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

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1answer
28 views

Is every field a Krull domain?

Background: A Krull domain is an integral domain $A$ with a family $(v_i)$ of valuations on the field of fractions $K$ for $A$ satisfying the following conditions: Each $v_i$ is discrete. The ...
2
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2answers
62 views

What is the actual definition of a function?

I am learning precalculus and my book defines the following: A function $f$ from a set $A$ to a set $B$ is a rule that assigns to every element $a$ in $A$ one and only one value in $B$. Well, I ...
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7answers
299 views

Why associativity $h \circ (g \circ f) = (h \circ g) \circ f$ is required in composition?

An introduction into category theory says that A category is a quadruple $A = (O, \mathrm{hom}, \mathrm{id}, \circ)$ consisting of blah-blah and is subject to the following conditions: (a) ...
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1answer
30 views

Definition of the vector cross product

As far as I understand the cross product between two vectors $\mathbf{a},\mathbf{b}\in\mathbb{R}^{3}$ is defined as a vector $\mathbf{c}=\mathbf{a}\times\mathbf{b}$ that is orthogonal to the plane ...
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2answers
83 views

Why are some branches of mathematics called 'theory' and others not? [on hold]

We say: graph theory , group theory, number theory , set theory, what is definition of theory? We also say abstract algebra, real analysis, but why we do not say abstract algebra theory or real ...
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2answers
73 views

Wrong pushforward of vector field definition on wikipedia

Wikipedia claims that for $$\mathrm d \varphi_x:T_xM\to T_{\varphi(x)}N\,$$ we have for $X \in T_pM$ and $f \in C^{\infty}(N,\mathbb{R})$ $$\mathrm d\varphi_x(X)(f) = X(f \circ \varphi)$$ whereas I ...
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2answers
81 views

Is defining an “$S$-scheme” this way just a stylistic choice?

I have a potentially pretty silly question, involving the definition of an "$S$-scheme". For a fixed scheme $S$, the definition of the category of $S$-schemes is a category with: objects: scheme ...
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0answers
98 views

Is there a special name for functors from a category C to a subcategory of C?

Is there a special name for functors from a category $C$ to a subcategory $S$ of $C$ where $S \ne C$? I'm using $\ne$ pretty informally above. I'd be happy with anything that reasonably captures the ...
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3answers
80 views

What is the definition of axiom (mathematically speaking)

I'm wondering about the exactly definition of axiom (mathematically speaking). The definition of this term seems a little blur in my mind. For example, the definition of point in the Euclidean ...
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0answers
22 views

Mathematical formal expression of find “subfunction” in function [on hold]

Imagine if I have a function $s(t)$ and $r(t)$. $s(t)$ may contain $r(t)$ one or more times as $s(t)$ is a quasi-period function. What is the correct expression if I want to say the $s(t)$ contains ...
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1answer
33 views

Weibel definition 1.4.1. understanding the indexes on splitting maps

The book says: Definition 1.4.1. A complex $C$ is a called split if there are maps $s_n : C_{n+1} \to C_{n+1}$ such that $d = dsd$. The maps $s_n$ are called splitting maps. If in addition $C$ ...
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1answer
56 views

About odd functions and improper integrals e.g. $\int^{\infty}_{-\infty}\sin x \; dx$

Does $\displaystyle \int^{\infty}_{-\infty}\sin x \; dx$ converge? Since $\sin x$ is an odd function, and we know that in definite integrals $\displaystyle \int^{a}_{-a}\sin x \; dx=0$ then does ...
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1answer
28 views

Question about negative value using the ratio convergence test for integrals

Find for what $p$, $\displaystyle \int ^{\infty}_0 x^p \arctan x dx$ converges. By parts, it's equal to: $\displaystyle \lim_{b\to \infty}\frac 1 {p+1}x^{p+1}\arctan x |^b_0- \int ^b _0\frac ...
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1answer
46 views

Metric Space Definition

From my book, the definition given is: Given a set $X$, a function $d: X \times X \to \mathbb{R}$ is a metric on $X$ if for all $x,y \in X \dots$ Then a metric space is a set $X$ together with a ...
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2answers
27 views

Is the co-domain needed if we have the range? [duplicate]

Why do we need the co-domain if we have the range? I know what both mean. Isn't it just better to use the range instead of the co-domain when defining a function? This question brought up to me when ...
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1answer
48 views

Uniqueness of a number $area(A)$

I have the following definition of area: Let $A$ be a bounded set from $\mathbb{R}^2$. We say that $A$ has area if there exist two sequences $(E_n)_{n\in \mathbb{N}}, (F_n)_{n\in \mathbb{N}}$ of ...
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3answers
115 views

(Is it a set?) Set of all months having more than 28 days.

Set is a well defined collection of distinct objects. Is the following is a set? Set of all months having more than 28 days. I'm confused here. Because on one hand I think that it is well ...
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2answers
24 views

Is there a relation eigenvectors and unitary operator.

I am trying the understand the spectral theorem as given in wikipedia link: https://en.wikipedia.org/wiki/Spectral_theorem I understand that eigenvectors are vectors and unitary operator is a ...
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1answer
36 views

Dually Dedekind Set and Weakly Dedekind set

$A$ is dually Dedekind infinite (dD-infinite), if there is a surjective non-injective map from $A$ onto $A$. $A$ is weakly Dedekind infinite (wD-infinite), if there is a surjective map from $A$ onto ...
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1answer
50 views

dual Dedekind-infinity may not imply Dedekind-infinite without AC

It is written in wikipedia: https://en.wikipedia.org/wiki/Dedekind-infinite_set It is not provable (in ZF without the AC) that dual Dedekind-infinity implies that A is Dedekind-infinite. (For ...
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1answer
16 views

Short question about modulo space $\mathbb Z^n_p$ and the zero vector

Say we have a vector in $\mathbb Z^3_5$: $v= (1,2,0)$ it looks like it isn't the zero vector but if we multiply it by a scalar: $5v=(5,10,0)\overset{mod5}=(0,0,0)$ so now it is the zero vector and we ...
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1answer
42 views

Why are all open subsets not infinite in extent?

I have been looking at the definition of an open set, for a metric space. I have come across the following definition, a few times: An open set $U$ of the metric space $(X,d)$ is a set given that ...
3
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1answer
17 views

What is the difference between functions and operations?

Wikipedia says that an operation $\omega$ is a function of the form $\omega: V \to Y$, where $V \subset X_1 \times\cdots\times X_k$. But as far as I know, every function's domain is a set, so ...
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0answers
23 views

Any relation between Kernel methods and Variational methods?

I am familiar with kernel method, which is defined in the link: https://en.wikipedia.org/wiki/Kernel_method On the other hand I am familiar with variational methods which is defined in the link: ...
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0answers
7 views

What is meant by factor of a (stationary) stochastic process?

Let $(X_n)$ be a stationary stochastic process. It is said, that a factor of a stationary stochastic process is itself stationary. But I did not find a definition of a factor of a stochastic ...
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1answer
55 views

What's so special about the grounded poset of cardinality $2$?

By a grounded poset, I mean a poset $P$ with a bottom element $0_P$. Arrows between grounded posets are monotone mappings that preserve the basepoint. This gives a self-enriched category; lets call it ...
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3answers
158 views

A lot of confusion in the “Polynomial Remainder Theorem”?

Lately I've been reading about Polynomial Remainder Theorem from various sources, mainly from the wikipedea article, this post and some high school books. Wikipedea says that if we divide a polynomial ...
3
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1answer
54 views

Positive-definite function and Positive-definite matrix

I am trying to understand Positive-definite function and read the wikipedia link: https://en.wikipedia.org/wiki/Positive-definite_function It has a relation to Positive-definite matrix and I did not ...
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0answers
12 views

Bruhat-Tits building of a non-split group

Where can one read about the definition of parahoric subgroups for a reductive group over a non-archimedean local field $k$ which is not split over $k$?
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0answers
7 views

definition of an affine F-variety where F is not algebraically closed

In the first chapter of his book "Linear Algebraic Groups", Springer considers a situation where $k$ is an algebraically closed field and $F$ is a subfield, and seeks to define the notion of an ...
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0answers
24 views

Help in understanding Bochner's theorem and Pontryagin duality theorem

I am trying to understand Bochner's theorem through wikipedia link https://en.wikipedia.org/wiki/Bochner's_theorem This refers to dual spaces of locally compact abelian group and leads to ...
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4answers
302 views

Meaning of finite, countably infinite, infinite?

Even after several attempts I could not find the motivation behind the finite, countable and infinite. Is there a simple way to look them differently? I have read the wikipedia definition several ...
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5answers
252 views

Theoretical function question

Suppose we have the function $f(x)= x^2 $. This function associates real numbers with real numbers ( $f:\mathbb{R}\rightarrow \mathbb{R}$). Now, what i get confused sometimes is what exactly the ...
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0answers
25 views

Definition of trigonal curves

I'm reading Fulton's book and I'm trying to understand the concept of trigonal in Hartshorne's book (page 345): On the other hand, Fulton's book define the $g_d^n$ in the following manner: So Can ...
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1answer
16 views

Functional limits and definition of continuity - difference and implications?

Continuity: A function $f : A → \mathbb{R}$ is continuous at a point $c ∈ A$ if, for all $\epsilon > 0$, there exists a $δ > 0$ such that whenever $|x − c| < δ$ (and $x ∈ A$) it follows that ...
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1answer
25 views

The definition of a series

Given definition: Given a sequence, $(a_n)$, then the series with terms $(a_1,a_2,...,)$ is a sequence $(s_n)$ of partial sums. Does this mean essentially the definition above can be re-stated to say ...
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2answers
35 views

What is “approximate compactness”? What is an example of an approximately compact set?

I read this: A property of a set $M$ in a metric space $X$ requiring that for any $x\in X$, every minimizing sequence $y_n\in M$ (i.e. a sequence with the property $\rho(x,y_n)\to\rho(x,M)$) has a ...
2
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1answer
23 views

Modules over a monoid: trouble with the definition.

I'm having trouble with a definition. We're working in the category of monoids. Take $A\in \mathfrak{Mon}$ and define a module over $A$ to be a set $M$ with an action: $A\times M \rightarrow M$ such ...
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2answers
28 views

How can I use a precise definition to find values of delta that correspond with given epsilon values

I have been given this problem: For the limit $$\lim_{x\to 2}({x^3-3x+4})=6$$ illustrate "Definition 2" (I have included this below) by finding values of $\delta$ that correspond to $\varepsilon=0.2$ ...
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1answer
30 views

Definition of finite-by-nilpotent [closed]

In group theory what are definitions of: finite-by-nilpotent, nilpotent-by-finite, abelian-by-finite, or in general: *-by- *?
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0answers
31 views

Is this an improper method of averaging grades? If so, what is a simple mathematical way of explaining it?

I have a professor who employs a unique method of averaging grades. On each assessment, the professor assigns a raw numerical score to each student based on performance. He then converts particular ...
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1answer
35 views

Definition of a sheaf: What is $s\rvert_{V_i}$ if $V_i\not\subseteq U$?

I am reading Hartshorne's book on algebraic geometry, which defines a sheaf to be a presheaf $\mathscr F$ on a topological space $X$ such that: For all open sets $U$ and open coverings $\{V_i\}$ of ...
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3answers
79 views

Numbers and reality [closed]

I have a question I don't really know how to formulate, so apologies for the cloudy mess. The topic is on the meaning of numbers, that is when I say 3, I am referring to, say, $3$ avocados, or $3$ ...
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13answers
5k views

What is a negative number?

I'm trying to get to an abstract definition of a negative number that would fit in with the basic concept of addition/subtraction. There are questions here about multiplying and dividing negative ...
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0answers
21 views

More on the $n$-dimensional cross product: Orientation

Wikipedia states: This formula is identical in structure to the determinant formula for the normal cross product in $\mathbb R^3$ except that the row of basis vectors is the last row in the ...
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2answers
27 views

What is the right hand side in this definition of $n$-dimensional cross product

Let ${\bf e_1}, \dots, {\bf e_n}$ be the standard basis for $\mathbb{R}^n$ and let ${\bf w_1} = (w_{11},\dots,w_{1n}), \dots, {\bf w_{n-1}}=(w_{n-1\;1},\dots,w_{n-1\;n}) \in \mathbb{R}^n$. Then one ...
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0answers
35 views

Simplification of a definition in Hartshorne's algebraic geometry book

I'm reading Hartshorne's book and on page 53 he defines intersection multiplicity of a projective variety and a hypersurface: I'm wondering if we can simplify this definition if we take $Y$ to be a ...
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0answers
14 views

Meaning and Underlying idea of a definition or a theorem

What does it mean by 'explain the meaning and underlying idea of a definition or a theorem'? For example, if we are asked to explain the Fundamental Theorem of Algebra, how should we explain its ...
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1answer
27 views

What is a linear embedding from a simplex $\Delta^n \to \mathbb{R}^n$?

As stated in the title, reading Milnor-Stasheff Characteristic classes, I encountered at page 95 the following sentence: let $\Delta^n$ be an $n$-simplex, linearly embedded in the $n$-dim vector ...
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2answers
176 views

Is the zero ring a domain?

Is the zero ring usually considered a domain or not? Wikipedia says: The zero ring is not an integral domain; this agrees with the fact that its zero ideal is not prime. Whether the zero ring ...