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Questions tagged [definition]

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

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Introduction of function symbols

According to this link, in order to introduce a new function symbol one needs to prove the formula $\forall x_1,...,x_n\exists!y:P(y,x_1,...,x_n)$. This allows for the introduction of a new $n$-ary ...
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two different orbit definition and their explanations

I am chemistry master student and had to interact with abstract algebra somehow. I am trying to learn concept of orbit. When I look at the book, I saw two orbit definition in two distinct but somehow ...
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How exactly are area integrals in the complex plane defined?

I've been thinking about integrals over open sets in the complex plane. This was a topic that was never discussed when I was in university. If $U$ is some connected open set in the complex plane, and $...
K.defaoite's user avatar
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4 votes
1 answer
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Making clear the definition of 'affine variety' in Mumford's book.

I am reading "The Red Book of Varieties and Schemes" by Mumford. In section 4 the author defines the term affine variety: An affine variety is a topological space $X$ plus a sheaf of $k$-...
Toni's user avatar
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"arbitrarily close" and "sufficiently close" confusion in a limit definition [closed]

After $2$ minutes $22$ seconds on this video, Dr. Linda Green said on an informal definition of the limit of a function that For any function $f(x)$ and for real numbers $a$ and $L$, $\lim_{x\to a} f(...
Aleph-null's user avatar
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Could we define a fifth arithmetic operation on real (or complex) numbers that is independent of addition, subtraction, multiplication, and division?

The four basic arithmetic operations with real (or complex) numbers are addition, subtraction, multiplication, and division. the first two being inverse operations and the last two being inverses of ...
Saaqib Mahmood's user avatar
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36 views

binary operations on a field [closed]

i am reading the start of a linear algebra book, and reached the part of the definition of a set. the book defines 2 binary operations $+_F$ and $\cdot_F$, my question is relating to their signs, do ...
thatpithere's user avatar
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Multifunction Measurability

Let $(\Omega, \mathcal{F}, \mu)$ be a probability space and $X$ a Polish Space. A multifunction (or set-valued) $F:\Omega \to 2^X$ is a map from $\Omega$ into the subsets of $X$. But when defining ...
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Is there a name for the arc $\mathbb{S}^1 / (x \sim360 - x)$

I was playing with some ideas in a vague way and I have encountered this structure that arises from taking the space of angles $\mathbb{S}^1$ and quotienting it by the relation $(x, 360-x)$ (here $360$...
Sidharth Ghoshal's user avatar
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Is the definition of inductive set in Discrete Mathematics the same as that in Set Theory?

James Heine in Discrete Mathematics (second edition), chapter 3, section 3.1, page 128 defines inductive set as "objects constructed, in some way, from objects that are already in the set." ...
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Definiiton of 'Annihilator' in the context of dual map

Could you tell me what the definiton of 'annihilator' in this context ? I know the definition of annihilator for modules over ring, but in the following context, any definition I know means nothing. ...
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Clarification on the definition of co-functions

The definition of co-functions(Wiki) is as follows: Definition: a function $f$ is cofunction of a function $g$ if $f(A) = g(B)$ whenever $A$ and $B$ are complementary angles. I would like to confirm ...
Reuben's user avatar
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Multiplying Two asymmetric Laurent series

The following result $$\left(\sum_{i=0}^{\infty} a_i x^i\right)\left(\sum_{j=0}^{\infty} b_j x^j\right) = \sum_{k=0}^{\infty}\left( \sum_{\substack{j+i=k\\ i,j \ge 0}} a_i b_j\right) x^k $$ is well ...
Sam's user avatar
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2 answers
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For a curve to be smooth, it is necessary that its derivative is never equal to $0$. Why? (Complex Analysis, Curves in the complex plane)

I have a question about curves in the complex plane. A parametrized curve is a function $z(t)$ which maps a closed interval $[a,b]\subset\mathbb{R}$ to the complex plane. We shall impose regularity ...
佐武五郎's user avatar
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1 answer
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In what sense are preadditive categories also enriched categories?

I'm confused about Wikipedia's definition of preadditive categories: In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that ...
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Proof strategies for dealing with limits and continuity questions.

Background The following Definitions, Theorem, Lemma and Example below are taken from How to Prove it A structured Approach 2nd Edition, by Daniel J. Velleman. The Rule of inference table is taken ...
Seth's user avatar
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2 votes
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How is differential notation defined for two functions?

I heard about the differential notation for ODEs before, but have not yet found any rigorous definition. So far I had the impression that it is define for e.g. rewriting the equation $\frac{dy}{dx}(x) ...
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On Wikipedia's definition of zero morphisms

Wikipedia defines $f : X \to Y$ to be a zero morphism if $(1)$ $gf = hf$ for any object $Z$ and $g, h:Y \to Z$, and $(2)$ $fg = fh$ for any object $W$ and any $g, h : W \to X$. It then defines a ...
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Regarding the ϵ-δ definition of limits [duplicate]

The ϵ-δ definition of limits states: Let $ƒ(x)$ be defined on an open interval about c, except possibly at $c$ itself. We say that the limit of $ƒ(x)$ as $x$ approaches c is the number $L,$ and write ...
sab's user avatar
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Prove this structure is a left shelf

Left shelf and maps are defined as follows. How can I prove that the structure in Prop2.5 is left shelf? I tried to prove it by definition but failed. enter image description here
Haoyuan Ma's user avatar
5 votes
3 answers
492 views

Why are Contour Integrals defined the way they are?

I have a question about Contour Integrals. Contour Integrals are defined as $$\int_{C} f(z)dz = \int_{a}^{b} f(z(t))z'(t)dt \tag{1}$$. But why define it this way? The textbook I'm using (Complex ...
Asterix's user avatar
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What does "point set" topology mean?

Is it an antonym for "function set" topology (topology of function spaces) or "point-free" topology (locale theory)? Or other?
Power's user avatar
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8 votes
2 answers
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How is the discriminant defined for $x^3+y^3+z^3+u^3+(ax+by+cz+du)^3+exyzu$?

Recently I am reading a book by Arnol'd et. al.. In this text I did not find the definition of discriminant $\Delta(a,b,c,d)$ : I wonder what is the meaning of the discriminant here ? The following ...
cbi's user avatar
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Forgot the name of a group property

I forgot what is the notation, or definition or the name of the number of subgroups (or elements), product of which is a subgroup $G$ (or element), I don't remember it is denoted $cov(G)$ or $con(G)$ ...
Jan Safronov's user avatar
1 vote
1 answer
53 views

Precise definition of derivative

I have read that the derivative of a function $f$ at a point $a$ is given by $f'(a) = \lim_{h \to 0}((f(a+h)-f(a))/h)$ provided that $f$ is continuous at $a$ (or in other words, that $\lim_{x \to a}(f(...
Lauren S's user avatar
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How to define $\Sigma \models \phi$ when $\phi$ is not a sentence? [duplicate]

Let $\Sigma$ be a theory and $\psi$ a sentence. I'm familiar with the notion of $\Sigma \models \psi$, however, lately, I've seen some authors using this notation when $\psi$ is a formula with free ...
Eduardo Magalhães's user avatar
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Definition of "interval of continuity" for function defined on sets

At the beginning of Chapter 8 of Kubilius's Probabilistic Methods in the Theory of Numbers, the author defines $Q=Q(E)$ to be a completely additive nonnegative function defined for all Borel subsets $...
Greg Martin's user avatar
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Definitions straight line

In wikipedia, the notion of straight line is described as a basic notion, a primitive that is not defined. I wonder if there're any formal definition for a straight line in any specificular context so ...
PermQi's user avatar
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The definition of action of $G/H$ on group cohomology $H^1(H,M)$

Let $G$ be a group and $H$ be a normal subgroup of $G$. Let $M$ be $G$-module. Let $H^1(H,M)$ be first group cohomology. $G$ acts on $H^1(H,M)$ by $(\sigma*f)(g):=gf({\sigma}^{-1}g{\sigma})$・・・①. My ...
Pont's user avatar
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Directional derivatives in the complex plane as a vector space.

Consider the directional derivative in the direction $v\in V$ on some vector space $V$ \begin{equation} f_v'(x)=\lim_{h\rightarrow 0}\frac{f(x+hv)-f(x)}{h}.\qquad(1) \end{equation} Consider $\mathbb{C}...
Bedge's user avatar
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Lee's definition of a consistently oriented basis for a real vector space $V$ of dimension $n\geq 1$

Let $V$ be a real vector space of dimension $n\geq 1$. Lee defines in his book Introduction to Smooth Manifolds chapter 15 that We say that two ordered bases $(E_1,\dots,E_n), (\tilde{E}_1,\dots,\...
Cartesian Bear's user avatar
1 vote
1 answer
62 views

A flaw in notation of the polynomial vector space $\mathcal{P}(\mathbf{F})$?

In Linear Algebra Done Right, Axler defines what it means for a function to be a polynomial over a field $\mathbf{F}$ (real or complex numbers) as follows: A function $p : \mathbf{F} \to \mathbf{F}$ ...
Paul Ash's user avatar
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40 views

In Euclidean Geometry how is "beyond" defined on a plane?

I am trying to do a construction on a plane. The beginning of the problem states "Let there be two lines and a point beyond them." I felt like I was making pretty good progress but when I ...
Heather Avera's user avatar
2 votes
1 answer
74 views

What are the cohomology groups $H^n(k, \mathbb{Q/Z})$ for a field $k$?

I'm reading a source that connects Brauer groups of fields $k$ to $H^1(k, \mathbb{Q/Z})$, but the source doesn't define the cohomology groups $H^\bullet(k, \mathbb{Q/Z})$. Googling the definition ...
mattematician 's user avatar
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62 views

In what sense is $\int (u \cdot \nabla) u \cdot u dx$ an energy flux?

Due to the nature of this question I have cross-listed it on physicsSE. Let $u$ be either a solution to either the Euler equations or Navier-Stokes equations over a domain $\Omega$. In fluid dynamics ...
CBBAM's user avatar
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0 votes
2 answers
63 views

Why is the relation $\{(1,1),(2,2),(3,3)\}$ antisymmetric and transitive? [duplicate]

Today, my teacher said that $R=\{(1,1),(2,2),(3,3)\}$ is reflexive, symmetric, antisymmetric and transitive on set $A=\{1,2,3\}$. I can see that it is reflexive clearly. Moreover when I think $x=1,y=1$...
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Are $\phi^{'}_i,\tau_i$ the coresponding compatible functions for equivalence relations constructed in the directed limit of groups/rings.

Background: The following is taken from: A Graduate Course In Algebra - Volume 1 by: Ioannis Farmakis and Martin Moskowitz, How to Prove it by: Dan Velleman, and An Invitation to Abstract Algebra by ...
Seth's user avatar
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1 vote
1 answer
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When given the Fourier Series Expansion, what is the "DC Component" of the Electrical Signal?

I have come across 2 different ways to calculate the DC Term in Fourier Series for Electrical Signal $x(t)$ : $$a_0 = \frac{2}{T}\int^T_0x(t)dt \tag{1}$$ And $$a_0 = \frac{1}{T}\int^T_0x(t)dt \tag{2}$$...
Meeth's user avatar
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1 vote
0 answers
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Constructing compatible functions $\phi'_i,\phi'_j$ for equivalence relation $\sim$ for directed limit of directed systems of groups.

Background: The following is taken from: A Graduate Course In Algebra - Volume 1 by: Ioannis Farmakis and Martin Moskowitz, and How to Prove it by: Dan Velleman. Definition 1: Let $(G_i)_{i\in I}$ be ...
Seth's user avatar
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Definition Gauge Field / Connection on principal bundle

Reading Division Algebras and Supersymmetry I by John C. Baez and John Huerta, one passage confused me a bit: A connection $A$ on a principal $G$ bundle over $M$. Since the bundle is trivial, we ...
anonymous250's user avatar
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1 answer
60 views

Quick question about the notation: $(g_i)\in \prod_{i\in I}G_i$ in inverse limit of the inverse system of groups.

Background: The following is taken from: A Graduate Course In Algebra - Volume 1 by: Ioannis Farmakis and Martin Moskowitz Definition 1: Let $(G_i)_{i\in I}$ be a family of groups. Then , the ...
Seth's user avatar
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1 vote
1 answer
35 views

Clarification needed on the notations for $g_i$-s and $f_{ji}(g_j)$ in the definition for the inverse limit for the inverse system of groups.

Background: The following is taken from: A Graduate Course In Algebra - Volume 1 by: Ioannis Farmakis and Martin Moskowitz Definition 1: Let $(G_i)_{i\in I}$ be a family of groups. Then , the ...
Seth's user avatar
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1 vote
0 answers
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Quick question about a possible typo concerning definition for inverse limit for the inverse systems of groups.

Background: The following is taken from: A Graduate Course In Algebra - Volume 1 by: Ioannis Farmakis and Martin Moskowitz Definition 1: Let $(G_i)_{i\in I}$ be a family of groups. Then , the ...
Seth's user avatar
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1 vote
0 answers
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Confusion on definition of fundamental solution for the heat equation

As mentioned on Wikipedia, a fundamental solution for a linear differential operator $L$ is a function (or distribution) $G$ such that $$LG = \delta$$ which by linearity of $L$ gives the following ...
CBBAM's user avatar
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1 vote
2 answers
143 views

Why is the definition of $\pi$ as integral by Weierstrass "inverted"?

Reading https://en.wikipedia.org/wiki/Pi#Definition I stumpled upon the following definition as an integral, presumably given by Weierstrass: $$ \pi = \int_{-1}^1 \frac{dx}{\sqrt{1-x^2}} $$ However I ...
asmaier's user avatar
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1 vote
1 answer
81 views

Is this correct notation for set of equivalence class and quotient sets for directed limit of directed system of groups.

Background: The following is taken from: A Graduate Course In Algebra - Volume 1 by: Ioannis Farmakis and Martin Moskowitz, and How to Prove it by: Dan Velleman. Definition 1: Let $(G_i)_{i\in I}$ be ...
Seth's user avatar
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1 vote
1 answer
44 views

Why is a resolution of portrait mode 480 x 640 described as 4 : 3 instead of 3 : 4?

As I've read that: The aspect ratio of an image is the ratio of its width to its height. For instance, if a rectangle has an aspect ratio of 2:1, then it is twice ...
Z0q's user avatar
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4 votes
2 answers
243 views

Are we assuming a relation to be transitive untill proven otherwise?

I do not know if the title would be correct title for the question but I think I am asking a valid question. While studying set theory and relations we were often asked about whether a relation $R$ is ...
madhurkant's user avatar
1 vote
0 answers
44 views

Is this the correct notation for equivalence relation for directed limit for the directed system of groups.

Background: The following is taken from: A Graduate Course In Algebra - Volume 1 by: Ioannis Farmakis and Martin Moskowitz Definition 1: Let $(G_i)_{i\in I}$ be a family of groups. Then, the ...
Seth's user avatar
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1 vote
0 answers
22 views

Clarification needed in direct product and sum in the directed limit for the directed system of groups.

Background: The following is taken from: Graduate Course In Algebra, A - Volume 1 by: Ioannis Farmakis and Martin Moskowitz Definition 1: Let $(G_i)_{i\in I}$ be a family of groups. Then, the ...
Seth's user avatar
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