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September 22, 2010

Statistics — Biography — Pierre-Simon Laplace

Filed under: History of Math, Statistics, Statistics Notes — Tags: — bowman @ 8:50 am

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Pierre Simon Laplace was born at Beaumont-en-Auge in Normandy on March 23, 1749. He died at Paris on March 5, 1827. He was the son of a farm laborer, and owed his education to some wealthy neighbors who were interested and excited by his abilities and engaging presence. Very little is known of his early years. When he became distinguished, he had the pettiness to hold himself aloof both from his relatives and from those who had assisted him.

Pierre-Simon Laplace’s father expected him to make a career in the Church. At the age of 16, Laplace entered Caen University to study theology. However, during his two years at the University of Caen, Laplace discovered his mathematical talents and his love of the subject. Consequently, he left Caen without taking his degree, and went to Paris.

He began producing a steady stream of remarkable mathematical papers. His first paper was on maxima and minima of curves where he improved on methods given by Joseph Louis Lagrange. His next paper concerned difference equations. He quickly wrote papers on the integral calculus, mechanics, physical astronomy, and mathematical astronomy.

Laplace’s reputation steadily increased during the 1770’s. The 1780’s were the period in which Laplace produced the depth of results which have made him one of the most important and influential scientists that the world has seen. It does appear however that Laplace was not modest about his abilities and achievements, and he probably failed to recognize the effect of his attitude on his colleagues. Laplace let it be known widely that he considered himself the best mathematician in France. The effect on his colleagues was only mildly eased by the fact that Laplace was right.

In 1780, Laplace and the chemist Lavoisier showed respiration to be a form of combustion. This work with Lavoisier marked the beginning of a third important area of research for Laplace, namely his work on the theory of heat, which he worked on towards the end of his career.

In 1784 Laplace was appointed as examiner at the Royal Artillery Corps, and in this role in 1785, he examined and passed the 16 year-old Napoleon Bonaparte. He also served on many of the committees of the Académie des Sciences. Laplace was promoted to a senior position in the Académie des Sciences in 1785. Two years later Lagrange also came to Paris. The two great mathematical geniuses, despite a rivalry between them, each was to benefit greatly from the ideas flowing from the other.

Laplace was a member of the committee of the Académie des Sciences to standardize weights and measures in May 1790. This committee worked on the metric system and advocated a decimal base.

In 1795, the Ecole Normale was founded with the aim of training school teachers and Laplace taught courses there, including one on probability. Laplace nowhere displayed the massiveness of his genius more than in the theory of probability. The theory of probability, which Laplace described as common sense expressed in mathematical language, engaged his attention from its importance in physics and astronomy. He applied his theories, not only to the ordinary problems of chances, but also in the causes of phenomena, vital statistics and future events.

The first edition of Laplace’s Théorie Analytique des Probabilités was published in 1812, covering generating functions, approximations to various expressions occurring in probability theory, Laplace’s definition of probability, Bayes’s rule, least squares, Buffon’s needle problem, inverse probability, and applications to mortality, life expectancy, the length of marriages, and legal matters.

January 15, 2010

Statistics Biography — Ronald A. Fisher

Filed under: History of Math, Statistics, Statistics Notes — Tags: — bowman @ 11:10 pm

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Sir Ronald Aylmer Fisher was a British geneticist and statistician. To create genetic experiments that yielded greater results with less effort, he pioneered the use of statistics in experimentation, and came up with the now widely used concepts of variance and randomization. Fisher wrote about 300 papers and seven books during his prodigious career. (more…)

Statistics Biography — William Sealy Gosset

Filed under: History of Math, Statistics, Statistics Notes — Tags: — bowman @ 10:41 pm

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William Sealy Gosset (June 13, 1876 – October 16, 1937) is famous as a statistician, best known by his pen name Student and for his work on Student’s t-distribution.

William Sealy Gosset was born in Canterbury, England, the latest descendant of an old Huguenot family that had left France after the revocation of the Edict of Nantes. He studied at Winchester, then at Oxford, where he focused on mathematics and natural sciences. Upon graduation, he joined Arthur Guinness and Son, a Dublin brewery, and he remained employed there throughout his life, ultimately becoming chief brewer at a new brewery in London. (more…)

December 21, 2009

Geometry Notes — History — The Golden Ratio

Filed under: Geometry, History of Math — Tags: — bowman @ 12:50 pm

The golden ratio has fascinated intellectuals of diverse interests for over two thousand years. Ancient Greek mathematicians first studied the golden ratio because of its frequent appearance in geometry. The Greeks usually attributed discovery of the ratio to Pythagoras and his followers. Later, Euclid’s Elements provides the first known written definition of the golden ratio: “A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.” Euclid explains a construction for dividing a segment in an extreme and mean, the golden ratio. Throughout Euclid’s masterpiece, several facts and their proofs used the golden ratio. Some of these factual statements show that the golden ratio is an irrational number. Luca Pacioli, a contemporary of Leonardo da Vinci, wrote of the golden ratio in his Divina Proportione of 1509. The golden ratio captured the imagination of artists, architects, scientists, and mystics with the properties, mathematical and otherwise.

Mark Barr was an American mathematician who, in about 1909, gave the golden ratio the name of phi φ , the first Greek letter in the name of Phidias, the Greek sculptor who lived around 450 BC. (Phidias, 480 BC – 430 BC, was an ancient Greek sculptor, painter and architect, universally regarded as the greatest of all Classical sculptors.)

Johannes Kepler once stated, “Geometry has two great treasures: one is the theorem of Pythagoras; the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of Gold; the second we may name a Precious jewel.” Johannes Kepler (December 27, 1571 – November 15, 1630) was a German mathematician, astronomer and astrologer, and key figure in the 17th century astronomical revolution. Kepler incorporated religious reasoning into his work, motivated by the religious conviction that God had created the world according to an intelligible plan that is accessible through the natural light of reason.

The number itself:

In Euclid’s words: A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.
Consider the segment BC .
A is a point between B and C and let’s say A is closer to C .
If the ratio of the lengths BC to AB is equal to the ratio of the lengths AB to AC, then the segment has been cut in the extreme and mean ratio, or in a golden ratio.

Let’s assume the length of segment BC is 1.
We can call AB “x” and AC “(1 – x).”
We can solve for x.

We will ignore the negative solution since we are solving for a length and we know by definition that a length is always positive. So by solving the above equation we know that AB is approximately 0.6180339887, which would give AC being approximately 1 – 0.6180339887 or 0.3819660113. Remember that these are approximate values because to solve our equation we were working with the irrational number, square root of 5.

Now Euclid’s extreme to mean ratio, or golden ratio:

If we divide the “whole line” by the “greater segment”, 1 / 0.6180339887, we obtain 1.618033989.

If we divide the “greater” by the “lesser”, 0.6180339887 / 0.3819660113, we obtain 1.618033988. The two values are different at the ninth decimal place, but that’s because my TI-84 is rounding numbers that I’m working with that I rounded earlier. Remember, when you work with rounded decimals, you are working with error, and if you continue to operate with decimals that you’ve rounded, you will magnify that error. Old school — no decimals unless there’s no choice.

φ = 1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204
189391137484754088075386891752126633862223536931793180060766726354433389086595939582905638322661
319928290267880675208766892501711696207032221043216269548626296313614438149758701220340805887954454…

A scratch at the history:

According to legend, the Greek Philosopher Pythagoras discovered the concept of harmony when he began his studies of proportion while listening to the different sounds given off when the blacksmith’s hammers hit their anvils. The weights of the hammers and of the anvils all gave off different sounds. From here he moved to the study of stringed instruments and the different sounds they produced. He started with a single string and produced a monochord in the ratio of 1:1 called the Unison. By varying the string, he produced other chords: a ratio of 2:1 produced notes an octave apart. In further studies of nature, he observed certain patterns and numbers reoccurring. Pythagoras believed that beauty was associated with the ratio of small integers. Astonished by this discovery and awed by it, the Pythagoreans endeavored to keep this a secret; declaring that anybody that broached the secret would get the death penalty. With this discovery, the Pythagoreans saw the essence of the cosmos as numbers and numbers took on special meaning and significance.

Phidias applied the golden ratio to the design of sculptures for the Parthenon.

Plato, in his views on natural science and cosmology presented in his “Timaeus,” considered the golden section to be the most binding of all mathematical relationships and the key to the physics of the cosmos.

The Egyptians thought that the golden ratio was sacred. Therefore, it was very important in their religion. They used the golden ratio when building temples and places for the dead. If the proportions of their buildings weren’t according to the golden ratio, the deceased might not make it to the afterlife or the temple would not be pleasing to the gods. As well, the Egyptians found the golden ratio to be pleasing to the eye. They used it in their system of writing and in the arrangement of their temples. The Egyptians were aware that they were using the golden ratio, but they called it the “sacred ratio.” The Egyptians used both pi and phi in the design of the Great Pyramids.

The Renaissance artists used the Golden Mean extensively in their paintings and sculptures to achieve balance and beauty.

Leonardo Da Vinci, for instance, used it to define all the fundamental proportions of his painting of “The Last Supper,” from the dimensions of the table at which Christ and the disciples sat to the proportions of the walls and windows in the background sculptures to achieve balance and beauty.

Leonardo da Vinci’s drawings of the human body emphasized its proportion. The ratio of the following distances is the Golden Ratio: (foot to navel) : (navel to head).

The proportions of Michelangelo’s David conform to the golden ratio from the location of the navel with respect to the height to the placement of the joints in the fingers.

The Ark of the Covenant is a Golden Rectangle. In Exodus 25:10, God commands Moses to build the Ark of the Covenant, in which to hold His Covenant with the Israelites, the Ten Commandments, saying, “Have them make a chest of acacia wood – two and a half cubits long, a cubit and a half wide, and a cubit and a half high.” The ratio of 2.5 to 1.5 is 1.666…, which is as close to phi as you can come with such simple numbers and is certainly not visibly different to the eye. The Ark of the Covenant is thus constructed using the Golden Section, or Divine Proportion.

December 7, 2009

Statistics — Biography — Abraham de Moivre

Filed under: History of Math, Statistics, Statistics Notes — Tags: — bowman @ 2:04 pm

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Abraham de Moivre (26 May 1667 in Vitry-le-François, Champagne, France – 27 November 1754 in London, England) was a French mathematician famous for de Moivre’s formula, which links complex numbers and trigonometry, and for his work on the normal distribution and probability theory.

Because of problems related to nationality and religion, Abraham de Moivre never had an opportunity to teach mathematics at a university. Nonetheless, he enjoyed fruitful interactions with Sir Isaac Newton and others and later published texts in which he advanced the understanding of probability theory and other areas of mathematics. (more…)

December 3, 2009

Statistics — Biography — Carl Friedrich Gauss

Filed under: History of Math, Statistics, Statistics Notes — Tags: — bowman @ 9:19 pm

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Johann Carl Friedrich Gauss (30 April 1777 – 23 February 1855) was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, electrostatics, astronomy, and optics. Sometimes known as the Prince of Mathematicians and the greatest mathematician since antiquity, Gauss had a remarkable influence in many fields of mathematics and science and is ranked as one of history’s most influential mathematicians.

Some mathematicians consider the German mathematician Gauss to be the greatest of all time, and almost all consider him to be one of the three greatest, along with Archimedes and Newton; in contrast, he is hardly known to the general public.

Gauss was a child prodigy. He was born in Braunschweig, now part of Lower Saxony, Germany, as the only son of poor working-class parents. He exhibited such early genius that his family and neighbors called him the “wonder child”. There are several stories of his early genius. When he was two years old, he gradually got his parents to tell him how to pronounce all the letters of the alphabet. Then, by sounding out combinations of letters, he learned (on his own) to read aloud. He also picked up the meanings of the number symbols and learned to do arithmetical calculations. According to another, his gifts became very apparent at the age of three when he corrected, mentally and without fault in his calculations, an error his father had made on paper while calculating finances. Another famous story has it that when Gauss was ten years old he was allowed to attend an arithmetic class that was being taught by a man who had a reputation for being cynical and having little respect for the peasant children he was teaching. This man tried to occupy pupils by making them add a list of integers. The young Gauss reputedly produced the correct answer within seconds.

As his father wanted him to follow in his footsteps and become a mason, he was not supportive of Gauss’s schooling in mathematics and science. Gauss was primarily supported by his mother in this effort and by the Duke of Braunschweig, who awarded Gauss a fellowship to a university, which he attended from 1792 to 1795, and subsequently he moved to the University of Göttingen from 1795 to 1798. While in university, Gauss independently rediscovered several important theorems.

Gauss’s personal life was overshadowed by the early death of his first wife in 1809, soon followed by the death of one child. Gauss plunged into a depression from which he never fully recovered.

Gauss was an ardent perfectionist and a hard worker. Gauss was also a loner.  He did not have many close friends.  He preferred to immerse himself in his work than to form relationships. He was never a prolific writer, refusing to publish works which he did not consider complete and above criticism. This was in keeping with his personal motto “few, but ripe.” His personal diaries indicate that he had made several important mathematical discoveries years or decades before his contemporaries published them. Mathematical historians estimate that had Gauss timely published all of his discoveries, Gauss would have advanced mathematics by fifty years.

Though he did take in a few students, Gauss was known to dislike teaching. Gauss usually declined to present the intuition behind his often very elegant proofs – he preferred them to appear “out of thin air” and erased all traces of how he discovered them.

The normal distribution, also called the Gaussian distribution, is an important family of continuous probability distributions, applicable in many fields. Each member of the family may be defined by two parameters, location and scale: the mean and variance. The standard normal distribution is the normal distribution with a mean of zero and a variance of one. Carl Friedrich Gauss became associated with this set of distributions when he analyzed astronomical data using them. It is often called the bell curve because the graph of its probability density resembles a bell.

November 18, 2009

Geometry Notes — History — The Platonic Solids

Filed under: Geometry, History of Math — Tags: — bowman @ 9:00 pm

The Platonic Solids

The Platonic solids are regular 3-D figures. A Platonic solid is a convex regular polyhedron. Regular means that all the edges are of equal length, all the angles of equal measure, and all faces are congruent shapes. They are unique in that the faces, edges and angles are all congruent. There are exactly five such figures. The name of each figure is derived from the number of its faces: 4, 6, 8, 12, and 20. Tetrahedron — 4 faces. Hexahedron or Cube — 6 faces. Octahedron — 8 faces. Dodecahedron — 12 faces. Icosahedron — 20 faces.

The following gives the name of the Platonic Solid along with the number of faces it has, the shape of each face (which is a regular polygon), the number of vertices, the number of edges, and the Platonic Solid’s Dual which is the Platonic Solid that can be inscribed within it by connecting the midpoints of the faces.

Platonic Solid       Faces     Shape of Faces      Vertices       Edges       Dual
Tetrahedron            4          Triangle            4             6          Tetrahedron
Cube                   6          Square              8             12         Octahedron
Octahedron             8          Triangle            6             12         Cube
Dodecahedron           12         Pentagon            20            30         Icosahedron
Icosahedron            20         Triangle            12            30         Dodecahedron 

The ancient Greeks studied the Platonic solids extensively.

Some sources credit Pythagoras and the Pythagoreans with their discovery. Other evidence suggests that they may have only been familiar with the tetrahedron, cube, and dodecahedron, and that the discovery of the octahedron and icosahedron belong to Theaetetus, a contemporary of Plato. Theaetetus gave a mathematical description of all five and may have been responsible for the first known proof that there are no other convex regular polyhedra.

The Platonic solids feature prominently in the philosophy of Plato. Plato theorized the classical elements were constructed from the regular solids. Plato was mightily impressed by these five definite shapes that constitute the only perfectly symmetrical arrangements of a set of non-planar points in space, and later expounded a complete “theory of everything” (in a writing called Timaeus) based explicitly on these five solids. To achieve perfect symmetry between the vertices, it’s clear that each face of a regular polyhedron must be a regular polygon, and all the faces must be identical. Plato speculated that these five solids were the shapes of the fundamental components of the physical universe. Plato associated each of the four classical elements (earth, air, water, and fire) with a regular solid. Fire was associated with the tetrahedron, earth with the cube, air with the octahedron, and water with the icosahedron. The fifth Platonic solid, the dodecahedron, Plato associated with the universe. Aristotle added a fifth element, aithêr (“ether” in English) and postulated that the heavens were made of this element, but he had no interest in matching it with Plato’s fifth solid.

The aesthetic beauty and symmetry of the Platonic solids have made them a favorite subject of geometers for thousands of years. The Platonic solids have been known since ancient times. Ornamented models of them can be found among the carved stone balls created by the late neolithic people of Scotland at least 1000 years before Plato.

Now for a technical point. There is a formula that associates the number of faces, the number of vertices, and the number of edges of any convex polyhedron. The adjective convex is very important. This formula works for the perfect Platonic solids as well as any convex polyhedron. Cubes are nice regular hexahedrons. They get all the credit and praise. They are pretty. But a shoe box, for example, is also a hexahedron. The formula F – E + V = 2 holds true for all polyhedron. F is the number of faces. E is the number of edges, and V is the number of vertices.

We call this formula Euler’s formula, named after Leonhard Euler. Leonhard was responsible for much of early work to prove various theorems about polyhedra.

Tetrahedron (recall 4 vertices, 6 edges, 4 equilateral triangles as faces) 4 – 6 + 4 = 2

Hexahedron (recall 8 vertices, 12 edges, 6 squares as faces) 6 – 12 + 8 = 2

Here is a nice website.  It shows the Platonic Solids in motion.  Click and hold on one, move the mouse.  You can even get them to roll around.  Kinda cool.

November 11, 2009

Math History — Chinese Mathematicians — Father Zu Chongzhi and Son Zu Gengzhi

Filed under: History of Math — bowman @ 6:14 pm

Zu Gengzhi was a Chinese mathematician. He lived from about 450 to 520 in China. He was the son of another famous Chinese mathematician, Zu Chongzhi. Zu Gengzhi discovered the “Zu Gengzhi’s Principle”, stating that “The volumes of two solids of the same height are equal if the areas of the plane sections at equal heights are the same.” This is same as Cavalieri’s principle, but was discovered about 1100 years earlier.

He is credited with the invention of the astronomical sighting tube, which Shen Kuo and Wei Pu would later improve during the 11th century.

The Zu family was an extremely talented one with successive generations being astronomers with special interests in the calendar. They handed their mathematical and astronomical skills down from father to son and, indeed, this was one of the main ways that such skills were transmitted in China at this time. Zu Gengzhi, in the family tradition, was taught a variety of skills as he grew up. Zu Gengzhi’s greatest achievement was to compute the diameter of a sphere of a given volume. He had demonstrated that a previously accepted formula was incorrect by using various constructions for comparison, but he was unable to derive the correct formula.

His father Zu Chongzhi and Zu Gengzhi wrote a mathematical text entitled Zhui Shu (Method of Interpolation). This book most likely contained astronomical calculation techniques due to the accuracy of his calendars. It most likely contained formulas for the volume of the sphere, cubic equations and the accurate value of pi. Unfortunately, this book didn’t survive to the present day, it has been lost since the Song Dynasty, which began in the middle to late tenth century.

Their mathematical achievements included:

  • the Da Ming calendar introduced by the father in 465. The midwinter was the starting point in making the calendar; therefore it was of extremely important to pinpoint the position of the sun on that day. However, the astronomers before Zu all believed the position was fixed, which caused errors in the calendar-making task from the very beginning. To solve the problem, Zu introduced the concept of procession (of the sun) in making the Da Ming Calendar, greatly enhancing the accuracy of calendar computation.
  • distinguishing the Sidereal Year and the Tropical Year. The sidereal year is the time taken for the Sun to return to the same position with respect to the stars of the celestial sphere. It is the orbital period of Earth. A tropical year (also known as a solar year) is the length of time that the Sun takes to return to the same position in the cycle of seasons.
  • calculating one year as 365.24281481 days, which is very close to 365.24219878 days as we know today. His estimate is only 46 seconds different from the modern estimate.
  • calculating the number of overlaps between sun and moon as 27.21223, which is very close to 27.21222 as we know today. Using this number he successfully predicted an eclipse four times during 23 years (from 436 to 459).
  • calculating the Jupiter year as about 11.858 Earth years, which is very close to 11.862 as we know of today.
  • deriving an approximation of pi that was correct to the seventh decimal place, which held as the most accurate approximation for π for over nine hundred years. His best approximation was between 3.1415926 and 3.1415927, with 355 ⁄ 113 (detailed approximation) and 22 ⁄ 7 (rough approximation) being the other notable approximations. He obtained the result by approximating a circle with a 12,288 sided polygon. This was an impressive feat for the time, especially considering that the only device he used for recording intermediate results were merely a pile of wooden sticks laid out in certain patterns. No one discovered more of pi for nearly 1000 years.
  • finding the volume of a sphere as 4πr³/3, where r is radius.
  • discovering the Cavalieri’s principle, 1000 years before Bonaventura Cavalieri in the West.

Not too shabby. Very talented mathematicians without technology. I guess talent, hard work, and perseverance count for something huh. Outstanding.

Geometry — Biography — Hippasus

Hippasus of Metapontum, ca. 500 B.C., was a Greek philosopher. He was a disciple of Pythagoras. Hippasus is attributed the discovery of the existence of irrational numbers. More specifically, he is credited with the discovery that the square root of 2 is irrational.

It is believed that he proved the existence of irrational numbers at a time when the Pythagorean belief was that whole numbers and their ratios could describe anything that was geometric. Not only that, they didn’t believe there was a need for any other numbers. (more…)

November 10, 2009

Statistics — Biography — Simeon Denis Poisson

Filed under: History of Math, Statistics, Statistics Notes — Tags: — bowman @ 3:16 pm

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Simeon Denis Poisson developed many novel applications of mathematics for statistics and physics. He was born at Pithviers on June 21, 1781, and died at Paris on April 25, 1840. His father had been a private soldier, and on his retirement was given a small administrative post in his native village. When the French revolution broke out, his father assumed the government of the village, and soon became a local dignitary.

He was educated by his father who prodded him to be a doctor. His uncle offered to teach him medicine, and began by making him prick the veins of cabbage-leaves with a lancet. When he had perfected this, he was allowed to practice on humans, but in the first case of bloodletting that he did this by himself, the patient died within a few hours. Although the other physicians assured him that this was not an uncommon occurance, he vowed he would have nothing more to do with the medical profession. His main weakness was the lack of coordination which had made a career as a surgeon impossible. This weakness followed him in some respects for drawing mathematical diagrams was quite beyond him.

Upon returning home, he discovered a copy of a question set from the Polytechnic school among the official papers sent to his father. This chance event determined his career. At the age of seventeen he entered the Polytechic. A memoir on finite differences which he wrote when only eighteen was so impressive that it was rapidly published in a prestigious journal. As soon as he had finished his studies he was appointed as a lecturer. Throughout his life he held various scientific posts and professorships. He made the study of mathematics his hobby as well as his business.

Over his life Simeon Poisson wrote between 300-400 manuscripts and books on a variety of mathematical topics, including pure mathematics, the application of mathematics to physical problems, the probability of random events, the theory of electrostatics and magnetism (which led the forefront of the new field of quantum mechanics), physical astronomy, and wave theory.

One of Simeon Poisson’s contributions was the development of equations to analyze random events, later dubbed the Poisson Distribution. The fame of this distribution is often attributed to the following story. Many soldiers in the Prussian Army died due to kicks from horses. To determine whether this was due to a random occurance or the wrath of god, the Czar commissioned the Russian mathematician Ladislaus Bortkiewicz to determine the statistical significance of the events. Fourteen corps were examined, each for twenty years. For over half the corps-year combinations there were no deaths from horse kicks; for the other combinations the number of deaths ranged up to four. Presumably the risk of lethal horse kicks varied over years and corps, yet the over-all distribution fit remarkably well to a Poisson distribution.

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