Wolfgang's tensor

You are the books📚 you read,the movies📽 you watch,the music🎧 you listen to, and the people👫 you spend time with.

[Wolfgang's tensor @Tensorwg006]

Nehruji is responsible for this. Right Modiji?

[Iffat Hasan Rizvi @IffatHasanRizvi]

She asked Modi a question. India went wild.

I interviewed @HelleLyng Norwegian journalist who question Modi in Oslo, Now trolls are after her.
Coincidence: We work on the same floor

Stay tuned for the interview https://t.co/0EDBGH3Ey3 !
#PressFreedom #Modi #Norway #India

[Math Lady Hazel 🇦🇷 @mathladyhazel]

Lissajous curve table. Best geometry gif ever.

[Oren Gottfried, MD @OGdukeneurosurg]

Eyes by medacope 360

[Erika  @ExploreCosmos_]

The Riemann hypothesis is one of the most famous unsolved problems in mathematics, but also one of the strangest: almost everyone agrees it is central, enormously important and probably true, yet very few mathematicians are actively trying to prove it.

The reason is not lack of interest. It is that the problem appears to sit far beyond the reach of current mathematical tools. It has resisted every serious attempt since Bernhard Riemann proposed it in 1859, became one of Hilbert’s great problems in 1900, and was later named one of the Clay Mathematics Institute’s Millennium Prize Problems, with a million-dollar reward attached to a proof.

At its core, the Riemann hypothesis is about prime numbers, the indivisible building blocks of arithmetic. Every whole number can be broken down into primes, which makes them fundamental to number theory in much the same way that forces are fundamental to physics.

But primes behave in a frustrating way: they are not distributed regularly along the number line. They become less frequent as numbers get larger, and there are statistical patterns in their distribution, but their exact locations still look irregular and difficult to predict.

Gauss noticed that the density of primes follows a broad trend, but that trend was only an approximation. The deeper question was how to understand the errors, the deviations, the fine structure behind where primes actually appear.

Riemann’s insight was to connect this problem to a complex mathematical object called the zeta function. This function takes complex numbers as inputs, meaning numbers with both a real and an imaginary part, and produces complex outputs.

The crucial points are the places where the function equals zero.

Riemann realized that these zeros encode information about the distribution of prime numbers. In a simplified way, Gauss’s approximation gives the broad shape of the prime distribution, while the zeros of the zeta function describe the corrections needed to make that picture precise.

It is comparable to decomposing a musical note into harmonics: each zero contributes one part of the “sound” of the primes.

The Riemann hypothesis says that all the important, nontrivial zeros of the zeta function lie on one specific vertical line in the complex plane, known as the critical line, where the real part is exactly one half. If this is true, it would mean that the hidden fluctuations in the primes are constrained in the cleanest possible way.

It would not make prime numbers simple, but it would show that their apparent randomness is governed by a deep and elegant order. That is why the hypothesis matters so much: it would give mathematicians a much sharper understanding of the primes and would confirm a structure that already underlies a huge amount of modern number theory.

The problem also matters because it has connections far beyond prime numbers. The same kind of mathematics appears in other L-functions, which are attached to many different mathematical objects. Versions of the Riemann hypothesis have become organizing principles across number theory and related fields.

There are also surprising links to physics, including patterns resembling energy levels in atomic nuclei, random systems, chaos theory and even black hole mathematics. This does not mean the hypothesis is “about” physics in a direct sense, but it shows that the same mathematical structures appear in very different parts of reality.

One of the striking points is that mathematicians already use the Riemann hypothesis as a conditional tool. Many papers prove results of the form: if the Riemann hypothesis is true, then something else follows. In other words, part of mathematics has already been built around the assumption that it is true, even though no one has proved it.

A proof would not simply settle an old question; it would lock into place a vast network of results and intuitions that mathematicians have been using for decades.

But proving it is another matter. The problem is so difficult that it falls outside the normal “productive zone” of mathematical research. Mathematicians usually work on problems that are hard enough to matter but not so hard that there is no visible path forward.

The Riemann hypothesis is different.

It is important precisely because solving it would probably require new mathematics, not just clever use of existing techniques. Even recent progress, such as work by James Maynard and Larry Guth, has only slightly improved known bounds on where the zeros can be. That progress is significant, but it does not look like a direct path to a proof.

So the paradox is that the Riemann hypothesis is both central and almost untouchable. It is a problem everyone recognizes, many mathematicians assume, and almost no one knows how to attack. Its real value may not lie only in the final answer, but in the kind of mathematics that would have to be invented to reach it.

A proof would likely reveal why the primes, despite their apparent disorder, obey such a profound hidden structure. That is why the hypothesis remains intimidating: not because mathematicians do not care, but because caring is not enough when the problem seems to demand a new way of seeing numbers.

[Dr Rupam Manna MD @DrRupamOncology]

What to see in a Breast Mammogram – A Systematic Approach 🔥

Step-by-step guide to reading mammograms like a pro:
✓ Basic details
✓ Density (A–D)
✓ Masses, calcifications, distortion
✓ Asymmetries, skin/nipple, axilla
✓ Prior comparison → BI-RADS

Fully aligned with the latest ACR BI-RADS® v2025 Manual (Dec 2025 update). Minor calcification terminology refinements only — core teaching points remain unchanged.

Perfect for residents, oncologists, radiologists & anyone wanting to understand breast imaging better.

Save & share! 👇

#BreastCancer #Mammogram #BIRADS #Radiology #BreastImaging #CancerAwareness #MedEd #Oncology

Follow for more: @DrRupamOncology

[Physics In History @PhysInHistory]

The most beautiful equations in Physics and Mathematics.

A visual thread 👇

[Internal Medicine @pgMedicine]

Localisation in ECG

[Math Files @Math_files]

Black holes seem mysterious, but just a few ideas describe them: their size depends on mass, they slowly emit radiation discovered by Stephen Hawking, and their entropy is linked to the surface area of the horizon, an idea by Jacob Bekenstein.

[Real World Physics @RWPhysics]

Relativity 😍

[CardiovascularCorner @TrackYourHeart]

Abdominal pain causes by location

[CardiovascularCorner @TrackYourHeart]

Critical ECG Findings Every Clinician Must Recognize.
Source: Rose C, Goodnough R.

[Mathematica @mathemetica]

Shannon Entropy: Measuring Uncertainty in Information

H(X) = - ∑ P(xᵢ) log P(xᵢ)
This is the legendary formula by Claude Elwood Shannon (1916–2001); the father of Information Theory.
Entropy quantifies how much uncertainty (or average information) is contained in the outcome of a random variable X. The more unpredictable the outcomes, the higher the entropy.

From data compression and cryptography to AI and communications; this concept powers the digital world.

[Cosmos Archive @cosmosarcive]

In 1972, physicist Jacob Bekenstein made a wild claim that black holes have entropy. They’re not just cosmic voids, they carry hidden information written on their event horizon. Years later Hawking proved they also have a temperature, meaning they glow faintly and eventually evaporate.

This image lays out the three beautiful equations that reveal black holes as thermodynamic objects, complete with size, heat, and memory.

[Vital Viz @VitalViz]

Oxygen Delivery Devices .

[Vital Viz @VitalViz]

Types of Strokes .

[Math Cafe @Riazi_Cafe_en]

"Solving Problems in Algebra and Trigonometry"

Archive link: https://t.co/5bdJo94xGX