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✍️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