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

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1answer
120 views

Why are there so many different definitions for differentiability?

I am studying differentiability for functions of several variables. Here is the first definition of differentiability I came across:$\quad$ A function $f:\Bbb R^n\to\Bbb R^m$ is differentiable at ...
2
votes
3answers
46 views

Definition: limit of a sequence

What is the purpose not to choose $|x_n-a|\leq\epsilon$ instead of $|x_n-a|<\epsilon$ in the definition of convergence? Is their a substancial difference (or a practical one)? Thanks in advance.
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1answer
50 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 ...
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2answers
554 views

Rigorous mathematical definition of “much greater than” symbol

What does $f(x) \gg g(x)$ mean mathematically? How can we characterize "much greater than"?
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1answer
68 views

Natural definitions of families of subgraphs

Cluster analysis is a vibrant area of applied mathematics, "used in many fields, including machine learning, pattern recognition, image analysis, information retrieval, and bioinformatics". A subfield ...
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2answers
33 views

What does $\text{mod}\ m$ in $a \equiv b (\text{mod}\ m)$ means

I am trying to do example 3.6 from this http://www.cs.fsu.edu/~lacher/courses/MAD3105/lectures/s1_3equivrel.pdf script, but I am not sure what does $(\text{mod}\ m)$ means. Can somebody explain it to ...
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1answer
269 views

Is a sequence a subsequence of itself?

I know that sets are subsets of themselves, so by that logic is that true for sequences?
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1answer
32 views

About definition of $a^b$ with $a \in \Bbb{R} \wedge b \in \Bbb{N}$

-- let $a,c \in \mathbb{R}$, and $b \in \Bbb{N}$, with $\Bbb{R}$ is a complete ordered field, $c \triangleq a^b$ if $c=\begin{cases} 1, & \mbox{if } a\neq 0 \wedge b=0\\ 0, & \mbox{if } a=0 ...
4
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1answer
195 views

Define positions of a set of points given (only) the distances between them

I have been thinking about spatial transforms. Given $n$ points, there are $\frac{n!}{(n-2)!2!}$ combinations of selecting two points, so for 64 points in space, there are 2016 single point-to-point ...
2
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1answer
61 views

Why do people not use “partially directed” graphs?

Are there structures in use which are a mix of directed and undirected graphs? I.e. the effective edge-set consists of both directed and undirected vertex pairs. In the case that the graph is simple, ...
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1answer
75 views

Is it correct to say $ x^3+2x+1=y^2 $ is an elliptic curve?

I'm a bit confused about the definition on elliptic curve. For example, can we say that $x^3+2x+1=y^2$ is an elliptic curve? My opinion is that it is not an elliptic curve as the definition given in ...
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1answer
64 views

$V, W$ vector fields, $\omega$ one-form, $d\omega(V, W)=?$

$V, W$ vector fields, $\omega$ one-form, what is the definition of $d\omega(V, W)$? I have seen some equalities and I suppose that $d\omega(V,W)=(V^jW^i-V^iW^j) \frac{\partial f_i}{\partial x^j}$ ...
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6answers
254 views

Why is exponentiation defined as $x^y=e^{\ln(x)\cdot y}$?

There are many curves that extend integer exponentiation to larger domains, so why was this one chosen?
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0answers
33 views

Difficulty with terminology/standard definition for Jordan-normal form.

(Perhaps this is something rather for MOverflow, don't know) I've understood the concept of the Jordan-normal-form such that it is similar to the idea of diagonalization of a matrix, but can be ...
9
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2answers
223 views

Additional ways of defining real powers.

I am familiar with the following 3 way of defining real powers: Given $x,y\in\mathbb{R}$, such that $x\gt0$, we can define $$ x^y = \begin{cases} \sup_{\,q\in\mathbb{Q}}\{x^q :q\le y\} ...
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1answer
56 views

Why the inverse of the matrix in this definition

We have this definition: I didn't understand why in this definition we just define $F^A$ to be $$F^A(X_0:X_1:X_2)=F\bigg((X_0:X_1:X_2)A\bigg)$$ $F^A$ is not the composition $F\circ c$? Why ...
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7answers
745 views

What is the square root of complex number i?

Square root of number -1 defined as i, then what is the square root of complex number i?, I would say it should be j as logic suggests but it's not defined in quaternion theory in that way, am I ...
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1answer
53 views

Can all real numbers be presented via a natural number and a sequence in the following way?

Is there (for each fixed base system with digits $0,1,\dots,m$) and then for each real number $r\in\mathbb R$, an integer $n\in\mathbb Z$ and a sequence $(a_i)_{i\in\mathbb{N}}$ with ...
0
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1answer
39 views

Is a relation$ R=\{(a,a):a\in I\}$ said to be reflexive, symmetric and transitive?

Consider a relation $R=\{(a,a): a\in A\}$ where $A=\{1,2,3\}$. i.e $R=\{(1,1),(2,2),(3,3)\}$ This clearly reflexive but is it necessary that all such relations are necessary to be symmetric and ...
0
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1answer
84 views

Definition of Limit Superior/Inferior of Real-Valued Functions

Let $f$ be a real valued function and let 1) $$\limsup_{r \to \infty}f(r) = \mu$$ 2) $$\liminf_{r \to \infty}f(r) = \phi$$ Does 1) $\iff$ $\exists\ r_n$ increasing, $r_n \to \infty$ such that for ...
4
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1answer
49 views

Transforming Nested Fixed-Point Formulas into Infinitary Logic Formulas with Finitely many Variables

There is a definition (actually a description of how it could be defined) of a fixed-point logic formula. The formula is in inflationary fixed point logic (IFP) in this case but it could also be ...
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2answers
55 views

Is this predicate expressing compactness?

I don't quite know why definitions of "compact" use the expression "arbitrary collection". Am I correct in thinking that the following predicate is a definition of compact? Let $X$ by a set with ...
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3answers
297 views

Definition of Door Space

Every reference I can find regarding (topological) door spaces gives the following definition almost verbatim: A door space is one in which every subset is either open or closed. [emphasis mine] ...
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3answers
215 views

Provocations on the existence of mathematical objects

The few Mathematics I have been studying so far is pure Mathematics. I happen to have some discussions with philosophers of Mathematics, but as they know I totally ignore their subject, we do not ...
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0answers
52 views

Is the empty matrix $M$ ($m \times n$) with $m$ or $n$ equal to zero in RREF ? Does it posses every matrix property one would wish?

Is the empty matrix $M$ ($m \times n$) with $m$ or $n$ equal to zero in RREF ? Reading a proof in Linear Algebra induction starts off by proving $P(n)$ holds for $n=0$, where $P(n)$ is the ...
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1answer
56 views

Definition of a binary operation is the same as definition of a closed binary operation?

I'm reading Wikipedia about operations and binary operations . Intuitively I always thought that a binary operation is a operation that takes two arguments. But Wikipedia defines a binary operation as ...
2
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1answer
43 views

Why do we want to define a $k$-scheme to be birational if the rational map (and its inverse) to $\Bbb A_k^n$ is over $k$?

Two varieties $X,Y$ are said to be birational if there exist rational maps in each direction such that either composition is the identity on a open dense subset. Note that here the morphisms aren't ...
1
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1answer
144 views

Algebraic definition or construction of real numbers

Is there any algebraic definition or construction of real numbers ? If not, why ?
1
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0answers
63 views

Definition: foci of a quadric

How are the foci of a quadric defined? By a quadric I mean a set $$ Q = \left\{ x \in \mathbb R^n \mid x^T A x + 2 b^T x + c = 0 \right\}, $$ where $A\in\mathbb R^{n\times n}$ is symmetric and ...
3
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2answers
93 views

What is an isomorphism?

I'm familiar with the concepts of group isomorphism, ring isomorphism, and graph isomorphism, but it's never been presented to me what an isomorphism is in general: given any X, what is an X ...
0
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1answer
26 views

A notation question on how to properly denote a function that takes inputs only of a certain form.

Suppose I have a set $B = \{n^2 + n + 1 : n \in \mathbf{N}\}$ and I want to define a function $g: B \rightarrow \mathbf{N}$ that only accepts as it's arguments numbers of the form $n^2 + n + 1$ for $n ...
1
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1answer
126 views

Questions on Killing form: its definition and a root space decomposition.

I have a question on Killing form. Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Consider the adjoint representation $(\mathrm{ad},\mathfrak{g})$ of $\mathfrak g$, i.e. $$ \mathrm{ad}: ...
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2answers
65 views

Double categories

So, I wanted to ask a question about double groupoids until I find myself having the answer, meanwhile I wrote a lot of stuff about double categories, so I decided to create a question and answer my ...
3
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2answers
179 views

Doubt: Proof of existence of the summation of natural numbers in Landau

With reference to the Definition of Addition in Landau's Foundations of Analysis, the author, in proving the existence of a natural number $(x+y)$, takes for granted that $x' + y = (x+y)'$ where ...
55
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19answers
11k views

What is the difference between a point and a vector

I understand that a vector has direction and magnitude whereas a point doesn't. However, the course note that I am using states that a point is the same as a vector. Also, can you do cross product ...
3
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1answer
273 views

Accuracy of the Newton-Cotes formulas for polynomials of degree $n+1$ and even $n$

Let $f$ be a polynomial of degree $n+1$. The Newton-Cotes formula is given by $$\int_{-t}^tf(x)\text{ d}x\approx\sum_{k=0}^nf(x_k)\int_{-t}^t\omega_{n+1}(x)\text{ d}x \tag{*}$$ where ...
0
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1answer
26 views

Transformation-Matrix (Definition & explanation)

I have to do a proof ("Let V be a vector space with basis A, B, C. Show that $T_{AC} = T_{BC}T_{AB}.$ Well, I really don't know what is menat by $"T_{AB}"$ or something like this. I thought about the ...
0
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1answer
35 views

Grammar question - defining something additionaly

Let's say that I have a function that is defined on some intervals, and on some it's not. I'd like to say that the interval which was not defined, was defined additionaly (because others were already ...
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2answers
748 views

What does the 'triple line equals' sign with a strikethrough mean?

I understand that the triple equal sign can mean "identical to" or "defined to be equal to", so intuitively, I assume that the triple equal sign with a strikethrough means "is not identical to" or ...
0
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1answer
200 views

Definition: $a \prec b$ , $ a \npreceq b$ , $a,b \in \mathbb{R}$ [duplicate]

let be $a,b \in \mathbb{R}$: $a \prec b$ if $a \preceq b \wedge a \neq b$ $a \npreceq b $ if $b \prec a$ is correct? Thanks in advance!
7
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2answers
161 views

Definition of truth values in a topos.

I'm trying to understand what is meant exactly by a "truth value" in a topos. Take for example the topos of irreflexive graphs. It is known that the classifying morphism can take nodes to 2 different ...
8
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3answers
168 views

What is a series?

This question is rather pedantic, but it is something that has been bothering me for some time. Summing up infinitely many terms of a sequence is something that is done in pretty much every subfield ...
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2answers
98 views

What does “flat hypersurface” mean?

If $S$ is a flat hypersurface with boundary in $\mathbb{R}^n$, what does it mean? Is it just a simple open domain (found in most PDE contexts)?
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1answer
101 views

Alternative definitions of functions requiring non-empty domains?

It is easy enough to prove in set theory, but it seems counter-intuitive to me that an empty set could be the domain of a function. Is there any literature requiring that functions have non-empty ...
2
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2answers
212 views

What is the difference between two real numbers?

Let $x$ and $y$ be real numbers. What does the difference between x and y mean? $|x-y|$ or $|y-x|$? $x-y$? $y-x$? To me, only the first case makes any sense whatsoever. However, I cannot find a ...
1
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1answer
53 views

Big-$O$ notation definition.

I have come across the notation $$...+\ O_R(x)$$ where $R$ is any positive real number, and $R<2$. What is the term big $O_R$ of $x$ referring to?
5
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3answers
205 views

What is the name of this function $f(x) = \frac{1}{1+x^n}$?

$f(x\in\mathbb{R}) = \frac{1}{1+x^n}$
1
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1answer
82 views

confusion about rank and nullity of matrix and rank-nullity theorem

I can see that rank + nullity = number of columns of the matrix. Does that mean the dimension of matrix is the number of columns? Isn't dimension of a $m\times n$ matrix $mn$? Thanks.
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5answers
862 views

$\epsilon, \delta$…So what?

Over the course of my studies I often encounter phrases in reference material of the type "and this avoids the need for using $\epsilon$, $\delta$ definitions" or "by this we can omit those ...
2
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1answer
50 views

What is the defenition of $\mathcal{c}$ and $\aleph_1$ if we assume ZFC without CH.

I am reading an intro to a chapter of Andreas Blass called "Combinatorial Cardinal Characteristics of the Continuum" and I am getting a bit confused. When I studied "Discrete Mathematics", it was ...