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Let $H_i$ be a subgroup of $G_i$ for $i=1,2,\dots,n.$ Prove that $H_1×\dots × H_n$ is a subgroup of $G_1 ×\dots × G_n.$

First I suuuuuck at proofs. I think I am on the right track but I need some fine tuning. Or if I am totally off let me know. First we show that $H_i$ is nonempty. Note that since $H_i$ is a subgroup of $G_i, H_i$ contains the identity element. So $e_G \in H_1, e_G \in […]

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Quartic function in four variables

$$\begin{array}{ll} \text{minimize} & x_1^2+ y_1^4+x_2^4+y_2^2+ 8x_1x_2+8y_1y_2\\ \text{subject to} & x_1+y_1=1\\ & x_2+y_2=1\end{array}$$ Is the function convex/ strictly convex? For critical points: I did $f_{x_1}= 2x_1+8x_2=0$; $f_{x_2}=4x_2^3+8x_1=0$; $f_{y_1}=4y_1^3+8y_2=0$; $f_{y_2}=2y_2+8y_1=0$ This gives $x_2=0, 2\sqrt{2}, x_1=0, -8\sqrt{2}$, same will be for $y_1, y_2$, could anyone tell me what next? Lagrange multiplier corresponding to the problem: $L(x_1,y_1,x_2,y_2, \lambda, \mu)= […]

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Let $Hi$ be a subgroup of $G$i for $i = 1, 2, . . . , n.$ Prove that $H1 × H2 × ⋯ × Hn$ is a subgroup of $G1 × G2 × ⋯ × Gn.$

First I suuuuuck at proofs. I think I am on the right track but I need some fine tuning. Or if I am totally off let me know. First we show that $H_i$ is nonempty. Note that since $H_i$ is a subgroup of $G_i, H_i$ contains the identity element. So $e_G \in H_1, e_G \in […]

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Vector of multivariate normal distribution

Let $\textbf{X} = (X_1, X_2, X_3)^T$ and $\textbf{Y} = (Y_1, Y_2, Y_3)^T$ be independent vectors with multivariate normal distribution, with means $\mu_X$ and $\mu_Y$ and covariance matrices $\Sigma_X$ and $\Sigma_Y$ with non-zero determinant. Let $A_{2 \times 3}$ and $B_{3 \times 3}$ be lineary independent matrices. Find distribution of $(\textbf{X}^TA^T, \textbf{Y}B^T)^T$. This is what I’ve done […]

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What is the distance from $G$ to centre of gravity set $S$?

Let: $$G=\left\{x\in \mathbb R^n: x=(x_1,…,x_{n-1},0) \right\},$$ $S$ – cone based on the set $A\subset G$ which is bounded set, $$A’=\left\{x’\in \mathbb R^{n-1}: x’=(x_1,…,x_{n-1}) \right\}, \lambda_{n-1}(A’)<\infty$$Moreover assume that the tip of the cone $S$ lies at a distance $g$ from $G$. What is the distance from $G$ to centre of gravity set $S$? If $A\subset \mathbb […]

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A question of conditional probability

I encountered a question about conditional probability in statistics. I want to figure out whether the equation below is true: $$P(x|y_1,y_2,y_3)=P\left[ (x|y_1,y_2) | y_3 \right],$$ where $y_1,y_2,y_3$ are some conditions and $x$ is an event that we concern. That’s meaning that if I could treat $(x|y_1,y_2)$ as a whole event?

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Representing math in a pesudocode algorithm

I want to write the pseudocode of my algorithm in a better abstract way but I want to simplify those math statements in a better elegant abstract way. $\mathcal{N}’ = \{n \in \mathcal{N} \mid \sum^M_{m = 1} x_m\geq k \; \text{where}\; x_m = 1 \; \text{if} \; a_{nm} = 2 \; \text{and} \; x_m = […]

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Show equivalence of two metric

Show that metrics d and $d_{2}$ are equivalent on $\mathbb{R}_{m}$, where $\mathbb{R}_{m}$ is the set of all bounded sequences of real numbers , where: $x=\left [ x_{1}, x_{2},….,x_{m},\right ]$, $y=\left [ y_{1}, y_{2},….,y_{m},\right ] \in \mathbb{R}_{m}$ $d(x,y)=sup\left \{ \left | x_{n}-y_{n} \right | \right \}$ $d_{2}(x,y)=\left\{\begin{matrix} 0, if x=y\\ \frac{1}{z}, if x\neq y > \end{matrix}\right.$, […]

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An infinite matrix define a bounded operator on $ \ell^2$ [closed]

I want to prove this: Let $(a_{ij})_{i,j=1}^{\infty}$ be an infinite matrix such that $\sum_{j=1}^{\infty }\sum_{i=1}^{\infty} \left | a_{ij} \right |<\infty $ and $A: \ell^2 \rightarrow \ell^2$ is a linear transformation such that $A(x)=y \Leftrightarrow (a_{ij})(x_1,x_2,\cdots )=(y_1,y_2,…)$ whit $y_{i}=\sum_{i=1}^{\infty}a_{ij}x_j$ show that A is a bounded operator in $ \ell^2$ and $ \left \| A \right \|^2 […]

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Handling Uncertainty with Budget Constraints in Robust Optimization

Let’s say I have a linear optimization problem with some uncertain parameters Case 1: Uncertain Parameters are present independently in different constraints $$maximize\ x_1+x_2$$ $$ax_1+bx_2\leq 6 $$ $$x_1,x_2\geq 0$$ $$\bar{a}=3, \bar{b}=5$$ where a & b are uncertain parameter $a\in [1.5,4.5 ]$($\pm 1.5$ from mean) and $b\in [3,7 ]$ ($\pm 2$ from mean). I solved this […]