Keinstein
  • Member for 10 years, 4 months
  • Last seen more than a month ago
The mathematics of music - why sine waves?
6 votes

Most of the answer has already been given with the reference to the auditory system. It dissolves the sound into a certain base functions. I don't know of a good description of their behaviour, but ...

View answer
understanding lattice in detailed
Accepted answer
4 votes

In general I would see the two main classes, that can are distinguished in German language. The term Verband refers to an algebra (in the sense of universal algbera) with a join and a meet operation. ...

View answer
The preorder of countable order types
Accepted answer
3 votes

Warning: This anwer should be treated with caution. For further information see the comments. As a partial anwer consider the set $\mathcal L$ of order types of linear orders. Then $\bar{\mathcal L}$ ...

View answer
Notation for "incommensurate" elements?
3 votes

If you don't want to use $\parallel$ you might try $\not\lessgtr$ (\not\lessgtr) or something that looks better, maybe $\,|\!\!\!\!\!\lessgtr$. This would complement the usage of $\lessgtr$ for ...

View answer
Order-Dense Set Definition
2 votes

Actually you should differentiate between a dense order relation and a dense subset of an ordered set. The order relation is dense when (3) holds. A $Y$ is dense in an ordered set $(X,≾)$ iff a ...

View answer
what kind of relationship is "is prefix of"?
Accepted answer
2 votes

The first thing you should keep in mind is that there is a difference between the Hasse diagram or the corresponding neighbourhood relation and the order itself. The next problematic term is the word “...

View answer
Proving well ordering is total relation
Accepted answer
1 votes

For the sake of contradiction, take an antichain $C \subseteq A$ in the well-ordered set $(A, \leq)$, that is a set of pairwise non-comparable elements ($\forall x,y \in C, x \not \leq y \text{ and } ...

View answer
How to easily create a polynomial function that gives a desired output?
1 votes

As Gabriel Romon commented: Lagrange interpolation does the trick. The polynomials form a vector space (which works very similar to the 3D coordinate space) where the coefficients work as ...

View answer
Introductions to posets on algerbaic structures (Everything I need to know about them)
1 votes

Just as you didn't mention groups: There has been done much work on po-groups as well. As far as I know, some people are working on quasiorders (preorders) on algeraic structures. Clone theory tells ...

View answer
total order on finite dimensional vector space over $\mathbb{R}$
1 votes

In case of an infinite dimensional vector space there exists a basis that represents the order. Nevertheless the separating “Hyperplane” may be the whole vector space itself. For example take $\...

View answer
How to show that a lattice is isomorphic to an other lattice?
1 votes

There are several methods to do that. The main question is how the lattices are given and which properties they have. In case the lattice is doubly founded, it is sufficient to consider all bijective ...

View answer
What relation does music have to math?
1 votes

Additionally to he two answers given so far: I think the main link between maths and music is hidden behind the scenes: Rules: Composition as well as Improvisation heavily relies on a reduced ...

View answer
Questions about relations on directed, and possibly cyclical graphs.
0 votes

Let $\equiv = \leq \cap \geq$ the equivalence relation of the preorder. Then, ${(P,\leq)_/}_\equiv$ is an ordered set. Let $C$ be a maximal chain in ${(P,\leq)_/}_\equiv$. And $c:C\to P$ a choice ...

View answer
Distributive lattice and embedding
Accepted answer
0 votes

The main idea is: Distributivity can be (theoretically) destroyed by a single counter example. Embedding means injectivity: if $x\neq y$ then $e(x)\neq e(y)$. So if an equation has a counterexample in ...

View answer
Quantify amount of oscillations in a time-series
0 votes

I interpret „being shaky“ as: The differences between nearby samples are constantly swapping their sign. This leads me to the suggestion: Model it using ARMA (autoregressive moving averages) models. ...

View answer
Describe a group $G$ that acts on a set $X$ of 4 elements such that the action of $G$ has 2 orbits.
0 votes

As you already stated there are orbit partitions into 2 2-element sets and into a singleton and a 3-element set. Furthermore a group action can be considered a group homomorphism of on group into ...

View answer
Join-irreducible elements in an infinite lattice.
0 votes

As Matt Samuel stated, $ℤ\cup\{-∞,∞\}$ is a lattice that consists only of irreducible elements. It is even a complete lattice with this property. On the other hand in the set of extended reals ...

View answer
Formal Definition of a Lattice
0 votes

As already said in the comments: every pair means every unordered pair, that can be formed of elements of the given base set, in our case the lattice or ordered set. An upper bound of a set $S$ is an ...

View answer
Weak and strong orders
0 votes

Well, I assume, you mean $(X,Y) ∈ R$ iff $X⊂Y$ or $X=Y$. Let us further assume that $⊆$ is a partial order relation, while $⊂$ is irreflexive ($∀X:X\not⊂X$). Then $⊂$ is called “strict (partial) order“...

View answer
Defining principal elements of every poset. Is this a new idea?
0 votes

To avoid confusion, you should provide a definition of a filter. I'm using the more general definition of an order filter that is sometimes called an upset. But your suggestion looks not very ...

View answer
Number of join-irreducible elements of a lattice: is it monotonic?
0 votes

Another proof is based on the fact that a join irreducible element has a unique lower neighbour. Let $A⊆X$ and $B⊆X$ with $A≠B$ be two irreducibles in $\mathcal L$. Let further for any element $A∈\...

View answer
Definition of finite direct decomposition of elements and indecomposable elements at arbitrary lattice
Accepted answer
0 votes

I'm not shure, if you are right. But if I'm right, you are looking for reducible and irreducible elements both exist in weaker versions: supremum irreducible elements and infimum irreducible elements. ...

View answer
Showing that a group is lattice-ordered
-1 votes

If it is a partially ordered group then it is also a lattice ordered group. The usual way is to prove $$a≤b⇒xay≤xby.$$ Edit: Your task is to prove that it is a partially ordered group. Your ...

View answer