Introductory Methods in Elementary Number Theory
Author: Delta Note: “introductory methods” does not mean “methods for getting started with elementary number theory.” It means “methods used at the introductory level of elementary number theory.” This distinction is stated upfront to avoid ambiguity. Before the article begins, let me ask a question: Is √2 an integer? There are only two possible answers: yes or no. I expect most readers to say no, because √2 is ordinarily understood as an irrational number, while the integers are 0, ±1, ±2, and so on. What I say next, however, may overturn some people’s assumptions. √2 really is an integer. To explain this, we must first understand the concept of an algebraic number: Clearly, every rational number and every irrational number expressible using radicals is algebraic, because ...
An Introduction to the Poincaré Conjecture (with Notes)
Author: Delta Introduction In 1904, in a paper titled Fifth Supplement to Analysis Situs, Henri Poincaré proposed the following conjecture: If every closed curve in a three-dimensional space can be contracted to a point, then that space must be a three-dimensional ball. What exactly does this statement mean? Is this really the famously difficult Poincaré conjecture? How could a claim that looks “obvious,” uses no mathematical notation, and is stated entirely in ordinary language trouble mathematicians for ninety-nine years? This article sets out to answer those questions. Main text The statement itself is not hard to understand. Let us first work in the opposite direction and consider a closed curve inside a three-dimensional ball D3. The following mathematical simulation sho ...
Leap Months in the Chinese Calendar and the 24 Solar Terms
Author: 斎藤信 Reviewer: 2333 At the end of the previous article, we learned that there will be no leap first month until at least 2060, so do not expect a two-month winter holiday. This year was purely an exception, though. Why is that? The end of the previous article offers a hint, but readers can think about it for themselves first. Have a wonderful start to the new semester (runs away
Immune Evasion in Five Minutes
Author: Shenzhou Reviewed by: Shiye The immune system has many ways to resist invasion by a wide range of pathogens. Pathogens, however, have strategies of their own: they evade immune attack so they can survive in the body and wait for an opportunity to spread. This is known as immune evasion. Viruses, bacteria, and parasites do not all evade immunity in the same way. For viruses, immune evasion is a major cause of persistent infection and treatment failure. Their tactics include establishing latent infections, changing their antigens, and disrupting immune function. One of the better-known strategies is antigenic variation, which is common among many RNA viruses. When the structure of a viral antigen changes, antibodies that the body has already produced may no longer bind to ...
An Introduction to the Gibbs Phenomenon
Author: phy Dongxi First-year students taking the second semester of calculus are probably wrestling with Fourier series by now, and many will be complaining about the complicated integrals. Anyone with plotting software, however, has probably tried approximating periodic functions with trigonometric functions. Careful observers may have noticed something: when a periodic function with a discontinuity is expanded as a Fourier series and reconstructed from finitely many terms, increasing the number of terms moves the peaks in the reconstructed waveform closer to the discontinuity. For a large number of terms, the overshoot approaches a constant—about 9% of the total jump. This is the Gibbs phenomenon. Increasing the order does not reduce the overshoot; it only moves its peak closer ...
A Guide to the Night Sky: Regions and Celestial Events
Author: Qingliu Reviewer: Saito Shin Astronomy is founded on observation. From the first human gaze at the stars to the first image of a black hole, the field was born and has advanced through what people observe. Modern astronomy has long since moved beyond relying on the unaided eye, but learning how to observe remains the first lesson for an amateur astronomer. This article introduces celestial events encountered in observational astronomy. Effective observation begins with knowing how the sky is organized and which important phenomena appear in each region. The image above shows the Big Dipper and Polaris. They are easy to locate by looking north after determining the latitude of your observing site. Polaris stands above the horizon by an angle equal to the observer’s geograph ...
The Zodiacal Signs and the Thirteen Zodiac Constellations
Author: Congyu The twelve zodiacal signs and the thirteen zodiac constellations have long been confused with each other, especially by people who are not astronomy enthusiasts. For a variety of reasons, the zodiac constellations used today differ greatly from the ancient zodiacal signs. This article offers a brief introduction and explains the distinction between them. The projection of Earth's orbit around the Sun onto the celestial sphere is the ecliptic (black in the figure), while the projection of Earth's equator is the celestial equator (green). The two points where the ecliptic and the celestial equator intersect are called the equinoxes. Source: Wikipedia Before discussing the twelve signs and the zodiac constellations, it helps to explain what the “ecliptic” is. The plane o ...
A Brief History of Humankind: The Sapiens Flood (Part II)
Author: Self-Control Reviewer: Guanfu Juntian P.S. First, an apology. In the article from two issues ago, “The Sapiens Flood, Part I,” I overlooked an error. Homo sapiens left Africa about 70,000 years ago, but I wrote that this happened when the species first appeared, 150,000 years ago. That date is itself uncertain: estimates of 160,000 or 200,000 years also appear, partly because some classifications treat Neanderthals as early Homo sapiens. Here, we will follow the view taken in the book. Fortunately, a friend on Qzone pointed out the mistake. My apologies. Two issues ago, we saw how Homo sapiens entered the Australian “level” and, in only a few thousand years, effortlessly swept away most of the continent’s medium-sized and large animals. Only the boxing kangaroos just managed ...
Storms Beyond the Atmosphere—A Conversation about Space Weather
Author: Abaiter Reviewed by: Lai Yuan When we hear the word “weather,” we usually think first of wind, rain, thunder, and similar phenomena. They are indeed an important part of our environment and the weather we experience most directly, but all of them occur within the thin troposphere, no more than a dozen or so kilometers above the ground. What does a dozen kilometers mean? Walking from one end of a moderately large city to the other can take you farther than that. The sky, as we know, is vast and boundless. The weather we experience directly in daily life is therefore only part of the whole. Above the troposphere, and especially beyond the atmosphere, another kind of turbulence unfolds: space weather. Air molecules carry weather in the troposphere, with water vapor playing an es ...
An Introduction to Deep-Sky Object Catalogues
Author: 丛雨 The Messier Catalogue Of all deep-sky catalogues, the Messier Catalogue is probably the best known. Compiled under the direction of French astronomer Charles Messier, it contains 110 deep-sky objects: 12 nebulae (M1, the Crab Nebula, is the only supernova remnant in the catalogue), 57 star clusters (28 globular and 29 open clusters), 40 galaxies, and one double star, M40, included by mistake. Many objects familiar to amateur astronomers are Messier objects, including the Andromeda Galaxy (M31), the Orion Nebula (M42), and the Pleiades (M45). Messier was in fact an enthusiastic comet hunter. During his searches, he encountered objects that looked like comets but were not, prompting him to compile the catalogue first published in 1771. His original list contained only 45 obje ...