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

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23 views

Alternative view of matrix inversion (explanation required)

We were taught in linear algebra that in order to try to find the inverse of a matrix we can create an augmented matrix $[AI]$ where $A$ is the original matrix and $I$ is the identity matrix. Then we ...
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1answer
31 views

Different definitions of a limit of a function in $\mathbb R$. Are they equivalent or not?

If a limit $L$ of a function $f:A\to\mathbb R$ exists at a point $a\in \mathbb R$, where $A\subset\mathbb R$ is a proper subset of the set of real numbers, is there any difference between the ...
4
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2answers
179 views

Is the reverse statement to “open” in Munkres topology true?

There is a problem: Given $(X, \tau), A \subseteq X, \forall x \in A, \exists U \in \tau, x \in U \text{ s.t. } U \subset A \implies A$ is open in $X$ So what I did was to show that $A$ is in ...
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1answer
25 views

Associated and non-associated polynomials

Studying algebraic geometry (if it makes any difference) I've found the following "... here the $F_i$ are irreducible and non-associated polynomials... ". It must be obvious what it means for ...
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0answers
24 views

Definition of regular map from aquasi projective variety

These are from the book Basic Algebraic geometry by Shafarevich Definition of regular function on a quasi projective variety is as follows : Let $X\subset \mathbb{P}^n $ be a quasi projective ...
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0answers
39 views

Is there an accepted definition of coprimality in commutative ring theory?

I can think of at least three possible definitions of coprimality in commutative ring theory: call $a,b \in R$ are coprime iff if $c \mid a$ and $c \mid b$, then $c \mid 1$. if $a \mid c$ and $b ...
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2answers
14 views

Is this triangulation consistent with Sperner's Lemma?

Since any two triangles which intersect have an edge or a vertex in common, the triangulation is simplicial. However, I am concerned about triangle $A$. Is every sub-triangle supposed to have a ...
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1answer
21 views

What is the definition of subcollection in topology?

I had thought that subcollection referred to a collection of sets in some topological space $(X,\tau)$, where each set is a subset of $X$ But I also see it being used in subbasis, and subspace ...
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3answers
96 views

Strange question about free abelian group

I was given the following question for homework, but it makes no sense to me. Since when are free abelian groups constructed w.r.t maps? Isn't the set $S$ all that matters? I don't understand what ...
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0answers
27 views

Clarification of Sperner's Lemma

From Graph Theory by Bondy, Murty Image from wikipedia I don't see how the picture holds according to the definition from the Graph Theory book. Specifically, the definition says to assign ...
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1answer
16 views

Definition of bounded in a metric space - confirmation

Is the following definition of a bounded metric space correct? $(M,d)$ is bounded if $\exists a \in M, r > 0$ such that $M = B(a,r)$. Looking around on the internet I instead see $M \subset ...
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2answers
190 views

Evaluating a definite integral by definition: $A(x) = \int_{-1}^x (t^2 + 1)\space dt$

I have an area function $A(x)$ defined as $$A(x) = \int_{-1}^{x} (t^2 + 1)\space dt$$ ... and I would like to use the definition of definite integral to evaluate it. I started this way $$A(x) = ...
4
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1answer
192 views

Terminology: Difference between Lemma, Theorem, Definition, Hypothesis, Postulate and a Proposition

Based on observation after reading few books and papers, I think that Lemma : Lemma contains some information that is commonly used to support a theorem. So, a Lemma introduces a Theorem and comes ...
4
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1answer
54 views

What is this structure involving a monad and a comonad?

Let $F$ be a monad on some category $\mathsf{C}$ and $G$ be a comonad on the same category. Assume further that they "commute" (see below): $FG \cong GF$. Then, for lack of a better name, one can ...
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2answers
34 views

Does “sphere” denote the surface or the entirety of a solid ball?

In everyday English, the word "sphere" denotes a 3-dimensional object, including the points inside the surface and its center. However, I get the sense that in mathematics, the sphere is used ...
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4answers
539 views

Why do they call it base 10?

Now, I know intuitively why it's called base 10: because there's 10 numbers. But see here's the thing, if we're working with numbers 0-9 (and of course we are), we use up our numerical artillery at ...
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2answers
1k views

Can a function be increasing *at a point*?

From what I understand we say that a function is increasing on an interval $I$ if $$ x_1 < x_2 \quad\Rightarrow\quad f(x_1) < f(x_2). $$ for all $x_1,x_2\in I$. I understand that some might call ...
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1answer
103 views

Can these definitions of the words “problem” and “solution” be formalized, and if so, has this been done? If so, where can I learn more about it?

I had a thought. Define that: Vague Definition 0. A problem consists of: a set $X$ a set $Y$ a function $f : X \rightarrow Y$ a way $\overline{X}$ of representing the elements of $X$ ...
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3answers
99 views

Is really $f(x)=\int g(x) dx$ a function?

I saw many of this kind of questions on some text/question books. Is there any other explanation of this, or is it really wrong as I thought? Here is a question of that kind: If $\displaystyle ...
4
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3answers
52 views

Vectors sometimes used in math just as arrays/lists of numbers, sometimes as concept of “change”

As a freshman in a small town college. Ive been getting mixed signals to what vectors (and matrices/tensors) are. Sometimes I get the feeling they are used just as containers/arrays for multiple ...
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2answers
58 views

Is continuity at a point only defined for points in the domain?

I'm using Michael Spivak's Calculus, 3rd edition textbook. Without ado, I'll state the definition given for continuity at a point: DEFINITION$\;\;\;\;$The function $f$ is continuous at $a$ if: ...
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1answer
42 views

How is a characteristic subgroup verified?

I know the definition of a characteristic subgroup: $\sigma (H)=H$ for all $\sigma \in \text{aut}\, G$ where $H \leq G$. But, I do not understand how $\sigma$ is defined. Surely we can map $H$ to $H' ...
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5answers
3k views

What does the symbol |_ mean?

For example, (6) The sequence of primes is endless. For, if $p$ is any prime, the number ${\begin{array}{|c}\color{red}p\\\hline\end{array} + 1}$ is greater than $p$ and is not divisible by ...
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3answers
2k views

The Degree of Zero Polynomial.

I wonder why the degree of the zero polynomial is $-\infty$ ? I heard that, it is $-\infty$ to make the formula $\deg(fg)=\deg(f)+\deg(g)$ hold when one of these polynomials is zero. However, if that ...
379
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21answers
66k views

What are imaginary numbers?

At school I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number which has something to do with the square root of -1. When I ...
2
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1answer
48 views

Is a hypersurface really defined by an arbitrary polynomial?

In An Invitation to Algebraic Geometry Karen Smith writes at the beginning of the book: The zero set of a single polynomial in arbitrary dimension is called a hypersurface in $\mathbb C^n$. The ...
2
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1answer
48 views

If $\lim_{x\to a} g(x) = M$, show that there exists a number $\delta > 0$ such that $0 < |x - a| < \delta \Rightarrow |g(x)| < 1 + |M|.$

The hint given was to take $\epsilon = 1$ for the formal definition of a limit. Doing this means that $|g(x) - M| < 1.$ Using the triangle inequality, you get $$|g(x)| = |(g(x) - M) + M| \le |g(x) ...
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0answers
19 views

What is characteristic time?

What is characteristic time? Where is it useful? From this answer by Joriki: The characteristic time is usually defined to be the time in which a quantity decreases by $1/e$. Why is ...
3
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1answer
30 views

Formalizing continuously indexed spaces in fiber bundles?

This MSE question asks for clarification of the local triviality condition imposed in the definition of a fiber bundle. As mentioned there, the point of local triviality seems to somehow ensure a ...
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0answers
22 views

Continuously variable *space* [on hold]

I'm trying to understand formally how and why a fiber bundle with fiber $F$ should be thought of as a gluing of homeomorphic copies of $F$ which varies continuously. I do not understand how this is ...
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4answers
475 views

Ambiguity in definition of compactness

I am struggling with the definition of compactness in a topological sense. Below is the definition presented in my lecture notes: A topological space $X$ is compact if every open cover has a ...
4
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3answers
8k views

The degree of a polynomial which also has negative exponents.

In theory, we define the degree of a polynomial as the highest exponent it holds. However when there are negative and positive exponents are present in the function, I want to know the basis that we ...
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0answers
27 views

Definition of Operation

What are operations in Mathematics? I do not find it formally defined anywhere. What is the difference between operation and function? Earlier I thoght operations are just binary operations. But later ...
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0answers
24 views

Rewording the definition of closure

In Munkres there was a statement: Given a topological space $(X, \tau)$ $x \in \overline A \iff \text{ for every open set } U \text{ containing } x, U \cap A \neq \varnothing$ Following from ...
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0answers
43 views

Projective and affine varieties: differences, advantages and why two definitions

I have recently started to learn algebraic geometry and this question has been bugging me. An affine variety is a zero set of a collection of polynomials in affine space and a projective variety is ...
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3answers
58 views

changing the order of the logical symbols $(\forall \epsilon) (\exists\delta)$ by $(\exists \delta)(\forall\epsilon) $ in limit definition

Some time ago a professor told the class, which I was in, to analyze why this definition of limit is not good (or if it is a good definition to argument why): There exists a $\delta>0$ for all ...
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1answer
50 views

What is the meaning of (resp. closed) in set theory?

I'm sure this a spectacularly basic question but I can't seem to find the definition of this anywhere. Here's some context: If $U$ and $V$ are open (resp. closed) then $U\cup V$ is open (resp. ...
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0answers
18 views

General definition of the discriminant

There is a notion of the discriminant of polynomials quadratic forms finite separable extensions of Dedekind domains (e.g., algebraic number fields) I don't know much about 2, but I think that 1 ...
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0answers
53 views

Why is the area of a rectangle given as $l\times h$? [duplicate]

While this may seem like a simple question, I have found it very difficult to answer. My question is "Why is the area of a rectangle given as $l\times h$?" We define area as the "size of a two ...
2
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1answer
30 views

Definition of component for a digraph?

I could find this in Wikipedia Component: A connected component of a graph is a maximal connected subgraph. The term is also used for maximal subgraphs or subsets of a graph's vertices that have ...
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1answer
31 views

Is $\forall x \in U: f(x) \in V$ the same as $x \in U \implies f(x) \in V$?

As the title says, do the following two statements have the same meaning? $$\forall x \in U: f(x) \in V \text{ (for all $x \in U$, $f(x) \in V$)}$$ $$x \in U \implies f(x) \in V \text{ ($x \in U$ ...
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1answer
19 views

Definition of vertex-cut for digraph?

I am trying to understand vertex cut for digraph. I could find this for graphs Vertex cut is a vertex whose removal increases the number of components in a graph. (D67, Handbook of Graph Theory by ...
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0answers
45 views

Meaning of alternation in definition of wedge product

Spivak defines the wedge product as $\omega \wedge \nu = \frac{(k+l)!}{k!l!} \text{Alt}(\omega \otimes \nu)$ and I have been running into some conceptual issues here. The alternation is defined as ...
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1answer
450 views

Tangent space to a surface at boundary points

Let $M$ be a $2$-dimensional compact oriented surface in $\mathbb R^3$ with boundary $\partial M$. For any $p\in M \setminus \partial M$ tangent vectors are defined as speed vectors of smooth curves ...
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1answer
49 views

Coordinate charts vs. coordinates on manifolds

I just realised that I'm confused what coordinates really means in the context of manifolds. For example, say $M=S^2$. Then we can define smooth charts as follows: Let the open sets be $U = S^2$ ...
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2answers
34 views

Interpretation of definitions and logical implication in Calculus - e.g. monotonic strictly increasing function

I read definitions in Calculus books that often confuse me from a logical perspective. For example, the definition of a monotonic function, e.g. a strictly increasing function, is defined as follows. ...
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1answer
51 views

When to use $\in$ and $\subseteq$ when talking about bases and topologies

Can someone demonstrate a concrete example of when to use $\in$ and $\subseteq$ when talking about topologies and bases? When is something $\subset$ of a basis or a topology and when is something ...
2
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0answers
23 views

Showing that an intersection of indexed sets is a subset of every individual indexed set

I am required to show that for every $k \in I$, $\bigcap_{i\in I}A_{i}\subseteq A_{k}$ where $I$ is an index for a collection of subsets $A_{i}\subseteq S$, $i \in I$. This seems obvious to me from ...
24
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5answers
2k views

Chain Rule Intuition

We know that the chain rule is used to differentiate a composite function ,say $$f(x) = h(g(x))$$ It's defined as the derivative of the outside function times the derivative of the inner function or ...
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0answers
25 views

w.r.t. which chain complex is $H^k_{sign}(M;R)$ computed?

This question is inspired by my previous question. People often write $H^k_{sign}(M;R)$ for the $k$th singular cohomology group with coefficients in $R$. However, I don't understand what this means. ...