Definitions are the core of mathematical precision; they come to answer "what is X" in mathematics. Into this category fit questions regarding equivalence of definitions; clarifications regarding complicated definitions; as well questions with purposed definitions for mathematical notions with ...

learn more… | top users | synonyms

2
votes
2answers
26 views

What is word reversal $w^R$?

In the following context, what is the formal meaning of "reversal of word $w$"? The free monoid on $A$ is the syntactic monoid of the language $\{ ww^R\ |\ w \in A^*\}$, where $w^R$ denotes the ...
1
vote
2answers
43 views

Definition question in algebraic topology.

Definition: The $p$th (de Rham) cohomology group is the quotient vector space $$H^p(U) = \frac{Ker(d:\Omega^{p}(U)\to \Omega^{p+1}(U)}{Im(d:\Omega^{p-1}(U)\to \Omega^{p}(U))}$$ where $U \in ...
1
vote
0answers
16 views

On closedness of $C^\ast$ subalgebras

By definition of a $C^\ast$ subalgebra it is a closed subalgebra. Why does it need to be closed? This is a restriction that is not required in the case of a Banach subalgebra. (although I can't ...
0
votes
2answers
63 views

Integral of $\frac{1}{x}$

The logarithm is defined as: $$ \ln x = \int_1^x \frac1{t} dt $$ Hence I am often told that for indefinite integrals, since $\frac1{x}$ is defined over $\mathbb{R} \setminus \{0\}$ (various sources ...
3
votes
1answer
30 views

What is the definition of contractible space? (It is not a duplicate)

I'm studying two algebraic topology texts (namely Munkres and Theodore) Here are definitions given in those texts Munkres Let $X$ be a topological space. If the identity map on $X$ is ...
1
vote
2answers
54 views

How is it called when one ellipse is “more elliptical” than another one?

Assume you have two ellipses, $A$ and $B$. Now $A$ looks "flatter" than $B$ because its ratio $\frac{\text{major axis}}{\text{minor axis}}$ is bigger. This means it "looks less" than a circle. How is ...
0
votes
0answers
20 views

Clarification on the definition of truth

So I am learning about the Godel's theorem. My instructor define truth and falsity as something arbitrary. he define $f$ to be true is $f(x)$ is $1$ and if $f(x) = 0$, it is false. I still dont have a ...
0
votes
0answers
24 views

Why an $R-algebra$ requires $R$ to be commutative?

Here is the definition of $R$-algebra. Let $R$ be a commutative ring Let $(A,+\cdot)$ be an $R$-module. Let $\ast$ be a binary operation on $A$. If $x\ast(y+z)=x\ast y + x\ast z$ ...
4
votes
0answers
52 views

Can we consider a hypergeometric function as a closed-form?

Let's say a calculus problem like an integral or a series has a solution that inevitably involving a hypergeometric function. It turns out that hypergeometric function cannot be expressed in term of ...
3
votes
1answer
37 views

Differentiability of chart functions

In the definition of a smooth (or $C^k$) manifold the charts $\varphi: U \to \mathbb R^n$ are assumed to have the property that for any two of them $\varphi \circ \psi^{-1}$ is smooth ($C^k$). Does ...
0
votes
0answers
63 views
+150

Functions $h$ such that $h(x*x') = f(x) * g(x').$

Definition 0. Call a magma $X$ surjective iff the distinguished binary operation of $X$ induces a surjective function $X \times X \rightarrow X$. Now for the main idea: Definition 1. Let: ...
0
votes
1answer
23 views

What is the name of this homotopy?

Here is my definition for homotopy: Let $X,Y$ be topological spaces. Let $f,g:X\rightarrow Y$ be continuous functions. If there is a function $F:X\times[0,1]\rightarrow Y$ such that ...
-1
votes
1answer
56 views

Measure Atoms: Definition?

Let $\Omega$ be a measure space with measure $\mu$. (Here, a measure is only meant to be countable additive!) Consider a subset $A\in\Sigma$. Then according to the wikipedia article it is an atom ...
1
vote
0answers
39 views

What's the best way to define $\mathbb{H}$?

I had once defined $\mathbb{C}$ as $\mathbb{R}\times\mathbb{R}$ as a set, then I found it extremely uncomfortable when doing complex analysis or measure theory. So I changed its definition so that ...
0
votes
0answers
8 views

Developing closed-form from recursive definition

Here is a recursively-defined function where c, d ∈ N. T(n) =    c, if n = 0 d, if n = 1 2T(n − 1) − T(n − 2) + 1, if n > 1 Carry out the five steps for repeated substitution to prove a ...
0
votes
1answer
20 views

What is the definition of finitely generated?

To be specific, here is an example. Note that $(\mathbb{Z},+)$ is a group. Definition 1. (Lang) Let $G$ be a group and $S\subset G$. If $G$ is the intersection of all subgroups $H$ of ...
2
votes
0answers
14 views

Integrable Systems-Description

I am relatively new to the topic of integrable systems and I came across the following description: "Integrable systems share a fundamental property which is related to the geometry of the initial ...
2
votes
0answers
30 views

Definition of “one element normalizes another element”

I understand that someone has asked the definition of "an element of a group normalizes a subgroup". Let $G$ be a group and $a,b\in G$. Then what does "$a$ normalizes $b$" mean? Does it make sense at ...
2
votes
1answer
35 views

What is an exact differential?

My book says "A differential expression $M(x, y)dx+N(x, y)dy$ is an exact differential in a region $R$ of the $xy$-plane if it corresponds to the differential of some function $f(x, y)$ defined on ...
2
votes
0answers
25 views

How to define a Holder seminorm of a section

I'm reading "Variational Problems in Geometry",Seiki Nishikawa, in the figure below. Let $(M,g)$ be a compact $m$ dimensional Riemannian manifold with no boundary. $T>0, 0<\alpha<1, ...
2
votes
0answers
19 views

What is $R$-algebra?

(To be clear, I mean a ring by a ring with unity.) Here is a definition in Wikipedia: Let $R$ be a commutative ring. Let $(M,+,\cdot)$ be an $R$-module. Let $\ast$ be a binary operation ...
-2
votes
2answers
52 views

Complex Measures: Integrability

Approaches A complex measure decomposes into: $$\mu=\Re_+\mu-\Re_-\mu+i\Im_+\mu-i\Im_-\mu=:\sum_{\alpha=0\ldots3}i^\alpha\mu_\alpha$$ This gives rise to integrability as: $$f\in L(\mu)\iff f\in ...
0
votes
0answers
10 views

Are there terminologies distinguishing modules over ring and rng?

Let $R$ be rng. Let $M$ be a left $R$-module. Let's say, after some verification, one realized that $R$ has a unity and it doesn't satisfy $1_R \cdot x$ for all $x\in M$. Hence, one cannot call $M$ ...
0
votes
1answer
26 views

Complex Functions: Integrability

Let $\Omega$ be a measure space with measure $\lambda$. Denote the space of simple functions by: ...
0
votes
2answers
24 views

Linear independence and grammar

Let $A$ be a commutative ring. Normally, the linear independence is a property of a subset or a family elements of an $A$-module. But one often sees statements like "$v$ and $w$ are linearly ...
0
votes
1answer
51 views

(Are there) subtleties in the definition of 'biproduct'

I always thought that a biproduct of two objects $A_1,A_2$ in some category $\mathcal{C}$ is an object $P$ with two maps $p_i:P\to A_i$ making it a product and two maps $j_i:A_i\to P$ making it a ...
0
votes
0answers
46 views

What makes a line “straight”?

In Euclidean space, there can be several definitions that makes a straght line: Line of shortest distance between two points Line that is linear, i.e with the points satisfying a linear equation ...
2
votes
2answers
129 views

Precise definition of affine, smooth, and irreducible

A book which I'm reading now says that "the Drinfeld curve $$ \mathbf{Y} = \{\, (x, y) \in \mathbf{A}^2(\mathbb{F}) \mid xy^q - yx^q = 1 \,\}$$ is affine, smooth, and irreducible." Here $p$ is an odd ...
0
votes
0answers
7 views

Intuitive explanation of “deterministic system”?

Wikipedia Definition: In mathematics and physics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will ...
0
votes
1answer
30 views

Two definitions of left-invariant vector fields of a Lie group

I am reading these lines from a text which shows why the bracket of two left-invariant vector fields is also a left-invariant vector field. But cannot easily get one of the lines. Let $L_a$ be the ...
-9
votes
7answers
265 views

How is greater than defined for real numbers?

I have an understanding of real numbers. For example, I can imagine real numbers as points on the line. The point which is more to the right represents bigger number. Or if I have decimal ...
1
vote
2answers
84 views

Algebraic Structures: Does Order Matter?

(I want to link to similar question with a very good answer: Question about Algebraic structure?) An algebraic structure is an ordered tuple of sets. One of these is called the underlying set, and ...
0
votes
1answer
181 views

The “Empty Tuple” or “0-Tuple”: Its Definition and Properties

(I would like to link to a previous discussion on the subject: What is A Set Raised to the 0 Power? (In Relation to the Definition of a Nullary Operation)) In axiomatic (ZFC) set theory, we define ...
1
vote
0answers
11 views

Weierstrass conditions, what does strong mean, and are both conditions required?

I have the Weierstrass condition: In order that the extremal $\bf{C^*}: x = x^*(t)$ give a strong local minimum to $\bf{J[x]}$ it is sufficient that: # ...
0
votes
0answers
17 views

Rotational invariant curve definition

I have a question about the definition of rotational invariant curve. The definition I have is the following "By an invariant curve for a twist map $F$ we mean a simple closed curve that is invariant ...
0
votes
1answer
20 views

Can a linear functional be infinite at a point?

On a Banach (or Hilbert) space $X$, when we define a linear functional (not necessarily bounded), we define it to be a linear function from the elements of $X$ to the field $\Bbb F$. (Say, $\Bbb R$). ...
3
votes
2answers
68 views

Definition of intersection multiplicity in Hartshorne VS Fulton for plane curves

In exercise I.5.2 in Harshorne there is the following definition of intersection multiplicity for two curves in $\mathbb{A}^2$: \begin{equation} \mathrm{length}_{\mathcal{O}_P}\mathcal{O}_P / (f,g) ...
2
votes
3answers
64 views

What is free product?

I have searched for it, but I found there are several many different definitions. Even wikipedia states just free product of $2$ sets, not an infinite product. I know what exactly free group of a ...
1
vote
1answer
27 views

Is my understanding of free group correct?

Let $(G,*)$ be a group. Let $S$ be a subset of $G$. Then, construct the free group $(F(S),*')$ on $S$. If there exists an isomorphism $\phi:(G,*)\rightarrow (F(S),*')$ such that $\phi(s)=(s)$ on ...
-1
votes
0answers
12 views

Integration over ordered random variables

I have a joint distribution function over random jointly distributed random variables $(X,Y)$ denoted by $f_{X,Y}(x,y)$. Assuming without loss of generality that $$X<Y$$ I would like to find ...
2
votes
2answers
24 views

Cauchy property of a series

Are these two definitions equivalent, even though the first one has an extra term: If we consider the series $\sum_{n=1}^{\infty}x_{n}$ and the formal definition of a Cauchy property defined in terms ...
1
vote
1answer
31 views

Exact definition of a circular helix

I know that a cylindrical helix is a curve $\alpha: I \to \Bbb R^3$, such that exists a constant vector $\bf u$ which makes a constant angle with the tangent vector $\bf T$ to the curve, at every ...
0
votes
0answers
32 views

Definition question.

Suppose we have $x \in \mathbb{R} / [0,1)$. We call $\lfloor x \rfloor$ the integer part of $x$. What do we call $\bar{x}=\frac{x}{\lfloor x \rfloor}$??? I would call it fractional part but it ...
0
votes
1answer
30 views

Confusion about the associative property and the mechanics of Parenthesis

This is a follow up question on my earlier post (Updated): Showing that a set $M$ with two elements classifies as a field. I feel this post is necessary because I realize that what confuses me is how ...
0
votes
2answers
26 views

Give the definition of S $\subseteq$ T for general sets S and T.

The answer I can come up with is; S is a subset of T, denoted by S $\subseteq$ T, or equivalently, T is a superset of S, denoted by T $\supseteq$ S. Can someone correct me if I am wrong, or provide ...
0
votes
0answers
40 views

Definition of a summation from minus infinity

Formally, an infinite series is defined as the limit of its partial sums: $$\sum_{n = 0}^\infty a_n \equiv \lim_{n \to N} \sum_{n = 0}^N a_n$$ However, how does this work for summations such as the ...
1
vote
1answer
38 views

Why is the structure group for lengths $\mathbb{R}^+$ and not automorphisms in $\mathbb{R}^+$?

While reading what is a gauge to understand what a gauge is, I got stuck at a point where Terry Tao wrote: This isomorphism group is called the structure group (or gauge group) of the class of ...
3
votes
3answers
121 views

Understanding of the formal and intuitive definition of a limit

The intuitive definition for $\lim\limits_{x \to a} f(x) = L$ is the value of $f ( x )$ can be made arbitrarily close to $L$ by making $x$ sufficiently close, but not equal to, $a$ . I can easily ...
2
votes
1answer
26 views

Concise description of Lebesgue measure in $\mathbb{R}^{n}$

I would like to confirm that the following is an acceptable description of the Lebesgue measure in $\mathbb{R}^{n}$. The outer Lebesgue measure $E \subset \mathbb{R}^{n}$: $$\lambda^{*}(E) = ...
0
votes
2answers
27 views

Regarding retractions of $X$ onto subspaces

Let $A \subset X$ be a subspace of $X$. Recall that a retraction of $X$ onto $A$ is a continuous map $r: X \to A$ such that $r(a) = a$ for every $a \in A$. Let $X = \bf R$ endowed with the standard ...