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

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5
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1answer
83 views

Why Riemann hypothesis and not Riemann's conjecture

I have a stupid question. We say Erdös's conjecture, Goldbach's conjecture, Beal's conjecture... and so on. But we don't say 'Riemann's conjecture.' Instead we use the word 'hypothesis'. Why?
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0answers
34 views

What does it mean for an automorphism to centralize a subgroup?

Suppose $H$ is a subgroup of a group $G$. What does it mean for an automorphism $\sigma\colon G\to G$ to centralize $H$? The context I have is that if $\sigma$ centralizes $H$, then the map ...
0
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1answer
51 views

Grothendieck Universes: Atoms

I'm very much beginner in set theory; my apologize for mistakes. Grothendieck universe: $$\forall X(X\in \mathcal{U}\Rightarrow\mathcal{P}(X)\in U\}$$ $$\forall X\forall x(x\in ...
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0answers
40 views

Supply/demand digraph - understanding problem

I'm dealing with multiflows and I found in "Combinatorial Optimization - Part C" by Schrijver in Chapter 70 a good source. The definition of the multiflow problem involves so-called supply- and ...
2
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1answer
23 views

Clarification of term in graph theory - about star polygon graphs

I was reading about star polygon graphs from the following link: http://mathworld.wolfram.com/StarPolygon.html. As far as I noticed I felt that whenever $d$ is a proper divisor of $n$, then we get ...
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0answers
14 views

The name of a polygon defined by multiple overlapping annuli

I am working on a problem in a metric space where points are partitioned into various annuli. If there exists multiple annuli that define a set of points then a polygon can be formed from their ...
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1answer
34 views

Name for sequence that begins and ends with the same element?

Is there a name for a sequence $(a_1,a_2,...,a_k)\in{A^k}$ where $a_1=a_k$?
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0answers
11 views

Definition of the sup-norm in the Qing/Fanghua

I am currently studying the book "Elliptic Partial Differential Equations" - Qing / Fanghua, COURANT, in chapter 2, section 2.3. A priori estimates , the autors define by $\alpha$ the sup-norm of ...
3
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1answer
68 views

What does Wolframalpha's definition of “contravariant vector” mean?

http://mathworld.wolfram.com/ContravariantVector.html Wolframalpha offered a one line definition to contravariant vector which is a bit confusing to me Contravariant Vector: The usual type of ...
2
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0answers
33 views

Proof $f(x,y)=x_1+e^{x_{2}}$ is strictly convex

I am trying to show that $f(x,y)=x_1+e^{x_2}$ is strictly convex. I can show this using the Hessian Matrix which is positive definite. However for some reason i can not put it together using algebra ...
2
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1answer
81 views

Why do Wikipedia and nLab seem to give completely incompatible definitions of the term “simplex”?

nLab seems to define that a simplex is an inhabited finite totally-ordered set. Wikipedia seems to define that simplex is a subset of real affine space satisfying certain conditions. Q. What's ...
3
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1answer
43 views

Why are Grothendieck's and Hartshorne's definitions of quasi-coherence equivalent?

Hartshorne's Algebraic Geometry defines an $\mathcal O_X$-module $\mathscr F$ to be quasi-coherent if there is an open affine cover $(U_i=\operatorname{Spec} A_i)_{i\in\mathcal I}$ of $X$ such that ...
1
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1answer
15 views

Valid method to obtain a basis of a topological subspace?

Let $(X,\tau)$ be a topological space and $Y \subset X.$ We know that if $\mathcal{B}$ is a basis for $\tau$ and $\tau_{\small{Y}}$ is the subspace topology on $Y$, then we can obtain a basis for ...
4
votes
1answer
77 views

How is exponentiation defined?

Here is how I think this work: We define $ a^b = \underbrace {a\cdots a}_b $ for $a \in R $ and $b \in N$. Since so far we have not defined what $ a^{-1}$ is, $a^{-4}$ makes no sense. (right?) We ...
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0answers
28 views

How to write this function in a “well-formed” way

Given an input $0 \lt x \lt 1$, find $x$'s Nearest Integer Continued Fraction with structure $$x = a_0 \pm \cfrac{1}{a_1 \pm \cfrac{1}{a_2 \pm \cdots}}.$$ Then $$f(c) = a_0 + 1 \mp \cfrac{1}{a_1 + ...
5
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1answer
95 views

Mathematical meaning of “may not”

Does "may not" mean that never allowed or sometimes not allowed? For example: the sequence may not converge. Does this mean that the sequence never converges or that there is no guarantee that the ...
3
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1answer
39 views

Difference between “real functions” and “real-valued functions”

According to my textbook: A function which has either $\mathbb R$ or one of its subsets as its range is called a real valued function. Further, if its domain is also either $\mathbb R$ or a ...
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3answers
100 views

If function $f$ is differentiable at a point $a$, does it imply that $f'(a)$ exists?

If function $f$ is differentiable at a point $a$, does it imply that $f'(a)$ exists? I want to proof that if function $f$ is differentiable at a point $a$, then the function is continious at this ...
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3answers
64 views

What is the difference between an indexed family and a sequence?

For indexed family wikipedia states: Formally, an indexed family is the same thing as a mathematical function; a function with domain J and codomain X is equivalent to a family of elements of X ...
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1answer
53 views

Function Definition Issue

I'm currently working through an real analysis text and came across a definition that seemed a little strange to me. When defining a function the text states: For a function, $$f:\,S\rightarrow ...
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0answers
25 views

Set of 2-tuples with indices

I want to define the set of two tuples. Actually, I want to describe the degree distribution (or histogram) in terms of ordered tuples. The first item denotes the degree of a node. The second item ...
1
vote
1answer
29 views

What does it mean to say that a vector is 'coordinate invariant'?

In my lecture notes, it says that a vector is 'coordinate invariant' if it's properties do not depend on the choice of basis used to represent them. I understand that the basis of a vector space is a ...
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0answers
29 views

What is the definition of Planar $3$-Sat problem?

I have some steps in my lecture-notes to make an instance of Planar $3$-Sat problem from an instance of $3$-Sat problem. The steps are as follows: Create one vertex for every literal $x_i$ Create ...
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0answers
10 views

What is the n-concavification of a Banach space?

I'm reading this paper about polynomials in Banach spaces and the authors use the notion of the n-concavification of a Banach space $X$ It is the first time that I encounter this concept. What is it? ...
4
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1answer
51 views

Is distance between two graphs defined somehow?

If the two graphs are isomorphic, then their distance is zero. And this distance increases, if vertices or edges are added or removed to/from one of the graphs. Does this "distance" have a special ...
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2answers
51 views

The formal definition of an interval

I is A real interval iff ∀ x,y ∈ I the segment [x,y] ⊂ I I can't understand why an interval is defined this way Why it isn't defined the same way segments are? how can the definition of an ...
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3answers
91 views

What am I missing in this argument for $\lim\limits_{x\rightarrow \infty} \ln x = \infty$?

In an appendix of Stewart's Calculus, the logarithmic and exponential functions are built up starting from the defnition $\ln x = \int_1^x \frac{1}{t}\,dt$. Having shown that $\ln(x^n) = n \ln(x)$ ...
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0answers
41 views

Understanding manifolds (asking just for confirmation)

In lecture we used the following definition of manifolds: A subset $ M \subset \mathbb{R}^n $ is called a k-dimensional manifold of the class $C^\alpha$, if $ \forall a \in M $ there is an open ...
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2answers
63 views

Support of a distribution, what does it mean?

In my course notes the support of a distribution (continous lineair functional) is defined as follows: Definitions First it defines something like open annihilation sets: An open annihilation ...
3
votes
3answers
88 views

Is there any way to universally define the notion of $\text{Isomorphism}$?

Suppose we want to give a very general definition of the term Isomorphism, first of all, we'll want an isomorphism to be a bijective function. Informally, we want our function to preserve whatever ...
3
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1answer
62 views

Absolutely continuous function on R

What is the definition of absolute continuity in whole $\mathbb{R}$. I know the definition on an interval $[a, b]$. I have a trouble with understanding the definition of absolute continuity in whole ...
2
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1answer
99 views

Lost in terminology: What is the meaning of the words “Constraint” and “Parameter” in a goodness of fit?

This is somewhat related to this previous question of mine. I need a clear distinction and/or definition of the words 'parameter' and 'constraint' in the following context which is the the only source ...
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0answers
32 views

Sup-norm of two functions

Suppose $\Omega$ is a bounded set and connected domain in $\mathbb{R}^n$. Consider the operator $L$ in $\Omega$ $L=a_{ij}(x)D_{ij}u+b_{i}(x)D_{i}u+c(x)u$ For $u \in C^{2}(\Omega)\cap ...
0
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0answers
23 views

Inverse Substitution ( Trigonometric Integration Rule Generalized )

I read that we can generalize the kind of substitution that is used in Trigonometric Substitution. The idea is that we replace the old variable with a new one (like we do in U-Substitution) but unlike ...
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2answers
92 views

Is an undefined equality vacuously true?

To quote Wikipedia, [..] equation is an equality containing one or more variables. Solving the equation consists of determining which values of the variables make the equality true. Consider a ...
0
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1answer
46 views

“Countable” and “Sequence” in Toplology

I'm having difficulty with the meanings of these terms in several references. "Countable" may mean either a set with cardinality = N (i.e. countably infinite) , or with cardinality $\le$ N (i.e. ...
4
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3answers
27 views

Alternative definition for Exponential and Logarithmic functions to prove identities (and by extension sin and cos related identities)

Edit : This question is about not about proving identities, but representations that are easier to work with than Taylor series or integral definition for the functions exp or ln functions. Please do ...
0
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1answer
16 views

Defining Compound Function

If $f(x) = (x+x)$ and $g(x) = 2(x-5)$, and we have the compound function $g(f(x))$, how can we "define what the resulting function does"? It's obvious what is happening, but I'm not quite sure how ...
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0answers
29 views

Why was this terminology of holomorphism used here?

page 6 in this notesays: A note on index convention: $i$,$j$, and $\bar{i}$, $\bar{j}$ index over the holomorphic and antiholomorphic components. I never knew there were holomorphic ...
3
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2answers
60 views

How to make sense of the binomial coefficient over $p$-adic integers?

I recently asked this question: Show that $y^2=x^3+1$ has infinitely many solutions over $\mathbb Z_p$. and now I'm trying to make sense of the first answer that was posted. It said that I should show ...
10
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1answer
156 views

Is there a reason for different nomenclature on Calculus of Variations?

While sightseeing aspects of Calculus of Variations, the following fact elludes me: there is a plethora of new definitions which seem redundant to me. This phenomenom happens, of course, with other ...
0
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1answer
46 views

Are math definitions iff statements? [duplicate]

I was wondering if definitions in mathematics are "if and only" statements? I know for sure that theorems are not "iff" statements. Thank you in advance for your help.
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1answer
37 views

Mathematical definitions of infill asymptotics

I am writing a paper that uses infill asymptotics and one of my reviewers has asked me to please provide a rigorous mathematical definition of what infill asymptotics is (i.e., with math symbols and ...
1
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1answer
33 views

Compact space - definition

I have a doubt at the definition of compact spaces. So if you have a topological space $X$, then $X$ is compact if every open cover of $X$ has a finite subcover. In other words, if $X$ is the union of ...
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2answers
27 views

Definition of linear/ affine/ etc transformation [duplicate]

A linear transformation is a function $f:V\to W$ from vector space $V$ to vector space $W$ such that linear combinations (alternatively vector addition and scalar multiplication) are preserved. An ...
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2answers
50 views

Terminology conventions in mathematics

I am writing a paper where I have more than one lemma (Lemma 1, Lemma 2, and Lemma 3) and when I cite them together I was wondering is it more appropriate to say, for example, because of Lemmas ...
0
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1answer
38 views

Is it ok to make an operation over some definition at very the first time you are defining it?

Suppose we have an expression and want refer to it with some symbol. Usually one can write something like "$X:= \text{Expression}$" to mean the symbol $X$ is defined to be the expression after the ...
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2answers
38 views

Definition for the complex exponential function

We define the exponential function as $\exp(z)=\sum\limits_{j=0}^\infty= \frac{z^j}{j!}$ for all $z\in \mathbb{C}$. Lets now compute $\exp(0)$, then we would have to calculate $0^0$ which undefined. ...
1
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1answer
43 views

Is $H_n(X)$ just a different way of writing$H_n(S_*(X))$?

While studying homology in algebraic topology, I sometimes see the notation $H_n(S_*(X))$, and sometime the notation $H_n(X)$. I think these are supposed to be the same, but I'm not sure. The first ...
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2answers
58 views

How can some integrals be zero?

This could be a nonsense but I have to try. We are used to know that an integration like $$\int_a^b F(x)\ \text{d}x$$ gives us the area under the curve $F(x)$ from $a$ to $b$. The question is then: ...