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The equality condition of $||A||_p \leq \left[ \sum_{j=1}^m || A_{j\cdot}^T ||_q^p \right]^{1/p}$

I want to show that the inequality below gets the equality $$||A||_p \leq \left[ \sum_{j=1}^m || A_{j\cdot}^T ||_q^p \right]^{1/p}$$ when $A$ is a rank 1 matrix. Here $A\in\mathcal M(m,n)$ and $p, q$...
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How can I prove that $10=2^{a}*3^{b}*7^{c}$ has infinite solutions?

Both in a unrescrited case and with the following restriction: $a+b+c=1$
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gradient of a transformation that uses an orthogonal matrix

If I have Y=QX, where Y and X are vectors of dimension n and belong to $R^{n}$, and Q is an orthogonal matrix. Then, why do we have $\nabla_{Y}f=Q\nabla_{X}f$? I know that orthogonal matrices have ...
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Prove that $C^m_{n+k+1}=$ $\sum_{i=0}^{m}$ $C^{i}_{k+i}$ $\times$ $C^{m-i}_{n-i}$ [duplicate]

Prove that $C^m_{n+k+1}=$ $\sum_{i=0}^{m}$ $C^{i}_{k+i}$ $\times$ $C^{m-i}_{n-i}$. I tried using double counting and Newton binom. Any idea? I don't now many identities......
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Inverse Laplace Transform - Pulling out the constant

If you refer to my picture: https://i.stack.imgur.com/lVsU1.png I'm having a hard time understanding why in the 2nd step the fraction is split up in two terms when 2 is a constant. I get why you ...
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How reliable is the linear problems like $\min \|Ax - b\|^2$?

The following linear optimization is common used $$\min_x \|Ax - b\|^2$$ Here, $A$ is the matrix; $x,b$ are the vectors. I am curious about how reliable is the solution $x$ by solving above ...
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Sobolev-Gagliardo-Nirenberg: Why is $|f|^q$ continously differentiable?

I wanna understand a proof of the Sobolev-Gagliardo-Nirenberg inequality. Therefore, I need to know why $|f|^q \in C_c^1(\mathbb{R}^n)$ for $f \in C_c^1(\mathbb{R}^n)$ and $q>1$. Can eventually ...
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What is $m_n$, when $(m_1,m_2,m_3) = (1,2,3)$ and $m_n =\sum_{i=1}^{3}(4-i)m_{n-i}$?

If $m_1, m_2, m_3$ are given then what will be the nth term of the series if $$m_{n}=\sum_{i=1}^{3}(4-i)m_{n-i}$$ ex if $m_1 ,m_2, m_3$ are $1, 2 ,3$ then $m_4 = 14$ $m_5= 15$ So what will be the ...
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How to prove that the derivative of a homogeneous equation is $y'=\frac{y}{x}$

I was solving a differential equation problem which required me to find out the derivative of. For the curve $x^2y^3=(2x+3y)^5$, $\frac{dy}{dx}=\frac{-y}{g(x)}$ The author simply stated that ...
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Showing $\langle x,y\mid x^2, y^3, xyxy^{-1}, (xy)^7\rangle$ is trivial.

I encountered this problem in Sims' "Computation with Finitely Presented Groups". Show that $\langle x,y\mid x^2, y^3, xyxy^{-1}, (xy)^7\rangle$ is trivial. The book uses coset enumeration or ...
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10th Grade Algebraic Rate of Bacteria Growth Problem

"A certain type of bacteria doubles every 6.5 hours. If there were 60 bacteria to start with, what is the hourly growth rate of the bacteria? How many bacteria will there be after a day and a half? ...
I am struggling to "visualize" the following equation. I would like to know how the following equation can be written in system-of-equation form in order to see that there are $K$ unknowns. The ...