👐#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)

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.

complete metric space is also called a Cauchy space, because sequences in such metric spaces #converge if and only if they are #Cauchy.

There is a reason higher math exists, you can't run away from it
His answer to what to do with contours of #Cauchy integral if you find a singularity
#homotopy

There is a reason higher math exists, you can't run away from it
His answer to what to do with contours of #Cauchy integral if you find a singularity
#homotopy

What to specify in which Boundary condition :
#Cauchy : function and normal derivative.
#Robin a weighted average of the two.

What to specify in which Boundary condition :
#Cauchy : function and normal derivative.
#Robin a weighted average of the two.

Although charged black holes with rQ ≪ rs are similar to the #Schwarzschild black hole, they have two horizons: the event horizon and an internal #Cauchy horizon.

Although charged black holes with rQ ≪ rs are similar to the #Schwarzschild black hole, they have two horizons: the event horizon and an internal #Cauchy horizon.

A #holomorphic function need not ,possess an antiderivative on its domain, w/o additional assumptions. converse does hold e.g. if domain is simply connected; this is #Cauchy's integral theorem, stating that line integral of a holomorphic function along a closed curve is zero.

A #holomorphic function need not ,possess an antiderivative on its domain, w/o additional assumptions. converse does hold e.g. if domain is simply connected; this is #Cauchy's integral theorem, stating that line integral of a holomorphic function along a closed curve is zero.

the sequence (x_nk)k ∈ N where x¥nk = 1/k for the first n entries (for k = 1, ..., n) and is zero everywhere else (i.e. (xnk)k ∈ N = (1, 1/2, ..., 1/(n−1), 1/n, 0, ...)) is #Cauchy w.r.t. infinity norm but not #convergent (to a sequence in c_00)