Johannes Kloos
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 Mar 17 comment Minimizing and maximizing a term by placement of parantheses Mar 17 comment Minimizing and maximizing a term by placement of parantheses If I understand you correctly, $a_1 * \cdots * a_n$ has two possible interpretations: $a_1 \cdots a_n$ (* is multiplication), and $a_1 / \ldots / a_n$ (* is division, with parentheses yet to be places). What have you tried so far, and how did this problem come up? Mar 17 comment Minimizing and maximizing a term by placement of parantheses In the latter case, note that parenthesis placement makes no difference if $*$ is multiplication (why?). Mar 17 comment Minimizing and maximizing a term by placement of parantheses Your question is not quite clear: What does "can be multiplication OR division" mean here? Can any single * be one of the both, or are all the * occuring in the term always the same? In other words, could $a * b * c$ be $(a \cdot b) / c$? Jan 12 comment Help with proving that $\pi$ is irrational $\beta_n \ell$ is not - why? This statement definitely needs proof. Note in particular that since $\beta_n$ is defined as a series, not a sum the "many primes" argument does not work here. Dec 22 comment Does there exist a neighborhood around a real number on the real line whose complement is finite? As for existence: The topological space being defined is given as the space where the closed sets are the finite sets and the whole space, or equivalently, the open sets are the co-finite sets and the empty set. This is a topology: - The empty set is finite, hence closed, and the whole space has finite complement. - If you intersect arbitrarily many finite sets, the result is still finite. Equivalently, taking the union of arbitrarily many co-finite sets gives you a co-finite set. - The union of two finite sets is finite, and hence, the intersection of two co-finite sets is co-finite. Dec 22 comment Combinatorial argument for an identity Hi, two things: 1. What have you tried so far? Have you noticed the connection to "$x_1 + \ldots + x_n \le n$"? 2. The notation "${}^n C_r$" is very outdated; in general, prefer writing $\binom{n}{r}$. Nov 14 comment What is $dx$ in integration? @Aerovistae: The $\Delta x$ describes the difference between consecutive values of $x$. Mar 31 comment Push Down Automata What have you tried so far? Also, do you know how to construct a PDA for $a^nb^n$? If you have one for $a^nb^n$ and one for $a^nc^n$, what can you do with that? Feb 22 comment When is matrix multiplication commutative? Here's an example why this condition is not neccesary: Take any non-diagonalizable matrix $A$. It will always commute with the unit matrix $1$, but $A$ and $1$ are clearly not simultaneously diagonalizable. Feb 12 comment When is matrix multiplication commutative? It's not neccesary, AFAICT. I can't think of a counter-example right know, though. Jan 22 comment $f(x) \in R[X]$ irreducible $\Rightarrow (f(x))$ ideal? Just to make sure: Do you mean by $(f)$ the ideal generated by $f$? Jan 11 comment Powers of $2\times 2$ matrices, such that $A^n = I$ Do you know about rotation matrices? Sep 13 comment What is a polynomially bounded function? Your argument would be correct if, say, $2^x-1$ was actually a polynomial. But by definition, a polynomial is of the form $a_n x^n + a^{n-1} x^{n-1} + \cdots + a_0$ for some parameters $a_0, \ldots, a_n$, and $2^x-1$ is not of this form. Jul 6 comment Help with function proof Indeed, my bad. Ignore my previous comment. Jul 6 comment Help with function proof @CameronWilliams: If you prefer, write $\mathbb{N}_0$. My natural numbers start at 0. Jul 6 comment Help with function proof To everybody trying to show a proof: Please make sure that what you want to prove is actually correct. In particular, consider $A = B = \mathbb{N}$, $f(x) = 0$, $Y = 2\mathbb{N}$ (the set of even numbers). Jul 6 comment Who named “Quotient groups”? Skimming the paper, Hölder uses the term "Faktorgruppe" (factor group), and refers to an expression "factor of composition" that he attributes to Jordan. May 31 comment How to extend an existing orthogonal set of vectors? Would guessing a linearly independent vector and then taking the orthogonal part work? You can make a vector orthogonal to a given set of vectors using methods similar to Gram-Schmidt... Mar 23 comment Why is this Goldbach's Conjecture Proof Wrong? As a first comment, I don't understand what your proof of Lemma 2 shows. In particular, how is "L cannot be a prime" a contradiction?