Category: On the constellations of weierstrass points

On the constellations of weierstrass points

The ecliptic, or the apparent path of the Sun, is defined by the circular path of the Sun across the sky, as seen from Earth. In other words, the Sun appears to pass through these constellations over the course of a year.

The passage of the Sun through the zodiac is a cycle that was used by ancient cultures to determine the time of year. Most of the planets in the solar system have orbits that take them near the ecliptic plane, within about 8 degrees above or below. The 12 constellations in the zodiac family can all be seen along the ecliptic. The Sun also passes through Ophiuchus and Cetusbut these constellations are not part of the zodiac, but belong to the Hercules and Perseus families respectively.

on the constellations of weierstrass points

The northern zodiac constellations — PiscesAriesTaurusGeminiCancer and Leo — are located in the eastern celestial hemisphere, while the southern — VirgoLibraScorpiusSagittariusCapricornus and Aquarius — are found in the west.

Today, the term zodiac is mostly associated with astrology, with the 12 signs of the western zodiac corresponding to the 12 constellations seen along the ecliptic.

The so-called cardinal signs AriesCancerLibra and Capricorn mark the beginning of the four seasons, i. The constellations represent the astrological signs of the zodiac. The image was taken from the Atlas Coelestis. The largest of the 12 zodiac constellations is Virgowhich covers Virgo is also the second largest of all 88 constellations, only slightly smaller than Hydra.

Covering an area of Also located in the southern celestial hemisphere, Aquarius represents Ganymede, the cup bearer to the Olympian gods in Greek mythology. Leothe third largest zodiac constellation, occupies an area of It represents the Nemean lion, a mythical monster killed by Heracles as part of his 12 labours.

Pisces comes in 4th with Libra In terms of brightness, several of the 12 constellations contain some of the brightest stars in the sky. Aldebaranthe brightest star in Taurusis the 14th brightest of all stars, followed by Spicathe brightest star in Virgo and 15th brightest star in the sky, Antaresthe bright red supergiant in Scorpius and 16th brightest star, Pollux in Geminithe 17th brightest of all stars, and Regulus in Leowhich comes in 21st overall.

Today, zodiac constellations are most commonly brought up in the context of western astrology, as the 12 constellations correspond to the 12 signs of the zodiac.

They are groups of stars that appear to be close to each other, arbitrarily named after different objects, animals, or figures from mythology by human observers at some point in history. Constellations make a two-dimensional map of the sky used for orientation, to make it easier for astronomers to find objects and explain their location and for navigators to use stars to determine their position. While even Carl Gustav Jung said that astrology holds some value as a theory of the personality, and it can use the scientific approach, it is in itself not based on any kind of science.

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It is mandatory to procure user consent prior to running these cookies on your website. Zodiac constellations are constellations that lie along the plane of the ecliptic. Zodiac signs, image: Wolfgang Rieger. This website uses cookies to personalise content and ads, and to analyse user traffic. By continuing to use the site, you agree to the use of cookies. Accept Reject Read More. Close Privacy Overview This website uses cookies to improve your experience while you navigate through the website.

Out of these cookies, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website. These cookies will be stored in your browser only with your consent.In mathematicsthe Weierstrass function is a pathological example of a real-valued function on the real line.

The function has the property that it is continuous everywhere but differentiable nowhere. It is named after its discoverer Karl Weierstrass. Historically, the Weierstrass function is important because it was the first published to challenge the notion that every continuous function was differentiable except on a set of isolated points.

The proof that this function is continuous everywhere is not difficult. Since the terms of the infinite series which defines it are bounded by and this has finite sum forconvergence of the sum of the terms is uniform by the Weierstrass M-test with.

Since each partial sum is continuous and the uniform limit of continuous functions is continuous, is continuous. To prove that it is nowhere differentiable, we consider a point and show that the function is not differentiable at that point for all x.

To do this, we construct two sequences of points and which both converge to x, having the property that. Naively it might be expected that a continuous function must have a derivative, or that the set of points where it is not differentiable should be "small" in some sense.

According to Weierstrass in his paper, earlier mathematicians including Gauss had often assumed that this was true. This might be because it is difficult to draw or visualise a continuous function whose set of nondifferentiable points is something other than a finite set of points.

Analogous results for better behaved classes of continuous functions do exist, for example the Lipschitz functionswhose set of non-differentiability points must be a Lebesgue null set Rademacher's theorem.

When we try to draw a general continuous function, we usually draw the graph of a function which is Lipschitz and has other nice properties. The Weierstrass function could perhaps be described as one of the very first fractals studied, although this term was not used until much later. The function has detail at every level, so zooming in on a piece of the curve does not show it getting progressively closer and closer to a straight line.

Rather between any two points no matter how close, the function will not be monotone. The Hausdorff dimension of the graph of the classical Weierstrass function is bounded above bywhere a and b are the constants in the construction above and is generally believed to be exactly that value, but this had not been proven rigorously.

Weierstrass function

The term Weierstrass function is often used in real analysis to refer to any function with similar properties and construction to Weierstrass' original example. For example, the cosine function can be replaced in the infinite series by a piecewise linear "zigzag" function. Hardy showed that the function of the above construction is nowhere differentiable with the assumptions.

It turns out that the Weierstrass function is far from being an isolated example: although it is "pathological", it is also "typical" of continuous functions:. This entry is from Wikipedia, the leading user-contributed encyclopedia. It may not have been reviewed by professional editors see full disclaimer.A constellation is a group of stars that appears to form a pattern or picture like Orion the Great Hunter, Leo the Lion, or Taurus the Bull.

Constellations are easily recognizable patterns that help people orient themselves using the night sky. Are the stars in a constellation near each other? Not necessarily. Each constellation is a collection of stars that are distributed in space in three dimensions — the stars are all different distances from Earth.

The stars in a constellation appear to be in the same plane because we are viewing them from very, very, far away. Stars vary greatly in size, distance from Earth, and temperature. Dimmer stars may be smaller, farther away, or cooler than brighter stars. By the same token, the brightest stars are not necessarily the closest. Of the stars in Cygnus, the swan, the faintest star is the closest and the brightest star is the farthest!

How are constellations named? Most of the constellation names we know came from the ancient Middle Eastern, Greek, and Roman cultures. They identified clusters of stars as gods, goddesses, animals, and objects of their stories. It is important to understand that these were not the only cultures populating the night sky with characters important to their lives. Cultures all over the world and throughout time — Native American, Asian, and African — have made pictures with those same stars.

In some cases the constellations may have had ceremonial or religious significance. In other cases, the star groupings helped to mark the passage of time between planting and harvesting. There are 38 modern constellations.

In the International Astronomical Union officially listed 88 modern and ancient constellations one of the ancient constellations was divided into 3 parts and drew a boundary around each. The boundary edges meet, dividing the imaginary sphere — the celestial sphere — surrounding Earth into 88 pieces. Astronomers consider any star within a constellation boundary to be part of that constellation, even if it is not part of the actual picture.

All stars, however, fall within the boundaries of one of the 88 constellation regions. You can see some of these stars by observing the sky on a dark night. If you look at the sky with binoculars, you will see even more stars. If you have a telescope, you will see even more!In mathematicsthe Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere.

It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass. The Weierstrass function has historically served the role of a pathological function, being the first published example specifically concocted to challenge the notion that every continuous function is differentiable except on a set of isolated points. The functions were impossible to visualize until the arrival of computers in the next century, so the proof of the result relied entirely on technically demanding theoretical steps.

The results did not gain wide acceptance until practical applications such as models of Brownian motion necessitated infinitely jagged functions nowadays known as fractal curves. In Weierstrass's original paper, the function was defined as a Fourier series :. Since each partial sum is continuous, by the uniform limit theoremit follows that f is continuous. Additionally, since each partial sum is uniformly continuousit follows that f is also uniformly continuous.

It might be expected that a continuous function must have a derivative, or that the set of points where it is not differentiable should be "small" in some sense. According to Weierstrass in his paper, earlier mathematicians including Gauss had often assumed that this was true. This might be because it is difficult to draw or visualise a continuous function whose set of nondifferentiable points is something other than a countable set of points.

Analogous results for better behaved classes of continuous functions do exist, for example the Lipschitz functionswhose set of non-differentiability points must be a Lebesgue null set Rademacher's theorem.

When we try to draw a general continuous function, we usually draw the graph of a function which is Lipschitz or otherwise well-behaved. The Weierstrass function was one of the first fractals studied, although this term was not used until much later. The function has detail at every level, so zooming in on a piece of the curve does not show it getting progressively closer and closer to a straight line. Rather between any two points no matter how close, the function will not be monotone.

The term Weierstrass function is often used in real analysis to refer to any function with similar properties and construction to Weierstrass's original example. For example, the cosine function can be replaced in the infinite series by a piecewise linear "zigzag" function. It turns out that the Weierstrass function is far from being an isolated example: although it is "pathological", it is also "typical" of continuous functions:. The derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument.

Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.

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In complex analysis, the Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex number plane C which is not all of Cthen there exists a biholomorphic mapping f from U onto the open unit disk. In mathematical analysis, the Weierstrass approximation theorem states that every continuous function defined on a closed interval [ ab ] can be uniformly approximated as closely as desired by a polynomial function.

What are stars and constellations? Why questions for kids. Educational cartoon

Because polynomials are among the simplest functions, and because computers can directly evaluate polynomials, this theorem has both practical and theoretical relevance, especially in polynomial interpolation. The original version of this result was established by Karl Weierstrass in using the Weierstrass transform.

In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions converges uniformly to a limiting function on a set if, given any arbitrarily small positive numbera number can be found such that each of the functions differ from by no more than at every point in. Described in an informal way, if converges to uniformly, then the rate at which approaches is "uniform" throughout its domain in the following sense: in order to determine how large needs to be to guarantee that falls within a certain distance ofwe do not need to know the value of in question — there is a single value of independent ofsuch that choosing to be larger than will suffice.

The fundamental theorem of algebra states that every non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part equal to zero. In mathematical analysis, Lipschitz continuitynamed after Rudolf Lipschitz, is a strong form of uniform continuity for functions.

Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the Lipschitz constant of the function.

on the constellations of weierstrass points

For instance, every function that has bounded first derivatives is Lipschitz continuous. Karl Theodor Wilhelm Weierstrass was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a school teacher, eventually teaching mathematics, physics, botany and gymnastics.

He later received an honorary doctorate and became professor of mathematics in Berlin.Thanks for helping us catch any problems with articles on DeepDyve. We'll do our best to fix them. Check all that apply - Please note that only the first page is available if you have not selected a reading option after clicking "Read Article". Include any more information that will help us locate the issue and fix it faster for you.

We investigate arithmetical properties of a class of semigroups that includesthose appearing as Weierstrass semigroups at totally ramified points of coveringof curves. Semigroup Forum — Springer Journals. Enjoy affordable access to over 18 million articles from more than 15, peer-reviewed journals. Get unlimited, online access to over 18 million full-text articles from more than 15, scientific journals.

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Open Advanced Search. DeepDyve requires Javascript to function. Please enable Javascript on your browser to continue. Read Article. Download PDF. Share Full Text for Free beta. Web of Science. Let us know here. System error. Please try again! How was the reading experience on this article?You've read 1 of 2 free monthly articles. Learn More. After four years at university spent drinking and fencing, Weierstrass had left empty handed.

He eventually took a teaching course and spent most of the s as a schoolteacher in Braunsberg. He hated life in the small Prussian town, finding it a lonely existence.

His only respites were the mathematical problems he worked on between classes. But he had nobody to talk to about mathematics, and no technical library to study in. Even his results failed to escape the confines of Braunsberg. Instead of publishing them in academic journals as a university researcher would, Weierstrass added them to articles in the school prospectus, baffling potential students with arcane equations. While his previous articles had made barely a ripple, this one created a flood of interest.

Weierstrass had found a new way to deal with a fiendish class of equations known as Abelian functions. The paper only contained an outline of his methods, but it was enough to convince mathematicians they were dealing with a unique talent. Despite having gone through the intellectual equivalent of a rags to riches story, many of his old habits remained.

on the constellations of weierstrass points

He would rarely publish papers, preferring instead to share his work among students. But when Weierstrass read its definitions, he found them to be wordy and vague. There was too much hand waving, and not enough detail.

Chief among this early work was the redefinition of a derivative. To calculate the gradient of a curve at a certain point—and hence its rate of change—Isaac Newton had originally considered a line that passed through the point of interest and a nearby point on the curve. He then moved that nearby point closer and closer, until the slope of the line was equal to the gradient of the curve. But it was difficult to define the concept mathematically.

If two statisticians were to lose each other in an infinite forest, the first thing they would do is get drunk. That way, they would walk more or less randomly, which would give them the best chance of finding each He wanted a more practical definition, so decided to convert the concept into a formula.

Rather than manipulating abstract ideas, mathematicians would instead be able to rearrange equations. In doing so, he was laying the foundations for his monster. A t the time, mathematicians drew much of their inspiration from nature. This thinking led to a geometrical intuition about mathematical structures. They should make sense in the same way that a physical object would. Conceptually, these are functions that can be drawn without taking pen from paper.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Weierstrass point

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on the constellations of weierstrass points

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