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

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4
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2answers
152 views

Is $e^x=\exp(x)$ and why?

In the comments to this question a discussion came up wether we have $e^x=\exp(x)$ by definition and what the "correct" definition of $\exp(x)$ is. Building on that, I want to line out the problem ...
1
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0answers
11 views

Can we have extension of Mercer theorem to interpolation?

This question is related to Mercer theorem, Reproducible kernel Hilbert space(RKHS) and interpolation. The wikipedia links are https://en.wikipedia.org/wiki/Mercer%27s_theorem and ...
1
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0answers
44 views

Is There a Common Definition of “Finite Support”?

I thought I understood this term, but when I tried to verify this I found three different and conflicting definitions, none corresponding to mine. Is there a generally agreed definition for this term ...
0
votes
1answer
67 views

Conditional Definition Rule(Theory of Definitions)

I am reading Patrick Suppes' "Axiomatic Set Theory". It defines "Conditional definitions of operations" as follows: An implication P is introducing a new operation symbol O is a conditional ...
2
votes
1answer
23 views

Pulling back a connection to a curve

What does it mean to pull a connection back to a curve? For example, if I take the connection $\nabla s = ds$ on the trivial bundle $\mathbb R^2 \times \mathbb R^2$ over $\mathbb R^2$, and the curve ...
3
votes
1answer
53 views

Am I right about this definition of submanifold?

Consider the following definition of submanifold: 1.5. $\ \bf Definition.\ $ A subset $M\subset\mathbf R^{n}$ is called a $\underline{\text{differentiable submanifold}}$ of $\mathbf R^n$ of ...
2
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0answers
36 views

When do you call something “a calculus” vs. “a logic”?

Similar to What do Algebra and Calculus mean?, what is the difference between a logic and a calculus? I am learning about the different kinds of logics, and often when I look them up in a different ...
4
votes
2answers
64 views

Does anybody know the definition of $C^{2+\alpha, 1+\frac{\alpha}{2}}(\Omega)$, where $0<\alpha<1$?

I hope someone can give me the definition of the following: $C^{2+\alpha, 1+\frac{\alpha}{2}}(\Omega)$ for some $0<\alpha<1$ and some domain $\Omega$. In this context they also talk about ...
0
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0answers
88 views

Meaning of $\mod{G'} in quasibasis

I took the definition used in Outer Automorphisms in Nilpotent $p$-Groups of Class 2, H. LlEBECK An abelian $p$-group $A$ with basic subgroup $B$ has a quasibasis $a_\lambda (\lambda \in \Lambda), ...
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2answers
26 views

Definition of associative algebra over a field

In the definition of an algebra over a field in the wiki entry , it states that an algebra over a field is a vector space equipped with a bilinear product. Question: Does anyone know how a bilinear ...
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0answers
65 views

Definition: vector or point belonging to an area

This is an applied problem, which I try to define mathematically. I have two vehicles, vehicle 1 is defined by the area, dependent on length $L$ and width $W$ of the vehicle, according to: ...
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0answers
16 views

Is there any parameter space of Cramér–Rao_bound

It is known that Cramér–Rao_bound is the lower bound of variance of a parameter. A useful link is https://en.wikipedia.org/wiki/Cram%C3%A9r%E2%80%93Rao_bound There is also a term called ...
2
votes
1answer
20 views

Softball here… what is the precise definition of a proper modification, in the context of complex geometry/manifolds?

It's a classic case of most papers using a concept and proving high-level mathematics about it, without ever stating the definition or some simple properties of the object. Which is just peachy for ...
2
votes
1answer
38 views

Math notation: What does $I$ mean in this context?

Sorry if this is a noobish question, but I don't know what $I$ means in this context: $$\hat{f}(X) = \sum_{m=1}^5 c_m I\{(X_1, X_2) \in R_m\}$$ I am reading about Decision Trees in The Elements of ...
3
votes
2answers
60 views

Are sequences properly denoted as $\subset$ of a set, or $\in$ a set?

Given some sequence $(x_n)$ of some subset $M \subset \mathbb{R^n}$, is it more appropriately to denote $(x_n) \subset M$, or $ (x_n)\in M$? This stems from confusion of using "in" i.e. whenever ...
1
vote
1answer
71 views

Are there expansions of the expression $(a+b)^{1/n}$? [duplicate]

Is there an expansion of the expression in the bracket such as $$ \sqrt{a + b} = (a + b)^{1/2}$$ If not do you know of a method that lets us solve such expression and ones with higher roots?
0
votes
0answers
28 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 ...
0
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0answers
29 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 ...
3
votes
3answers
136 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 ...
1
vote
1answer
29 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?
1
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0answers
29 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 ...
2
votes
1answer
37 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 ...
13
votes
2answers
397 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 ...
1
vote
1answer
47 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
vote
1answer
34 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 ...
1
vote
1answer
18 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?
4
votes
1answer
61 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 ...
-2
votes
1answer
43 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
votes
2answers
79 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
votes
0answers
32 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
votes
2answers
75 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
votes
1answer
55 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
votes
1answer
30 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 ...
1
vote
1answer
46 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. ...
1
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0answers
58 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
votes
1answer
16 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 ...
2
votes
3answers
119 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 ...
0
votes
0answers
22 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
votes
1answer
35 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 ...
2
votes
4answers
79 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 ...
1
vote
2answers
32 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 ...
1
vote
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
33 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 ...
1
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1answer
47 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 ...
1
vote
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 ...
4
votes
1answer
48 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 ...
0
votes
2answers
35 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, ...
3
votes
2answers
86 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 ...
0
votes
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
60 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 ...