Tagged Questions

Definitions are the core of mathematical precision; they come to answer "what is X" in mathematics. Into this category fit questions regarding equivalence of definitions; clarifications regarding complicated definitions; as well questions with purposed definitions for mathematical notions with ...

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Mathematics Defined

I was having a discussion with a family member regarding the definition of "Mathematics" and it's categories and what is studied under each category. I read within in a forum post here, that there's ...
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1answer
28 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
16 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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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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11 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
42 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
53 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 $$\delta_a(x)=\frac{1}{a\sqrt{\pi}}e^{-\frac{x^2}{a^2}}\text{ as }a\to\infty.$$ Obviously, this sequence of functions ...
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Specific terms for partitioning a set

I need specific terms for those definitions. Given a set $S$ with $n$ elements and $k$ partitions what is it called if: Each partition has at least one element and the partitions are not ...
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74 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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44 views
+50

What is the definiton for “best probability measure”?

I'm looking for this definition is notes that use the phrase and elsewhere, but it just isn't there. Does anyone else recognize the phrase?
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33 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
22 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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23 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
95 views

Borel Measures: Atoms vs. Point Masses

Let a measure be $\mu:\Sigma\to\mathbb{R}_+$. Call a measurable $A\in\Sigma$ an atom if: $$\mu(A)>0:\quad\mu(E)<\mu(A)\implies\mu(E)=0\quad(E\subseteq A)$$ and a singleton $\{a\}\in\Sigma$ a ...
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3answers
44 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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25 views

Wronskian, Linear Independence

Show that the following functions are linearly independent: $$e^t\begin{bmatrix}1\\1\\0\end{bmatrix}$$ $$e^{2t}\begin{bmatrix}0\\2\\1\end{bmatrix}$$ $$e^{-t}\begin{bmatrix}1\\0\\1\end{bmatrix}$$ ...
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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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1answer
60 views

Borel Measures: Discrete & Continuous Definition? [closed]

Here, the focus lies on discrete & continuous - not atomic & atomless!!! What is the rigorous definition for a Borel measure to be continuous? (The definition for discrete measure can be ...
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2answers
25 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
17 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
103 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
29 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
53 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 ...
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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
64 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
32 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
59 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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20 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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0answers
24 views

Why an $R-algebra$ requires $R$ to be commutative?

Here is the definition of $R$-algebra. Let $R$ be a commutative ring Let $(A,+\cdot)$ be an $R$-module. Let $\ast$ be a binary operation on $A$. If $x\ast(y+z)=x\ast y + x\ast z$ ...
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60 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
39 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 ...
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109 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: ...
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1answer
24 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 ...
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1answer
93 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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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 ...
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Developing closed-form from recursive definition

Here is a recursively-defined function where c, d ∈ N. T(n) =    c, if n = 0 d, if n = 1 2T(n − 1) − T(n − 2) + 1, if n > 1 Carry out the five steps for repeated substitution to prove a ...
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1answer
21 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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15 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 ...
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0answers
34 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 ...
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1answer
36 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 ...
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25 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, ...
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19 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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Complex Measures: Integrability

Approaches 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 integrability as: $$f\in L(\mu)\iff f\in ...
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Are there terminologies distinguishing modules over ring and rng?

Let $R$ be rng. Let $M$ be a left $R$-module. Let's say, after some verification, one realized that $R$ has a unity and it doesn't satisfy $1_R \cdot x$ for all $x\in M$. Hence, one cannot call $M$ ...
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1answer
28 views

Complex Functions: Integrability

Let $\Omega$ be a measure space with measure $\lambda$. Denote the space of simple functions by: ...
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24 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 ...
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1answer
53 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 ...
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47 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 ...
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2answers
131 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 ...
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Intuitive explanation of “deterministic system”?

Wikipedia Definition: In mathematics and physics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will ...