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.

In this section we mainly deal with moduli of continuity that are concave, or subadditive, or uniformly continuous, or sublinear.

Other sublinear functions can be useful as well, especially Minkowski functionals of convex sets.

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 ).

** Search for a regular expression P in time expected sublinear in.

* is sublinear, that is, there are constants and such that for all ;

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.

Given a vector space V over the field R of real numbers, a function is called sublinear if

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

In computer science, a function is called sublinear if in asymptotic notation ( Notice the small ).

Every seminorm on V ( in particular, every norm on V ) is sublinear.

Several important space complexity classes are sublinear, that is, smaller than the size of the input.

The generalized version of the conjecture is equivalent to the statement that the number of unexpandable fractions is not just sublinear but bounded.

Such generalizations are useful for constructing different modulus of 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.

* Lévy's modulus of continuity

If and has modulus of continuity,

We define the local modulus of continuity at the point t by

The global modulus of continuity ( or simply the modulus of continuity ) is defined by

In mathematical analysis, a modulus of continuity is a function

So, a function admits as a modulus of continuity if and only if

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.

Formally, a modulus of continuity is any real-extended valued function

A function admits as ( local ) modulus of continuity at the point if and only if,

Also, admits as ( global ) modulus of continuity if and only if,

One equivalently says that is a modulus of continuity ( resp., at ) for, or shortly, is-continuous ( resp., at ).

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.

Young's modulus can vary somewhat due to differences in sample composition and test method.

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.

These analyzers can test higher modulus materials than torsional analyzers.

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.