User Fei Cao

13 votes

1 answer

422 views

prove a challenging inequality or find a counterexample to it

asked Dec 30, 2020 at 22:49

7 votes

2 answers

307 views

Prove that $0\le\int_0^1\log(u){\rm d}x+\frac1{2\pi^2}\int_0^1\frac1{u^2}\left(\frac{{\rm d}u}{{\rm d}x}\right)^2{\rm d}x$

asked Feb 28, 2021 at 19:50

7 votes

0 answers

373 views

A possible error in Villani's monograph “Hypocoercivity”

asked Mar 20, 2021 at 5:53

7 votes

1 answer

404 views

Modified Energy Method for Transformed Fokker-Planck Equation (Tricky Integration by parts...)

asked Aug 6, 2020 at 21:47

7 votes

1 answer

399 views

A possible bug in a highly cited paper (Adam gradient descent)?

asked Oct 9, 2021 at 5:44

6 votes

5 answers

228 views

Show that $\frac{1}{n}\sum_{j=1}^\infty \left( 1 - (1-p_j)^n\right) \to 0$ as $n \to \infty$

asked Dec 23, 2021 at 3:47

6 votes

1 answer

181 views

Proper linearization of ODEs of the form $\dot{x}(t) + f(x(t)) + \sigma(t) = 0$?

asked Aug 6 at 23:56

6 votes

1 answer

214 views

Showing $\frac{f(a+2b)+f(a-2b)}{2f(a)}\leq\sqrt{1-\left(\frac{b}{a(1-a)}\right)^2}$ for $f(x)=\sqrt{x(1-x)}$ with some constraints

asked Nov 20, 2020 at 18:42

5 votes

3 answers

346 views

A tough definite integral using contour integration

asked Jul 15, 2020 at 19:36

5 votes

2 answers

254 views

Geometric implication of the Sobolev embedding

asked Jul 6, 2021 at 21:52

4 votes

1 answer

134 views

Bounding Euclidean norm by slanted 1-norm

asked Jul 21, 2021 at 23:08

4 votes

1 answer

104 views

Parseval-Plancherel type identity for probability generating function

asked Jul 30 at 15:45

4 votes

1 answer

206 views

Poincare inequality for Poisson random variables

asked Dec 25, 2021 at 18:05

4 votes

1 answer

200 views

Optimal Poincare constant (with constraints)

asked Apr 11, 2022 at 0:50

4 votes

1 answer

182 views

exact meaning of uniform integrability for empirical distributions

asked Oct 27, 2020 at 6:44

4 votes

0 answers

82 views

Understanding the fast sweeping algorithm for Eikonal equations

asked Apr 25, 2021 at 22:22

3 votes

0 answers

111 views

Meaning of $K'[\cdot]$ when $K$ is an symmetric Onsager (matrix) operator

asked Apr 17, 2021 at 5:17

3 votes

1 answer

134 views

Non-uniqueness of a Dirichlet's problem

asked Jan 19, 2021 at 6:20

3 votes

1 answer

86 views

A basic question on martingales and filtrations

asked Dec 16, 2020 at 4:50

3 votes

1 answer

92 views

A discrete Poincare inequality

asked Feb 4, 2022 at 16:05

3 votes

1 answer

27 views

Show that $K(t) = \int_0^t \mathrm{e}^{-Cs} \, D \, \mathrm{e}^{-C^\intercal s} \,\mathrm{d} s$ satisfies a given equation involving $K(\infty)$

asked Aug 4 at 17:41

3 votes

0 answers

78 views

Generalized second derivative of a concave and piecewise $C^2$ function

asked Jul 25, 2021 at 19:46

3 votes

1 answer

25 views

Lipschitz continuity of $\sqrt{f}$ for $f(x) = \sup_{\alpha \in T} \sum_{i=1}^d \left(\sum_{j=1}^D \alpha_j x_{ij}\right)^2$

asked Dec 31, 2021 at 0:32

3 votes

0 answers

99 views

Deriving inequalities from other inequalities

asked Jan 1, 2022 at 0:14

2 votes

1 answer

124 views

Show $\mathbb{E}\left[\left|\sum_{i=1}^n X_iY_i\right|\right] \leq 2\mathbb{E}\left[\left|\sum_{i=1}^n Y_i\right|\right]$

asked Jan 3, 2022 at 23:55

2 votes

1 answer

220 views

Random time change from Oksendal's SDE textbook

asked Jul 31, 2021 at 22:54

2 votes

1 answer

737 views

Wasserstein distance between two empirical measures

asked Jun 13, 2021 at 0:23

2 votes

0 answers

132 views

Show that $\int_{-\infty}^{a_1} (a_1-x)^r f(x) \mathrm{d} x$ and $\int_{a_n}^\infty (x - a_n)^r f(x) \mathrm{d} x$ are of order $\mathcal{O}(n^{-r})$

asked Aug 21 at 0:01

2 votes

1 answer

98 views

Show that $x^2 + 3\sin^2 x$ satisfies the Polyak–Łojasiewicz condition

asked Mar 30 at 21:30

2 votes

1 answer

75 views

Gronwall inequality for a $2$-dimensional system of linear differential inequalities

asked May 26 at 16:05

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83 Questions

prove a challenging inequality or find a counterexample to it

Prove that $0\le\int_0^1\log(u){\rm d}x+\frac1{2\pi^2}\int_0^1\frac1{u^2}\left(\frac{{\rm d}u}{{\rm d}x}\right)^2{\rm d}x$

A possible error in Villani's monograph “Hypocoercivity”

Modified Energy Method for Transformed Fokker-Planck Equation (Tricky Integration by parts...)

A possible bug in a highly cited paper (Adam gradient descent)?

Show that $\frac{1}{n}\sum_{j=1}^\infty \left( 1 - (1-p_j)^n\right) \to 0$ as $n \to \infty$

Proper linearization of ODEs of the form $\dot{x}(t) + f(x(t)) + \sigma(t) = 0$?

Showing $\frac{f(a+2b)+f(a-2b)}{2f(a)}\leq\sqrt{1-\left(\frac{b}{a(1-a)}\right)^2}$ for $f(x)=\sqrt{x(1-x)}$ with some constraints

A tough definite integral using contour integration

Geometric implication of the Sobolev embedding

Bounding Euclidean norm by slanted 1-norm

Parseval-Plancherel type identity for probability generating function

Poincare inequality for Poisson random variables

Optimal Poincare constant (with constraints)

exact meaning of uniform integrability for empirical distributions

Understanding the fast sweeping algorithm for Eikonal equations

Meaning of $K'[\cdot]$ when $K$ is an symmetric Onsager (matrix) operator

Non-uniqueness of a Dirichlet's problem

A basic question on martingales and filtrations

A discrete Poincare inequality

Show that $K(t) = \int_0^t \mathrm{e}^{-Cs} \, D \, \mathrm{e}^{-C^\intercal s} \,\mathrm{d} s$ satisfies a given equation involving $K(\infty)$

Generalized second derivative of a concave and piecewise $C^2$ function

Lipschitz continuity of $\sqrt{f}$ for $f(x) = \sup_{\alpha \in T} \sum_{i=1}^d \left(\sum_{j=1}^D \alpha_j x_{ij}\right)^2$

Deriving inequalities from other inequalities

Show $\mathbb{E}\left[\left|\sum_{i=1}^n X_iY_i\right|\right] \leq 2\mathbb{E}\left[\left|\sum_{i=1}^n Y_i\right|\right]$

Random time change from Oksendal's SDE textbook

Wasserstein distance between two empirical measures

Show that $\int_{-\infty}^{a_1} (a_1-x)^r f(x) \mathrm{d} x$ and $\int_{a_n}^\infty (x - a_n)^r f(x) \mathrm{d} x$ are of order $\mathcal{O}(n^{-r})$

Show that $x^2 + 3\sin^2 x$ satisfies the Polyak–Łojasiewicz condition

Gronwall inequality for a $2$-dimensional system of linear differential inequalities