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

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7
votes
4answers
264 views

The interest rate last year was 2%, this year it is 3% - did interest rates go up 1% or 50%

I've heard some experts say 1% and other experts say 50% to describe this same scenario. Can both be correct? Which one is more mathematically correct? How do you remove ambiguity when trying to ...
2
votes
3answers
57 views

What does the sentence “The only sub-algebras of $\mathbb{R}^{2}$ are $0,\mathbb{R}^{2},\mathbb{R}(0,1),\mathbb{R}(1,0),\mathbb{R}(1,1)$” mean?

I started studying functional analysis a couple of days ago, I have reached the Stone-Weierstrass theorem which is stated in my lecture notes as Let $X$ be a compact metric space, $A\subseteq ...
0
votes
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 ...
1
vote
1answer
42 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 ...
0
votes
2answers
138 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
votes
1answer
107 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
votes
2answers
68 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
votes
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
votes
1answer
150 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
votes
2answers
822 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". ...
3
votes
1answer
91 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 ...
0
votes
2answers
621 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
votes
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 ...
1
vote
1answer
184 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 ...
-1
votes
2answers
172 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 ...
0
votes
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
votes
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 $\mathbb{N}$ ...
0
votes
0answers
59 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
votes
1answer
75 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 $\overline{\mathbb{Q}}$? ...
0
votes
2answers
123 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
votes
1answer
285 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
votes
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 ...
0
votes
1answer
18 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 ...
0
votes
1answer
128 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 $B = \{1,2,3\}$ ...
0
votes
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
votes
1answer
604 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 ...
1
vote
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 ...
1
vote
1answer
64 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 ...
0
votes
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
votes
1answer
63 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..
1
vote
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 ...
2
votes
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
votes
1answer
283 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: ...
1
vote
1answer
66 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 $A$ & $B$. $ ...
1
vote
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
votes
2answers
456 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
votes
2answers
224 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 ...
0
votes
1answer
77 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 $\preceq$ ...
5
votes
2answers
70 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
votes
1answer
216 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 ...
40
votes
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
votes
4answers
281 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 ...
8
votes
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
votes
1answer
182 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. ...
1
vote
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
votes
1answer
253 views

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

I am not sure what does the "topological" imply. Thanks.
3
votes
2answers
151 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, ...
1
vote
1answer
76 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
votes
2answers
177 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
votes
2answers
122 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, ...