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

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1answer
64 views

What does it mean if $P^n$ is irreducible for every $n\in\mathbb{N}$?

If $P$ is the transition matrix belonging to a markov chain, then what does it mean that $P^n$ is irreducible for every $n\in\mathbb{N}$? For $n=1$ it means that all states communicate with each ...
2
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0answers
52 views

Conceptual question on showing properties of the absolute value function on $\mathbb{Q}$

I have a rather conceptual question about showing certain small lemmas regarding the absolute value function on $\mathbb{Q}$. I want to only give one example: Let $a,b \in \mathbb{Q}$ and $|.|$ ...
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3answers
80 views

Contrapositive of a Definition

I have a problem (in real analysis class) that states "What is the contrapositive of the definition of "closed"?" The definition in our class of closed is: "a set E is closed iff the set contains all ...
59
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5answers
8k views

Do we have negative prime numbers?

Do we have negative prime numbers? $..., -7, -5, -3, -2, ...$
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0answers
111 views

What does $p\mathbb{Z}_p$ mean?

I am looking at Hensel's Lemma: Let $F(x)=a_0+a_1x+ \dots + a_n x^n \in \mathbb{Z}_p[x]$. We suppose that there is a $p$-adic number ($p>2$) $\alpha_1 \in \mathbb{Z}_p$, such that: $$F(\alpha_1) ...
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2answers
47 views

X and Y have the same cardinality if and only if bijection from X to Y? [duplicate]

My textbook says "Let X and Y be sets. We say X and Y have the same cardinality if there is a bijection f: X --> Y." I was wondering why the text does not say "if and only if." A bijection implies ...
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1answer
24 views

Codomain confusion

I'm confused about the codomain of a linear transformation. If we have a linear transformation which maps from $\mathbb{R}^n$ to $\mathbb{R}^m$ and the range of the linear transformation is only the ...
2
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1answer
53 views

Family of “something very close to be a curve” over a curve $C$

Hartshorne (IMHO restrictive) definition of a curve: Definition of (complex) curve: A curve is an integral separated scheme of finite type over $\mathbb C$ of dimension $1$. (The definition of a ...
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1answer
107 views

What is the word for the numerical portion of a numeric value with units?

When I see 3 Miles the word "miles" represents the units. What is the word which 3 represents? I want to say "scalar value" but I know that's not correct. Or if it is correct is 4 also the scalar ...
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2answers
56 views

Deriving the analytical properties of the logarithm from an algebraic definition.

Definition: The base $a$ logarithm ($a\in]0,1[\cup]1,+\infty[$) is the continuous function defined by: $\log_a(xy)=\log_a(x)+\log_a(y)~~\forall x,y>0$ and $\log_a(a)=1$ If I used this definition ...
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0answers
21 views

Definition of Adic-filtration

I am reading On the Associated Graded Ring of a Group Ring by DANIEL G. QUILLEN and this is the beginning of the article: I have no idea what is $\overline{KG}$-adic filtration, as well as the ...
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1answer
139 views

What is the definition of “Augmentation Ideal Filtration”?

Let $A$ be an algebra. What is the definition of the Augmentation Ideal Filtration of $A$? Any answer with reference will be greatly appreciated.
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1answer
145 views

When is it insufficient to treat the Dirac delta as an evaluation map?

The Dirac delta "function" is often introduced as a limit of normal distributions $$\delta_a(x)=\frac{1}{a\sqrt{\pi}}e^{-\frac{x^2}{a^2}}\text{ as }a\to0^+.$$ Obviously, this sequence of functions ...
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0answers
214 views

Borel Measures: Atoms (Summary)

Disclaimer: The question here has been solved, now: Finest Measurable Partition (For jeapardy it is stated below, anyway. Have fun! ;) ) Summary: This is a summary of the discussions: ...
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2answers
53 views

Are there definition of percent?

In a school I was taught that percent is the same as 1/100. But I think that definition is not rigorous enough because that would imply for example that $5+4\%=5+4/100=5.04$ but this seems weird. So ...
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1answer
45 views

Alternative Definition for Injective Function?

I came up with an alternative definition of an injective function and would like to know if it's correct and how to prove it if it is, or why it's not correct if it isn't. f:A→B is injective if ...
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2answers
32 views

I don't understand the definition of a base of a metric space

Definition: A collection {$v_n$} of subsets is said to be a base for X if for every x $\in$ $X$ and every open set $G$ $\subset$ $X$, such that x $\in$ $G$ we have x $\in$ {$V_n$} $\subset$ $G$ for ...
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2answers
309 views

Borel Measures: Atoms vs. Point Masses

Let a measure be $\mu:\Sigma\to\mathbb{R}_+$. Call a measurable $A\subset\Sigma$ an atom if: $$\mu(A)>0:\quad\mu(E)<\mu(A)\implies\mu(E)=0\quad(E\subseteq A)$$ and a singleton $\{a\}\in\Sigma$ ...
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3answers
144 views

Is this formula satisfiable?

I am confused whether or not my explanation for whether or not this formula is satisfiable is correct. Note that the question state it should be Brief and it should not be necessary to write down a ...
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7answers
2k views

Are there mathematical contexts where “finite” implicitly means “nonzero?”

I recently gave my students in a discrete math class the following problem, a restatement of the heap paradox: Let's say that zero rocks is not a lot of rocks (surely, 0 is not a lot of rocks) and ...
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2answers
151 views

Uniform convergence - definition / notation clarification

My professor gave the following definition in class for uniform convergence: $(f_{n}: A \subset \mathbb{R}^{k} \rightarrow \mathbb{R}^{l})_{n=1}^{\infty}$ converges uniformly to $f$ on $A$ if and ...
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1answer
37 views

Subalgebra Definition

If A is an algebra over a field K, and B is a subalgebra, must B be an algebra over K or can it also be an algebra over some subfield of K? For example, if you take $\mathbb{R}$ as an algebra over ...
6
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1answer
163 views

Why limit $\sqrt{\frac{\sin(x)}{x}}$ as $x \rightarrow \infty$ is not a real number?

Let $f(x)=\sqrt{\frac{\sin(x)}{x}}$. Why isn't the $\lim\limits_{x\rightarrow \infty} f(x)$ equals to some $l \in \mathbb{R}$? The definition of a finite limit at inifinity is: $$\forall ...
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2answers
43 views

What is word reversal $w^R$?

In the following context, what is the formal meaning of "reversal of word $w$"? The free monoid on $A$ is the syntactic monoid of the language $\{ ww^R\ |\ w \in A^*\}$, where $w^R$ denotes the ...
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2answers
86 views

Definition question in algebraic topology.

Definition: The $p$th (de Rham) cohomology group is the quotient vector space $$H^p(U) = \frac{Ker(d:\Omega^{p}(U)\to \Omega^{p+1}(U)}{Im(d:\Omega^{p-1}(U)\to \Omega^{p}(U))}$$ where $U \in ...
1
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1answer
38 views

On closedness of $C^\ast$ subalgebras

By definition of a $C^\ast$ subalgebra it is a closed subalgebra. Why does it need to be closed? This is a restriction that is not required in the case of a Banach subalgebra. (although I can't ...
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2answers
76 views

Integral of $\frac{1}{x}$

The logarithm is defined as: $$ \ln x = \int_1^x \frac1{t} dt $$ Hence I am often told that for indefinite integrals, since $\frac1{x}$ is defined over $\mathbb{R} \setminus \{0\}$ (various sources ...
5
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1answer
67 views

What is the definition of contractible space? (It is not a duplicate)

I'm studying two algebraic topology texts (namely Munkres and Theodore) Here are definitions given in those texts Munkres Let $X$ be a topological space. If the identity map on $X$ is ...
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2answers
89 views

How is it called when one ellipse is “more elliptical” than another one?

Assume you have two ellipses, $A$ and $B$. Now $A$ looks "flatter" than $B$ because its ratio $\frac{\text{major axis}}{\text{minor axis}}$ is bigger. This means it "looks less" than a circle. How is ...
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0answers
26 views

Clarification on the definition of truth

So I am learning about the Godel's theorem. My instructor define truth and falsity as something arbitrary. he define $f$ to be true is $f(x)$ is $1$ and if $f(x) = 0$, it is false. I still dont have a ...
5
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0answers
109 views

Can we consider a hypergeometric function as a closed-form?

Let's say a calculus problem like an integral or a series has a solution that inevitably involving a hypergeometric function. It turns out that hypergeometric function cannot be expressed in term of ...
3
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1answer
46 views

Differentiability of chart functions

In the definition of a smooth (or $C^k$) manifold the charts $\varphi: U \to \mathbb R^n$ are assumed to have the property that for any two of them $\varphi \circ \psi^{-1}$ is smooth ($C^k$). Does ...
2
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0answers
135 views

Functions $h$ such that $h(x*x') = f(x) * g(x').$

Definition 0. Call a magma $X$ surjective iff the distinguished binary operation of $X$ induces a surjective function $X \times X \rightarrow X$. Now for the main idea: Definition 1. Let: ...
0
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1answer
28 views

What is the name of this homotopy?

Here is my definition for homotopy: Let $X,Y$ be topological spaces. Let $f,g:X\rightarrow Y$ be continuous functions. If there is a function $F:X\times[0,1]\rightarrow Y$ such that ...
0
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1answer
186 views

Measures: Atom Definitions

Let $\Omega$ be a measure space with measure $\mu$. (Here, a measure is only meant to be countable additive!) Consider a subset $A\in\Sigma$. Then according to the wikipedia article it is an atom ...
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0answers
42 views

What's the best way to define $\mathbb{H}$?

I had once defined $\mathbb{C}$ as $\mathbb{R}\times\mathbb{R}$ as a set, then I found it extremely uncomfortable when doing complex analysis or measure theory. So I changed its definition so that ...
0
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1answer
26 views

What is the definition of finitely generated?

To be specific, here is an example. Note that $(\mathbb{Z},+)$ is a group. Definition 1. (Lang) Let $G$ be a group and $S\subset G$. If $G$ is the intersection of all subgroups $H$ of ...
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0answers
31 views

Integrable Systems-Description

I am relatively new to the topic of integrable systems and I came across the following description: "Integrable systems share a fundamental property which is related to the geometry of the initial ...
2
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0answers
83 views

Definition of “one element normalizes another element”

I understand that someone has asked the definition of "an element of a group normalizes a subgroup". Let $G$ be a group and $a,b\in G$. Then what does "$a$ normalizes $b$" mean? Does it make sense at ...
2
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1answer
55 views

What is an exact differential?

My book says "A differential expression $M(x, y)dx+N(x, y)dy$ is an exact differential in a region $R$ of the $xy$-plane if it corresponds to the differential of some function $f(x, y)$ defined on ...
2
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0answers
49 views

How to define a Holder seminorm of a section

I'm reading "Variational Problems in Geometry",Seiki Nishikawa, in the figure below. Let $(M,g)$ be a compact $m$ dimensional Riemannian manifold with no boundary. $T>0, 0<\alpha<1, ...
2
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0answers
314 views

What is $R$-algebra?

(To be clear, I mean a ring by a ring with unity.) Here is a definition in Wikipedia: Let $R$ be a commutative ring. Let $(M,+,\cdot)$ be an $R$-module. Let $\ast$ be a binary operation ...
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2answers
129 views

Complex Measures: Integrability

Problem On the one hand, a complex measure decomposes into: $$\mu=\Re_+\mu-\Re_-\mu+i\Im_+\mu-i\Im_-\mu=:\sum_{\alpha=0\ldots3}i^\alpha\mu_\alpha$$ This gives rise to the integrability condition: ...
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1answer
100 views

Complex Functions: Integrability

Let $\Omega$ be a measure space with measure $\lambda$. Denote the space of simple functions by: ...
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2answers
39 views

Linear independence and grammar

Let $A$ be a commutative ring. Normally, the linear independence is a property of a subset or a family elements of an $A$-module. But one often sees statements like "$v$ and $w$ are linearly ...
2
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2answers
213 views

(Are there) subtleties in the definition of 'biproduct'

I always thought that a biproduct of two objects $A_1,A_2$ in some category $\mathcal{C}$ is an object $P$ with two maps $p_i:P\to A_i$ making it a product and two maps $j_i:A_i\to P$ making it a ...
2
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1answer
236 views

What makes a line “straight”?

In Euclidean space, there can be several definitions that makes a straght line: Line of shortest distance between two points Line that is linear, i.e with the points satisfying a linear equation ...
2
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2answers
152 views

Precise definition of affine, smooth, and irreducible

A book which I'm reading now says that "the Drinfeld curve $$ \mathbf{Y} = \{\, (x, y) \in \mathbf{A}^2(\mathbb{F}) \mid xy^q - yx^q = 1 \,\}$$ is affine, smooth, and irreducible." Here $p$ is an odd ...
0
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1answer
196 views

Two definitions of left-invariant vector fields of a Lie group

I am reading these lines from a text which shows why the bracket of two left-invariant vector fields is also a left-invariant vector field. But cannot easily get one of the lines. Let $L_a$ be the ...
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2answers
180 views

Algebraic Structures: Does Order Matter?

(I want to link to similar question with a very good answer: Question about Algebraic structure?) An algebraic structure is an ordered tuple of sets. One of these is called the underlying set, and ...