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

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1answer
33 views

Question about theorem involving GCDs?

From http://www.math.niu.edu/~beachy/abstract_algebra/study_guide/11.html, it says Theorem 1.1.4. Let I be a nonempty set of integers that is closed under addition and subtraction. Then I either ...
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1answer
43 views

Terminology for a generalization of multilinearity to any algebraic structure.

Let $S$ and $T$ be sets with the same algebraic structure. Let $\Phi:S^n\to T$ such that for any $i\leq n$, and $s_1,\ldots,s_n\in S$, the aplication $s\in S\mapsto ...
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2answers
149 views

Donald Knuth's notations on multiple sum

I am reading Donald Knuth's Concrete Mathematics (2nd Edition) and I am on chapter 2 (Sums). I have problems in understanding his some notations on multiple sums. I quote his explanations I can't ...
4
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1answer
117 views

Why doesn't the definition of (model-theoretic) conservative extension need strengthening?

In the wikipedia page dedicated to conservative extensions, we find the following sentence: $T_2$ is a model-theoretic conservative extension of $T_1$ if every model of $T_1$ can be expanded to ...
2
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2answers
76 views

definition of rectangle

I am wondering whether it is fine to define a rectangle with right angles like Wikipedia's page of rectangle. In Euclidean plane geometry, a rectangle is any quadrilateral with four right angles. ...
2
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1answer
65 views

What is the function that is not a binary function called?

A binary operation is a calculation involving two elements of the set and returning another element of the set. Suppose it doesn't return an element of the set. What is the function called? For ...
4
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1answer
169 views

Multiple-valued analytic functions

Although our definition requires all analytic functions to be single-valued, it is possible to consider such multiple-valued functions as $\sqrt{z}$, $\log z$, or $\arccos z$, provided that they ...
4
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2answers
885 views

What does “characteristic” mean in mathematics?

In Germany, I have heard "ABC is 'characteristic' for XYZ" sometimes from math students. It was used like "if you know ABC, then you know you're talking about XYZ" or "ABC describes XYZ completely". ...
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1answer
93 views

What's the intuition behind this definition of ordered pair in the $\lambda$-calculus?

On this page, we have the following definitions. pair = λabf.fab first = λp.p(λab.a) second = λp.p(λab.b) So I tried computing "first (pair a b)," and sure ...
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2answers
719 views

Understanding the definition of domain in Complex Analysis

I have a definition in my book which states, "a nonempty open set that is connected is called a domain." I understand what an open set is (a set containing none of its boundary points and I know what ...
4
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1answer
53 views

Is there a specification language that covers all of these classes of structures?

Classes of mathematical structures abound in modern math. Examples include: The class of all groups. The class of all partially-ordered sets. The class of vector spaces. The class of ordered fields ...
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1answer
210 views

If we have an embedding $f:X \rightarrow A$, where $A \subset Y$, do we have to show $f^{-1}$ is continuous?

If we have an embedding $f:X \rightarrow A$, where $A \subset Y$, do we have to show $f^{-1}$ is continuous? I'm looking at a proof where they only show that $f$ is continuous and 1-1. Then I looked ...
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2answers
176 views

Example of a manifold?

Why is this picture an example of a $1$-dimensional manifold? My thought process is: the circle must have a point removed from it because otherwise it would be self-intersecting, and self-intersection ...
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1answer
52 views

Does this definition say what I want it to?

I don't have a real background in math but I should still be able to define stuff in my MSc thesis, although the thesis does not involve a lot of math. I want to define an object $o$'s ...
3
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1answer
93 views

How to define $a^x$?

It's so common that we use the function $f(x)=a^x$. But actually how do we define it? In simple language we can say $a^n$ is the number $a$ multiplied with $a$ $n$ times for any $n$ in $\mathbb{N}$ ...
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0answers
60 views

About definition of inductive set (with sets or ur-elements)!!

---let $A$ a set, $A$ is inductive if $\emptyset \in A$ and $\forall x \in A (x^+ \in A) $ ---let $A$ a set, $A$ is inductive if $\emptyset \in A$ and $\forall Y \in A (Y^+ \in A) $ an example of ...
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1answer
77 views

Group of finite ideles

A simple question: If $\overline{\mathbb{Q}}$ denotes the algebraic closure of $\mathbb{Q}$ in $\mathbb{C}$ then what is the definition of the group of finite ideles of $\overline{\mathbb{Q}}$? ...
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2answers
142 views

Definition of dense set

Is this definition is correct?: Let $\preceq$ an order in $A$, $B \subset A$, with $B \neq \emptyset $. Then $B$ is a dense set in $A$ if $$\forall x,y \in A ( x \prec y \to \exists b \in B( x ...
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1answer
349 views

Definition of the Young Symmetrizer

I have a question regarding the equivalence of two definitions of the young symmetrizer. First, some notation: let $\lambda$ be a partition of $n$. Given a $\lambda$-tableau $T$ (that is, a tableau of ...
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1answer
41 views

doubt about my last question

I have a very basic doubt. If we talk about rooted graph, can we consider any graph whose one vertex is labeled in a special way to distinguished it from other vertices or only rooted tree. this doubt ...
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1answer
20 views

Subseqeunce convergence definition

Definition: A subsequence $(a_{n_k})$ of $(a_n)$ is convergent if given any $\epsilon >0$, there is an $N$ such that $\forall k\geq N \implies\vert a_{n_k} - \ell\vert < \epsilon$ Why do we ...
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1answer
134 views

Is this particular function not well defined?

I was looking more into what it means for a function to be well defined, and I believe I understand it. Suppose we have a function $f:A \rightarrow B$ where $A = \{1,2,3,4\}$ and $B = \{1,2,3\}$ ...
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1answer
91 views

Equivalent definitions of piecewise affine convex function over a convex set of $\mathbb{R}^n$.

Let $C\subset\mathbb{R}^n$ be a convex set and $f:C\to\mathbb{R}$ a convex function. I want to show that the following are equivalent: $\mathrm{epi}(f)=\{(x,y)\in C\times\mathbb{R}\mid y\geq f(x)\}$ ...
3
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1answer
654 views

Formal definition for indexed family of sets

Essentially I'd like to know the formal definition of the object $\{A_{i}|i\in I\}$ .This is my context: 1.- From Wikipedia (Here) I understand that a family of elements in $S$ indexed by $I$ and ...
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3answers
119 views

Why is $\infty-\infty$ undefined in measure theory?

Some additions to the title: I stumbled over this problem going through my measure theory lecture notes; the author explicitly mentions that he leaves $\infty-\infty$ undefined. I would like to know ...
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1answer
76 views

Definition of a function's domain and co-domain with subscript in name

I want to define a function that takes a parameter (lets say a real number) and returns a number (lets say a natural number). However, the function makes use of a 'global environment constant ...
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0answers
86 views

$\mathbb{R}$ as set of Dedekind cuts on $\mathbb{Q}$

-- "let $A,B \in \mathbb{Q}$, $A < B$ if $A\leq B$ and $A \neq B$ -- "let $\preceq$ be a ordering of a set $A$, and $B \subsetneqq A$, $B$ is initial segment of $A$ under $\preceq $ if $\forall a ...
6
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1answer
64 views

Definition of $\Omega$-group and $\Omega$-composition series

What are the definitions of $\Omega$-group and $\Omega$-composition series? No luck searching on the internet..
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1answer
50 views

Median based on number of entries instead of values

I’m writing a computer program that provides some useful statistical information about files. Calculating the mean is trivial, and the mode at least has a simple definition, but the median is proving ...
2
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1answer
61 views

Is Wikipedia's definition of $\omega$-inconsistency problematic in this way?

I could be wrong, but the definition of $\omega$-inconsistency given at over Wikipedia seems slightly problematic. In particular, Wikipedia claims that $\omega$-inconsistency is a property of a theory ...
3
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1answer
320 views

Definition of multiplication of real numbers from product of positive dedekind cuts and absolute value

--let $A, B \in \mathbb{R}$, with $0\leq A$ and $0 \leq B$, $A \star B :=\{q\in\Bbb Q\mid q<0\}\cup\{a\cdot b\mid a\in A\wedge b\in B\wedge a\ge 0\wedge b\ge 0\}$" this definition is correct: ...
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1answer
68 views

Is it legal to define a piecewise define a function like this?

I'm trying to piecewise define a function $h$ using two other functions $f$ and $g$. I want to use $h$ to draw conclusions on a certain set $T$ that's a union of two other sets $A$ & $B$. $ ...
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1answer
83 views

Definition of Real Absolute Value

This definition is correct: "let $A,B \in \mathbb{R}$, $B$ is absolute value of $A$, $B \triangleq|A|$, if $B=\begin{cases} A, & \mbox{if }A \geq0 \\ (-A), & \mbox{if }A \leq 0 \end{cases}$" ...
2
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2answers
483 views

distance between point and empty set

While playing with my little sister earlier we where inventing distances form earth to sun/stars/planets and who had the bigger distance wins. Now at some point she said "from earth to jesus" and ...
2
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2answers
226 views

What are authoritative publications regarding foundational mathematics?

I have a computer science background. In our world, there usually is an organization publishing standard documents for certain areas (e.g. W3C has Web standards, IETF publishes Internet-related ...
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1answer
83 views

Definition of Dedekind cut as initial segment

the definition of Dedekind cut by initial segment is correct: "let $\preceq$ be a linear ordering of a set $A$, and $B \subsetneqq A$, $B \neq \emptyset$, $B$ is Dedekind cut of $A$ under $\preceq$ ...
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2answers
73 views

Why this definition for “symmetry transformation”?

This question concerns section 8.5.1 in these notes: I don't understand why a symmetry transformation is defined as such. What implications is there if $\delta \mathcal L$ is a total ...
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1answer
254 views

Definition of initial segment

the definition of initial segment is correct: "let $\preceq$ be a linear ordering of a set $A$, and $B \subsetneqq A$, $B$ is initial segment of $A$ under $\preceq$ if $\forall a \in A, \forall b \in ...
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8answers
5k views

Why is there never a proof that extending the reals to the complex numbers will not cause contradictions?

The number $i$ is essentially defined for a property we would like to have - to then lead to taking square roots of negative reals, to solve any polynomial, etc. But there is never a proof this cannot ...
4
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4answers
289 views

Using the definition of the derivative prove that if $f(x)=x^\frac{4}{3}$ then $f'(x)=\frac{4x^\frac{1}{3}}{3}$

So I have that $f'(x)=\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ and know that applying $f(x)=x^\frac{4}{3}=f\frac{(x+h)^\frac{4}{3} -x^\frac{4}{3}}{h}$ but am at a loss when trying to expand ...
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5answers
3k views

difference between theorem, lemma and corollary

Can anybody explain what is the basic difference between theorem, lemma and corollary. We have been using it for a long time but I never paid any attention. I am just curious to know.Thanks a lot.
3
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1answer
195 views

Conglomerate in mathematical literature.

I rememeber I downloaded a pdf in category theory and after talking about classes it defined something called conglomerates, but I can't remember what that is and wikipedia has no article on it. ...
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0answers
72 views

How to Define an Infinite Anti-Diagonal Matrix

We can define an infinite diagonal matrix with some ease, and then say that a finite diagonal matrix is the top left sub-matrix of our infinite one. Can we define an infinite anti-diagonal matrix? It ...
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1answer
289 views

What does “topological dual of a Banach space” mean?

I am not sure what does the "topological" imply. Thanks.
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2answers
160 views

to clear doubt about basic definition in graph theory

Can anybody help me in clearing the doubt about hierarchical product of graphs. Its quite different from other graph products. Here is the screenshot and link how it is done. I know the rooted graph, ...
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1answer
79 views

Analogous to convolution

A convolution of two functions $f$ and $g$ is defined as $$[f*g] = \int_{-\infty}^{\infty} f(\tau) g(t-\tau) d\tau.$$ I am interested on an analogous transformation of the form $$[f\star g] = ...
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2answers
180 views

Apostol question on alternative definition of dot product

The problem says: Suppose we define the dot product by $A\cdot B = \sum_{k=1}^n |a_kb_k|$. Which of the following properties hold with this new defition? Does the Cauchy-Schwarz inequality still ...
2
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2answers
126 views

Ted Sider's Definition of a Total Relation over a Set D

I'm working through Ted Sider's book "Logic for Philosophy," and I'm noticing some discrepancies between the definition of a "Total Relation" that he uses and the definition used in other places, ...
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0answers
64 views

Is a single nontrivial convex set in a topological vector space enough to make it locally convex?

This is sort of a definition question. While a tvs is locally bounded if it contains a bounded neighborhood of the origin, a tvs is called locally convex if it contains a fundamental system of ...
3
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1answer
164 views

What is meant by tangential operator?

The context is as follows : Let $\Omega$ be a open set of $\mathbb{R}^n$, with smooth boundary $\Gamma$. Then there is the claim that for $\Psi \in C^1(\bar{\Omega})$, we have on $\Gamma$ $$ ...