For requesting, clarifying, and comparing definitions of mathematical terms.

learn more… | top users | synonyms

0
votes
1answer
22 views

Are these partial order or total orders?

I just want to know if the following are considered partially ordered sets or total set. Here is the definition: $\mathbb{R}^2$ : (a,b) R (c,d) iff $a\leqslant c$ and $b\leqslant d$ $\mathbb{R}^2$ ...
3
votes
0answers
19 views

Transitive relation vs. transitive action

A transitive relation $\circ$ is a relation with the property that: $$ (a\circ b \wedge b \circ c) \Rightarrow a \circ c.$$ A transitive group action is a group action $$\phi : G \times X \rightarrow ...
0
votes
1answer
364 views

The Definition of Consistency and Compactness in FOL

First order logic: "consistency," "compactness"? Consistency: A set $\Sigma\subseteq\text{WFF}$ is consistent iff there is no $\varphi\in\text{WFF}$ such that $\Sigma\vdash\varphi$ and ...
1
vote
1answer
67 views

How the branch cut make a multi-valued function several branches of single valued function?

In the wikipedia article, it describe the branch points and branch cuts: A branch cut is a curve in the complex plane such that it is possible to define a single analytic branch of a multi-valued ...
0
votes
0answers
11 views

What is the definition of ``2nd-order quasilinear parabolic'' ? for partial differential systems?

I have to know why the mean curvature flows are 2nd-order quasilinear parabolic. Let $\Omega\subset\mathbb{R}^n$ be a bonded domain (or a smooth manifold of $n$ dimensional) and $N\geq 2$. When the ...
0
votes
1answer
7 views

Terminology for probability matrix.

I have two related questions about terminology. If a matrix contains probabilities such that each column (or row or both) sums to $1$ , is this matrix always called a stochastic matrix i.e. even if ...
1
vote
1answer
43 views

Is my understanding of a limit correct?

When i first learned about limit i was taught that the limit $\lim \limits_{x \to a}$ f(x) = L exists if f(x) $\rightarrow$ L when x $\rightarrow$ a. Or that f(x) gets arbitrarily close to L when x ...
0
votes
0answers
21 views

It is correct this definition of limit of a function?

I have a definition of the limit of a function in some point $\alpha$ for metric spaces on this manner: We have two metric spaces $(E,d)$ and $(F,p)$; $A\subset E$, $f:A\to F$. Then ...
1
vote
1answer
44 views

Rephrasing the definition of a limit

Is $\large{\lim_{x \to a}f(x)=L}$ The same as saying "one can get $\large{f(x)}$ as close as imaginable to $\large{L}$, by setting $\large{x}$ close enough to $\large{a}$" and vice versa "one can ...
1
vote
0answers
24 views

What is a projective rational function in Fulton's book?

I'm reading Fulton's algebraic curves and I don't know if I understood correctly this definition on the page 46: What does the author mean by $f,g\in \Gamma_h(V)$ being of the same degree? Since ...
1
vote
5answers
47 views

How is “smaller than” defined on $\mathbb{R}$?

According to http://en.wikipedia.org/wiki/Binary_relation Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", and "divides" in ...
0
votes
1answer
23 views

What is the definition of a binary relation?

According to http://en.wikipedia.org/wiki/Binary_relation it is first defined as "a collection of ordered pairs of elements of A" and then as "an ordered triple (X, Y, G) where X and Y are arbitrary ...
2
votes
1answer
32 views

Epsilon Delta Limit Intuition

I am having a hard time grasping the definition of a limit. I initially learned this loose definition of a limit: $\lim\limits_{x \to a}f(x)=L$ iff $f(x)$ approaches L when $x$ approaches $a$ from ...
0
votes
0answers
12 views

Definition of curvilinear coordinates?

Please can someone give me a formal definition of curvilinear coordinates, preferably with as source. The once that I have found don't seem to be very formal.
0
votes
1answer
20 views

What is a Galois closure and Galois group?

I was reading wikipedia for Galois groups and this term suddenly appears and there is no definition for it. What is a Galois closure of a field $F$? Does this mean a maximal Galois extension of $F$ ...
0
votes
2answers
36 views

Is this definition correct for the inverse of a function?

Is this definition correct for the inverse of a function? Let $f:X\to Y$ be a function. The inverse of $f$ is the function $g:Y\to X$ such that $g\circ f=i_X$ and $f\circ g=i_Y$. We denote the ...
1
vote
1answer
376 views

Meanings of the terms “conjunct” and “disjunct” in a logic?

This sentential logic problem is stated as: Suppose that $A \models B$, where $A$ is a conjunction of literals and $B$ is a disjunction of literals. Show that $ \models \neg A$, $ \models B$, or a ...
0
votes
0answers
35 views
+50

Characterizing affine subspaces order-theoretically

Let $V$ denote a real vectorspace and $\mathrm{Con}(V)$ denote the poset of convex subsets of $V$. The goal is to identify those elements of $\mathrm{Con}(V)$ that happen to be affine subspaces of $V$ ...
1
vote
1answer
43 views

simplicial homology definition

I am revising for my algebraic topology exam based on Hatchers 2nd chapter of algebraic topology and I have this question regarding the definition of simplicial homology on pages 104-105: Hatcher ...
-2
votes
1answer
49 views

Question about the definition of a supremum

I want to know if the next definition is correct or I should fix it: Lets say that we have set $K$ and $a\in K$. Then, if $a\sup K$, for every $x\in K\Rightarrow x<a$? Or I shold cahge it to $x\le ...
1
vote
1answer
15 views

Minimal planar domain

I am recently studing minimal surfaces on my own. I have meet in many places the fallowing statement: The only connected, properly embedded, minimal planar domains in $\mathbb{R}^3$ are a plane, a ...
0
votes
0answers
19 views

What is a purely inseparable extension?

There are many different definitions of purely inseparable extension, and below is what I have chosen for my definition. (Since I don't know what is a standard one, if you know please tell me what ...
1
vote
0answers
10 views

On random functions taking values in the space of continous functions.

I upload the image of the passage that is unclear to me (Theoretical statistics by Keener): I do not understand how $W_1, W_2, \dots$ take values in $C(K)$. For every different realization of the ...
1
vote
0answers
10 views

What is a separable closure? And why the standard definition of algebraic closure is defind as so?

Instead of defining it as a subfield of algebraic closure, what would be the natural way to define it? Here's what I figured out: Let $F$ be a field and $E$ be an extension field of $F$. ...
3
votes
3answers
252 views

Intuition behind independence & conditional probability

This is a basic question. I have a good intuition that $A$ is independent of $B$ if $P(A \vert B) = P(A)$, and see how you can easily derive from this that it must hold that $P(A,B) = P(A)P(B)$. ...
3
votes
1answer
17 views

Is normal extension algebraic?

Let $E/F$ be a field extension. Then $E/F$ is called normal iff there exists $\mathscr{A}\subset F[X]\setminus\{0\}$ such that $\forall f\in\mathscr{A}$, $f$ splits over $E$ and ...
5
votes
4answers
83 views

Constructing the natural numbers without set theory.

As I understand it the idea of defining everything as sets is a relatively new idea in mathematics. Does that mean there's a non-set theoretic definition of the natural numbers? Could there be?
-1
votes
1answer
26 views

In an augmented matrix representing a system of equations, why is it a contradiction when the LHS isn't zero and RHS is zero but not when flipped?

In an augmented matrix representing a system of equations, say a $1\times 3$ matrix: $(a,b \mid c)$, why is it a contradiction when $a=b=0, c\neq 0$ but not when $a,b\neq 0, c=0$ ?
1
vote
2answers
32 views

Vector bundle and its definition

Take this definition of vector bundle: its a triple $(V,\pi,M)$ where $V$ is a set, $\pi:V \rightarrow M$ a sutjective map such that for every $m \in M$ the set $\pi^{-1}(m)$ has the structure of real ...
10
votes
3answers
824 views

What is meant by a stopping time?

TL;DR: is a stopping time some sort of event, or is it a point in discrete time, or something else entirely what is an example of something which is not a stopping time? is my understanding of the ...
3
votes
1answer
110 views

If $C=M \times [0,1]$, what is an integral over $\partial C$?

Let $M$ be a compact Riemannian manifold. Define $C=M \times [0,1]$ so that $\partial C = M \times \{0\} \cup M \times \{1\}$. If $f:\partial C \to \mathbb{R}$, is then the integral over $\partial C$ ...
1
vote
1answer
35 views

Clarification on some definitions in Operator Theory

I'm trying to read this paper http://arxiv.org/abs/1206.3325 , but I'm having a lot of difficulty making sense of two phrases. The setting is $L_2(\mathbb R^d)$. i) He mentions that for a function ...
1
vote
0answers
43 views

On definition and usefulness of “Cayley table”

Dummit-foote p.21 Let $G=\{g_1,\cdots,g_n\}$ be a finite group with $g_1=1$. The Cayley table of $G$ is the $n\times n$ matrix whose (i,j)-entry is $g_ig_j$. This definition is based on an ...
0
votes
0answers
17 views

(Partial) derivatives exist vs. are finite?

Is there a difference between the following two statements or do they mean the same? The (partial) derivatives of $f$ exist. The (partial) derivatives of $f$ are finite. I believe that it ...
0
votes
1answer
50 views

How to define a b-open set relative to a larger set?

In the paper of D. Andrijevic entitled "On b-open Sets", it is defined that A subset $S$ of a topological space $(X,\tau)$ is $b$-open if $$ S\subseteq\bar{\operatorname{int} S}\cup ...
14
votes
4answers
721 views

Vector Spaces: Redundant Axiom?

Question Why are the axioms for vector space independent? More precisely $1x=x$ seems redundant... (I take the axioms from: Wikipedia) Explanation One has for zero vector: ...
13
votes
4answers
2k views

What is a point?

In geometry, what is a point? I have seen Euclid's definition and definitions in some text books. Nowhere have I found a complete notion. And then I made a definition out from everything that I know ...
2
votes
3answers
59 views

Injective or one-to-one? What is the difference?

What is the difference between the terms 'injective' and 'one-to-one', 'surjective' and 'onto', and 'bijective' and 'isomorphic'?
0
votes
0answers
6 views

Definition of a kind of section

Let $L$ a line bundle of a complex algebraic surface. What is a rational function of $L$? (Or if you want you can take the case of the Riemann surfaces)
0
votes
1answer
34 views

On the definition of sphere in analytic geometry…

Last year, when I was teaching mathematics (analytic geometry) for one of my clever freands, I arrived to the definition of sphere. I said Fix $r>0$, An sphere is the set of all triples ...
0
votes
1answer
20 views

Definition of linear chains

I am reading Hatcher books in algebraic topology and he defined by $LC_n(Y)$ as the subgroup of $C_n(Y)$ generated by linear maps $\Delta^n$ $\to Y$ where $C_n(Y)$ is the abelian group of the $n$ ...
0
votes
0answers
13 views

Name for maximal number of pre-images under a function

Let $E$ and $F$ be sets and $f:E\to F$ be any function. I want a name for the quantity $$\displaystyle\sup_{y\in F}\operatorname{Card}(f^{-1}(y)).$$ I am currently calling it the "maximal ...
0
votes
4answers
40 views

What is the exact definition of an Injective Function

Am I right to believe that a function is injective, if some elements of the first set are mapped to some elements of the second set? It is also possible to 4 elements of the first set, are mapped to ...
3
votes
2answers
67 views

Holomorphic function definition. Am I missing something very obvious?

I'm reading a book of complex analysis in which the definition of holomorphic function is given as follows: Definition: If $V$ is an open set of complex numbers, a function $f:V \to \mathbb C$ is ...
1
vote
1answer
42 views

Number of orbits in a graph.

I am confused with this concept. Consider for instance the graph $G$ with $V=\{v_1,\dots, v_{10} \}$, $E=\{12, 15, 16, 23, 27, 34, 38, 45, 49, 67, 78, 89, 910, 510, 610 \}$. This is a 3-regular ...
1
vote
4answers
112 views

Are all groups algebraic?

I know the definition of a group as a set with an operation that satisfies certain axioms. I have heard that there is something called an algebraic group and that this is a group with a topology such ...
2
votes
1answer
159 views

Tetration and its inverse to various exponents

I've recently seen in my studies tetration, or the next operation in the addition, multiplication, exponentation... series. I've also heard much discussion about how to extend this operation to ...
2
votes
0answers
24 views

What vector field property means “is the curl of another vector field?”

I know that a vector field $\mathbf{F}$ is called irrotational if $\nabla \times \mathbf{F} = \mathbf{0}$ and conservative if there exists a function $g$ such that $\nabla g = \mathbf{F}$. Under ...
3
votes
0answers
24 views

Gradient vector interpretation

I have trouble understanding the gradient vector $\nabla$. For a surface, when we find it's $\nabla$, why is the resultant vector the normal vector when $\nabla$ means gradient vector?? Thanks
0
votes
0answers
5 views

Spherical representation on locally compact group

What is the definition of a spherical representation of the the pair $(G \times G, G)$, where $G$ is a locally compact group?