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

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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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2answers
235 views

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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2answers
103 views

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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2answers
59 views

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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3answers
652 views

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 ...
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164 views

Definition of Base-point

Consider the family of curves defined by $f(x,y)=g(x)+h(y)+a$, where $a$ is a free parameter. Now, it states that the family of curves intersect at $\infty$ and that $\infty$ is a base-point of these ...
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57 views

Motivation for defining a functions codomain

Why not always refer to a functions image, what is the point in specifying some super set of that image and then naming that super set the functions "codomain"? What does explicitly defining a set ...
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1answer
75 views

On Constant mean curvature surfaces.

I have two involved questions, firstly, I know that the gauss map sends a surface to the unit sphere, so for a surface $\Sigma\subset\Bbb R^3$, parametrised by $u:U\subset\Bbb R^2\to \Bbb R^3$. Would ...
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1answer
126 views

Is the function $f(x)=1^x=1$ considered an exponential function?

I am confused about the following: The exponential function (by definition) is a function of the form $f(x)=a^x$ where $a>0$. However, when $a=1$, we get the constant function $f(x)=1^x=1$. Is the ...
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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
68 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
94 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
31 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
150 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
133 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
54 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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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
434 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
82 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
137 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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207 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
63 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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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
51 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
101 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
83 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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55 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
764 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
79 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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196 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
53 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
44 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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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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101 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
218 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
59 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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65 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
87 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
137 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
273 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
41 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
70 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
38 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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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
3k 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
3k 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
124 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?