Less formally, the equivalence relation ker on X, takes each function f: X → X to its kernel ker f. Likewise, ker ( ker ) is an equivalence relation on X ^ X.

Less formally, people also use loan words within day-to-day speech, although this has been on the decline in recent decades and among younger generations.

Less formally, a fly-front or button down shirt may be substituted.

Less formally, on occasions of political controversy in many countries and times the usage of marking the face of a political opponent on a poster with an " X " is a commonly understood way of expressing strong disapproval, sometimes even the wish to kill the person whose picture is so marked.

Less formally than the theoretical distinctions mentioned above, two main types of signals encountered in practice are analog and digital.

Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a " weighting factor " from a given ring.

Less than three months after the two territories were formally merged in December 1956, the Gold Coast gained independence as Ghana in March 1957.

" Less than a week later, however, McAllister announced his plan to formally retire from the NFL at the conclusion of the playoffs.

Less formally, M ( n ) is the count of square-free integers up to n that have an even number of prime factors, minus the count of those that have an odd number.

Less formally, such a person may be referred to as a refugee, although that term has a specific narrower legal definition.

Less formally, it is the number of linear combinations included in the sum of squares defining Q < sub > i </ sub >, provided that these linear combinations are linearly independent.

Less formally, this means that from the system's outputs it is possible to determine the behaviour of the entire system.

Less formally, " second-in-command " is a term used, usually unofficially and sometimes jocularly, for any deputy.

Less than a month later, on May 1, the Army formally charged MacDonald with the murder of his family.

Less formally, verse endings are usually also indicated by a two horizontal dots following the word with a silluq.

Less formally, hot and high can also describe a botched landing approach, where a fixed-wing aircraft is too fast ( hot ) and too high above the glidepath: since the only way to increase the rate of descent is to increase speed, and the aircraft is already too fast, it is unlikely that the approach can be salvaged.

Less formally, one can think of it as a conduit between the two entities ; one entity can insert bytes into the conduit, and the other entity then receives them.

Less formally structured games are also played out in internet forums, where players manage governments and nations according to a set of agreed rules, such as the MOCpages-based 21st Century World group.

Less than a week later, on April 4, 2005, the Third Appellate District ruled more explicitly, in Knight v. Superior Court, that Prop 22 also included within the initiative's ambit marriages licensed within the state:

Less than a month after the DC Circuit Court of Appeals upheld the redesignation of Kach and Kahane Chai as Foreign Terrorist Organizations, Kahane Net sent out a fundraising appeal, crossing out with a black marker the URL www. kahane. org printed on the envelope.

Less than a year later, he received his first personal applause for dancing a Hungarian solo at the Court Theatre.

Less than fifty years later ( 1339 – 43 ), Kitabatake Chikafusa ( 1293 – 1354 ), the chief commander of the Southern Court forces, wrote the Jinnō Shōtōki.

Less than three weeks later the Court issued its decision ; the justices struggled to write an opinion that all eight could agree to.

Less than two years later, the Supreme Court decided Nectow v. City of Cambridge.

Ball culture: Less frequently, the Imperial Court has had overlapping membership and joint ventures with drag houses of the type documented in Paris is Burning.

Less than two months after the 1781 defeat of Cornwallis at Yorktown, the 88 year old Fairfax died at his seat at Greenway Court.

Less ambitious freeway plans may be more successful -- especially when the roadways and interchanges are raised, allowing for cross access at many points and providing parking areas below the ramp.

Less dramatic improvements are possible.

Less common are tandems, lowriders, tall bikes, fixed gear, folding models, amphibious bicycles and recumbents ( one of which was used to set the IHPVA Hour record ).

Less obvious is the fact that Puzzle Bobble also features all the enemies from Bubble Bobble, which are trapped inside the bubbles and fly out when the bubbles pop.

Less common are cardiac problems including cardiomyopathy and cardiac arrhythmia disorders.

Less illustrious uses refer to folklore and urban legend and a variety of explanatory narratives which are constructed with methodological flaws or biases.

Less than 1 % of all runway models are known to make more than $ 1000 dollars USD for every fashion showcase.

Less politically, most faiths have any number of people that are believed to have been touched by divine forces: saints, prophets, heroes, oracles, martyrs, and enlightened beings, among others.

Less common species such as Exmoor Horn and Cheviot are also to be found on Dartmoor.

Less common are well-documented cases with enough evidence to support the accusation.

Less than one-half of Ethiopia ’ s towns and cities are connected to the national grid.

Less than 10 % are Christians, mostly Roman Catholics.

Less than 65 % of children of primary school age are actually enrolled.

Less common are bara — larger, often heavily muscled and sometimes hairy males, the yaoi counterpart of the " bear " in gay pornography — as well as oyaji ( meaning " daddy " or " uncle "), featuring middle-aged and elderly men ; these types are mainly found in material aimed at gay men, which may be called bara or ' men's love ' ( ML ), and is considered to be distinct from yaoi.

Less likely is the commonly proposed derivation from Welsh arth " bear " + ( g ) wr " man " ( earlier * Arto-uiros in Brittonic ); there are phonological difficulties with this theory — notably that a Brittonic compound name * Arto-uiros should produce Old Welsh * Artgur and Middle / Modern Welsh * Arthwr and not Arthur ( in Welsh poetry the name is always spelled Arthur and is exclusively rhymed with words ending in-ur-never words ending in-wr-which confirms that the second element cannot be wr " man ").

Less common are proportional pneumatic valves or even various electrical actuators.

Less radical protests are also discussed, such as the various movements aimed at putting an end to sweatshop labour.

Less often seen are noun-verb combinations of two words, such as the line " The silence whistles " from Nathan Alterman's Summer Night, or in a record album title like Sounds of Silence.

* Less code required when applications are object oriented.

Less commonly, postage stamps are made of materials other than paper, such as embossed foil ( sometimes of gold ).

Less labor-intensive tools are more common now: a microplane or fine grater can be used to grind small amounts ; a coffee grinder is useful for larger amounts.

Less common are cherrywood, olivewood, maple, mesquite, oak, and bog-wood.

Less common are stems made of reeds, bamboo, or hollowed out pieces of wood.

Less frequently used types of materials include stag antler, buffalo horn and bone, as the time and effort needed to create one is often too much when conventional slides are available.