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

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3
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2answers
231 views

Definition of “Up to” (homeomorphism,isotopy, etc), and Examples?

I've tried googling this usage and understanding the results but I'm struggling to make intuitive sense of it. So my question is, what is the phrase "up to" understood to mean, and what are some ...
2
votes
3answers
602 views

Positive constant scalar definition

In French when we say "$k$ est une constante positive", that means $k\geq 0$. But I remarked that using the same sentence in English, "$k$ is a positive constant", means that $k>0$. Can one explain ...
0
votes
2answers
39 views

Defined amount and value amount of a function

What is the defined amount and value amount for this function: $$f(x)=\sqrt{(x+7)(1-x)}?$$ The defined amount is all the x-values the function can be and the value amount is all the y-values the ...
1
vote
1answer
149 views

Differentiable manifold in dimension 1 and its critical point

Please, I want to know how to define a differentiable manifold in dimension 1, and if the circle is a differentiable manifold in dimension $1$, and what is its critical point. Thank you.
1
vote
1answer
48 views

If a ring $R$ is a field, must $R$ be a unitary ring?

If a ring $R$ is a field, then does it automatically imply that $R$ is a unitary ring? Thank you.
4
votes
2answers
63 views

Question on the definition of a ring.

A ring $\langle R,+, \cdot\rangle $ is a set $R$ with two binary operations such that: $\langle R,+\rangle $ is an abelian group. Multiplication is associative. Left and right distributive laws ...
2
votes
0answers
260 views

Different Definitions on the Differentiability of Functions on a closed set.

I have encountered three different definitions on the differentiablity of functions on a closed set. In the following, suppose that $\Omega\subset M$ is a (open) domain, where $M$ is a manifold. The ...
3
votes
2answers
551 views

Formal Dirichlet-Bourbaki definition of function

What is the formal Dirichlet-Bourbaki definition of a function? I have come across this in this essay: http://www.k-12prep.math.ttu.edu/journal/contentknowledge/meel01/article.pdf on page 1. I know ...
0
votes
1answer
137 views

Operations with planar graphs

I think in the graph theory operations such as decomposition into cycles, union, intersection, difference, subdivision can be done. If I am given a planar graph (for e.g. see figure), then can the ...
1
vote
2answers
2k views

What is a direct correlation?

I have two contrary definitions of for the direct correlation between two variables $X$ and $Y$ Their correlation coefficient is close to $1$. There is a direct causal relationship between the ...
0
votes
4answers
313 views

Definition of subgroup generated by a subset

I'm confused about the definition of a subgroup $(W)$ generated by a subset $W$ of a group $G$. My textbook gives: Let $(W)$ be the set of all elements of $G$ representable as a product of ...
2
votes
1answer
430 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 ...
3
votes
3answers
1k views

Definition of degree of finite morphism plus context

Let $f: X \rightarrow Y$ be a finite morphism of schemes, defined here, http://en.wikipedia.org/wiki/Finite_morphism I always assumed that the degree of $f$ was the degree of the induced field ...
6
votes
0answers
258 views

Understanding the roots of homomorphism and homeomorphism

I understand the formal (i.e. mathematical) definitions of the terms "homomorphism" and "homeomorphism" as they relate to functions, but I am curious as to the origin of these terms. I don't know ...
3
votes
3answers
4k views

What's the difference between direction, sense, and orientation?

I'm trying to understand the difference between the sense, orientation, and direction of a vector. According to this, sense is specified by two points on a line parallel to a vector. Orientation is ...
0
votes
2answers
175 views

Difference between Spanning set and Postitive Spanning Set

I do understand the difference as mentioned in the texts about spanning set and positive spanning set, im somehow missing how if $v_1.. v_r$ is a positive spanning set for $R^n$, then $v_2 ... v_r$ ...
2
votes
2answers
320 views

How to extend definition of n-tuple to the case $n=0$?

The classical definition of n-tuple $(x_i)_{i < n}$ starts at $n=2$. In this case $$(x_0,x_1) := \{\{x_0\},\{x_0,x_1\}\}$$(1). For $2<n=k+1$, $(x_i)_{i < n}:=((x_i)_{i < ...
9
votes
4answers
444 views

Is the free group on an empty set defined?

I'm guessing that the free group on an empty set is either the trivial group or isn't defined. Some clarification would be appreciated.
1
vote
1answer
47 views

What is an evaluation operator and what is its use?

In a lecture about numerical mathematics, mostly about ODEs, we were given the following definition: The two-parametric family $\mathbf \Phi^{s,t}$ of maps $\mathbf \Phi^{s,t}: D \mapsto D$ is ...
2
votes
1answer
362 views

What is a tangential gradient?

If $\mathbb{R}_{+}^{n}$ is the half space $\{x\in\mathbb{R}^n\vert\ x_n>0\}$ and $u$ is a twice differentiable function in $\mathbb{R}^{n}$. If we write: \begin{equation} ...
11
votes
6answers
10k views

Is the empty set a subset of itself?

Sorry but I don't think I can know, since it's a definition. Please tell me. I don't think that $0=\emptyset\,$ since I distinguish between empty set and the value $0$. Do all sets, even the empty ...
1
vote
1answer
69 views

Harmonic conjugate of $u,v$ in $f=u+iv$

Do I understand correctly the definition of being harmonic conjugate if I understand it that: $v$ is the harmonic conjugate of $u$ but $u$ is not the harmonic conjugate of $v$, but rather $-u$ ?
2
votes
2answers
574 views

two notation: semi-metric and pesudometric

There are two notations: semi-metric and pesudometric make me unclear. Are they the same thing, or they are different? Thanks ahead.
1
vote
3answers
131 views

Precise differences in meaning of Power Series, Taylor Series

Being an physicist/artist, not a real mathematician, I often toss around the terms "Taylor Series" and "Power Series" without any concern. Are these terms be considered interchangeable by ...
1
vote
3answers
199 views

Formula for Product of Subgroups of $\mathbb Z$, Problem

What is the product of $\mathbb{Z}_2$ and $\mathbb{Z}_5$ as subgroups of $\mathbb{Z}_6$? Since $\mathbb{Z}_n$ is abelian, any subgroup should be normal. From my understanding of the subgroup product, ...
1
vote
3answers
2k views

What does it mean to have no proper non-trivial subgroup

I am reading a first course in abstract algebra and there is a claim that says a group $G$ with no proper non trivial subgroups is cyclic. But I don't understand what does it mean to have no proper ...
3
votes
2answers
146 views

Definition of complete in the context of Lebesgue measurable sets

I came across this statement on Lebesgue measurable sets. The Lebesgue measurable sets are said to be complete because every subset of a null set is again measurable and the lebesgue measurable ...
2
votes
3answers
125 views

Example of a (dis)continuous function

The following thought came to my mind: Given we have a function $f$, and for arbitrary $\varepsilon>0$, $f(a+\varepsilon)= 100\,000$ while $f(a) = 1$. Why is or isn't this function continuous? I ...
5
votes
1answer
107 views

Is uncountably summation defined?

We know that finite and countably summation is defined. But How about uncountably summation, say $$\sum_{i\in \mathbb{R}}0$$ Is it defined?
2
votes
1answer
201 views

Limit point definition

I have read the definition of a limit point of a set in Real Analysis. The definition goes like: A number $p$ is said to be a limit point of a set of reals, $S$, if every neigbourhood of $p$ has at ...
-3
votes
3answers
137 views

Do dihedral groups $D_n$ for $n\geq 5$ exist?

I know we can generate dihedral group of order three ($D_3$) and four ($D_4$) but my question is whether we can generate dihedral group of order five?
8
votes
4answers
960 views

Congruent Modulo $n$: definition

In an Introduction to Abstract Algebra by Thomas Whitelaw, he gives examples of the congruence mod operation, such as $13 \equiv5 \pmod4$, and $9 \equiv -1 \pmod 5$. But when I first learned about ...
1
vote
2answers
193 views

What does relax mean in the mathematical context

Here is a direct citation from wikipedia: The assumptions were further relaxed in the works of Terence Tao and Van H. Vu, Friedrich Götze and Alexander Tikhomirov. Finally, in 2010 Tao and Vu ...
3
votes
0answers
73 views

Define composition of small cyles and making a big graph

I am having following sub graphs and wish to compose all and make a one bigger graph (say G). After that, I want to select the closed path where it is passing along the outer vertices of that ...
4
votes
2answers
142 views

Standard definition of group isomorphism

ProofWiki defines a group isomorphism as a bijective homomorphism. In Topics in Algebra 2$\varepsilon$, Herstein defines a group isomorphism as an injective homomorphism: Definition. A ...
3
votes
2answers
318 views

Basic question about analyticity vs. differentiability in complex analysis.

In chapter $V$ of Palka, "Consequences of the Local Cauchy Integral Formula," 3.1. If a function $f$ is analytic in an open set $U$, then $f'$ is analytic in $U$. In particular, $f$ belongs to ...
3
votes
1answer
108 views

What is Absolute convergence?

Take: $$ (u*v)(k) = \sum_{i=-\infty}^\infty u(i)v(k-i). $$ $k$ is there, it's because you want to define $$ \ldots\ldots, (u*v)(-3), (u*v)(-2), (u*v)(-1), (u*v)(0), (u*v)(1), (u*v)(2), (u*v)(3), ...
1
vote
1answer
32 views

Change Along A Tangent Line

I am taking some time to review differentials. What I don't quite get is why the change along the tangent line is $f'(x) \Delta x$, and how it leads to $f'(x)dx$
0
votes
1answer
218 views

Different formulation of a Traveling Salesman Problem

Given a undirected, weighted, complete graph $(V,E,c)$ with $c \to \mathbb{N}$ and $v_0 \in V$ we are looking for a set $E' \subset E$ minimal with respect to $c$ with the following conditions: for ...
2
votes
2answers
190 views

What is the meaning of the expression $\liminf f_n$?

I am a little confused as to what $\liminf f_n$ means for a sequence $f_n$ of functions converging to $f$. I can not locate a definition anywhere.
6
votes
2answers
106 views

Collecting definitions of continuity.

Let $X$ and $Y$ denote topological spaces and consider a function $f : X \rightarrow Y$. I'm collecting possible definitions/characterizations of the statement "$f$ is continuous." Here's two to get ...
7
votes
3answers
249 views

What's Geometry?

I am a grad student. I am writing an article on geometry and relativity theory and trying to start with discussing basic ideas of topology. In my article I tried very hard to motivate the idea of ...
2
votes
1answer
930 views

What is the definition of a geometric progression?

If the first term in our geometric progression (GP) is $k$, and the common ratio is 0, then our sequence is $\{k, 0, 0, 0, 0,\ldots\}$. Is there anything wrong with this statement? So, is $\{0, 0, ...
1
vote
5answers
1k views

Definition Of Symmetric Difference

The definition of a symmetric difference of two sets, that my book provides, is: Set containing those elements in either $A$or $B$, but not in both $A$ and $B$. So, in set builder notation, I figured ...
2
votes
1answer
70 views

Why Can't we define the differentiation of vector fields in the same way as in $\mathbb{R^{n}}$

In $\mathbb{R^{n}}$, if $X$ is a vector field on $\mathbb{R^{n}}$, and $X=$$\sum_{i=1}^{i=n}$ $X^{i}$ $\frac{\partial}{\partial x^{i}}$, $X^{i}$ $\in$$C^{\infty}(p)$. Then 1.The differentiation of ...
1
vote
4answers
209 views

Definition of a metric

I'm having a hard time understanding what the definition of a metric is. From what I think I understand, it's just a method of measurement between $2$ points in $\mathbb R^n$? Is that somewhere along ...
0
votes
1answer
1k views

How to find Df in functions

Well, I do understand what Df is and how you find it in simple equations, however, I am kinda confused in "complex" functions. For example, the following functions: 1* f(x)=x^3+x^2-x-1 , Df=R ...
9
votes
4answers
701 views

A question on definition of field of fractions

Wikipedia defines the field of fractions of a domain as The field of fractions or field of quotients of an integral domain is the "smallest" field in which it can be embedded. What does ...
0
votes
2answers
69 views

How to make precise the notion of “the multiset of roots of a polynomial function”?

A (real) polynomial function can be defined as a function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that there exists a sequence $a : \mathbb{N} \rightarrow \mathbb{R}$ such that the terms of $a$ ...
0
votes
2answers
612 views

What is a bijective linear mapping called?

Friedberg - Linear Algebra p.102 This book states that "a bijective linear map from a vector space to another vector space is called an isomorphism". As far as know, generally isomorphism means ...