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V.S.e.H.
  • Member for 7 years, 1 month
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7 votes

Prove that $\sum \frac{a^3}{a^2+b^2}\le \frac12 \sum \frac{b^2}{a}$

6 votes

Prove $\sqrt{x_1^2+4x_1x_2+5x_2^2+2x_1+6x_2+5}$ is convex

5 votes
Accepted

Prove that $\displaystyle\ln\left(1+\frac{1}{x} \right) < \frac{2x-1}{x^2-x}$ when $x > 1$

5 votes

Is the following generalization of Cauchy-Schwarz inequality true?

4 votes

How do I solve $\min_x \max(c_1^Tx, c_2^Tx, \dots, c_k^Tx)$ for $\lVert x \rVert_2 = 1$.

3 votes

Symmetric positive definitine matrix $I-X^TX$ => $X^TX<1$

3 votes
Accepted

For a given function $f: \mathbb{R}^n \rightarrow \mathbb{R}$, find the direction of least change

3 votes

Prove that $x_1+2x_2+3x_3+\cdots+nx_n \leq \frac{n(n-1)}{2}+x_1+x_2^2+x_3^3+\cdots+x_n^n$ where $x_i > 0$ for all $i$ from $1$ to $n$ inclusive.

3 votes
Accepted

Let $z_1, z_2, z_3 \in\mathbb C$ such that $|z_1|=|z_2|=|z_3|=1$. If $z_1+z_2+z_3\neq0$ and ${z_1}^2+{z_2}^2+{z_3}^2=0$. Then find $|z_1+z_2+z_3|$.

3 votes
Accepted

Prove: if $\bf{AB^T}$ is skew-symmetric and $\bf A$ full-rank, then $\bf{AX}=\bf B$ has unique solution $\bf X$

3 votes
Accepted

If $a$, $b$, $c$, $d>0$, prove that $\sum\limits_{\rm cyc}\sqrt[3]{\frac{abc}{(b+c+d)(c+d+a)(d+a+b)}}\le\frac43$

2 votes

Show that if $\sum\limits_{k=1}^nx_k=\sum\limits_{k=1}^n{x_k}^2=\dots=\sum\limits_{k=1}^n{x_k}^n$ where $x_k\in\mathbb{R}$ then $x_1x_2\dots x_n\le1$.

2 votes
Accepted

Prove that $e^{x_1+x_2+\dots+x_n}\geq\frac{1}{2024}\sqrt[2024]{(x_1+1)(x_2+1)\dots(x_n+1)}\big(2023x_1+2023x_2+\dots+2023x_n+2024\big)$

2 votes
Accepted

Prove that $J=JH=HJ$

2 votes

Show that $(A+B)^{-1}-A^{-1}=-(I+A^{-1}B)^{-1}A^{-1}BA^{-1}$

2 votes
Accepted

Finding the right function to use in Jensen's Inequality.

2 votes
Accepted

Solution verification: Let $A$ be a matrix; $A^TA=AA^T$. If $x$ is an eigenvector of $A$ with eigenvalue $\lambda$, then $A^Tx=\lambda x$

2 votes
Accepted

Why $\|(A + cI)^{-1}x\|\leq \frac{\|x\|}{\lambda_{\min}(A)}$

2 votes
Accepted

Reflect a point without matrix math

2 votes

Show that a matrix has eigenvalues at zero

2 votes

Fenchel-Young inequality for matrices

2 votes
Accepted

Randomly generate Hurwitz matrices?

2 votes

Why does SVD solve $\underset{U,V}{\min}\| A - UV^T\|_F^2$

2 votes

$\sqrt{a^2+5b^2}+\sqrt{b^2+5c^2}+\sqrt{c^2+5a^2}\geq\sqrt{10(a^2+b^2+c^2)+8(ab+ac+bc)}$ for any real numbers.

2 votes

Complicated Inequality Proof, Variables Subject To Constraint

2 votes

“More rigorous” way to minimize $\sqrt{1^2+a_1^2}+\sqrt{2^2+a_2^2}+\cdots+\sqrt{8^2+a_8^2}$, subject to $a_1+a_2+a_3+\cdots+a_8=27$, for $a_n>0$

2 votes
Accepted

Gradient descent minimum step size

1 vote
Accepted

Problem about maximize profit

1 vote
Accepted

$a\sqrt{b+c}+b\sqrt{a+c}+c\sqrt{a+b}\leq \sqrt\frac{2}{3}.$

1 vote

$a^2b^2+b^2c^2+c^2a^2+2abc\left(a+b+c\right) \geq 3\left(a^3b+b^3c+c^3a\right)$