Questions tagged [definition]

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

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If for every positive number $\varepsilon$ there exists a positive number $\delta$, why not the other way back? [duplicate]

Definition. Let $f(x)$ be a function of $x$. If for every positive number $\varepsilon$, however small it may be, there exists a number $\delta$ such that whenever 0 $<$ |x - a| $<$ $\delta$ we ...
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Definition of a Lipschitz hypersurface

In the last days the expression Lipschitz hypersurface came up a lot. I didn't find a definition. I think a Lipschitz hypersurface is just a hypersurface which can locally be described by Lipschitz ...
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32 views

Strange but simple question

Is there a name for the type of trick where we do something like $s=(1-p)r+pq$ where we are making a sum of complementary ratios of two numbers? This kind of reminds me of convolution but I thought ...
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contradiction in the definition of exponents of congruence class in modular arithmatics

I know that if $a\equiv b\pmod n $ ,then $a^k\equiv b^k\pmod n$ where $k \in Z^{\geq0}$.However , i saw a general definition in Discrete mathematics and applications by Sussanna 's book. It says that ...
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61 views

Why does a definition not have a proof? [duplicate]

Hi I have a question that in general in mathematics: Why does a definition not have a proof? I mean how did we reach it without proving it? I know that the proposition, or theorem, proof comes ...
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1answer
41 views

Pedagogical arguments for/against the “defineorem” (simultaneously definition and theorem)

Often times, we like to define things in ways that are not obviously possible. For example, we may define something to be the unique object satisfying a certain property, or we may want to state many ...
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24 views

Definition of expectation of measure on Banach space

For a real-valued random variable $X$ on a probability space $(\Omega, \mathbb{P})$ and distribution $\mu$, the expectation is defined by $$ E[X] := \int_{\Omega} X(\omega) ~ d \mathbb{P}(\omega) = \...
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61 views

Why define $\bigcup_{n\in I} A_n=\emptyset,\bigcap_{n\in I} A_n=\Omega$ when $I$ is the empty set [duplicate]

In the textbook I used on Probability and measure theory, a definition catches my eye: If $I=\emptyset$, we define $\bigcup_{n\in I} A_n=\emptyset,\bigcap_{n\in I} A_n=\Omega$ I didn't draw ...
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13 views

Understanding the definition of a Tagged Poisson point process

I've been struggling to unwind the following definition and understand it heuristically. Definition 1 (Tagged Poisson point processes). Let $\Omega$ be a locally compact metric space. The tagged ...
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22 views

Defining terms in Regression [closed]

What does it mean when we say we regress $X_i$ on $u^i$ by OLS? How does this affect the regression equation and how do we go about finding the estimated coefficient on $u^i$?
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1answer
32 views

Defining Long Line like $\mathbb R \times [0,1)$ (cartesian product of real line with unit interval) instead of using ordinals?

I was trying to understand the long line, and came across this reddit thread: https://www.reddit.com/r/math/comments/apfzi/in_topology_the_long_line_or_alexandroff_line_is/, in which there's a comment ...
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1answer
25 views

Is there a difference between the three best-reply (best-response) functions in game theory?

I recent became aware that in game theory there are not one but three definitions of the best response function, which is unfortunately routinely referred to interchangeably depending on the context. ...
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What does 'fintie order of contact with any hyperplnae' mean?

I am reading through an article and defines a manifold $M$ is nondegenerate at $x \in M$ if $M$ has at most finite order of contact with any hyperplane that passes through $x$. I have tried searching ...
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56 views

How to understand $\lim_{x \to a} f(x) = \lim_{h \to 0} f(a + h)$? [closed]

I have two interpretations for this equation: If $a, l \in \mathbb{R}$, $f$ is a real function, $(a-c, a+c) \setminus \{a\} \subseteq \mbox{Dom }f$ for some $c > 0$, and $\lim_{x \to a} f(x) = l$, ...
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What is the opposite of a “discrete set”?

$\mathcal{N} = \{1, \ldots, p\}$ or $\{1, 2, 3, 4,\ldots\}$ are discrete sets. So what is the opposite of a discrete set? There is no such thing as a "continuous set". I know there is convex ...
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The semidirect product of $G$ with ${\rm Aut}(G)$ in the canonical way: its name and its implementation in GAP and/or Magma

Let $G$ be a finite group with automorphism group ${\rm Aut}(G)$. Let $A_G$ denote the semidirect product of $G$ with ${\rm Aut}(G)$ in the canonical way. Question 1: Is there are name for $A_G$ in ...
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Definition of an End Block

Can someone explain to me what is an End Block in graph theory. I can not find an exact definition for this.
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Formally write a definition using quantifiers and predicate logic

I quote a passage from Ted Sundstrom's Mathematical Reasoning: Writing and Proof: A note about Definitions: Technically, a definition in Mathematics should almost always be written using "if and ...
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Is there a name for these subgroups?

Let $F$ be a free group. Let $F_1$ be a subgroup with basis $B$. Assume that $F_1$ has the property that for every $\alpha\in F-F_1$, the set $\{\alpha\}\cup B$ is still free. Is there a name for ...
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Switching quantifies for the definition of a limit.

For my analysis homework I am asked to prove or disprove that if you switch the quantifiers of the definition of a limit, then the definitions are equivalent. More specifically we are asked to prove ...
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1answer
53 views

Formal definition of limit (epsilon delta)

DEF.$$ \lim_{x \rightarrow x_0} f(x)=L \Leftrightarrow \forall \varepsilon >0: \exists \delta >0: 0<\left|x-x_0 \right|< \delta \Longrightarrow \left|f(x)-L \right|<\varepsilon  $$ Why ...
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1answer
52 views

Definition of countable dimension of vector space

It's a well known that Hausdorff locally convex space $(X, P)$, where $P$ is a family of seminorms which generate topology on $X$ is metrizable iff $P$ is equivalent to an at most countable subfamily $...
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The graph of units and definitions

Definitions of units (like other definitions) induce dependency graphs between the defined units. For instance, Wikipedia displays the following ones for the International System of Units (SI), before ...
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1answer
28 views

How is a linear time invariant system a dynamic system

This question may be a bit elementary, but I am having some confusion with understanding the basics of a linear time invariant system. My goal was trying to understand what an impulse response was, ...
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1answer
36 views

How do you call a graph which is a simplification of another one

The upper graph ($G_1$) seems to be "inside" the lower one ($G_2$). If I understand the definition of a subgraph correctly, $G_1$ is not a subgraph of $G_2$ because there is no A->B and C-...
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Are the definitions of embedded equivalence of planar graph in these two books consistent?

I am learning following famous theorem about planar graph in Bondy’s textbook published in 2008 . Theorem(due to Whitney(1983)) Every simple 3-connected planar graph has unique planar embedding. A ...
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1answer
48 views

Differentiability of a function at a point where a vertical tangent can be drawn

I know that a function is said to be diffentiableat a point if a tangent exists at that point . So, when I saw a example where there was a vertical tangent and then also it's answer said it's non ...
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23 views

What is a fairly strong function?

I am reading the Chapter 7 where one can find the Eq. (7.17): $$ \frac{Rn^0 Z(\rho_{sys}, T_{ref})T_{ref}}{\rho_{sys}(T_{ref}, T_s)} - V_{ref}=V_1 + \frac{Z(\rho_{sys}, T_{ref})T_{ref}}{Z(\rho_{sys}, ...
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1answer
29 views

Definition of transitive group

I need a definition of a transitive group that's accessible to someone who's just started learning group theory (so won't know about actions and orbits etc.). I've written the following: A ...
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1answer
79 views

How might “a linear mapping is its own differential” be stated better?

This is Example II-2.3 of https://www.scribd.com/read/282634061/Advanced-Calculus-of-Several-Variables?mode=standard If $F:\mathbb{R}^n\to\mathbb{R}^m$ is linear, then $F$ is differentiable ...
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Definition of characters in the context of twists of elliptic curves

In the Wikipedia article about Twists of curves, I encountered the term character several times and I do not know what the definition of it is (and there is no reference to another Wikipedia article ...
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1answer
41 views

The meaning of a closed differential inequality

I'm currently reading Villani's notes on hypocoercivity and on pp87 it states that the differential inequality (14.10): $$ \frac{d}{dt}(\mathcal E (f) - \mathcal E(f_\infty)) = -\mathcal D(f) \leq -K_\...
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1answer
54 views

What is min and max in $\delta -\epsilon$ proof?

So I've been going through limit proofs and I came across the definition of $MIN$ and $MAX$ I saw a definition say that $MAX(a,b)$ picks the smaller of the 2 numbers my question is what if $a =b$ ...
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1answer
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What is a substochastic measure?

I read this from Huber's "Robust Statistics" p76. I cannot find the definition anywhere. I can only find the definition of a substochastic matrix on the internet. Huber did not provide a ...
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24 views

Ask clarification definition of two types of error

In George Casella textbook page 383, it said we have: $P_\theta(\textbf{X}\in R) = $ probability of a type I error if $\theta\in \Theta_0$; = one minus the probability of a Type II error if $\theta\in ...
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35 views

How is this representation defined? [closed]

Given a Lie algebra $\mathfrak{g}$ with a representation $V$, how do we define the representation $\bigwedge^n\text{Sym}^m(V) $?
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37 views

What does $R_I$ for a ring $R$ and ideal $I$

In this paper, I am confused by the third sentence of the third paragraph, which reads If $A \subset B$ is a minimal ring extension, it follows from [2 (sic)] that there exists a unique maximal ideal ...
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35 views

Motivation for the definition of quotient space

I'm having some trouble understanding the concept of quotient space of a vector space. The definition my teacher gave us is the following: Let $V$ be a finite-dimensional vector space over a field $\...
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1answer
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Confusion about definition of faithful group action.

Here are some excerpts from my lecture note: An action from a group $G$ to a set $X$ is a homomorphism $\alpha: G \to \text{Sym}(X)$, where $\text{Sym}(X)$ is the group of bijections $X \to X$. ...
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1answer
49 views

Why does $(g\circ f)'(x_0)=g'(f(x_0))\circ f'(x_0)$?

I had a Multivariable calculus class this morning and the teacher showed us a proposition that I'm not quite sure I understand. The proposition is the following: Let: $D\subset \mathbb R^n $ $D^* \...
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55 views

Derivative of $f(x) = \sqrt[3]{(x-1)(x-2021)}|\sin(\pi x)|$ at $x = 1$

So I got this really strange contradiction that I couldn't explain to myself and I'd really love some clarification on the pitfalls that I'm potentially facing. Let $$f(x) = \sqrt[3]{(x-1)(x-2021)}|\...
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Definition of a formula in propositional logic

The set of formulae in propositional logic is characterised by ( at least in Derek Goldreis's book) Every propositional variable is a formula If $a$ is a formula then so is $\neg a$ If $a,b$ are ...
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Trouble understanding the definition of a parabola: Spivak Calculus Chapter 4-21b

Spivak's problem reads as follows: Given the horizontal line $L$ defined by $g(x)=\gamma$ and a point $P=(\alpha, \beta)$ not on $L$ so that $\gamma \neq \beta$, show that the set of all points $(x,y)...
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Ph.D. Reviewer complains: “(Syntax) Error in function definition”

Next week I will defend my Ph.D. thesis in the field of Scientific Visualization. I have a question that is at the end related to syntax and personal preference, I think. I use bold letters for ...
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1answer
63 views

Is there a non-standard definition of an absolute value?

So I was thinking about how we can define a number $x$ such that $|x|=-1$. This topic was already talked about so many times as we can see here or there. So I wanted to consider it from a different ...
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35 views

differences between Ideals and principal ideals in math notational descriptions

In the following post: (Confusion between principal ideal and ideal) on clarification between the concepts of ideals and principal ideals, @Yury stated: "... if $I$ is a principal ideal then ...
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1answer
81 views

Law of composition in a group

Most textbooks on group theory define the law of composition, say $\star$, as a function $\star: S \times S \to S$. The next part of the definition is sending $(a,b) \mapsto a \star b$. This confuses ...
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36 views

Empty function finite support

If $I=\emptyset$ is the empty set and $G$ a group with trivial element 1. Then $G^I=\lbrace f:I\to G \rbrace$ is the trivial group with the empty function as an element. Am I right that $G^{(I)}=\...
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1answer
67 views

Is my understanding of backward stability correct?

An algorithm $\tilde{f}$ of a problem $f$ is said to be backward stable if for any input $x$, $$ \tilde{f}(x) = f(\tilde{x}) \ \text{ for some } \tilde{x} \text{ such that } \frac{||\tilde{x} - x||}{||...
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Confused about the definition of $\mathcal O_{\operatorname{Spec}A}(U)$.

Consider $\operatorname{Spec} A$ for some ring $A$. When defining the structure sheaf, Hartshorne defines For an open subset $U \subset \operatorname{Spec}(A)$, we define $\mathcal O(U)$ to be the ...

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