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* The case n = 1 can be illustrated by the claim that there always exist a pair of opposite points on the earth's equator with the same temperature, this can be shown to be true much more easily using the intermediate value theorem.

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## Some Related Sentences

case and n

But humans

**can**do something equally useful**,**in**the**__case__**of**certain enumerably infinite sets: They**can**give explicit instructions for determining**the**nth member**of****the**set**,**for arbitrary finite__n__**.**Such instructions are**to****be**given quite explicitly**,**in**a**form in which they could**be**followed**by****a**computing machine**,**or**by****a**human who is capable**of**carrying out only very elementary operations**on**symbols**.**
( Expressed

**more**technically**,**in each__case__**the****pair**( m**,**__n__) decreases in**the**lexicographic order**on**pairs**,**which is**a**well-ordering**,**just like**the**ordering**of**single non-negative integers ;**this**means one cannot go down in**the**ordering infinitely many times in succession**.**
A binary relation is

**the**special__case__**of**an n-ary relation R ⊆ A < sub >**1**</ sub > × … × A < sub >__n__</ sub >,**that**is**,****a**set**of**n-tuples where**the**jth component**of**each n-tuple is taken from**the**jth domain A < sub > j </ sub >**of****the**relation**.*******

**The**

__case__

__n__

**=**2 is often

**illustrated**

**by**saying

**that**at any moment

**there**is

**always**

**a**

**pair**

**of**antipodal

**points**

**on**

**the**Earth's surface

**with**equal temperatures and equal barometric pressures

**.**

For

**the**__case__**of****a**non-commutative base ring R and**a**right module M < sub > R </ sub > and**a**left module < sub > R </ sub > N**,**we**can**define**a**bilinear map**,**where T is an abelian group**,**such**that**for any__n__in N**,**is**a**group homomorphism**,**and for any m in M**,**is**a**group homomorphism too**,**and which also satisfies**The**properties

**of**binomial coefficients have led

**to**extending

**the**meaning

**of**

**the**symbol beyond

**the**basic

__case__where

__n__and k are nonnegative integers

**with**; such expressions are then still called binomial coefficients

**.**

In

**the**__case__**of**integer order__n__**,****the**function is defined**by**taking**the**limit as**a**non-integer α tends**to**__n__:
Bennett

**,**Bernstein**,**Brassard**,**and Vazirani proved in 1996**that****a**brute-force key search**on****a**quantum computer cannot**be**faster than roughly 2 < sup >__n__/ 2 </ sup > invocations**of****the**underlying cryptographic algorithm**,**compared**with**roughly 2 < sup >__n__</ sup > in**the**classical__case__**.**
If

**the**conjecture were**true****,**it would**be****a**generalization**of**Fermat's last**theorem****,**which could**be**seen as**the**special__case____n__**=**2: if**,**then**.****The**Earley parser executes in cubic time in

**the**general

__case__

**,**where

__n__is

**the**length

**of**

**the**parsed string

**,**quadratic time for unambiguous grammars

**,**and linear time for almost all LR ( k ) grammars

**.**

This is

**the**use**of****the**factor n-1 in**the**denominator**of****the**formula**,**rather than just__n__**.**This occurs when**the**sample mean rather than**the**population mean is used**to**centre**the**data and since**the**sample mean is**a**linear combination**of****the**data**the**residual**to****the**sample mean overcounts**the**number**of**degrees**of**freedom**by****the**number**of**constraint equations — in**this**__case__one**.**
It is not

**true****that**if it**always**has been**the**__case__**that****a**God-like being exists then**a**God-like being exists and it is**always**going**to****be****the**__case__**that****a**God-like being exists in (__n__).
If G

**=**GL < sub >*</ sub >( K ), then**the**set**of**natural numbers is**a**proper subset**of**G < sub > 0 </ sub >, since for each natural number__n__**,****there**is**a**corresponding identity matrix**of**dimension__n__**.**G ( m**,**__n__) is empty unless m**=**__n__**,**in which__case__it is**the**set**of**all nxn invertible matrices**.**
In

**the**__case__where V is**of**finite dimension__n__it is common**to**choose**a**basis for V and identify GL ( V )**with**GL (__n__**,**K )**the**group**of**n-by-n invertible matrices**on****the**field K**.**

case and =

Over 99

**.**94 %**of**an atom's mass is concentrated in**the**nucleus ,< ref group__=__note > In**the**__case__**of**hydrogen-1**,****with****a**single electron and nucleon**,****the**proton is**,**or 99**.**95 %**of****the**total atomic mass**.**
If

**a**is algebraic over K**,**then K**,****the**set**of**all polynomials in**a****with**coefficients in K**,**is not only**a**ring but**a**field: an algebraic extension**of**K which has finite degree over K**.**In**the**special__case__where K__=__Q is**the**field**of**rational numbers**,**Q is an example**of**an algebraic number field**.**
This is not

**always****the**__case__:**the**trivial equation x__=__x specifies**the**entire plane**,**and**the**equation x < sup > 2 </ sup > + y < sup > 2 </ sup >__=__0 specifies only**the**single point ( 0**,**0 ).
If

**the**ratio**a**'/**a****be**sufficiently constant**,**as is often**the**__case__**,****the**above relation reduces**to****the**condition**of**Airy**,**i**.**e**.**tan w '/ tan w__=__**a**constant**.**
In

**the**first__case__**,**ƒ ( x ) has y__=__c as asymptote when x tends**to**−∞, and in**the**second**that**ƒ ( x ) has y__=__c as an asymptote as x tends**to**+∞
Every Boolean algebra ( A

**,**∧, ∨) gives rise**to****a**ring ( A**,**+, ·)**by**defining**a**+ b := (**a**∧ ¬ b ) ∨ ( b ∧ ¬**a**)__=__(**a**∨ b ) ∧ ¬(**a**∧ b ) (**this**operation is called symmetric difference in**the**__case__**of**sets and XOR in**the**__case__**of**logic ) and**a**· b :=**a**∧ b**.****The**zero element**of****this**ring coincides**with****the**0**of****the**Boolean algebra ;**the**multiplicative identity element**of****the**ring is**the****1****of****the**Boolean algebra**.****a**− λ1 is not invertible ( because

**the**spectrum

**of**

**a**is not empty ) hence

**a**

__=__λ1:

**this**algebra A is naturally isomorphic

**to**C (

**the**complex

__case__

**of**

**the**Gelfand-Mazur

**theorem**).

3-satisfiability is

**a**special__case__**of**k-satisfiability ( k-SAT ) or simply satisfiability ( SAT ), when each clause contains exactly k__=__3 literals**.**
In

**the**__case__**of**SWNT**,**covalent functionalization will break some C__=__C double bonds**,**leaving " holes " in**the**structure**on****the**nanotube and**,**thus**,**modifying both its mechanical and electrical properties**.**
Indeed

**,**following**,**suppose ƒ is**a**complex function defined in an open set Ω ⊂ C**.**Then**,**writing for every z ∈ Ω**,**one**can**also regard Ω as an open subset**of**R < sup > 2 </ sup >, and ƒ as**a**function**of**two real variables x and y**,**which maps Ω ⊂ R < sup > 2 </ sup >**to**C**.**We consider**the**Cauchy – Riemann equations at z__=__0 assuming ƒ ( z )__=__0**,**just for notational simplicity –**the**proof is identical in general__case__**.**
Since

**the**separation**of**variables in**this**__case__involves dividing**by**y**,**we must check if**the**constant function y__=__0 is**a**solution**of****the**original equation**.**
One reason for

**the**popularity**of****the**dissociation constant in biochemistry and pharmacology is**that**in**the**frequently encountered__case__where x__=__y__=__**1****,**K < sub > d </ sub > has**a**simple physical interpretation: when__=__K < sub > d </ sub >,__=__or equivalently /(+)=**1**/ 2**.**
In

**this**__case__**,**we might compute T__=__A *( B*****D ) only once and simply blend T*****C**to**produce F**,****a**single operation**.**

case and 1

**The**reader will find it helpful

**to**think

**of**

**the**special

__case__when

**the**primes are

**of**degree

__1__

**,**and even

**more**particularly

**,**

**to**think

**of**

**the**proof

**of**Theorem 10

**,**

**a**special

__case__

**of**

**this**

**theorem**

**.**

Pierre-Simon Laplace and Antoine Lavoisier

**,**in their 1780 treatise**on**heat**,**arrived at values ranging from__1__**,**500**to**3**,**000 below**the**freezing-point**of**water**,**and thought**that**in any__case__it must**be**at least 600 below**.**
It may

**be**applied**,**as it is stated in article__1__**of****the**convention in__case__**of**: offenses against penal law ; acts which**,**whether or not they are offenses**,**may or do jeopardize**the**safety**of****the**aircraft or**of**persons or property therein or which jeopardize good order and discipline**on**board**.*******If

**the**balance factor

**of**L is +

__1__

**,**

**a**single right rotation (

**with**P as

**the**root ) is needed ( Left-Left

__case__).

Idealists are skeptics about

**the**physical world**,**maintaining either:__1__)**that**nothing exists outside**the**mind**,**or 2 )**that**we would have no access**to****a**mind-independent reality even if it may**exist**;**the**latter__case__often takes**the**form**of****a**denial**of****the**idea**that**we**can**have unconceptualised experiences ( see Myth**of****the**Given ).
In

**the**__case__**of**parallelization**,**Amdahl's law states**that**if P is**the**proportion**of****a**program**that****can****be**made parallel ( i**.**e., benefit from parallelization ), and (__1__− P ) is**the**proportion**that**cannot**be**parallelized ( remains serial ), then**the**maximum speedup**that****can****be**achieved**by****using**N processors is
A court

__case__allowing**the**União do Vegetal**to**import and use**the**tea for religious purposes in**the**United States**,**Gonzales v**.**O Centro Espirita Beneficente Uniao do Vegetal**,**was heard**by****the**U**.**S**.**Supreme Court**on**November__1__**,**2005 ;**the**decision**,**released February 21**,**2006**,**allows**the**UDV**to**use**the**tea in its ceremonies pursuant**to****the**Religious Freedom Restoration Act**.**
Therefore

**,****a**base**on**balls does not increase nor decrease**a**player's batting average**,**but it does increase his on-base percentage**,**unless**a**player's on-base percentage is__1__**.**000**,**in which__case__it remains**the****same****.****The**normal boiling point ( also called

**the**atmospheric boiling point or

**the**atmospheric pressure boiling point )

**of**

**a**liquid is

**the**special

__case__in which

**the**vapor pressure

**of**

**the**liquid equals

**the**defined atmospheric pressure at sea level

**,**

__1__atmosphere

**.**

Whenever

**a**confusion between**the**two kinds**of**definitions might arise it**can****be**avoided**by**referring**to****the****more**general definition and**by**reintroducing**the**erased parameter: writing B < sub > m </ sub >( 0 ) in**the**first__case__and B < sub > m </ sub >(__1__) in**the**second will unambiguously denote**the****value**in question**.**
In

**the**__case__**of****the**Creme Brulee**,**placing**the**ramekins in**a**roasting pan and filling**the**pan**with**hot water until it is__1__/ 2**to**2 / 3**of****the**way up**the**sides**of****the**ramekins transfers**the**heat**to****the**custard gently**,**which prevents**the**custard from curdling**.**
With an insurance

**of**$__1__**,**500 promised in__case__**of**failure**,**Sennett also allowed Chaplin**to**direct his own film**.**
As

**a**special__case__**,****the**three naturally occurring isotopes**of****the**element hydrogen are often specified as H for < sup >__1__</ sup > H ( protium ), D for < sup > 2 </ sup > H ( deuterium ), and T for < sup > 3 </ sup > H ( tritium ).
Under

**the**federal Holidays Act**,**Canada Day is observed**on**July__1__unless**that**date falls**on****a**Sunday**,**in which__case__July 2 is**the**statutory holiday**,**although celebratory events generally take place**on**July__1__even though it is not**the**legal holiday**.****The**fastest impacts occur at

**more**than 80 km / s in

**the**" worst

__case__" scenario which

**the**asteroid hits

**the**earth in

**a**retrograde parabolic orbit ( because kinetic energy scales as velocity squared

**,**

**earth's**gravity only contributes

__1__km / s

**to**

**this**figure

**,**not 11 km / s ).

Since these stock additions retain

**the**short pistol barrel ( as short as 100 mm ( 4 inches )) they are highly restricted under**the**NFA unless**the**shoulder stocks are**of**original manufacture for**the**gun and**the**gun has been " delisted " as outlined in links__1__& 2 in**the**paragraph above**,**as is**the**__case__**with**so-equipped " Broomhandle " Mausers and Lugers**.**
In

**this**__case__**,****the**ordinals 0**,**__1__**,****,****,**and are regular**,**whereas 2**,**3**,****,**and ω < sub > ω · 2 </ sub > are initial ordinals which are not regular**.**
Most canids bear young once

**a**year**,**from__1__**to**16 or**more**( in**the**__case__**of****the**African wild dog ) at**a**time**.**0.199 seconds.