Gutenberg–Richter law: Difference between revisions

Revision as of 14:41, 20 June 2013

In mathematics, there are at least two results known as "Weyl's inequality".

Weyl's inequality in number theory

In number theory, Weyl's inequality, named for Hermann Weyl, states that if M, N, a and q are integers, with a and q coprime, q > 0, and f is a real polynomial of degree k whose leading coefficient c satisfies

| c - a / q | \leq t q^{- 2},

for some t greater than or equal to 1, then for any positive real number $ε$ one has

\sum_{x = M}^{M + N} \exp (2 π i f (x)) = O (N^{1 + ε} {(\frac{t}{q} + \frac{1}{N} + \frac{t}{N^{k - 1}} + \frac{q}{N^{k}})}^{2^{1 - k}}) as N \to \infty .

This inequality will only be useful when

q < N^{k},

for otherwise estimating the modulus of the exponential sum by means of the triangle inequality as $\leq N$ provides a better bound.

Weyl's inequality in matrix theory

In linear algebra, Weyl's inequality is a theorem about the changes to eigenvalues of a Hermitian matrix that is perturbed. It is useful if we wish to know the eigenvalues of the Hermitian matrix H but there is an uncertainty about the entries of H. We let H be the exact matrix and P be a perturbation matrix that represents the uncertainty. The matrix we 'measure' is $M = H + P$ .

The theorem says that if M, H and P are all n by n Hermitian matrices, where M has eigenvalues

μ_{1} \geq \dots \geq μ_{n}

and H has eigenvalues

ν_{1} \geq \dots \geq ν_{n}

and P has eigenvalues

ρ_{1} \geq \dots \geq ρ_{n}

then the following inequalties hold for $i = 1, \dots, n$ :

ν_{i} + ρ_{n} \leq μ_{i} \leq ν_{i} + ρ_{1}

More generally, if $j + k - n \geq i \geq r + s - 1, \dots, n$ , we have

ν_{j} + ρ_{k} \leq μ_{i} \leq ν_{r} + ρ_{s}

If P is positive definite (that is, $ρ_{n} > 0$ ) then this implies

μ_{i} > ν_{i} \forall i = 1, \dots, n .

Note that we can order the eigenvalues because the matrices are Hermitian and therefore the eigenvalues are real.

References

Matrix Theory, Joel N. Franklin, (Dover Publications, 1993) ISBN 0-486-41179-6

"Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen", H. Weyl, Math. Ann., 71 (1912), 441–479

@@ Line 1: / Line 1: @@
-Nestor is the title my parents gave me but [http://extended-car-warranty-services-review.toptenreviews.com/ I don't] like when people use  [http://www.obsession-circle.de/index.php?mod=users&action=view&id=24823 auto warranty] my full title. Her [http://www.Kbb.com/extended-auto-warranty/ family life]  [http://www.undaheaven.de/index.php?mod=users&action=view&id=11659 extended car warranty] in Idaho. The preferred hobby for him  auto warranty and his children is to perform  car warranty badminton but he is having difficulties to discover time for it. The occupation I've been occupying for many years is a bookkeeper but I've currently [http://En.wikipedia.org/wiki/Warranty applied] for another 1.<br><br>my weblog :: [http://www.gettingtherefromhere.info/User-Profile/userId/13851 automobile extended warranty]
+In mathematics, there are at least two results known as '''"Weyl's inequality"'''.
+==Weyl's inequality in number theory==
+In [[number theory]], '''Weyl's inequality''', named for [[Hermann Weyl]], states that if ''M'', ''N'', ''a'' and ''q'' are integers, with ''a'' and ''q'' [[coprime]], ''q''&nbsp;>&nbsp;0, and ''f'' is a [[real number|real]] [[polynomial]] of degree ''k'' whose leading coefficient ''c'' satisfies
+:<math>|c-a/q|\le tq^{-2},\,</math>
+for some ''t'' greater than or equal to 1, then for any positive real number <math>\scriptstyle\varepsilon</math> one has
+:<math>\sum_{x=M}^{M+N}\exp(2\pi if(x))=O\left(N^{1+\varepsilon}\left({t\over q}+{1\over N}+{t\over N^{k-1}}+{q\over N^k}\right)^{2^{1-k}}\right)\text{ as }N\to\infty.</math>
+This inequality will only be useful when
+:<math>q < N^k,\,</math>
+for otherwise estimating the modulus of the [[exponential sum]] by means of the [[triangle inequality]] as <math>\scriptstyle\le\, N</math> provides a better bound.
+==Weyl's inequality in matrix theory==
+In linear algebra, '''Weyl's inequality''' is a theorem about the changes to [[eigenvalues]] of a [[Hermitian matrix]] that is perturbed.  It is useful if we wish to know the eigenvalues of the Hermitian matrix ''H'' but there is an uncertainty about the entries of ''H''.  We let ''H'' be the exact matrix and ''P'' be a perturbation matrix that represents the uncertainty.  The matrix we 'measure' is <math>\scriptstyle M \,=\, H \,+\, P</math>.
+The theorem says that if ''M'', ''H'' and ''P'' are all ''n'' by ''n'' Hermitian matrices, where ''M'' has eigenvalues
+:<math>\mu_1 \ge \cdots \ge \mu_n\, </math>
+and ''H'' has eigenvalues
+:<math>\nu_1 \ge \cdots \ge \nu_n\, </math>
+and ''P'' has eigenvalues
+:<math>\rho_1 \ge \cdots \ge \rho_n\, </math>
+then the following inequalties hold for <math>\scriptstyle i \,=\, 1,\dots ,n</math>:
+:<math>\nu_i + \rho_n \le \mu_i \le \nu_i + \rho_1\, </math>
+More generally, if  <math>\scriptstyle j+k-n \,\ge\, i \,\ge\, r+s-1,\dots ,n</math>, we have
+:<math>\nu_j + \rho_k \le \mu_i \le \nu_r + \rho_s\, </math>
+If ''P'' is positive definite (that is, <math>\scriptstyle\rho_n \,>\, 0</math>) then this implies
+:<math>\mu_i > \nu_i \quad   \forall i = 1,\dots,n.\,</math>
+Note that we can order the eigenvalues because the matrices are Hermitian and therefore the eigenvalues are real.
+==References==
+* ''Matrix Theory'', Joel N. Franklin, (Dover Publications, 1993) ISBN 0-486-41179-6
+* "Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen", H. Weyl, Math. Ann., 71 (1912), 441–479
+{{DEFAULTSORT:Weyl's Inequality}}
+[[Category:Diophantine approximation]]
+[[Category:Inequalities]]
+[[Category:Linear algebra]]

Gutenberg–Richter law: Difference between revisions

Revision as of 14:41, 20 June 2013

Weyl's inequality in number theory

Weyl's inequality in matrix theory

References

Navigation menu

Search