad version for their study. It features fundamental "processes of concurrent change in at least six dimensions, including demography, economy, infrastructure, environment, governance and culture," they wrote. "Together they give rise to outcomes, measurable results of the process, such as population growth, urban land expansion, GDP growth, and innovation." Their chosen metrics reflect this dynamic view: Cities are not static grids but "living, adaptive ecosystems."

“For decades, we had just been capturing the outcome of urbanization—a house that’s been built, or a road expansion,” said Zhu. “But you don’t really see the dynamics within an urban area. This is going to be a very impactful tool influencing not only top-down policy decisions from governments but also bottom-up decisions from everyday people navigating their cities.” One day we may be able to check a neighborhood's "urban pulse" while house-hunting, for instance, or while scouting potential locations for a new business.

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© Zhe Zhu/GERS Lab

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What Exactly is Dirac’s Delta Function?

Introduction: “Convenient Notation” In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred  to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta.  The Kronecker delta is simply the indexed components of the identity operator in matrix algebra: [tex]\delta^j_k =\left\{\begin{array}{lcl}1&\text{ if }...
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Fixing Things Which Can Go Wrong With Complex Numbers

Abstract This article will build on the hints about treating the complex numbers as a branched surface, briefly described and pictured in section 4.2 of https://www.physicsforums.com/insights/views-on-complex-numbers/#The-Radish. Using a particular set of conventions, all the problems described in https://www.physicsforums.com/insights/things-can-go-wrong-complex-numbers/  can be removed, and the rules described there as applying only to reals generalized to complex numbers. A...
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Series in Mathematics: From Zeno to Quantum Theory

Introduction Series play a decisive role in many branches of mathematics. They accompanied mathematical developments from Zeno of Elea (##5##-th century BC) and Archimedes of Syracuse (##3##-th century BC), to the fundamental building blocks of calculus from the ##17##-th century on, up to modern Lie theory which is crucial for our understanding of quantum theory....
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