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Bass, T. Houghton-Mifflin, Boston Google Scholar. Feller, W. Wiley, New York Google Scholar. Jaynes, E. Kerrich, J. Jorgensen, Copenhagen Google Scholar. Harris, M. Benjamini, I. Howard, D. Haken, H. Springer, Berlin Google Scholar. Rocha, A. Kmet, A. Petkovsek, M. Itoh, Y. Harik, G.

Duan, Z. Howard, L. Cooper, C. Balzass, N. E52 , Google Scholar. Diaconis, P. Grunbaum, B. George Carre, Paris Google Scholar. Barnhart, R. Hopf, E. Lindley, T. Philosophy 56 , CrossRef Google Scholar. Stevens, M. Engel, E. Springer, New York Google Scholar. Hansen, L. McDonald, S. Sport betting has been popular for centuries. The earliest forms of contemporary sports betting revolved around cock fights in the late 17 th century, while wagering on horse races became highly popular between the 19 th to 20 th centuries.

In s, the advent of the Internet facilitated online sports betting, creating an increasing need for sophisticated statistical tracking, like regression analysis, to develop winning strategies for wagering. There are several key differences between traditional gambling and sports betting online with the help of sites like SportsbookNavigator.

In traditional gambling, the likelihood of any event can be calculated accurately even though the number of possible results may differ for example: the odds of picking a specific card out of a standard deck is always one in However, the likely result of a sporting event cannot be calculated with such precision. While the odds of winning in traditional gambling are derived from a known probability, sports betting relies on parsing a large number of variables, usually with more than one potential outcome.

Online gambling includes all Internet-based portals that provide lotteries, poker, casino games, bingo, and sports betting. Figure: Worldwide gross gambling revenue. Various methods can be used to generate a sports betting system, although most experts agree that the most widely used method is regression analysis.

Multivariate linear regression, logistic regression, and multiple regression analysis can all be used to calculate the probability of any outcome, and since determining the outcome of a sporting event requires analyzing a high number of variables, regression analysis provides a suitable framework for defining and assigning a value to these variables. Recent movies and bestseller titles like Moneyball have delved into the world of statistical analysis, driving increased interest in the use of regression analysis for sports betting.

Logistic regression is a forecasting technique that provides a probability percentage for a given variable. Margin of victory is a statistical term which indicates difference between the number of points scored by the winning team and the number of points scored by the losing team. A smaller MV represents a close match, and by using statistical software like SPSS, the following equation can provide the percentage chance that the team will win, based on MV scores:.

A percentage chance of winning can also be determined in Microsoft Excel by using this equation:. Multiple regression systems are widely considered the most reliable modern sports betting system. This means that one must know the past to know the future. To create a multiple regression betting system, one must have reliable data regarding past information of the players and teams, meaning that trustworthy historical data is crucial to building an effective multiple regression system.

Regression 1: Bettor finds that Team A won the regular series against Team B by during the first match of the year. Since both teams have scored a victory, bettor determines that the key variable is the presence of Player X, and decides that Team B will win the match. Thus, by using multiple regression analysis, bettor is able to analyze the events of the past and extrapolate the most probable future. To develop a multiple regression system, mining data from an online sports book that can offer accurate historical sports data in a format that is easily accessible and actionable is highly recommended.

These sports books also provide step by step rules for implementing regression analysis techniques in sports betting. Note that regression analysis methodology is also employed by most casinos in an effort to generate probabilities that favor the house — for similar reasons, sports books use regression analysis to provide sports betting enthusiasts with the same advantage.

There is one glaring problem in using regression analysis to predict outcomes of sporting events: the differentiation between correlation and causation.

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The Euler-Mascheroni constantsometimes also called 'Euler's constant' or 'the Euler constant' but not to be confused with the constant is defined as the limit of the sequence.

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He served on the committee of the Academy dealing with the library and of scientific publications. He served as an advisor to the government on state lotteries, insurance, annuities and pensions and artillery. On top of this his scientific output during this period was phenomenal. During the twenty-five years spent in Berlin, Euler wrote around articles. He wrote books on the calculus of variations; on the calculation of planetary orbits; on artillery and ballistics extending the book by Robins ; on analysis; on shipbuilding and navigation; on the motion of the moon; lectures on the differential calculus; and a popular scientific publication Letters to a Princess of Germany 3 vols.

In Maupertuis died and Euler assumed the leadership of the Berlin Academy , although not the title of President. The king was in overall charge and Euler was not now on good terms with Frederick despite the early good favour. Euler, who had argued with d'Alembert on scientific matters, was disturbed when Frederick offered d'Alembert the presidency of the Academy in However d'Alembert refused to move to Berlin but Frederick's continued interference with the running of the Academy made Euler decide that the time had come to leave.

In Euler returned to St Petersburg and Frederick was greatly angered at his departure. Soon after his return to Russia, Euler became almost entirely blind after an illness. In his home was destroyed by fire and he was able to save only himself and his mathematical manuscripts. A cataract operation shortly after the fire, still in , restored his sight for a few days but Euler seems to have failed to take the necessary care of himself and he became totally blind.

Because of his remarkable memory he was able to continue with his work on optics, algebra, and lunar motion. Amazingly after his return to St Petersburg when Euler was 59 he produced almost half his total works despite the total blindness. Euler of course did not achieve this remarkable level of output without help. He was helped by his sons, Johann Albrecht Euler who was appointed to the chair of physics at the Academy in St Petersburg in becoming its secretary in and Christoph Euler who had a military career.

Fuss , who was Euler's grandson-in-law, became his assistant in Yushkevich writes in [ 1 ] For example Euler credits Albrecht, Krafft and Lexell for their help with his page work on the motion of the moon, published in Fuss helped Euler prepare over articles for publication over a period on about seven years in which he acted as Euler's assistant, including an important work on insurance which was published in Yushkevich describes the day of Euler's death in [ 1 ] :- On 18 September Euler spent the first half of the day as usual.

He gave a mathematics lesson to one of his grandchildren, did some calculations with chalk on two boards on the motion of balloons; then discussed with Lexell and Fuss the recently discovered planet Uranus. About five o'clock in the afternoon he suffered a brain haemorrhage and uttered only "I am dying" before he lost consciousness. He died about eleven o'clock in the evening. After his death in the St Petersburg Academy continued to publish Euler's unpublished work for nearly 50 more years.

Euler's work in mathematics is so vast that an article of this nature cannot but give a very superficial account of it. He was the most prolific writer of mathematics of all time. He made large bounds forward in the study of modern analytic geometry and trigonometry where he was the first to consider sin, cos etc.

He made decisive and formative contributions to geometry, calculus and number theory. He integrated Leibniz 's differential calculus and Newton's method of fluxions into mathematical analysis. He introduced beta and gamma functions , and integrating factors for differential equations.

He studied continuum mechanics, lunar theory with Clairaut , the three body problem , elasticity, acoustics, the wave theory of light, hydraulics, and music. He laid the foundation of analytical mechanics, especially in his Theory of the Motions of Rigid Bodies Let us examine in a little more detail some of Euler's work. Firstly his work in number theory seems to have been stimulated by Goldbach but probably originally came from the interest that the Bernoullis had in that topic.

Perhaps the result that brought Euler the most fame in his young days was his solution of what had become known as the Basel problem. The problem had also been studied unsuccessfully by Leibniz , Stirling , de Moivre and others. References show. Biography in Encyclopaedia Britannica. R Fueter, Leonhard Euler Basel, G du Pasquier, Leonhard Euler et ses amis Paris, O Speiss, Leonhard Euler R Thiele, Leonhard Euler Leipzig, A P Yushkevich and E Winter eds.

G E Andrews, Euler's pentagonal number theorem, Math. R Ayoub, Euler and the zeta function, Amer. Monthly 81 , - Storia Sci. Theory Contin. Fractions 3 , 4 - D Cameron, Euler and Maclaurin made easy, Math. The past, Annuaire Univ. Sofia Fac. The present, Annuaire Univ. The future, Annuaire Univ. Monthly 66 , - Exact Sci. Histoire Sci. J Dutka, On the summation of some divergent series of Euler and the zeta functions, Arch. H M Edwards, Euler and quadratic reciprocity, Math.

J Ewing, Leonhard Euler, Math. Intelligencer 5 3 , 5 - 6. Acta 56 6 , - B F Finkel, Biography. Leonard Euler, Amer. Monthly 4 , - C G Fraser, The concept of elastic stress in eighteenth-century mechanics : some examples from Euler, in Hamiltonian dynamical systems New York, , 1 - C G Fraser, Mathematical technique and physical conception in Euler's investigation of the elastica, Centaurus 34 3 , - C Grau, Leonhard Euler und die Berliner Akademie der Wissenschaften, in Ceremony and scientific conference on the occasion of the th anniversary of the death of Leonhard Euler Berlin, , - J Gray, Leonhard Euler - , Janus : archives internationales pour l'histoire de la medecine et pour la geographie medicale 72 , - J Gray, Leonhard Euler : - , Janus 72 1 - 3 , - D R Green, Euler, Math.

M Kline, Euler and infinite series, Math. E Knobloch, Leibniz and Euler : problems and solutions concerning infinitesimal geometry and calculus, Conference on the History of Mathematics Rende, , - H Loeffel, Leonhard Euler - , Mitt. Nauk 36 , 66 - Maupertuis et Euler, - , Rev. Todestag Leonhard Eulers, Praxis Math. Graph Theory 12 1 , - H Samelson, In defense of Euler, Enseign.

This line has come to be named after him - the Euler line. His name is pronounced the German way - "oiler". The three centers that have this surprising property are the triangle's centroid , circumcenter and orthocenter. In the figure above press 'reset' first if necessary the centroid is the black middle point on the line.

The circumcenter is the magenta point on the left, and the orthocenter is the red point on the right. As you drag any of the triangle's vertices around, you can see that these points remain collinear , all lying on the green Euler line.

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The XYZ system rotates, while xyz is fixed. Starting with XYZ overlapping xyz , a composition of three intrinsic rotations can be used to reach any target orientation for XYZ. Euler angles can be defined by intrinsic rotations. The rotated frame XYZ may be imagined to be initially aligned with xyz , before undergoing the three elemental rotations represented by Euler angles.

Its successive orientations may be denoted as follows:. For the above-listed sequence of rotations, the line of nodes N can be simply defined as the orientation of X after the first elemental rotation. Moreover, since the third elemental rotation occurs about Z , it does not change the orientation of Z. This allows us to simplify the definition of the Euler angles as follows:. Extrinsic rotations are elemental rotations that occur about the axes of the fixed coordinate system xyz.

Starting with XYZ overlapping xyz , a composition of three extrinsic rotations can be used to reach any target orientation for XYZ. For instance, the target orientation can be reached as follows note the reversed order of Euler angle application :. In sum, the three elemental rotations occur about z , x and z. Indeed, this sequence is often denoted z - x - z or Sets of rotation axes associated with both proper Euler angles and Tait—Bryan angles are commonly named using this notation see above for details.

Angles are commonly defined according to the right-hand rule. Namely, they have positive values when they represent a rotation that appears clockwise when looking in the positive direction of the axis, and negative values when the rotation appears counter-clockwise. The opposite convention left hand rule is less frequently adopted. About the ranges using interval notation :. These ambiguities are known as gimbal lock in applications. There are six possibilities of choosing the rotation axes for proper Euler angles.

In all of them, the first and third rotation axes are the same. The six possible sequences are:. Precession , nutation , and intrinsic rotation spin are defined as the movements obtained by changing one of the Euler angles while leaving the other two constant. These motions are not expressed in terms of the external frame, or in terms of the co-moving rotated body frame, but in a mixture.

They constitute a mixed axes of rotation system, where the first angle moves the line of nodes around the external axis z , the second rotates around the line of nodes N and the third one is an intrinsic rotation around Z , an axis fixed in the body that moves. As an example, consider a top.

The top spins around its own axis of symmetry; this corresponds to its intrinsic rotation. It also rotates around its pivotal axis, with its center of mass orbiting the pivotal axis; this rotation is a precession. Finally, the top can wobble up and down; the inclination angle is the nutation angle.

The same example can be seen with the movements of the earth. Though all three movements can be represented by a rotation operator with constant coefficients in some frame, they cannot be represented by these operators all at the same time. Given a reference frame, at most one of them will be coefficient-free. Only precession can be expressed in general as a matrix in the basis of the space without dependencies of the other angles.

These movements also behave as a gimbal set. If we suppose a set of frames, able to move each with respect to the former according to just one angle, like a gimbal, there will exist an external fixed frame, one final frame and two frames in the middle, which are called "intermediate frames". The two in the middle work as two gimbal rings that allow the last frame to reach any orientation in space.

It is the convention normally used for aerospace applications, so that zero degrees elevation represents the horizontal attitude. Tait—Bryan angles represent the orientation of the aircraft with respect to the world frame. When dealing with other vehicles, different axes conventions are possible.

The definitions and notations used for Tait—Bryan angles are similar to those described above for proper Euler angles geometrical definition , intrinsic rotation definition , extrinsic rotation definition. The only difference is that Tait—Bryan angles represent rotations about three distinct axes e. This implies a different definition for the line of nodes in the geometrical construction. In the proper Euler angles case it was defined as the intersection between two homologous Cartesian planes parallel when Euler angles are zero; e.

In the Tait—Bryan angles case, it is defined as the intersection of two non-homologous planes perpendicular when Euler angles are zero; e. The three elemental rotations may occur either about the axes of the original coordinate system, which remains motionless extrinsic rotations , or about the axes of the rotating coordinate system, which changes its orientation after each elemental rotation intrinsic rotations.

There are six possibilities of choosing the rotation axes for Tait—Bryan angles. Tait—Bryan convention is widely used in engineering with different purposes. There are several axes conventions in practice for choosing the mobile and fixed axes, and these conventions determine the signs of the angles. Therefore, signs must be studied in each case carefully. These angles are normally taken as one in the external reference frame heading , bearing , one in the intrinsic moving frame bank and one in a middle frame, representing an elevation or inclination with respect to the horizontal plane, which is equivalent to the line of nodes for this purpose.

For an aircraft, they can be obtained with three rotations around its principal axes if done in the proper order. A yaw will obtain the bearing, a pitch will yield the elevation and a roll gives the bank angle.

Therefore, in aerospace they are sometimes called yaw, pitch and roll. Notice that this will not work if the rotations are applied in any other order or if the airplane axes start in any position non-equivalent to the reference frame. A common problem is to find the Euler angles of a given frame. The fastest way to get them is to write the three given vectors as columns of a matrix and compare it with the expression of the theoretical matrix see later table of matrices.

Hence the three Euler Angles can be calculated. Nevertheless, the same result can be reached avoiding matrix algebra and using only elemental geometry. Notice that any other convention can be obtained just changing the name of the axes. Assuming a frame with unit vectors X , Y , Z given by their coordinates as in the main diagram, it can be seen that:.

Assuming a frame with unit vectors X , Y , Z given by their coordinates as in this new diagram notice that the angle theta is negative , it can be seen that:. Note that the inverse sine and cosine functions yield two possible values for the argument. In this geometrical description, only one of the solutions is valid. When Euler angles are defined as a sequence of rotations, all the solutions can be valid, but there will be only one inside the angle ranges.

This is because the sequence of rotations to reach the target frame is not unique if the ranges are not previously defined. For computational purposes, it may be useful to represent the angles using atan2 y , x. For example, in the case of proper Euler angles:. Euler angles are one way to represent orientations. There are others, and it is possible to change to and from other conventions.

Three parameters are always required to describe orientations in a 3-dimensional Euclidean space. They can be given in several ways, Euler angles being one of them; see charts on SO 3 for others. The most used orientation representation are the rotation matrices , the axis-angle and the quaternions , also known as Euler—Rodrigues parameters , which provide another mechanism for representing 3D rotations.

This is equivalent to the special unitary group description. Regardless, the rotation matrix calculation is the first step for obtaining the other two representations. Any orientation can be achieved by composing three elemental rotations, starting from a known standard orientation. Equivalently, any rotation matrix R can be decomposed as a product of three elemental rotation matrices. For instance:. However, both the definition of the elemental rotation matrices X , Y , Z , and their multiplication order depend on the choices taken by the user about the definition of both rotation matrices and Euler angles see, for instance, Ambiguities in the definition of rotation matrices.

Unfortunately, different sets of conventions are adopted by users in different contexts. The following table was built according to this set of conventions:. For the sake of simplicity, the following table of matrix products uses the following nomenclature:. The financial improvement which came from this appointment allowed Euler to marry which he did on 7 January , marrying Katharina Gsell, the daughter of a painter from the St Petersburg Gymnasium.

Katharina, like Euler, was from a Swiss family. They had 13 children altogether although only five survived their infancy. Euler claimed that he made some of his greatest mathematical discoveries while holding a baby in his arms with other children playing round his feet. We will examine Euler's mathematical achievements later in this article but at this stage it is worth summarising Euler's work in this period of his career. This is done in [ 24 ] as follows The core of his research program was now set in place: number theory ; infinitary analysis including its emerging branches, differential equations and the calculus of variations ; and rational mechanics.

He viewed these three fields as intimately interconnected. Studies of number theory were vital to the foundations of calculus, and special functions and differential equations were essential to rational mechanics, which supplied concrete problems. The publication of many articles and his book Mechanica - 37 , which extensively presented Newtonian dynamics in the form of mathematical analysis for the first time, started Euler on the way to major mathematical work.

Euler's health problems began in when he had a severe fever and almost lost his life. However, he kept this news from his parents and members of the Bernoulli family back in Basel until he had recovered. In his autobiographical writings Euler says that his eyesight problems began in with overstrain due to his cartographic work and that by he had [ 24 ] However, Calinger in [ 24 ] argues that Euler's eyesight problems almost certainly started earlier and that the severe fever of was a symptom of the eyestrain.

He also argues that a portrait of Euler from suggests that by that stage the sight of his left eye was still good while that of his right eye was poor but not completely blind. Calinger suggests that Euler's left eye became blind from a later cataract rather than eyestrain. By Euler had a very high reputation, having won the Grand Prize of the Paris Academy in and On both occasions he shared the first prize with others. Euler's reputation was to bring an offer to go to Berlin, but at first he preferred to remain in St Petersburg.

However political turmoil in Russia made the position of foreigners particularly difficult and contributed to Euler changing his mind. Accepting an improved offer Euler, at the invitation of Frederick the Great, went to Berlin where an Academy of Science was planned to replace the Society of Sciences. In a letter to a friend Euler wrote:- I can do just what I wish [ in my research ] The king calls me his professor, and I think I am the happiest man in the world.

Even while in Berlin Euler continued to receive part of his salary from Russia. For this remuneration he bought books and instruments for the St Petersburg Academy , he continued to write scientific reports for them, and he educated young Russians. Maupertuis was the president of the Berlin Academy when it was founded in with Euler as director of mathematics. He deputised for Maupertuis in his absence and the two became great friends.

Euler undertook an unbelievable amount of work for the Academy [ 1 ] The king also charged Euler with practical problems, such as the project in of correcting the level of the Finow Canal At that time he also supervised the work on pumps and pipes of the hydraulic system at Sans Souci, the royal summer residence. This was not the limit of his duties by any means. He served on the committee of the Academy dealing with the library and of scientific publications.

He served as an advisor to the government on state lotteries, insurance, annuities and pensions and artillery. On top of this his scientific output during this period was phenomenal. During the twenty-five years spent in Berlin, Euler wrote around articles.

He wrote books on the calculus of variations; on the calculation of planetary orbits; on artillery and ballistics extending the book by Robins ; on analysis; on shipbuilding and navigation; on the motion of the moon; lectures on the differential calculus; and a popular scientific publication Letters to a Princess of Germany 3 vols.

In Maupertuis died and Euler assumed the leadership of the Berlin Academy , although not the title of President. The king was in overall charge and Euler was not now on good terms with Frederick despite the early good favour. Euler, who had argued with d'Alembert on scientific matters, was disturbed when Frederick offered d'Alembert the presidency of the Academy in However d'Alembert refused to move to Berlin but Frederick's continued interference with the running of the Academy made Euler decide that the time had come to leave.

In Euler returned to St Petersburg and Frederick was greatly angered at his departure. Soon after his return to Russia, Euler became almost entirely blind after an illness. In his home was destroyed by fire and he was able to save only himself and his mathematical manuscripts. A cataract operation shortly after the fire, still in , restored his sight for a few days but Euler seems to have failed to take the necessary care of himself and he became totally blind.

Because of his remarkable memory he was able to continue with his work on optics, algebra, and lunar motion. Amazingly after his return to St Petersburg when Euler was 59 he produced almost half his total works despite the total blindness. Euler of course did not achieve this remarkable level of output without help. He was helped by his sons, Johann Albrecht Euler who was appointed to the chair of physics at the Academy in St Petersburg in becoming its secretary in and Christoph Euler who had a military career.

Fuss , who was Euler's grandson-in-law, became his assistant in Yushkevich writes in [ 1 ] For example Euler credits Albrecht, Krafft and Lexell for their help with his page work on the motion of the moon, published in Fuss helped Euler prepare over articles for publication over a period on about seven years in which he acted as Euler's assistant, including an important work on insurance which was published in Yushkevich describes the day of Euler's death in [ 1 ] :- On 18 September Euler spent the first half of the day as usual.

He gave a mathematics lesson to one of his grandchildren, did some calculations with chalk on two boards on the motion of balloons; then discussed with Lexell and Fuss the recently discovered planet Uranus. About five o'clock in the afternoon he suffered a brain haemorrhage and uttered only "I am dying" before he lost consciousness. He died about eleven o'clock in the evening. After his death in the St Petersburg Academy continued to publish Euler's unpublished work for nearly 50 more years.

Euler's work in mathematics is so vast that an article of this nature cannot but give a very superficial account of it. He was the most prolific writer of mathematics of all time. He made large bounds forward in the study of modern analytic geometry and trigonometry where he was the first to consider sin, cos etc.

He made decisive and formative contributions to geometry, calculus and number theory. He integrated Leibniz 's differential calculus and Newton's method of fluxions into mathematical analysis. He introduced beta and gamma functions , and integrating factors for differential equations. He studied continuum mechanics, lunar theory with Clairaut , the three body problem , elasticity, acoustics, the wave theory of light, hydraulics, and music.

He laid the foundation of analytical mechanics, especially in his Theory of the Motions of Rigid Bodies Let us examine in a little more detail some of Euler's work. Firstly his work in number theory seems to have been stimulated by Goldbach but probably originally came from the interest that the Bernoullis had in that topic.

Perhaps the result that brought Euler the most fame in his young days was his solution of what had become known as the Basel problem. The problem had also been studied unsuccessfully by Leibniz , Stirling , de Moivre and others. References show. Biography in Encyclopaedia Britannica. R Fueter, Leonhard Euler Basel, G du Pasquier, Leonhard Euler et ses amis Paris, O Speiss, Leonhard Euler R Thiele, Leonhard Euler Leipzig, A P Yushkevich and E Winter eds.

G E Andrews, Euler's pentagonal number theorem, Math. R Ayoub, Euler and the zeta function, Amer. Monthly 81 , - Storia Sci. Theory Contin.