For requesting, clarifying, and comparing definitions of mathematical terms.

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1answer
57 views

Some troubles about topology and definition of a Vector Bundle

Disclaimer: It's heavily related to my old question : Visualizing the Topology of a Vector Bundle but I wanted to open a new question because the former had already got an answer and this time my ...
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1answer
30 views

Formal notation for finite intersection

How to state following sentence formally? Let $\tau$ be family of sets (possibly infinite). Any finite intersection of sets in $\tau$ is in $\tau$. My attempt is: $\tau$ is family of sets and ...
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0answers
37 views

A question about the names of the components of an interval

Suppose you have the interval $[n,10n]$. How do you call the $n$ in this interval? I think the $n$ can be called the "independent variable of the interval" (since the interval $[n,10n]$ can be written ...
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2answers
751 views

What is the meaning of an “objective function” ?

I have studied Mathematics in french. Right now I am in front of this expression "objective function" and "objective functionals". I don't know its meaning, some one can help please ?
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3answers
89 views

Why is $ |z|^2 = z z^* $?

I've been working with this identity but I never gave it much thought. Why is $ |z|^2 = z z^* $ ? Is this a definition or is there a formal proof?
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1answer
30 views

Is there any definition for homogeneous rotations?

Most of the geometric transformations can only be represented into square matrices via homogeneous coordinates, e.g., translation and 3D rotations with axes not through coordindate system origin. ...
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1answer
27 views

Reusing Variables First Order Logics

Assume we have a parametrized FO formula of this form: $$\varphi(x_1,x_2, y_1, \dots ,y_m) := \xi(x_1,x_2) \land \psi(y_1,\dots,y_m)$$ We want to use as few additional bound (quantified by $\exists$ ...
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1answer
1k views

Bounded Set: definition

I'm having some trouble with the definition of "Bounded Set". I have a pretty good idea of what "Limited" means: a Set with a Upper and a Lower bounds. Now i have a quiz in which I must choose the ...
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1answer
114 views

How is addition on N formally defined in textbooks on real analysis?

This is a follow-up question to Why does the definition of addition require proofs? In Landau's Foundations of Analysis, his definition of addition on the natural numbers seems a bit strange to me -- ...
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0answers
16 views

Borel - Regular elements

In Borel's Linear Algebraic Groups (2ed) page 160 a regular element is defined in terms of its semisimple part, “thus $g$ is regular if and only if $g_s$ is regular.” A unipotent element $g$ has ...
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1answer
31 views

How would you define the square of the linear operator

If you define the linear operator norm of $A:X\to Y$ to be $$\|A\|_{op} = \inf\{C>0: \|Ax\|_Y \leq C\|x\|_X \text{ for all } x \in X \}$$ Then how would you define $\|A\|_{op}^2$? My guess is you ...
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1answer
66 views

Equivalence between submodular function definitions.

I am trying to show that the definitions, given by wikipedia, of a submodular set function are equivalent. See section definition of: http://en.wikipedia.org/wiki/Submodular_set_function. Mainly I ...
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1answer
735 views

Graph Theory: What is the definition of the “Sorted Edge” algorithm?

I've been googling for a while and can't find a clear definition of the "sorted edge" algorithm--can anyone provide it please? A description would be helpful, but a simple statement of the algorithm ...
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0answers
64 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 $m\cdot r=r'\bullet m.$ Examples of this bimodules can be seen ...
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3answers
347 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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0answers
47 views

Connected sum of surfaces with boundary

The connected sum of closed surfaces (2-manifolds) is defined by removing a disk from each and gluing the exposed edges together. When defining the connected sum of surfaces with boundary, is the ...
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3answers
163 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
2k views

Two definitions of $\limsup$

Here are two equivalent definitions of $\limsup_{n\rightarrow\infty} a_n$: Let $u_n=\sup\{a_n, a_{n+1}, a_{n+2},\ldots\}$. Then $$\limsup_{n\rightarrow\infty} a_n = \lim_{n\rightarrow\infty} u_n = ...
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1answer
236 views

Characterization of lim sup, lim inf

If $(a_n)$ is a real sequence, in lecture we had: $\limsup_{n\to\infty} a_n=a \iff (i)\forall \epsilon >0 \,\exists n_0\in \mathbb{N} :a_n<a+\epsilon$ $ \forall n\ge n_0$ and $(ii) \forall ...
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1answer
133 views

What is the intuition behind the name “Flat modules”?

I am studying Atiyah and MacDonald's book "Introduction to Commutative Algebra" and I have just read the definition of a flat module. It seems to me that if they have called that kind of modules ...
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3answers
368 views

Codomain of a function

At high school we were told that a function has a domain and a range, the function maps from the domain to the range. Such that the domain contains all and only the possible inputs and the range ...
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1answer
222 views

Precise definition of oscillation behavior of functions like $\sin(\frac1{x})$

I tried today precisely defining the oscillation behavior present in functions like $\sin(\frac1{x})$ i.e: To do this, I started with the domain of function as in limits, and let $f(x)-L$ have ...
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1answer
445 views

Definition of the nth derivative? [First post]

If the definition of the derivative is $$ f^\prime(x) = \lim_{\Delta x \to 0} \dfrac{f(x+\Delta x) - f(x)}{\Delta x} $$ Would it make sense that the nth derivative would be (I know that the 'n' in ...
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1answer
28 views

What is the Support of a permuation

Is this definition of the support of a permutation correct: let $\pi\in S_{\Omega}$ for $\Omega$ a finite set, and $S_\Omega$ the set of all permutations (bijections) on $\Omega$. Ie ...
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3answers
543 views

How to define the magnitude of a rotation in $\mathbb{R}^n$?

Is there some well established way on how to quantify rotations in $\mathbb{R}^n$? To say which rotation is greater and which is smaller? In $\mathbb{R}^2$ the rotation is characterized by a single ...
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1answer
66 views

Is the function $f(x)=x$ on $\{\pm\frac1n:n\in\Bbb N\}$ differentiable at $0$?

This is really a question of definitions. If a function $f$ is not defined on an open set containing $x$, how do we define the derivative of $f$? Is it sufficient to be locally approximable by linear ...
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1answer
67 views

Why is the length of one of the segments negative in the Menelaus' theorem? Aren't all distances by definition positive?

Why is the length of one of the segments negative in the Menelaus' theorem? Aren't all distances by definition positive? I think we all know the Menelaus' theorem, which claims that the following ...
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0answers
43 views

How is this function continuous?

The textbook stated that the following function whose domain is $\mathbb{R}$is continuous for every point in the domain: $g(x)=1, 0\le x\le1$ $=2, 2\le x \le3$, and it continues this patern. What ...
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2answers
60 views

A problem with the domain of function in the defintion of limits

My Stewart's Calculus gives the following definition of limit: $f(x)$ is defined on some open interval containing $a$, except at possibly $a$. So, $\lim_{x\to a} f(x) = L $ if and only if for ...
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1answer
51 views

Simple closed curve definition of genus

The genus of a connected surface can be defined as the maximum number of disjoint simple closed curves that can be removed from it without disconnecting it. Why must the simple closed curves be ...
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1answer
84 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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0answers
68 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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2answers
57 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
135 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
138 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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1answer
21 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
71 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
167 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
26 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
36 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
47 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
64 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
452 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
73 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
126 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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1answer
49 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
194 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
93 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
205 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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6answers
1k 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 ...