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Questions tagged [definition]

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

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Definition of the probabiliy for reaching an information set in a game when neglecting one player

This question concerns behavioral strategies in extensive form games. To this end, suppose a behavioral strategy $\sigma = (\sigma_0, \dots, \sigma_n)$ where each $\sigma_i$ maps every information set ...
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What is the definition of this $L^p$-space?

I am reading a book by E. Zehnder and I am confused about an $L^p$-space he is using. What is the definition of the space $$ L^p(S^1,\mathbb{R}^{2n}) $$ Thank you for your kind help.
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Trying to get it right regarding “ inverse” applied to relations, functions and operations…

[ The discussion ( see below) tends to show that my question was based on the erroneous assumption that the expression " inverse operation" is standard. Apparently, it is an informal expression used ...
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Formal definition of “ inverse operations” on a set?

Suppose f1 and f2 are two operations on A, that is functions from A² to A. My question is : how to express in general and formally the fact that each operation is the inverse of the other? Is it ...
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Are the endpoints of the domain of a function counted as critical points? [duplicate]

Do the end points of a domain come under critical points? I know we say critical point is a point where the derivative is zero or the derivative doesn't exist. For example: $$ f:[0,\pi] \to [-1,1], ...
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What is the formal definition of change of variable?

Change of variable is a very common and elementary technique in introductory mathematics. But it do know the formal definition of it. First of all What is the formal definition of variable in ...
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Is it correct to say that every one-to-one mapping is a bijection between its domain (preimage) and its image?

To me this seems tautological. But I've had someone challenge me on it. Here $\mathbb{R}$ is the set of real numbers. What I call the domain of a mapping is the set of all arguments for which there ...
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Generalization of graph connectivity to edge cases (null graph, singleton graph)

I am looking for advice on what would be a reasonable or useful generalization of vertex- and edge-connectivity to the graphs with 0 and 1 vertices (null graph and singleton graph). Motivation: ...
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What do we mean by $\lim_{x\to a} f(x)$? [duplicate]

Do we mean by that $f($number that approach from $a)$ , OR what is the number that the function is getting close to when we are getting close to $a$ OR non of the above ? if non of the above please ...
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Definition of “of odd order”

Could someone point me to a definition of "odd order"? The definition of group order I found here seems to refer to a group as a set of numbers. The context that I'm reading about 'odd order' is: "$p\...
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$L^p$-space on the circle, question about the definition

I am reading a book by E. Zehnder and I am confused about an $L^p$-space he is using. Here's what is written in the book: Start by considering integrable functions $f \in L^1(S^1)$ which are ...
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Is “$\lim\limits_{n \to \infty}f(x_0+\frac{1}{n})=l$” another way of expressing the right-sided limit?

Let $f:\mathbb{R} \to \mathbb{R}$. Can we say that $\lim\limits_{n \to \infty}f(x_0+\frac{1}{n})=l$ is another way of expressing the right-sided limit at $x_0$? I tried to use the definition of the ...
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Why is the subject of math overall considered an essential for compulsory education/non-scientific circles/society overall? [on hold]

It would seem that math is pushed both scientifically and culturally as some sort of plethora of necessity. What really does math do for the layperson? Aside from basic counting and stuff, I can't ...
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Clarification on normality and separability in the context of field extensions. [duplicate]

I'm a little confused in regards to the definitions of normality and separability of field extensions in the context of Galois theory . The definitions seem very similar . In class they were defined ...
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Some notational question related to Haar measure (what do $d g^{-1}$ or $d (hg)$ mean?)

I am looking at some brief introduction to Haar measures and since I'm not understanding basic notion, I would greatly appreciate any clarification. Let $G$ be a locally compact group and say we ...
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Does the function which unpacks singletons have a name?

Let $X$ be any set and $P_1(X) = \{ \{x\} : x\in X\}$. Does the function which unpacks the singleton, i.e. $$f\colon P_1(X) \to X, \{x\} \mapsto x$$ have any special name? On a related note, ...
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Why are these two definitions of conditional expectation equivalent?

From Rick Durrett's book Probability: Theory and Examples: We define the conditional expectation of $X$ given $\mathcal{G}$, $E(X | \mathcal{G})$ to be any random variable $Y$ that has (1) $Y \...
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Definition of transient state

Consider the following definition Transient States It is often useful to talk about whether a process entering a state will ever return to this state. Here is one possibility. A state ...
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1answer
56 views

Understanding definition of Periodicity of Markov chain

Consider the following example that is used to understand the definition of periodicity property. Why does it says that: starting in state $1$, it is possible for the process to enter ...
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4answers
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In Peano's Axioms are the uniqueness of the successor and $x^{\prime}=y^{\prime}\implies{x=y}$ redundant?

In Peano's Axioms are the uniqueness of the successor and the property $x^{\prime}=y^{\prime}\implies{x=y}$ redundant? This seems obvious to me, but I may be missing something. In the various forms ...
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Is there a formal definition for $f(x)$ ~ g(x)?

I was looking to see if curved asymptotes were possible and came across an answer that referred to an end behavior of a function as being $f(x)$ ~ $x^2$. I'm assuming this either means the end ...
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Question about linear algebraic groups split vs isotropic

I am reading notes on linear algebraic groups and I'm getting confused with some definitions and I would appreciate any clarification. They define $G$ to be split if there exists a maximal torus $T$ ...
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What does it mean for a map to factorize over something?

In lectures about ring theory my professor says that a map factorizes over a ring and then draws commutative diagrams. I have never heard this expression before. What exactly does it mean for a map ...
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is a null set the same as {}

Ø vs set {}: Ø has no elements whereas {} has the null set as an element; that is, say you are making power set of {}: Ø would be an element of the power set or in symbols, P({}): {Ø} right? I feel ...
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Is the exclusion of infinite decimal expressions of the form $a_0.a_1\dots{a_n}\bar{9}$ logically necessary?

My question really is as simple as: Is the exclusion of infinite decimal expressions of the form $a_0.a_1\dots{a_n}\bar{9}$ logically necessary? The obvious alternative would be $a_0.a_1\dots{a_n}\...
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Definition of operator $x\otimes y$

Suppose that $\mathbb H$ is a separable Hilbert space. I am interested in the tensor $x\otimes y$, where $x,y\in\mathbb H$. From what I understand, the tensor $x\otimes y$ can be viewed as a bounded ...
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1answer
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What is $l$-th division polynomial of $E$?

I am reading some papers about elliptic curves and I come across the term $l$-th division polynomial of $E$. I don't really know much about field theory and I tried to look for this definition but I'm ...
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Erroneous Definition of Stem Field

Here's the definition of stem field provided by the book I am using (https://www.jmilne.org/math/CourseNotes/FT.pdf): Let $f$ be a monic irreducible polynomial in $F[X]$. A pair $(E,\alpha)$ ...
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Proving Trigonometric “Definitions” [closed]

The expression trigonometric “definitions” refers here, rather narrowly, to statements expressing stable relations between the sides of right triangle. Thus, for instance, the traditional definition ...
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1answer
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Definition of holomorphic functions in multiple dimensions

What is the definition of a function $$f:U\rightarrow\mathbb{C}^n$$ being holomorphic? Where $U\subseteq\mathbb{C}^n$. When I look around online all I can see is the definition for $$f:U\rightarrow\...
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Why this is a well defined definition? Integrals

There's a definition that says that if we have $\int_{a}^{c}f = \int_{a}^{b}f + \int_{b}^{c}f$ and at least one of $\int_{a}^{b}f, \int_{b}^{c}f$ diverges, then $\int_{a}^{c}f$ diverges. Why is this ...
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1answer
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Is there a more common term for “hemigroup”?

In The Number System by Thurston the author introduces an algebraic structure he calls a "hemigroup". It doesn't appear to be a very common usage. The laws of a hemigroup are: (i) $\left(x*y\...
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1answer
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How to expand a vector to its monomial basis?

What exactly is monomial basis? Consider a vector of $n$ dimension. How can I write all monomials up-to a order say $k$ of this vector? If I manged to write the monomials of the vector up-to order $...
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What is a real or a complex eigenvector?

In my course of ODE, they talk about real or complex eigenvectors. What is it exactly ? Is a an eigenvector s.t. the eigenvalue is real or complex ? Or is it an eigenvector s.t. the component of the ...
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1answer
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Notation for a connectivity matrix using k closest points

Considering a set of n points $\{P_i\}_{i \in [1,n]}$, I've defined a distance matrix $\{D_{ij}\}_{i,j \in [1,n]} = \lVert Pi - Pj \lVert$ and I would like to define a connectivity matrix $\{C^p_{ij}\}...
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Notation: for strongly-regular graph $(v,k;\lambda, \mu)$, how to interpret “$k(k-\lambda-1)$” and “$\mu(v-k-1)$”?

Proposition: Let $G=(V,E)$ denote a $(v,k;\lambda, \mu)$ strongly-regular graph. Then $$k(k-\lambda-1)=\mu(v-k-1)$$ The notation is quite overloaded for me. Since $k$ and $\mu$ are parameters, then ...
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Is $y=mx$ direct or linear — or both?

Suppose I have the equation $y=4x$ that I wish to graph. Is this a direct relation or a linear relation? I know it is a direct relation because the relationship between $x$ and $y$ is defined by ...
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2answers
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What does “passage to the limit” means?

this might be a very simple question, so I apologize beforehand. I am new to calculus and while I was investigating a bit more about it, I found this expression: "passage to the limit". I suspect it ...
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1answer
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Linearly independent set definition

The definition of a linearly independent set I have been given is: A set S is linearly independent if every finite subset of S is linearly independent. Do there exist any sets S such that all finite ...
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Is this the correct definition for the standard topology in $\Bbb R^2$

Is this the correct definition for the standard topology in $\Bbb R^2$; $T_{st}=\{(x_1,y_1)\times(x_2,y_2)|x_i,y_i\in \Bbb R\}$
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How are the intersections and quotients of algebraic groups defined?

Let $k$ be a field and we define $Spec(A)$ to be the functor from $Alg_k$ to $Sets$ such that it sends $$ R \rightarrow Hom_{Alg_k}(A, R). $$ We let functors of this form where $A$ is a $k$-algebra, ...
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What is a faithful representation?

In the notes I am reading, it states: A representation fo an affine group scheme $G$ is a morphism $r: G \rightarrow GL_V$. It is faithful if it is injective. In this notes they are defining schemes ...
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What does it mean for a function $f$ to be defined on a disk? (Clairaut's theorem)

Clairaut's theorem states: "Suppose $f$ is defined on a disk $D$ that contains the point $(a,b)$. If the functions..." My question is just about the first part of this. What does it mean for a ...
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1answer
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$KK$-groups definitions

In $K$ Theory of Operator Algebras, page 144 and a paper by Skandalis, page 35 the $KK$ groups are defined differently: Both are triples $(E,\phi, F)$ but Skandalis does not require the condition ...
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What does “an immersed sub manifold is in general not a submanifold as a subset” mean?

According to Wikipedia https://en.wikipedia.org/wiki/Submanifold : An immersed submanifold of a manifold M is the image S of an immersion map f: N → M; in general this image will not be a submanifold ...
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What is a simply presented group?

I have some background in commutative ring theory. At the moment I am going through factorization theory of integral domains. I found out that it is a conjecture, that every Abelian group is the ...
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1answer
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About the definition of divergence

The divergence is defined as: $\nabla . \mathbf{A}=\lim \limits_{V \to 0} \dfrac{ \unicode{x222F}_{\partial V} \mathbf{A}.d\mathbf{S}}{V}$ My question is of two parts: $(1)$ If we are using ...
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Why do we need continuity for convergence in distribution?

Convergence in Distribution Let $P_{n}$ and $P$ be distributions on $(\mathbb{R},\mathscr{B}(\mathbb{R}))$ with corresponding cdf's $F_n(x)=P_{n}((-\infty,x])$ and $F(x)=P((-\infty,x])$ then we say ...
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what's the meaning of “informally” in this context

This is the context: "The actual proof is a proof by contradiction. In order for Cantor to show that no such correspondence between N and all of (0,1) exists, he assumes (wrongly) that such a ...
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Permutation representation of a group

Consider the action of $\operatorname{Aut}(\Gamma)$ of a graph $\Gamma$ on the edges of $\Gamma$. My book states that this is not always a permutation group. Question 1: Why is this not always a ...