[permalink] [id link]

Every second-order linear ODE on the extended complex plane with at most four regular singular points, such as the Lamé equation or the hypergeometric differential equation, can be transformed into this equation by a change of variable.

from
Wikipedia

## Some Related Sentences

Every and second-order

For example

**,**__second-order__arithmetic**can**express**the**principle "__Every__countable vector space has**a**basis " but it cannot express**the**principle "__Every__vector space has**a**basis ".
Many principles that imply

**the**axiom**of**choice in their general form (**such****as**"__Every__vector space has**a**basis ") become provable in weak subsystems**of**__second-order__arithmetic when they are restricted**.**
* ( Soundness )

__Every__provable__second-order__sentence is universally valid**,**i**.**e., true in all domains under standard semantics**.**
* ( Completeness )

__Every__universally valid__second-order__formula**,**under standard semantics**,**is provable**.**__Every__

__second-order__

**linear**

**ODE**

**with**three

**regular**

**singular**

**points**

**can**

**be**

**transformed**

**into**

**this**

**equation**

**.**

Every and linear

__Every__module over

**a**division ring has

**a**basis ;

__linear__maps between finite-dimensional modules over

**a**division ring

**can**

**be**described

**by**matrices

**,**and

**the**Gaussian elimination algorithm remains applicable

**.**

__Every__time

**a**diode switches from

**on**to off

**or**vice versa

**,**

**the**configuration

**of**

**the**

__linear__network changes

**.**

__Every__vector v in determines

**a**

__linear__map from R to taking 1 to v

**,**which

**can**

**be**thought

**of**

**as**

**a**Lie algebra homomorphism

**.**

__Every__smooth (

**or**differentiable ) map φ: M → N between smooth (

**or**differentiable ) manifolds induces natural

__linear__maps between

**the**corresponding tangent spaces:

*

__Every____linear__combination**of**its components Y =**a**< sub > 1 </ sub > X < sub > 1 </ sub > + … +**a**< sub > k </ sub > X < sub > k </ sub > is normally distributed**.**__Every__continuous function in

**the**function space

**can**

**be**represented

**as**

**a**

__linear__combination

**of**basis functions

**,**just

**as**every vector in

**a**vector space

**can**

**be**represented

**as**

**a**

__linear__combination

**of**basis vectors

**.**

__Every__sedenion is

**a**real

__linear__combination

**of**

**the**unit sedenions 1

**,**< var > e </ var >< sub > 1 </ sub >, < var > e </ var >< sub > 2 </ sub >, < var > e </ var >< sub > 3 </ sub >, ..., and < var > e </ var >< sub > 15 </ sub >,

__Every__physical quantity has

**a**Hermitian

__linear__operator associated to it

**,**and

**the**states where

**the**value

**of**

**this**physical quantity is definite are

**the**eigenstates

**of**

**this**

__linear__operator

**.**

__Every__nontrivial proper rotation in 3 dimensions fixes

**a**unique 1-dimensional

__linear__subspace

**of**R < sup > 3 </ sup > which is called

**the**axis

**of**rotation (

**this**is Euler's rotation theorem ).

__Every__finite-dimensional normed space is reflexive

**,**simply because in

**this**case

**,**

**the**space

**,**its dual and bidual all have

**the**same

__linear__dimension

**,**hence

**the**

__linear__injection J from

**the**definition is bijective

**,**

**by**

**the**rank-nullity theorem

**.**

__Every__finite-dimensional Hausdorff topological vector space is reflexive

**,**because J is bijective

**by**

__linear__algebra

**,**and because there is

**a**unique Hausdorff vector space topology

**on**

**a**finite dimensional vector space

**.**

__Every__bounded

__linear__transformation from

**a**normed vector space to

**a**complete

**,**normed vector space

**can**

**be**uniquely

**extended**to

**a**bounded

__linear__transformation from

**the**completion

**of**to

**.**

__Every__vector

**a**in three dimensions is

**a**

__linear__combination

**of**

**the**standard basis vectors i

**,**j

**,**and k

**.**

__Every__lattice in

**can**

**be**generated from

**a**basis for

**the**vector space

**by**forming all

__linear__combinations

**with**integer coefficients

**.**

__Every__

__linear__program has

**a**dual problem

**with**

**the**same optimal solution

**,**but

**the**variables in

**the**dual problem correspond to constraints in

**the**primal problem and vice versa

**.**

Every and on

__Every__day

**,**when

**the**President took his nap

**,**Rob Roy would stretch out

__on__

**the**window seat near him

**,**like

**a**perfect gentleman

**,**and stare thoughtfully out

**the**window

**,**

**or**he would take

**a**little nap himself

**.**

__Every__man who dabbles in

**the**market to make

**a**little easy money

__on__

**the**side and suffers losses could

**at**

**the**time hardly face his wife who was wondering how her husband could

**be**so dumb

**.**

__Every__few minutes she would awaken for

**a**moment to review things: Stowey

**,**yes

**,**was

__on__his way south

**,**and

**the**two boys were away in school

**,**and nothing was burning

__on__

**the**stove

**,**and Lucretia was coming for dinner and bringing three guests

**of**hers

**.**

__Every__night they all went to Mama and Papa Albright's

**,**and sat

__on__

**the**open front porch

**,**where they could get

**the**breeze

**.**

__Every__few days

**,**in

**the**early morning

**,**

**as**

**the**work progressed

**,**twenty men would appear to push it ahead and to shift

**the**plank foundation that distributed its weight widely

__on__

**the**Rotunda pavement

**,**supported

**as**it is

**by**ancient brick vaulting

**.**

__Every__night when he wanted

**a**drink

**of**water

**,**didn't he practice being fearless

**by**not turning

__on__

**the**bathroom light??

__Every__single instance

**of**altruistic behavior need not always increase inclusive fitness ; altruistic behaviors would have been selected for if

**such**behaviors

__on__average increased inclusive fitness in

**the**ancestral environment

**.**

*

__Every__continuous functor__on__**a**small-complete category which satisfies**the**appropriate solution set condition has**a**left-adjoint (**the**Freyd adjoint functor theorem ).__Every__year

**,**

__on__

**the**last Sunday in April

**,**there is an ice fishing competition in

**the**frozen estuarine waters

**of**

**the**Anadyr River's mouth

**.**

__Every__actor who has opened in

**the**role

**of**Pseudolus

__on__Broadway ( Zero Mostel

**,**Phil Silvers and Nathan Lane ) won

**a**Best Actor Tony Award for their performance

**.**

* Revised Code

**of**Washington 9**.**12**.**010: "__Every__person who brings__on__his**or**her own behalf**,****or**instigates**,**incites**,****or**encourages another to bring**,**any false suit**at**law**or**in equity in any court**of****this**state**,****with**intent thereby to distress**or**harass**a**defendant in**the**suit**,****or**who serves**or**sends any paper**or**document purporting to**be****or**resembling**a**judicial process**,**that is not in fact**a**judicial process**,**is guilty**of****a**misdemeanor ; and in case**the**person offending is an attorney**,**he**or**she may**,**in addition thereto**be**disbarred from practicing law within**this**state**.**__Every__year

**,**

**the**European Joint Conferences

__on__Theory and Practice

**of**Software ( ETAPS ) sponsors

**the**International Conference

__on__Compiler Construction ( CC ),

**with**papers from both

**the**academic and industrial sectors

**.**

__Every__Halloween

**,**Dabney House conducts

**the**infamous " Millikan pumpkin-drop experiment " from

**the**top

**of**Millikan Library

**,**

**the**highest point

__on__campus

**.**

__Every__person should

**be**identified uniquely

**as**resident in one place but where they happen to

**be**

__on__census day

**,**their de facto residence

**,**may not

**be**

**the**best place to count them

**.**

0.293 seconds.