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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

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 ).
sublinear and for

** 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
.
sublinear and function

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 ).
sublinear and is

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
.
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

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 ).
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
.

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
.
0.244 seconds.