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Pi (letter)
Pi (uppercase Π, lowercase π and ϖ; el, πι ) is the sixteenth letter of the Greek alphabet, representing the voiceless bilabial plosive . In the system of Greek numerals it has a value of 80. It was derived from the Phoenician letter Pe (). Letters that arose from pi include Latin P, Cyrillic Pe (П, п), Coptic pi (Ⲡ, ⲡ), and Gothic pairthra (𐍀). Uppercase Pi The uppercase letter Π is used as a symbol for: * In textual criticism, '' Codex Petropolitanus'', a 9th-century uncial codex of the Gospels, now located in St. Petersburg, Russia. * In legal shorthand, it represents a plaintiff. In science and engineering: * The product operator in mathematics, indicated with capital pi notation (in analogy to the use of the capital Sigma as summation symbol). * The osmotic pressure in chemistry. * The viscous stress tensor in continuum mechanics and fluid dynamics. Lowercase Pi The lowercase letter π is used as a symbol for: * The mathematical re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Greek Alphabet
The Greek alphabet has been used to write the Greek language since the late 9th or early 8th century BCE. It is derived from the earlier Phoenician alphabet, and was the earliest known alphabetic script to have distinct letters for vowels as well as consonants. In Archaic and early Classical times, the Greek alphabet existed in many local variants, but, by the end of the 4th century BCE, the Euclidean alphabet, with 24 letters, ordered from alpha to omega, had become standard and it is this version that is still used for Greek writing today. The uppercase and lowercase forms of the 24 letters are: : , , , , , , , , , , , , , , , , , /ς, , , , , , . The Greek alphabet is the ancestor of the Latin and Cyrillic scripts. Like Latin and Cyrillic, Greek originally had only a single form of each letter; it developed the letter case distinction between uppercase and lowercase in parallel with Latin during the modern era. Sound values and conventional transcriptions for some ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plaintiff
A plaintiff ( Π in legal shorthand) is the party who initiates a lawsuit (also known as an ''action'') before a court. By doing so, the plaintiff seeks a legal remedy. If this search is successful, the court will issue judgment in favor of the plaintiff and make the appropriate court order (e.g., an order for damages). "Plaintiff" is the term used in civil cases in most English-speaking jurisdictions, the notable exceptions being England and Wales, where a plaintiff has, since the introduction of the Civil Procedure Rules in 1999, been known as a "claimant" and Scotland, where the party has always been known as the "pursuer". In criminal cases, the prosecutor brings the case against the defendant, but the key complaining party is often called the "complainant". In some jurisdictions, a lawsuit is commenced by filing a summons, claim form or a complaint. These documents are known as pleadings, that set forth the alleged wrongs committed by the defendant or defendants with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime-counting Function
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number ''x''. It is denoted by (''x'') (unrelated to the number ). History Of great interest in number theory is the growth rate of the prime-counting function. It was conjectured in the end of the 18th century by Gauss and by Legendre to be approximately : \frac x where log is the natural logarithm, in the sense that :\lim_ \frac=1. This statement is the prime number theorem. An equivalent statement is :\lim_\pi(x) / \operatorname(x)=1 where li is the logarithmic integral function. The prime number theorem was first proved in 1896 by Jacques Hadamard and by Charles de la Vallée Poussin independently, using properties of the Riemann zeta function introduced by Riemann in 1859. Proofs of the prime number theorem not using the zeta function or complex analysis were found around 1948 by Atle Selberg and by Paul Erdős (for the most ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Geometry
Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the ''Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions ( theorems) from these. Although many of Euclid's results had been stated earlier,. Euclid was the first to organize these propositions into a logical system in which each result is '' proved'' from axioms and previously proved theorems. The ''Elements'' begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the ''Elements'' states results of what are now called algebra and number theory, explained in geometrical language. For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diameter
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for the diameter of a sphere. In more modern usage, the length d of a diameter is also called the diameter. In this sense one speaks of diameter rather than diameter (which refers to the line segment itself), because all diameters of a circle or sphere have the same length, this being twice the radius r. :d = 2r \qquad\text\qquad r = \frac. For a convex shape in the plane, the diameter is defined to be the largest distance that can be formed between two opposite parallel lines tangent to its boundary, and the is often defined to be the smallest such distance. Both quantities can be calculated efficiently using rotating calipers. For a curve of constant width such as the Reuleaux triangle, the width and diameter are the same beca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circumference
In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out to a line segment. More generally, the perimeter is the curve length around any closed figure. Circumference may also refer to the circle itself, that is, the locus corresponding to the edge of a disk. The is the circumference, or length, of any one of its great circles. Circle The circumference of a circle is the distance around it, but if, as in many elementary treatments, distance is defined in terms of straight lines, this cannot be used as a definition. Under these circumstances, the circumference of a circle may be defined as the limit of the perimeters of inscribed regular polygons as the number of sides increases without bound. The term circumference is used when measuring physical objects, as well as when considering abstr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irrational Number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being '' incommensurable'', meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself. Among irrational numbers are the ratio of a circle's circumference to its diameter, Euler's number ''e'', the golden ratio ''φ'', and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational. Like all real numbers, irrational numbers can be expressed in positional notation, notably as a decimal number. In the ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transcendental Number
In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and . Though only a few classes of transcendental numbers are known—partly because it can be extremely difficult to show that a given number is transcendental—transcendental numbers are not rare. Indeed, almost all real and complex numbers are transcendental, since the algebraic numbers comprise a countable set, while the set of real numbers and the set of complex numbers are both uncountable sets, and therefore larger than any countable set. All transcendental real numbers (also known as real transcendental numbers or transcendental irrational numbers) are irrational numbers, since all rational numbers are algebraic. The converse is not true: not all irrational numbers are transcendental. Hence, the set of real numbers consists of non-overlapping rational, alg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Viscous Stress Tensor
The viscous stress tensor is a tensor used in continuum mechanics to model the part of the stress at a point within some material that can be attributed to the strain rate, the rate at which it is deforming around that point. The viscous stress tensor is formally similar to the elastic stress tensor (Cauchy tensor) that describes internal forces in an elastic material due to its deformation. Both tensors map the normal vector of a surface element to the density and direction of the stress acting on that surface element. However, elastic stress is due to the ''amount'' of deformation (strain), while viscous stress is due to the ''rate'' of change of deformation over time (strain rate). In viscoelastic materials, whose behavior is intermediate between those of liquids and solids, the total stress tensor comprises both viscous and elastic ("static") components. For a completely fluid material, the elastic term reduces to the hydrostatic pressure. In an arbitrary coordinate system, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Osmotic Pressure
Osmotic pressure is the minimum pressure which needs to be applied to a solution to prevent the inward flow of its pure solvent across a semipermeable membrane. It is also defined as the measure of the tendency of a solution to take in a pure solvent by osmosis. Potential osmotic pressure is the maximum osmotic pressure that could develop in a solution if it were separated from its pure solvent by a semipermeable membrane. Osmosis occurs when two solutions containing different concentrations of solute are separated by a selectively permeable membrane. Solvent molecules pass preferentially through the membrane from the low-concentration solution to the solution with higher solute concentration. The transfer of solvent molecules will continue until equilibrium is attained. Theory and measurement Jacobus van 't Hoff found a quantitative relationship between osmotic pressure and solute concentration, expressed in the following equation: :\Pi = icRT where \Pi is osmotic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Summation
In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined. Summations of infinite sequences are called series. They involve the concept of limit, and are not considered in this article. The summation of an explicit sequence is denoted as a succession of additions. For example, summation of is denoted , and results in 9, that is, . Because addition is associative and commutative, there is no need of parentheses, and the result is the same irrespective of the order of the summands. Summation of a sequence of only one element results in this element itself. Summation of an empty sequence (a sequence with no elements), by convention, results in 0. Very often, the elem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sigma
Sigma (; uppercase Σ, lowercase σ, lowercase in word-final position ς; grc-gre, σίγμα) is the eighteenth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 200. In general mathematics, uppercase Σ is used as an operator for summation. When used at the end of a letter-case word (one that does not use all caps), the final form (ς) is used. In ' (Odysseus), for example, the two lowercase sigmas (σ) in the center of the name are distinct from the word-final sigma (ς) at the end. The Latin letter S derives from sigma while the Cyrillic letter Es derives from a lunate form of this letter. History The shape (Σς) and alphabetic position of sigma is derived from the Phoenician letter ( ''shin''). Sigma's original name may have been ''san'', but due to the complicated early history of the Greek epichoric alphabets, ''san'' came to be identified as a separate letter in the Greek alphabet, represented as Ϻ. Herodotus reports that "sa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
