Definitions are the core of mathematical precision; they come to answer "what is X" in mathematics. Into this category fit questions regarding equivalence of definitions; clarifications regarding complicated definitions; as well questions with purposed definitions for mathematical notions with ...

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2
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1answer
175 views

define simultaneous substitution recursively

Can you help me with my approach to the following task: Define simultaneous substitution $\phi[\psi_1,...,\psi_k/p_1,...,p_k]$ recursively. Usually we have recursive definitions about formulas, but ...
6
votes
1answer
406 views

What do these arrows mean? (Froda's Thm)

I was reading "A Course in Probability Theory" by Kai Lai Chung, and in the book he was discussing discontinuity of monotonic functions, and after doing some searching online to learn more about the ...
4
votes
1answer
185 views

intrinsic and geometric definition of blow-up

Suppose I am given an algebraic variety $X$ and a (closed) point $x \in X$. I know of two descriptions of the blow-up of $X$ at $x$. One is intrinsic but not geometric: if $\mathcal{I}_x$ denotes the ...
3
votes
1answer
512 views

Definition of Separability Degree

For an assignment, I am trying to determine the separability degree of some algebraic field extension $L/K$. The definition of the separability degree of polynomial is not difficult to find at all, ...
6
votes
1answer
276 views

Are these definitions of submanifold equivalent?

Let $M$ be a manifold – by which I mean a second countable Hausdorff smooth manifold. Here's an "obviously" correct definition of (embedded) submanifold: Definition A. A subspace $S$ of $M$ is a ...
9
votes
3answers
2k views

Question about basis and finite dimensional vector space

I have seen the statement "Every finite dimensional vector space has a basis." (Here on page 5) I'm confused about what this tells me. It seems to tell me nothing: by definition, the dimension of a ...
3
votes
2answers
971 views

Local definition of Hölder continuity

What does it mean for a continuous function $ f $ on $ \mathbb{R} $ to be Hölder continuous with exponent $ \alpha $ at a point $ x_0 $ ? I only now the global definition: A function $ f $ on $ ...
2
votes
2answers
298 views

What does it mean for a stochastic process to be independent of a sigma algebra?

Does anybody know what it means for a stochastic process $ X = (X_t)_{t \geq 0} $ on a filtered probability space $ (\Omega, \mathcal{F}, \mathbb{F}, P) $ to be independent of a sigma-algebra $ ...
1
vote
1answer
176 views

What is the nature of the definition symbol?

A question about definitions and also notation, illustrated with a trivial example: Let $a,b,c\in\mathbb{N}$ and $$a:=2,$$ and $$b:=1$$ I postulate the the following formula holds ...
0
votes
1answer
191 views

Locally Euclidean

A quick definition check: Would someone please tell me what "locally Euclidean" means when applied to a Riemannian metric? I have kind of an idea about it, for example when we multiply by a ...
9
votes
1answer
355 views

How to motivate the axioms for the inner product

Typically, one doesn't just write down lists of axioms and then sees if there are enough interesting examples that satisfy them; they evolve over time, usually from a couple of very ...
3
votes
3answers
849 views

What are the “whole numbers”?

Just recently, I attempted to answer a question involving "whole numbers", but discovered that my long-held assumption (that they're the same as the integers), is not universal. [In fact, it seems I ...
3
votes
0answers
83 views

Closed / embedded surface

Given a closed surface in $\mathbb R^3$, is it necessarily an "embedded surface"? I think it is true, but that is just because I can't think of a closed surface for which we cannot construct a smooth ...
0
votes
1answer
308 views

MATLAB's implementation of cross correlation

Wikipedia gives the cross-correlation as $$ \begin{align*} (f \star g)[n] = \sum^{\infty}_{m = -\infty} f^{*}[m] g[n+m] \end{align*} $$ MATLAB's documentation gives ...
0
votes
1answer
142 views

What is the X, Y, Z “resolution” of a three-dimensional grid of points?

I came accross a software which requires the X, Y and Z resolution of a three-dimensional grid of points as Integer. What is a "3D grid resolution" and how do I find it? From what I understand, the ...
7
votes
1answer
470 views

Why $\lim\limits_{n\to \infty}\left(1+\frac{1}{n}\right)^n$ doesn't evaluate to 1?

I am trying to identify what the flaw is exactly when reasoning about a limit such as the definition of $\mathbf e$: $$ \lim_{n\rightarrow \infty}\left(1+\frac{1}{n}\right)^n={e} $$ Now, I know ...
1
vote
1answer
77 views

In category theory, is there any such thing as “compatibility” for arrow composition?

Is this a properly defined category? Objects $\{P, R, S\}$ Arrows $f_{1} : P \rightarrow R$ $f_{2} : P \rightarrow R$ $g : R \rightarrow S$ $h_{1} : P \rightarrow S$ $h_{2} : P \rightarrow S$ ...
2
votes
3answers
109 views

Proving the relationship $1 + r \leq \left(1 +\frac{r}{m}\right)^m$ for any $m \geq 1.$

I am currently trying to prove the following relationship $$1 + r \leq \left(1 +\frac{r}{m}\right)^m\quad \text{for any }m \geq 1.$$ Would you be so kind and provide some hints/solutions to the ...
2
votes
4answers
111 views

What are possible variations of the definition of $\sigma$-additivity?

From what I've read in Wikipedia, $\sigma$-additivity is defined in the following way: Let $\mathcal{A}$ be a $\sigma$-algebra of some underlying set $X$ and let $f$ be a mapping ...
3
votes
1answer
625 views

Free group and universal property

I'm trying to understand universal properties. An example is the definition of a free group (as I understand it so far): Revised definition: A free group $F_S$ over a set $S$ is a pair $(g,F_S)$ ...
19
votes
3answers
618 views

Conflicting definitions of “continuity” of ordinal-valued functions on the ordinals

I've encountered the following definition in Kunen, Levy, and other places: A function $\mathbf{F}:\mathbf{ON}\to\mathbf{ON}$ is continuous iff for every limit ordinal $\lambda$, we have ...
2
votes
1answer
131 views

The precise definition of a “sheaf of rings”

Let $X$ be a topological space. Let $Op(X)$ be the category of open sets. A sheaf of abelian groups is a (contravariant) functor $F:\mathcal{C} \to Ab$ satisfying the sheaf condition: the following ...
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0answers
65 views

Definition of fragility

What does it mean for a solution to a system of differential equations to be fragile? A context for the term can be found here: This is taken from here in Mathematical Methods for Mechanics: A ...
0
votes
2answers
1k views

real numbers and number line

While reading some articles, I got a bit confused by the definitions of numbers. Specifically, Can the number line contain decimal values? I read that Real numbers = All numbers on the number ...
18
votes
5answers
17k views

What is the difference between Singular Value and Eigenvalue?

I am trying to prove some statements about singular value decomposition, but I am not sure what the difference between singular value and eigenvalue is. Is singular value just another name for ...
5
votes
1answer
188 views

What set operation is this?

Given two sets $ A = \{\{1\} , \{2 , 6\} \}$ and $ B = \{\{2\} , \{3\} , \{4 , 5\} \}$, what set operation can produce $$ C = \{ \{ 1 , 2 \} , \{ 1 , 3 \} , \{ 1 , 4 , 5 \} , \{ 2 , 6 , 2 \} , \{ 2 , ...
0
votes
1answer
79 views

Which distribution can be abbreviated as “LD”?

Which distribution can be abbreviated as LD and which PDF is expressed as a formula with sum of erfc() functions? $$p(o)=\frac{1}{4\ell} ...
5
votes
2answers
333 views

the definition of the area of a surface

When we say the area of a rectangle is the product of the length by the width is it a definition based on geometric intuition or is it a result? I know it is a result that we can find after defining ...
0
votes
1answer
99 views

Definition of the 1-dimensional $\mathbb{C}GL(V)$ module “$\det ^n$”

I'm reading through my notes on representation theory of $S_n$ and $GL(V)$, and have come unstuck on a definition which I can't understand - furthermore I can't seem to find any information on it ...
1
vote
1answer
124 views

Definition of “succession of central extensions of abelian groups”

What is the meaning of the phrase: "A group $G$ can be realized as a succession of central extensions of abelian groups"?
3
votes
2answers
1k views

A simple explanation of differential calculus and its link to geometry?

The wikipedia articles on differential calculus and differential geometry are quite long and not so straightforward for a layman like me. Is there a master of math vulgarization out there that could ...
0
votes
1answer
51 views

Math vocab: operator on $S$ and into $S =$?

Is there a special name for a binary operation on the set $S$ that is also into $S$, that is unambiguous with other uses. I.e. if it's "operator on $S$", I've heard that in other places meaning the ...
3
votes
1answer
117 views

What do mathematicians call the Two's Complement on 8-bits group?

It is isomorphic to $\mathbb{Z}_{2^8},$ only difference is the symbols usually identifying the elements of the set are from $\{-128, \ldots, 127 \}$ and not $\{0, \ldots, 256\}.$ What is an elegant ...
5
votes
2answers
320 views

Is 2+2=4 an identity?

I know this seems like a silly question, but someone was trying to debate with me about how 2+2=4 should be called an identity and not an equation. I mentioned how it has no variables and isn't true ...
5
votes
5answers
467 views

What's the difference between tuples and sequences?

Both are ordered collections that can have repeated elements. Is there a difference? Are there other terms that are used for similar concepts, and how are these terms different?
3
votes
2answers
998 views

$\{0,1\}^n$ and $[0,1]^n$ notations

Can someone please help me clarify the notations/definitions below: Does $\{0,1\}^n$ mean a $n$-length vector consisting of $0$s and/or $1$s? Does $[0,1]^n$ ($(0,1)^n$) mean a $n$-length vector ...
4
votes
2answers
132 views

Defining a Differentiable Function in a Non-Pointwise Manner

Is there a way to define a differentiable function in a non-pointwise manner? That is without defining function differentiable at a point first. Just like we define a continuous function through open ...
1
vote
0answers
58 views

Are equidimensional ideals and unmixed ideals the same?

Zariski-Samuel define an unmixed/equidimensional ideal to be one whose associated primes have the same dimension. At other places I have seen definitions saying unmixed=all associated primes have same ...
1
vote
1answer
151 views

Name of binary relation: if $(x, y)\in R$ then there is no $z$ such that $(y, z)\in R$

Is there a term for a binary relation $R\subset A^2$ on some set $A$ such that if $(x, y)\in R$ then there is no $z$ such that $(y, z)\in R$ ? Are there any examples of it? Are there any related ...
0
votes
1answer
375 views

Pairwise non-integral numbers

I have a set of complex numbers a_1 through a_n which are said to be "pairwise non-integral numbers". Could someone explain to me exactly what this means? Thanks. From comment below: I should also ...
2
votes
2answers
481 views

What are the primitive notions of real analysis?

My dad introduced my to primitive notions in geometry in high school. It's come back to haunt me as I study real analysis; I find myself wondering, Have we given this a formal definition? ...
2
votes
3answers
249 views

A question about a certain way to define mathematical objects

It is common in mathematics to see definitions of the following form: we begin with a certain object $A$. we perform some construction depending on a choice of some parameter $\lambda\in\Lambda$ for ...
1
vote
1answer
66 views

Does this function have a “name”, somewhat linked to Euler totient

If $\varphi$ denotes the Euler totient and $n=p_1^{k_1}\cdots p_r^{k_r}$ is the prime factorization of $n>1$ we have $ \varphi(n)= \varphi(p_1^{k_1}) \varphi(p_2^{k_2}) \cdots\varphi(p_r^{k_r})= ...
1
vote
2answers
125 views

Elaborate on $A^{c}:=\{p\in\mathbb Q : 0<p<\sqrt{2}\}$ not open and not closed in $\mathbb R$

I know that $\sqrt{2}\not\in\mathbb Q$ and $\sqrt{2}\in\mathbb R$ but it is not obvious to me why $\{p\in\mathbb Q : 0<p<\sqrt{2}\} \subset \mathbb R$ is not open. If it is not open, it means ...
1
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3answers
73 views

A parametrized surface

If I am given a map $f:U\subset \mathbb R^2 \to \mathbb R^3$ where $(x,y)\to (f(x,y),g(x,y),h(x,y))$. Is this necessarily a "parametrized surface"?  Am I right in thinking that any map of the above ...
-1
votes
2answers
232 views

Definition of a point and object

Is there any theory in which a point has a definition? What is the definition of "object" as seen in category theory?
51
votes
9answers
3k views

What makes elementary functions elementary?

Is there a mathematical reason (or possibly a historical one) that the "elementary" functions are what they are? As I'm learning calculus, I seem to focus most of my attention on trigonometric, ...
5
votes
1answer
523 views

Understanding various definitions of TREE($n$) in Friedman's finite form of Kruskal's tree theorem.

I was reading the Wikipedia article on Friedman's finite form of Kruskal's tree theorem, and am interested in the large numbers TREE(n). I would like to verify TREE(2)=3 myself, but find conflicting ...
1
vote
3answers
2k views

why do we use 'non-increasing' instead of decreasing?

In english based math language it seems that non-increasing $\Longleftrightarrow$ less or equal (non-strict decreasing) decreasing $\Longleftrightarrow$ strict less ( strict decreasing) ...
1
vote
1answer
244 views

What is the definition of an induced cut

I am reading A simple min-cut algorithm (Stoer & Wagner, 1997), and the proof uses some terms I don't understand. Specifically I am unclear on what is meant by "$C_v$ the cut of $A_v \cup \{v\}$ ...