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for some real-valued functions a and b depending, respectively, on x and y. Conversely, given any such pair of real-valued functions, there exists a vector field X satisfying 1. and 2.

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## Some Related Sentences

for and some

I had

__for____some__time been hoping**,**in vain**,**__for__one**of**the dim figures to pass between the fan vents**and**myself**.**
Later

**,**riding in__for____some__lusty enjoyment**of**the liquor**and**professional ladies**of**Cheyenne**,**he laid claim to the killing with the vague insinuations he made**.**
How lightly her `` eventshah-leh '' passed into the crannies where I was storing dialect material

__for____some__vaguely dreamed opus**,****and**how the word would echo**.**
Sometimes I wondered vaguely what he did about women

__for__my Aunt**,**by blood**,**had died__some__years ago**,**but neither**of**us said anything**.**
Although New Orleans was not to learn

**of**it__for__**a**spell**,**she also was**a**sadist**,****a**nymphomaniac**and**unobtrusively mad -- the perpetrator**of**__some__**of**the worst crimes against humanity ever committed**on**American soil**.**
At the same time

**,**all suggestions that__some__sort**of**societal responsibility existed__for__the welfare**of**the people within the territorial state was strongly resisted**.**
By subduing disparate lesser groups the nation has

**,**to__some__degree at least**,**broadened the capacity__for__individual liberty**.**
I persuaded an Australian friend who had lived `` outback ''

__for__years to take me to see__some__aborigines living in the bush**.**
There were two rubbing sticks

__for__making fire**,**two stones shaped roughly like knives**,****a**woven-root container which held**a**few pounds**of**dried worms**and**the dead body**of**__some__rodent**.**
He must construct transitions so that

**a**dancer who is told to lie prone one second**and**to leap wildly the next will have__some__physical preparation__for__the leap**.**
When it comes to this

**,**I shall prefer emigrating to__some__country where they make no pretence**of**loving liberty -- to Russia**,**__for__instance**,**where despotism can be taken pure**,**without the base alloy**of**hypocrisy '' ( His emphasis )
This almost trivial example is nevertheless suggestive

**,**__for__**there**are__some__elements in common between the antique fear that the days would get shorter**and**shorter**and**our present fear**of**war**.**
In agriculture

**,**__for__example**,**despite the advances in biology**,**elaborate rituals tend to persist along with**a**continued sense**of**the imminence**of**__some__natural disaster**.**
And the life they lead is undisciplined

**and**__for__the most part unproductive**,**even though they make**a**fetish**of**devoting themselves to__some__creative pursuit -- writing**,**painting**,**music**.**
It is worth dwelling in

__some__detail**on**the crisis**of**this story**,**because it brings together**a**number**of**characteristic elements**and**makes**of**them**a**curious**,**riddling compound obscurely but centrally significant__for__Mann's work**.**
not less strikingly so

__for__being mysterious**,**as though__some__deeply hidden constatation**of**thoughts were enciphered in**a**single image**,****a**single moment**.**
His name is Praisegod Piepsam

**,****and**he is rather fully described as to his clothing**and**physiognomy in**a**way which relates him to**a**sinister type in the author's repertory -- he is**a**forerunner**of**those enigmatic strangers in `` Death In Venice ''**,**__for__example**,**who represent__some__combination**of**cadaver**,**exotic**,****and**psychopomp**.**
These desires presuppose

**a**sense**of**causally efficacious powers in which one is involved**,**__some__working__for__one's good**,**others threatening ill**.**
When

__some__question arises in the medical**field**concerning cancer**,**__for__instance**,**we do not turn to free**and**open discussion as in**a**political campaign**.**
If he had been `` liquidated '' in

__some__way**,**he would have become**a**martyr**,****a**rallying point__for__people who shared his ideas**.**
As Helion's work showed more

**and**more nostalgia__for__the world**of**man**and**nature**,**the pure abstractionists expressed__some__disapproval ; ;
Actually

**,**you could wish__for____some__passion**,**now**and**then**,**but when you look around the world**and**see the little volcanos**of**current history which partisan social passions have wrought**,**you are glad that in these pamphlets**there**is at least__some__civilized calm**.**

for and real-valued

This interpolation does not minimize the slope

**,****and**is not generally__real-valued____for__real ; its use is**a**common mistake**.**
Equations

__for__coefficients corresponding to ratios**of**the electric**field**complex-valued amplitudes**of**the waves ( not necessarily__real-valued__magnitudes ) are also called " Fresnel equations ".
In numerical analysis

**,**Newton's method ( also known as the Newton – Raphson method ), named after Isaac Newton**and**Joseph Raphson**,**is**a**method__for__finding successively better approximations to the roots ( or zeroes )**of****a**__real-valued__function**.**
If the random variable is

__real-valued__( or more generally**,**if**a**total order is defined__for__its possible values ), the cumulative distribution function gives the probability that the random variable is no larger than**a****given**value ; in the__real-valued__case it is the integral**of**the density**.**
A constructive proof

**of**this theorem (__for__ƒ__real-valued__) using Bernstein polynomials is outlined**on**that page**.**
The set C

**of**continuous__real-valued__**functions****on****,**together with the supremum norm**,**is**a**Banach algebra**,**( i**.**e**.**an associative algebra**and****a**Banach space**such**that__for__all f**,**g ).
Let be the space

**of**__real-valued__continuous**functions****on****X**which vanish at infinity ; that is**,****a**continuous function f is in if**,**__for__every**,****there****exists****a**compact set**such**that**on**
A

__real-valued__function ƒ: M → R belongs to C < sup >∞</ sup >( M ) if ƒ ∘ φ < sup >−**1**</ sup > is infinitely differentiable__for__every chart φ: U → R < sup > n </ sup >.
For instance

**,****a**__real-valued__measurable function is**a**function__for__which the preimage**of**each Borel set is measurable**.**
There is

**a**convention among engineers**,**climatologists**,****and**others to reserve “ negative binomial ” in**a**strict sense or “ Pascal ”__for__the case**of**an integer-valued stopping-time parameter r**,****and**use “ Polya ”__for__the__real-valued__case**.**
* Suppose that is

**a**sequence**of**Lipschitz continuous mappings between two metric spaces**,****and**that all have Lipschitz constant bounded by**some**K**.**If ƒ < sub > n </ sub > converges to**a**mapping ƒ uniformly**,**then ƒ is also Lipschitz**,**with Lipschitz constant bounded by the same K**.**In particular**,**this implies that the set**of**__real-valued__**functions****on****a**compact metric space with**a**particular bound__for__the Lipschitz constant is**a**closed**and**convex subset**of**the Banach space**of**continuous**functions****.**
An extended

__real-valued__function f is upper (**respectively****,**lower ) semi-continuous at**a**point**x**< sub > 0 </ sub > if**,**roughly speaking**,**the function values__for__arguments near**x**< sub > 0 </ sub > are either close to f (**x**< sub > 0 </ sub >) or less than (**respectively****,**greater than ) f (**x**< sub > 0 </ sub >).
Topological rings occur in mathematical analysis

**,**__for__examples as rings**of**continuous__real-valued__**functions****on****some**topological space ( where the topology is**given**by pointwise convergence ), or as rings**of**continuous linear operators**on****some**normed**vector**space ; all Banach algebras are topological rings**.**
A pseudometric space is

**a**set together with**a**non-negative__real-valued__function ( called**a**pseudometric )**such**that**,**__for__every**,**
If we restrict attention to

__real-valued__W then the relation is defined only__for__**x**≥ −**1**/ e**,****and**is double-valued**on**(−**1**/ e**,**0 ); the additional constraint W ≥ −**1**defines**a**single-valued function W < sub > 0 </ sub >(**x**).
In certain technical fields ( especially computer programming

**and**statistics ),**a**data type is**a**classification identifying one**of**various types**of**data**,****such**as__real-valued__**,**integer or Boolean**,**that determines the possible values__for__that type ; the operations that can be done**on**values**of**that type ; the meaning**of**the data ;**and**the way values**of**that type can be stored**.**
Other examples are regression

**,**which assigns**a**__real-valued__output to each input ; sequence labeling**,**which assigns**a**class to each member**of****a**sequence**of**values (__for__example**,**part**of**speech tagging**,**which assigns**a**part**of**speech to each word in an input sentence );**and**parsing**,**which assigns**a**parse tree to an input sentence**,**describing the syntactic structure**of**the sentence**.**
Often

**,**categorical**and**ordinal data are grouped together ; likewise__for__integer-valued**and**__real-valued__data**.**
Markov's inequality states that

__for__**any**__real-valued__random variable Y**and****any**positive number**a****,**we have Pr (| Y | >**a**) ≤ E (| Y |)/**a****.**
This might require quite bizarre laws

**of**physics (__for__example**,****a**measurable physical constant with an oracular value**,****such**as Chaitin's constant ),**and**would at minimum require the ability to measure**a**__real-valued__physical value to arbitrary precision despite thermal noise**and**quantum effects**.**
In order

__for__this to be well-defined**,**we must show that it is the unique operation**on**bounded__real-valued__Borel**functions****satisfying****a**number**of**conditions**.**
Let U be an open set in R < sup > n </ sup >

**and**φ: U → R < sup > n </ sup > an injective differentiable function with continuous partial derivatives**,**the Jacobian**of**which is nonzero__for__every**x**in U**.**Then__for__**any**__real-valued__**,**compactly supported**,**continuous function f**,**with support contained in φ ( U ),0.091 seconds.