Instead of focusing merely on the individual objects ( e. g., groups ) possessing a given structure, category theory emphasizes the morphisms – the structure-preserving mappings – between these objects ; by studying these morphisms, we are able to learn more about the structure of the objects.

Instead, the Rockies dropped even further, finishing 72 – 90 and in last place in the West as the Diamondbacks won the division in just their second year of existence.

Instead, European sets contain seven additional dominoes, with six of these representing the values that result from throwing a single die with the other half of the tile left blank, and the seventh domino representing the blank-blank ( 0 – 0 ) combination.

Instead of asking for three words that contain " gry " – or had " gry " at either end – the flawed version asks for three words that end in " gry.

Instead, women's coats of arms were shown on a lozenge – a rhombus standing on one of its acute corners, an oval or a cartouche.

" Instead of coaching his 1926 team against Carnegie Tech, Rockne traveled to Chicago for the Army – Navy Game to " write newspaper articles about it, as well as select an All-America football team.

Instead this mitochondrial genome is arranged in 18 minicircular chromosomes, each of which is 3 – 4 kb long and has one to three genes.

Instead materials scientists manipulate the defects in crystalline materials such as precipitates, grain boundaries ( Hall – Petch relationship ), interstitial atoms, vacancies or substitutional atoms, to create materials with the desired properties.

Instead he resigned his right of primogeniture and from Pope Innocent XI ( 1676 – 89 ) he purchased, according to the custom of the time, for 30, 000 scudi, a position of prelatial rank and devoted his wealth and leisure to the enlargement of the library bequeathed to him by his uncle.

Instead, Smith dropped the pass and the Cowboys had to settle for a field goal from kicker Rafael Septien, cutting their deficit to 21 – 17.

Instead, she later married Herluin de Conteville, with whom she had two sons – Odo of Bayeux and Robert, Count of Mortain – and a daughter whose name is unknown.

Instead, application software – such as window managers, GUI widget toolkits and desktop environments, or application-specific graphical user interfaces – define and provide such details.

Instead, it was released as a B-side of the " Go Your Own Way " single, and would remain in some obscurity until it appeared on the four-disc Fleetwood Mac retrospective 25 Years – The Chain in 1992.

Instead, the ruler needs only to make himself respectful, and the people will be induced and enlightened by his superior virtues to follow his example – an ideal known as wuwei.

Instead, Ivano Balić scored the 23 – 22 goal with twenty seconds to spare, and only a late equaliser gave Norway third place in the group.

The German princes refused to give the crown to his nephew, the duke of Swabia, for fear he would try to regain the imperial power held by Henry V. Instead, they chose Lothair III ( 1125 – 1137 ), who found himself embroiled in a long-running dispute with the Hohenstaufens, and who married into the Welfs.

Instead, extensive experimentation with the tablebases – combined with an intensive study of tablebase-answers to well designed problems, and with knowledge of prior art ( i. e. pre-tablebase knowledge ) – is used to yield insightful patterns.

Instead, in April 1819 he was allowed to live in the French capital – as English critic George Saintsbury describes it – " in a garret furnished in the most Spartan fashion, with a starvation allowance and an old woman to look after him ", while the rest of the family moved to a house twenty miles outside Paris.

Instead of dialing directly “ into ” the host computer – which would require the host to have its own pool of modems and phone lines, and require non-local callers to make long-distance calls – the host could have an X. 25 connection to a network service provider.

One example is the Banach – Tarski paradox which says that it is possible to decompose (" carve up ") the 3-dimensional solid unit ball into finitely many pieces and, using only rotations and translations, reassemble the pieces into two solid balls each with the same volume as the original.

It is possible to prove many theorems using neither the axiom of choice nor its negation ; such statements will be true in any model of Zermelo – Fraenkel set theory ( ZF ), regardless of the truth or falsity of the axiom of choice in that particular model.

* 1973 – The American League of Major League Baseball begins using the designated hitter.

* Spatial analysis – the study of entities using geometric or geographic properties

* Lithic analysis – the analysis of stone tools using basic scientific techniques

* 1958 – Pioneer 0, America's first attempt at lunar orbit, is launched using the first Thor-Able rocket and fails.

* Lo mein – The term means " stirred noodles "; these noodles are frequently made with eggs and flour, making them chewier than simply using water.

* Beijing beef – This dish exists in native Chinese form, but using gai-lan ( Chinese broccoli ) rather than American broccoli.

* 1944 – The 1st Air Commando Group using Sikorsky R-4 helicopters stage the first use of helicopters in combat with CSAR operations in the China-Burma-India theater.

* 1894 – The first ever commercial motion picture house opened in New York City using ten Kinetoscopes, a device for peep-show viewing of films.

* 1919 – Leslie Irvin of the United States makes the first successful voluntary free-fall parachute jump using a new kind of self-contained parachute.

Abba Arikka ( 175 – 247 ) ( Talmudic Aramaic: ; born: Abba bar Aybo, Hebrew: רבי אבא בר איבו ) was a Jewish Talmudist who lived in Sassanid Babylonia, known as an amora ( commentator on the Oral Law ) of the 3rd century who established at Sura the systematic study of the rabbinic traditions, which, using the Mishnah as text, led to the compilation of the Talmud.

Alkynes can be prepared from aldehydes using the Corey-Fuchs reaction and from aldehydes or ketones by the Seyferth – Gilbert homologation.

Sensitivity of the instrument was increased by using additional turns of wire to multiply the effect – the instruments were called " multipliers ".

The United States Embassy in Sarajevo, Bosnia and Herzegovina produces the Country Commercial Guide – an annual report that delivers a comprehensive look at Bosnia and Herzegovina ’ s commercial and economic environment, using economic, political, and market analysis.

Bluetooth is a proprietary open wireless technology standard for exchanging data over short distances ( using short-wavelength radio transmissions in the ISM band from 2400 – 2480 MHz ) from fixed and mobile devices, creating personal area networks ( PANs ) with high levels of security.

During grand jury testimony in December 2003 – which was illegally leaked to the San Francisco Chronicle and published in December 2004 – Giambi allegedly admitted to using many different steroids, including fertility drugs ( which could account for his declining health in the past few years ).

In combinatorial game theory dots and boxes is very close to being an impartial game and many positions can be analyzed using Sprague – Grundy theory.

In 1995 the first gaseous condensate was produced by Eric Cornell and Carl Wieman at the University of Colorado at Boulder NIST – JILA lab, using a gas of rubidium atoms cooled to 170 nanokelvin ( nK ) ().

* Bogofilter – A free / open source spam filter that saves its wordlists using Berkeley DB.

To break out of the limitations caused by river valleys, the first summit level canals were developed with the Grand Canal of China in 581 – 617 AD whilst in Europe the first, also using single locks, was the Stecknitz Canal in Germany in 1398.

The contributions of Kurt Gödel in 1940 and Paul Cohen in 1963 showed that the hypothesis can neither be disproved nor be proved using the axioms of Zermelo – Fraenkel set theory, the standard foundation of modern mathematics, provided ZF set theory is consistent.

In Provincial Letters ( 1656 – 7 ) he scolded the Jesuits for using casuistic reasoning in confession to placate wealthy Church donors, while punishing poor penitents.

The first ab initio Hartree – Fock calculations on diatomic molecules were carried out in 1956 at MIT, using a basis set of Slater orbitals.