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

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Synonyms for “Theorem”

Some mathematical results, despite being formally proven, are not actually called "theorem". Examples include: Bertrand's postulate Pigeonhole principle Law of large numbers Do these names imply ...
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1answer
172 views

Why is an open interval needed in this definition? (definition of a limit of a function)

Here's a part of the definition Ross' Elementary Analysis states for limits of a function: 20.3 Definition (a) For $a\in\mathbb R$ and a function $f$ we write $\lim_{x\to a} f(x)=L$ provided $\...
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1answer
32 views

On the definition of transition maps

When defining a manifold the domain and codomain of the transition maps is usually denote like this: $$\varphi_\eta \circ \varphi_\lambda^{-1}: \varphi_\lambda(U_\lambda \cap U_\eta) \to \varphi_\eta(...
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0answers
205 views

Anti-Diagonal Matrix

Hi just a quick question. Can a anti-diagonal have zero in its diagonal? I have this definition in my paper. "An anti-diagonal matrix is a square matrix where all the values are zeroes except those ...
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1answer
125 views

What is Bourbaki's definition of subfield? or categorical definition of subfield?

Let $F$ be a field. Let $K$ be a subset of $F$ which is closed under two binary operations $+,\cdot$ Assume $(K,+,\cdot)$ is a field. Is $K$ called a subfield of $F$ in Bourbaki's definition? Or, ...
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1answer
319 views

What is a sparse subset?

In a work about fully homomorphic encryption I found usage of the expression: "sparse subset", as in: Our hint will consist of a set of vectors that has a (secret) sparse subset of vectors whose ...
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1answer
72 views

Examples of Free Lie Algebra

In wikipedia, free Lie algebras are defined using the universal property. Can anyone give some concrete examples of free Lie algebras? "In mathematics, a free Lie algebra, over a given field $K$, is ...
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8answers
1k views

Why is the absolute sign needed in the definition of a bounded function

A function $f$ is bounded if there exists a real number $M$ such that $|f(x)| \le M$ for all $x \in \operatorname{dom}(f)$. Why is the absolute sign needed?
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1answer
67 views

On the definitions of $n$-manifold etc.

I'm doing 3rd year undergraduate geometry (an introductory subject), and we've been given formal definitions of terms like "$n$-manifold" and "smooth $n$-manifold." However, I tend to think about ...
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3answers
77 views

When finding the derivative using its definition, why do we plug $0$ for $h$?

If $\lim h\to 0$, when finding the derivative of the function, why do you plug in the limit that is being approached. Like why would you plug in $0$ in the function $4x+2h$ (which is the derivative of ...
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2answers
71 views

What is the polarization identity?

Hi I am studying stochastic calculus and my professor often mentions "Polarization Identity" but I do not know how it is defined. I tried googling it but could not find the right definition and ...
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0answers
66 views

Two definitions of “affine stratification”

I see two different definitions of "affine stratification" in the literature: A stratification where each stratum is isomorphic to $\mathbb{A}^n$ for some $n$. A stratification where each stratum is ...
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2answers
115 views

Is the angle between a vector and a line defined?

Is the angle between a vector and a line defined? The angle between two lines $a,b$ is defined as the smallest of the angles created. The angle between two vectors $\vec{a},\vec{b}$ is the smallest ...
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2answers
69 views

Concept: The graph component

I have the following definition for a Component of a graph: A subgraph $H$ of a graph $G$ is a component of $G$ if $H$ is a maximal connected subgraph of $G$, ...
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1answer
45 views

Characteristics of $f(x,y)=\sqrt{xy}$

$f(x,y)=\sqrt{xy}$ (i) Determine the maximal domain of $f$, thus that the range is real. (ii) Is $f$ continuous its domain? (iii) Determine and describe all Level sets of $f$ To (i) $D:=\{(x,y)\in ...
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1answer
161 views

Examples of the Geometric Realization of a Semi-Simplicial Complex

I am reading The Geometric Realization of a Semi-Simplicial Complex by J. Milnor and here are the definitions: I find it difficult to visualize without specific examples. Can anyone help to provide ...
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3answers
333 views

What is in clear mathematical terms the definition for a sequence of integers, to be called *random*?

Sequences of integers might be ordered, totally ordered,... For all such attributes we find definitions in clear mathematical terms. (1) But what is in clear mathematical terms the established and ...
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2answers
627 views

Is the Euler characteristc defined wrong? If not, why not?

Ever since learning that $$\chi(S_0\# S_1) = \chi(S_0)+\chi(S_1)-2$$ (where $\chi$ denote the Euler characteristic), I've wondered whether $\chi$ isn't "defined wrong." If we let $\chi' = 2-\chi,$ ...
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1answer
85 views

On the Definition of multiplication in an abelian group

In class we had the following Definition: Let $(A,+)$ be an abelian Group with $a \in G$. We define: $$na:= \begin{cases}na, \ \forall n \in \mathbb{N} \\ |n|(-a), \ \forall n \in \mathbb{Z}\...
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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 ...
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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
82 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 ...
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6answers
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Do we have negative prime numbers?

Do we have negative prime numbers? $..., -7, -5, -3, -2, ...$
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0answers
112 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
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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 ...
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1answer
54 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
115 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
57 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
142 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
150 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
219 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
46 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 ∀y∈f(...
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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
318 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
147 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
157 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
38 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 $\...
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1answer
165 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 \epsilon&...
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2answers
44 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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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 \tau_{...
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1answer
39 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 ...
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1answer
68 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
90 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 ...
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1answer
127 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 ...