All Questions

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Why isn't the identity $\sqrt{ab}$ = $\sqrt{a}$.$\sqrt{b}$ always true?

If we take $a=b=1$ the the L.H.S. is $1$ but the R.H.S. is $-1$. Is this identity not applicable for complex numbers? How to prove this and prove that this is not applicable for some complex ...
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Galois group of $X^{5}-2X+7$ over $\mathbb{Q}$

Is there any way to determine the Galois group of $X^{5}-2X+7$ over $\mathbb{Q}$ not using the discriminant? Thanks!
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Laurent series of $\psi(-2z)$ at positive integers and positive half integers.

The laurent series of $\psi(-z)$ at $z=n$ is $$\psi(-z)=\frac{1}{z-n}+H_n-\gamma-(H_n^{(2)}+\zeta(2))(z-n)+\cdots$$ However, may I ask how does one determine the laurent series expansion for ...
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A tricky complex numbers if and only if proof

For complex numbers $z$ and $w$ prove that $$|z|^2w -|w|^2z = z-w\quad \iff\quad z=w\quad\text{or}\quad z\bar{w}=1.$$ How would you go about solving this problem?
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How to find max and min bounds of a uncertain function

First I would like to say that I have searched the for uncertain fitting, robust fitting, linear optimization, convex optimization, etc. But I'm lacking the knowledge to solve this problem, and I need ...
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Integral Dependence relations for $x+y$ and $xy$

It is a question from Miles Reid's Undergraduate Commutative Algebra: Let $B$ be a $k$ -algebra and let $y,z \in B$ and both $y, z$ satisfies the following integral dependence relations over $k$ ...
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How to Prove an Algorithm is $O(n \log n)$ Using Substitution Method and a Substitution?

Consider the following: Show that $T(n) = 2T(\left \lfloor n/2 \right \rfloor + 17) + n$ is $O(n\log n)$. Here is what I have come up with: Restrict the domain of $n$ to $n \geq 1$ and define ...
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Re-arranging formula

I was asked to re-arrange the formula in terms of V for: NI = (SP-VC)(V)-FC These are the steps I have done and it was wrong. Can anyone explain (1) NI/(SP-VC)=V-FC (1) FC+[NI/(SP-VC)]
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Inverse of a Rotation matrix

If $R$ is a rotation matrix (determinant 1,orthonormal) can we say that $R^{-1}$ is also a rotation matrix? If yes how do we prove it?
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Why do we interpolate - no guarantee of success

this is somewhat of a general question about interpolation, I don't fully understand how can we be confident that our approximation is good, even if we know a lot of points. An example would be: ...
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Way to Show that a limit does not exist

how can I show that the limit of the function: $\displaystyle f(x)=2x\sin(1/x)-\cos(1/x)$ does not exist as $xto0$?
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Binomial Theorem on a Matrix

Does the expression follow binomial theorem? (A + I) ^n where A is matrix ,I is identity matrix. I know the binomial theorem but do not know whether it is applicable to matrices also.
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Is this statement meaningful if one of the elements is undefined?

Am I allowed to say a statement like $\max\left\lbrace a,b\right\rbrace$ if it turns out that the element $b$ is undefined, or simply does not exist? Would the result be $a$, or is the whole ...
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Alternative proof: Matrix $A$ is similar to $B$ iff $\lambda I - A$ is equivalent to $\lambda I - B$

We have this theorem for square matrices: If $\lambda I - A$ is equivalent to $\lambda I - B$, then $A$ is similar to $B$. ($A$, $B$ are matrices in $K^{n\times n}$, $K$ is a number field, ...