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

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1answer
26 views

Order of cardinal number

I am puzzled. Wikipedia says: "|X| ≤ |Y| means that there exists an injective function from X to Y." Let's see sets A and B: A = {1,2,3} and B = {1,2}. f: A → B: 1 ↦ 1, 2 ↦ 2. f is injective, but |B| ...
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4answers
69 views

Why to see that $\overline{B}(x;r)$ is closed if it was just defined?

I'm reading Conway's A Course in Point Set Topology. He defines open and closed balls and then he introduces some examples, one of these examples is this: (c) For any $r>0$, $\overline{B}(x;r)$ ...
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1answer
97 views

Halmos on Definability and Luzin on Division by 0

For a successful introduction of a new symbol (e.g. '$\emptyset$') into a mathematical discourse it is necessary and sufficient that the symbol refer to something (e.g. Existence + Specification in ...
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1answer
32 views

Critical points of a function of absolute value

Say I have the function $f(x) = |x|$ I believe that $x = 0$ is a critical point, although not I'm not positive. As the function is decreasing and increasing each side of $x = 0$ does that alone make ...
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2answers
166 views

The Definition of the Indicative Conditional

From Wikipedia, we have: In natural languages, an indicative conditional is the logical operation given by statements of the form "If A then B". Unlike the material conditional, an indicative ...
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1answer
157 views

Hessian matrix as derivative of gradient

From a text: For a real-valued differentiable function $f:\mathbb{R}^n\rightarrow\mathbb{R}$, the Hessian matrix $D^2f(x)$ is the derivative matrix of the vector-valued gradient function $\nabla ...
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1answer
59 views

Definition of a continuous function

I am struggling to understand a basic definition of a continuous function from a textbook: A function f is continuous if for all x, and for all $\epsilon>0$, there exists $\delta>0$ such that ...
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0answers
31 views

Definition review: how to make this geometric definition clearer?

In a paper I am writing, I rely on the following definition Given a geometric shape $C$ and a family of geometric shapes $S$, The division number of $C$ relative to $S$, denoted $DivNum(C,S)$, is ...
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1answer
456 views

A circular proof in Rudin that $\mathbb{R}$ is a field.

Today I'm afraid, I found a circular reasoning in Rudin's Principles of Mathematical Analysis( I found no errata that mentions this). Before actually going through the actual question, I have ...
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3answers
85 views

Set question - $ ℤ^+ = ℕ$ [duplicate]

I am not sure whether the following statement is true: $ ℤ^+ = ℕ$ if not, why? Thank you in advance! I appreciate your help!
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1answer
151 views

Am I understanding Eccentricity, Radius and Diameter right?

Eccentricity, radius and diameter as defined in "Graph Theory and Complex Networks: An Introduction" (van Steen, 2010): Consider a connected graph G and let d(u,v) denote the distance between ...
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0answers
113 views

Meaning of $A^A$

Let $A$ be a matrix. I know that there is definition for $A^k$ power of $A$, and $e^A$ expontential of $A$. Is there any meaning of $A^A$?
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3answers
223 views

Why must an interior point of $E$ be an element of $E$?

This question takes place in a general metric space $X$. Let $x$ be an interior* point of $E \subset X$ iff there exists a deleted neighborhood of $x$ that is contained in $E$. This is like the ...
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3answers
64 views

Definition of homogeneous ODE

In my lecture notes, it gives this following definition of a homogeneous ODE: A differential equation is called homogeneous if it can be written in the form $x′=f(\frac{x}{t})$ Then in one of ...
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0answers
31 views

Is there a term for an endomorphism defined up to conjugation by an automorphism?

Is there a standard term to designate the equivalence class of endomorphisms where two endomorphisms $\phi$ and $\psi$ are considered equivalent if there exists an automorphism $\alpha$ such that ...
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1answer
55 views

Branched covering of a manifold [duplicate]

What would be the definition of a branched covering of a manifold? I am not familiar with branched coverings at all.
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2answers
102 views

In Mathematical Logic, What is a Language?

I've been reading about mathematical logic and computability theory, but I'm somewhat confused on one note: what exactly is a language? What does it mean when I am told "let $\langle 0, +, \leq ...
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1answer
87 views

Definition of intersection of a family of sets

The definition of an intersection of family($F$) is: $$\cap F=\bigl\{x\mid\forall A: (A\in F \implies x\in A)\bigr\} $$ If my understanding serves me correctly this notation means that all the $x$ ...
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1answer
1k views

Upper Triangular Matrix Definition

Is \begin{bmatrix} 1 & 1 & 2 & 3 \\ 0 & 0 & 4 & 4 \\ 0 & 0 & 4 & 4 \end{bmatrix} an upper triangular matrix? My linear algebra teacher says that the main ...
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0answers
56 views

“Filters” of associative rings

Does there exist some good notion of "filters" for arbitrary associative rings with unity that would generalize filters of Boolean rings? For me, if $R$ is an associative ring with unity $1$, it ...
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1answer
958 views

Kronecker product and outer product confusion

I have two column vectors: \begin{equation} u = \left[\matrix{ 1 \cr 2\cr }\right] \end{equation} \begin{equation} v = \left[\matrix{ 4 \cr 4\cr }\right] \end{equation} I'm trying to ...
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0answers
83 views

Binary operation (english) terminology

Foreword: I have read R.H. Bruck's A Survey of binary systems, where the notion of halfoperation is given. A halfoperation $\ast$ differs from a (binary) operation since $a\ast b$ may not be defined ...
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0answers
225 views

Definition of smooth (variety)

I don't understand the motivation for the definition of smoothness of a variety: A variety $V(f_1,...,f_m)$ in $n$-space is smooth iff $\mbox{rank}$ = $n-\mbox{dim} V$. Could you please give me ...
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1answer
57 views

Should I use supremum or maximum in the definition?

Let $U$ be a certain function on sets ($U(S)$ represents the 'value' of the set $S$). If a set $S$ is partitioned to two disjoint subsets $A$ and $B$ with $A \cup B=S$, then: $V(A,B):=U(A)+U(B)$ ($V$ ...
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1answer
46 views

Definition of Limit of a Function

I'm a little confused by the definition of the limit of a function-on one hand I feel the definition suggests that your limiting variable is shrinking into a little delta ball- on the other hand when ...
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0answers
17 views

acclivity of differential equation not defined

the differential equation y'*y=cos(x) has the acclivity of y'=cos(x)/y. Obviously the acclivity is not defined for points (x=r;y=0). r= real numbers Is there a way to give a short but logic argument ...
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0answers
103 views

Is it false that the complement of an open set is closed?

Let $f:\mathbb{C}\rightarrow \mathbb{C}$ be a continuous function. Let $Z(f)$ be the zero of $f$. Prove that $Z(f)$ is closed. This is one of problems in my mid-term exam. I have used ...
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1answer
234 views

Stokes' Theorem Explanation

Can someone explain what Stokes' Theorem is measuring? What would taking the integral of a vector on a surface give you? When would you use it? This is the only definition I have and I don't really ...
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1answer
75 views

Does a bijection between two sets $A$ and $B$ implies $P(a \in C ) = P(b \in C)$, if $A,B \subset C$?

I'm just thinking about it. For example, a bijection between $\mathbb{Z^*_+}$ and $\mathbb{Q}$ implies that the probability of a random real number being rational or positive integer is the same (in ...
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0answers
69 views

What is a Complete Set of Weights of a Representation of a Lie Subalgebra?

In relation to Lie Group and Lie Algebra theory, I am studying about the weights of representations. I have come across the terminology "a complete string of weights" in my lecture course, but it is ...
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1answer
93 views

Why does $0/0$ have to be undefined? [duplicate]

Why can't it no be $\pm$ Infinity? If $x/1$ is $x$ then $x/0$ should be $\pm$ Infinity.
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0answers
174 views

Difference between a vector space and an algebra

I'm new to the subject of algebras and I would like to get a better understanding of what they are exactly. Am I right to say the follwing: ...
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2answers
294 views

Word for “openness”/“closedness” of an interval

What word properly completes the phrase the radius of convergence does not depend on the $\text{______}$ of the interval to mean that it doesn't matter whether $(a, b)$, $[a, b)$, $(a, b]$, or ...
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1answer
42 views

Convergence of vector spaces

I'm considering vector spaces over $\mathbb{R}$ or $\mathbb{C}$ : $(E_n)_n$ is a sequence of vector space (all included in $\mathbb{R}^p$ for example, with $p$ fixed). What meaning(s) do we usually ...
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1answer
78 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
39 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
38 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
4k 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
1k views

The difference between hermitian, symmetric and self adjoint operators.

I am struggling with the concept of hermitian operators, symmetric operators and self adjoint operators. All of the relevant material seems quite self contradictory, and the only notes I have to go ...
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9answers
4k views

Positive and negative complex numbers?

Can there be such a thing as positive and negative complex numbers? Why or why not? What about positive or negative imaginary numbers? It seems very tempting to say $+5i$ is a positive number ...
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3answers
144 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
34 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
29 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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0answers
21 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
166 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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1answer
38 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
77 views

Defining the integral of differential $1$-forms

Assume $M$ is a smooth manifold, $g:[0,1]\to M$ is a smooth curve on $M$, and $w$ is $1$-form on $M$. Definition: $$\int_gw=\lim_{\Delta\to 0}\sum_{i=1}^nw(x_i)$$ The tangent vectors $x_i$ are ...
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1answer
107 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
1k 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
85 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 ...