A-erh-chin Mountains also called Altyn Tagh, Chinese (Wade-Giles) A-erh-chin Shan, or (Pinyin) Altun Shan, mountain range in southern Uighur Autonomous Region of Sinkiang, China. Branching off from the Kunlun Mountains, the range runs from southwest to northeast to form the boundary between the Tarim Basin to the north and the minor basin of Lake A-ya-k'o-k'u-mu and the Tsaidam Basin areas of interior drainage to the south.
The range falls into three divisions. The southwest section, bordering the Kunlun, is extremely rugged and complex; some ranges and peaks rise to heights of more than 20,000 feet (6,100 m) and are covered with perpetual snows. The central portion, forming the border of the western Tsaidam Basin, is lower, averaging about 13,000 feet (4,000 m) in height, and is much narrower. The eastern section, in which the range joins the Nan Mountains, is again higher, with peaks of 16,500 feet (5,000 m); it is structurally more complicated, consisting of a series of short ranges, the axes of which gradually adapt to the main northwest-to-southeast axis of the Nan system.
There are very few rivers, because the area is one of extreme aridity, particularly in its central section. In the west various small streams run off into the Takla Makan Desert in the north, into Lake A-ya-k'o-k'u-mu to the south, or into the western Tsaidam Basin. The main pass is the Tang-chin Pass at the eastern end, which is crossed by a motor road between eastern Sinkiang (via Kansu province), the Tsaidam Basin, and the Tibet Autonomous Region (via Tsinghai province).
Pinyin Acheng, formerly (until 1909) Ashihho, city in Heilungkiang sheng (province), China. Located southeast of Harbin (Ha-erh-pin) in the basin of the A-shih River, it is a collecting and commercial centre for a rich agricultural district that provides part of Harbin's food supply. A-ch'eng is also an industrial center with brickworks, engineering works specializing in electrical equipment and other machinery, and various industries based on local agricultural production, such as breweries, sugar refineries, and a flax-processing plant. A-ch'eng also developed a small iron industry in the late 1950s. Most of its industries are closely integrated with those in Harbin.
To the south of A-ch'eng are the remains of an ancient walled city known as Pai-ch'eng, or Pei-ch'eng. This site is thought to be the remains of Hui-ning, which was the capital of the Chin (Juchen) dynasty from 1122 to 1234 and a subsidiary capital after 1173. Pop. (1989 est.) 188,600.
Arabic:“the Night-Blind”, born before 570,Durna,Arabia
died c.625,Durna
In full Maymun Ibn Qays Al-a'sha( pre-Islamic poet whose qasldah (“ode”) is included by the critic Abu Ubaydah (d. 825) in the celebrated Mu'allaqat, a collection of seven pre-Islamic qasldahs, each of which was considered by its author to be his best; the contents of the collection vary slightly, according to the views of several compilers.
Al-A'sha spent his youth in travels through Mesopotamia, Syria, Arabia, and Ethiopia. He continued to travel, even after becoming blind, particularly along the western coast of the Arabian Peninsula. It was then that he turned to the writing of panegyrics as a means of support. His style, reliant on sound effects and full-bodied foreign words, tends to be artificial.
born 614, Mecca, Arabia [now in Saudi Arabia]
died July 678, Medina
In full Aishah Bint Abi Bakr the third and most favored wife of the Prophet Muhammad (the founder of Islam, who played a role of some political importance after the Prophet's death.
All Muhammad's marriages had political motivations, and in this case the intention seems to have been to cement ties with Aishah's father, Abu Bakr, who was one of Muhammad's most important supporters. Aishah's physical charms, together with the genuine warmth of their relationship, secured her a place in his affections that was not lessened by his subsequent marriages. It is said that in 627 she accompanied the Prophet on an expedition but became separated from the group. When she was later escorted back to Medina by a man who had found her in the desert, Muhammad's enemies claimed that she had been unfaithful. Muhammad, who trusted her, had a revelation asserting her innocence and publicly humiliated her accusers. She had no important influence on his political or religious policies while he lived.
When Muhammad died in 632, Aishah was left a childless widow of 18. She remained politically inactive until the time of Uthman (644–656; the third caliph, or leader of the Islamic community , during whose reign she played an important role in fomenting opposition that led to his murder in 656. She led an army against his successor, Ali, but was defeated in the Battle of the Camel. The engagement derived its name from the fierce fighting that centered around the camel upon which Aishah was mounted. Captured, she was allowed to live quietly in Medina.
In Western philosophy since the time of Immanuel Kant, knowledge that is independent of all particular experiences, as opposed to a posteriori knowledge, which derives from experience alone. The Latin phrases a priori (“from what is before”) and a posteriori (“from what is after”) were used in philosophy originally to distinguish between arguments from causes and arguments from effects.
The first recorded occurrence of the phrases is in the writings of the 14th-century logician Albert of Saxony. Here, an argument a priori is said to be “from causes to the effect” and an argument a posteriori to be “from effects to causes.” Similar definitions were given by many later philosophers down to and including G.W. Leibniz, and the expressions still occur sometimes with these meanings in non philosophical contexts. It should be remembered that medieval logicians used the word “cause” in a syllogistic sense corresponding to Aristotle's aitia and did not necessarily mean by prius something earlier in time. This point is brought out by the use of the phrase demonstratio propter quid (“demonstration on account of what”) as an equivalent for demonstratio a priori and of demonstratio quia (“demonstration that, or because”) as an equivalent for demonstratio a posteriori. Hence the reference is obviously to Aristotle's distinction between knowledge of the ground or explanation of something and knowledge of the mere fact.
Latent in this distinction for Kant is the antithesis between necessary, deductive truth and probable, inductive truth. The former applies to a priori judgments, which are arrived at independently of experience and hold universally; the latter applies to a posteriori judgments, which are contingent on experience and therefore must acknowledge possible exceptions. In his Critique of Pure Reason Kant used these distinctions, in part, to explain the special case of mathematical knowledge, which he regarded as the fundamental example of a priori knowledge.
Although the use of a priori to distinguish knowledge such as that which we have in mathematics is comparatively recent, the interest of philosophers in that kind of knowledge is almost as old as philosophy itself. No one finds it puzzling that one can acquire information by looking, feeling, or listening, but philosophers who have taken seriously the possibility of learning by mere thinking have often considered that this requires some special explanation. Plato maintained in his Meno and in his Phaedo that the learning of geometrical truths was only the recollection of knowledge possessed in a previous existence when we could contemplate the eternal ideas, or forms, directly. Augustine and his medieval followers, sympathizing with Plato's intentions but unable to accept the details of his theory, declared that the ideas were in the mind of God, who from time to time gave intellectual illumination to men. René Descartes, going further in the same direction, held that all the ideas required for a priori knowledge were innate in each human mind. For Kant the puzzle was to explain the possibility of a priori judgments that were also synthetic (i.e., not merely explicative of concepts), and the solution that he proposed was the doctrine that space, time, and the categories (e.g., causality), about which we were able to make such judgments, were forms imposed by the mind on the stuff of experience.
In each of these theories the possibility of a priori knowledge is explained by a suggestion that we have a privileged opportunity for studying the subject matter of such knowledge. The same conception recurs also in the very un-Platonic theory of a priori knowledge first enunciated by Thomas Hobbes in his De Corp ore and adopted in the 20th century by the logical empiricists. According to this theory, statements of necessity can be made a priori because they are merely by-products of our own rules for the use of language.
(Spanish “in the sacred style” or “in sacred terms”)
in Spanish literature, the recasting of a secular work as a religious work, or, more generally, a treatment of a secular theme in religious terms through the use of allegory, symbolism, and metaphor. Adaptations a lo divino were popular during the Golden Age of Spanish literature during the 16th and 17th centuries.
(Italian: “in the church style”), performance of a polyphonic (multipart) musical work by unaccompanied voices. Originally referring to sacred choral music, the term now refers to secular music as well.
The a cappella style arose about the time of the composer Josquin des Prez, in the late 15th century, and reached preeminence with Palestrina in the late 16th century in the music that he wrote for the Sistine Chapel of the Vatican. Because no independent instrumental parts were written, later scholars assumed that the choir sang unaccompanied, but the evidence is now that an organ or other instruments exactly “doubled” some or several of the vocal parts. By the 17th century, a cappella music was giving way to the cantata, for which parts were written specifically for instruments as well as for voices.
In mathematics and logic, a proposition or statement that is demonstrated. In geometry, a proposition is commonly considered as a problem (a construction to be effected) or a theorem (a statement to be proved). The statement “If two lines intersect, each pair of vertical angles is equal,” for example, is a theorem. The so-called fundamental theorem of algebra asserts that every (complex) polynomial equation in one variable has at least one complex root or solution. The Greeks also recognized a proposition lying between a theorem and a problem, the porism, directed to producing or finding what is proposed.
In formal settings, an necessary property of theorems is that they are derivable using a fixed set of supposition rules and axioms without any additional assumptions. This is not a matter of the semantics of the language: the expression that results from a source is a syntactic consequence of all the expressions that precede it. In mathematics, the derivation of a theorem is often interpreted as a proof of the fact of the resulting expression, but dissimilar deductive systems can yield other interpretations, depending on the meanings of the derivation rules.
The proofs of theorems have two components, called the hypotheses and the conclusions. The proof of a mathematical theorem is a logical argument signifying that the conclusions are a necessary outcome of the hypotheses, in the sense that if the hypotheses are true then the conclusions must also be true, without any further assumptions. The concept of a theorem is therefore fundamentally deductive, in contrast to the concept of a scientific theory, which is empirical.
Formal and informal notions
Logically most theorems are of the form of an indicative conditional: if A, then B. Such a theorem does not state that B is always true, only that B must be true if A is true. In this case A is called the hypothesis of the theorem (note that "hypothesis" here is something very different from a conjecture) and B the conclusion. The theorem "If n is an even natural number then n/2 is a natural number" is a typical example in which the hypothesis is that n is an even natural number and the conclusion is that n/2 is also a natural number.
In order to be proven, a theorem must be expressible as a precise, formal statement. Nevertheless, theorems are usually expressed in natural language rather than in a completely symbolic form, with the intention that the reader will be able to produce a formal statement from the informal one. In addition, there are often hypotheses which are implicit in context, rather than explicitly stated.
It is common in mathematics to choose a number of hypotheses that are assumed to be true within a given theory, and then proclaim that the theory consists of all theorems provable using those hypotheses as assumption. In this case the hypotheses that form the foundational basis are called the axioms (or postulates) of the theory. The field of mathematics known as proof theory studies formal axiom systems and the proofs that can be performed within them.
Relation with scientific theories
Theorems in mathematics and theories in science are basically different in their epistemology. A scientific theory cannot be proven; its key attribute is that it is falsifiable, that is, it makes predictions about the natural world that are testable by experiments. Any disagreement between prediction and experiment demonstrates the incorrectness of the scientific theory, or at least limits its exactness or domain of validity. Mathematical theorems, on the other hand, are purely abstract formal statements: the proof of a theorem cannot involve experiments or other empirical evidence in the same way such evidence is used to support scientific theories.
Nonetheless, there is some degree of empiricism and data collection involved in the discovery of mathematical theorems. By establishing a pattern, sometimes with the use of a powerful computer, mathematicians may have an idea of what to prove, and in some cases even a plan for how to set about doing the proof. For example, the Collatz conjecture has been verified for start values up to about 2.88 × 1018. The Riemann hypothesis has been verified for the first 10 trillion zero's of the zeta function. Neither of these statements is considered to be proven.
Such evidence does not constitute proof. For example, the Mertens conjecture is a statement about natural numbers that is now known to be false, but no explicit counterexample (i.e., a natural number n for which the Mertens function M(n) equals or exceeds the square root of n) is known: all numbers less than 1014 have the Mertens property, and the smallest number which does not have this property is only known to be less than the exponential of 1.59 × 1040, which is approximately 10 to the power 4.3 × 1039. Since the number of particles in the universe is generally considered to be less than 10 to the power 100 (a googol), there is no hope to find an explicit counterexample by exhaustive search at present.
Note that the word "theory" also exists in mathematics, to denote a body of mathematical axioms, definitions and theorems, as in, for example, group theory. There are also "theorems" in science, particularly physics, and in engineering, but they often have statements and proofs in which physical assumptions and intuition play an important role; the physical axioms on which such "theorems" are based are themselves falsifiable.
Theorems in logic
Logic, especially in the field of proof theory, considers theorems as statements (called formulas or well formed formulas) of a formal language. A set of deduction rules, also called transformation rules or a formal grammar, must be provided. These deduction rules tell exactly when a formula can be derived from a set of premises.
Different sets of derivation rules give rise to different interpretations of what it means for an expression to be a theorem. Some derivation rules and formal languages are intended to capture mathematical reasoning; the most common examples use first-order logic. Other deductive systems describe term rewriting, such as the reduction rules for λ calculus.
The definition of theorems as elements of a formal language allows for results in proof theory that study the structure of formal proofs and the structure of provable formulas. The most famous result is Gödel's incompleteness theorem; by representing theorems about basic number theory as expressions in a formal language, and then representing this language within number theory itself, Gödel constructed examples of statements that are neither provable nor disprovable from axiomatizations of number theory.
Terminology
Theorems are often indicated by several other terms: the actual label "theorem" is reserved for the most important results, whereas results which are less important, or distinguished in other ways, are named by different terminology.
1) A proposition is a statement not associated with any particular theorem. This term sometimes connotes a statement with a simple proof.
2) A lemma is a "pre-theorem", a statement that forms part of the proof of a larger theorem. The distinction between theorems and lemmas is rather arbitrary, since one mathematician's major result is another's minor claim. Gauss's lemma and Zorn's lemma, for example, are interesting enough that some authors present the nominal lemma without going on to use it in the proof of a theorem.
3) A corollary is a proposition that follows with little or no proof from one other theorem or definition. That is, proposition B is a corollary of a proposition A if B can readily be deduced from A.
4) A claim is a necessary or independently interesting result that may be part of the proof of another statement. Despite the name, claims must be proved.
In mathematics, the ratio of the circumference of a circle to its diameter. The symbol π was popularized by the Swiss mathematician Leonhard Euler in the early 18th century to represent this ratio. Because pi is irrational (not equal to the ratio of any two whole numbers), an approximation, such as 22/7, is often used for everyday calculations; to 31 decimal places pi is 3.1415926535897932384626433832795.
The Babylonians (c. 2000 BC) used 3.125 to approximate pi, a value they obtained by calculating the perimeter of a hexagon inscribed within a circle. The Rhind papyrus (c. 1650 BC) indicates that ancient Egyptians used a value of 256/81 or about 3.16045. Archimedes (c. 250 BC) took a major step forward by devising a method to obtain pi to any desired accuracy, given enough patience. By inscribing and circumscribing regular polygons about a circle to obtain upper and lower bounds, he obtained 223/71 π 22/7, or an average value of about 3.1418. Archimedes also proved that the ratio of the area of a circle to the square of its radius is the same constant.
Over the ensuing centuries, Chinese, Indian, and Arab mathematicians extended the number of decimal places known through tedious calculations, rather than improvements on Archimedes' method. By the end of the 17th century, however, new methods of mathematical analysis in Europe provided improved ways of calculating pi involving infinite series. For example, Sir Isaac Newton used his binomial theorem to calculate 16 decimal places quickly. Early in the 20th century, the Indian mathematician Srinivasa Ramanujan developed exceptionally efficient ways of calculating pi that were later incorporated into computer algorithms. By the end of the 20th century, computers had calculated pi to more than 200,000,000,000 decimal places.
Pi occurs in various mathematical problems involving the lengths of arcs or other curves, the areas of ellipses, sectors, and other curved surfaces, and the volumes of many solids. It is also used in various formulas of physics and engineering to describe such periodic phenomena as the motion of pendulums, the vibration of strings, and alternating electric currents.
S.A.Ramanujan - Died April 26, 1920, Kumbakonam,Born Dec. 22, 1887, Erode, India in Erode, a small village about 400 km southwest of Madras (Chennai). He was one of India's greatest mathematical geniuses. Using one of his identities, mathematics have been able to calculate the value of Π (pi) correct to millions of places of decimal.
Early life
Ramanujan was born on 22 December 1887 in Erode, Tamil Nadu, India, at the place of residence of his maternal grandparents. His father, K. Srinivasa Iyengar worked as a clerk in a sari shop and hailed from the district of Thanjavur. His mother, Komalatammal was a housewife and also a singer at a local temple. They lived in Sarangapani Street in a south-Indian-style home (now a museum) in the town of Kumbakonam. When Ramanujan was a year and a half old, his mother gave birth to a son named Sadagopan. The newborn died less than three months later. In December 1889, Ramanujan had smallpox and fortunately recovered, unlike the thousands in the Thanjavur district who had succumbed to the disease that year. He moved with his mother to her parents' house in Kanchipuram, near Madras. In November 1891, and again in 1894, his mother gave birth, but both children died before their first birthdays.
When he was 15 years old, he obtained a copy of George Shoobridge Carr's Synopsis of elementary results in Pure and Applied Mathematics, 2 vol. (1880–86).This collection of some 6,000 theorems (none of the material was newer than 1860) aroused his genius. Having verified the results in Carr's book, Ramanujan went beyond it, developing his own theorems and ideas. In 1903 he secured a scholarship to the University of Madras but lost it the following year because he neglected all other studies in pursuit of mathematics.
Ramanujan continued his work, without employment and living in the poorest circumstances. After marrying in 1909 he began a search for permanent employment that culminated in an interview with a government official, Ramachandra Rao. Impressed by Ramanujan's mathematical prowess, Rao supported his research for a time, but Ramanujan, unwilling to exist on charity, obtained a clerical post with the Madras Port Trust.
In 1911 Ramanujan published the first of his papers in the Journal of the Indian Mathematical Society. His genius slowly gained recognition, and in 1913 he began a correspondence with the British mathematician Godfrey H. Hardy that led to a special scholarship from the University of Madras and a grant from Trinity College, Cambridge. Overcoming his religious objections, Ramanujan traveled to England in 1914, where Hardy tutored him and collaborated with him in some research.
Ramanujan's knowledge of mathematics (most of which he had worked out for himself) was startling. Although almost completely ignorant of what had been developed, his mastery of continued fractions was unequaled by any living mathematician. He worked out the Riemann series,the elliptic integrals, hyper geometric series, the functional equations of the zeta function, and his own theory of divergent series. On the other hand, the gaps in his knowledge were equally startling. He knew nothing of doubly periodic functions, the classical theory of quadratic forms, or Cauchy's theorem, and had only the most nebulous idea of what constitutes a mathematical proof. Though brilliant, many of his theorems on the theory of prime numbers were completely wrong.
In England Ramanujan made further advances, especially in the partition of numbers. His papers were published in English and European journals, and in 1918 he became the first Indian to be elected to the Royal Society of London.
In 1917 Plagued by health problems all through his life, living in a country far away from home, and obsessively involved with his mathematics, Ramanujan's health worsened in England, perhaps exacerbated by stress, and by the scarcity of vegetarian food during the First World War. He was diagnosed with tuberculosis and a severe vitamin deficiency and was confined to a sanatorium. Ramanujan returned to Kumbakonam, India in 1919 and died soon thereafter at the age of 32,generally unknown to the world at large but recognized by mathematicians as a phenomenal genius, without peer since Leonhard Euler (1707–83)and Karl Jacobi (1804–51).His wife, S. Janaki Ammal, lived in Chennai (formerly Madras) until her death in 1994.
A 1994 analysis of Ramanujan's medical records and symptoms by Dr. D. A. B. Young concluded that it was much more likely he had hepatic amoebiasis, a parasitic infection of the liver. This is supported by the fact that Ramanujan had spent time in Madras, where the disease was widespread. He had two episodes of dysentery before he left India. When not properly treated, dysentery can lie dormant for years and lead to hepatic amoebiasis.It was a difficult disease to diagnose, but once diagnosed would have been readily curable.
Personality
Ramanujan has been described as a person with a somewhat shy and quiet disposition, a dignified man with pleasant manners. He lived a rather spartan life while at Cambridge.
Spiritual life
Ramanujan's first Indian biographers describe him as rigorously orthodox. Ramanujan credited his acumen to his family goddess, Namagiri, and looked to her for inspiration in his work. He often said that an equation for me has no meaning, unless it represents a thought of God.
G. H. Hardy cites Ramanujan as remarking that all religions seemed equally true to him. Hardy further argued that Ramanujan's religiousness had been overstated in the point of belief, not practice by his Indian biographers, and romanticized by Westerners. At the same time, he remarked on Ramanujan's strict observance of vegetarianism.
Mathematical achievements
In mathematics, there is a distinction between having an insight and having a proof. Ramanujan's talent suggested a plethora of formulae that could then be investigated in depth later. It is said that Ramanujan's discoveries are unusually rich and that there is often more in it than what initially meets the eye. As a by-product, new directions of research were opened up. Examples of the most interesting of these formulae include the intriguing infinite series for π, one of which is given below
This result is based on the negative fundamental discriminant d = −4×58 with class number h(d) = 2. Note that 5×7×13×58 = 26390 and is related to the fact that,
Ramanujan's series for π converges extraordinarily rapidly (exponentially) and forms the basis of some of the fastest algorithms currently used to calculate π. Truncating the sum to the first term also gives the approximation 
for π, which is correct to six decimal places.
One of his remarkable capabilities was the rapid solution for problems. He was sharing a room with P.C.Mahalanobis who had a problem, "Imagine that you are on a street with houses marked 1 through n. There is a house in between (x) such that the sum of the house numbers to left of it equals the sum of the house numbers to its right. If n is between 50 and 500, what are n and x." This is a bivariate problem with multiple solutions. Ramanujan thought about it and gave the answer with a twist: He gave a continued fraction. The unusual part was that it was the solution to the whole class of problems. Mahalanobis was astounded and asked how he did it. "It is simple. The minute I heard the problem, I knew that the answer was a continued fraction. Which continued fraction, I asked myself. Then the answer came to my mind", Ramanujan replied.
His intuition also led him to derive some previously unknown identities, such as
for all θ, where Γ(z) is the gamma function. Equating coefficients of θ0, θ4, and θ8 gives some deep identities for the hyperbolic secant.
In 1918, G. H. Hardy and Ramanujan studied the partition function P(n) extensively and gave a very accurate non-convergent asymptotic series that permits exact computation of the number of partitions of an integer. Hans Rademacher, in 1937, was able to refine their formula to find an exact convergent series solution to this problem. Ramanujan and Hardy's work in this area gave rise to a powerful new method for finding asymptotic formulae, called the circle method.
One example of his intuition is his discovery of Mock theta functions, in the last year of his life. This was no surprise to some mathematicians as they remarked, "He has his own creativity and the collaboration with Hardy to back it up. So, his finding these is no surprise to the mathematical community." This has gained some interest recently due to proof of the exact formula for the coefficients of any Mock Theta function. Many mathematicians have cited it as the most significant among his discoveries.
Hillary, Obama, Giuliani & more play paintball for the USA Presidency!
Read more Entry>>Amazing real-life 3D trials game. Negotiate obstacles in the quickest time!
Moves
Up and down cursor keys - Accelerate or brake |left and right cursor keys - Lean forward or back
Obama and Hillary fight it out in hand-to-hand combat.Moves
Up cursor key - Jump | down cursor key - Crouch | left and right cursor keys - Move left and right | A - Punch | S - Smack | D - Kick | W - Bonus Move | T - Taunt
This cartoon is about two ants atom ant and karate ant.At first two bad guys convince karate ant that atom ant is a bad guy and must get rid of him.When karate ant realized that atom ant is the good guy and the bad guys wanted to kill atom ant with the help of him he got angry and thrashed the bad guys.
