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

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0answers
106 views

Bad and good definitions in mathematics [on hold]

The topic was brought up several times on StackExchange, but no satisfactory answer was given in any of them. One of similar topics on MO It’s about mathematical definitions – can a definition be ...
-1
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0answers
26 views

Signature of a complex matrix

If $A\in M_n(\Bbb R)$, we define its signature as the triple $(s^+,s^-,s^0)$, where $s^+,s^-,s^0\in\Bbb Z_{\ge0}$ denotes respectively the number of eigenvalues $>0,<0,=0$. How can we define ...
3
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3answers
120 views

Undetermined vs. Undefined [duplicate]

This often comes up in precalculus and calculus, that is sometimes an expression will be said to undefined while at other times undetermined. What is the fundamental difference between the two? For ...
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0answers
22 views

formal definition of a linear programming formulation

Despite having done operations research for several years, and being familiar with linear programming formulations, I am having difficulty giving a formal definition of what it means for something to ...
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1answer
25 views

Is there a name for embedding of Lebesgue spaces $L^q(a,b) ⊂ L^p(a,b)$

Let $1 ≤ p ≤ q ≤ ∞$ then $L^q(a,b) ⊂ L^p(a,b)$ where $L^q(a,b), L^p(a,b)$ are Lebesgue spaces Is there a name for this relationship? Is this the Sobolev embedding theorem?
11
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3answers
749 views

Differentiable at a point

My roommates and I have an argument you guys can help to settle (peace is at stake, don't let us down!) In undergrad calculus courses, one usually explains what it means for a function to be ...
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2answers
296 views

What is a third proportional?

I searched online, couldn't find anything clear. If I had two numbers, $a,b$, what is their third proportional? Apparently it can be either $c$ such that $a/b=b/c$ or $b/a=a/c$, but obviously these ...
0
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2answers
33 views

Splitting condition of forcing posets

I was looking at Wikipedia for brief reminders of what I learned in my elementary set theory class, and discovered the forcing page (which I did not learn): A forcing poset is an ordered triple, ...
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0answers
24 views

How Kriging, Bochner theorem and Positive definite (PD) function are related?

This question referes to the link: https://en.wikipedia.org/wiki/Kriging I can understand the relation between Bochner's theorem and PD function. But could not properly understand and connect all ...
13
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3answers
381 views

Is a space compact iff it is closed as a subspace of any other space?

I am trying to come up with an alternate definition of a compact topological space that coincides with the usual one. Sorry if my topology is a little rusty. My proposed alternative definition is ...
2
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1answer
35 views

Can we state the triangle inequality as $|\int_D f(x) dx| \leq \int_D |f(x)| dx$

$|\int_D f(x) dx| \leq \int_D |f(x)| dx$ is just the infinitestimal version of the triangle inequality commonly presented in any book on vector spaces Can we replace the definition of triangle ...
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3answers
1k views

Why do functions with compact support include those that vanish at infinity?

The support of a function is defined in Wikipedia as "the set of points where the function is not zero-valued, or the closure of that set". Functions with compact support in $X$ are defined in ...
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1answer
44 views

A question about combinatorial commutative diagrams.

In the comments of this question the directed graphs that can appear as commutative diagrams are axiomatized: A graph is a combinatorial commutative diagram iff it's nonempty and such that any ...
1
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1answer
32 views

A subset on which a measure is strictly positive

There is a "cake" which is represented by the interval $[0,1]$. There is a non-atomic value measure $V$ defined on the cake. I would like to define an algorithm for dividing the cake between two ...
3
votes
2answers
6k views

The degree of a polynomial which also has negative exponents.

In theory, we define the degree of a polynomial as the highest exponent it holds. However when there are negative and positive exponents are present in the function, I want to know the basis that we ...
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2answers
302 views

Rudin's definition of continuity in terms of pre-images (inverse images). Is this simple function continuous or not?

I am reading W. Rudins book ``Principles of Mathematical Analysis''. I find it hard to exactly understand the definition of continuity in terms of pre-images. Rudins definition of a continuous ...
4
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1answer
43 views

definitions of order-dense Riesz spaces

I would like to get an intuitive idea as to why the definitions of order-dense and locally order-dense Riesz subspaces were chosen in the following way: A Riesz subspace $F$ of $E$ is order-dense if ...
1
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1answer
15 views

Lifting a principal G-bundle to a principal bundle with structure group a covering of G

Let $P\to $ be a principal $G$-bundle. Suppose $U$ covers $G$. What do we mean by a lift of $P$ with respect to $U$? Can we take $P,M,G,U$ such that no lift exists?
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2answers
31 views

Possible definition of the matrix representation of a linear transformation with respect to given bases

Let $E$, $F$ be vector spaces with basis $\{e_1,\dots,e_m\}$, $\{f_1,\dots,f_n\}$. Let $T:E\to F$ be a linear transformation. We say that the matrix $A\in\mathbb{R}^{m\times n}$ represents $T$ with ...
4
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1answer
58 views

Is there a name for this simple structure?

Is there a name for $(X,S)$ where $X$ is a set and $S\subseteq X$ and a morphism $(X,S)\overset{\alpha}{\longrightarrow}(X^\prime,S^\prime)$ is a function $\alpha:X\rightarrow X^\prime$ such that ...
-1
votes
1answer
37 views

Is 1 coprime to itself? [closed]

Is $\{1,1\}$ a pair of co-prime numbers? According to the definition, two numbers are coprime if $\gcd(a,b)=1$, and for $\{1,1\}$ it is true that $\gcd(1,1)=1$.
3
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2answers
77 views

Clarification on the two assumptions of Lebesgue integral?

The Lebesgue measure has the following properties: $\mu(0) = 0$; $\mu( C) = \operatorname{vol} C$ for any $n$-cell $ C$; if $\{M_1, M_2,\ldots \}$ is a collection of mutually disjoint sets in ...
0
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0answers
31 views

Are permutation group block only defined in the context of finite sets?

From Dummit and Foote (emphasis mine): Let $G$ be a transitive permutation group on the finite set $A$. A block is a nonempty subset $B$ of $A$ s.t. $\forall \sigma\in G:$ either $\sigma(B)\cap B ...
3
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2answers
71 views

Why do we care if a function is uniformly continuous? [duplicate]

There are a lot of question regarding whether a function is or is not uniformly continuous or just continuous and there are a lot of $\epsilon_s$ and $\delta_s$ trying to show whether a function is ...
3
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1answer
50 views

What is the precise difference between functions and operators?

I have heard affirmatively that all operators are functions, but not all functions are operators. But at the same time I have heard that functions map numbers to numbers, whereas operators map ...
2
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1answer
25 views

How do we define discriminant over a commutative ring?

Let $f$ be a nonconstant polynomial over a field $F$. Since there exists a splitting field of $f$ over $F$, $f$ can be decomposed as $f=c\prod_{i=1}^n (X-\alpha_i)$ Hence, it is possible to define ...
2
votes
3answers
113 views

What is the precise definition of 'between'?

I'm wondering what the precise definition of 'between' is in a mathematical context. For example, many statements of the intermediate value theorem state that a value $k$ is 'between' $f(a)$ and ...
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1answer
43 views

Is there a specific mathematical term for a shape whose dimensions are defined?

When I say the word "circle", I know that I have described a "shape". Specifically, a "circle" is the shape formed by the set of all points in a plane that are at a given distance from a given point. ...
2
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4answers
75 views

Are there relations between elements of $L^p$ spaces?

I have read about dual spaces and the relation $1/p+1/q=1$ as mentioned in the Wikipedia page. Are there any more theorems or relations that connect elements between the $L^p$ spaces for different ...
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0answers
55 views

Operator form $L^2$ space to$L^1$

Can we have an operator such that it transforms an element of $L^2$ to $L^1$? Is this a valid question or this is incorrect? We can consider the measure space as finite.
2
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1answer
15 views

Simplicial homology: chain group with basis open n-simplices vs. chain group with basis closed n-simplices

In his Algebraic Topology book, Hatcher defines the chain groups for simplicial homology as free abelian groups with basis the open $n$-simplices of some simplicial complex X. Is there any ...
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0answers
21 views

Is there one to one relation Positive definite(PD) matrix and PD function?

Is it correct to say that a PD matrix can be built from a PD function? For example circulant matrix or Toeplitz seems to be built from a positive definite function. Positive definite function is ...
0
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1answer
34 views

How to prove this two separations of connectedness is equivalent?

Definition 1$\quad$ A metric space $E$ is connected if it cannot be written as the union of two nonempty separated sets (in $E$). Definition 2$\quad$ A metric space $E$ is connected if it cannot be ...
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2answers
28 views

Why does the Big Oh (and similar) notations needs $n_0$?

The generally agreed definition of the Big Oh notation (afaik) is as follows: The function $f(n)$ is $O(g(n))$ if there exists constants $c$ and $n_0$ such that for all $n \ge n_0$, $f(n) \le c ...
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2answers
69 views
1
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1answer
18 views

Why should the matrix $A$ in an ILP be integer?

Almost everywhere I read about integer linear programming (ILP), I found that the matrix has to be integer (by definition). More precisely, an ILP is defined as follows: An ILP in canonical form is ...
1
vote
1answer
32 views

General meaning of the term “Functional”

In the most general is a "functional" simply a function which can accept a function as input? So, is it natural to describe: $f: \mathbb{N} \rightarrow \mathbb{N}$ as a function. Whereas it is more ...
3
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4answers
1k views

What precisely is the difference between Euclidean Geometry, and non-Euclidean Geometry?

I was wondering, what it is precisely which defines the difference between Euclidean and non-Euclidean Geometry, in a few words/equations/diagrams? Would I be correct in understanding that ...
1
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1answer
45 views

Priority of the 3 axioms of groups [duplicate]

In my book about Abstract Algebra, it is stated that A group $\langle G,*\rangle$ is a set $G$, closed under a binary operation $*$, such that these 3 axioms are satisfied: $g_1$: For all ...
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1answer
13 views

Name for a collection of subsets of a set $E$ such that every element of $E$ is in some member?

Let $X$ be a set. A partition of $X$ is a collection of subsets $\{E_\alpha\}\subset\mathcal{P}(X)$ such that the following holds. For every $x\in X$, there is some $\alpha$ such that $x\in ...
3
votes
2answers
85 views

Notation for Tautologies

I've been stuck for a while in this question and so far I don't understand the flaw of my reasoning please if you guys could help me out. See, this is my context. From the definition of argument we ...
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0answers
31 views

$\mathbf{A}$ is unimodular $\Rightarrow$ $\mathbf{A}$ has entry in $\{-1, 0, 1\}$?

Is it true that $$\mathbf{A}\;\text{is unimodular}\;\Rightarrow\mathbf{A}\;\text{has entry in}\; \{-1, 0,1\}?$$ Also can an unimodular matrix $\mathbf{A}$ has entry in $\mathbb{R}$?
0
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2answers
56 views

Rudin's Chain Rule

Rudin's chain rule theorem goes like this: Suppose $f$ is continuous on ${[a,b]}$, $f'(x)$ exists at some point $x\in [a,b], g$ is defined on an interval $I$ which contains the range of $f$, and ...
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1answer
24 views

Nurbs parametric coordinate span

I am using the Nurbs definition of Wikipedia. I might have missed something in the definition but I cannot understand how to know on which interval does the parametric coordinate span. Particularily ...
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4answers
363 views

Why do Topologies get “finer”?

Why are topologies with many elements called "fine" and topologies with few elements called "coarse"? It seems as though the finer a topology is, the more likely it is for a function defined from that ...
3
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1answer
62 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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1answer
48 views

Help in understanding Bochner's theorem

This relates to the Bochner's theorem stated in the link: https://en.wikipedia.org/wiki/Bochner%27s_theorem My question is related to the unique probability measure μ on G. I want to express the ...
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1answer
109 views

Why are some branches of mathematics called 'theory' and others not?

We say: graph theory , group theory, number theory , set theory, what is definition of theory? We also say abstract algebra, real analysis, but why we do not say abstract algebra theory or real ...
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1answer
29 views

Ambient Isotopy of Knots

Statement 1: Knots of opposite chirality have ambient isotopy, but not regular isotopy. Statement 2: We can then define two such knots to be equivalent if they are ambient isotopic, meaning that ...
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0answers
32 views

Isotopy: Definition

An isotopy is a homotopy from one embedding of a manifold $M$ in $N$ to another such that at every time, it is an embedding. In this definition, I am wondering why $M$ and $N$ are required to be ...