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

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Name for “relative difference”

If we want to express that two numbers $x, y$ are not so far away from each other absolutely we use the absolute value function $|\cdot|$ with $0 < \epsilon \ll 1$: $$|x-y| < \epsilon$$ ...
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1answer
60 views

definition of limit of function on topological spaces

Def.: let be $(A,\tau)$,$(C,\zeta)$ two topological spaces, $f \in C^E$, with $E \subseteq A$, and $x_0$ an accumulation point of $E$, a point $l \in C$ is limit of $f$ as $x$ approaches $x_0$ if ...
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2answers
223 views

An “elementary” approach to complex exponents?

Is there any way to extend the elementary definition of powers to the case of complex numbers? By "elementary" I am referring to the definition based on $$a^n=\underbrace{a\cdot a\cdots ...
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1answer
22 views

Forms - unitary group?

If $Id$ is the $n$ by $n$ identity matrix and $J$ is the $2n$ by $2n$ matrix with $Id$ in the upper right corner and $-Id$ in the lower left corner, then $Sp_{2n} = \{G\in Gl_{2n} : G^{tr}JG = J\}$ is ...
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1answer
11 views

Characters only on commutative unital algebras?

I saw the following definition of a character: Let $A$ be a commutative unital Banach algebra. Then a non-zero homomorphism $\chi : A \to \mathbb C$ is called a character. For this definition to ...
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11 views

Category of a PDE and its properties

Now I am working on numerical method for a PDE. I am considering the following PDE: $$ u_t+a^2u_{xx}=f\\ u(x,0)=u_0\\ u(x,t)|_\Gamma=u_g $$ That equation seems very like heat equation which only ...
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2answers
72 views

The sequence in the definition of the integral

In my high school Calculus class, we learned this definition of the definite integral: $$\int_a^b f(x)dx = \lim_{n\to \infty} \sum_{i=1}^n f(x_i) \frac{b-a}{n}$$ Now that I know more about sequences ...
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0answers
47 views

What is real periodic function

I would like to know what is real periodic function. I understand what is periodic function, but I do not understand what is "real" periodic function.
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68 views

What is “Triangulable Triad”?

I am reading George W. Whitehead's Homotopy Theory; Corollary 1.0.2 mentioned the term "Triangulable triad" without definition. May I know how it is defined?
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1answer
122 views

Perpendicular to Z axis or Skew to Z axis? (Definition of Perpendicular)

Question Part 1. Consider the following, where the point is the intersection of the sphere and a tangent plane. Consider a Euclidean coordinate system where: Blue dot is the origin (0,0,0). ...
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1answer
82 views

Formalizing the idea of a set $A$ *together* with the operations $+,\cdot$''.

I will illustrate my question in the case of the definition of vector spaces. It is custom to define a vector space in the following way: "Let $K$ be a field. Then a $K$-vector space is a set $V$ ...
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66 views

What is the formal meaning of “determine” in Baby Rudin 2.40?

In Theorem 2.40, Rudin talks about a $k$-cell $I$ formed by the intervals $[a_1, b_1], \ldots, [a_k, b_k]$. We split each interval at its midpoint $c_j = \frac{a_j + b_j}{2}$ and end up with $2k$ ...
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2answers
73 views

Should “together with” be taken as slang for an n-tuple?

When an algebraic structure is defined, it is often defined as a set $S$ "along with"/"together with"/"having" operations $\circ_1, \circ_2, \ldots, \circ_n$, and "denoted" by $(S, \circ_1, \circ_2, ...
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3answers
124 views

Why do we formalize conceptions?

Why do we always try to formalize conceptions? Let's take the naive conception of sets, why do we try to write down a list of axioms? what do we earn in doing so? I'm looking especially for ...
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1answer
100 views

Limit of vector-valued function is equal to the limit of its components

Let $f: \Bbb R^m \to \Bbb R^n$. Express $f(x)$ in terms of components: $$f(x)=(f_1(x), f_2(x), ... , f_n(x))$$ I need to prove that $f$ is continuous at $a$ if and only if each $f_i$ is continuous ...
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1answer
60 views

Example of a uniformly convex domain in $\Bbb R^n$

I am trying to understand the differences between a convex domain, and a uniformly convex domain. Intuitively, to my knowledge, a convex domain is one where any line between any two points in the ...
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3answers
138 views

Alternative to epsilon-delta definition

Back in Multivariable Calculus, I remember my professor, when explaining why he decided to skip a rigorous definition of limit (the reason was that those of us that would continue in math would go ...
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27 views

Definition of standard functions [duplicate]

In many texts and books about calculus we see There are functions $f$ for which the anti-derivative cannot be expressed in terms of standard functions or there are many integrals that cannot ...
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1answer
44 views

On the definition of commutators

We know that given a group $G$, for every $x,y\in G$ we define $[x,y]:=x^{-1}y^{-1}xy$, the commutator of $x$ and $y$. I saw something more general, commutators involving more than two elements, like ...
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1answer
52 views

Has this topology a name?

let $(X,\tau )$ be the topological space where $\tau =\{\emptyset, X, \{x\}, X-\{x\}\}$ , $ x \in X$. Does this topology have a name? Thanks in advance!
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3answers
4k views

What's the difference between direction, sense, and orientation?

I'm trying to understand the difference between the sense, orientation, and direction of a vector. According to this, sense is specified by two points on a line parallel to a vector. Orientation is ...
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1answer
86 views

Is there way to formalize the idea that a category can be “cocomplete from the inside”?

Let $\mathrm{KSet}$ denote the category of all countable sets, including the finite ones. Then $\mathrm{KSet}$ is finitely complete. Furthermore, $\mathrm{KSet}$ admits all countable colimits, or, ...
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2answers
33 views

Formalize definition of subbase of a topology

Def.: let be $(A,B)$ a topological space, and $C \subseteq B$, "$C$ is subbasis of $B$ if $$\{X|\exists X_1,X_2,...,X_n \in C(X=\bigcap_{i=1}^n X_i)\} \text{ is basis of } B$$ Is it correct?
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112 views

Definition of a principal ultrafilter

I just started trying to read through Lectures on the Hyperreals: An Introduction to Nonstandard Analysis, by Robert Goldblatt. I'm near the beginning in the section on filters. He's defined an ...
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1answer
67 views

On Constant mean curvature surfaces.

I have two involved questions, firstly, I know that the gauss map sends a surface to the unit sphere, so for a surface $\Sigma\subset\Bbb R^3$, parametrised by $u:U\subset\Bbb R^2\to \Bbb R^3$. Would ...
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203 views

What is a “subscheme”?

Every source I've looked at defines open subschemes and closed subschemes, but the definitions always look ad-hoc and not closely related to one another. Are there other kinds of subschemes? If not, ...
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1answer
98 views

Semantics and Logical structure in Definitons

Continuation of Free and bound variables in "if" statements definitions: A number is even if it is divisible by $2$. The number is even if it is divisible by $2$. Is the usage of the ...
2
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1answer
66 views

formalize definition of topology

In my studies I used this definition of topology, but I am reading on wikipedia a different definition... I thought to formalize: Def. let be $A$ a set and $B \in \mathcal{P}(\mathcal{P}(A))$, ...
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1answer
62 views

Logical structure of definitions

Here are some "concepts" that are confusing me: The assertion $x>2$ is true when we replace $x$ by a number greater than $2$($3$ for example) and false if we replace it by $1$. But the assertion ...
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1answer
40 views

A special kind of metric-spaces

Is there a special name for those metric-spaces or topological spaces in which every non-empty open set is uncountable ?
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1answer
111 views

Mystery of non-vanishing derivative

Studying complex line integrals.. I can't see why we include "non-vanishing derivative" in the definition of a smooth curve. And although not related -- is complex line integral a type of ...
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299 views

Is the set {a,b} uniquely defined?

First answer to this question would be yes, but consider the following question: How many elements has the set $\{a,\, b\}$? The answer to this question depends on $a$ and $b$: If $a=b$, then ...
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1answer
58 views

Please verify this definition of locally convex vector space

A locally convex vector space is a vector space $V$ equipped with a family $P$ of separating semi-norms. Is this a correct definiton? So to determine whether a norm induces a locally convex topology ...
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0answers
42 views

Terminology: Name for general, non-differential equations over $\mathbb{R}^n$

Let $x \in \mathbb{R}^n$ be a variable, and $f_1, \ldots , f_n : \mathbb{R}^n \mapsto \mathbb{R}$ be a sequence of functions, each having a closed form expression. Assume that we want to solve the ...
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2answers
96 views

Meaning of “There exists a proper class of…”

How a statement of the form "There exists a proper class of..." can be formalized in $\sf ZFC$? It sounds a bit like an oxymoron to me, because it's the very essense of a proper class that it does not ...
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3answers
476 views

2+2=4; Not in the Z3 algebraic group

I was reading the article/wiki here When I came across this quote ObviousFact?: examples: 2+2=4 for most people Those with higher mathematical knowledge may disagree - not in the Z3 ...
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2answers
113 views

Definition of Base-point

Consider the family of curves defined by $f(x,y)=g(x)+h(y)+a$, where $a$ is a free parameter. Now, it states that the family of curves intersect at $\infty$ and that $\infty$ is a base-point of these ...
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2answers
570 views

two notation: semi-metric and pesudometric

There are two notations: semi-metric and pesudometric make me unclear. Are they the same thing, or they are different? Thanks ahead.
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9answers
5k 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, ...
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0answers
52 views

Motivation for defining a functions codomain

Why not always refer to a functions image, what is the point in specifying some super set of that image and then naming that super set the functions "codomain"? What does explicitly defining a set ...
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1answer
122 views

Is the function $f(x)=1^x=1$ considered an exponential function?

I am confused about the following: The exponential function (by definition) is a function of the form $f(x)=a^x$ where $a>0$. However, when $a=1$, we get the constant function $f(x)=1^x=1$. Is the ...
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1answer
25 views

Order of cardinal number

I am puzzled. Wikipedia says: "|X| ≤ |Y| means that there exists an injective function from X to Y." Let's see sets A and B: A = {1,2,3} and B = {1,2}. f: A → B: 1 ↦ 1, 2 ↦ 2. f is injective, but |B| ...
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4answers
67 views

Why to see that $\overline{B}(x;r)$ is closed if it was just defined?

I'm reading Conway's A Course in Point Set Topology. He defines open and closed balls and then he introduces some examples, one of these examples is this: (c) For any $r>0$, $\overline{B}(x;r)$ ...
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1answer
24 views

Critical points of a function of absolute value

Say I have the function $f(x) = |x|$ I believe that $x = 0$ is a critical point, although not I'm not positive. As the function is decreasing and increasing each side of $x = 0$ does that alone make ...
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371 views

A circular proof in Rudin that $\mathbb{R}$ is a field.

Today I'm afraid, I found a circular reasoning in Rudin's Principles of Mathematical Analysis( I found no errata that mentions this). Before actually going through the actual question, I have ...
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2answers
118 views

The Definition of the Indicative Conditional

From Wikipedia, we have: In natural languages, an indicative conditional is the logical operation given by statements of the form "If A then B". Unlike the material conditional, an indicative ...
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1answer
97 views

Hessian matrix as derivative of gradient

From a text: For a real-valued differentiable function $f:\mathbb{R}^n\rightarrow\mathbb{R}$, the Hessian matrix $D^2f(x)$ is the derivative matrix of the vector-valued gradient function $\nabla ...
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31 views

Definition review: how to make this geometric definition clearer?

In a paper I am writing, I rely on the following definition Given a geometric shape $C$ and a family of geometric shapes $S$, The division number of $C$ relative to $S$, denoted $DivNum(C,S)$, is ...
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1answer
43 views

Definition of a continuous function

I am struggling to understand a basic definition of a continuous function from a textbook: A function f is continuous if for all x, and for all $\epsilon>0$, there exists $\delta>0$ such that ...
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2answers
363 views

Definition of logarithm in complex domain

My first question is: What is the proper definition of logarithmic function $f(z)=\ln{z}$. where $z\in \mathbb{C}$. quoting Wikipedia. a complex logarithm function is an "inverse" of the ...