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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Abbreviate a tuple with a variable?

In a book I found the following notation $M = \{(x_1,y_1,z_1), \ldots, (x_n,y_n,z_n)\}$ for a set of 3-tuples. The author always refers to a tuple by writing $(x_i,y_i,z_i) \in M$. I'm wondering if I ...
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Determinant of a non-square matrix

I wrote an answer to this question based on determinants, but subsequently deleted it because the OP is interested in non-square matrices, which effectively blocks the use of determinants and thereby ...
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0answers
41 views

How do you call a scale that starts at $∞$, has $1/n$ divisions and tends to $0$?

A linear scale $2n$ divisions: 0 2 4 6 8 Logarithmic scales $10^n$ divisions: 1 10 100 1000 10000 ...
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73 views

Why do we define functions to be set theoretic objects?

Why do we define functions to be set theoretic objects? Functions are so intuitive, why do we define it in complicated set theory language?
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1answer
30 views

What does it mean for two conjectures to be incompatible?

What does it mean for two conjectures to be incompatible? I read about Incompatibility of two Hardy-Littlewood Conjectures. http://mathworld.wolfram.com/Hardy-LittlewoodConjectures.html What does ...
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1answer
52 views

Collection vs set in this textbook about category theory, and some related questions.

What is the meaning of collection in this context ? Is it here a synonym of set ? Can someone please explain what the author means by "A moment's though shows that, as sets of functions, these two ...
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2answers
184 views

Definition of forgetful functor [duplicate]

Is there an actual definition of "forgetful functor?" Wolfram MathWorld defines it in terms of functors from algebraic categories to the category of sets, but then says, "Other forgetful functors..." ...
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3answers
498 views

Can asymptotes be curved?

When I was first introduced to the idea of an asymptote, I was taught about horizontal asymptotes (of form $y=a$) and vertical ones ( of form $x=b$). I was then shown oblique asymptotes-- slanted ...
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91 views

How to define the disciplines of mathematics

Would you say that it is possible to give a unified, general definition of the different structures of mathematics and draw a clear distinction between them? I have been repeatedly trying to come up ...
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2answers
124 views

Categorial definition of free products?

If $X$ and $Y$ are objects of a concrete category $\mathcal{C}$, is there an accepted definition of "free product of $X$ and $Y$," generalizing the in the special case where $\mathcal{C}$ is the ...
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4answers
127 views

Why is the derivative at a jump undefined even if the slope remains the same?

I've searched online and found almost nothing. What in the mathematical definition of a derivative makes it so that the derivative of the following is undefined at ...
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1answer
25 views

Definition of a ring S being finitely generated as an R-algebra

What is the definition of a ring S being finitely generated as an R-algebra, where R is a ring?
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35 views

$\mathbb{Z}^{+}$ includes zero or not?

Does $\mathbb{Z}^{+}$ includes zero or not? I think that $0$ is not involved in the set of positive integers, but my book included zero in the set of positive integers in an answer.
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2answers
29 views

Limit definition of a sequence question

I'm trying to use the limit definition of a sequence to prove that the limit as $ n \rightarrow \infty$ of $\frac{1}{10^n}$ is equal to $0$. It is evident to see that this limit approaches 0,this is ...
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0answers
29 views

Meaning of proper antichain

I saw this sequence on the Online Encyclopedia of Integer Sequences, which describe the number of 3-element proper antichains of an n-element set. What does it mean to be a 3-element proper antichain ...
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1answer
48 views

When is a binary operation bipotent?

I learnt that $\max(-,-)$ is a bipotent binary operation but I'm not able to find a definition of bipotent operation. QUESTION A binary operation $*:M\times M \rightarrow M$ is bipotent if ...
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1answer
55 views

Definition singular manifold

I'm looking for the definition of a singular manifold. I haven't found it yet. For instance, in $\mathbb{R}^4$, with $f(x,y,z,t)=xy-zt$, $f^{-1}(0)$ is a singular submanifold. I only found a few ...
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1answer
15 views

What means $df(\tilde x) \in {\mathcal{L}(\mathbb{R}^n)}$

I'm trying to learn math on my own. The bad thing is, I can't google latex letters and they often have multiple meanings. For exmaple ${\mathcal{L}}$ could stand for lagrangian or something else. The ...
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3answers
55 views

Definition of Equals

DISCLAIMER: This is a first time Math.SE post from a 30-something who is only now learning math. I have read the Rules and this may not satisfy the "Questions with too many answers" criteria or ...
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30 views

Notation problem with a set of tuples and a metric

The first question: Assume we have tuples $T_i = (x_i, \vec{c}_i)$ ($x_i$ is the name of the object which is characterized by $\vec{c}_i$ in a d-dimensional space) and define a set of them $TS = ...
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2answers
36 views

A question about the boundedness theorem

I have a question about the boundedness theorem: http://en.wikipedia.org/wiki/Extreme_value_theorem The boundedness theorem which states that a continuous function $f$ in the closed interval ...
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1answer
68 views

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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8answers
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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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1answer
29 views

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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1answer
225 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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1answer
51 views

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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1answer
30 views

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
23 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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0answers
42 views

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
387 views

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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0answers
17 views

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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4answers
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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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1answer
45 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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1answer
27 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
11 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
9 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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0answers
8 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 ...
4
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2answers
115 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
67 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
41 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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1answer
23 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
35 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
87 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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0answers
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
39 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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28 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
51 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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1answer
28 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
27 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?