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

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0
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2answers
31 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
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1answer
230 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
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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
votes
1answer
101 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
59 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
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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 ...
0
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1answer
63 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 $d\omega(V,W)=(V^jW^i-V^iW^j) \frac{\partial f_i}{\partial x^j}$ ...
0
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1answer
54 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 ...
1
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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 ...
1
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1answer
50 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 ...
4
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1answer
48 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
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1answer
38 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 ...
2
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2answers
183 views

Is everything an expression?

Is everything that you can write in math (that makes mathematical sense) an expression? If not, what would be examples of non-expressions? And would all expressions be composed of expressions ...
0
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1answer
76 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 ...
0
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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 ...
2
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0answers
49 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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2answers
1k views

What is the difference between isomorphism and homeomorphism?

I have some questions understanding isomorphism. Wikipedia said that isomorphism is bijective homeomorphism I kown that $F$ is a homeomorphism if $F$ and $F^{-1}$ are continuous. So my question ...
1
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0answers
63 views

With infinite size, we can have $P \cdot M = M \cdot D $ (D diagonal) but where $M^{-1}$ does not exist. Can we say “P is diagonalizable”?

(I had this question in mind for longer time, but it is just triggered now by some comments at that recent question in mse) (Background) I was looking at properties of the Pascal-matrix: ...
1
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0answers
59 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 ...
46
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10answers
3k views

Given real numbers: define integers?

I have only a basic understanding of mathematics, and I was wondering and could not find a satisfying answer to the following: Integer numbers are just special cases (a subset) of real numbers. ...
2
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1answer
42 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 ...
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0answers
126 views

Algebraic definition or construction of real numbers

Is there any algebraic definition or construction of real numbers ? If not, why ?
3
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2answers
91 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 ...
0
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2answers
63 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
votes
2answers
175 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 ...
-1
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2answers
172 views

Example of a manifold?

Why is this picture an example of a $1$-dimensional manifold? My thought process is: the circle must have a point removed from it because otherwise it would be self-intersecting, and self-intersection ...
3
votes
1answer
260 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
34 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 ...
7
votes
2answers
152 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 ...
1
vote
2answers
556 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 ...
3
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0answers
187 views

Exercise on measure theory

I have this exercise: Let $\mathscr {S}$ be a collection of subsets of the set $X$. Show that for each $A$ in $\sigma(\mathscr S)$ there is a countable subfamily $\mathscr C_0$ of $\mathscr S$ ...
0
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1answer
196 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!
2
votes
2answers
94 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)?
7
votes
3answers
167 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 ...
-4
votes
1answer
98 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 ...
5
votes
1answer
273 views

Question about definition of regular projection of a knot

One can define a knot in two ways: (1) A knot is a closed polygonal curve in $\mathbb R^3$ (2) A knot is an equivalence class of embeddings $S^1 \hookrightarrow \mathbb R^3$ And perhaps also: (3) ...
2
votes
2answers
166 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
vote
1answer
52 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?
1
vote
1answer
74 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.
4
votes
2answers
300 views

What are some physical, geometric, or otherwise useful interpretations of divergent sums?

Is there any practical application to discussing the 'sum' of sequences that are not convergent under Cesàro summation? Why would we want to assign a value to an otherwise divergent sequence and ...
2
votes
1answer
48 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 ...
3
votes
0answers
138 views

Zariski cotangent space, as defined in Arapura's “Algebraic Geometry over the Complex Numbers”

In "Algebraic Geometry over the Complex Numbers", Arapura gives the following definition: Definition 2.5.8. When $(R, m, k)$ is a local ring satisfying the tangent space conditions, we define its ...
3
votes
2answers
157 views

Radius of Convergence and its application to a Power Series including $x^{2n}$ rather than $x^n$

(Radius of Convergence) Consider the Power Series $f(x)=\sum_{n=0}^{+ \infty}a_n x^n$, the radius of convergence $\rho$ can be found using $$\rho = \displaystyle \lim_{n \to + \infty} \left| ...
1
vote
1answer
82 views

Definition of functions on metric spaces.

In the post Definition of functions, it is stated in the accepted answer that one way to define a function is to define it as the triple $(f, X, Y)$ where $f \subset X \times Y$. My question is what ...
0
votes
1answer
105 views

A question concerning Ulam's Theorem from Oxtoby's “Measure and Category”

I am reading the following theorem from Oxtoby's Measure and Category Theorem 5.6 (Ulam). A finite measure $\mu$ defined for all subsets of a set $X$ of power $\aleph_1$ vanishes identically if ...
2
votes
1answer
55 views

Axiom or Postulate?

In wikipedia we see that the words “axiom” and “postulate” are synonyms: “An axiom, or postulate, is a premise or starting point of reasoning”. But in A Friendly Introduction to Numerical ...
0
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2answers
43 views

In a commutative algebraic theory, do all constant symbols necessarily represent the same value?

Let $T$ denote a commutative algebraic theory with two nullary function symbols $a$ and $b$ (i.e. constants). Is it an automatic consequence of the definitions that $a=b$ is a theorem of $T$? My ...