In the second formula, the inner summation in the right hand side is only extended over indices within bounds, 0 ≤ i ≤ m and 0 ≤ j ≤ n. Alternative expressions of addition and multiplication, without using explicit bounds in the sums, are as follows:

Alternative expressions of the three theorems are as follows:

Alternative expressions:

Alternative names for this breed include Belgian Blue-White ; Belgian White and Blue Pied ; Belgian White Blue ; Blue ; and Blue Belgian.

Alternative notations include C ( n, k ), < sub > n </ sub > C < sub > k </ sub >, < sup > n </ sup > C < sub > k </ sub >, C < sup > k </ sup >< sub style =" position: relative ; left :-. 5em ;"> n </ sub >, C < sup > n </ sup >< sub style =" position: relative ; left :-. 5em ;"> k </ sub >, in all of which the C stands for combinations or choices.

Alternative start codons ( depending on the organism ), include " GUG " or " UUG "; these codons normally represent valine and leucine, respectively, but, as a start codon, they are translated as methionine or formylmethionine.

Alternative names of the order include the Christ Knights, Sword Brethren, and The Militia of Christ of Livonia.

Alternative American names for the mojo bag include hand, mojo hand, conjure hand, lucky hand, conjure bag, trick bag, root bag, toby, jomo, and gris-gris bag.

Alternative mechanisms of action that have been proposed include the activation of glutamatergic circuits while inhibiting GABAergic neurotransmission.

Alternative names for police force include constabulary, gendarmerie, police department, police service, crime prevention, protective services, law enforcement agency, civil guard or civic guard.

Alternative methods of scanning include x-ray computed tomography ( CT ), magnetic resonance imaging ( MRI ) and functional magnetic resonance imaging ( fMRI ), ultrasound and single-photon emission computed tomography ( SPECT ).

Alternative names include spaced rehearsal, expanding rehearsal, graduated intervals, repetition spacing, repetition scheduling, spaced retrieval and expanded retrieval.

Alternative theories include that the stripes coincide with fat patterning beneath the skin, serving as a thermo-regulatory mechanism for the zebra, or that wounds sustained disrupt the striping pattern to clearly indicate the fitness of the animal to potential mates.

Alternative names for a gender changer include " gender bender ", " gender blender ", " gender mender " and " homosexual adapter ".

Alternative and complementary algorithms include evolution strategies, evolutionary programming, simulated annealing, Gaussian adaptation, hill climbing, and swarm intelligence ( e. g.: ant colony optimization, particle swarm optimization ) and methods based on integer linear programming.

Alternative historical English spellings of the Sanskrit phonological words with different meanings to include suastika, swastica, and svastica.

As a counterproposal to this initiative, Chávez has promoted the Bolivarian Alternative for the Americas ( Alternativa Bolivariana para las Américas, ALBA ), vaguely based on the model of the European Union, which makes emphasis on energy and infrastructure agreements that are gradually extended to other areas finally to include the total economic, political and military integration of the member states

Water-powered funiculars include the Lynton and Lynmouth Cliff Railway in North Devon, England ; the CAT Funicular at the Centre for Alternative Technology in Gwynedd, Wales ; the Nerobergbahn in Wiesbaden, Germany ; and Bom Jesus funicular in Braga, Portugal ( the oldest, still working, in the world ).

Alternative remote file access protocols include the Server Message Block ( SMB, also known as CIFS ), Apple Filing Protocol ( AFP ), NetWare Core Protocol ( NCP ), and OS / 400 File Server file system ( QFileSvr. 400 ).

Alternative names include the United Provinces ( Verenigde Provinciën ), Federated Dutch Provinces ( Foederatae Belgii Provinciae ), and Dutch Federation ( Belgica Foederata ).

Alternative transliterations of the last element include Naaser, Nasir and others, with the whole name subject to variable spacing and hyphenation.

Alternative defenses include a range of antipredator adaptations, including alarm signals.

Alternative classifications of life so far proposed include:

Alternative treatments under investigation include the inhibition of zonulin, an endogenous signalling protein linked to increased permeability of the bowel wall and hence increased presentation of gliadin to the immune system, and other modifiers of other well-understood steps in the pathogenesis of coeliac disease, such as the action of HLA-DQ2 or tissue transglutaminase and the MICA / NKG2D interaction that may be involved in the killing of enterocytes.

Alternative methods of classification include: method of killing ( pore forming, dnase, nuclease, murein production inhibition, etc.

Alternative terms for the fetish subculture include fetish scene and fetish community.

Alternative names include Benweed, Tansy Ragwort, St. James-wort, Ragweed, Stinking Nanny / Ninny / Willy, Staggerwort, Dog Standard, Cankerwort, Stammerwort and Mare's Fart.

Alternative civil remedies include restitution or transfer of property, or an injunction to restrain or order certain actions.

Alternative rock grew out of the grunge scene of the late 1980s and early 1990s and is particularly favored by college radio and adult album alternative stations ; there is a strong focus on songwriters and bands with an outsider sound or a more sophisticated sound than the " three chord wonder " cliché.

The Board of Education also operates the Quitman Mitchell Opportunity Center, located diagonally across from the Board on 5th Avenue and 17th Street, which includes an adult learning center, Even Start child care center, and New Horizon Alternative School.

Alternative definitions have evolved to include family units headed by same-sex parents, and perhaps additional adult relatives who take on a cohabiting parental role ; in this later case it also receives the name of conjugal family.

The station, which now had the slogan of " New York's Rock Alternative ", evolved to an eclectic mix of adult rock by the end of 1995.

The radio format evolved into what came to be known as " adult contemporary " or " Adult Album Alternative ", a format that has less overt rock bias than its forebear radio categorisation.

By April 2006, KYSR readopted the modern adult contemporary music format with the tag line of " Today's Music Alternative " playing more alternative rock hits at the time and Marino relinquished his duties at KYSR to solely focus on KHHT.

A participant in the Refugee Action Collective ( Victoria ) | Refugee Action Collective, Socialist Alternative took part in the 2002 protest at the Woomera Detention Centre in which several refugees, with the aid of demonstrators outside, tore down the fences of the facility and Woomera_Immigration_Reception_and_Processing_Centre # The Woomera Breakout | broke out.

Alternative attempts have been made to identify her with Esther, although Esther is an orphan whose father was a Jew named Abihail.

The Jenkins Commission reported in favour of a new uniquely British mixed-member proportional system called " Alternative vote top-up " or " limited AMS " in October 1998, although no action was taken on this recommendation.

Alternative means of analysis have thus been devised in order to provide some insight into problems that do not fall under the scope of queueing theory, although they are often scenario-specific because they generally consist of computer simulations or analysis of experimental data.

In 1997 the party adopted its current name, although in elections it fields candidates as Socialist Alternative, due to the right to stand under the name Socialist Party being held by the Socialist Party of Great Britain.

Alternative names for the spitball are spitter, mud ball, shine ball, supersinker, vaseline ball ( because originally, Vaseline was used to give the ball a little more break ), and emery ball, although technically, an emery ball is one where the ball has been abraded in much the same way that the original cut ball had been physically cut ( an emery ball is also known as a scuff ball ).

His top legislative priorities have been to reform the Alternative Minimum Tax ( AMT ) and close tax loopholes, although his success has been mixed.

On February 3, 2011, " Mona Lisa " was released on The Alternative Press website to be streamed for free, although it was played previously at an Acoustic set for 104. 5 Radio with " On Love, On Life ".

He had previously called for cooperation between the right-wing Reform Party of Canada and the centre-right Progressive Conservative Party of Canada, although he did not endorse the United Alternative initiative when it was first announced in 1999.

Alternative renderings of the name, appearing in manuscripts of the Gospel of John, include Beth-zatha and Bethsaida ( not to be confused with Bethsaida, a town in the Galilee ), although the latter is considered to be a metathetical corruption by Biblical scholars.

Alternative names for RBs are " deep backs " and " setbacks ", although the latter is disfavored because it has also been used for backs in general, both on offense and defense.

Sorensen has written the following books, published by Alternative Comics, which are generally compilations of the weekly comic strip, although the first book contained entirely new material.

Nellist stood as a candidate for TUSC in the 2010 general elections in the constituency Coventry North East, although he stood under the label Socialist Alternative.