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### A proof of divisor of Polynomial by Polynomial's GCD with its derivative have the same roots as itself

Problem If $Q(x)$ is the greatest common divisor of the polynomials $P(x)$ and $P'(x)$, where $P'(x)$ is the derivative of $P(x)$, then the polynomial $\frac{P(x)}{Q(x)}$ has the roots of $P(x)$ as ...
• 370
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### Jacobian of the row de-meaning of a matrix X, with respect to matrix X

Let $\mathbf{X}$ $\in \mathbb{R}^{M \times N}$ be a long matrix, with $M<N$. Let $\mathbf{Y}$ $\in \mathbb{R}^{M \times N}$ represent $\mathbf{X}$ after each row of $\mathbf{X}$ has been "de-...
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### Show that $\sum_{k=1}^n{2^{2k-1}\binom{2n+1}{2k}B_{2k}(0)}=n$

Lately, I've been working on a proof (whose context is not necessary to discuss) and I only need one last thing in order to finish it. To be more specific, for completeness it would suffice to show ...
1 vote
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### Conservative idempotent magma - proof attempt

I need help with checking proof about idempotent and conservative magmas. Let magma be any ordered pair $(M, \odot)$, where $M$ is nonempty set and $\odot$ binary operation on $M$. Now I need to ...
1 vote
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### Asymptotic growth of the Fibonacci primes

Define $\kappa(n)$ to be the number of prime Fibonacci numbers less than $F_n$. Is it the case that $\kappa(n) \sim a\log(n)$ for some constant $a$? I know that $F_n$ prime $\implies n$ prime (except ...
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1 vote
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### Trouble Understanding Difference in Epsilon-Delta Arguments: Why One Works But The Other Fails (Spivak Calculus Problem 5-10c)

In the problems for the limits chapter (5) of Spivak's Calculus, we are asked to prove: $\lim\limits_{x \to 0} f(x) = \lim\limits_{x \to 0} f(x^3)$. The relevant to the question proof alternative is: ...
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### Spivak's proof of the parallelogram law of forces

In Physics for Mathematicians, Spivak offers a "proof" of the parallelogram law for forces, i.e. the rule that forces combine according to the rule of standard vector addition. I say "...
• 28k
1 vote
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### Orthocenter: The "Bad Boy" of Distinguished Points in a Triangle

It is a well-known fact that the altitudes of a triangle $ABC$ (with vertices $A,B,C)$ intersect at exactly one point, the orthocenter. The proof known to me (see eg here) involves the construction of ...
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### Simplifying Sentences that Precede a Conditional

Suppose we have the following proposition: Suppose that $x \in \mathbb Z$, that $a$ is even, and that $b$ is odd. If $x^2-ax+b$ is even, then $x$ is odd. I am not interested in proving the proposition;...
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