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

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5
votes
1answer
131 views

An alternative definition of finite?

Does the following definition adequately characterize the notion of finite? Is it equivalent to, say, Dedekind-finiteness? A set $S$ is finite if and only if for all $x_0\in S$ and all $f:S\to S$, if ...
2
votes
3answers
3k views

What is a formal definition of “predicate logic”?

I'm currently trying to get clear about some terms that are often used in computer science (I'm a computer science student), but were never formally introduced. Especially, I would like to know what a ...
3
votes
1answer
261 views

Alternative definition of strong/weak operator topology.

Given two normed spaces $(X, ||\cdot||_X)$ and $(Y,||\cdot||_Y)$ the space of bounded linear maps $\mathcal{B}(X,Y)$ can be equipped with the strong operator topology (SOT) as follows: The ...
0
votes
1answer
126 views

Definition of “symmetric bilinear (real) form indefinite”

In my studies I use these definition: Def.: $f \in \mathscr{B}_ \Bbb{R}((e \times e), \Bbb{R}) $, $f $ is symmetric bilinear (real) form positive definite if 1) $\forall x \in e(f(x,x)\geq0)$ 2) ...
0
votes
1answer
76 views

Equivalent definitions of isometry

Consider a map $T:\mathbb{R}^2\to\mathbb{R}^2$ such that $\lVert T(x)\rVert=\lVert x\rVert$. Is this equivalent to stating that $\langle x, y\rangle=\langle T(x), T(y)\rangle$ for all ...
1
vote
2answers
146 views

Meaning of variables and applications in lambda calculus

The wikipedia definition of lambda terms is: The following three rules give an inductive definition that can be applied to build all syntactically valid lambda terms: a variable, $x$, is ...
1
vote
1answer
130 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
48 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.
-2
votes
1answer
54 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
2answers
964 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"?
0
votes
1answer
79 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 ...
0
votes
2answers
39 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 ...
1
vote
1answer
640 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?
0
votes
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
votes
1answer
229 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
votes
1answer
78 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, ...
0
votes
1answer
77 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 ...
0
votes
1answer
70 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 $d\omega(V,W)=(V^jW^i-V^iW^j) \frac{\partial f_i}{\partial x^j}$ ...
5
votes
6answers
278 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?
1
vote
0answers
37 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
votes
2answers
232 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\} ...
0
votes
1answer
59 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 ...
-2
votes
7answers
1k 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 ...
3
votes
2answers
138 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
votes
1answer
42 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
votes
1answer
109 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 ...
5
votes
1answer
56 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 ...
0
votes
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 ...
4
votes
3answers
441 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] ...
3
votes
3answers
246 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 ...
2
votes
0answers
59 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 ...
1
vote
1answer
67 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
votes
1answer
48 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
vote
1answer
223 views

Algebraic definition or construction of real numbers

Is there any algebraic definition or construction of real numbers ? If not, why ?
1
vote
0answers
73 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
votes
2answers
100 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
votes
1answer
27 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
vote
1answer
152 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}: ...
0
votes
2answers
68 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 ...
4
votes
2answers
208 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 ...
63
votes
19answers
17k 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
votes
1answer
338 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
votes
1answer
28 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
votes
1answer
38 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 ...
1
vote
2answers
2k 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
votes
1answer
213 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
votes
2answers
196 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
votes
3answers
176 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 ...
2
votes
2answers
113 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)?
-4
votes
1answer
121 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 ...