[permalink] [id link]

Conversely, if ƒ: C → C is a function which is differentiable when regarded as a function on R < sup > 2 </ sup >, then ƒ is complex differentiable if and only if the Cauchy – Riemann equations hold.

from
Wikipedia

## Some Related Sentences

Conversely and if

__Conversely__

**,**

__if__statistics were uncovered

**which**contradicted

**a**cherished theory

**,**

**the**sources were denounced

**as**faulty

**.**

__Conversely__

**,**experimenters would consider

**as**impressive such statements

**as**

**the**following

**,**

**which**

**,**

__if__they turned out to be hits

**,**are so unusual

**as**to be really significant

**:**

__Conversely__

**,**

__if__he gives

**a**heavy rating to his own reading

**,**

**and**finds more accurate facts in it than in

**the**others

**,**

**a**point

**is**chalked up for

**the**intrinsic

**,**objective meaningfulness of this type of mediumistic material

**.**

__Conversely__

**,**

**a**subset

**R**defines

**a**binary

**function**

__if__

**and**

**only**

__if__

**,**for any x in X

**and**y in Y

**,**there exists

**a**unique z in Z such that ( x

**,**y

**,**z ) belongs to

**R**

**.**

__Conversely__

**,**

__if__

**a**Boolean ring A

**is**given

**,**we can turn it into

**a**Boolean algebra by defining x ∨ y := x + y + ( x · y )

**and**x ∧ y := x · y

**.**

__Conversely__

**,**

__if__

**the**second theory

**is**correct

**,**

**then**

**the**Athenians were merely reacting to

**the**Persians attacking them

**.**

__Conversely__

**the**equilibrium position

**is**said to be " far to

**the**left "

__if__hardly any product

**is**formed from

**the**reactants

**.**

__Conversely__

**,**

__if__

**the**amount of fat

**is**increased

**,**increasing

**the**number of calories ingested will

**only**add weight

**.**

__Conversely__

**,**

__if__

**the**source of waves

**is**moving away from

**the**observer

**,**each wave

**is**emitted from

**a**position farther from

**the**observer than

**the**previous wave

**,**so

**the**arrival time between successive waves

**is**increased

**,**reducing

**the**frequency

**.**

__Conversely__every partition of comes from an equivalence relation in this way

**,**according to

**which**

__if__

**and**

**only**

__if__

**and**belong to

**the**same set of

**the**partition

**.**

If neither A nor B includes

**the**idea of existence**,****then**" some A are B " simply adjoins A to B**.**__Conversely__**,**__if__A or B do include**the**idea of existence in**the**way that " triangle " contains**the**idea " three angles equal to two right angles ",**then**" A exists "**is**automatically true**,****and**we have an ontological proof of A's existence**.**__Conversely__

**,**

__if__

**the**measured potential

**is**negative

**,**

**the**electrode

**is**more oxidizing than

**the**SHE ( such

**as**Zn in ZnSO

**<**sub

**>**4

**</**sub

**>**where

**the**standard electrode potential

**is**− 0

**.**76 V ).

__Conversely__

**,**

__if__

**the**firm

**is**able to get bulk discounts of an input

**,**

**then**it could have economies of scale in some range of output levels even

__if__it has decreasing returns in production in that output range

**.**

__Conversely__

**,**daylight stock shot in daylight will look normal

**,**but orange

__if__shot under tungsten lights

**.**

__Conversely__

**,**

__if__

**the**data are sparse — that

**is**

**,**

__if__

**only**K out of N Fourier coefficients are nonzero —

**then**

**the**complexity can be reduced to O ( K log N log ( N / K )),

**and**this has been demonstrated to lead to practical speedups compared to an ordinary FFT for N / K

**>**32 in

**a**large-N example ( N =

**2**

**<**

**sup**

**>**22

**</**

**sup**>) using

**a**probabilistic approximate algorithm (

**which**estimates

**the**largest K coefficients to several decimal places ).

Conversely and ƒ

__Conversely__every positive functional φ gives

**a**corresponding inner product

**<**

__ƒ__

**,**g >< sub

**>**φ

**</**sub

**>**= φ ( g *

__ƒ__).

**

__Conversely__**,****if**__ƒ__**:**I**→****R****is**absolutely continuous**and**thus**differentiable**almost everywhere**,****and**satisfies |__ƒ__′( x )| ≤ K for almost all x in I**,****then**__ƒ__**is**Lipschitz continuous with Lipschitz constant at most K**.**__Conversely__

**,**

**if**F has characteristic zero

**,**

**then**given an homogeneous polynomial

__ƒ__of degree n

**on**V

**,**

**the**polarization of

__ƒ__

**is**

**a**multilinear

**function**

**on**

**the**n-th Cartesian product of V

**.**The polarization

**is**defined by

for all compactly supported functions

__ƒ__**and**h**.**__Conversely__**,**() characterizes Δ completely**,**in**the**sense that it**is****the****only**operator with this property**.**

Conversely and C

__Conversely__

**,**paintings

**which**came out of his workshop that were not necessarily physically worked

**on**by Cuyp but merely overseen by him technically

**,**were marked with A

**.**

__C__

**.**to show that it was his instruction

**which**saw

**the**paintings ’ completion

**.**

* The spectrum of any bounded linear operator

**on****a**Banach space**is****a**nonempty compact subset of**the****complex**numbers__C__**.**__Conversely__**,**any compact subset of__C__arises in this manner**,****as****the**spectrum of some bounded linear operator**.**
We define Thm (

__C__) to be**the**set of all formulas that are valid in__C__**.**__Conversely__**,****if**X**is****a**set of formulas**,**let Mod ( X ) be**the**class of all frames**which**validate every formula from X**.**
As stated earlier

**,**an adjunction between categories__C__**and**D gives rise to**a**family of universal morphisms**,**one for each object in__C__**and**one for each object in D**.**__Conversely__**,****if**there exists**a**universal morphism to**a**functor G**:**__C__**→**D from every object of D**,****then**G has**a**left adjoint**.**__Conversely__

**,**by

**the**inverse

**function**theorem

**,**there can be at most one value for σ that

**,**

**when**applied

**as**an input to

**,**will result in

**a**particular value for

__C__

**.**

__Conversely__

**,**July

**is**

**the**warmest month

**,**with an average temperature of 21

**.**4

__C__

**.**The annual temperature ranges between-12

__C__

**as**an annual low to an annual high of 38

__C__

**.**

__Conversely__

**,**Oymyakon

**on**July 28

**,**2010 recorded

**a**record high temperature of

**,**yielding

**a**temperature range of 105

**.**8 °

__C__( 190

**.**4 ° F ).

__Conversely__

**,**

**if**

**a**program adheres to standards such

**as**POSIX

**and**ANSI

__C__

**,**it will be easier to port to

**the**USS environment

**.**

__Conversely__

**,**CPython applications that depend

**on**extensions to

**the**language that are implemented in

__C__( e

**.**g

**.**NumPy ) are not compatible with IronPython

**,**although

**a**commercially-supported open source project to address this

**is**underway

**.**

__Conversely__two widely recognized standards for industrial RTDs IEC 6075

**and**ASTM E-1137 specify an alpha of 0

**.**00385 ohms / ohm /°

__C__

**.**

__Conversely__

**,**in National Collegiate Athletic Association v

**.**Smith

**,**

**the**Supreme Court has found that

**the**National Collegiate Athletic Association

**is**not

**a**state actor for

**the**purposes of 28 U

**.**S

**.**

__C__

**.**

__Conversely__

**,**several cable news channels exist that carry news information specifically geared toward

**a**particular metropolitan area

**,**state or region such

**as**NY1 in New York City

**,**Ohio News Network (

**which**

**is**based in Columbus

**,**Ohio

**,**but

**is**carried throughout

**the**state of Ohio ), Bay News 9 in Tampa

**,**Florida

**and**NewsChannel 8 in Washington

**,**D

**.**

__C__

**.**

__Conversely__

**,**

**the**V

**.**

__C__

**.**s

**,**all colonists from different worlds

**,**often used

**the**term ' earthworm ' to describe Smith

**and**his compatriots

**.**

Conversely and →

__Conversely__

**,**starting from this equivalence we can recover U

**as**

**the**composite Rel

__→__Sup

__→__Set of

**the**forgetful functor for Sup with this embedding of Rel in Sup

**.**

__Conversely__

**,**

**a**complete variety

**is**close to being

**a**projective variety

**:**Chow's lemma states that

**if**X

**is**

**a**complete variety

**,**there

**is**

**a**projective variety Z

**and**

**a**birational morphism Z

__→__X

**.**

__Conversely__

**,**

**a**

**function**u

**:**E

**<**sub

**>**1

**</**sub

**>**

__→__E

**<**sub

**>**

**2**

**</**sub

**>**defined

**on**

**a**basis of E

**<**sub

**>**1

**</**sub

**>**can be uniquely extended to

**a**linear

**function**

**.**

__Conversely__

**,**one can show that given

**a**manifold M

**and**

**a**principal H-bundle P over M

**,**

**and**

**a**1-form η with these properties

**,**

**then**P

**is**locally isomorphic

**as**an H-bundle to

**the**principal homogeneous bundle G

__→__G / H

**.**

__Conversely__

**,**

**the**following characterization of derived functors holds

**:**given

**a**family of functors

**R**

**<**

**sup**

**>**i

**</**

**sup**>: A

__→__B

**,**satisfying

**the**above

**,**i

**.**e

**.**mapping short exact sequences to long exact sequences

**,**such that for every injective object I of A

**,**

**R**

**<**

**sup**

**>**i

**</**

**sup**>( I )= 0 for every positive i

**,**

**then**these functors are

**the**right derived functors of

**R**

**<**

**sup**

**>**0

**</**

**sup**>.

__Conversely__

**,**an Ehresmann connection H ⊂ TP ( or v

**:**TP

__→__V )

**on**P defines

**a**principal G-connection ω

**if**

**and**

**only**

**if**it

**is**G-equivariant in

**the**sense that

**.**

A section Γ of dπ

**:**TP / G__→__TM such that Γ**:**TM__→__TP / G**is****a**linear morphism of vector bundles over M**,**can be identified with**a**principal connection in P**.**__Conversely__**,****a**principal connection**as**defined above gives rise to such**a**section Γ of TP / G**.**__Conversely__

**,**given

**a**left kG module M

**,**

**then**M

**is**

**a**k vector space

**,**

**and**multiplication with an element g of G yields

**a**k-linear automorphism of M ( since g

**is**invertible in kG ),

**which**describes

**a**group homomorphism G

__→__GL ( M ).

Suppose S

**is**an immersed submanifold of M**.**If**the**inclusion map i**:**S__→__M**is**closed**then**S**is**actually an embedded submanifold of M**.**__Conversely__**,****if**S**is**an embedded submanifold**which****is**also**a**closed subset**then****the**inclusion map**is**closed**.**__Conversely__

**,**given any element u ∈ F ( A ) we may define

**a**natural transformation Φ

**:**Hom ( A ,–)

__→__F via

__Conversely__

**,**

**if**G contains

**a**proper non-trivial normal subgroup N

**,**

**then**

**the**composition of

**the**natural surjective group homomorphism G

__→__G / N with

**the**regular representation of G / N produces

**a**representation π of G

**which**has kernel N

**.**Taking χ to be

**the**character of some non-trivial subrepresentation of π

**,**we have

**a**character satisfying

**the**hypothesis in

**the**direct statement above

**.**

0.210 seconds.