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

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28 views

Unique Union Problem

Given a the set $a = \{1,...,n\}$ and let $b$ denote a set of subsets of $a$. Find a subset of $c$ of $b$ so that the union of all subsets in $c$ is equal to $a$ and the intersection of any of the ...
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2answers
21 views

definition of negative binomial in probability karr

The book defines the probability of the negative binomial as: $$P\{X=k\}={{k-1}\choose{n-1}} p^k (1-p)^{k-n}$$ but where does the ${k-1}\choose{n-1}$ come from? It's quite different to wikipedia's ...
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18 views

Definition of some terms of transformation geometry

Recently I was studying transformation geometry from problem solving strategies . I liked the subject but could not understand some terms. Please anyone help me--- What is isometry? What is ...
2
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1answer
48 views

How to call “equivalent-looking” vertices in graph..?

In the above figure, the vertices expressed as blue dots are "equivalent-looking." Although my expression is somewhat ambiguous, I believe one can simply answer it. How can we call such vertices? ...
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1answer
23 views

Intuition behind a particular definition regarding cycles of partitions

I'm reading this paper on cycles of partitions, and was wondering if anyone could motivate the last condition in the definition of the sets $M_n$ in terms of the partitions being examined. In ...
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38 views

What is the “minimal” structure in which points and lines are defined?

Points, straight lines and planes are fundamental concept of geometry. Usually this entities are defined in a structure. We can easily define points in a vector space, or in a affine or projective ...
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2answers
52 views

how to understand the definition of continuity in analysis?

Please have a look at the picture above. This is about the continuity in analysis. I don't really understand how to utilize this definition? It says that is statement is equivalent to f is ...
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1answer
33 views

How to prove that homology is a functor?

Is a homology operator $H_k:(cKom) \rightarrow (Ab)$ a functor? I know this is a really simple question, but I'm not familiar with category theory and do not know how to prove this... (I'm even not ...
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2answers
108 views

Definition of a Minimal Set

A few times while studying math I have encountered the notion of a "minimal set". For example, given some set of subsets, what is the "minimal" sigma algebra generated by that set of subsets? Or, in ...
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1answer
21 views

interpreting the curve of intersection

I would like to understand the idea of a 'curve of intersection' in $\mathbb{R}^{3}$. Say we are given a surface $z = x^{\frac{1}{3}}y^{\frac{1}{3}}$ and a plane $y = x$. Then the curve of ...
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2answers
55 views

What's a local angle?

When I was trying to understand the definition of conformal map I got confused. A conformal map is a function $f: U \to \mathbb C$ where $U \subset \mathbb C$ such that $f$ preserves local angles. ...
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4answers
79 views

Find the derivative of $f(x) = \sec(x)$ without the quotient rule

Part of an analysis assignment I have: " Given $f(x) = \sec(x)$, compute the derivative of $f(x)$ by using the definition of derivative. (Note that $\sec(x) = 1/\cos(x)$ and $(\cos(x))' = ...
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2answers
39 views

About a norm : $p(uv)=p(u)p(v)$ all the time? [closed]

Say, $p$ is a norm on a vector space. Then can we say that $$p(uv)=p(u)p(v)$$ all the time? Thanks.
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1answer
18 views

Definition for trigonometric function with a different “system”

For fun, I decided to create a sort of "intuitive" (for me, anyhow) approach to degrees and such. As I can recall, degrees are based on (the Mesopotamians?)'s base $60$ math. I've read that radians ...
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8answers
139 views

How can expressions like $x^2+y^2 = 4$ be defined?

I'm wondering how to define the expression(?) $x^2+y^2 = 4$, because I realised it's not a function because it cannot be expressed in terms of $x$ or $y$ alone. Is it even called an expression? Of ...
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1answer
28 views

How do I define a block of a $(0,1)$-matrix as one that has no proper sub-blocks?

I'm struggling to come up with a definition of a "block" in a $(0,1)$-matrix $M$ such that we can decompose $M$ into blocks, but the blocks themselves don't further decompose. This is what I've got ...
2
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1answer
38 views

Extending the Ordinals analogously to the Integers

I have been (recreationally) trying to expand the notion of ordinal numbers in the same way that the natural numbers $\mathbb N$ are extended to the integers $\mathbb Z$. My objective is to be able ...
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3answers
36 views

Does Stars and Bars or the binomial coefficient represent binary sequences?

Does Stars and Bars or the binomial coefficient represent binary sequences? With the binomial coefficient we can calculate all the paths on a grid with moving up or right, that's like defining up to ...
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1answer
29 views

Slice, projection, contour: A terminology question.

Consider a multivariate function, say $y=f(x_1,x_2,\dots,x_n)$, and suppose that $z=f(x_1,x_2,\dots,x_{n-1},g(x_1,x_2,\dots,x_{n-1}))$. What do we call $z$ with respect to $y$? Projection, level set, ...
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1answer
54 views

Cotangent space explicit definition

Given a tangent space $T_xM$, where $M$ is a differentiable manifold homeomorphic to $\mathbb{R}^n$, we have the cotangent space $T^{*}_xM$ defined as being the set of linear functionals $\eta: T_xM ...
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0answers
19 views

Definition of “axes”

I'm working on a question which goes like this: "23. A variable rectangle is inscribed in a circle of radius 13 cm. At a certain instant one side is 10 cm long and is increasing at the rate of 3 cm ...
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1answer
44 views

What is a “right” automorphism?

Let $B_n$ be the braid group with $n$ strands and let $F_n$ be the free group of rank $n$ generated by $x_1,\ldots,x_n$. The classical Artin Representation Theorem reads: If an automorphism of ...
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1answer
25 views

A kind of planar figures

Studying issues related to the planar shapes I've found some attribute, useful for my investigations: Any segment with origin in mass center and end point on figure's boundary is contained within ...
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0answers
41 views

Hausdorff Distance on Fuzzy Sets

I'm trying to define the Hausdorff distance between two fuzzy sets in terms of non-fuzzy sets. Is this a viable definition? How can I show that this reduces to the Hausdorff Distance for non-fuzzy ...
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1answer
25 views

n-tuple function definition

I've read this definition for an hour now and I cannot piece it together abstractly. To define an n-tuple as a function $F$, where $X$ is the index set and domain, and $Y$ is the set containing the ...
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1answer
47 views

How many teams of $5$ players out of $15$ girls and $10$ boys can be formed with at least $2$ boys and $2$ girls [with complement]

How many teams of $5$ players out of 15 girls and 10 boys can be formed with at least 2 boys and 2 girls? The solution has to be with complement. This is related to: How many ways to assemble a ...
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1answer
12 views

Definition of the tensor product of finite sequence of modules

I have posted several questions about the tensor product of modules before and this post would be the final one. I have read wikipedia,mathSE,Dummit&Foote and Bourbaki for the definition of the ...
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1answer
64 views

What is the difference between an antiderivative and an integral?

In my textbook, it states the fundamental theorem of calculus as follows: If $f(z) $ has an antiderivative $F(z)$, then $\int^{z_2}_{z_1} f(z)dz=F(z_2)-F(z_1)$. There isn't a definition of what an ...
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18 views

Definition of multimodule

Reference : Bourbaki - Algebra I p.224 Let $\{A_i\}_{i\in I}$ and $\{B_j\}_{j\in J}$ be families of rings. Let $M$ be a set such that for each $i\in I, M$ is a left $A_i$-module and for each ...
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1answer
38 views

Definition of tensor product

Here is the standard dedonition of the tensor product of two modules: Definition for tensor products of two modules: Let $R$ be a ring and $M$ be a right module and $N$ be a left module. ...
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36 views

Use of the word “if” in mathematical definitions [duplicate]

I'm looking at the following definition The random variables $X_{1}, \ldots, X_{d}$ are said to be comonotonic if they admit as copula the Frechet upper bound. I am however not quite sure how to ...
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1answer
49 views

Is free abelian group a “free” abelian group or “free abelian group”?

Let $G$ be an abelian group. What does it mean that $G$ is a free abelian group? Does this mean that $G$ is a free group or a free-$\mathbb{Z}$-module with the operation $n•a=a+...+a (n-times)$? Or ...
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56 views

Relations in a finitely complete category with images

I can't use tikz inside MathStackExchange; so my question is on an image below.
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1answer
27 views

Dirac Delta Function definition with ksi (ξ)

The dirac delta function has a definition $$f(0)=\int_∞^∞f(x)δ(x)dx$$ and $$ f(x)=\int_∞^∞f(x-ξ)δ(ξ)dξ $$ (the lower bound is minus infinity but I couldn't add a minus :/) I do understand the ...
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1answer
84 views

If for any $\varepsilon$ exists $\delta$, does that mean that for every $\delta$ exists $\varepsilon$? [closed]

For any $\varepsilon \gt 0$ exists $\delta \gt 0$, does that mean that for any $\delta \gt 0$ exists $\varepsilon \gt 0$? If $\delta$ depends on $\varepsilon$ such as $\delta = \frac 1 \varepsilon$, ...
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2answers
300 views

Why use open sets in definitions?

I've been wondering why do we use mostly open sets in mathematics to define numerous things. For example continuity is defined using (open) neighbourhoods, differentiability and Taylor's theorem is ...
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1answer
36 views

What do these definitions of conjugacy have in common?

Here are four (seemingly) different uses of the word conjugate: Complex conjugates are a concrete instance of the idea of conjugacy in field extensions. In group theory, there's the idea of ...
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2answers
82 views

Memorising lots of maths theorems/lemmas

In a few weeks I'll have my summer exams in a Senior Freshman Mathematics course. Two of my modules have a huge number of theorems, lemmas and definitions - over 200 definitions and 250 proofs by my ...
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1answer
31 views

Definition: What is a two-sided Lie ideal of a Lie algebra?

Let $\mathfrak{g}$ be a Lie algebra and let $\mathfrak{h}$ be a subalgebra. According to wikipedia, $\mathfrak{h}$ is called an ideal of $\mathfrak{g}$ if it satisfies the condition that ...
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0answers
32 views

What is the standard definition of Torsion element?

Here are two different definitions of torsion element. Let $M$ be an $R$-module and $m\in M$. Wikipedia: $m$ is a torsion element iff there exists a nonzero regular element $r$ (i.e. Not a zero ...
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1answer
16 views

Different percent-valued Majority gates

It has been defined that a Majority gate follows this formula concerning its behavior about outputs: $$\operatorname{Maj} \left ( p_1,\dots,p_n \right ) = \left \lfloor \frac{1}{2} + ...
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0answers
25 views

Definition of a topology on a set $X$: may $I$ contain uncountable many elements or is it restricted to be finite or countable?

I've the following definition of a topology $\mathcal I$ on a set $X$: (T1) $U_a \in \mathcal I, a \in I \Rightarrow \cup_{a \in I} U_a \in \mathcal I$ (T2) $U_1, U_2 \in \mathcal I \Rightarrow U_1 ...
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1answer
55 views

About the definition of universal covering space

There are some references (for instance in Greenberg & Harper) that consider the universal covering space to be not only simply connected but also locally path connected. This definition seems to ...
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23 views

Why do we define the ordered pair in this way? [duplicate]

When we define an ordered pair (x,y) in a set, why, in many textbook, do we define it as {x,{x,y}} or {{x},{x,y}} instead of {x,{y}} or {{x},{y}} which obviously makes more intuitive sense if we ...
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What is a “nonparametric function”?

I am looking for a formal definition of the term "nonparametric function." I understand the term and I use nonparametric regressions http://en.wikipedia.org/wiki/Nonparametric_regression but I ...
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3answers
102 views

Field axioms: Why do we have $ 1 \neq 0$?

In the definitions of a field, we have $ 1 \neq 0$. I know that in regular multiplication $0 \times 1=0$ but for reciprocal we don't have inverse of $0$. But all the spaces and different definitions ...
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0answers
17 views

definition of limit at a point and relationship with gradient

Why is $\lim\limits_{t\to\infty}\frac{f(x^*+td)-f(x^*)}{t}=\nabla f(x^*)^Td$ for $t>0$ and sufficiently small?
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2answers
58 views

Proper divisors of 1?

What are the proper divisors of 1? I understand proper divisors do not include the number itself, but is 1 an exception or does it have no proper divisors?
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2answers
78 views

Logical consequence in all structures in Kripke semantics

I read* the following definition of logical consequence in all structures within Kripke semantics:$$X\models A\iff\text{ for every } (W,R),\text{ if }(W,R)\models X,\text{ then }(W,R)\models A$$ ...
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25 views

What the meaning 'directed'

What's the meaning of 'directed' and 'cofinal'.It's about a partially ordered set. Please give me an example? A preordered set (I, ≤) is directed if every finite subset F of I has an upper bound. A ...