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A sublinear modulus of continuity can easily found for any uniformly function which is a bounded perturbations of a Lipschitz function: if is a uniformly continuous function with modulus of continuity, and is a Lipschitz function with uniform distance from, then admits the sublinear module of continuity Conversely, at least for real-valued functions, any bounded, uniformly continuous perturbation of a Lipschitz function is a special uniformly continuous function ; indeed more is true as shown below.

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## Some Related Sentences

sublinear and modulus

For

**a****function**between metric spaces**,**it**is**equivalent to admit**a**__modulus__**of****continuity**that**is**either concave**,**or subadditive**,**or**uniformly****continuous****,**or__sublinear__( in**the**sense**of**growth ).
Thus

**,****for****a****function**between metric spaces it**is**equivalent to admit**a**__modulus__**of****continuity****which****is**either concave**,**or subadditive**,**or**uniformly****continuous****,**or__sublinear__**.**

sublinear and continuity

In this section we mainly deal

**with**moduli**of**__continuity__that are concave**,**or subadditive**,**or**uniformly****continuous****,**or__sublinear__**.**

sublinear and can

It

__can__find all**the**primes up to N in time O ( N ), while**the**sieve**of**Atkin**and**most wheel sieves run in__sublinear__time O ( N / log log N ).

sublinear and for

Note that

**as**an immediate consequence**,****any****uniformly****continuous****function**on**a**convex subset**of****a**normed space has**a**__sublinear__growth**:**there are constants**and**such that__for__all**.**
One-way

**functions**are necessary**,**but not known to be sufficient**,**__for__nontrivial ( i**.**e.,**with**__sublinear__communication ) single database computationally private information retrieval**.**
In functional analysis

**the**name Banach functional**is**used__for____sublinear__**function****,**especially when formulating Hahn – Banach theorem**.**
Even

**more**strongly**,**__for__**any**fixed k**,**only**a**__sublinear__number**of**values**of**n need**more**than two terms in their Egyptian fraction expansions**.**

sublinear and function

The Hahn – Banach theorem states that

**if****is****a**__sublinear____function__**,****and****is****a**linear functional on**a**linear subspace U ⊆ V**which****is**dominated by on U**,****A**

__sublinear__

__function__

**,**in linear algebra

**and**related areas

**of**mathematics

**,**

**is**

**a**

__function__on

**a**vector space V over F

**,**an ordered field ( e

**.**g

**.**

**the**real numbers ),

**which**satisfies

sublinear and is

Several important space complexity classes are

__sublinear__**,**that__is__**,**smaller than**the**size**of****the**input**.**

sublinear and bounded

The generalized version

**of****the**conjecture**is**equivalent to**the**statement that**the**number**of**unexpandable fractions**is**not just__sublinear__but__bounded__**.**

modulus and continuity

Intuitively

**,****a****Lipschitz****continuous****function****is**limited in how fast it**can**change**:****for**every pair**of**points on**the**graph**of**this**function****,****the**absolute value**of****the**slope**of****the**line connecting them**is**no greater than**a**definite real number**;**this bound**is**called**the**function's "**Lipschitz**constant " ( or "__modulus__**of****uniform**__continuity__").
For example according to Errett Bishop's definitions

**,****the**__continuity__**of****a****function**( such**as**sin x ) should be proved**as****a**constructive bound on**the**__modulus__**of**__continuity__**,**meaning that**the**existential content**of****the**assertion**of**__continuity__**is****a**promise that**can**always be kept**.**
Since moduli

**of**__continuity__are required to be infinitesimal**at**0**,****a****function**turns out to be**uniformly****continuous****if****and**only**if**it**admits****a**__modulus__**of**__continuity__**.**
Moreover

**,**relevance to**the**notion**is**given by**the**fact that sets**of****functions**sharing**the**same__modulus__**of**__continuity__are exactly equicontinuous families**.**
For instance

**,****the**__modulus__describes**the**k-Lipschitz**functions****,****the**moduli describe**the**Hölder__continuity__**,****the**__modulus__describes**the**almost**Lipschitz**class**,****and**so on**.**
However

**,****a****uniformly****continuous****function**on**a**general metric space**admits****a**concave__modulus__**of**__continuity__**if****and**only**if****the**ratios are**uniformly****bounded****for**all pairs**bounded**away**from****the**diagonal**of****.**
One equivalently says that

**is****a**__modulus__**of**__continuity__( resp.,**at**)**for****,**or shortly**,**is-continuous ( resp.,**at**).

modulus and can

The bulk

__modulus____can__be expressed in terms**of****the**density**and****the**speed**of**sound in**the**medium ()**as**
You

__can__also compute**the**absolute magnitude**of**an object given its apparent magnitude**and****distance**__modulus__**:**
It improves Dirichlet's theorem on prime numbers in arithmetic progressions

**,**by showing that by averaging over**the**__modulus__over**a**range**,****the**mean error**is**much less than__can__be proved in**a**given case**.**
Decryption

**is**possible because**the**multiplier**and**__modulus__used to transform**the**easy**,**superincreasing knapsack into**the**public key__can__also be used to transform**the**number representing**the**ciphertext into**the**sum**of****the**corresponding elements**of****the**superincreasing knapsack**.**
Stronger than

**the**determinant restriction**is****the**fact that an orthogonal matrix__can__always be diagonalized over**the**complex numbers to exhibit**a**full set**of**eigenvalues**,**all**of****which**must have ( complex )__modulus__1**.**
Assuming

**the**truth**of****the**GRH**,****the**estimate**of****the**character sum in**the**Pólya – Vinogradov inequality__can__be improved to**,**q being**the**__modulus__**of****the**character**.**
The area compression

__modulus__K < sub >**a**</ sub >, bending__modulus__K < sub > b </ sub >,**and**edge energy**,**__can__be used to describe them**.**
The carbon

__can__become further enhanced**,****as**high__modulus__**,**or high strength carbon**,**by heat treatment processes**.**
If we compare this

**with****the**formula we derived before**the**environment introduced decoherence we__can__see that**the**effect**of**decoherence has been to move**the**summation sign**from**inside**of****the**__modulus__sign to outside**.**
The multiplication

**of**two complex numbers__can__be expressed most**easily**in polar coordinates –**the**magnitude or__modulus__**of****the**product**is****the**product**of****the**two absolute values**,**or moduli**,****and****the**angle or argument**of****the**product**is****the**sum**of****the**two angles**,**or arguments**.**
Young's

__modulus__**,**E**,**__can__be calculated by dividing**the**tensile stress by**the**tensile strain in**the**elastic ( initial**,**linear ) portion**of****the**stress-strain curve**:**
The Young's

__modulus__**of****a**material__can__be used to calculate**the**force it exerts under specific strain**.**
In terms

**of**Young's__modulus__**and**Poisson's ratio**,**Hooke's law**for**isotropic materials__can__**then**be expressed**as**
This relationship only applies in

**the**elastic range**and**indicates that**the**slope**of****the**stress vs**.**strain curve__can__be used to find Young's__modulus__**.**
The temperature

**of****the**sample or**the**frequency**of****the**stress are often varied**,**leading to variations in**the**complex__modulus__**;**this approach__can__be used to locate**the**glass transition temperature**of****the**material**,****as**well**as**to identify transitions corresponding to other molecular motions**.**
Increasing

**the**amount**of**SBR in**the**blend decreased**the**storage__modulus__due to intermolecular**and**intramolecular interactions that__can__alter**the**physical state**of****the**polymer**.**
Thermal analysis

**of**composite materials**,**such**as**carbon fibre composites or glass epoxy composites are often carried out using DMA or DMTA**,****which**__can__measure**the**stiffness**of**materials by determining**the**__modulus__**and**damping ( energy absorbing ) properties**of****the**material**.**
For multi-loop integrals that will depend on several variables we

__can__make**a**change**of**variables to polar coordinates**and****then**replace**the**integral over**the**angles by**a**sum so we have only**a**divergent integral**,**that will depend on**the**__modulus__**and****then**we__can__apply**the**zeta regularization algorithm**,****the**main idea**for**multi-loop integrals**is**to replace**the**factor after**a**change to hyperspherical coordinates so**the**UV overlapping divergences are encoded in variable r**.**In order to regularize these integrals one needs**a**regulator**,****for****the**case**of**multi-loop integrals**,**these regulator__can__be taken**as**so**the**multi-loop integral will converge**for**big enough's ' using**the**Zeta regularization we__can__analytic continue**the**variable's ' to**the**physical limit where s = 0**and****then**regularize**any**UV integral**.**0.240 seconds.