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
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The definition of the logarithm.

One usually gets several definitions of the logarithm along his studies. You might be first introduced to the exponential and then told that the logarithm is its inverse. You might be given $$\log ...
2
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1answer
39 views

Terminology clarification: ***exchanges***

I need help with a terminology definition. If we say "R is a reflection that exchanges the sides a and b in some triangle", does it mean sides a and b have the same length and the reflection maps one ...
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0answers
93 views

Definition of stereoprojection and Möbius maps

@WillieWong has kindly pointed out that there are 2 definitions of stereographic projection. One with the unit sphere placed on top of the plane, the other where the plane is at the equator of the ...
2
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1answer
45 views

Reducts of categories

There are several ways to reduce a category. The skeleton of a category is the category with isomorphic objects collapsed into one i.e. the only isomorphisms that remain are the identities. What's ...
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1answer
143 views

Motivation behind the definition of reflections in affine hyperplane

What is the motivation behind the definition of the reflection map in affine hyperplanes? $R: x \to x-2(x\cdot u-c)u$ where $u\cdot x=c$ defines the affine plane. Of course one requirement is for it ...
2
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1answer
73 views

Definition clarifications : on adjectives of functions

Could somebody please explain what are the differences between a differentiable function and a holomorphic function analytic function and conformal function? (Am I right to think that all analytic ...
2
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1answer
291 views

Perfect Hash Function just an Injection?

I just read up on the concept of perfect hash functions on a set $S$. I am quoting: "A perfect hash function for a set S is a hash function that maps distinct elements in S to a set of integers, with ...
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2answers
247 views

Solvability and Simplicity

I have read that Burnside's theorem implies that a group with order $p^aq^b$ cannot be simple. So I looked up Burnside's theorem and saw that it doesn't mention "simple" explicitly, rather it says ...
7
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1answer
503 views

Higher Order Trigonometric Function

Once in a time, I had to work with functions that have the following Taylor series expansion: $$ t_m(x)=1-\frac{x^m}{m!}+\frac{x^{2m}}{(2m)!}+\cdots =\sum_{k=0}^\infty \frac{(-1)^k x^{km}}{(km)!}. $$ ...
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2answers
685 views

How to write “let” in symbolic logic

How do I write let in symbolic logic? For example, if I am in the middle of a proof and there is a variable which I can assign to an arbitrary value, what would I write? My best guess is: $$ x := a ...
5
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1answer
353 views

Understanding the homotopy extension property

I'm reading Chapter 0 of Hatcher right now, and there's something in the definition of homotopy extension property that I don't understand. Suppose one is given a map $f_0:X\to Y$, and on a ...
2
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2answers
627 views

What is the differentiation operator

On Wikipedia the Differential Operator is described as an operator defined as a function of the differentiation operator. The link that underlies the words "differentiation operator" in fact gives ...
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3answers
101 views

Oddities in the Definition of IntegralCosinus ${\rm Ci}(x)$

Reading the defintion of the IntegralCosinus $$ {\rm Ci}(x) = \gamma + \ln x + \int_0^x\frac{\cos t-1}{t}\,dt $$ I wonder what happens, if I to split the function in the integral: $$ ...
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2answers
558 views

How many 'supremum(s)' and 'infimum(s)' can a set have?

I am learning calculus/real analysis with Apostol's Calculus (2nd Edition). I have a doubt about the grammer of this book. Apostol, everywhere, uses a supremum (or a least upper bound) and an infimum ...
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2answers
170 views

On the definition of jets

I have some problems with the definition of jets and it would be great if someone could help me here: In many books it is written, that the $r-th$ order jet $j^r_xf$ of a smooth function $f:M ...
0
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1answer
200 views

definition: dual of a vector field

Let $X:\mathbb{R}^3\rightarrow T\mathbb{R}^3$ be a vector field, what is the definition of its dual ? I know that the set of vector fields on $\mathbb{R}^3$ forms an $\mathbb{R}$-vector space.
15
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2answers
541 views

What is it to be normal?

I'm interested to find out why the word 'normal' crops up so many different areas of mathematics, especially but not exclusively in abstract algebra, and how the definitions are related, if at all. ...
0
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1answer
262 views

Why is this definition of an additive inverse significant

In the process of learning Real Analysis, I encountered a definition of an additive inverse of a cut $\alpha$ to be $$\text {add inv of } \alpha \colon= \{p:\exists r>0 \text{ s.t.} (-p-r)\notin ...
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1answer
2k views

Formal definition of big-O when multiple variables are involved?

(My apologies if this is a duplicate; I did some searching but didn't turn up anything else like this on the site. Please let me know if it's a duplicate and I'll gladly delete it.) I was reading up ...
2
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0answers
210 views

How to prove that $e = \lim_{n \to \infty} (\sqrt[n]{n})^{\pi(n)} = \lim_{n \to \infty} \sqrt[n]{n\#} $?

While reading this post, I stumbled across these definitions (Wiki_german) $$e = \lim_{n \to \infty} \sqrt[n]{n\#}$$ and $$e = \lim_{n \to \infty} (\sqrt[n]{n})^{\pi(n)}.$$ The last one seems ...
3
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1answer
3k views

Relationships among the terms “slope”, “parameter”, and “coefficient”?

In $y=mx$, is $m$, are there different implications of referring to $m$ as a "slope", a "coefficient", a "parameter"? Or perhaps the "slope coefficient" or "slope parameter"? For context, I am ...
2
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3answers
1k views

Definition of “maximal” and “minimal” [duplicate]

Possible Duplicate: difference between maximal element and greatest element When I first encountered the terms maximal and minimal, I confused them with maximum and minimum. Many of my ...
2
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1answer
2k views

What does it mean for a functional equation to have a unique solution?

It is my thinking that unique conventionally means special or one of its kind. But in the context of solving functional equations*, I am confused what it means to have a unique solution... *e.g. Find ...
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2answers
2k views

Clarifying the definition of “unstable”

I would appreciate a definition clarification. if a numerical method is "unstable", does it mean that if we introduce a small random error in one of the steps, the error would be magnified greatly ...
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8answers
3k views

Why does the Dedekind Cut work well enough to define the Reals?

I am a seventeen year old high school student and I was studying some Real Analysis on my own. In the process, I encountered the Dedekind Cut being used to construct the Reals. I just can't get the ...
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1answer
165 views

Rate of convergence of random variables for weak convergence

Suppose $X_{n}$ be a sequence of random variable that converges to $X$ in distribution. How can we define the rate of convergence? What would be the reference?
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1answer
77 views

Rate of convergence of double sequences

Suppose $ \{X_{n,m} \}$ be a double sequence of real numbers and suppose $\lim_{n}\lim_{m}X_{n,m}=X$. What is the definition and reference for the rate of convergence of double sequences?
2
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0answers
55 views

terminology clarification needed: projection of a measure to an open set

I'm reading in Doobs "Classical Potential Theory and its Probabilistic Counterpart" and I'm having trouble with terminology. Specifically, in Part I chapter 6, he talks about the projection of a ...
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2answers
887 views

What is which distinguishes Pure Mathematics from Applied Mathematics? Please explain [closed]

I like to know in depth what really differs Pure Maths from Applied Maths. What are their respective applications? Also when this distinction was made in the history of Mathematics and why?
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5answers
434 views

Can series start with $-∞$?

So, I decided to dig a little deeper into numerical integration because we hardly had any of that in my analysis class. I've come across this method for improper integrals: Метод Самокиша (not in ...
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1answer
1k views

What does “refinement” mean?

I was reading a book and it had the following sentence: $A$ is a refinement of $B$ where $A$ and $B$ are sets. What does this mean? Perhaps $A \subseteq B$ ?
4
votes
1answer
97 views

Sequences convergent to 'cycles'

Consider sequences $(x_n)_{n=1}^\infty\subset\mathbb R$. Is there a name for the following property? There exists $L\in\mathbb N$ such that: ...
3
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0answers
192 views

Can GCD be called an operator?

Can $\gcd(a,b)$ be called a binary operator which takes operands $a$ and $b$ and returns their greatest common divisor. And if for some operator —say $\bigotimes$ —$(a_1\bigotimes a_2 ...
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2answers
480 views

Does *finitely many* include the option *none*?

Does finitely many include the option none? Say I have a sequence $(x_n)$ and I want to say that there can only be $0$ or $n\in \mathbb N$ non-zero terms. Can I say that the sequence has finitely ...
3
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3answers
1k views

Definition of open and closed sets for metric spaces

For metric spaces, the definition of an open set $U\subset X$ is that it is a set which for any point $u\in U$ in the set there exists some $\epsilon>0$ such that the open ball ...
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5answers
218 views

Why is there implied an equality between vectors and $n$-tuples?

Are they considered equal in some sense? For instance, I always write "...for vector $\mathbf{x} \in {\mathbb{R}}^n$ we have ...". I have a small problem with this (not a big one). The problem comes ...
0
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1answer
58 views

Restricted bilinear map

Suppose $\phi$ is a bilinear form on the vector space $V$. What does it mean (perhaps in matrix form) for $\phi|_X$ to be non-singular, where $X\leq V$? This question is probably elementary, but I ...
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2answers
2k views

Rank and signature of a quadratic form

Is the rank and signature of a quadratic form $x^TAx$ basically the rank and signature of $A$? Thanks.
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2answers
376 views

Exact sequence in a nonabelian category [previously: “Exact sequence for topological groups?”]

If $A$, $B$, and $C$ are topological groups, and $f: A \to B$ and $g: B \to C$ are two continuous group homomorphisms, what does it usually mean for $$1 \to A \stackrel{f}{\to} B \stackrel{g}{\to} C ...
12
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2answers
663 views

What's the name for the property of a function $f$ that means $f(f(x))=x$?

I can think of several examples of functions such that twice application of the function is equivalent to no application of it. Additive inverse Multiplicative inverse Fourier transform Complex ...
0
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1answer
205 views

Differentiability of a matrix function

To prove the differentiability of a function defined on $M\in M_n$, how should I adapt the definition of differentiability? $h^{-1}[f(x+h)-f(x)-hL(x)]\to 0$  How do I take the reciprocal of a ...
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2answers
27 views

Definition clarifications on functions of vectors and their derivatives

Definition clarifications would be appreciated: How do I interpret the following ? For $f: R^n\to R^m$, $Df(\vec\xi)(\vec{x})$ in differentiation of a vector function. I know it as a function that ...
4
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1answer
91 views

Definition of a groupoid of fractions

The title sums it up : I am looking for a definition of "a groupoid of fractions for a category". I would be interested in any example someone might have as well...
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3answers
166 views

What is the proper term for a function where domain and codomain coincide?

What is the proper term for a function where domain and codomain coincide? E.g. in programming languages a function f : Int => Int or f : Double => Double. Thanks.
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1answer
577 views

Time-invariant IVP

How does one know that a system (of differential equation and initial value constraints) is time-invariant (perhaps by inspection...)? What are the implications of a system with this property (esp. ...
0
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1answer
97 views

Definition of $\ell_\infty$

What does the $\ell_\infty$ space stand for? Thank you.
1
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1answer
163 views

Definition of “or”

A quick definition clarification: Does the set $\{(x,y):x =0 \,\,\,\,\text{or} \,\,\,\,y=1 \}$ include the element $(0,1)$? (Sorry, English is not my first language, I get confused sometimes... Also ...
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2answers
661 views

Definition: Eigenspace of a matrix

If I am given a matrix and told to find a basis for its eigenspace, does that just mean find the eigenvectors of the matrix? In my understanding, an eigenspace of an eigenvalue $\lambda$ is the set of ...
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5answers
511 views

What is a Real Number?

I believe I'm over thinking it, but I want to be 100% sure. A Real Number is any number, correct? Whether it be an integer or something else. It's the set $\mathbb R$ from $(-\infty, +\infty)$ ...
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2answers
699 views

Definition: transient random walk

What exactly does a "transient random walk on a graph/binary tree" mean? Does it mean that we never return to the origin (assuming there is one as for the tree) or just any vertex of the graph or ...