Questions tagged [definition]
For requesting, clarifying, and comparing definitions of mathematical terms.
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Help with definition on regular graph
What does it mean $\frac{n-1}{2}K_{2}$?
In general which graph am I referring to if I write $l\times K_{2}$?
To add a little bit of contest, we are talking about undirected graph on n-1 vertices with ...
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In the definition of outer measure, can we replace "open intervals" by "disjoint open intervals"
The definition of the outer measure of a set $A\subseteq\mathbb{R}$ is as follows:
$$
|A| = \inf \left\{ \Sigma_{k=1}^{\infty}\ \mathscr{l}(I_k): I_1, I_2,\dots\text{ are open intervals such that }A\...
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What is meant by "find the eigenvectors of a matrix" [closed]
Let's say I have a question asking me to find the eigenvectors associated with $\lambda = 2$ for $A$ (a $3 \times 3$ matrix).
I find that the eigenvectors associated with $\lambda = 2$ are all vectors ...
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Definition of a weak-star weak-star continuous function
I have seen the phrase "weak-star weak-star continuous" many times. But I'm don't know what it means for a function to be weak-star weak-star continuous. I just assumed it means that the ...
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Is $\forall x\exists x(x < x)$ a sentence?
Going through my notes on predicate logic, I read the following inductive definitions:
Definition: An atomic term is either a variable or a constant. If $f$ is an $n$-place function symbol and $t_1, . ...
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Are the "intramural" and "extramural" definitions of "set" from category theory equivalent?
Note: This question takes a pluralistic / "multiverse" view of sets (including "classes" and "collections") and set theories.
My understanding (from reading many nLab ...
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Continuous analogue of the discrete simple continued fraction
Background
The classical Riemann integral of a function $f : [a,b] \to \mathbb{R}$ can be defined by setting $$\int_{a}^{b} f(x) \ dx := \lim_{\Delta x \to 0} \sum f(x_{i}) \ \Delta x. $$ Here, the ...
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Is there a term for the idea that mathematical objects are defined by their relationships?
In a recent Veritasium video discussing Euclid's Elements, Alex Kontorovich comments that Euclid's definitions of primitive objects (e.g. "A point is that which has no part.") are absurd and ...
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Question about definition of Sequences in Analysis I by Tao.
Here's the definition of a sequence as laid out in the text:
Let $m$ be an integer. A sequence $(a_n)_{n=m}^\infty$ of rational
numbers is any function from the set $\{n \in \mathbf{Z} : n \geq m\}$ ...
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Definition of algebraic curves
I'm doing a module on algebraic curves which follows Fulton's book and I'm very confused.
Let K be a field.
Right at the beginning of chapter 3 he defines an affine plane curve to be an equivalence ...
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Equivalence between two definitions of winding number
I've noticed that the definition of a winding number is rather different in Stewart & Tall's Complex Analysis than in Ahlfors' Complex Analysis:
Let $\gamma:[a,b]\to\mathbb{C}$ be an arc and let $...
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An exact definition of multiplication
I am looking into repeated operations, and it seems really hard to precisely define multiplication.
Of course, for integer $b$ and real number $a$, we use the grade school definition we all know:
$$ab ...
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explain the definition of the book about one of the generator of integers
The book:
The notation ka in additive notation does not represent a product of k and a but, rather, a sum
$ka=a+a+a+\cdots+a$
with k terms
I confused about the above definition, because we say that $...
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Lebesgue Integral on Rough Paths
In a nutshell: how can we integrate rough paths using Lebesgue integration?
What I mean by a rough path is a continuous map $[a,b]\to\mathbb{R}^n$, or $[a,b]\to\mathbb{C}$ in the complex case.
(The ...
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Is the complement of a triangle on the sphere also a triangle?
In the sphere above, the shaded area defined by the points A, B, C clearly makes a triangle. My question is, can the complement of this area, that is everything on the sphere that is white, also be ...
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Ordinary functor being E-indexed
I am learning parts of topos theory and it feels like I am missing something regarding indexed functors.
This comes from the definition of a locally connected geometric morphism $f : E \rightarrow F$ ...
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What is the indeterminate in the set of all symbols in $R[x]$ and what does the elements in $R[x_1,x_2]$ looks like?
I was studying polynomial rings over commutative rings from the book Topics in Algebra by I.N Herstein.
From there what I understood was, that:
If $R$ be a commutative ring with a unit element then $R[...
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Sequential divergence criterion for functional limit, diverging function evaluated at converging sequence
According to Abbott 4.2.5 "Divergence Criterion for Functional Limits",
Let $f$ be a function defined on $A$, and $c$ be a limit point of $A$. If there exist two sequences $(x_n)$, $(y_n)$ ...
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Use of equivalent and equal
Really simple question.
Is valid to do this:
$$ 2x + 2 = 1 \iff x = \frac{1 - 2}{2} = -\frac{1}{2} $$
I mean is anything wrong when solved for $x$ to not use $\iff$ symbol more and just use $=$.
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Equivalence of optimization problems
In Boyd and Vandenberghe (https://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf), section 4.1.3, they say:
We call two problems equivalent if from a solution of one, a solution
of the other is ...
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Which is the correct definition of covectors?
Some says covectors are linear map that maps $ V \mapsto R $ (which means it's just a row vector considering vectors are $ n $ x $ 1 $ matrix and mapping is matrix multiplication), while some say it's ...
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Do Wikipedia, nLab and several books give a wrong definition of categorical limits?
It seems unlikely that all these sources are wrong about the same thing, but I can’t find a flaw in my reasoning – I hope that either someone will point out my error or I can go fix Wikipedia and ...
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Prove a function from a set of linear operators of a hilbert space to a set of linear operators of a hilbert space is well defined
$\mathscr{L}(\mathcal{H})=$ Set of linear operators from $\mathcal{H}\to \mathcal{H}$. For $T\in \mathscr{L}(\mathcal{H}_A\otimes \mathcal{H}_B)$ specified through $T=\sum\limits_{i,j}\gamma_{i,j}A_i\...
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convex functions, is it enough to check if convex on all intervals
Let $f: X \to \mathbb{R}$ be some function defined on a convex set $X \subset \mathbb{R}^n$. Now assume that for all $x \in X$ and $y \in \mathbb{R}^n$ such that $x+ty \in X$ for all $t \in [0,1]$, ...
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unbiased estimator of sample mean
The question: Given a random sample $X_1,...,X_n$ show that $\frac{1}{n}\sum_{i=1}^n X_i$ is an unbiased estimator for $E(X_1)$.
My confusion: Given a statistical model $(\Omega,\Sigma,p_{\theta})$, ...
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Are these definitions of the total derivative equivalent?
The total derivative of a function $F:\mathbb{R}^n \to \mathbb{R}^m$ is defined in this way:
If a linear map $L:\mathbb{R}^n\to\mathbb{R}^m$ exists such that
\begin{equation}
\tag{*}
\lim_{\boldsymbol{...
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Is my informal understanding of probability definitions correct?
I am struggling a little bit on probability and I was hoping someone could clear things up in a slightly informal way. I am new to probability so I am in an awkward position where I need to learn how ...
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Can adjacent points exist in geometric space?
My question is going to focus on quite a counterintuitive thing.
A couple of preliminaries. I understand geometric space as a set of points. A point, in turn, is an abstract idealization of an exact ...
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Is "up to natural isomorphism" crucial?
Let $\mathcal{C}$ be a category. By diagram I mean a covariant functor $F\colon\mathcal{J}\to\mathcal{C}$ for some category $\mathcal{J}$. In this source it is said that a commutative diagram is a ...
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Is this an adequate definition of the natural numbers in terms of elementary set theory?
Is this an adequate definition of the natural numbers?
The term number is synonymous with natural number in this context. The terms front and back are used in order to avoid using numbers first and ...
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Doubt in van Oosten's Topos Theory notes
I came across this definition while reading Jaap van Oosten's Topos Theory lecture notes (pg 29, def 1.3).
In a category with finite limits, an equivalence relation on
an object $X$ is a subobject $R$ ...
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Difference between different definitions of diagram in a category
I'm currently reading the book "Topoi: The Categorial Analysis of Logic" by Robert Goldblatt, and in chapter 3.11, in order to define limits and co-limits he defines a diagram in a category ...
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Question about definition of upper derivative and lower derivative.
I study the definition of upper and lower derivative in the book Real Analysis H.L. Royden 4th edition.
$\overline{D}f(x)=\lim\limits_{h \to 0}\left[\sup\limits_{0<|t|\leq h}\frac{f(x+t)-f(x)}{t}\...
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Topology in the context of Pontryagin dual
Let $A$ be an abelian group.
The definition of the Pontryagin dual of $A$ is $\text{Hom}_{\text{conti}}(A, \mathbb{Q}/\mathbb{Z})$.
In this context, what are the topologies on $A$ and $\mathbb{Q}/\...
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Showing that the two definitions of conditional expectations are equivalent.
How do I show that $\displaystyle\sum_{x\in\text{Im}(X)}xP[X=x\mid A] = \dfrac{E[X\cdot1_A]}{P[A]}$, for $A \subseteq \Omega$ a sub-sigma algebra and $X$ a discrete random variable from $\mathcal{F} \...
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Why we consider the dual space when defining tensors
My question is very simple: A type $(m,n)$ tensor is an element of $V^{\otimes m}\otimes (V^*)^{\otimes n}$. Is there a reason/motivation, beyond more general definitions, to consider the dual space ...
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Show that $\lim_{K \to \infty} \sum_{k=-K}^{K} \hat{f}(k) e(k \alpha) = \dfrac{f(\alpha^+)+f(\alpha^-)}{2}.$
I am studying Multiplicative number theory I: Classical theory by Hugh L. Montgomery, Robert C. Vaughan. The following is the beginning of Appendix D:
I could not understand the last sentence (the ...
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Among morphisms of morphisms, what makes commutative squares special?
Given two (1-)categories $\mathcal{C}, \mathcal{D}$, and given the 0-category (class) of funtors $\mathcal{C} \to \mathcal{D}$, denoted $Func(\mathcal{C} \to \mathcal{D})$, let's say we want to make ...
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An extremely rigorous and formal definition of differential equation.
I actually asked a version of this question before, here: What is the formal, rigorous definition of a differential equation?. However, I should have asked what an equation is, first, which I did here:...
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Why isn't the third property in the definition of vector bundles redundant?
I am studying Manifold theory and it is essential for me to know vector bundles.The usual definition of vector bundles as given in the standard texts is a follows:
Suppose $M$ is a topological ...
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Subobject in the category of topological spaces
Given an object $X$, we can define an equivalence relation on the monomorphisms with range $X$: $u:S\to X,v:T\to X$ are equivalent iff exists an isomorphism $\phi:S\to T$ such that $u=v\circ \phi$. By ...
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What does "convex class of probability measures" mean in the definition of scoring rules?
Taken from Wikipedia (here), a scoring rule has the following definition
Let $\Omega$ be a sample space, and $\mathcal{A}$ is a $\sigma$-algebra of subsets of $\Omega$. Let $\mathcal{P}$ be a convex ...
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How to formally define this square matrix?
I was wondering if there is formal name for a 3x3 matrix where the third column is the sum of the first two columns and the third row is the sum of the first two rows. In other words, a square matrix ...
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What are $0$ and $1$, the elements of $\mathbb Z/2\mathbb Z$? How do they relate to $\mathbb Z$? [duplicate]
I often see field $\mathbf{Z}/\mathbf{2Z}=\{0,1\}$. Without other indication we might see elements of field $\mathbf{Z}/\mathbf{2Z}$ as a subset of $\mathbf{Z}$. Operations such as $2+1=3=1$ are also ...
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Definition of hypergraph homomorphism
W.Dörfler and D.A.Waller's paper "A category-theoretical approach to hypergraphs" gives the following definitions:
A hypergraph is a triple $H = (V,E,f)$ where $V$ is the set of vertices, $...
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Can a non-holomorphic function have a pole?
As far as my studies have brought me, I've only see so far the definition of a "pole" for a complex valued function $f:\Omega \rightarrow \mathbb C$ if we assume that the function is ...
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Definition of non-monotonic sequence
I am working on other proofs, namely for so-called "quasi-increasing" but non-monotonic sequences.
But, my question: is the following a sufficient definition of non-monotonic?
For all $N \...
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On the definition of an Interpretation Isomorphism
I wish to check my understanding of an interpretation isomorphism, as defined (although paraphrased) in Boolos' Computability and Logic:
Two interpretations $P$ and $Q$ are isomorphic iff there is a ...
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How can numerator and denominator of any variable-containing algebraic irrational term be uniquely defined in general?
My question is for formulation or notation of numerators and denominators of variable-containing algebraic irrational terms (radicals) in general.
I'm looking for formulations to talk about variable-...
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Definition of Tensor Product in Hungerford’s Algebra
Let $A$ be a right module and $B$ a left module over a ring $R$. Let $F$ be the free abelian group on the set $A\times B$. Let $K$ be the subgroup of $F$ generated by all elements of the following ...
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