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Metric Entropy for Functions of Bounded Total Generalized Variation
SIAM Journal on Mathematical Analysis ( IF 2 ) Pub Date : 2021-02-22 , DOI: 10.1137/20m1310953
Rossana Capuani , Prerona Dutta , Khai T. Nguyen

SIAM Journal on Mathematical Analysis, Volume 53, Issue 1, Page 1168-1190, January 2021.
We establish a sharp estimate for a minimal number of binary digits (bits) needed to represent all bounded total generalized variation functions taking values in a general totally bounded metric space $(E,\rho)$ up to an accuracy of $\varepsilon>0$ with respect to the ${\bf L}^1$-distance. Such an estimate is explicitly computed in terms of doubling and packing dimensions of $(E,\rho)$. The obtained result is applied to provide an upper bound on the metric entropy for a set of entropy admissible weak solutions to scalar conservation laws in one-dimensional space with weakly genuinely nonlinear fluxes.


中文翻译:

有界总广义变化函数的度量熵

SIAM数学分析期刊,第53卷,第1期,第1168-1190页,2021年1月。
我们对表示所有有界总广义变异函数所需的最小二进制数字(位)进行了最小估计,以总的总和为准。有界度量空间$(E,\ rho)$相对于$ {\ bf L} ^ 1 $距离的精度为$ \ varepsilon> 0 $。这种估计是根据$(E,\ rho)$的倍增和打包维数明确计算的。所获得的结果可用于为一维具有弱真正非线性通量的一维空间中的标量守恒律的熵可容许弱解集提供度量熵的上限。
更新日期:2021-02-23
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