👐#call4reading #highlycitedpaper

✍️Quantum #algorithm for matrix functions by #Cauchy's integral formula #by Souichi Takahira, Asuka Ohashi, Tomohiro Sogabe, and Tsuyoshi S. Usuda

🔗https://doi.org/10.26421/QIC20.1-2-2 (#arXiv:2106.08075)

Rinton Press - Publisher in Science and Technology

Rinton Press, a science and technology publisher, is founded by a team of scholars and publication professionals, and is based in Princeton, New Jersey. Rinton Press serves the scientific and academic community by publishing, marketing and distributing journals, books and proceedings, through a progressively wide variety of media such as CD-ROM and Internet in addition to conventional print on paper.

LINEAR TRANSPORT EQUATION
The linear transport equation (LTE) models the variation of the concentration of a substance flowing at constant speed and direction. It's one of the simplest partial differential equations (PDEs) and one of the few that admits an analytic solution.

Given \(\mathbf{c}\in\mathbb{R}^n\) and \(g:\mathbb{R}^n\to\mathbb{R}\), the following Cauchy problem models a substance flowing at constant speed in the direction \(\mathbf{c}\).
\[\begin{cases}
u_t+\mathbf{c}\cdot\nabla u=0,\ \mathbf{x}\in\mathbb{R}^n,\ t\in\mathbb{R}\\
u(\mathbf{x},0)=g(\mathbf{x}),\ \mathbf{x}\in\mathbb{R}^n
\end{cases}\]
If \(g\) is continuously differentiable, then \(\exists u:\mathbb{R}^n\times\mathbb{R}\to\mathbb{R}\) solution of the Cauchy problem, and it is given by
\[u(\mathbf{x},t)=g(\mathbf{x}-\mathbf{c}t)\]

#LinearTransportEquation #LinearTransport #Cauchy #CauchyProblem #PDE #PDEs #CauchyModel #Math #Maths #Mathematics #Linear #LinearPDE #TransportEquation #DifferentialEquations

Some useful inequalities:

1. Cauchy–Schwarz inequality

\[\displaystyle\sum_{k=1}^na_kb_k\leq\sqrt{\sum_{k=1}^na_k^2}\sqrt{\sum_{k=1}^nb_k^2}\]

2. Hölder's inequality

\[\displaystyle\sum_{k=1}^n\left|a_kb_k\right|\leq\left(\sum_{k=1}^n|a_k|^p\right)^{1/p}\left(\sum_{k=1}^n|b_k|^q\right)^{1/q}\]

3. Minkowski's inequality

\[\displaystyle\left(\sum_{k=1}^n\left|a_k+b_k\right|^p\right)^{1/p}\leq\left(\sum_{k=1}^n|a_k|^p\right)^{1/p}+\left(\sum_{k=1}^n|b_k|^p\right)^{1/p}\]

4. Hardy's inequality

\[\displaystyle\sum_{k=1}^\infty\left(\dfrac{a_1+a_2+\cdots+a_k}{k}\right)^p\leq\left(\dfrac{p}{p-1}\right)^p\sum_{k=1}^\infty a_k^p\]

#Inequality #Cauchy #Schwarz #Hölder #Minkowski #Hardy #Maths #Mathstodon #Mastodon #Mathematics

@board
美國核心PCE年增率經過迴歸後的殘差分布是柯西分布(平均數未定義)。

這一點都不是美好的事情。這和美國個人儲蓄率經過迴歸後的殘差分布一樣,只是參數不同。

#經濟 #財經 #通貨膨脹 #通膨 #通脹 #儲蓄 #economics #inflation #saving #Cauchy #econdon #econmastodon #econtwitter #AI經濟

Lebniez was working on continuum, when #Cauchy satisfied his own research needs by introducing the convergence criterion for sequences
- Without using countable choice, it is not possible to constructively prove the fundamental theorem of algebra for complex numbers based on the #Dedekind real numbers (which are not constructively equivalent to #Cauchy real numbers without countable choice).
- a #Cauchy space is a set equipped with a class of filters declared to be Cauchy.
c. f is just Fourier transform of PDF - parameters of the #Cauchy distribution do not correspond to a mean and variance, attempting to estimate the parameters of the Cauchy distribution by using a sample mean and a sample #variance will not succeed.
On the other hand #probability theory can use some #spivak when it comes to calculate characteristic function of #Cauchy distribution
#Residuetheorem
complete metric space is also called a Cauchy space, because sequences in such metric spaces #converge if and only if they are #Cauchy.