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Let $ X_{1},…,X_{d} \in \{-1,1\}^d$ be random variables, with $ E[X_j]=\mu_j$ . Having $ n$ i.i.d. samples $ x^{(i)}_1,x^{(i)}_2,….,x^{(i)}_d$ , $ i=1,…,n $ , let $ \hat{\mu}_{j}=\frac{1}{n}\sum^{n}_{i=1}x^{(i)}_j$ Then we would like to find an upper bound for $ \text{Pr}[|\prod^{d}_{i=1}\hat{\mu}_{j}-\prod^{d}_{i=1}\mu_{j}|\geq \epsilon]$ where $ \epsilon>0$ . We have $ \prod^{d}_{j=1}\hat{\mu}_{j}=\prod^{d}_{j=1}(\frac{1}{n}\sum^{n}_{i=1}x^{(i)}_j)=\frac{1}{n^d}\prod^{d}_{j=1}\sum^{n}_{i_j=1}x^{(i_j)}_j=\frac{1}{n^d}\sum^{n}_{i_1,…,i_d=1}\prod^{d}_{j=1}x^{(i_j)}_j$ . The random variables $ \prod^{d}_{j=1}x^{(i_j)}_j$ areRead more

Let $ 0<a<1,\; \psi_a(x)=\displaystyle \prod_{j=0}^\infty (1-a^jx).$ For each $ k\in \mathbb{N},$ set $ $ f_k(a;x):=\frac{x^k}{(1-a)(1-a^2)\dots (1-a^k)}\,\psi_a(x).$ $ Question. Is it true that, for mutually prime $ j$ and $ m$ , where $ 1<j<m,$ the following inequality holds: $ $ f_{mk}(a^j;x)\leq f_{jk}(a^m;x) \;\;\mathrm{for\;\;all}\;\;k\in \mathbb{N}\;\;\mathrm{and\;\;all}\;\;x\in [0,1]?$ $ Remark. When $ j=1, m=2,$ the answer in providedRead more

Let $ A, B$ be Hermitian matrices. Is it true $ $ \det(A^{2}+B^{2}+|AB+BA|)\leq \det(A^{2}+B^{2}+|AB|+|BA|)?$ $ As usual, $ |X|=(X^*X)^{1/2}$ . Clearly, quantities on both sides are no less than $ \det(A+B)^2$ .Read more

Let $ \Omega$ be a $ C^1$ bounded domain in $ \mathbb{R}^3$ . Suppose that $ u:\Omega \rightarrow \mathbb{R}$ satisfies $ -\Delta u \leq |u|^\gamma $ on $ \Omega$ , $ u=0$ on $ \partial \Omega$ . Can we obtain that all the functions $ u$ enjoy some uniform bound from above? I’m thinking aboutRead more