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

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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 ...
2
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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
128 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 ...
3
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1answer
304 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 $\lambda$-tableaux $T$, we define the row ...
2
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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
131 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
89 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: $\mathrm{epi}(f)=\{(x,y)\in C\times\mathbb{R}\mid y\geq f(x)\}$ ...
2
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1answer
623 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
118 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
74 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 ...
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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
48 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 ...
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1answer
58 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
294 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
67 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
463 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
225 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
78 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
71 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 ...
0
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1answer
226 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 ...
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4answers
285 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
191 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
71 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 ...
0
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1answer
264 views

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

I am not sure what does the "topological" imply. Thanks.
3
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2answers
153 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] = ...
2
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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
125 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
61 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
155 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 $\Gamma$ $$ ...
4
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4answers
3k views

Continuity on open interval

A function is said to be continuous on an open interval if and only if it is continuous at every point in this interval. But an open interval $(a,b)$ doesn't contain $a$ and $b$, so we never ...
2
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0answers
2k views

Continuity on open and closed intervals

I will be taking Calculus I soon, and I just want to make sure I understand some concepts correctly. So far, reading my book for Calculus I, I've encountered the definition of continuity as being ...
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1answer
85 views

The use of = vs := for definitions

I have seen the following conventions, e.g. We define $K=\mathbb C$ ... We define $K$ to be $\mathbb C$ ... We define $K:=\mathbb C$ ... I prefer number 3, because it is concise and it is clear ...
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0answers
68 views

With infinite size, we can have $P \cdot M = M \cdot D $ (D diagonal) but where $M^{-1}$ does not exist. Can we say “P is diagonalizable”?

(I had this question in mind for longer time, but it is just triggered now by some comments at that recent question in mse) (Background) I was looking at properties of the Pascal-matrix: ...
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1answer
93 views

What is an operator norm?

I came across this symbol where they are mentioning operator norms. I am not sure how it is different from the Frobenius norm. It was something like this: $|||\Omega-\hat{\Omega} |||_2$ where ...
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3answers
556 views

What is the epsilon-delta definition of limits, exactly?

I am a bit confused with infinitesimals, and want to know why they were discarded and the epsilon-delta definition is being used? What is the epsilon-delta definition of limit? What is the intuition ...
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1answer
145 views

Is taking inverse automatically well-defined?

By the usual definition, Lie group is a manifold $G$ with a group structure on it such that the multiplication $m\colon G\times G\to G$ and taking inverse $i\colon G\to G$ are both smooth maps. But it ...
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1answer
57 views

exponent mod $n$

What does the term "exponent mod $n$" refer to? I guessed that this refers to the multiplicative order mod n, but it doesn't look like this is the case in what I'm reading right now.
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1answer
99 views

Is it true that the definition of an open subset in a metric space is different from the combination of the definitions of subsets and opens sets?

Dear reader of this post, I have a question concerning the equivalence of two definitions of open subsets (in metric spaces). To avoid confusion, I will state the two definitions and then ask my ...
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1answer
644 views

Norm of a vector-valued function?

When studying commutator estimates, I have encountered the following problem. Consider $f\in C^1(\mathbb{R}^d,\mathbb{R})$ with $\nabla f\in L^p$. So $\nabla f(x)\in\mathbb{R}^d$. My question is ...
2
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1answer
97 views

What is $\lim_{x=0\rightarrow1}x^\infty$?

I know it's all about definition... But still I want to know whether the answer is $0$, $1$, impossible to say or something else, like that the mathematical statement is wrong. However, to clarify, ...
2
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2answers
168 views

A definition of algebraic expression

Definition of algebraic expression An algebraic expression is a collection of symbols; it may consist of one or more than one terms separated by either a $+$ or $-$ sign. If by symbol we only mean ...
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1answer
74 views

Triangle of Multinomial Coefficients

What is the "Triangle Of Multinomial Coefficients" seen here: http://oeis.org/A036038 (OEIS: A036038) I can see that the diagonals of this triangle are just factorials... for example the last number ...
3
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3answers
141 views

Calculating derivative by definition vs not by definition

I'm not entirely sure I understand when I need to calculate a derivative using the definition and when I can do it normally. The following two examples confused me: $$ g(x) = \begin{cases} x^2\cdot ...
4
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1answer
277 views

What is the definition of “formal identity”?

In Ahlfors' Complex Analysis he remarks that harmonic $u(x,y)$ can be expressed as $$ u(x,y) = \frac{1}{2}[f(x + i y) + \overline{f}(x - i y)] $$ when $x$ and $y$ are real. He then writes "It is ...