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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What actually constitutes a *definition* for a function?

I'm reading Enderton's text on logic trying to justify ( to myself ) the use of induction on recursively defined sets. This is of course used repeatedly in trying to prove results about well-formed ...
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Compact Set: Cover by Merely Neighborhoods

Disclaimer: This thread is just a record of thoughts and written in Q&A style. A subset is compact if every open cover admits a finite subcover. What if one replaces open covers with covers by ...
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What does it mean to categorify something? [on hold]

What does it mean to categorify something? Is there any simple illustrative example of this process? How it is done and what is it useful for?
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Why is there a 'missing' $1$ in the Euler–Mascheroni constant?

It is easy to show that: $$ \sum_{k=1}^n \frac{1}{k} > \ln(n+1), $$ but the Euler–Mascheroni constant is defined as: $$ \gamma = \lim_{n \to \infty} \left( \sum_{k=1}^n \frac{1}{k} - \ln(n) ...
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Discussion on Measures: Sigma-Additivity

Disclaimer: Though this thread is written in a Q&A style any new thoughts are really welcome! What reasons are there to restrict measures to countable additivity rather than uncountable ...
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polynomials and minimality

Could someone explain the concept of minimal polynomials? It seems like these are polynomials which cant be reduced further, but at the same time I am confused cause when we consider $\mathbb Z_2[x]$ ...
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Definition of quotient category

Is there any reason why only gluing of morphisms sharing domain and codomain is usually allowed in the definition of quotient category?
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What is a convex optimisation problem? Objective function convex, domain convex or codomain convex?

My teacher in the course Mat-2.3139 did not want to answer this question because it would take too much time. So what does a convex optimisation problem actually mean? Convex objective function? ...
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Basic question about analyticity vs. differentiability in complex analysis.

In chapter $V$ of Palka, "Consequences of the Local Cauchy Integral Formula," 3.1. If a function $f$ is analytic in an open set $U$, then $f'$ is analytic in $U$. In particular, $f$ belongs to ...
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What is $1 / \mathbf{Set}$ if $1$ is a one-element set and $\mathbf{Set}$ a category?

What does $1 / \mathbf{Set}$ denote? A pointed set is a set $X$ equipped with an element (a basepoint) $x \in X$. Let $\mathbf{Set_*}$ be the category of pointed sets and basepoint-preserving ...
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Can only one ordered pair be a relation?

I'm sorry, but I really can't find an answer to this no matter how deep I dig. A relation is defined as any set of ordered pairs. But what about a set of only one ordered pair? Is it still a ...
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Fountain code and online code

Are fountain code and online code the same? It seems to me they have the same property, which is used in lossy channel and generate unlimited encoded block. If they are the same, then what encoding ...
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What is the desirable function identification when setting up arrows in the category of types?

My question is which functions can not be allowed in a statically typed programming language, so that the "canonical" category is less coarse than what you get if you define it's arrows to be ...
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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 ...
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138 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 ...
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35 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 ...
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2answers
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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 $a^0$ ($a\neq 0$) ...
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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 ...
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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 ...
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1answer
53 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 ...
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Is the proof of non-totality of a function $f$ trivial if $f$ is recursively defined only on a subset of $\Bbb N$?

I define a fuction $*$ in a recursive way on a subset of natural numbers (with subset I mean that is $*$ define in the second argument only for $\Bbb N-\{0,1\}$ in other words we have $*:\Bbb N ...
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1answer
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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 ...
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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 ...
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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 ...
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1answer
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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 ...
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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 ...
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$361$ 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 ...
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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 ...
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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 ...
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1answer
71 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 ...
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2answers
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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 ...
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3answers
159 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 ...
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Precise definition of a “game of incomplete information” (Game Theory)

Question: In game theory, what is the precise definition of a "game of incomplete information"? What I've found so far: In the standard first year graduate economics textbook on microeconomics ...
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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 ...
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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 ...
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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} ...
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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 ...
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1answer
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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 ...
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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 ...
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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 ...
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972 views

Embedding, immersion

Could someone please explain what "embedding" means? (Maybe a more intuitive definition) I read that the Klein bottle and real projective plane cannot be embedded in ${\mathbb R}^3$ but is embedded in ...
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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| ...
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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 ...
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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?
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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 $\arg(z)$. ...
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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 ...
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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.
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Construction of the Hyperreal numbers

Several times I have seen questions/answers here about using the correct definition of derivatives. There are also questions about whether or not $1/0$ is defined. Sometimes there is a discussion ...
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Definition of determinant [closed]

Determinant is a certain function from the set of all $n\times n$ matrices to the set of scalars. How is the determinant defined? What characterizes the determinant function?
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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 ...