More precisely, a binary operation on a non-empty set S is a map which sends elements of the Cartesian product S × S to S:

More abstractly, one talks about the product in category theory, which formalizes these notions.

More than 100 atoms of hassium have been synthesized to date in various cold and hot fusion reactions, both as a parent nucleus and decay product.

More generally, the direct product of two cyclic groups Z < sub > m </ sub > and Z < sub > n </ sub > is isomorphic to Z < sub > mn </ sub > if and only if m and n are coprime.

More generally any quadratically closed subfield of or will suffice for this purpose, e. g., the algebraic numbers, but when it is a proper subfield ( i. e., neither nor ) even finite-dimensional inner product spaces will fail to be metrically complete.

: More generally, any Euclidean space < sup > n </ sup > with the dot product is an inner product space

More substantive basis for this designation is that the vocabulary, grammar, and orthography of Interlingua are natural ; they have been standardized and presented by a linguistic research body, but they predated it and are not themselves considered a product of human invention.

More generally, a parabola is a curve in the Cartesian plane defined by an irreducible equation — one that does not factor as a product of two not necessarily distinct linear equations — of the general conic form

More generally, if κ is any infinite cardinal, then a product of at most 2 < sup > κ </ sup > spaces with dense subsets of size at most κ has itself a dense subset of size at most κ ( Hewitt – Marczewski – Pondiczery theorem ).

More recently, the loosely coupled, self-organizing network of businesses that cooperate to provide product and service offerings has been called the Extended Enterprise.

More generally, the torque on a particle ( which has the position r in some reference frame ) can be defined as the cross product:

More generally, if S is a subset of a group G, then < S >, the subgroup generated by S, is the smallest subgroup of G containing every element of S, meaning the intersection over all subgroups containing the elements of S ; equivalently, < S > is the subgroup of all elements of G that can be expressed as the finite product of elements in S and their inverses.

More generally we consider a Hermitian map A on a finite-dimensional real or complex inner product space V endowed with a positive definite Hermitian inner product.

More recently, some lambic brewers have added sugar to the final product of their fruit beers, in order to make them less intense and more approachable to a wider audience.

More generally, the number of points in the intersection of 3 algebraic surfaces in projective space is, counting multiplicities, the product of the degrees of the equations of the surfaces, and so on.

More generally, the magnitude of the product equals the area of a parallelogram with the vectors for sides ; in particular for perpendicular vectors this is a rectangle and the magnitude of the product is the product of their lengths.

More generally the coefficient ( i, j ) of a product of matrices is the dot product of the transpose of row i of the first matrix and column j of the second matrix.

More precisely, if V is a vector space over &# x211D ;, the inner product is a function

More examples of the use of the empty product in mathematics may be found in the binomial theorem, factorial, fundamental theorem of arithmetic, birthday paradox, Stirling number, König's theorem, binomial type, difference operator, Pochhammer symbol, proof that e is irrational, prime factor, binomial series, and multiset.

More recently, the loosely coupled, self-organizing network of businesses that cooperates to provide product and service offerings has been called the Extended Enterprise.

More sophisticated routers, such as enterprise routers, connect large business or ISP networks up to the powerful core routers that forward data at high speed along the optical fiber lines of the Internet backbone.

; One Day More: Valjean gets the first few lines in the song, singing about his hopes to live in peace at last, once he is in England.

More complex line moiré patterns are created if the lines are curved or not exactly parallel.

" More recently the Green Party has revived the Metropolitan's plan of connecting the Northern City and Waterloo & City lines as a Crossrail route.

More precisely, defining two linear functions: and, the graph of the functions will be perpendicular and will make four right angles where the lines intersect if and only if.

More than 30, 000 were organized along conventional lines in external units that were stationed in Moroccan and Tunisian sanctuaries, where they served primarily to divert some French manpower from the main theaters of guerrilla activity to guard against infiltration.

More importantly the council and the movement lead to not only ecumenism but to the forming of councils amongst the denominations that connected churches across continental lines.

More recently, some strips have served as an homage of sorts for The Big Lebowski, featuring Mooch as the Dude, with different characters quoting lines from other characters from the film.

More than 15 boat lines operate on the rivers and canals of the city, including commuter lines.

More detailed designations usually include the line wavelength and may include a multiplet number ( for atomic lines ) or band designation ( for molecular lines ).

Similar measures, from Frederic the Great's camp at Bunzelwitz, to Arthur Wellesley's with his defense lines at Torres Vedras, to the French lines of Weissenburg, were frequently used .< ref > George Ripley, Charles Anderson Dana, The New American Cyclopaedia: A Popular Dictionary of General Knowledge, D. Appleton & Company, 1859, p. 622, ( a work in the public domain ) More than 80, 000 of the best shooters received the semi-automatic RSC 1917 rifle, allowing them to rapid fire at waves of attacking soldiers.

More recent translations are those by Giovanni Giudici ( a first version in 1975, a second one in 1990, in lines of unequal length ) and by Pia Pera ( 1996 ).

More abstractly and more precisely, it may be taken to ask whether specified axioms of Euclidean geometry concerning the existence of lines and circles entail the existence of such a square.

More precisely, the surface of a sphere is a model of elliptic geometry if lines are modeled by great circles, and points at each other's antipodes are considered to be the same point.

More specifically, all north-south running lines ( all range lines and half of all section lines ), as with the prime meridian, are always established with reference to true, geodetic north.

More generally, a finite affine plane of order n has n < sup > 2 </ sup > points and n < sup > 2 </ sup > + n lines ; each line contains n points, and each point is on n + 1 lines.

More abstractly, it can be defined as an incidence structure ( P, L, I ) consisting of a set P of points, a set L of lines, and an incidence relation I stating which points lie on which lines.

More subtle is the slight over-representation of string operations, which is largely language-related: both Ada and Pascal have strings as normal variables in the language, whereas C does not, so what was simple variable assignment in reference benchmarks became buffer copy operations in the C library.

More of his plays have survived intact than those of Aeschylus and Sophocles together, partly due to mere chance and partly because his popularity grew as theirs declinedhe became, in the Hellenistic Age, a cornerstone of ancient literary education, along with Homer, Demosthenes and Menander.

Records like " Play At Your Own Risk " by Planet Patrol, " One More Shot " by C Bank, " Al-Naafiyish ( The Soul )" by Hashim, and " I. O. U. " by Freeez became hits.

More significantly, the HR fitted standard front seat belts ; Holden thus became the first Australian automaker to provide the safety device as standard equipment across all models.

More controversially, he also defended the provisions for emergency presidential powers that became Article 48 of the Weimar Constitution.

More than likely chocolate, vanilla and strawberry became the standard for the reason that they were the most popular flavors in the United States at the time of introduction.

Likewise, after 1965 when Leone's second Western For a Few Dollars More brought a still larger box office bonanza, bounty killer suddenly became the choice profession of Spaghetti Western heroes in films like Arizona Colt, Vengeance is Mine, 10. 000 Dollars Blood Money, The Ugly Ones, Cry for Revenge, Any Gun Can Play.

Erasmus reported that More had taken an interest early on in giving his young wife a better education than she had previously received at home, and became a personal tutor to her in the areas of music and literature.

More became the guardian of a young girl named Anne Cresacre, who would eventually marry his son, John More.

More became Master of Requests in 1514, the same year in which he was appointed as a Privy Councillor, a member of His Majesty's Most Honourable Privy Council.

As secretary and personal adviser to King Henry VIII, More became increasingly influential in the government, welcoming foreign diplomats, drafting official documents, and serving as a liaison between the King and his Lord Chancellor: Thomas Wolsey, the Cardinal Archbishop of York.

The band's early albums, Rattus Norvegicus, No More Heroes and Black and White were highly successful with the record-buying public and singles such as " Peaches ", " Something Better Change " and " No More Heroes " became instant punk classics.

More than two million people were displaced from their homes because of the conflict ( well over one-third of the population ) many of whom became refugees in neighboring countries.

More recently, Alexander Basilevsky related an account where on May 9, 1973, the rover's open lid touched a crater wall and became covered with dust.

More profoundly, Shakespeare became a source, by way of its dramatic truth, for Berlioz ' fundamental notion of expressive truth ; this was how he could call Romeo and Juliet " the supreme drama of my life.

More versatile groups like War, the Commodores and Earth, Wind and Fire became popular around this time.

More controversially, gold and black became the primary logo colors, with red, white and blue being reduced to accent colors only.

More importantly, Dutch merchants, especially those from Amsterdam, became involved in the supply of arms and munitions to the rebels soon after the start of the American Revolution.

More certain is his development of the cathedral school in Bayeux, and his patronage of a number of younger men who later became prominent prelates.

More activity and building occurred and the Center area became a trading center.

More settlers from New England came to the area in the late 18th century, gradually establishing mills and other small businesses in a village that became known as Slocum Hollow.

More residents meant farming and fishing became staples in the area, with milk, poultry and berry farming becoming the main grown food staples in the area.

Faith No More became underground, metal, and alternative superstars, and are hailed as " the kings of alternative metal " by fans.