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; and Algebraically
; Algebraically closed field: A field F is algebraically closed if every polynomial in F has a root in F ; equivalently: every polynomial in F is a product of linear factors.

; and independent
the temperature of our homes is independent of the seasons ; ;
Large planes juxtaposed with other large planes tend to assert themselves as independent shapes, and to the extent that they are flat, they also assert themselves as silhouettes ; ;
On June 7, 1776, Richard Henry Lee tabled a resolution before the Continental Congress declaring the colonies independent ; at the same time he also urged Congress to resolve “ to take the most effectual measures for forming foreign Alliances ” and to prepare a plan of confederation for the newly-independent states.
The Declaration announced the states ' entry into the international system ; the model treaty was designed to establish amity and commerce with other states ; and the Articles of Confederation, which established “ a firm league ” among the thirteen free and independent states, constituted an international agreement to set up central institutions for the conduct of vital domestic and foreign affairs.
* 1784 – Western North Carolina ( now eastern Tennessee ) declares itself an independent state under the name of Franklin ; it is not accepted into the United States, and only lasts for four years.
This led to the first of a number of periods in which an outside power controlled Athens ; Often the outside power set up a local agent as political boss in Athens ; but when Athens was independent, it operated under its traditional form of government ; even the bosses, like Demetrius of Phalerum, kept the traditional institutions in formal existence.
An independent Athens was a minor power in the Hellenistic age ; it rarely had much in the way of foreign policy ; it generally remained at peace, allied either with the Ptolemaic dynasty, or later, with Rome ; when it went to war, the result ( as in the Lamian, Chremonidean, and Mithridatic War ) was usually disastrous.
* 1529 – The Second Diet of Speyer bans Lutheranism ; a group of rulers ( German: Fürst ) and independent cities ( German: Reichsstadt ) protests the reinstatement of the Edict of Worms, beginning the Protestant Reformation.
This is a pedagogical movement with over 1000 Steiner or Waldorf schools ( the latter name stems from the first such school, founded in Stuttgart in 1919 ) located in some 60 countries ; the great majority of these are independent ( private ) schools.
Like all early Germanic rulers, he was heavily involved in ecclesiastical disputes ; in 895, at the Diet of Tribur, he presided over a dispute between the Episcopal sees of Bremen, Hamburg and Cologne over jurisdictional authority, which saw Bremen and Hamburg remain a combined see, independent of the see of Cologne.
Many of the independent agencies operate as miniature versions of the tripartite federal government, with the authority to " legislate " ( through rulemaking ; see Federal Register and Code of Federal Regulations ), " adjudicate " ( through administrative hearings ), and to " execute " administrative goals ( through agency enforcement personnel ).
One of these corrections is that for lamp flicker noise, which is independent of wavelength, resulting in measurements with very low noise level ; other corrections are those for background absorption, as will be discussed later.
From this it follows that correctness of drawing depends solely upon the principal rays ; and is independent of the sharpness or curvature of the image field.
Clinton won the 1992 presidential election ( 43. 0 % of the vote ) against Republican incumbent George H. W. Bush ( 37. 4 % of the vote ) and billionaire populist Ross Perot, who ran as an independent ( 18. 9 % of the vote ) on a platform focusing on domestic issues ; a significant part of Clinton's success was Bush's steep decline in public approval.
The bill passed the house with 234 votes against 200 opposed ( 132 Republicans and 102 Democrats voting in favor ; 156 Democrats, 43 Republicans, and 1 independent against ).
Freundel Stuart, and his Cabinet ; ( Parliament ) as legislature, divided into two chambers ( Senate and House of Assembly ), and an independent judiciary.
In Gregory ’ s day, history was not recognized as an independent field of study ; it was a branch of grammar or rhetoric, and historia ( defined as ‘ story ’) summed up the approach of the learned when they wrote what was, at that time, considered ‘ history .’ Gregory ’ s Dialogues Book Two, then, an authentic medieval hagiography cast as a conversation between the Pope and his deacon Peter, is designed to teach spiritual lessons.
This does not mean that the episcopate, in the sense of the holder of the order or office of bishop, must have developed only later, or have been plural, because in each church the college or presbyter-overseers ( also called " presbyter-bishops ") did not exercise an independent supreme power ; it was subject to the Apostles or to their delegates.

; and elements
Internal illumination, protected from the elements, gives them powerful visual appeal at night ; ;
Toward the turn of the twentieth century, a number of anthropologists became dissatisfied with this categorization of cultural elements ; historical reconstructions also came to seem increasingly speculative to them.
The word Nepōhualtzintzin comes from the Nahuatl and it is formed by the roots ; Ne-personal -; pōhual or pōhualli-the account -; and tzintzin-small similar elements.
This is not the most general situation of a Cartesian product of a family of sets, where a same set can occur more than once as a factor ; however, one can focus on elements of such a product that select the same element every time a given set appears as factor, and such elements correspond to an element of the Cartesian product of all distinct sets in the family.
The axiom of choice asserts the existence of such elements ; it is therefore equivalent to:
Three elements have been important in holding the Communion together: First, the shared ecclesial structure of the component churches, manifested in an episcopal polity maintained through the apostolic succession of bishops and synodical government ; second, the principle of belief expressed in worship, investing importance in approved prayer books and their rubrics ; and third, the historical documents and standard divines that have influenced the ethos of the Communion.
His epistemology can be regarded as primitive materialist empiricism ; he believed that human cognition ought to be based on one's perceptions – one's sensory experiences, such as sight and hearing – instead of imagination or internal logic, elements founded on the human capacity for abstraction.
A contraction of a word is made by omitting certain letters or syllables and bringing together the first and last letters or elements ; an abbreviation may be made by omitting certain portions from the interior or by cutting off a part.
Alfred's military reorganisation of Wessex consisted of three elements: the building of thirty fortified and garrisoned towns ( burhs ) along the rivers and Roman roads of Wessex ; the creation of a mobile ( horsed ) field force, consisting of his nobles and their warrior retainers, which was divided into two contingents, one of which was always in the field ; and the enhancement of Wessex's seapower through the addition of larger ships to the existing royal fleet.
In the film, the monster is called a Kraken, although it is depicted as a lizard-like creature rather than a squid ; and combining two elements of the myth, Perseus defeats the sea monster by showing it Medusa's face, turning the monster into stone.
All but one of the actinides are f-block elements, corresponding to the filling of the 5f electron shell ; lawrencium, a d-block element, is also generally considered an actinide.
Another was unveiled at St Mildred's Church, Bread St, London, in 1932 ; that church was destroyed in the London Blitz in 1940, but the principal elements of the monument were re-erected in St Mary-le-Bow at the west end of Watling Street, near Saint Paul's Cathedral, in 1968.
The advantage of this technique is that only one radiation source is used ; among the disadvantages are that the high-current pulses reduce lamp lifetime, and that the technique can only be used for relatively volatile elements, as only those exhibit sufficient self-reversal to avoid dramatic loss of sensitivity.
; Identity element: There exists an element e in A, such that for all elements a in A, the equation holds.
Bromine is rarer than about three-quarters of elements in the Earth's crust ; however, the high solubility of bromide ion has caused its accumulation in the oceans, and commercially the element is easily extracted from brine pools, mostly in the United States, Israel and China.
The core ideas of the Big Bang — the expansion, the early hot state, the formation of helium, and the formation of galaxies — are derived from many observations that are independent from any cosmological model ; these include the abundance of light elements, the cosmic microwave background, large scale structure, and the Hubble diagram for Type Ia supernovae.
There was an element naming controversy as to what the elements from 104 to 106 were to be called ; the IUPAC adopted unnilseptium ( symbol Uns ) as a temporary, systematic element name for this element.
The first was the late 7th century Deuteronomistic reform of official Judean religion under king Josiah, who banned many elements of the old polytheistic cult from the Temple, and the sudden collapse of Assyria and the rise of Babylon to take its place ; the second was exile of the royal court, the priests and other members of the ruling elite following the Babylonian destruction of Jerusalem c. 586 BCE.
Key elements in the " blitzkrieg warfare " at the decisive Battle of Megiddo included concentration, surprise and speed ; success depending on attacking only in favourable terrain, where the terrain favoured the movement of large formations around the battlefield, and tactical improvements in the British artillery and infantry attack.
) Thus a set is completely determined by its elements ; the description is immaterial.
It was the final stage of the reformers ' work of removing all elements of sacrificial offering from the Latin Mass ; so that it should cease to be seen as a ritual at which the priest, on behalf of the faithful offered Christ's body and blood to God ; and might rather be seen as a ritual whereby Christ shared his body and blood, according to a different sacramental theology, with the faithful.

; and Elements
This trend has often been ascribed to the cult of the Five Elements itself, as though they had served as the base for all the rest ; ;
* Guide to the Elements – Revised Edition, Albert Stwertka, ( Oxford University Press ; 1998 ) ISBN 0-19-508083-1
* Naturalism, the School of Naturalists or the Yin-yang school, which synthesized the concepts of yin-yang and the Five Elements ; Zou Yan is considered the founder of this school.
The School of Naturalists or the School of Yin-yang ( 陰陽家 / 阴阳家 ; Yīnyángjiā ; Yin-yang-chia ; " School of Yin-Yang ") was a Warring States era philosophy that synthesized the concepts of yin-yang and the Five Elements ; Zou Yan is considered the founder of this school.
Elements of the Oral Torah were committed to writing and edited by Judah HaNasi in the Mishnah in 200 CE ; much more of the Oral Torah were committed to writing in the Babylonian and Jerusalem Talmuds, which were edited around 600 CE and 450 CE, respectively.
* Data deals with the nature and implications of " given " information in geometrical problems ; the subject matter is closely related to the first four books of the Elements.
The very first geometric proof in the Elements, shown in the figure above, is that any line segment is part of a triangle ; Euclid constructs this in the usual way, by drawing circles around both endpoints and taking their intersection as the third vertex.
* Guide to the Elements – Revised Edition, Albert Stwertka, ( Oxford University Press ; 1998 ) ISBN 0-19-508083-1
In particular, he gave an algorithm for computing the greatest common divisor of two numbers ( the Euclidean algorithm ; Elements, Prop.
While Greek astronomy — thanks to Alexander's conquests — probably influenced Indian learning, to the point of introducing trigonometry, it seems to be the case that Indian mathematics is otherwise an indigenous tradition ; in particular, there is no evidence that Euclid's Elements reached India before the 18th century.
Elements from the Spanish language were also integrated into the Palauan lexicon ; the Palauan word for hammer, " martiliong ", is derived from the Spanish " martillo ".
* Guide to the Elements – Revised Edition, Albert Stwertka, ( Oxford University Press ; 1998 ) ISBN 0-19-508083-1
The third edition of The Elements of Style ( 1979 ) features 54 points: a list of common word-usage errors ; 11 rules of punctuation and grammar ; 11 principles of writing ; 11 matters of form, and 21 reminders for a better style, in Chapter V. The final reminder, the 21st, " Prefer the standard to the offbeat ", is thematically integral to the subject of The Elements of Style, yet does stand as a discrete essay about writing lucid prose.
However, by Proposition 2 of Book 1 of Euclid's Elements, no computational power is lost by using such a collapsing compass ; there is no need to transfer a distance from one location to another.
" ( Trevor Hastie, Robert Tibshirani, and Jerome Friedman, co-authors of The Elements of Statistical Learning in their Preface to the Second Edition have a footnote which reads: " On the Web, this quote has been widely attributed to both Deming and Robert W. Hayden ; however Professor Hayden told us that he can claim no credit for this quote, and ironically we could find no ' data ' confirming Deming actually said this.
* De Elementen (" The Elements ", 1988 ; novel )

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