Ryan
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Here are some different cases I can think of: $A=B$. Either $A=cI$ or $B=cI$, as already stated by Paul. $A$ and $B$ are both diagonal matrices. There exists an invertible matrix $P$ such that $P^{-1}... View answer Accepted answer 7 votes We could say that there are two desirable goals for a base system: A minimized number of symbols (e.g. there are two symbols in binary, 0 and 1). Minimal digital lengths for each number (e.g. the ... View answer 5 votes As others have already mentioned, it doesn't make much sense to "alter" mathematics. You can however invent "new" mathematics, by starting from a different set of axioms. If what you are looking for ... View answer 4 votes The solution to$e^a=a^3$is given by the Lambert W function:$a=-3 \text{W} \left( - \frac{1}{3} \right)$Now, take a look at the summation definition of the Lambert W function. If you change your ... View answer 4 votes If there is a rational solution for$x$, we know that its numerator must divide$88$evenly. This leaves us with$1$,$2$,$4$,$8$,$11$,$22$,$44$, and$88$as possible choices of numerator for$x$... View answer 3 votes Consider any rational number$2^x 3^y 5^{-z} 7^{-w}$, where$x$,$y$,$z$,$w \in \mathbb{Z}^+$. Størmer's theorem guarantees that there are a finite number of such fractions where the numerator and ... View answer Accepted answer 2 votes Questions of "intuition" can be fairly subjective, so here is an example of my thought process when I try to visualize$x^2-y^2=\text{constant}$. First, rewrite it as$y^2=x^2-\text{constant}$. ... View answer 2 votes We can simplify the problem by substituting$x=y-\frac{1}{2}$. $$x^6 = \left(x+1\right)^6$$ $$\left(y-\frac{1}{2}\right)^6 = \left(y+\frac{1}{2}\right)^6$$ If we expand both sides and then collect ... View answer Accepted answer 2 votes As already stated by Darth Geek,$1$,$2$, and$5$are the proper divisors of$10$. This means that all sufficiently large powers of$10$are achievable by using a finite number of any chosen coin/... View answer 2 votes This is a result of a theorem due to Carl Størmer. His theorem can be stated as follows: Let$S=\left\{ p_1^{e_1} p_2^{e_2} \cdots p_j^{e_i} \mid e_i \in \mathbb{N}_0 \right\}$, where$p_1$,$p_2 \...

$\dfrac{152 \mathrm{e}}{180n^4}<0.0001$ $152 \mathrm{e} < 0.018n^4$ $\dfrac{152 \mathrm{e} }{0.018} < n^4$ $n > \sqrt[4]{22954.38}$ $n > 12.31$ This still doesn't explain why the ...

Let $c=d=e$, and let $f=g$. This gives us the following system of equations: $2 \left( a^2-c^2\right)=b^2$ $a^2-c^2=2f^2$ If you further make the substitution $c=a-2$ and $b=2f$, we can combine ...

Consider the following sequence: $a(n) = \displaystyle \frac{1}{2} \sum \limits_{d \mid n} \mu(d) 2^{n/d}$ It is known that $a(n)$ is divisible by $3$ for all $n \ge 3$. See A000740 for more ...

Hint: $\lceil A \rceil \ge A \ge \lfloor A \rfloor$ $\lceil A \rceil - \lfloor A \rfloor = 0 \text{ or }1$ Another hint: Divide both sides by $N$. $Nx - \lfloor Nx \rfloor \le c$ $\dfrac{Nx}{N} - \... View answer Accepted answer 1 votes A bit of division does the trick. I assume this is what you are looking for?$n-i=n \left(1-\dfrac{i}{n} \right)n-i=n \left( \dfrac{a}{b} \right)n-i=\dfrac{n}{\left( \frac{b}{a} \right)}$... View answer Accepted answer 1 votes Let's rewrite your sum like this:$\dfrac{a+ b\sum_{i=1}^ni!}{n!}=\dfrac{a+ b \left( 1!+2!+ \cdots + \left( n-2\right)! + \left( n-1\right)! +n! \right)}{n!}=\dfrac{a}{n!}+b \left( \dfrac{1!}{n!} ...

Consider the fact that any positive rational number $q$ can be rewritten as $2^{e_1}3^{e_2}5^{e_3}7^{e_4}… p_n^{e_n}$. Then, taking the exponents $e_i$ of each prime, we can construct a vector as ...

This form might make it easier to see what is happening: $\dfrac{x^3+y^3}{x^2+y^2}=\dfrac{x}{1+\frac{y^2}{x^2}}+\dfrac{y}{1+\frac{x^2}{y^2}}$

Notice how $x \left( t \right)$ is composed of a combination of $\sin{t}$, $\sin{2t}$ and $\sin{3t}$ terms. The period of $x \left( t \right)$ is simply the least common multiple (lcm) of each of the ...
Suppose $b=a$ and $c=3a$. The inequality reduces to the following: $\frac{a}{4a}+\frac{a}{4a}+\frac{3a}{2a}+\frac{a^2}{2a^2}+\frac{3a^2}{10a^2}+\frac{3a^2}{10a^2} \leq 3$ $\frac{1}{4}+\frac{1}{4}+\... View answer 0 votes This notation has always confused me before, because computer science professors always seem to teach the concept incorrectly. No doubt textbooks teach it wrong too. If a function$f(x)$satisfies ... View answer 0 votes Octave-equivalence is a learned phenomena. There is no mathematical reasoning for why we should hear a frequency ratio of 2:1 as more "equivalent" than 3:1 or 5:1, for instance. Check out the Bohlen-... View answer -2 votes I don't think this question has a well defined answer. As far as I know, all proofs of the uniqueness of the Laplace transform are essentially corollaries of this one statement:$\displaystyle \int_{...