My thought on (higher) education

I believe in college/graduate school, it is the students' responsibility to study and learn, they are their inner drive to further explore and establish themselves intellectually and spiritually.

Unlike K-12 education, where teachers cultivate wholeheartedly for the character, good learning habit of the students, prepare them to college/society. While in higher education, the role of professors are guidance and advisors, teaching and research are compliments of each other, their basic and applied research will fundamentally change the society in the long run.

Higher education and K-12 are totally two different stages. Please make the distinction. Thank you!




Grit坚毅、Zest激情、Self-control自制力、Optimism乐观态度、Gratitude感恩精神、Social intelligence社交智力、Curiosity 好奇心。



















Are Singularities Real? (ZT)

Are Singularities Real?

It’s hard to imagine infinity: something that is, by definition, larger than everything you can imagine. Physicists have to deal with the unimaginable every day, and have the tools to do so. But does their math describe reality?

Mathematicians have found a way to pack infinity into manageable equations and theorems as part of a class of mathematical oddities called “singularities.” To a mathematician, a singularity is simply a point where a function breaks down, as 1/x does when x gets close to zero. The defining property of a singular point is that it’s impossible to predict what happens beyond it. But are the singularities in mathematicians’ equations just an abstract concept? Or do they occur in nature?


Flickr user slightly-less-random, adapted under a Creative Commons license.

The word “singularity” was popularized in a 2005 book by Ray Kurzweil, who uses it to refer to an impending revolution in artificial intelligence (AI). According to Kurzweil, once artificial intelligences become smart enough to improve their own kin, a feedback loop will lead to a runaway process. After that, all bets are off: nobody knows what will happen. But Kurzweil’s technological singularity, if it comes to pass, is not a true singularity. There is no law of nature that limits our ability to predict what might happen once AI evolves past the “singularity” point; we’re confined instead by the limits of the human mind.

Though they sound exotic, mathematical singularities are actually common in solutions to all but the simplest equations in physics. The formation of shock waves and cracks, and even the motion of a billiard ball bouncing off a hard wall, can contain singularities. While these singularities fulfill the mathematical definition, they aren’t physically real either: they arise from idealized assumptions that physicists make to force the messy world of reality into the neat one of mathematics. In reality, no crack is perfectly sharp, no wall is perfectly hard, no shock wave is perfectly localized.

Here is another example: Turn down the water on your kitchen tap until it starts dripping. The hydrodynamical equation describing the surface of the drops has a singularity at the pinching point: you cannot from one drop predict where the next will be. But this singularity, too, can be avoided by applying a more suitable theory. Using atomic physics, you could, in principle, calculate exactly how the water stream breaks apart on the level of single atoms. All these singularities are thus artifacts of using a theory outside its range of applicability, on distances so short that a more precise theory would be needed.

The one type of singularity that might be real—that physicists don’t know how to resolve—is the one that appears in Einstein’s theory of General Relativity when matter collapses under the gravitational pull of its own weight. There is nothing in General Relativity that then stands in the way of this collapse. It will continue until all the matter is located at a single point of infinite matter density and infinite space-time curvature: a singularity.

The singularities that appear inside black holes pose a big problem for physicists. Crossing a black hole’s event horizon is like jumping into a river upstream of a waterfall, at a place where the water flows faster than you can swim. Whatever you do, you’ll end up being pulled down the waterfall. Likewise, whatever falls into a black hole is pulled down into the singularity. And once there, it reaches its end.

At the black hole singularity every particle’s path seems to dead-end. Space-time stops at the singularity, and nobody knows what happens at this point.  Yet we cannot imagine how something can just end. The black hole space-time is for this reason referred to as “incomplete,” but in General Relativity there is no way to complete it. This is the origin of the black hole information loss problem: It is the horizon that makes the information irretrievable, but it is the singularity that ultimately annihilates it. This is a big headache for physicists because such annihilation of information is incompatible with quantum mechanics.

The Big Bang is a singularity, too. If you run the expansion of the universe which we observe today backwards, then the density of matter must have been larger the younger the universe was, all the way back to an initial moment where the density must have been infinitely high: it must have been singular.

Are these singularities real, or just vestiges of the gap between math and reality? Based on their experience with other systems, physicists suspect that the singularities in General Relativity are a warning, a tip-off that we need another theory to describe the physics in the extreme situations when gravity is very strong and its quantum effects are very large. Physicists still don’t know how to describe the quantum effects of gravity, but we hope that by doing so we will one day resolve the singularities.

None of our measuring devices can show an infinite value. Not only have we never observed it, we don’t know how to observe it—we don’t know how to even give meaning to such an observation. Physicists therefore treat singularities as symptoms of an ill theory in need of a cure. It would be possible to mathematically deal with the singularities. But based on past experience and intuition, for all we know so far, nature doesn’t like sudden ends. Does nature make an exception for black holes? Right now, we just don’t know.

Why Is Our Universe Filled With Spirals? (ZT)

Why Is Our Universe Filled With Spirals?


The spiral galaxy M81 as seen through a combination of X-ray, optical, ultraviolet and infrared imaging techniques. 

Can you tell me why a spiral design seems omnipresent in our natural world and in the universe? I see this pattern in plant tendrils, flowers and leaves, pinecones, the unfurling of needles, as well as in astronomy. 

— Dorothea Fox Jakob, Toronto

The short answer is, sadly, we don’t really know. Nature does seem to have quite the affinity for spirals, though. In hurricanes and galaxies, the body rotation spawns spiral shapes: When the center turns faster than the periphery, waves within these phenomena get spun around into spirals.

The florets in a sunflower head also form two spirals, but there’s no rotation here — it’s simply an efficient packing solution for the plant. With 55 florets running clockwise and 34 counterclockwise, the sunflower is an example of a pattern of numbers called the Fibonacci sequence, named after the medieval mathematician who popularized it. It’s a simple pattern with complex results, and it is often found in nature.

The Fibonacci sequence starts with 0 and 1 and increases based on the sum of the previous two numbers: 0, 1, 1, 2, 3, 5, 8, 13, 21 and so on. You’ll notice that both groups of florets in a sunflower match Fibonacci numbers — as do the number of rows in a pinecone, the arrangement of leaves on a stem and many other natural formations. In fact, the spiral shape itself is built upon the rapidly increasing pattern of the Fibonacci sequence.

But since nature’s swirly patterns result from a few different mechanisms, the phenomenon is likely coincidence more than some underlying physical property of the universe. Nevertheless, it is striking that many natural examples follow this number sequence, either broadly in their curling shape or with their actual numerical values.

[This article originally appeared in print as “Ask Discover.”]


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