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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3
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3answers
130 views

Has the opposite category exactly the same morphisms as the original?

This is actually a question about categories; not only about the category that I mention here specifically. I only use category $\mathsf{Rel}$ as an example. How to describe a morphism that ...
0
votes
0answers
15 views

Definition of weak convergence of an uncountable collection of measures

I am having trouble finding a definition for weak convergence of an uncountable collection of measures $\{ \mu_t\}_{t \in I}$ to a measure $\mu$, where $I$ is some index set ($\bf{R}$ in my case). The ...
4
votes
2answers
34 views

What is a transformation?

I am not a native English speaker and I have been pointed out that the word "transformation" as a synonym of "function" is grammatically incorrect. However, I even found a wikipedia and a mathworld ...
1
vote
0answers
28 views

Definition - Logarithmic Zeros and Algebraic Vector Fields

In Algebraic Geometry, what is the difference between an algebraic vector field and a derivation (are they completely different or synonymous)? What does it mean to say an algebraic vector field has ...
0
votes
1answer
15 views

Maximal rank definition?

What would it be to say that a linear map $T:\mathbb R^{m+n}\longrightarrow \mathbb R^n$ has maximal rank? I'd like a precise definition of it, I skimmed several linear algebra books after the ...
2
votes
2answers
140 views

Why we can't define $\frac{1}{0}$ to be $1$ (or anything else), but we can define $1^0$ to be $1$?

We know that we can't define division by zero "in any mathematical system that obeys the axioms of a field", because it would be inconsistent with such axioms. (1) Why can we define $a^0$ ($a\neq 0$) ...
1
vote
1answer
50 views

Definition of a parallelizable manifold

My text that I am self studying from says that a manifold $M$ is parallelizable if it has a trivial tangent bundle which means that there is an isomorphism $\varphi:M\times \mathbb{R}^n\rightarrow ...
0
votes
1answer
54 views

Help in this equivalence in Fulton's book

I'm studying Fulton's algebraic curves book and some of its notation look like confusing. I need help to understand this equivalence in the page 35. The preceding paragraph he mentioned I put ...
5
votes
4answers
81 views

Question on definitions

I was going through some basic recap of complex numbers and in the book (M. Boas. Mathematical Methods in the Physical Sciences) she says we define $e^{ix}$ by the Taylor series with $x$ replaced by ...
1
vote
1answer
24 views

What does it mean for a function to be uniquely determined by another function?

In munkres topology, I went through an exercise which asks me to show that a function is uniquely determined by another function. I wonder, What does this mean? I googled it but No answer! Here is ...
0
votes
1answer
14 views

Definitions of hemicontinuity

can anyone see the equivalence or relation between the following two definitions of hemicontinuity that I encountered: Assume that $K$ is a closed, convex subset of Banach space $X$. Let $X^{*}$ be ...
2
votes
2answers
100 views

$361$ degrees: acute or obtuse?

Recently I encounter a problem (of trigonometry) where $\sin{x}$ was asked and it was also told that $x$ is acute. So, I like anyone else, found the general solution, however my general solution ...
1
vote
1answer
35 views

What is the definition of Lindelöf space?

My definition for "countable set" is a set with the cardinal $\aleph_0$ and "at most countable set" is a set $A$ such that $|A|≦\aleph_0$. Till now, my definition for Lindelöf space is a topological ...
2
votes
1answer
70 views

A better definition of Polynomial

Usually, we define a polynomial as $a_n x^n + \cdots + a_1 x + a_0$ where $x$ is called indeterminate. Would it be better to define it as $a_n x^n + \cdots + a_1 x + a_0 x^0$ where $x^0$ means ...
2
votes
2answers
46 views

Characterizing a circle.

Is it correct to characterize a circle by saying that it is a closed curve in $\mathbb{R}^2$ such that all points on the curve are equidistant from a single fixed point? I am familiar with shapes ...
2
votes
0answers
42 views

Path-independent contour integrals and how to define them

If a contour $C$ is parameterized by $z(t): [a, b] \to \mathbb{C}$, then we define $$ \int_C f(z) \, dz= \int_a^b f(z(t)) \, z'(t) \, dt.$$ If the contour integral on the left side is equal to some ...
1
vote
5answers
78 views

What is the reason to introduce and study logarithmic functions?

I don't understand why logarithms exist when we have exponential functions. Exponential functions seem to be an easier and less convoluted way to write something. Why invent logarithms to do something ...
0
votes
0answers
60 views

Confused by definition of an open set in “All the Mathematics You Missed”

On page 66 of Thomas Garrity's "All the Mathematics You Missed", Garrity gives the following definition of an open set in $\mathbb{R}^n$: A set $U$ in $\mathbb{R}^n$ will be open if given any $a ...
2
votes
1answer
71 views

Do those 'groups' have a name?

Is there a name for 'groups' that only have a neutral element on the right and an inverse for each element on the right ? If there is, does that name also hold for a neutral elt on the right and an ...
4
votes
1answer
48 views

what are and why are sine and cosine modulated integrals used?

I have found the definition of the following formulas in a paper regarding active vibration control, where they are called sine and cosine modulated integrals. $y$ is measurement signal with a strong ...
1
vote
0answers
15 views

Non Borel Spaces: Gauge Integral

Is there a gauge integral over non Borel spaces? (My interest lies in finite measure spaces.) It seems as the definition of the gauge integral crucially depends on the existence of open sets for a ...
1
vote
0answers
29 views

Types of definitions

My question is about terminology. Consider the following two types of definitions of a circle: A circle is the collection of all points at the same distance from a given point. A circle is the ...
1
vote
2answers
89 views

Why in open balls is radius $r>0$?

the usual definition is the following: Def.1: let be $(a,f)$ a metric space, $c \in a$ and $r \in \Bbb{R}_{>0}$, the open ball of radius $r$ about $c$ is the set $$\mathcal{B}_f(c,r)=\{x \in a| ...
1
vote
1answer
35 views

Localization of the Integer Ring

Let $\mathbb{Z}$ be the ring of integers and let $p$ be a prime, then the $p$-localization of $\mathbb{Z}$ is defined as $\mathbb{Z}_{(p)}=\{\displaystyle\frac{a}{b}|a,b\in\mathbb{Z},p\nmid b\}$. I ...
0
votes
1answer
42 views

Is there a name for a property defined in terms of open sets?

We know that if a property is defined in terms of open sets then the property is preserved under a homeomorphism. Is there a name for such a property?
3
votes
1answer
33 views

Usage of the term $\arg(z)$

Consider the complex number $z = -1 - i$. Is it mathematically correct to say that $\arg(z) = 5\pi/4$? Sure, $5\pi/4$ is not the principle argument of $z$, but it is an element of the set $\arg(z)$. ...
0
votes
0answers
16 views

Conjugate actions

What is the definition for two group actions to be conjugate? For example a smooth action of a finite group on a manifold is locally conjugate to an orthogonal action.
0
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2answers
33 views

Could anybody provide a more detailed explanation of a tangent equation in its general form?

In my textbook I'm currently at the topic of a tangent line to an ellipsis and hyperbola. And there I've encountered this statement: If a curve has an equation $$ y = f(x) $$ then an equation of a ...
3
votes
0answers
64 views

Why not defining a measure as a function on functions?

A measure $\mu$ is a function to $\left[0,\infty\right]$ on the sets belonging to a $\sigma$-algebra. Then for integrable functions $f$ the integral $\int fd\mu$ comes in, having nice properties ...
0
votes
1answer
41 views

Definition of elliptic pde

http://en.wikipedia.org/wiki/Elliptic_partial_differential_equation Very confusing. So are there non-linear elliptic pdes? It seems to be the case. And what about $$A u_{xx} + C u_{yy} + F u = 0$$ ...
3
votes
1answer
87 views

Has anyone succeeded in formalizing Leibniz notation in such a way that the chain rule and inversion rule “work”?

The notation $\frac{\partial}{\partial x}$ is ubiquitous and totally useful, but also kind of weird. It seems to be doing the following: Bind $x$ Compute the derivative Evaluate at $x$ To ...
-1
votes
2answers
80 views

Definition of dimension

Let us consider Euclidean space $\mathbb{R}^n$. We say it is $n$-dimensional because each vector in it is an $n$-tuple $(x_1,...,x_n)$. However, it is possible to represent this exact same space using ...
1
vote
0answers
44 views

Rapidly Decreasing Functions

Can someone explain the notion of a rapidly decreasing function? Namely, a function in the Schwartz space: $$\mathscr{S}(\mathbb{R}^n):= \{ f \in C^{\infty} (\mathbb{R}^n) : ||f||_{\alpha, \beta} ...
0
votes
1answer
31 views

Definition of global field

This is very embarrassing thing to ask but what is a definition of global field? Every text or internet sources says either a number field or function field over finite field. (Yes, I understand there ...
0
votes
0answers
15 views

Definition of Non-characteristic Manifold

What is the definition of non-characteristic manifold in the following context. "Since $ f (x, y)=0 $ is assumed to be a non-characteristic singularity manifold, we have $ f_{x}\neq 0 $." Thanks, ...
0
votes
1answer
46 views

Why we use ANY in the definition of a maximal element?

I am confused about the following definition: "a maximal element of a subset S of some partially ordered set is an element of S that is not smaller than any other element in S." I do not understand ...
6
votes
3answers
126 views

Functions with different codomain the same according to my book?

My book gives the following definition: A function $f$ from $A$ to $B$ is defined as $f\subseteq A\times B$ such that if $(a,b)\in f$ and $(a,b_1)\in f$ then $b=b_1$ and there exists a $(a,b)\in ...
0
votes
1answer
40 views

Is a definition either intensional or extensional?

Is a definition either intensional or extensional? Can a definition be neither? Can a definition be both? How about this definition? when there is only one object that satisfies a definition, e.g. ...
2
votes
1answer
55 views

What's the difference between a cyclic and periodic function?

I've seen the words "cyclic" and "periodic" used to describe characteristics of a given function. What do they mean? I can't seem to find a difference. Wikipedia says a periodic function is one that ...
1
vote
1answer
25 views

Solutions depending on something continuously

Let $V$ be a a real Banach space, $K \subset V$ a closed convex set, $A: K \rightarrow V^{*}$ a (nonlinear) operator and $F \in V^{*}$. Then the variational inequality is the following problem: find ...
2
votes
4answers
59 views

What is a complex constant and how do I use it?

I have a question I am trying to understand: "Let $b$ and $c$ be complex constants such that $z^2+bz+c=0$ has two different real roots. Show that $b$ and $c$ are real." My biggest problem here is ...
4
votes
3answers
92 views

Why is this definition of complex numbers “informal”?

I'm reading the proofwiki page about complex number: https://proofwiki.org/wiki/Definition:Complex_Number According to proofwiki there is an informal and formal definitions of complex numbers. The ...
0
votes
1answer
52 views

How is this definition of a constant divided by zero called?

I divide a constant by zero. One example is the following: 2/0 My father told me he learned at school earlier that the result is "not defined". If I enter this arithmetic problem in Wolfram Alpha, I ...
0
votes
1answer
32 views

About definition of lexicographical order

Def. let be $\preceq_A$ a total order, $(a_1,a_2,...,a_n),(b_1,b_2,...,b_n) \in A^n$, $(a_1,a_2,...,a_n) \leq^d (b_1,b_2,...,b_n)$ if one and only one of the following holds is true: $$a_1 \prec_A ...
2
votes
1answer
61 views

Definition of disc and open ball

I have the following definitions in my notes for arbitrary discs and open balls - $$D^n = \{x \in \mathbb{R^{n+1}}: ||x|| \le 1\}$$ $$B^n = \{x \in \mathbb{R^{n+1}}: ||x|| < 1\}$$ The ...
0
votes
0answers
19 views

Vanishes of second order

I encountered the following statement in an article: Let $f(X) = f(X_1, ..., X_N) = \sum_{|\alpha| \geq 2} {a_{\alpha}X^{\alpha}}$ be a convergent power series which vanishes of second order at ...
5
votes
2answers
460 views

What is an ordered pair actually?

What does $(a,b)$ mean actually? I saw this in the 'formal defintion' of functions, and it tripped me up. We haven't even defined what an ordered pair is, before using it. Is it just a notation of ...
0
votes
1answer
34 views

What are the differences between a collation and a rule of formation?

I'm a beginner in mathematical logic, and currently studying(myself, without any colleague, which is sad and so asking in here) basics of formal system. Before asking a question, I'll introduce my ...
1
vote
0answers
57 views

Different definitions of uniform integrability

In different books and resources, I saw different definitions of uniformly integrable. For example, in some books the definition is like: Definition 1: $\{f_k\}\in L^1(E)$ is uniformly integrable ...
1
vote
3answers
107 views

Why isn't an injection an iif?

Suppose that $f:X \rightarrow Y$ is a function. Then an injection can be defined as: $\forall x_1,x_2 \in X, f(x_1) = f(x_2) \Rightarrow x_1=x_2$ Why isn't it defined instead as follows: $\forall ...