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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Uniform Space: Neighborhood System [closed]

Disclaimer: This thread is meant informative and therefore written in Q&A style. Of course everybody is encouraged to give an answer as well! As the idea behind uniform spaces is to represent a ...
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If A = B, then B = A… Not Always True? Definition of “=”

A friend and I recently got into a silly argument where I stated A = B so B = A. He stated this was not always true. After asking for an example he stated ...
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Contractible objects in model categories [closed]

Please, what is the definition of contractible object in a (closed) model category (if it exists)?
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235 views

Topology: Opens vs Neighborhoods

Disclaimer: This thread is meant informative and therefore written in Q&A style. The problems are highlighted in bold face. The axiomatization of topology can be done in various ways all of ...
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Defining principal elements of every poset. Is this a new idea?

Fix an arbitrary complete lattice $\mathfrak{A}$ with order $\sqsubseteq$. I call elements $a,b\in\mathfrak{A}$ intersecting and denote $a\not\asymp b$ iff there is a non-least element ...
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definition of unitary operator

Wiki says " A bounded linear operator $U: H \to H$ on a Hilbert space $H$ is called a unitary operator if it satisfies $U^{*}U=UU^{*}=I$ , where $U^{*}$ is the adjoint of $U$, $I$ is the identity ...
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Equivalent Definitions of Negative Order Sobolev Spaces

Ignoring fractional sobolev spaces, if we restrict ourselves to $k>0$ when $k$ is an integer, then the Sobolev space of order $k$, for $W^{k,p}(\mathbb{R})$ is the space of functions $f$ such that ...
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2answers
24 views

definition or property of logarithms

I've seen a lot of complicated logarithm definitions on this StackExchange and I have a rather simple question: $$a^{b}=c \leftrightarrow \log_a{c}=b$$ Is this a definition of logarithms, which all ...
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What's the name of this type operator?

If $H$, is a seperable Hilberspace, $E$ seperable Banach space. $(h_n)$ orthonormal basis of $H$. Let $T\in B(H,E)$. If we have the condition on $T$ that $$\sum_k \left\|Th_k\right\|^p <\infty,$$ ...
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2answers
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Difference between being faithful and being injective on arrows

I'm studying the concept of faithful functors, but I cannot grasp the difference between being faithtful and being injective on arrows. Could someone explain the difference and provide some examples? ...
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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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Why can't we define more elementary functions?

$\newcommand{\lax}{\operatorname{lax}}$ Liouville's theorem is well known and it asserts that: The antiderivatives of certain elementary functions cannot themselves be expressed as elementary ...
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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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1answer
33 views

contact point and point of intersection

I am just unable to understand the definitions of contact point and point of intersection.My doubts can be summed up into the following two questions : 1) Suppose $f(x)=x^2$ and $g(x)=0$ are two ...
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1answer
13 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
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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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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
124 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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2answers
68 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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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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1answer
47 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
46 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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3answers
107 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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40 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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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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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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33 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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2answers
29 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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78 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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421 views

What is the definition of 'within one' in mathematics

I need help with the definition of "within 1": If $x = 8$ and $y = 7$, then $x$ is "within 1" of $y$. If $x = 8$ and $y = 9$, then $x$ is "within 1" of $y$. If $x = 8$ and $y = 8$, is $x$ still ...
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91 views

Topology: Interior

The neighborhood filters satisfy: $$\forall N\in\mathcal{N}(x):\qquad x\in N$$ $$\forall N\in\mathcal{N}(x)\exists M_0\in\mathcal{N}(x):\qquad N\in\mathcal{N}(m)\text{ for all }m\in M_0$$ Define the ...
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3answers
111 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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76 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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Strong Notion of Integral

Is there a strong(!) notion of integral that can face all of those issues: Singularities Decay Modes Oscillations Measure Spaces Locally Convex Spaces For example combining decay modes with ...
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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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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
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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 ...
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1answer
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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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39 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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85 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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“Identify” terminology

What does it mean to "identify" two objects in algebra, as distinct from having an isomorphism or automorphism between them? I was led to think about this while reading Stewart's Galois Theory, where ...
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282 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
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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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Dyadic expansion: Definition of $F^{i-1}\omega$?

In Billingsley probability and measure under Dyadic expansion it is stated: dinfe a mapping $F$ from $\Omega=(0,1]$ into itself given by $$F\omega=\begin{Bmatrix} 2\omega & \mbox{if } 0< ...
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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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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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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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1answer
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Ordered tuples of proper classes

From time to time I encounter notation like this: A triple $\langle \mathbf{No}, \mathrm{<}, b \rangle$ is a surreal number system if and only if ... The confusing part is that a proper class ...
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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 ...