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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 The notation "${}^n C_r$" is very outdated; in general, prefer writing $\binom{n}{r}$.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?
Mar
17
comment Program with no intermediary states
Oh, wait. You may need higher-order functions; see the answer.
Mar
17
comment Program with no intermediary states
As long as you can give effective semantics to your model/programming language, yes. Does this answer your question?
Mar
17
comment Program with no intermediary states
To be more concrete, suppose you have big-step semantics for an imperative programming language without function calls. Then for each statement, we get a (computable) function mapping a variable context to a variable context. Chain all these functions, and you have a representation of your program as a sequence of function applications.
Mar
17
comment Program with no intermediary states
What is your computation model? If you allow any model and provide denotational semantics, the answer is "trivially, yes": Just take the function induced by the denotational semantics. Edit: This also works with other formal semantics, e.g., small-step operational semantics or natural semantics.