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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.
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 ),