- #1
swampwiz
- 571
- 83
They all seem to mean the same thing. I personally have been using locus.
Have you looked these terms up? Your other thread, about rings and fields could also have been answered by just a bit of research on your part.swampwiz said:They all seem to mean the same thing. I personally have been using locus.
jedishrfu said:Where are you getting these questions from? Are you taking a course in Abstract Algebra or self studying math in general?
WWGD said:What kind of graph are you referring to? A collection of vertices, edges with an incidence relation or the graph of a function, etc?
Mark44 said:Have you looked these terms up? Your other thread, about rings and fields could also have been answered by just a bit of research on your part.
The difficulties with your questions are that they are not precise and confuse different things, so they would need an entire lecture to explain them, and that Wikipedia should already have solved much of them.swampwiz said:What I mean is the collection of points that correspond to some continuous function (whether it is known or unknown) in some domain space.
I think I know what you are talking about in terms of a discrete graph such as would be encountered in computer science; this is not what I am referring to here.
Rings: ##\mathbb{Z}\, , \,\mathbb{Z}_n\, , \,\mathbb{F}[x_1,\ldots,x_n]\, , \,\mathbb{M}_n(\mathbb{F})## and a lot more.swampwiz said:I did research it, but it just confused me since they seem to mean almost the same thing. And reading through "Galois theory" by Stewart, the discussion seems to go back & forth between fields & rings, which just totally confused me. I have not had any coursework in abstract algebra.
It sounds, but it isn't: "whose location satisfies or is determined by one or more specified conditions".swampwiz said:This sounds like the proper definition. It sounds like a manifold is some abstract form of a locus, which is deeper than the knowledge I am seeking.
fresh_42 said:Rings: ##\mathbb{Z}\, , \,\mathbb{Z}_n\, , \,\mathbb{F}[x_1,\ldots,x_n]\, , \,\mathbb{M}_n(\mathbb{F})## and a lot more.
Fields: ##\mathbb{Q}\, , \,\mathbb{R}\, , \,\mathbb{C}\, , \,\mathbb{Z}_p\, , \,\mathbb{F}(x_1,\ldots,x_n)## and many more.
Fields allow divisions, rings usually do not. Some rings (integral domains like ##\mathbb{Z}## or ##\mathbb{F}[x]##) allow an extension into fields (##\mathbb{Q}##, resp. ##\mathbb{F}(x)##), some do not, e.g. ##\mathbb{Z}_6##. Galois theory is based on finite (automorphism) groups and field extensions (e.g. ##\mathbb{Q}\subsetneq \mathbb{Q}[\sqrt{3}] =\mathbb{Q}(\sqrt{3})##).
WWGD said:A manifold can be seen as a collection of pieces of Euclidean space glued together smoothly.
I don't think that is a good picture because it sounds like a chain complex, not a manifold, at least not a differentiable one. A cube is a manifold in its most abstract sense, yes, but what you described sounds as if only such objects were manifolds, but they are not what is normally associated with the term.WWGD said:A manifold can be seen as a collection of pieces of Euclidean space glued together smoothly.
No. no, no. Firstly, the comparison is a very bad one, and secondly, read my posts: analytical, not topological. Although what @WWGD described was more a topological object rather than a manifold.swampwiz said:So a manifold is like a locus that mathematicians do topological stuff to.
fresh_42 said:No. no, no. Firstly, the comparison is a very bad one, and secondly, read my posts: analytical, not topological. Although what @WWGD described was more a topological object rather than a manifold.
No, they actually do have structure, otherwise they would just be a set.swampwiz said:OK, a locus is a just the most basic of manifolds - a bunch of points with nothing special going on.
No. Closed alone makes no sense, i.e. needs a specification by an adverb or by context. You used closed as closed under arithmetic operations, but this is trivially given by definition of a field. In Galois theory we consider algebraically closed fields, which are fields in which any polynomial ##p(x)=a_n+a_{n-1}x+\ldots + a_1x^{n-1}+x^n## can be written as ##p(x)=(x-\alpha_1)\cdot \ldots \cdot (x-\alpha_n)##.swampwiz said:So a field basically means a set of numbers that can have arithmetic operations done on them such that the results stay in the field; is this what is known as "closed field"?
These are conventions. A theorem is a major statement, a Lemma something which helps to prove theorems but often have a meaning on its own. Some Lemmata are called as such for historic reasons. A corollary is Latin and means a "gift", i.e. it is something which drops out of a theorem for free, or with only few additional arguments.BTW, what is the difference between a theorem, lemma & corollary? Is it that a theorem requires a major proof, while a lemma or corollary only requires a minor proof that the reader could figure out on his own?
Mark44 said:Have you looked these terms up? Your other thread, about rings and fields could also have been answered by just a bit of research on your part.
I provided definitions of rings and fields in the other thread. If your research had been more careful, you would have seen that these terms didn't mean the same thing.swampwiz said:I did research it, but it just confused me since they seem to mean almost the same thing.
A locus is just a collection of points determined by some condition. This is way too general. There is a very quick diminishing return of doing things so casually. There is a good reason why precise technical definitions are used.swampwiz said:OK, a locus is a just the most basic of manifolds - a bunch of points with nothing special going on.
fresh_42 said:No. Closed alone makes no sense, i.e. needs a specification by an adverb or by context. You used closed as closed under arithmetic operations, but this is trivially given by definition of a field. In Galois theory we consider algebraically closed fields, which are fields in which any polynomial ##p(x)=a_n+a_{n-1}x+\ldots + a_1x^{n-1}+x^n## can be written as ##p(x)=(x-\alpha_1)\cdot \ldots \cdot (x-\alpha_n)##.
fresh_42 said:
Sure. That was an illustration of the basic usage. I could as well have given you the correct definitions, butswampwiz said:A locus can be more than 1 dimension. Perhaps it is that a surface manifold has a 2-dimensional locus of points for the surface and 1-dimensional locus for the edge of the surface? This sounds like the idea of the surface integral used in Maxwell's Equations.
Mark44 said:I provided definitions of rings and fields in the other thread. If your research had been more careful, you would have seen that these terms didn't mean the same thing.
This is also half baked and needed precision. The notation ##G^{1/n}## should be avoided, it is misleading. Maybe you should first examine how we constructed the following tower:swampwiz said:OK, I think I understand this now. If I have an established field F and I add a new number that is a root G(1/n) as a field extension to F, because I can do any of the arithmetic operations, adding the new number G is equivalent to adding any number that would result from using that number G(1/n) as the argument for a polynomial having coefficients that are in F. This is so because the only extra numbers that are added are those that have terms that contain the original numbers in F and any power of G(1/n), or G(k/n), with k being in the range [0,n) since the power of a root repeats as per a DeMoivre root around a circle.
A graph is a visual representation of data that shows the relationship between two or more variables. It consists of points or vertices connected by lines or edges.
A locus is a set of points that satisfy a given condition or equation. It is often represented as a curve or surface in a graph.
A manifold is a mathematical concept that describes a space that locally resembles Euclidean space. It can be thought of as a higher-dimensional version of a curve or surface in a graph.
A graph is a visual representation of data, while a locus and manifold are mathematical concepts. A locus is a set of points that satisfy a given condition, while a manifold is a space that locally resembles Euclidean space.
Graphs are commonly used to display and analyze data in various scientific fields. Loci and manifolds are used in mathematics and physics to describe and study complex systems and phenomena.