Reversi

Reversi is a strategy board game for two players, played on an 8×8 uncheckered board. There are sixty-four identical game pieces called disks, which are light on one side and dark on the other. Players take. During a play, any disks of the opponent's color that are in a straight line and bounded by the disk just placed and another disk of the current player's color are turned over to the current player's color; the object of the game is to have the majority of disks turned to display your color when the last playable empty square is filled. Reversi was most marketed by Mattel under the trademark Othello; the game Reversi was invented in 1883 by either of two Englishmen, Lewis Waterman or John W. Mollett, gained considerable popularity in England at the end of the nineteenth century; the game's first reliable mention is in the 21 August 1886 edition of The Saturday Review. Mention includes an 1895 article in The New York Times: "Reversi is something like Go Bang, is played with 64 pieces." In 1893, the German games publisher Ravensburger started producing the game as one of its first titles.

Two 18th-century continental European books dealing with a game that may or may not be Reversi are mentioned on page fourteen of the Spring 1989 Othello Quarterly, there has been speculation, so far without documentation, that the game has older origins. The modern version of the game — the most used rule-set, the one used in international tournaments — is marketed and recognized as Othello, it was patented in Japan in 1971 by Goro Hasegawa a 38-year-old salesman. There is one difference from the original game: The first four pieces go in the center, but in a standard diagonal pattern, rather than being placed by players. According to Ben Seeley, another difference of Reversi from Othello is that in the first one the game ends as soon as either player cannot make a move, while in the latter the player without a move passes. Hasegawa established the Japan Othello Association on March 1973, held the first national Othello championship on April 4, 1973 in Japan; the Japanese game company Tsukuda Original launched Othello in late April, 1973 in Japan under Hasegawa’s license, which led to an immediate commercial success.

The name was selected by Hasegawa as a reference to the Shakespearean play Othello, the Moor of Venice, referring to the conflict between the Moor Othello and Iago, more controversially, to the unfolding drama between Othello, black, Desdemona, white. The green color of the board is inspired by the image of the general Othello, valiantly leading his battle in a green field, it can be likened to a jealousy competition, since players engulf the pieces of the opponent, thereby turning them to their possession. Othello was first launched in the U. S. in 1975 by Gabriel Industries and it enjoyed commercial success there. Othello game sales have exceeded $600 million and more than 40 million classic games have been sold in over 100 different countries. Hasegawa wrote How to Othello in Japan in 1974, translated into English and published in the U. S. in 1977 as How to Win at Othello. Kabushiki Kaisha Othello, owned by Hasegawa, registered the trademark "OTHELLO" for board games in Japan and Tsukuda Original registered the mark in the rest of the world.

All intellectual property regarding Othello outside Japan is now owned by MegaHouse, a Japanese toy company that acquired PalBox, the successor to Tsukuda Original. Each of the disks' two sides corresponds to one player; the game may for example be played with a chessboard and Scrabble pieces, with one player letters and the other backs. The historical version of Reversi starts with an empty board, the first two moves by each player are in the four central squares of the board; the players place their disks alternately with their color facing up and no captures are made. A player may choose to not play both pieces on the same diagonal, different from the standard Othello opening, it is possible to play variants of Reversi and Othello wherein the second player's second move may or must flip one of the opposite-colored disks. For the specific game of Othello, the rules state that the game begins with four disks placed in a square in the middle of the grid, two facing white side up, two pieces with the dark side up, with same-colored disks on a diagonal with each other.

Convention has initial board position such that the disks with dark side up are to the north-east and south-west, though this is only marginally meaningful to play. If the disks with dark side up are to the north-west and south-east, the board may be rotated by 90° clockwise or counterclockwise; the dark player moves first. Dark must place a piece with the dark side up on the board, in such a position that there exists at least one straight occupied line between the new piece and another dark piece, with one or more contiguous light pieces between them. In the below situation, dark has the following options indicated by translucent pieces: After placing the piece, dark turns over all light pieces lying on a straight line b

Parity (mathematics)

In mathematics, parity is the property of an integer's inclusion in one of two categories: or odd. An integer is if it is divisible by two and odd if it is not even. For example, 6 is because there is no remainder when dividing it by 2. By contrast, 3, 5, 7, 21 leave a remainder of 1 when divided by 2. Examples of numbers include −4, 0, 82 and 178. In particular, zero is an number; some examples of odd numbers are −5, 3, 29, 73. A formal definition of an number is that it is an integer of the form n = 2k, where k is an integer, it is important to realize that the above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings; the sets of and odd numbers can be defined as following: Even = Odd = A number expressed in the decimal numeral system is or odd according to whether its last digit is or odd.

That is, if the last digit is 1, 3, 5, 7, or 9 it is odd. The same idea will work using any base. In particular, a number expressed in the binary numeral system is odd if its last digit is 1 and if its last digit is 0. In an odd base, the number is according to the sum of its digits – it is if and only if the sum of its digits is even; the following laws can be verified using the properties of divisibility. They are a special case of rules in modular arithmetic, are used to check if an equality is to be correct by testing the parity of each side; as with ordinary arithmetic and addition are commutative and associative in modulo 2 arithmetic, multiplication is distributive over addition. However, subtraction in modulo 2 is identical to addition, so subtraction possesses these properties, not true for normal integer arithmetic. Even ± = even; the division of two whole numbers does not result in a whole number. For example, 1 divided by 4 equals 1/4, neither nor odd, since the concepts and odd apply only to integers.

But when the quotient is an integer, it will be if and only if the dividend has more factors of two than the divisor. The ancient Greeks considered 1, the monad, to be neither odd nor even; some of this sentiment survived into the 19th century: Friedrich Wilhelm August Fröbel's 1826 The Education of Man instructs the teacher to drill students with the claim that 1 is neither nor odd, to which Fröbel attaches the philosophical afterthought, It is well to direct the pupil's attention here at once to a great far-reaching law of nature and of thought. It is this, that between two different things or ideas there stands always a third, in a sort of balance, seeming to unite the two. Thus, there is here between odd and numbers one number, neither of the two. In form, the right angle stands between the acute and obtuse angles. A thoughtful teacher and a pupil taught to think for himself can scarcely help noticing this and other important laws. Integer coordinates of points in Euclidean spaces of two or more dimensions have a parity defined as the parity of the sum of the coordinates.

For instance, the face-centered cubic lattice and its higher-dimensional generalizations, the Dn lattices, consist of all of the integer points whose sum of coordinates is even. This feature manifests itself in chess, where the parity of a square is indicated by its color: bishops are constrained to squares of the same parity; this form of parity was famously used to solve the mutilated chessboard problem: if two opposite corner squares are removed from a chessboard the remaining board cannot be covered by dominoes, because each domino covers one square of each parity and there are two more squares of one parity than of the other. The parity of an ordinal number may be defined to be if the number is a limit ordinal, or a limit ordinal plus a finite number, odd otherwise. Let R be a commutative ring and let I be an ideal of R whose index is 2. Elements of the coset 0 + I may be called while elements of the coset 1 + I may be called odd; as an example, let R = Z be the localization of Z at the prime ideal.

An element of R is or odd if and only if its numerator is so in Z. The numbers form an ideal in the ring of integers, but the odd numbers do not — this is clear from the fact that the identity element for addition, zero, is an element of the numbers only. An integer is if it is congruent to 0 modulo this ideal, in other words if it is congruent to 0 modulo 2, odd if it is congruent to 1 modulo 2. All prime numbers are odd, with one exception: the prime number 2. All known perfect numbers are even. Goldbach's conjecture states that every integer greater than 2 can be represented as a sum of two prime numbers. Modern computer calculations have shown this conjecture to

Machine learning

Machine learning is the scientific study of algorithms and statistical models that computer systems use to perform a specific task without using explicit instructions, relying on patterns and inference instead. It is seen as a subset of artificial intelligence. Machine learning algorithms build a mathematical model of sample data, known as "training data", in order to make predictions or decisions without being explicitly programmed to perform the task. Machine learning algorithms are used in a wide variety of applications, such as email filtering, computer vision, where it is infeasible to develop an algorithm of specific instructions for performing the task. Machine learning is related to computational statistics, which focuses on making predictions using computers; the study of mathematical optimization delivers methods and application domains to the field of machine learning. Data mining is a field of study within machine learning, focuses on exploratory data analysis through unsupervised learning.

In its application across business problems, machine learning is referred to as predictive analytics. The name machine learning was coined in 1959 by Arthur Samuel. Tom M. Mitchell provided a quoted, more formal definition of the algorithms studied in the machine learning field: "A computer program is said to learn from experience E with respect to some class of tasks T and performance measure P if its performance at tasks in T, as measured by P, improves with experience E." This definition of the tasks in which machine learning is concerned offers a fundamentally operational definition rather than defining the field in cognitive terms. This follows Alan Turing's proposal in his paper "Computing Machinery and Intelligence", in which the question "Can machines think?" is replaced with the question "Can machines do what we can do?". In Turing's proposal the various characteristics that could be possessed by a thinking machine and the various implications in constructing one are exposed. Machine learning tasks are classified into several broad categories.

In supervised learning, the algorithm builds a mathematical model from a set of data that contains both the inputs and the desired outputs. For example, if the task were determining whether an image contained a certain object, the training data for a supervised learning algorithm would include images with and without that object, each image would have a label designating whether it contained the object. In special cases, the input may be only available, or restricted to special feedback. Semi-supervised learning algorithms develop mathematical models from incomplete training data, where a portion of the sample input doesn't have labels. Classification algorithms and regression algorithms are types of supervised learning. Classification algorithms are used. For a classification algorithm that filters emails, the input would be an incoming email, the output would be the name of the folder in which to file the email. For an algorithm that identifies spam emails, the output would be the prediction of either "spam" or "not spam", represented by the Boolean values true and false.

Regression algorithms are named for their continuous outputs, meaning they may have any value within a range. Examples of a continuous value are the length, or price of an object. In unsupervised learning, the algorithm builds a mathematical model from a set of data which contains only inputs and no desired output labels. Unsupervised learning algorithms are used to find structure in the data, like grouping or clustering of data points. Unsupervised learning can discover patterns in the data, can group the inputs into categories, as in feature learning. Dimensionality reduction is the process of reducing the number of "features", or inputs, in a set of data. Active learning algorithms access the desired outputs for a limited set of inputs based on a budget, optimize the choice of inputs for which it will acquire training labels; when used interactively, these can be presented to a human user for labeling. Reinforcement learning algorithms are given feedback in the form of positive or negative reinforcement in a dynamic environment, are used in autonomous vehicles or in learning to play a game against a human opponent.

Other specialized algorithms in machine learning include topic modeling, where the computer program is given a set of natural language documents and finds other documents that cover similar topics. Machine learning algorithms can be used to find the unobservable probability density function in density estimation problems. Meta learning algorithms learn their own inductive bias based on previous experience. In developmental robotics, robot learning algorithms generate their own sequences of learning experiences known as a curriculum, to cumulatively acquire new skills through self-guided exploration and social interaction with humans; these robots use guidance mechanisms such as active learning, motor synergies, imitation. Arthur Samuel, an American pioneer in the field of computer gaming and artificial intelligence, coined the term "Machine Learning" in 1959 while at IBM; as a scientific endeavour, machine learning grew out of the quest for artificial intelligence. In the early days of AI as an academic discipline, some researchers were interested in having machines learn from data.

They attempted to approach the problem with various symbolic methods, as well as what were termed "neural networks". Probabilistic reasoning was employed in automated medical

Potty parity

Potty parity is equal or equitable provision of public toilet facilities for females and males within a public space. Parity may be defined in various ways in relation to facilities in a building; the simplest is as equal floorspace for female washrooms. Since men's and boys' bathrooms include urinals, which take up less space than stalls, this still results in more facilities for males. An alternative parity is by number of fixtures within washrooms. However, since females on average spend more time in washrooms more males are able to use more facilities per unit time. More recent parity regulations therefore require more fixtures for females to ensure that the average time spent waiting to use the toilet is the same for females as for males, or to equalise throughputs of male and female toilets. Women/girls spend more time in washrooms than men/boys, both for physiological and cultural reasons; the requirement to use a cubicle rather than a urinal means urination takes longer and hand washing must be done more thoroughly.

Females make more visits to washrooms. Urinary tract infections and incontinence are more common in females. Pregnancy, menstruation and diaper-changing increase usage; the elderly, who are disproportionately female, take more frequent bathroom visits. A variety of female urinals and personal funnels have been invented to make it easier for females to urinate standing up. None has become widespread enough to affect policy formation on potty parity. John F. Banzhaf III, a law professor at George Washington University, calls himself the "father of potty parity." Banzhaf argues that to ignore potty parity, that is, to have equal facilities for males and females, constitutes a form of sex discrimination against women. In the 1970s the Committee to End Pay Toilets in America made a similar point: that allowing toilet providers to charge for the use of a cubicle while urinals required no money was unfair to women. Several authors have identified potty parity as a potential rallying issue for feminism, saying all women can identify with it.

The first bathroom for congresswomen in the United States Capitol was opened in 1962. Segregation of toilet facilities by race was outlawed in the United States by the Civil Rights Act of 1964. Provision of disabled-access facilities was mandated in federal buildings by the Architectural Barriers Act of 1968 and in private buildings by the Americans with Disabilities Act of 1990. No federal legislation relates to provision of facilities for women; the banning of pay toilets came about because women had to pay to urinate whereas men only had to pay to defecate. In many older buildings, little or no provision was made for women because few would work in or visit them. Increased gender equality in employment and other spheres of life has impelled change; until the 1980s, building codes for stadiums in the United States stipulated more toilets for men, on the assumption that most sports fans were male. In 1973, to protest the lack of female bathrooms at Harvard University, women poured jars of fake urine on the steps of the University's Lowell Hall, a protest Florynce Kennedy thought of and participated in.

The first "Restroom Equity" Act in the United States was passed in California in 1989. It was introduced by then-Senator Arthur Torres after several long waits for his wife to return from the bathroom. Facilities for female U. S. senators on the Senate Chamber level were first provided in 1992. LP Field in Nashville, Tennessee was built in 1999 in compliance with the Tennessee Equitable Restrooms Act, providing 288 fixtures for men and 580 for women; the Tennessean reported fifteen-minute waits compared to none at women's rooms. The Act was amended in 2000 to empower the state architect to authorize extra men's rooms at stadiums, horse shows and auto racing venues. In 2011 the U. S. House of Representatives got its first women’s bathroom near the chamber, it is only open to women lawmakers, not the public. Current laws in the United Kingdom require a 1:1 female–male ratio of restroom space in public buildings; the International Building Code requires a 2:1 female–male ratio of toilets. New York City Council passed a law in 2005 requiring this in all public buildings.

An advisory ruling had been passed in 2003. U. S. state laws vary between 1:1, 3:2, 2:1 ratios. The Uniform Plumbing Code specifies a 4:1 ratio in movie theaters. Gender-neutral toilets are common in some contexts, including on aircraft, on trains or buses, portable toilets, accessible toilets. In parts of Europe they are common in buildings. In the United States, they began to appear in the 2000s on university campuses and in some upmarket restaurants; as of late 2013 gender-neutral toilets are still limited to some public restrooms. Although gender-neutral toilets were introduced to the U. S. in 2000 it wasn't until 2013 that the state of California passed bill 1266 recognized as the "transgender bathroom bill" which enforces gender-neutral toilets at universities. Bill 1266 only targets a small group of the population in the United States. Most range from 18–24 years old In 2011 a "Right to Pee" campaign began in Mumbai, India's largest city. Women, but not men, have to pay to urinate despite regulations against this practice.

Women have been sexually assaulted while urinating in fields. Thus, activists have collected more than 50,000 signatures supporting their demands that the local government stop charging women to urinate, build more toilets, keep them clean, provide sanitary napkins and a trash can, hire female attendants. In response, city officials have agreed to build hundreds of public t

Parity (physics)

In quantum mechanics, a parity transformation is the flip in the sign of one spatial coordinate. In three dimensions, it can refer to the simultaneous flip in the sign of all three spatial coordinates: P: ↦, it can be thought of as a test for chirality of a physical phenomenon, in that a parity inversion transforms a phenomenon into its mirror image. All fundamental interactions of elementary particles, with the exception of the weak interaction, are symmetric under parity; the weak interaction thus provides a means for probing chirality in physics. In interactions that are symmetric under parity, such as electromagnetism in atomic and molecular physics, parity serves as a powerful controlling principle underlying quantum transitions. A matrix representation of P has determinant equal to −1, hence is distinct from a rotation, which has a determinant equal to 1. In a two-dimensional plane, a simultaneous flip of all coordinates in sign is not a parity transformation. In quantum mechanics, wave functions which are unchanged by a parity transformation are described as functions, while those which change sign under a parity transformation are odd functions.

Under rotations, classical geometrical objects can be classified into scalars and tensors of higher rank. In classical physics, physical configurations need to transform under representations of every symmetry group. Quantum theory predicts that states in a Hilbert space do not need to transform under representations of the group of rotations, but only under projective representations; the word projective refers to the fact that if one projects out the phase of each state, where we recall that the overall phase of a quantum state is not an observable a projective representation reduces to an ordinary representation. All representations are projective representations, but the converse is not true, therefore the projective representation condition on quantum states is weaker than the representation condition on classical states; the projective representations of any group are isomorphic to the ordinary representations of a central extension of the group. For example, projective representations of the 3-dimensional rotation group, the special orthogonal group SO, are ordinary representations of the special unitary group SU.

Projective representations of the rotation group that are not representations are called spinors, so quantum states may transform not only as tensors but as spinors. If one adds to this a classification by parity, these can be extended, for example, into notions of scalars and pseudoscalars which are rotationally invariant. Vectors and axial vectors. One can define reflections such as V x: ↦, which have negative determinant and form a valid parity transformation. Combining them with rotations one can recover the particular parity transformation defined earlier; the first parity transformation given does not work in an number of dimensions, because it results in a positive determinant. In odd number of dimensions only the latter example of a parity transformation can be used. Parity forms the abelian group Z 2 due to the relation P ^ 2 = 1 ^. All Abelian groups have only one-dimensional irreducible representations. For Z 2, there are two irreducible representations: one is under parity, P ^ ϕ = + ϕ, the other is odd, P ^ ϕ = − ϕ.

These are useful in quantum mechanics. However, as is elaborated below, in quantum mechanics states need not transform under actual representations of parity but only under projective representations and so in principle a parity transformation may rotate a state by any phase. Newton's equation of motion F → = m a → equates two vectors, hence is invariant under parity; the law of gravity involves only vectors and is therefore, invariant under parity. However, angular momentum L → is an axial vector, L → = r → × p → {\displaystyle =\times {\vec