“If you pass all the thinking to GenAI, then the result is that the developer isn’t doing any thinking.”

It is a famous result of Kolmogorov that there exists a (Lebesgue) integrable function on the torus such that the partial sums of Fourier series of $f$ diverge almost everywhere (a.e.). More specifically, he exhibited an $f\in L^{1}(\mathbb{T})$ such that.

\begin{align*} \sup_{N\geq 1}\left|S_{N}f(x)ight|=\sup_{N\geq 1}\left|(f\ast D_{N})(x)ight|=\infty, \qquad\forall \text{ a.e. } x\in\mathbb{T}, \end{align*} where $S_{N}$ is the $N^{th}$ partial sum and $D_{N}$ is the $N^{th}$ Dirichlet kernel given below. \begin{align*} S_{N}f(x):=\sum_{\left|night|\leq N}\widehat{f}(n)e^{2\pi inx}, \quad D_{N}(x):=\dfrac{\sin 2\pi(N+\frac{1}{2})x}{\sin \pi x} \end{align*} and we identify $\mathbb{T}$ with the unit interval $[0,1]$.

I have read, for example pg. 118 in [Pinsky], that Kolmogorov’s counterexample can be replicated in the context of the Fourier transform on the real line $\mathbb{R}$, showing that $L^{1}$ pointwise Fourier inversion can fail quite horribly. If my understanding is correct, then the following claim is true:

Dr Alan D. Thompson

The largest review of “gold standard” antidepressant withdrawal studies to date has identified the type and incidence of symptoms experienced by people discontinuing antidepressants, finding most people do not experience severe withdrawal.

“Incidence and Nature of Antidepressant Discontinuation Symptoms, A Systematic Review and Meta-analysis” was published in JAMA Psychiatry.

In a systematic review and meta-analysis of previous randomized controlled trials relating to antidepressant withdrawal, a team of researchers led by Imperial College London and King’s College London concluded that, while participants who stopped antidepressants did experience an average of one more symptom than those who continued or were taking placebos, this was not enough to be judged as significant.

In England alone, one in six properties are at risk of flooding. Approaches of the past are no longer enough and Wildlife Trusts are leading the way in delivering Natural Flood Management Solutions which slow the flow of water through catchments, reducing flood risk downstream.

Some species of fig trees store calcium carbonate in their trunks—essentially turning themselves (partially) into stone, new research has found. The team of Kenyan, U.S., Austrian, and Swiss scientists found that the trees could draw carbon dioxide (CO₂) from the atmosphere and store it as calcium carbonate ‘rocks’ in the surrounding soil.

The research was presented at the Goldschmidt conference in Prague.

The trees —native to Kenya—are one of the first fruit trees shown to have this ability, known as the oxalate carbonate pathway.

What happens when you mix the tiniest molecule on Earth with your water bottle? Maybe… more than you’d expect.

Daniel Kleppner, an experimental physicist who helped to develop an atomic clock that became an essential part of global positioning systems, or GPS, and who also helped to discover a rare fundamental state of matter predicted by Albert Einstein and his fellow theoretical physicist Satyendra Nath Bose, died on June 16 in Palo Alto, Calif. He was 92.

His wife, Beatrice, confirmed the death. She said he collapsed while visiting their daughter, Sofie Kleppner, and her son, Darwin, who was graduating from high school.

It was in the mid-1950s, while he was doing a fellowship at the University of Cambridge in England, that Dr. Kleppner learned something surprising: It was possible, a tutor told him, to build a clock precise enough to detect the effects of gravity on time. Curious, he went in search of more information and read Norman Ramsey’s 1953 book “Nuclear Moments.”

A new instrument has enabled the first measurement of ultra-low backscatter in hollow-core fibers, validating their potential to outperform traditional fiber optics. Researchers from the University of Southampton and Université Laval, Canada, have successfully measured for the first time back-refle