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Questions tagged [definition]

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

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Definition of Path in Graph Theory

One of the definitions for a path in Graph theory is : A path (of length r) in a graph G = (V,E) is a sequence $v_0,...,v_r ∈ V$ of vertices such that $v_{i-1} −v_i ∈ E$ for all $i = 1,...,r$ ...
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Question regarding the definition of ideal/subalgebra of a Lie algebra

Let $L$ be a finite dimensional Lie algebra. I am getting rather confused with the notation/definition and I would appreciate any clarification. 1) When one talks about the commutator subalgebra (or ...
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Is this a free variable? Something else?

I am struggling to understand the terminology behind free variables. If we have $\forall x \ P(x)$ I would believe $x$ is bound and not free. If we have $\forall x \ P(x, y)$ I believe $y$ is free, ...
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Definition of a network in Graph Theory (Complex Networks)

I'm trying to understand a more precise definition of a graph by Bollobas: "A graph $G$ is an ordered pair of disjoint sets $(V, E)$ such that $E$ (the edges) is a subset of the set $V_2$ of ...
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Almost normal subgroups: Is there any notion which is weaker than normal subgroup?

Let $G$ be a group then $N$ a subgroup of $G$ is said to be normal subgroup of $G$ if $\forall g \in G$, $g^{-1}Ng = N$. Is there any notion which is weaker than normal subgroup? I mean something ...
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Equivalent definition of $\text{limsup}$

Let $\{x_n\}_{n\in\mathbb{N}}\subseteq\overline{\mathbb{R}}$ a sequence. On some texts the definition of $\text{limsup}$ is as follows: Definition 1. $$\text{limsup}x_n=\inf_{k\ge1}\bigg(\sup_{n\...
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On the notion of periodicity

I'm working with the following definition: $Def1$: We say that a number $τ ∈ ℝ$ is a period for the function $f : ℝ → ℝ$ if $x ∈ ℝ ⇒ f(x+τ) = f(x)$. A function with a positive period is said to be ...
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Are quantifiers only required because of infinite proposition chains?

Quantifiers such as $\forall x \ P(x)$ and $\exists x \ P(x)$ are in some ways equivalent to a long conjunction chain being true versus at least one statement being true in a long disjunction chain. ...
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Centered family in a Boolean algebra

I do not understand the definition of a centered family in a Boolean algebra. Here's the definition A family $R\subset A$ is said to be centered, if for every finite set $\{a_1,\ldots, a_n\}\...
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Translating set syntax in FOL

Even though the formal syntax rules for first order logic talk about $\forall x$ or $\exists x$ without necessarily including any kind of $\in Y$ part for some domain/set $Y$, sometimes we'll see ...
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What does orthogonal mean? [closed]

Say I have a node in a grid that effects all other nodes "orthogonally". If a node has a value of "4", than it increases the value stored in the other orthogonal nodes next to it by "1". So If the ...
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Why is the definition of cardinal number as the set of all sets equivalent to a given set “problematical”?

In Fundamentals of Mathematics, Volume 1 Foundations of Mathematics: The Real Number System and Algebra, after defining set equivalence as the ability to put the elements of the related sets in one-to-...
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Direct sum and normal sum [closed]

What is the differences between $(\hat{e_1}+2\hat{e_2})(e_1+e_2)$ and $(\hat{e_1}\oplus 2\hat{e_2})(e_1+e_2,e_2)\\$ $e_1=(1,0,0), e_2=(0,1,0)$
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Must an equation contain at least one variable? Can we call 1+1=2 an equation?

According to the Wikipedia and Encyclopædia Universalis, an equation must contain at least one variable but there is no such condition mentioned in other definitions. Thus can we call the following ...
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Alternate definition of prime numbers

The prime numbers are usually defined to be positive integers with exactly two distinct divisors—one and the number itself. There are plenty of variations on this definition. Just out of sheer ...
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Computing the matrix of a Linear Transformation

For a matrix $A\in M_n(\mathbb{F})$ consider the linear transformation $T_A:\mathbb{F}^n\rightarrow \mathbb{F}^n$. Denote the $B_{st}=\{e_1,...,e_n\}$ the standard basis of $\mathbb{F}^n.$ Compute ...
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Is a matrix with zero entries on the diagonal triangular?

The definition of triangular I've been working with is a square matrix is lower triangular if all the entries above the main diagonal are zero or upper triangular if all the entries below the main ...
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Defining the complex numbers as the algebraic closure of the real numbers [closed]

It is of course possible to define the complex numbers as a quotient ring of real polynomials: https://math.stackexchange.com/a/1083130/359302. But is it possible to prove all of their properties -- ...
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Trouble understanding topological groups.

I understand that a topological group is a group $G$ endowed with a topology $\tau$ on $G$ such that addition and inverse are continuous on $\tau$. Now, the definition of continuity is that for all $...
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1answer
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Definition of closed surface/manifold

This question might appear silly, I was reading on wikipedia that a closed surface (or manifold in general) is a surface without a boundary, I'd like to elaborate a bit on such definition. Assuming ...
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2answers
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$\frac{1}{\sqrt{-1}}=\sqrt{-1}$? [duplicate]

I have trouble to comprehend what my mistake is in the following calculation: If we set $\sqrt{-1}$ to be the new number with the property that $(\sqrt{-1})^2 = -1$ then I can write $$\frac{1}{\sqrt{-...
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Meaning of “universal” in category theory (equalizer definition)

In Awodey's Category Theory on page 62, equalizers are defined like so: In any category C, given parallel arrows $f, g : A \rightarrow B$, an equalizer of $f$ and $g$ consists of an object $E$ and ...
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1answer
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Definition of sigma algebra of a group

The Wikipedia article on measurable groups says "let $(G, \circ)$ be a group and further let $\mathcal{G}$ be a $\sigma$-algebra on the set $G$." I am confused because they assume a sigma-algebra ...
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How to say that $\epsilon$ should be small enough in a mathematical statement?

Assuming I approximate $x$ by $y$ throughout minimizing $\|x-y\|_2^2$, I want to define $x$ as the approximate of $y$ when $\|x-y\|_2^2< \epsilon$ in a mathematical definition statement. Then, ...
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Do we need the axiom of replacement (ZFC) to define a product of structures (Universal Algebra)?

Do we need the axiom of replacement (ZFC) to define a product of structures (Universal Algebra)? Here $\mathrm{V}$ is the class of all sets. From my perspective here is how the product of structures ...
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2answers
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Lie algebra: intuition of “Lie Algebra is tangent space of corresponding Lie Group”?

I am an engineering student and learned of Lie Group/Lie Algebra recently. I can follow and understand all the formula derivation of Lie Algebra from Lie Group. But I cannot grasp the meaning of "...
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How to define the antiderivative of a function with singularities?

It seems like there are a few ways of describing the antiderivative of a partial real function with singularities. What are the different ways of nailing down more specifically what an antiderivative ...
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2answers
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Finding $\lim\limits_{x\to 0} \frac{\lfloor x \rfloor}{x}$ and the different definitions of fractional part function.

I understand that $\lim\limits_{x\to 0} \frac{\lfloor x \rfloor}{x}$ does not exist because RHL is $0$ and LHL is $\infty$. However, when I tried to calculate the limit of the equivalent expression $1-...
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1answer
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Rudin's functional analysis, section 3.22 (Extreme points)

I struggle to understand the definition of "Extreme set" and "Extreme point". Let $K$ a subset of a vector space $X$. A non-empty set $S \subset K$ is called an extreme set of $K$ if no point of $S$...
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Turning the Ring of Polynomials, $\mathbb{F}[\lambda]$, into a Module.

Modules associated to a linear operator. Suppose that $\mathbb{F}$ is a field and $V$ a vector space over $\mathbb{F}$( i.e. an $\mathbb{F}$-module). Let $T:V\rightarrow V$ be a linear operator on $V$ ...
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What is a ring algebra? [duplicate]

Let $R$ be a ring. What does it mean to be an $R$-algebra? I missed the definition in my lectures and when I search online, I can only find definitions of what a ring is.
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nonconstructive characterization of definite integral

Is there a collection of properties that characterizes a notion of a definite integral of real-valued functions? The determinant can be uniquely defined using an explicit definition (via cofactor ...
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What is the correct definition of a Cunningham chain of length $n$?

According to Wikipedia, A Cunningham chain of the first kind of length n is a sequence of prime numbers $(p_1, \ldots, p_n)$ such that for all $1 ≤ i < n, p_i+1 = 2p_i + 1$. Hence each term of ...
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Calculus based or calculus-based?

I've noticed something strange in the literature. Regardless of context, there seems to be a trend where some authors will say 'calculus-based solution' or 'calculus-based physics' whereas others will ...
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Lang's *Algebra*: definition of $F[\alpha]$ and why it's an integral domain?

I am reading Lang's Algebra, namely the chapter about Fields. The first thing which confused me is the following: how he defines $F[\alpha]$? Later he defines this as the smallest subfield of $E$ ...
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If $A-B\ne\emptyset$ and $A,B$ are countably infinite how can they be the same size? [duplicate]

The subject line isn't exactly what I wish to ask, but I ran out of space. Basically my conundrum goes like this: If we have two finite sets $X,Y$ such that $Y\subset X$ and $X-Y\ne\emptyset$ then ...
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Question about Semigroups, Monoids, and Groups

My book defines semigroups, monoids and groups in a traditional way. Where semigroups are defined with associativity, monoids with associativity and identity and groups with associativity, identity ...
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Finding the residue of a function at $z_0$?

I encountered a particular question that led me to question the definition that I was given for a residue, after reviewing the literature I simply want to confirm that my understanding is correct. ...
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The Action of a Linear Transformation $T$ on a subspace.

Problem Statement: Let $T:V\rightarrow V$ be a linear transformation. The following are equivalent: ($i$) $\exists \mathcal{B}=\{v_1,...,v_n\}\subset V$ such that $$[T]_\mathcal{B}=\begin{pmatrix}...
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Possible description/definition of exponential function (proof verification/review)

I wish to describe the exponential function as unique continuous group isomorphism $f:(\mathbb{R},+)\rightarrow(\mathbb{R}^+,\cdot)$ satisfying $f(1)=a$, $f\equiv a^x$. Lemma 1: Assume $f:G\...
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symmetric functions vs symmetric polynomials

I am doing my thesis related with symmetric functions and representations. It is for this reason that I am reading MacDonald's book Symmetric Functions and Hall Polynomials. When reading chapter 1....
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Are definitions axioms?

I just want to ask a very elementary question. When we introduce a "definition" in a first order logical system. For example when we say Define: $Empty(x) \iff \not \exists y (y \in x) $ Isn't ...
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Definition of Lebesgue integral as a supremum

Let $X$ be a measure space and $f\colon X\to R$ a Lebesgue-measurable non-negative function. Wikipedia claims that $$\int_X f=\sup_{s\le f} \int_Xs$$ with $s$ running over all step functions bounded ...
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Is “linear function” in linear algebra different from “linear function” in calculus? If so, why not use different words?

Taking a single-variable function as an example, it seems to me linear function in linear algebra (also called linear transformation) is a function $f(x)$ that satisfies $f(x + y) = f(x) + f(y)$ and ...
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What is it called when the definition of “<adjective> <thing>” does not imply that it is a special case of “<thing>”?

There are a couple of terminologies that I find mildly confusing: Using the definition of graph on wikipedia, a graph is defined as having a finite number of vertices and edges. It states that there ...
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What is meant by “free choice” in mathematics?

The following quoted text was written before 1963, so the authors didn't have the benefit of decades of automated computing machines to inform their statements. The essential feature of an ...
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Why is the risk equal to the empirical risk when taking the expectation over the samples?

From Understanding Machine Learning: From theory to algorithms: Let $S$ be a set of $m$ samples from a set $Z$ and $w^*$ be an arbitrary vector. Then $\Bbb E_{S \text{ ~ } D^m}[L_S(w^*)] = L_D(w^*)...
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How could a scalar be equal to the trace of the same scalar?

A scalar is supposed to be a matrix and a trace is suppose to be a number; how could the two be equal in the picture below where $Y$ is an $n \times 1$ random vector and $A$ is any $n \times n$ ...
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What is $\Bbb{R}^n$ when $n=0$?

My teacher ask this question, though I thought $\mathbb R^n$ is Euclidean space with $n\geq 1$ but I didn't find this $\mathbb R^0$ space in web search. Can anybody help me out? Thanks in advance.
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Atom in first order programming

Consider the following statements regarding atom in first order programming An atom is a predicate applied to a tuple of objects. Atoms: An atom evaluates to a number. A scalar, a ...