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Discrete

Discrete

The word discrete comes from the Latin word discretus which means separate. It is used with different meanings in different contexts:
- In perception, a discrete entity is something that can be perceived individually and not as connected to, or part of something else.
- In topology, a branch of mathematics, a discrete space is a topological space in which all sets are open, and a discrete set is a set of isolated points.
- In discrete mathematics, without notion of continuity, a discrete set is a countable set; this concept is also important for combinatorics, probability theory, and statistical theory.
- In discrete mathematics and in theoretical computer science, the abstract world is usually modeled as a discrete space with discrete time.
- In electrical engineering, discrete means having separate electronic components, such as individual resistors and inductors. This is the opposite of integrated circuitry.
- In audio engineering, discrete means having separate and independent channels of audio, as opposed to matrixed stereo or quadrophonic, or other multi-channel sound.
- A discrete signal in information theory and signal processing
- In music, a discrete pitch is one with a steady frequency, rather than an indiscrete gliding, glissando or portamento, pitch. Discrete is not the same thing as discreet.

Latin

Latin is an ancient Indo-European language originally spoken in the region around Rome called Latium. It gained great importance as the formal language of the Roman Empire. All Romance languages, those being most notably Spanish, French, Portuguese, Italian, and Romanian, are descended from Latin, and many words based on Latin are found in other modern languages such as English. The Latin alphabet, derived from the Greek, remains the most widely-used alphabet in the world. It is said that 80 percent of scholarly English words are derived from Latin (in a large number of cases by way of French). Moreover, in the Western world, Latin was a lingua franca, the learned language for scientific and political affairs, for more than a thousand years, being eventually replaced by French in the 18th century and English in the late 19th. Ecclesiastical Latin remains the formal language of the Roman Catholic Church to this day, and thus the official national language of the Vatican. The Church used Latin as its primary liturgical language until the Second Vatican Council in the 1960s. Latin is also still used (drawing heavily on Greek roots) to furnish the names used in the scientific classification of living things. The modern study of Latin, along with Greek, is known as Classics.

Main features

Latin is a synthetic inflectional language: affixes (which usually encode more than one grammatical category) are attached to fixed stems to express gender, number, and case in adjectives, nouns, and pronouns, which is called declension; and person, number, tense, voice, mood, and aspect in verbs, which is called conjugation. There are five declensions (declinationes) of nouns and four conjugations of verbs. There are six noun cases: #nominative (used as the subject of the verb or the predicate nominative), #genitive (used to indicate relation or possession, often represented by the English of or the addition of s to a noun), #dative (used of the indirect object of the verb, often represented by the English to or for), #accusative (used of the direct object of the verb, or object of the preposition in some cases), #ablative (separation, source, cause, or instrument, often represented by the English by, with, from), #vocative (used of the person or thing being addressed). In addition, some nouns have a locative case used to express location (otherwise expressed by the ablative with a preposition such as in), but this survival from Proto-Indo-European is found only in the names of lakes, cities, towns, small islands, and a few other words related to locations, such as "house", "ground", and "countryside". Latin itself, being a very old language, is far closer to Proto-Indo-European than are most modern Western European languages; it has, in fact, about the same relationship with PIE as modern Italian or French has to Latin. There are six general tenses in Latin (technically they are tense/aspect/mood complexes). The indicative mood can be used with all of them. The subjunctive mood, however, has only present, imperfect, perfect, and pluperfect tenses. These tenses in the subjunctive mood do not completely correlate in meaning to the tenses in the indicative. The following examples are of the first conjugation verb "laudare" ("to praise") in the indicative mood and the active voice:
Primary sequence tenses
# present (laudo, "I praise") # imperfect (laudabam, "I was praising") # future (laudabo, "I shall praise," "I will praise")
Secondary sequence tenses
# perfect (laudavi, "I praised", "I have praised") # pluperfect (laudaveram, "I had praised") # future perfect (laudavero, "I shall have praised," "I will have praised") The future perfect tense can also imply a normal future idea (like in "When I will have run...") and so may also sometimes be included in the primary sequence.
Latin and Romance
After the collapse of the Roman Empire, Latin evolved into the various Romance languages. These were for many centuries only spoken languages, Latin still being used for writing. For example, Latin was the official language of Portugal until 1296 when it was replaced by Portuguese. The Romance languages evolved from Vulgar Latin, the spoken language of common usage, which in turn evolved from an older speech which also produced the formal classical standard. Latin and Romance differ (for example) in that Romance had distinctive stress, whereas Latin had distinctive length of vowels. In Italian and Sardo logudorese, there is distinctive length of consonants and stress, in Spanish only distinctive stress, and in French even stress is no longer distinctive. Another major distinction between Romance and Latin is that all Romance languages, excluding Romanian, have lost their case endings in most words except for some pronouns. Romanian retains a direct case (nominative/accusative), an indirect case (dative/genitive), and vocative. In Italy, Latin is still compulsory in secondary schools as Liceo Classico and Liceo Scientifico which are usually attended by people who aim to the highest level of education. In Liceo Classico Ancient Greek is a compulsory subject.
Latin and English
See Latin influence in English for a more complete exposition. English grammar is independent of Latin grammar, though prescriptive grammarians in English have been heavily influenced by Latin. Attempts to make English grammar follow Latin rules — such as the prohibition against the split infinitive — have not worked successfully in regular usage. However, as many as half the words in English were derived from Latin, including many words of Greek origin first adopted by the Romans, not to mention the thousands of French, hundreds of Spanish, Portuguese and Italian words of Latin origin that have also enriched English. During the 16th and on through the 18th century English writers created huge numbers of new words from Latin and Greek roots. These words were dubbed "inkhorn" or "inkpot" words (as if they had spilled from a pot of ink). Many of these words were used once by the author and then forgotten, but some remain. Imbibe, extrapolate, dormant and inebriation are all inkhorn terms carved from Latin words. In fact, the word etymology is derived from the Greek word etymologia, meaning "true sense of the word." Latin was once taught in many of the schools in Britain with academic leanings - perhaps 25% of the total [http://www.channel4.com/history/microsites/T/teachem2/thennow/]. However, the requirement for it was gradually abandoned in the professions such as the law and medicine, and then, from around the late 1960s, for admission to university. After the introduction of the Modern Language GCSE in the 1980s, it was gradually replaced by other languages, although it is now being taught by more schools along with other classical languages.
Latin education
The linguistic element of Latin courses offered in high schools or secondary schools, and in universities, is primarily geared toward an ability to translate Latin texts into modern languages, rather than using it in oral communication. As such, the skill of reading is heavily emphasized, whereas speaking and listening skills are barely touched upon. However, there is a growing movement, sometimes known as the Living Latin movement, whose supporters believe that Latin can, or should, be taught in the same way that modern "living" languages are taught, that is, as a means of both spoken and written communication. One of the most interesting aspects of such an approach is that it assists speculative insight into how many of the ancient authors spoke and incorporated sounds of the language stylistically; without understanding how the language is meant to be heard it is very difficult to identify patterns in Latin poetry. Institutions offering Living Latin instruction include the Vatican and the University of Kentucky. In Britain the Classical Association encourages this approach, and there has been something of a vogue for books describing the adventures of a mouse called Minimus. In the United States there is a thriving competitive organization for high school Latin students, the National Junior Classical League (the second-largest youth organization in the world after the Boy Scouts), backed up by the Senior Classical League for college students. Many would-be international auxiliary languages have been heavily influenced by Latin, and the moderately successful Interlingua considers itself to be the modernized and simplified version of the language (le latino moderne international e simplificate). Latin translations of modern literature such as Paddington Bear, Winnie the Pooh, Harry Potter and the Philosopher's Stone, Le Petit Prince, Max und Moritz, and The Cat in the Hat have also helped boost interest in the language.
See also

About the Latin language

- Latin grammar
- Latin spelling and pronunciation
- Latin declension
- Latin conjugation
- Latin alphabet
- List of Latin words with English derivatives
- Latin verbs with English derivatives
- Latin nouns with English derivatives
- ablative absolute
- Word order in Latin
About the Latin literary heritage

- Latin literature
- Romance languages
- Loeb Classical Library
- List of Latin phrases
- List of Latin proverbs
- Brocard
- List of Latin and Greek words commonly used in systematic names
- List of Latin place names in Europe
- Carmen Possum
Other related topics

- Roman Empire
- Internationalism
References

- Bennett, Charles E. Latin Grammar (Allyn and Bacon, Chicago, 1908)
- N. Vincent: "Latin", in The Romance Languages, M. Harris and N. Vincent, eds., (Oxford Univ. Press. 1990), ISBN 0195208293
- Waquet, Françoise, Latin, or the Empire of a Sign: From the Sixteenth to the Twentieth Centuries (Verso, 2003) ISBN 1859844022; translated from the French by John Howe.
- Wheelock, Frederic. Latin: An Introduction (Collins, 6th ed., 2005) ISBN 0060784237
External links

- [http://www.jambell.com/latin.html Latin Phrases for after dinner conversation (Thanks to Elaine Poole)]
- [http://www.ethnologue.com/show_language.asp?code=lat Ethnologue report for Latin]
- [http://forumromanum.org/literature/index.html Corpus Scriptorum Latinorum] is a comprehensive webography of Latin texts and their translations.
- [http://www.perseus.tufts.edu/ The Perseus Project] has many useful pages for the study of classical languages and literatures, including [http://www.perseus.tufts.edu/cgi-bin/resolveform?lang=Latin an interactive Latin dictionary].
- [http://lysy2.archives.nd.edu/cgi-bin/words.exe words by William whitaker] is a dictionary program online capable of looking up various word forms.
- [http://retiarius.org/ Retiarius.Org] includes a Latin text search engine.
- [http://www.nd.edu/~archives/latgramm.htm Latin-English dictionary and Latin grammar from U of Notre Dame]
- [http://latin-language.co.uk/ Latin language] History of Latin language, Latin texts with English translation and a collection of dictionaries.
- [http://augustinus.eresmas.net/scl/ Societas Circulorum Latinorum] gathers together Latin Circles all over the world.
- [http://www.learnlatin.tk LearnLatin.tk] - Free online course in Latin
- [http://www.latintests.net/ LatinTests.net] - Lets Latin learners test their grammar and vocabulary with self-checking quizzes.
- [http://thelatinlibrary.com/ The Latin Library] contains many Latin etexts
- [http://www.textkit.com/ Textkit] has Latin textbooks and etexts.
- [http://www.websters-online-dictionary.org/definition/Latin-english/ Latin–English Dictionary]: from Webster's Rosetta Edition.
- [http://www.language-reference.com/ Language reference] Cross-foreign-language lexicon powered by its own search engine. All cross combinations between Latin and French, German, Italian, Spanish.
- [http://comp.uark.edu/~mreynold/rhetor.html Rhetor by Gabriel Harvey] was originally published in 1577 and never again reprinted.
- [http://freewebs.com/omniamundamundis omniamundamundis] Latin hypertexts from fourteen ancient Roman authors.
- [http://www.saltspring.com/capewest/pron.htm Pronunciation of Biological Latin, Including Taxonomic Names of Plants and Animals]
- [http://www.yleradio1.fi/nuntii Nuntii Latini (News in Latin)], written and spoken (RealAudio) news in latin. Weekly review of world news in Classical Latin, the only international broadcast of its kind in the world, produced by YLE, the Finnish Broadcasting Company.
- [http://www.tranexp.com:2000/InterTran?url=http%3A%2F%2F&type=text&text=Replace%20Me&from=eng&to=ltt InterTran Latin], Translate from Latin to ENGLISH or vice versa.
- [http://www.latinvulgate.com Latin Vulgate] The Latin and English of the Old & New Testaments in parallel, along with the Complete Sayings of Jesus in parallel Latin and English. Category:Classical languages Category:Ancient languages Category:Fusional languages Category:Languages of Italy Category:Languages of Vatican City als:Latein zh-min-nan:Latin-gí ko:라틴어 ja:ラテン語 simple:Latin language th:ภาษาละติน

Perception

In psychology and the cognitive sciences, perception is the process of acquiring, interpreting, selecting, and organizing sensory information. Methods of studying perception range from essentially biological or physiological approaches, through psychological approaches to the often abstract 'thought-experiments' of mental philosophy.

History of the study of perception

Perception is one of the oldest fields within scientific psychology, and there are correspondingly many theories about its underlying processes. The oldest quantitative law in psychology is the Weber-Fechner law, which quantifies the relationship between the intensity of physical stimuli and their perceptual effects. It was the study of perception that gave rise to the Gestalt school of psychology, with its emphasis on holistic approaches.

Perception and reality

Many cognitive psychologists hold that, as we move about in the world, we create a model of how the world works. That is, we sense the objective world, but our sensations map to percepts, and these percepts are provisional, in the same sense that scientific hypotheses are provisional (cf. in the scientific method). As we acquire new information, our percepts shift. Abraham Pais' biography refers to the 'esemplastic' nature of imagination. In the case of visual perception, some people can actually see the percept shift in their mind's eye. Others who are not picture thinkers, may not necessarily perceive the 'shape-shifting' as their world changes. The 'esemplastic' nature has been shown by experiment: an ambiguous image has multiple interpretations on the perceptual level. Just as one object can give rise to multiple percepts, so an object may fail to give rise to any percept at all: if the percept has no grounding in a person's experience, the person may literally not perceive it. This confusing ambiguity of perception is exploited in human technologies such as camouflage, and also in biological mimicry, for example by Peacock butterflies, whose wings bear eye markings that birds respond to as though they were the eyes of a dangerous predator. Cognitive theories of perception assume there is a poverty of stimulus. This (with reference to perception) is the claim that sensations are, by themselves, unable to provide a unique description of the world. Sensations require 'enriching', which is the role of the mental model. A different type of theory is the perceptual ecology approach of James J. Gibson. Gibson rejected the assumption of a poverty of stimulus by rejecting the notion that perception is based in sensations. Instead, he investigated what information is actually presented to the perceptual systems. He (and the psychologists who work within this paradigm) detailed how the world could be specified to a mobile, exploring organism via the lawful projection of information about the world into energy arrays. Specification is a 1:1 mapping of some aspect of the world into a perceptual array; given such a mapping, no enrichment is required and perception is direct.

Discrete space

In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are "isolated" from each other in a certain sense.

Definitions

Given a set X:
- the discrete topology on X is defined by letting every subset of X be open, and X is a discrete topological space if it is equipped with its discrete topology;
- the discrete uniformity on X is defined by letting every superset of the diagonal in X × X be an entourage, and X is a discrete uniform space if it is equipped with its discrete uniformity.
- the discrete metric on X is defined by letting the distance between any distinct points x and y be , and X is a discrete metric space if it is equipped with its discrete metric. There are many other types of discrete structures that can be placed on a set, but only these cases (to the knowledge of the authors so far of this article) are generally called "spaces".

Properties

The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on a discrete uniform space is the discrete topology. Thus, the different notions of discrete space are compatible with one another. On the other hand, the underlying topology of a non-discrete uniform or metric space can be discrete; an example is the metric space X := (with metric inherited from the real line and given by d(x,y) = |x − y|). Obviously, this is not the discrete metric; also, this space is not complete and hence not discrete as a uniform space. Nevertheless, it is discrete as a topological space. We say that X is topologically discrete but not uniformly discrete or metrically discrete. Additionally:
- A topological space is discrete if and only if its singletons are open, which is the case if and only if it doesn't contain any accumulation points.
- The singletons form a basis for the discrete topology.
- A uniform space X is discrete if and only if the diagonal is an entourage.
- Every discrete topological space satisfies each of the separation axioms; in particular, every discrete space is Hausdorff, that is, separated.
- A discrete space is compact iff it is finite.
- Every discrete uniform or metric space is complete.
- Combining the above two facts, every discrete uniform or metric space is totally bounded iff it is finite.
- Every discrete metric space is bounded.
- Every discrete space is first-countable, and a discrete space is second-countable iff it is countable.
- Every discrete space is totally disconnected.
- Every non-empty discrete space is second category. Any function from a discrete topological space to another topological space is continuous, and any function from a discrete uniform space to another uniform space is uniformly continuous. That is, the discrete space X is free on the set X in the category of topological spaces and continuous maps or in the category of uniform spaces and uniformly continuous maps. These facts are examples of a much broader phenomenon, in which discrete structures are usually free on sets. With metric spaces, things are more complicated, because there are several categories of metric spaces, depending on what is chosen for the morphisms. Certainly the discrete metric space is free when the morphisms are all uniformly continuous maps or all continuous maps, but this says nothing interesting about the metric structure, only the uniform or topological structure. Categories more relevant to the metric structure can be found by limiting the morphisms to Lipschitz continuous maps or to short maps; however, these categories don't have free objects (on more than one element). However, the discrete metric space is free in the category of bounded metric spaces and Lipschitz continuous maps, and it is free in the category of metric spaces bounded by 1 and short maps. That is, any function from a discrete metric space to another bounded metric space is Lipschitz continuous, and any function from a discrete metric space to another metric space bounded by 1 is short. Going the other direction, a function f from a topological space Y to a discrete space X is continuous if and only it if is locally constant in the sense that every point in Y has a neighborhood on which f is constant.

Uses

A discrete structure is often used as the "default structure" on a set that doesn't carry any other natural topology, uniformity, or metric. For example, any group can be considered as a topological group by giving it the discrete topology, implying that theorems about topological groups apply to all groups. Indeed, analysts may refer to the ordinary, non-topological groups studied by algebraists as "discrete groups" . In some cases, this can be usefully applied, for example combined with Pontryagin duality. A 0-dimensional manifold (or differentiable or analytical manifold) is nothing but a discrete topological space. In the spirit of the previous paragraph, we can therefore view any discrete group as a 0-dimensional Lie group. While discrete spaces are not very exciting from a topological viewpoint, one can easily construct interesting spaces from them. For instance, a product of countably infinitely many copies of the discrete space of natural numbers is homeomorphic to the space of irrational numbers, with the homeomorphism given by the continued fraction expansion. A product of countably infinitely many copies of the discrete space is homeomorphic to the Cantor set; and in fact uniformly homeomorphic to the Cantor set if we use the product uniformity on the product. Such a homeomorphism is given by ternary notation of numbers. (See Cantor space.) In the foundations of mathematics, the study of compactness properties of products of is central to the topological approach to the ultrafilter principle, which is a weak form of choice.

Indiscrete spaces

In some ways, the opposite of the discrete topology is the trivial topology (also called the indiscrete topology), which has the least possible number of open sets (just the empty set and the space itself). Where the discrete topology is initial or free, the indiscrete topology is final or cofree: every function from a topological space to an indiscrete space is continuous, etc.

Quotation

- Stanislaw Ulam characterized Los Angeles, California as "a discrete space, in which there is an hour's drive between points".

Notes

- Stanislaw Ulam's autobiography, Adventures of a Mathematician. Category:Topology Category:General topology

Open set

In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can "wiggle" or "change" any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U; it can't be on the edge of U. As a typical example, consider the open interval (0,1) consisting of all real numbers x with 0 < x < 1. If you "wiggle" such an x a little bit (but not too much), then the wiggled version will still be a number between 0 and 1. Therefore, the interval (0,1) is open. However, the interval (0,1] consisting of all numbers x with 0 < x ≤ 1 is not open; if you take x = 1 and wiggle a tiny bit in the positive direction, you will be outside of (0,1]. Note that whether a given set U is open depends on the surrounding space, the "wiggle room". For instance, the set of rational numbers between 0 and 1 (exclusive) is open in the rational numbers, but it is not open in the real numbers. Note also that "open" is not the opposite of "closed". First, there are sets which are both open and closed (called clopen sets); in R and other connected spaces, only the empty set and the whole space are clopen, while the set of all rational numbers smaller than √2 is clopen in the rationals. Also, there are sets which are neither open nor closed, such as (0,1] in R.

Definitions

The concept of open sets can be formalized in various degrees of generality.

Function-analytic

A point set in Rⁿ is called open when every point P of the set is an inner point.

Euclidean space

A subset U of the Euclidean n-space Rⁿ is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in Rⁿ whose Euclidean distance from x is smaller than ε, y also belongs to U. (Equivalently, U is open if every point in U has a neighbourhood contained in U) Intuitively, ε measures the size of the allowed "wiggles". An example of an open set in R² (on a plane) would be all the points within a circle radius r, which satisfy the equation

r>\sqrt

. Because the distance of any point p in this set from the edge of the set is greater than zero:

r-\sqrt>0

, we can set ε to half of this distance, which means ε is also greater than zero, and all the points that are within a distance of ε to p are also in the set, thus satisfying the conditions for an open set.

Metric spaces

A subset U of a metric space (M,d) is called open if, given any point x in U, there exists a real number ε > 0 such that, given any point y in M with d(x,y) < ε, y also belongs to U. (Equivalently, U is open if every point in U has a neighbourhood contained in U) This generalises the Euclidean space example, since Euclidean space with the Euclidean distance is a metric space.

Topological spaces

In topological spaces, the concept of openness is taken to be fundamental. One starts with an arbitrary set X and a family of subsets of X satisfying certain properties that every "reasonable" notion of openness is supposed to have. (Specifically: the union of open sets is open, the finite intersection of open sets is open, and in particular the empty set and X itself are open.) Such a family T of subsets is called a topology on X, and the members of the family are called the open sets of the topological space (X,T). Note that infinite intersections of open sets need not be open. Sets that can be constructed as the intersection of countably many open sets are denoted G_δ sets. The topological definition of open sets generalises the metric space definition: If you start with a metric space and define open sets as before, then the family of all open sets will form a topology on the metric space. Every metric space is hence in a natural way a topological space. (There are however topological spaces which are not metric spaces.)

Uses

Every subset A of a topological space X contains a (possibly empty) open set; the largest such open set is called the interior of A. It can be constructed by taking the union of all the open sets contained in A. Given topological spaces X and Y, a function f from X to Y is continuous if the preimage of every open set in Y is open in X. The map f is called open if the image of every open set in X is open in Y. An open set on the real line has the characteristic property that it is a countable union of disjoint open intervals.

Manifolds

A manifold is called open if it is a manifold without boundary and if it is not compact. This notion differs somewhat from the openness discussed above. Category:general topology ja:開集合

Discrete mathematics

Discrete mathematics, sometimes called finite mathematics, is the study of mathematical structures that are fundamentally discrete, in the sense of not supporting or requiring the notion of continuity. Most, if not all, of the objects studied in finite mathematics are countable sets, such as the integers. Discrete mathematics has become popular in recent decades because of its applications to computer science. Concepts and notations from discrete mathematics are useful to study or express objects or problems in computer algorithms and programming languages. In some mathematics curricula, finite mathematics courses cover discrete mathematical concepts for business, while discrete mathematics courses emphasize concepts for computer science majors. See also the list of basic discrete mathematics topics. For contrast, see continuum, topology, and mathematical analysis. Discrete mathematics usually includes :
- logic - a study of reasoning
- set theory - a study of collections of elements
- number theory
- combinatorics - a study of counting
- graph theory
- algorithmics - a study of methods of calculation
- information theory
- the theory of computability and complexity - a study on theoretical limitations on algorithms
- elementary probability theory and Markov chains
- linear algebra - a study of related linear equations Some applications: game theory — queuing theory — graph theory — combinatorial geometry and combinatorial topology — linear programming — cryptography (including cryptology and cryptanalysis) — theory of computation — analysis of atonal music

CombinaTorics

Combinatorics

Discrete mathematics

Theoretical computer science

Computer science, an academic discipline (abbreviated CS or compsci), is a body of knowledge generally about computer hardware, software, computation and its theory. The discipline itself includes, but is not limited to, the fundamentals of computer languages, operating systems and mathematical foundations of computer science. The study of these fundamentals may lead to a wide variety of topics, such as algorithms, formal grammars, programming languages, program design, artificial intelligence and computer engineering. There exist a number of technical definitions of computer science. The status of computer science as a science is often challenged, typically arguing that it is more like mathematics and that it does not follow the scientific method, however these facts are not unanimously accepted. In popular language, the term computer science is often confusingly used to denominate anything related to computers.

History of computer science

Evolutionary

Before the 1920s, computers were human clerks that performed calculations. They were usually under the lead of a physicist. Many thousands of computers were employed in commerce, government, and research establishments. Most of these computers were women, and they were known to have a degree in calculus. After the 1920s, the expression computing machine refered to any machine that performed the work of a human computer, especially those in accordance with effective methods of The Church-Turing Thesis. The thesis states that a mathematical method is effective if it could be set out as a list of instructions able to be followed by a human clerk with paper and pencil, for as long as necessary, and without ingenuity or insight. Machines that computed with discrete values became known as the analog kind. They used machinery that represented discrete numeric quantities, like the angle of a shaft rotation or difference in electrical potential. Digital machinery, in contrast to analog, were able to render a state of a numeric value and store each individual digit. Digital machinery used difference engines or relays before the invention of faster memory devices. The phrase computing machine gradually gave away, after the late 1940s, to just computer as the onset of electronic digital machinery became common. These computers were able to perform the calculations that were performed by the previous human clerks. Since the values stored by digital machines were not bound to physical properties like the analog device, a logical computer, based on digital equipment, was able to do anything that could be described "purely mechanical." Alan Turing, known as the Father of Computer Science, invented such a logical computer, also known as a Turing Machine, that evolved into the modern computer from the tasks performed by the previous human clerks. These new computers were also able to perform non-numeric computations, like music. Computability, by logical computers, began a science by being able to make evident which was not explicitly defined into ordinary sense more immediate.

Academic discipline

Computer science has roots in electrical engineering, logic, mathematics, and linguistics. In the last third of the 20th century computer science emerged as a distinct discipline and developed its own methods and terminology. Originally, CS was taught as part of mathematics or engineering departments, for instance at the University of Cambridge in England and at the Gdansk University of Technology in Poland, respectively. Cambridge claims to have the world's oldest taught qualification in computing. The first computer science department in the United States was founded at Purdue University in 1962, while the first college entirely devoted to computer science was founded at Northeastern University in 1982. Most universities today have specific departments devoted to computer science, while some conjoin it with engineering, with applied mathematics, or other disciplines. Most research in computer science has focused on von Neumann computers or Turing machines (computation models that perform one small, deterministic step at a time). These models resemble, at a basic level, most real computers in use today. Computer scientists also study other models of computation, which includes parallel machines and theoretical models such as probabilistic, oracle, and quantum computers.

Careers

Some of the potential careers for those who study computer science are listed below:

Demographics

- Nearly half of all computer programmers held a bachelor’s degree in 2002; about 1 in 5 held a graduate degree. [http://www.bls.gov/oco/ocos110.htm]
- Computer programmer employment is expected to grow much more slowly than that of other computer specialists.
- Education requirements range from a 2-year degree to a graduate degree. [http://www.bls.gov/oco/ocos042.htm]
- Employment, other than computer programmer, is expected to increase much faster than the average as organizations continue to adopt increasingly sophisticated technologies.

Sub-disciplines of computer science

Computer Science has a number of major sub-fields which can be classified by a number of means (for example the [http://www.acm.org/class/1998/overview.html ACM classification system]).

Algorithms

The study of algorithms is aimed at creating techniques that will enable a computer to perform a certain task in an efficient manner. An algorithm is a set of well-defined instructions for accomplishing some task, often explained by analogy with a culinary recipe. Algorithms are often implemented in software, and advancing the state of the art in algorithms is responsible for many of the most spectacular successes in computing. An algorithms specialist may come up with methods to accomplish new tasks, but just as often, they will work on improving the efficiency of an existing algorithm. These improvements can come in "time" (the length of time it takes for the algorithm to work) and "space" (the amount of computer memory the algorithm consumes. The field of algorithms is highly formal and many things can be proved about a given algorithm (using complexity theory), including roughly how long it will take to complete, as compared to the size of its input (the number of options it must consider). One interesting open question in algorithms concerns the complexity classes P and NP, and whether or not P = NP. If it does, then a whole range of seemingly difficult algorithms can in theory be performed quickly.

Data structures

A data structure is a way to store data so that it can be remembered and retrieved efficiently. A well-designed data structure means that an algorithm can be performed using as little memory space and time as possible. Some data structures are very simple: the simplest is an array which is simply a numbered list of items. Since the memory of a computer is usually modelled as an array (of bytes), this is also one of the most important data structures in computer science. For example, strings of text are usually modelled as arrays. But there are many other data structures such as linked lists, trees, hash tables and many others that are quite different but critical to the science. Type theory classifies data at a most basic (mathematical) level into different types, such as integers, complex numbers, strings, etc. and deals with how those types can interact. Abstract data types, at a more concrete level, deal with how types and data structures are used in software programming.

Listing of sub-disciplines

Computer science is closely related to a number of fields. These fields overlap considerably, though important differences exist:

Major fields of importance for computer science

External links

- [http://www.dmoz.org/Computers/Computer_Science/ Open Directory Project: Computer Science]
- [http://www.techbooksforfree.com/science.shtml Downloadable Science and Computer Science books]
- [http://liinwww.ira.uka.de/bibliography/ Collection of Computer Science Bibliographies]
- [http://www.geocities.com/tablizer/science.htm Belief that title "science" in "computer science" is inappropriate] Category:Computer science ko:컴퓨터 과학 ja:情報工学 simple:Computer science th:วิทยาการคอมพิวเตอร์

Discrete space

Definitions

Properties

Uses

Indiscrete spaces

Quotation

- Stanislaw Ulam characterized Los Angeles, California as "a discrete space, in which there is an hour's drive between points".

Notes

- Stanislaw Ulam's autobiography, Adventures of a Mathematician. Category:Topology Category:General topology

Electrical engineering

Electrical engineering is an engineering discipline that deals with the study and application of electricity and electromagnetism. Its practitioners are called electrical engineers. Electrical engineering is a broad field that encompasses many subfields including those that deal with power, control systems, electronics and telecommunications. Electrical engineering is sometimes distinguished from electronics engineering. Where this distinction is made, electrical engineering is considered to deal with the problems associated with large scale electrical systems such as power transmission and motor control where as electronics engineering is considered to deal with the problems associated with small scale electronic systems such as printed circuit board design and very-large-scale integration. However for the purposes of this article electronics engineering is considered to be a subfield of electrical engineering (see note).

History

Early developments in electricity

Electricity has been a subject of scientific interest since at least the seventeenth century. However it was not until the nineteenth century that research into the subject started to intensify. Notable developments in this century include the work of Georg Ohm who in 1827 quantified the relationship between the electric current and potential difference in a conductor and the work of Michael Faraday who in 1831 discovered electromagnetic induction. However during these years the study of electricity was largely considered to be a subfield of physics and hence the domain of physicists. It was not until the late nineteenth century that universities started to offer degrees in electrical engineering. The Darmstadt University of Technology established the first chair of electrical engineering worldwide in 1882 and offered a four year study course of electrical engineering in 1883. In 1882, MIT offered the first course on electrical engineering in the United States. This course was organized by Professor Charles Cross who was head of the Physics department and who later became a founder of the American Institute of Electrical Engineers (which later became the Institute of Electrical and Electronics Engineers). In 1885, the University College London founded the first chair of electrical engineering in the United Kingdom and, in 1886, the University of Missouri established the first department of electrical engineering in the United States. During this period, work in the area increased dramatically. In 1882, Edison switched on the world's first large-scale electrical supply network that provided 110 volts direct current to fifty-nine customers in lower Manhattan. In 1887, Nikola Tesla filed a number of patents related to a competing form of power distribution known as alternating current. In the following years a bitter rivalry between Tesla and Edison, known as the "War of Currents", took place over the preferred method of distribution. Tesla's work on induction motors and polyphase systems influenced electrical engineering for years to come. Edison's work on telegraphy and his development of the stock ticker proved lucrative for his company (which eventually became one of the world's largest companies, General Electric). As well as the contributions of Edison and Tesla, a number of other figures played an equally important role in the progress of electrical engineering at this time.

The emergence of radio and electronics

In 1896, Guglielmo Marconi made the world's first wireless radio transmission. In 1905, John Fleming invented the first radio tube, the diode. One year later, in 1906, Robert von Lieben and Lee De Forest independently developed the amplifier tube, called the triode. In 1928, the first successful transatlantic television transmission was made from London to New York. Manfred von Ardenne then introduced the cathode ray tube and thus the electronic television in 1931. In 1942, Konrad Zuse presented the Z3, the world's first functional computer. In 1946, the ENIAC (Electronic Numerical Integrator and Computer) of John Presper Eckert and John Mauchly followed, beginning the computing era. The arithmetic performance of these machines allowed engineers to develop completely new technologies and achieve new objectives. Early examples include the Apollo missions and the NASA moon landing. The invention of the transistor in 1947 by William B. Shockley, John Bardeen and Walter Brattain opened the door for more compact devices and led to the development of the integrated circuit in 1959 by Jack Kilby and independently in 1961 by Robert Noyce. In 1958, G.C. Devol and J. Engelberger invented and built in the USA the world's first industrial robot. Such a robot was used for the first time in 1960 in industrial production by General Motors. In 1968, Marcian Hoff at Intel invented the microprocessor and thus ignited the development of the personal computer. Hoff's invention was part of an order by a Japanese company for a desktop computer, which Hoff wanted to build as cheaply as possible. The first realization of the microprocessor was the Intel 4004, a 4-bit processor, in 1969, but only in 1973 did the Intel 8080, an 8-bit processor, make the building of the first personal computer, the Altair 8800, possible.

Education

Electrical engineers typically possess an academic degree with a major in electrical engineering. The length of study for such a degree is usually three or four years and the completed degree may be designated as a Bachelor of Engineering, Bachelor of Science or Bachelor of Applied Science depending upon the university. The degree generally includes units covering physics, mathematics, project management and specific topics in electrical and electronics engineering. Initially such topics cover most, if not all, of the subfields of electrical engineering. Students then choose to specialize in one or more subfields towards the end of the degree. Some electrical engineers also choose to pursue a postgraduate degree such as a Master of Engineering, a Doctor of Philosophy in Engineering or an Engineer's degree. The Master and Engineer's degree may consist of either research, coursework or a mixture of the two. The Doctor of Philosophy consists of a significant research component and is often viewed as the entry point to academia. In the United Kingdom and various other European countries, the Master of Engineering is often considered an undergraduate degree of slightly longer duration than the Bachelor of Engineering.

Training and certification

In most countries, a Bachelor's degree in engineering represents the first step towards certification and the degree program itself is certified by a professional body. After completing a certified degree program the engineer must satisfy a range of requirements (including work experience requirements) before being certified. Once certified the engineer is designated the title of Professional Engineer (in the United States and Canada), Chartered Engineer (in the United Kingdom, Ireland, India, South Africa and Zimbabwe), Chartered Professional Engineer (in Australia) or European Engineer (in much of the European Union). The advantages of certification vary depending upon location. For example, in the United States and Canada "only a licensed engineer may...seal engineering work for public and private clients". This requirement is enforced by state and provincial legislation such as Quebec's Engineers Act. In other countries, such as Australia, no such legislation exists. Practically all certifying bodies maintain a code of ethics that they expect all members to abide by or risk expulsion. In this way these organizations play an important role in maintaining ethical standards for the profession. Even in jurisdictions where certification has little or no legal bearing on work, engineers are subject to contract law. In cases where an engineer's work fails he or she may be subject to the tort of negligence and, in extreme cases, the charge of criminal negligence. An engineer's work must also comply with numerous other rules and regulations such as building codes and legislation pertaining to environmental law. Significant professional bodies for electrical engineers include the Institute of Electrical and Electronics Engineers (IEEE) and the Institution of Electrical Engineers (IEE). The IEEE claims to produce 30 percent of the world's literature on electrical engineering, has over 360,000 members worldwide and holds over 300 conferences anually. The IEE publishes 14 journals, has a worldwide membership of 120,000, certifies Chartered Engineers in the United Kingdom and claims to be the largest professional engineering society in Europe.

Tools and work

From the global positioning system to electric power generation, electrical engineers are responsible for a wide range of technologies. They design, develop, test and supervise the deployment of electrical systems and electronic devices. For example, they may work on the design of telecommunication systems, the operation of electric power stations, the lighting and wiring of buildings, the design of household appliances or the electrical control of industrial machinery. control Fundamental to the discipline are the sciences of physics and mathematics as these help to obtain both a qualitative and quantitative description of how such systems will work. Today most engineering work involves the use of computers and it is commonplace to use computer-aided design programs when designing electrical systems. Nevertheless, the ability to sketch ideas is still invaluable for quickly communicating with others. Although most electrical engineers will understand basic circuit theory, the theories employed by engineers generally depend upon the work they do. For example, quantum mechanics and solid state physics might be relevant to an engineer working on VLSI but are largely irrelevant to engineers working with macroscopic electrical systems. Even circuit theory may not be relevant to a person designing telecommunication systems that use off-the-shelf components. Perhaps the most important technical skills for electrical engineers are reflected in university programs, which emphasize strong numerical skills, computer literacy and the ability to understand the technical language and concepts that relate to electrical engineering. For most engineers technical work accounts for only a fraction of the work they do. A lot of time is also spent on tasks such as discussing proposals with clients, preparing budgets and determining project schedules. Many senior engineers manage a team of technicians or other engineers and for this reason project management skills are important. Most engineering projects involve some form of documentation and strong written communication skills are therefore very important. The workplaces of electrical engineers are just as varied as the types of work they do. Electrical engineers may be found in the pristine lab environment of a fabrication plant, the offices of a consulting firm or on site at a mine. During their working life, electrical engineers may find themselves supervising a wide range of individuals including scientists, electricians, computer programmers and other engineers. Obsolescence of technical skills is a serious concern for electrical engineers. Membership and participation in technical societies, regular reviews of periodicals in the field and a habit of continued learning are therefore essential to maintaining proficiency.

Demographics

computer programmers There are around 366,000 people working as electrical engineers in the United States constituting 0.25% of the labour force (2002). This makes electrical engineering the largest engineering discipline in the United States with the exception of software engineering. In Australia, there are around 24,000 constituting 0.23% of the labour force (2005) and in Canada, there are around 34,600 constituting 0.21% of the labour force (2001). All of these countries expect employment in the field to grow, but not rapidly, in the near future. Outside of these countries, it is difficult to gauge the demographics of the profession due to less meticulous reporting on labour statistics. One way to estimate the relative size of the profession in each country is to compare graduation statistics. In 2002, the National Science Foundation published statistics on the number of degrees granted in engineering by various countries. A summary of these statistics is shown on the right though the foundation notes that the numbers "may not be strictly comparable". In the United States and, to a lesser extent, throughout the western world there is a perception that a large number of technical jobs including those concerned with electrical engineering are being outsourced to countries such as India and China. To illustrate this claim statistics are often misrepresented (see note). Overall probably one of the best summaries of the effect of outsourcing on the United States is given by the U.S. Department of Labor which notes that "increasing use of engineering services performed in other countries will act to limit employment growth" but that overall the profession "is expected to grow more slowly than the average for all occupations through 2012". Other statements on the profession are less controversial. In the United States, the number of electrical engineers graduating has fallen from a peak in the mid-1980's. In 2000, engineering degrees formed less than 20% of the degrees granted in the United States and Australia, compared to just over 25% for the United Kingdom and Japan and over 30% for Germany and South Korea. Also widely accepted is that the profession is male dominated. This is illustrated by the statistical sources in the first paragraph that show 96% of electrical engineers in Australia and 89% of electrical engineers in Canada are male.

Subfields

Electrical engineering has many subfields. This section describes seven of the most popular subfields in electrical engineering. Although there are engineers who focus exclusively on one subfield, there are also many who focus on a combination of subfields. As explained in the lead, electronics engineering is not always considered to be a subfield of electrical engineering.

Related disciplines

One discipline related to electrical engineering is that of mechatronics. Mechatronics is an engineering discipline, which deals with the convergence of electrical and mechanical systems. Such combined systems are known as electromechanical systems and have widespread adoption. Examples include automated manufacturing systems, heating, ventilation and air-conditioning systems and various subsystems of aircrafts and automobiles. The term mechatronics is typically used to refer to macroscopic systems but futurists have predicted the emergence of very small electromechanical devices. Already such small devices, known as micro electromechanical systems (MEMS), are used in automobiles to tell airbags when to deploy, in digital projectors to create sharper images and in inkjet printers to create nozzles for high-definition printing. In the future it is hoped the devices will help build tiny implantable medical devices and improve optical communication. Another related discipline is that of biomedical engineering, which is concerned with the design of medical equipment. This includes fixed equipment such as ventilators, MRI scanners and electrocardiograph monitors as well as mobile equipment such as cochlear implants, artificial pacemakers and artificial hearts.

References

Notes :Note I - Whether or not electronics engineering is distinguished from electrical engineering must be interpreted from the context in which the term is used. Some have suggested that in places such as the United States the distinction is less common than in places such as the United Kingdom. However both usages can be found throughout the world. For example, the Institute of Electrical Engineers (which also includes electronics engineers) is a U.K. based organization but the Institute of Electrical and Electronics Engineers is a U.S. based organization. Conversely the Massachusetts Institute of Technology names its electrical and electronics engineering department as the "Department of Computer Science and Electrical Engineering" where as the University of Sheffield refers to its deparment as the "Department of Electronic and Electrical Engineering". :Note II - In October 2002, Cadence Design Systems CEO Ray Bingham announced that "China produces 600,000 engineers a year, and 200,000 are electrical engineers." The United States branch of the IEEE disputed this pointing out that it was triple the figure reported for 1999 by the National Science Foundation. Other sources draw comparisons using the number of engineering graduates reported by the All India Council for Technical Education (350,000) with that reported by the National Science Foundation (60,000) . But this comparison is dubious because the National Science Foundation excludes software engineers from its statstics. A more reasonable comparison is probably given by U.S. News who suggest the Indian figure is around 82,000. Citations # # # # # # # # # # (see here regarding copyright) # Trevelyan, James; (2005). What Do Engineers Really Do?. University of Western Australia. (seminar with [http://www.mech.uwa.edu.au/jpt/Engineering%20Roles%20050503.pdf slides]) # # and # # # National Science Foundation (2002), [http://www.nsf.gov/sbe/srs/seind02/append/c2/at02-18.pdf Science and Engineering Indicators 2002], Appendix 2-18. # # # Department of Education, Science and Training (2004), [http://www.dest.gov.au/sectors/science_innovation/publications_resources/other_publications/documents/7_x7_pdf.htm Australian Australian Science and Technology at a glance 2004 - Human Resources in Science and Technology], slide 10. # # IEEE-USA, [http://www.ieeeusa.org/communications/releases/2003/013003pr.html IEEE-USA Seeks to Substantiate Information in the H-1B Guest Worker Visa Policy Debate], January 30, 2003. # # #

External links

- [http://www.ieee.org/portal/site/mainsite/menuitem.818c0c39e85ef176fb2275875bac26c8/index.jsp?&pName=corp_level1&path=about/whatis&file=index.xml&xsl=generic.xsl History of the IEEE Electrical Engineering Professional Society at its website]
- [http://www.allaboutcircuits.com All About Circuits] Learn the nuts and bolts about building electrical circuits, and to build appliances based on electrical circuits
- [http://www.ieee-virtual-museum.org/ IEEE Virtual Museum] A virtual museum that illustrates many of the basic electrical engineering and electricity concepts through examples, figures, and interviews.
- [http://www.careercornerstone.org/eleceng/eleceng.htm Sloan Career Center: Electrical Engineering] This is an excellent resource for anyone that is interested in electrical engineering as a career. Learn what electrical engineers do on a daily basis, where they work, how much they earn, and much more. Category:Engineering
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- Category:Electronics ja:電気工学 th:วิศวกรรมไฟฟ้า

Electronics

The field of electronics is the study and use of systems that operate by controlling the flow of electrons or other electrically charged particles in devices such as thermionic valves and semiconductors. The design and construction of electronic circuits to solve practical problems is part of the fields of electronic engineering, and the hardware design side of computer engineering. The study of new semiconductor devices and their technology is sometimes considered as a branch of physics.

Electronic devices today

Electronic devices are used to perform a wide variety of tasks. The main uses of electronic circuits are the controlling, processing and distribution of information, and the conversion and distribution of electric power. Both of these uses involve the creation or detection of electromagnetic fields and electric currents. While electrical energy had been used for some time to transmit data over telegraphs and telephones, the development of electronics truly began in earnest with the advent of radio.

CAD/CAM of electronic circuits

Today's electronics engineers enjoy the ability to design circuits using premanufactured building blocks such as power supplies, resistors, capacitors, semiconductors (such as transistors), and integrated circuits. Electronic design automation software programs include schematic capture programs such as ORCAD , used to make circuit diagrams and printed circuit board layouts.

Electronic systems

One way of looking at an electronic system is to divide it into the following parts: # Inputs – Electronic or mechanical sensors (or transducers), which take signals (in the form of temperature, pressure, etc.) from the physical world and convert them into current/voltage signals. # Signal processing circuits – These consist of electronic components connected together to manipulate, interpret and transform the signals. # Outputs – Actuators or other devices (also transducers) that transform current/voltage signals back into useful physical form. One example is a television set. Its input is a broadcast signal received by an antenna or fed in through a cable. Signal processing circuits inside the television extract the brightness, colour and sound information from this signal. The output devices are a cathode ray tube that converts electronic signals into a visible image on a screen and magnet driven audio speakers.

Electronic test equipment

- Ammeter, e.g. Galvanometer (Measure current)
- Ohmmeter, e.g. Wheatstone bridge (Measure resistance)
- Voltmeter (Measures voltage)
- Multimeter (Measures all of the above)
- Oscilloscope (Measures all of the above, except Ohm, as they change over time)
- Logic analyzer (Tests digital circuits)
- Spectrum analyzer (SA) (Measures spectral energy of signals)
- Vector signal analyzer (VSA) (Like the SA but it can also perform many more useful digital demodulation functions)
- Electrometer (Measures charge)
- Frequency counter (Measures frequency)
- Time-domain reflectometer for testing integrity of long cables

Electronic components

- Electronic components
- Electronic Devices and Circuits

Analog circuits

Most analog electronic appliances, such as radio receivers, are constructed from arrays of a few types of circuits.
- Analog computer
- Analog multipliers
- electronic amplifiers
- electronic filters
- electronic oscillators
- Phase-locked loops
- electronic mixers
- Power conversion
- Electronic Power Supply
- impedance matchers
- operational amplifiers
- comparators

Digital circuits

Computers, electronic clocks, and programmable logic controllers (used to control industrial processes) are constructed of digital circuits. Digital Signal Processors are another example. Building-blocks:
- logic gates
- flip-flops
- counters
- registers
- multiplexers
- Schmitt triggers Highly integrated devices:
- microprocessors
- microcontrollers
- DSP
- Field Programmable Gate Array

Mixed-signal circuits

Mixed-signal circuits, also known as hybrid circuits, are becoming i

Discrete

Latin

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Perception

History of the study of perception

Perception and reality

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Discrete space

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Properties

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Indiscrete spaces

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See also

Notes

Open set

Definitions

Function-analytic

Euclidean space

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Theoretical computer science

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Major fields of importance for computer science

See also

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Discrete space

Definitions

Properties

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Indiscrete spaces

Quotation

See also

Notes

Electrical engineering

History

Early developments in electricity

The emergence of radio and electronics

Education

Training and certification

Tools and work

Demographics

Subfields

Related disciplines

References

See also

External links

Electronics

Electronic devices today

CAD/CAM of electronic circuits

Electronic systems

Electronic test equipment