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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1answer
37 views

What does “weakly compact” mean when applied to subsets $X \subset Y$?

Let $X$ be a subset of a Banach space $Y$. Please can you give me a definition of what "$X$ is weakly compact" means? I want one which is in terms of sequences and boundedness, as opposed to one with ...
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3answers
94 views

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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0answers
16 views

Question about definition of some classes of bimodules.

Suppose that we have a ring $R$ and a $R$-$R$ bimodule $M$ such that: For every $r\in R$ and $m\in M$ there exists $r'\in R$ such that $mr=r'm.$ I want to know if there is a name for this kind of ...
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0answers
33 views

Confused about the definitions of atom

I went though half a dozen books on measure theory, and it occurs to me that the definition of atom is not particularly unified. Version $1$: A set $E$ in a $\sigma$-algebra $\Sigma$ is called ...
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1answer
57 views

Multiplication of positive fractional numbers.

I am reading this answer. We know that multiplication of two positive rational number $\dfrac{a}{b}$ and $\dfrac{r}{s}$ respectivly is defined as follows: $\dfrac{a}{b}\times\dfrac{r}{s} = ...
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2answers
54 views

A question in Isomorphism

Let G be a cyclic group. Soppose G and G' are isomorphic groups. Show that G' is also cyclic. Can Someone Solve this pleaase? I have an exam 2 hours later!
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1answer
33 views

“internal” definition of complete regularity?

There is something strange (I think) about the complete regularity separation axiom. Consider the definitions. T0 means for every two distinct points there is an open set containing exactly one of ...
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1answer
47 views

Nonorientable surfaces: genus or demigenus?

The genus $g$ of a closed, orientable surface is the maximum number of disjoint simple closed curves that can be drawn on the surface without disconnecting it. In terms of the Euler characteristic, ...
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2answers
63 views

On the canonical isomorphism between $V$ and $V^{**}$

I am trying to understand more about the Bidualspace (or double dual space). The whole idea is that $V$ and $V^{**}$ are canonically isomorphic to one another, which means that they are isomorphic ...
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3answers
154 views

Definition of General Associativity for binary operations

Let's say we are talking about addition defined in the real numbers. Then, by induction we define $\sum_{i=0}^{0}a_i=a_0$ and $\sum_{i=0}^{n}a_i=\sum_{i=0}^{n-1}a_i+a_n$ for $n> 1.\:$ Now, how do ...
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2answers
67 views

Notation for Partial Functions

Suppose we have sets $f$, $A$ and $B$ such that $f\subset A\times B$ and $\forall x\in A\space \forall y,z\in B: [(x,y)\in f \land (x,z)\in f \implies y=z]$ i.e. $f$ is a partial function mapping ...
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1answer
17 views

Beginning Proof on functions and Sum of functions

I've been having a hard time with this proof because I do not know where to go from where I am. The proof we were assigned to in class is as follows: Let $f : \mathbb Z \rightarrow \mathbb Z$ be a ...
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2answers
39 views

Notation - Transpose of Block Matrices [Lay P121 Q2.4.12]

Definition of Transpose is $(A^T)_{ij} = A_{ji}$ $1.$ Why $\begin{bmatrix} M & N \end{bmatrix}^T = \begin{bmatrix} M^T \\ N^T \end{bmatrix}$, and NOT $\begin{bmatrix} M \\ N\end{bmatrix}$? ...
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1answer
46 views

What are some practical uses of functions? [on hold]

Functions are basically formal equations that relate a set of inputs to output. What are some practical uses for functions and inverse functions?
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1answer
26 views

Different ways to formally define trigonometric functions

When I first learnt trigonometric functions I was in highschool and obviously the explanation they gave me was mostly intuitive. Now that I have taken my first curse of calculus I learnt a formal ...
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1answer
24 views

What is the definition of sigma field generated by random variable $X$? [on hold]

What is the definition of $\sigma$-field generated by a random variable $X$? And what does it mean?
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1answer
21 views

Definition of an integrand

General Question: Say we have integral $$ \int f(z)\ dz $$ Is the integrand in this context (i) $f(z)$ or (ii) $f(z)\ dz$? In any case, is $f(z)\ dz$ a formally defined mathematical object in its ...
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1answer
15 views

what is the definition of linearly independent subset of abelian group?

WHat is the definition of linearly independent subset of abelian group? I know the concept of this, but i don't know how to define this term precisely. Below is what i tried to formulate: Let ...
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1answer
43 views

Linear maps and matrix coefficients

I am currently working through this page in my script: Can somebody explain what this means and how it works in practice? Perhaps if I saw an example I could follow it. Thanks for your help!
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1answer
24 views

What is the definition of “disjoint cycles”?

I'm the one who thinks clear definition(clear with meta-language) is very important for doing mathematics. Below, i list my definitions for cycle and orbit. Let $X$ be a nonempty set. Let ...
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3answers
358 views

A doubt in the rigorous definition of limits.

I studied the definition of limits today, and I think I mostly understood it, but I have a little doubt. In the definition: $f(x)$ is defined on some open interval containing $a$, except at possibly ...
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2answers
31 views

What is “the orbit of a permutation”? Is the term “orbit” consistent with that for group action?

reference: What is the orbit of a permutation? To be honest, i don't understand the answer in the link. The orbit of a group action is defined as follows: Let $G$ be a group acting on a set $X$. ...
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1answer
66 views

Integral definition of e

I know that $e$ can be defined via a convergent series: $$ e = \sum_{n=0}^\infty {1\over n!}$$ Or as a limit: $$ e = \lim_{n \to \infty} { \left(1 + {1 \over n}\right)^n }$$ Or as the value which ...
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3answers
103 views

What is the “Principle of permanence”?

While reading the book "The Number-System of Algebra (2nd edition)." term "Principle of permanence" occurred to me. I remember I had read this in the book "Beginning algebra for college students.". I ...
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1answer
27 views

Lipschitz continuity of $f(x,y)=4x^2+xy-\frac{1}{y-1}$ on an open set $U \subset \mathbb{R} \times (\mathbb{R} \setminus \lbrace 1 \rbrace)$

Problem: Find an open set $U \subset \mathbb{R} \times (\mathbb{R} \setminus \lbrace 1 \rbrace )$ which includes the points $(0, 1/2$) and $(0,3/2)$ such that the function ...
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4answers
228 views

What is the definition of a positive integer?

I am reading the book "The Number-System of Algebra (2nd edition)". At the starting of page-4 the author writes: A positive integer is a symbol for the number of things in a group of distinct ...
4
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1answer
34 views

Definition of tangent vector

I have a small bit of confusion with the definition my text is providing me with for a tangent vector. Given a manifold $M$, it is first stated that to define a tangent vector, a curve $c:(a,b) ...
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1answer
112 views

Conglomerate in mathematical literature.

I rememeber I downloaded a pdf in category theory and after talking about classes it defined something called conglomerates, but I can't remember what that is and wikipedia has no article on it. ...
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1answer
74 views

What is an operator norm?

I came across this symbol where they are mentioning operator norms. I am not sure how it is different from the Frobenius norm. It was something like this: $|||\Omega-\hat{\Omega} |||_2$ where ...
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6answers
114 views

Orthogonal Projections: Definition + Characterization

Hey there I'm hanging at so many little steps when proving: $P\text{ orthogonal projection}\iff X\text{ orthogonal decomposable}$ My problem is there is simply missing a precise definition of what an ...
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0answers
24 views

What is the definition of a norm in the context of rings?

On several places in ring-theory I encountered so-called 'norms'. For instance on integral domain $\mathbb{Z}\left[i\right]$ with prescription $a+bi\mapsto a^{2}+b^{2}$ where it also serves as a ...
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6answers
539 views

What does “homomorphism” require that “morphism” doesn't?

I'm starting to learn category theory, but there's one thing I don't get: all morphisms seem to be homomorphisms; the definition seems to be the same. What's the difference between these two? Can you ...
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0answers
31 views

The term $rank$ in methematics

Reading wikipedia's disambiguation page about the "rank" word I see many concept of rank of many different matematical object. I only know about the rank of a graded poset and the rank of a set that ...
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1answer
35 views

What is the meaning of an algebra?

An algebra $A(*,\hat{} ,\sim)$ is said to be Boolean algebra if it satisfies some conditions...In this statement what is the meaning of starting word an algebra?
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2answers
195 views

The phrases “has … in ” vs. “contains … of” in Baby Rudin

Consider the following two statements. (Assume $E \subseteq K$.) $E$ has a limit point in $K$. vs. $E$ contains a limit point of $K$. What do they each mean and how are they different?
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2answers
54 views

Tensor product is o?

$K$ generated by $(ar,rb)$ for every $a\in A$, $b\in B$, $r\in R$. $F$ generated by $r(a,b)$ for every $a\in A$, $b\in B$, $r\in R$. $(a,rb)=(ar,b)=r(a,b)$,then $K=F$, $F/K=0$ Apparently I ...
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Fixed point - definitions (asymptotically stable, repelling)

Given the following fixed point $(\widehat{x},\widehat{y} )$. If this point is not asymptotically stable, can I then conclude it is a repellor? Is a repellor/repeller defined to be a not ...
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2answers
29 views

Rank of a Matrix under certain conditions

I am a little confused about the rank of a matrix. When does the rank of a matrix equals to zero? Is rank of a matrix equal to zero when it is a zero matrix or the matrix has no elements in it? Thank ...
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1answer
8 views

Writing order of geospacial coordinates

Is there a defined order in which to write geospacial coordinates? When looking at GPX files generated by my GPS, latitude is named first but when looking at a KML file generated by google earth ...
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1answer
24 views

Definition of Derivative for $sgn(x)$

When using the definition of the second derivative for $sgn(x)$, I'm a little confused on evaluating something like $sgn(x+h)$. Since $h\rightarrow 0$ does that mean that I should treating $sgn(x+h)$ ...
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2answers
49 views

What is the definition of 'line' in $\hat{\mathbb{C}}$?

What is the definition of straight line in $\hat{\mathbb{C}}$? Is it defined as $\{x\in\mathbb{C}: \frac{Re(x-a)}{Re(b)} = \frac{Im(x-a)}{Im(b)}\}\cup \{\infty\}$? ($a,b$ are complex numbers and ...
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14answers
3k views

Are “if” and “iff” interchangeable in definitions?

In some books the word "if" is used in definitions and it is not clear if they actually mean "iff" (i.e "if and only if"). I'd like to know if in mathematical literature in general "if" in definitions ...
2
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2answers
50 views

Odd and even functions.

I have a book which says: If a function $f$ satisfies $f(-x)=f(x)$ for all $x$ in its domain, then $f$ is called an even function. However, if $f(-x)=-f(x)$ for every $x$ in the domain of $f$, ...
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3answers
75 views

Orthonormal Bases

I am struggling to get my head around orthonormal bases, this is the defintion in my course notes: If anyone could clarify/explain the concept to me, it would be much appreciated. I am a university ...
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1answer
483 views

normalized subgroup by another subgroup

Let $A$ and $B$ be two subgroups of the same group $G$. What does it mean for the subgroup $A$ to be normalized by the subgroup $B$?
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Computing the differential (not the Jacobi-Matrix) independent of a basis choice

I am eager to learn more about the differential. I was told that it is a good practice to compute the differential independent of a choice of a basis using the following definition Definition: ...
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1answer
23 views

Derivatives from the left and right

I need to give a definition of the derivative from right and from left. (I know it doesn't make a whole lot of sense, but it is supposed to be similar to the same thing as the limits from the right ...
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8answers
1k views

Why do we accept Kuratowski's definition of ordered pairs?

I've been struggling understanding Kuratowski's definition of ordered pairs. I understand what it means but I don't see why I should accept it. I've seen this question and this one, most importantly ...
2
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1answer
45 views

The differential is NOT the Jacobi Matrix?

In the book Analysis II by C.T. Michaels the differential is introduced as the Jacobi-Matrix. In class we had the following definition: Definition: Let $U \subset \mathbb{R}^m$ be open, $f: U \to ...
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2answers
24 views

Homomorphisms, Linear Transformations, and Distributivity

What are the differences between homomorphisms, linear transformations, and distributive operations? To me, they all seem essentially the same, they just are different names for the same phenomenon ...