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300 | "Counting"
Counting sometimes involves numbers other than one; for example, when counting money, counting out change, "counting by twos" (2, 4, 6, 8, 10, 12,...), or "counting by fives" (5, 10, 15, 20, 25,...). There is archaeological evidence suggesting that humans have been counting for at least 50,000 years. [1] Cou... |
301 | "Counting"
[2] The development of counting led to the development of mathematical notation, numeral systems, and writing. Counting can occur in a variety of forms. Counting can be verbal; that is, speaking every number out loud (or mentally) to keep track of progress. This is often used to count objects that are presen... |
302 | "Counting"
Tallying is base 1 counting; normal counting is done in base 10. Computers use base 2 counting (0s and 1s), also known as Boolean algebra. Counting can also be in the form of finger counting, especially when counting small numbers. This is often used by children to facilitate counting and simple mathematical... |
303 | "Counting"
By using finger binary (base 2 counting), it is possible to keep a finger count up to 1023 = 2 10 − 1. Various devices can also be used to facilitate counting, such as hand tally counters and abacuses. Inclusive counting is usually encountered when dealing with time in Roman calendars and the Romance languag... |
304 | "Counting"
In contrast, the English word "fortnight" itself derives from "a fourteen-night", as the archaic " sennight " does from "a seven-night"; the English words are not examples of inclusive counting. In exclusive counting languages such as English, when counting eight days "from Sunday", Monday will be day 1, Tue... |
305 | "Counting"
[5] Names based on inclusive counting appear in other calendars as well: in the Roman calendar the nones (meaning "nine") is 8 days before the ides; [4] and in the Christian calendar Quinquagesima (meaning 50) is 49 days before Easter Sunday. Musical terminology also uses inclusive counting of intervals betw... |
306 | "Counting"
However, some cultures in Amazonia and the Australian Outback do not count, [6] [7] and their languages do not have number words. Many children at just 2 years of age have some skill in reciting the count list (that is, saying "one, two, three,..."). They can also answer questions of ordinality for small num... |
307 | "Counting"
[8] Research suggests that it takes about a year after learning these skills for a child to understand what they mean and why the procedures are performed. [9] [10] In the meantime, children learn how to name cardinalities that they can subitize. In mathematics, the essence of counting a set and finding a re... |
308 | "Counting"
This is the fundamental mathematical theorem that gives counting its purpose; however you count a (finite) set, the answer is the same. In a broader context, the theorem is an example of a theorem in the mathematical field of (finite) combinatorics —hence (finite) combinatorics is sometimes referred to as "t... |
309 | "Counting"
Infinite sets cannot be counted in the usual sense; for one thing, the mathematical theorems which underlie this usual sense for finite sets are false for infinite sets. Furthermore, different definitions of the concepts in terms of which these theorems are stated, while equivalent for finite sets, are inequ... |
310 | "Counting"
This kind of counting differs in a fundamental way from counting of finite sets, in that adding new elements to a set does not necessarily increase its size, because the possibility of a bijection with the original set is not excluded. For instance, the set of all integers (including negative numbers) can be... |
311 | "Counting"
Sets for which there exists a bijection between them are said to have the same cardinality, and in the most general sense counting a set can be taken to mean determining its cardinality. Beyond the cardinalities given by each of the natural numbers, there is an infinite hierarchy of infinite cardinalities, a... |
312 | "Counting"
A related fact is known as the pigeonhole principle, which states that if two sets X and Y have finite numbers of elements n and m with n > m, then any map f: X → Y is not injective (so there exist two distinct elements of X that f sends to the same element of Y ); this follows from the former principle, sin... |
313 | "Counting"
Similar counting arguments can prove the existence of certain objects without explicitly providing an example. In the case of infinite sets this can even apply in situations where it is impossible to give an example. [ citation needed ] The domain of enumerative combinatorics deals with computing the number ... |
314 | "Calculation"
A calculation is a deliberate process that transforms one or more inputs into one or more results. The term is used in a variety of senses, from the very definite arithmetical calculation of using an algorithm, to the vague heuristics of calculating a strategy in a competition, or calculating the chance o... |
315 | "Calculation"
To calculate means to determine mathematically in the case of a number or amount, or in the case of an abstract problem to deduce the answer using logic, reason or common sense. [1] The English word derives from the Latin calculus, which originally meant a small stone in the gall-bladder (from Latin calx ... |
316 | "Measurement"
Measurement is the quantification of attributes of an object or event, which can be used to compare with other objects or events. [1] [2] In other words, measurement is a process of determining how large or small a physical quantity is as compared to a basic reference quantity of the same kind. [3] The sc... |
317 | "Measurement"
[2] However, in other fields such as statistics as well as the social and behavioural sciences, measurements can have multiple levels, which would include nominal, ordinal, interval and ratio scales. [1] [4] Measurement is a cornerstone of trade, science, technology and quantitative research in many disci... |
318 | "Measurement"
The science of measurement is pursued in the field of metrology. Measurement is defined as the process of comparison of an unknown quantity with a known or standard quantity. The measurement of a property may be categorized by the following criteria: type, magnitude, unit, and uncertainty. [ citation need... |
319 | "Measurement"
Artifact-free definitions fix measurements at an exact value related to a physical constant or other invariable phenomena in nature, in contrast to standard artifacts which are subject to deterioration or destruction. Instead, the measurement unit can only ever change through increased accuracy in determi... |
320 | "Measurement"
[7] With the exception of a few fundamental quantum constants, units of measurement are derived from historical agreements. Nothing inherent in nature dictates that an inch has to be a certain length, nor that a mile is a better measure of distance than a kilometre. Over the course of human history, howev... |
321 | "Measurement"
For example, the metre was redefined in 1983 by the CGPM in terms of the speed of light, the kilogram was redefined in 2019 in terms of the Planck constant and the international yard was defined in 1960 by the governments of the United States, United Kingdom, Australia and South Africa as being exactly 0.... |
322 | "Measurement"
unit is known or standard quantity in terms of which other physical quantities are measured. Before SI units were widely adopted around the world, the British systems of English units and later imperial units were used in Britain, the Commonwealth and the United States. The system came to be known as U.S.... |
323 | "Measurement"
units. Many Imperial units remain in use in Britain, which has officially switched to the SI system—with a few exceptions such as road signs, which are still in miles. Draught beer and cider must be sold by the imperial pint, and milk in returnable bottles can be sold by the imperial pint. Many people mea... |
324 | "Measurement"
Similarly, gasoline is sold by the gallon in many countries that are considered metricated. The metric system is a decimal system of measurement based on its units for length, the metre and for mass, the kilogram. It exists in several variations, with different choices of base units, though these do not a... |
325 | "Measurement"
It is the world's most widely used system of units, both in everyday commerce and in science. The SI was developed in 1960 from the metre–kilogram–second (MKS) system, rather than the centimetre–gram–second (CGS) system, which, in turn, had many variants. The SI units for the seven base physical quantitie... |
326 | "Measurement"
Other physical properties may be measured in compound units, such as material density, measured in kg/m 3. The SI allows easy multiplication when switching among units having the same base but different prefixes. To convert from metres to centimetres it is only necessary to multiply the number of metres b... |
327 | "Measurement"
Strictly speaking, the ruler is the instrument used to rule straight lines and the calibrated instrument used for determining length is called a measure, however common usage calls both instruments rulers and the special name straightedge is used for an unmarked rule. The use of the word measure, in the s... |
328 | "Measurement"
Some non-systematic names are applied for some multiples of some units. The Australian building trades adopted the metric system in 1966 and the units used for measurement of length are metres (m) and millimetres (mm). Centimetres (cm) are avoided as they cause confusion when reading plans. For example, t... |
329 | "Measurement"
A link is abbreviated "lk", and links "lks", in old deeds and land surveys done for the government. The Standard Method of Measurement (SMM) published by the Royal Institution of Chartered Surveyors (RICS) consisted of classification tables and rules of measurement, allowing use of a uniform basis for mea... |
330 | "Measurement"
[12] NRM2 has been in general use since July 2013. SMM7 was accompanied by the Code of Procedure for the Measurement of Building Works (the SMM7 Measurement Code). Whilst SMM7 could have a contractual status within a project, for example in the JCT Standard form of Building Contract), the Measurement Code... |
331 | "Measurement"
It is also used to denote an interval between two relative points on this continuum. Mass refers to the intrinsic property of all material objects to resist changes in their momentum. Weight, on the other hand, refers to the downward force produced when a mass is in a gravitational field. In free fall, (n... |
332 | "Measurement"
A spring scale measures force but not mass, a balance compares weight, both require a gravitational field to operate. Some of the most accurate instruments for measuring weight or mass are based on load cells with a digital read-out, but require a gravitational field to function and would not work in free... |
333 | "Measurement"
As all other measurements, measurement in survey research is also vulnerable to measurement error, i.e. the departure from the true value of the measurement and the value provided using the measurement instrument. [15] In substantive survey research, measurement error can lead to biased conclusions and wr... |
334 | "Measurement"
For example, consider the problem of measuring the time it takes an object to fall a distance of one metre (about 39 in ). Using physics, it can be shown that, in the gravitational field of the Earth, it should take any object about 0.45 second to fall one metre. However, the following are just some of th... |
335 | "Measurement"
[18] Quantity and measurement are mutually defined: quantitative attributes are those possible to measure, at least in principle. The classical concept of quantity can be traced back to John Wallis and Isaac Newton, and was foreshadowed in Euclid's Elements. [18] In the representational theory, measuremen... |
336 | "Measurement"
A property is quantitative if such structural similarities can be established. In weaker forms of representational theory, such as that implicit within the work of Stanley Smith Stevens, [20] numbers need only be assigned according to a rule. The concept of measurement is often misunderstood as merely the... |
337 | "Measurement"
Likewise, computing and assigning arbitrary values, like the "book value" of an asset in accounting, is not a measurement because it does not satisfy the necessary criteria. Three type of Representational theory 1) Empirical relation In science, an empirical relationship is a relationship or correlation b... |
338 | "Measurement"
Thus the definition of measurement is: "A set of observations that reduce uncertainty where the result is expressed as a quantity." [21] This definition is implied in what scientists actually do when they measure something and report both the mean and statistics of the measurements. In practical terms, on... |
339 | "Measurement"
This also implies that there is not a clear or neat distinction between estimation and measurement. In quantum mechanics, a measurement is an action that determines a particular property (position, momentum, energy, etc.) of a quantum system. Before a measurement is made, a quantum system is simultaneousl... |
340 | "Measurement"
[ citation needed ] In biology, there is generally no well established theory of measurement. However, the importance of the theoretical context is emphasized. [23] Moreover, the theoretical context stemming from the theory of evolution leads to articulate the theory of measurement and historicity as a fu... |
341 | "Shape"
A shape or figure is a graphical representation of an object or its external boundary, outline, or external surface, as opposed to other properties such as color, texture, or material type. A plane shape or plane figure is constrained to lie on a plane, in contrast to solid 3D shapes. A two-dimensional shape or... |
342 | "Shape"
Each of these is divided into smaller categories; triangles can be equilateral, isosceles, obtuse, acute, scalene, etc. while quadrilaterals can be rectangles, rhombi, trapezoids, squares, etc. Other common shapes are points, lines, planes, and conic sections such as ellipses, circles, and parabolas. Among the ... |
343 | "Shape"
A geometric shape consists of the geometric information which remains when location, scale, orientation and reflection are removed from the description of a geometric object. [1] That is, the result of moving a shape around, enlarging it, rotating it, or reflecting it in a mirror is the same shape as the origin... |
344 | "Shape"
Many three-dimensional geometric shapes can be defined by a set of vertices, lines connecting the vertices, and two-dimensional faces enclosed by those lines, as well as the resulting interior points. Such shapes are called polyhedrons and include cubes as well as pyramids such as tetrahedrons. Other three-dime... |
345 | "Shape"
For instance, the letters " b " and " d " are a reflection of each other, and hence they are congruent and similar, but in some contexts they are not regarded as having the same shape. Sometimes, only the outline or external boundary of the object is considered to determine its shape. For instance, a hollow sph... |
346 | "Shape"
Simple shapes can often be classified into basic geometric objects such as a point, a line, a curve, a plane, a plane figure (e.g. square or circle ), or a solid figure (e.g. cube or sphere ). However, most shapes occurring in the physical world are complex. Some, such as plant structures and coastlines, may be... |
347 | "Shape"
In other words, the shape of a set of points is all the geometrical information that is invariant to translations, rotations, and size changes. Having the same shape is an equivalence relation, and accordingly a precise mathematical definition of the notion of shape can be given as being an equivalence class of... |
348 | "Shape"
Shapes of physical objects are equal if the subsets of space these objects occupy satisfy the definition above. In particular, the shape does not depend on the size and placement in space of the object. For instance, a " d " and a " p " have the same shape, as they can be perfectly superimposed if the " d " is ... |
349 | "Shape"
Even though they have the same size, there's no way to perfectly superimpose them by translating and rotating them along the page. Similarly, within a three-dimensional space, a right hand and a left hand have a different shape, even if they are the mirror images of each other. Shapes may change if the object i... |
350 | "Shape"
Objects that can be transformed into each other by rigid transformations and mirroring (but not scaling) are congruent. An object is therefore congruent to its mirror image (even if it is not symmetric), but not to a scaled version. Two congruent objects always have either the same shape or mirror image shapes,... |
351 | "Shape"
Thus, congruent objects are always geometrically similar, but similar objects may not be congruent, as they may have different size. A more flexible definition of shape takes into consideration the fact that realistic shapes are often deformable, e.g. a person in different postures, a tree bending in the wind o... |
352 | "Shape"
An often-repeated mathematical joke is that topologists cannot tell their coffee cup from their donut, [4] since a sufficiently pliable donut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while preserving the donut hole in a cup's handle. A described shape ha... |
353 | "Shape"
This shape has a outline and boundary so you can see it and is not just regular dots on a regular paper. The above-mentioned mathematical definitions of rigid and non-rigid shape have arisen in the field of statistical shape analysis. In particular, Procrustes analysis is a technique used for comparing shapes o... |
354 | "Shape"
Lester [5] and Rafael Artzy. For example, an equilateral triangle can be expressed by the complex numbers 0, 1, (1 + i √3)/2 representing its vertices. Lester and Artzy call the ratio the shape of triangle ( u, v, w ). Then the shape of the equilateral triangle is For any affine transformation of the complex pl... |
355 | "Shape"
For instance, Combining these permutations gives S ( v, w, u ) = ( 1 − p ) − 1. {\displaystyle S(v,w,u)=(1-p)^{-1}.} Furthermore, The shape of a quadrilateral is associated with two complex numbers p,q. If the quadrilateral has vertices u,v,w,x, then p = S( u,v,w ) and q = S( v,w,x ). Artzy proves these proposi... |
356 | "Shape"
{\displaystyle S(z_{j},z_{j+1},z_{j+2}),\ j=1,...,n-2.} The polygon bounds a convex set when all these shape components have imaginary components of the same sign. [6] Human vision relies on a wide range of shape representations. [7] [8] Some psychologists have theorized that humans mentally break down images i... |
357 | "Motion (physics)"
ct as a reference point}} In physics, motion is the phenomenon in which an object changes its position with respect to time. Motion is mathematically described in terms of displacement, distance, velocity, acceleration, speed and frame of reference to an observer and measuring the change in position ... |
358 | "Motion (physics)"
Modern physics holds that, as there is no absolute frame of reference, Newton's concept of absolute motion cannot be determined. [1] As such, everything in the universe can be considered to be in motion. [2]: 20–21 Motion applies to various physical systems: objects, bodies, matter particles, matter ... |
359 | "Motion (physics)"
In physics, the motion of massive bodies is described through two related sets of laws of mechanics. Classical mechanics for super atomic (larger than an atom) objects (such as cars, projectiles, planets, cells, and humans ) and quantum mechanics for atomic and sub-atomic objects (such as helium, pro... |
360 | "Motion (physics)"
Classical mechanics is used for describing the motion of macroscopic objects moving at speeds significantly slower than the speed of light, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. It produces very accurate results with... |
361 | "Motion (physics)"
Newton's three laws are: Newton's three laws of motion were the first to accurately provide a mathematical model for understanding orbiting bodies in outer space. This explanation unified the motion of celestial bodies and the motion of objects on Earth. Modern kinematics developed with study of elec... |
362 | "Motion (physics)"
This part of mechanics is special relativity. Efforts to incorporate gravity into relativistic mechanics were made by W. K. Clifford and Albert Einstein. The development used differential geometry to describe a curved universe with gravity; the study is called general relativity. Quantum mechanics is... |
363 | "Motion (physics)"
[4] In classical mechanics, accurate measurements and predictions of the state of objects can be calculated, such as location and velocity. In quantum mechanics, due to the Heisenberg uncertainty principle, the complete state of a subatomic particle, such as its location and velocity, cannot be simul... |
364 | "Motion (physics)"
Many of these "imperceptible motions" are only perceivable with the help of special tools and careful observation. The larger scales of imperceptible motions are difficult for humans to perceive for two reasons: Newton's laws of motion (particularly the third) which prevents the feeling of motion on ... |
365 | "Motion (physics)"
This motion is the most obscure as it is not physical motion, but rather a change in the very nature of the universe. The primary source of verification of this expansion was provided by Edwin Hubble who demonstrated that all galaxies and distant astronomical objects were moving away from Earth, know... |
366 | "Motion (physics)"
This frame of reference indicates that the Milky Way is moving at around 582 kilometres per second (1,300,000 mph). [9] [ failed verification ] The Milky Way is rotating around its dense Galactic Center, thus the Sun is moving in a circle within the galaxy 's gravity. Away from the central bulge, or ... |
367 | "Motion (physics)"
[11] The Earth is also orbiting around the Sun in an orbital revolution. A complete orbit around the sun takes one year, or about 365 days; it averages a speed of about 30 kilometres per second (67,000 mph). [12] The Theory of Plate tectonics tells us that the continents are drifting on convection cu... |
368 | "Motion (physics)"
At the other extreme, the slowest-moving plate is the Eurasian Plate, progressing at a typical rate of about 21 millimetres (0.83 in) per year. The human heart is constantly contracting to move blood throughout the body. Through larger veins and arteries in the body, blood has been found to travel at... |
369 | "Motion (physics)"
Though different foods travel through the body at different rates, an average speed through the human small intestine is 3.48 kilometres per hour (2.16 mph). [17] The human lymphatic system is also constantly causing movements of excess fluids, lipids, and immune system related products around the bo... |
370 | "Motion (physics)"
[20] Vesicles propelled by motor proteins have been found to have a velocity of approximately 0.00000152 m. [21] According to the laws of thermodynamics, all particles of matter are in constant random motion as long as the temperature is above absolute zero. Thus the molecules and atoms which make up... |
371 | "Motion (physics)"
[22] Within the standard atomic orbital model, electrons exist in a region around the nucleus of each atom. This region is called the electron cloud. According to Bohr's model of the atom, electrons have a high velocity, and the larger the nucleus they are orbiting the faster they would need to move.... |
372 | "Motion (physics)"
[23] Inside the atomic nucleus, the protons and neutrons are also probably moving around due to the electrical repulsion of the protons and the presence of angular momentum of both particles. [24] Light moves at a speed of 299,792,458 m, or 299,792.458 kilometres per second (186,282.397 mi ), in a va... |
373 | "Motion (physics)"
In addition, the speed of light is an invariant quantity: it has the same value, irrespective of the position or speed of the observer. This property makes the speed of light c a natural measurement unit for speed and a fundamental constant of nature. In 2011, the speed of light was redefined alongsi... |
374 | "Motion (physics)"
[25] This implicit change to the speed of light was one of the changes that was incorporated in the 2019 redefinition of the SI base units, also termed the New SI. [26] Some motion appears to an observer to exceed the speed of light. Bursts of energy moving out along the relativistic jets emitted fro... |
375 | "Motion (physics)"
When measuring the movement of distant objects across the sky, there is a large time delay between what has been observed and what has occurred, due to the large distance the light from the distant object has to travel to reach us. The error in the above naive calculation comes from the fact that whe... |
376 | "Mathematician"
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. One of the earliest known mathematicians were Thales of Miletus (c. 624–c.54... |
377 | "Mathematician"
He was the first to calculate the value of Pi. [ citation needed ] Pythagoras's theorem today is already found in Baudhayana sutras which was written several years before the age of Pythagoras. The number of known mathematicians grew when Pythagoras of Samos (c. 582–c. 507 BC) established the Pythagorea... |
378 | "Mathematician"
She succeeded her father as Librarian at the Great Library and wrote many works on applied mathematics. Because of a political dispute, the Christian community in Alexandria punished her, presuming she was involved, by stripping her naked and scraping off her skin with clamshells (some say roofing tiles... |
379 | "Mathematician"
As these sciences received wider attention from the elite, more scholars were invited and funded to study particular sciences. An example of a translator and mathematician who benefited from this type of support was al-Khawarizmi. A notable feature of many scholars working under Muslim rule in medieval ... |
380 | "Mathematician"
As time passed, many mathematicians gravitated towards universities. An emphasis on free thinking and experimentation had begun in Britain's oldest universities beginning in the seventeenth century at Oxford with the scientists Robert Hooke and Robert Boyle, and at Cambridge where Isaac Newton was Lucas... |
381 | "Mathematician"
Thus, seminars and laboratories started to evolve. [6] British universities of this period adopted some approaches familiar to the Italian and German universities, but as they already enjoyed substantial freedoms and autonomy the changes there had begun with the Age of Enlightenment, the same influences... |
382 | "Mathematician"
[8] According to Humboldt, the mission of the University of Berlin was to pursue scientific knowledge. [9] The German university system fostered professional, bureaucratically regulated scientific research performed in well-equipped laboratories, instead of the kind of research done by private and indiv... |
383 | "Mathematician"
In some universities, a qualifying exam serves to test both the breadth and depth of a student's understanding of mathematics; the students, who pass, are permitted to work on a doctoral dissertation. Mathematicians involved with solving problems with applications in real life are called applied mathema... |
384 | "Mathematician"
[ citation needed ] The discipline of applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry; thus, "applied mathematics" is a mathematical science with specialized knowledge. The term "applied mathematics" also describes the... |
385 | "Mathematician"
Another insightful view put forth is that pure mathematics is not necessarily applied mathematics: it is possible to study abstract entities with respect to their intrinsic nature, and not be concerned with how they manifest in the real world. [13] Even though the pure and applied viewpoints are distinc... |
386 | "Mathematician"
Many professional mathematicians also engage in the teaching of mathematics. Duties may include: Many careers in mathematics outside of universities involve consulting. For instance, actuaries assemble and analyze data to estimate the probability and likely cost of the occurrence of an event such as dea... |
387 | "Mathematician"
As another example, mathematical finance will derive and extend the mathematical or numerical models without necessarily establishing a link to financial theory, taking observed market prices as input. Mathematical consistency is required, not compatibility with economic theory. Thus, for example, while... |
388 | "Mathematician"
[14] There is no Nobel Prize in mathematics, though sometimes mathematicians have won the Nobel Prize in a different field, such as economics or physics. Prominent prizes in mathematics include the Abel Prize, the Chern Medal, the Fields Medal, the Gauss Prize, the Nemmers Prize, the Balzan Prize, the C... |
389 | "Conjecture"
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. [1] [2] [3] Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles ), have shaped much of mathematic... |
390 | "Conjecture"
This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica, where he claimed that he had a proof that was too large to fit in the margin. [5] The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort... |
391 | "Conjecture"
[6] In mathematics, the four color theorem, or the four color map theorem, states that given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map—so that no two adjacent regions have the same color. Two reg... |
392 | "Conjecture"
Möbius mentioned the problem in his lectures as early as 1840. [8] The conjecture was first proposed on October 23, 1852 [9] when Francis Guthrie, while trying to color the map of counties of England, noticed that only four different colors were needed. The five color theorem, which has a short elementary ... |
393 | "Conjecture"
The four color theorem was ultimately proven in 1976 by Kenneth Appel and Wolfgang Haken. It was the first major theorem to be proved using a computer. Appel and Haken's approach started by showing that there is a particular set of 1,936 maps, each of which cannot be part of a smallest-sized counterexample... |
394 | "Conjecture"
Showing this with hundreds of pages of hand analysis, Appel and Haken concluded that no smallest counterexample exists because any must contain, yet do not contain, one of these 1,936 maps. This contradiction means there are no counterexamples at all and that the theorem is therefore true. Initially, their... |
395 | "Conjecture"
It was originally formulated in 1908, by Steinitz and Tietze. [13] This conjecture is now known to be false. The non-manifold version was disproved by John Milnor [14] in 1961 using Reidemeister torsion. The manifold version is true in dimensions m ≤ 3. The cases m = 2 and 3 were proved by Tibor Radó and E... |
396 | "Conjecture"
A variety V over a finite field with q elements has a finite number of rational points, as well as points over every finite field with q k elements containing that field. The generating function has coefficients derived from the numbers N k of points over the (essentially unique) field with q k elements. W... |
397 | "Conjecture"
The conjecture states that:.mw-parser-output.templatequote{overflow:hidden;margin:1em 0;padding:0 40px}.mw-parser-output.templatequote.templatequotecite{line-height:1.5em;text-align:left;padding-left:1.6em;margin-top:0} Every simply connected, closed 3- manifold is homeomorphic to the 3-sphere. An equivale... |
398 | "Conjecture"
An analogous result has been known in higher dimensions for some time. After nearly a century of effort by mathematicians, Grigori Perelman presented a proof of the conjecture in three papers made available in 2002 and 2003 on arXiv. The proof followed on from the program of Richard S. Hamilton to use the ... |
399 | "Conjecture"
Several teams of mathematicians have verified that Perelman's proof is correct. The Poincaré conjecture, before being proven, was one of the most important open questions in topology. In mathematics, the Riemann hypothesis, proposed by Bernhard Riemann ( 1859 ), is a conjecture that the non-trivial zeros o... |
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