# Tagged Questions

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

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### Understanding the condition of $\diamondsuit^+$

I try to understand $\diamondsuit^+$, a variant of the diamond principle. It says that there is a sequence $\langle \mathcal{A}_\alpha\rangle_{\alpha<\omega_1}$ of collection of subsets which ...
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### Proving rigorously that if $3\mid(a+b)$ then $3\mid(a^3+b^3)$ using divisibility definition

Let $a,b\in \mathbb Z$. Prove rigorously using divisibility definition that if $3\mid(a+b)$ then $3\mid(a^3+b^3)$ After a bit of algebra I get that $$3\overset{?}{\mid}(a+b)^3-3ab(a+b)$$ So now how ...
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### Clarification of some doubts on the definition of submatrix

I don't fully understand how I can choose a submatrix in a matrix. Judging from this definition and picture (http://mathworld.wolfram.com/Submatrix.html), I would assume that you can't pick as a ...
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### What abstract structures allows us to describe “nets that converge toward each other”?

Topological spaces are equipped with a bare minimum of structure to allow for a formalization of the statement "the net $a$ converges to the point $x$." Actually this isn't strictly true, but its true ...
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### Definition of complex conjugate in complex vector space

I am starting reading about Hodge theory and while reading the definition of abstract Hodge structure a very basic question came to my mind... What is the definition of the conjugate of a subspace of ...
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### Elliptic boundary value problems and elliptic partial differential equations

I am interested in the relation between the definition of an 'elliptic boundary value problem' and an 'elliptic partial differential equation'. From the wiki entries it seems that 'elliptic boundary ...
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### Is there a term for something equally distributed around zero?

Let's say X is uniformly distributed in [-1 , 1] Then what can we call the distribution of X³ ? It is not uniform, but it "mirrors" around 0 as well. Is there a simple word describing X³ that would ...
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### Why are the two definitions of covariance equal? [duplicate]

For example in Wolfram mathworld, you get these two definitions of covariance. http://mathworld.wolfram.com/Covariance.html cov(X,Y) = E[ (X - E[X])(Y - E[Y]) ] = E[XY] - E[X]E[Y] ...
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### How can we think and/or write rigorously about integration by substitution?

Define a function $I:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ as follows. $$I(a,b)=\int_a^b \sin t \cos t \,d t$$ Then we can find a more explicit description of $I$ using integration by ...
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### What is the difference between assuming for $\forall k< n$ and proving for $n$ than assuming for $n$ and proving for $n+1$

What is the difference between assuming for $\forall k< n$ and proving for $n$ than assuming for $n$ and proving for $n+1$ (or $n-1$ and proving for $n$)? both with induction. The first one is ...
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### The definition of continuously differentiable functions

When we say $f \in C^1$, we mean that f is continuously differentiable. Isn't the continuity a redundant word? I mean, we have a theorem that says if $f$ is differentiable then it is continuous. So ...
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Reference: A primer of nonlinear analysis - Antonio & Giovanni Let $H$ be a hilbert space over $\mathbb{K}$ and $U$ be open in $H$ and $p\in U$ and $f:U\rightarrow \mathbb{K}$ be a functional ...
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### What does rigoruous but non-technical mean?

Hi I find the above expression a bit confusing. I am considering buying a book and it says that it's a rigorous but non-technical introduction to optimal stochastic control. Could someone explain ? ...
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### What is the definition of differentiation in normed space?

I'm trying to generalize implicit&inverse function theorems in Euclidean spaces to the context of Banach spaces. I'm wondering what would be the definition of differentiation in Banach space and ...
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### What is an omega model?

I went to a seminar and a side question was if a theory had an omega model, however from the context I could not deduce the exact meaning. Does an omega model have a general meaning in mathematical ...
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### Proper usage for term, addend, factor, multiplicand, expression, formula

The definitions and usage of the following words seem to vary, depending on the source text: term addend factor multiplicand expression formula The words are being used in the context of pre-...
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### Clarification on definition of a basis

Quick question; lets say that $S$ is a basis of $V$. I understand that this means all vectors in $S$ are linearly independent, and that every vector in $V$ is an element of $\text{span} \ S$. Is it ...
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### Is this relation transitive? $R=\{(1,2),(1,1),(2,1),(2,2)\}$ over $A=\{1,2,3\}$

Is this relation $R$ over $A$ transitive?$$A=\{1,2,3\}$$ $$R=\{(1,2),(1,1),(2,1),(2,2)\}$$ Since from the definition a relation is transitive if $\forall x,y,z\in A (xRy,yRz\to xRz)$, so since $3$ ...
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### What is the “correct” way of making $\mathcal{P}(X)$ into a topological space?

If $X$ is a topological space, I want to know the "correct" way of making the powerset $\mathcal{P}(X)$ into a topological space. So let $\mathsf{SupLat}$ denote the infinitary Lawvere theory whose ...
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### Are 'vectors' vectors?

Let us say I have a 'vector' $\vec v$ for which I can do the following operation on $A\vec v$ where $A$ is a matrix. Now most people (i think) would say that $\vec v \in R^n$ however $\vec v$ is not a ...
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### Given a field extension $K\colon F$, $K$ is an $F$-vector space

I'm having a hard time understanding fields. Could someone help with the following I need to show that if $F$ $\subseteq$ $K$ are both fields and addition and multiplication on F are the ...
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### If $f\circ g = g \circ f$ does that mean that both functions are to and from the same set and both are bijections? Does it tell us anything else?

If $f\circ g = g \circ f$ does that mean that both functions are to and from the same set and both are bijections? Does it tell us anything else?
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### Question about defintion of inner product space

While practising I came across the following easy question: "Is the space $B(0,1):=\{f:[0,1]\rightarrow\mathbb{R}$ bounded$\}$ an inner product space?" But I'm not quite sure what the correct answer ...
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### If the empty set is a subset of every set, why isn't $\{\emptyset,\{a\}\}=\{\{a\}\}$?

If the empty set is a subset of every set, why it isn't written with the elements of a set? like so $\{1,2,3,\emptyset\}$ Or why isn't $\{\emptyset,\{a\}\}=\{\{a\}\}$? I know one has two elements ...
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### Find the right cosets of $H$ in $G$ simple example

Question: Let $G$ be a group and $H<G$ a subgroup with $|G:H|=2$ Show that the right cosets of $H$ in $G$ are $H$ and $G\backslash H$ Answer given: There are two right cosets, they are disjoint ...
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### Give an example of a set which is not transitive

Transitive set: set $x$ is transitive if $\forall y\in x(y\subseteq x)$ I think $\{\varnothing\}$ is not transitive since $\varnothing\in\{\varnothing\}$ but $\varnothing\not\subseteq\{\varnothing\}$ ...
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### Why can we write a uncontinual function continual?

Let's consider $$f(x) = \frac{(x-1)(x-2)(x-3)}{x^2-3x+2}$$ with definition $D_{f} = \mathbb{R} \setminus \lbrace 1, 2 \rbrace$. This means we are allowed to set $x$ to every value of $\mathbb{R}$ ...
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### What does bimeasurable mean? Is an invertible transformation bimeasurable?

What exactly is the meaning of a bimeasurable transformation? I did not find a very clear answer to that. As far as I see it means that Borelsets are maped to Borelsets. So an invertible ...
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### Corresponding partition in equivalence relation

The relation $R$ on the set $A=\{2,4,6,8,10\}$ is defined by $$R=\{(2,2),(2,6),(2,10),(4,4),(4,8),(6,2),(6,6),(6,10),(8,4),(8,8),(10,2),(10,6),(10,10)\}$$ Question 1 Verify if $R$ is an ...
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### Is a function defined at a single point continuous?

Is a function defined at a single point continuous? For example $f:\{0\}\to\{0\}$ defined by $f(x)=\sqrt{x}+\sqrt{-x}$ is a sum of two continuous functions and is therefore continuous, however for $f$...
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### What is a regular homotopy?

The definition of regular homotopy from Wikipedia says that two immersion $f,g:M\to N$ are regularly homotopic if they represent points in the same path-component of $\text{Imm}(M,N)$. What does "...
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### Questions which have false conditions

There are many "questions" on the internet like If $$1=5$$ $$2=6$$ $$3=7$$ $$4=8$$ then how many is $5$? With one "logic" answer is $9$ because $n=n+4$, then $5=9$. With other "logic" ...