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

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1answer
116 views

Multiplication of positive fractional numbers.

I am reading this answer. We know that multiplication of two positive rational number $\dfrac{a}{b}$ and $\dfrac{r}{s}$ respectivly is defined as follows: $\dfrac{a}{b}\times\dfrac{r}{s} = ...
3
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3answers
298 views

Equality of positive rational numbers.

I am reading the second article Rational Numbers of the book "A Treatise on Advanced Calculus" by Philip Franklin. I have mainly 3 questions regarding this article. I am writing all these $3$ ...
2
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3answers
244 views

An example of an ordered, uncountable set in $\Bbb R$?

Is there an example of an ordered, uncountable set in $\Bbb R$? My Calculus professor, who likes to keep things simple, defined a sequence in $\Bbb R$ as an "ordered and infinite list of real ...
2
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0answers
797 views

Proper functions

I'm a bit confused with the definition of proper function, i.e.: "$f: X \to Y$ is proper if the inverse image of every compact set in $Y$ is compact in $X$." Can anyone provide a more intuitive ...
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2answers
122 views

How to proof that $\lim_{h \to 0}\frac{e^h-1}{h} = 1$ using the definition $e = \lim_{n \to \infty}(1+\frac{1}{n})^n$?

In other words, how I can prove that these two definitions of $e$ is equal? I saw these two definitions while trying to find proofs for $\frac{d}{dx}e^x$ and $\frac{d}{dx}\ln x$; some use the former ...
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2answers
236 views

Integral of a differential 1-form along a curve (clarification on the definition)

Let's denote with $(e_1,\dots,e_d)$ the usual basis of $\Bbb R^d$, and with $({e_1}^*,\dots,{e_d}^*)$ the dual basis of its dual space $\Bbb {(R^d)}^*$. Let $U$ be an open subset of $\Bbb R^d$ and ...
2
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1answer
62 views

Embedded and Non-Parametric Surface definition

What does it mean for a minimal surface to be embedded? For example the Scherk surfaces? How would I define what 'an embedded surface' is? And also what does it mean for a surface to be ...
14
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1answer
460 views

Definition of the nth derivative? [First post]

If the definition of the derivative is $$ f^\prime(x) = \lim_{\Delta x \to 0} \dfrac{f(x+\Delta x) - f(x)}{\Delta x} $$ Would it make sense that the nth derivative would be (I know that the 'n' in ...
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1answer
58 views

About definition of recursive sequence

Can I define the recursive sequence in the following? Let $f: \Bbb{N} \to \Bbb{R}$ a sequence and $k \in \Bbb{N}$ and $\forall x \in \Bbb{N}(f_x:=f(x))$ , $f$ is recursive sequence if ...
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1answer
747 views

Definition of “the surface measure”?

Let $\mu_n$ be the $n$-dimensional Lebesgue measure. Let $||\cdot||$ be a norm on $\mathbb{R}^n$. Define $S^{n-1}=\{x\in\mathbb{R}:||x||=1\}$. I have proven that $\forall A\in\mathscr{B}_{S^{n-1}}, ...
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0answers
46 views

Indecomposable groups vs. indecomposable objects

An object $X$ in a category $\cal C$ with an initial object is called indecomposable if $X$ is not the initial object and $X$ is not isomorphic to a coproduct of two noninitial objects. A group $G$ ...
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4answers
336 views

$\hom(V,W)$ is canonic isomorph to $\hom(W^*, V^*)$

Introduction My Semester just started and we have a new Professor for Linear Algebra II (replacing our former Professor). Apparently we are behind our schedule and thus we only had a brief ...
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2answers
47 views

Point as an element of an affine space vs point as an element of a topological space?

I am searching for the "most natural" definition of a (geometrical/space) point as an element of "something" in mathematics (I am trying to design a small computational geometry library on strong ...
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0answers
58 views

definition of a fiber bundle

I came across the definition of a fiber bundle in May's "A Concise Course in Algebraic Topology" (http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf, Chapter 7, section 4). There were two ...
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1answer
267 views

Tangent space to a surface at boundary points

Let $M$ be a $2$-dimensional compact oriented surface in $\mathbb R^3$ with boundary $\partial M$. For any $p\in M \setminus \partial M$ tangent vectors are defined as speed vectors of smooth curves ...
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1answer
54 views

Definition of “point” and “vector” in $\Bbb{R}^n$, and a model for $\Bbb{A}^n(\Bbb{R})$..

Can I use the following definitions? let $(a_1,a_2,..,a_{n+1}) \in \Bbb{R}^{n+1}$, $(a_1,a_2,..,a_{n+1})$ is a point of $\Bbb{R}^n$ if $$a_1=1 \wedge \forall i \in \{2,...,n+1\} (a_i \in \Bbb{R})$$ ...
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4answers
233 views

Have any definitions in mathematics been redefined

Based on certain intuitions and motivations we make certain definitions and then proceed to use these concepts in further developing our intuition. For example, we have an intuition that a line has ...
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2answers
312 views

What is the zero subscheme of a section

Let $X$ be a scheme, $\mathcal F$ a locally free sheaf of rank $r$ and $s \in \Gamma(X, \mathcal F)$ a global section of $\mathcal F$. Question: What is the zero subscheme of $s$? I can't believe ...
2
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3answers
62 views

I'm confused about the definition of poset.

The definition of poset : $\qquad$A set with a partial ordering. Partial ordering is a binary relation $\preceq$ over a set ($P$). If I understood the definition of relation correctly, then ...
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1answer
30 views

What is the name for the property that a subset of a set follows the same rules as the set?

I have a set that follows a certain property and I want to say that the subsets of this set also follows the property. What is this called? I know that closure under an operation means that performing ...
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2answers
87 views

About definition of “direct sum of $p$-vector subspaces”

In the books 1 and 2, in Somme directe d'une famille de sous-espaces vectoriels, I am reading the following: 1) let $E,F$ two vector subspaces of $V$, $E+F$ is direct sum, $E+F \doteq E\oplus F$, if ...
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1answer
66 views

One definition of the determinant of a matrix

Suppose you define as follows : for $(a,b,c,d)\in \mathbb{R}^4$, $\det \begin{pmatrix} a & b \\ c & d\end{pmatrix} = ad-bc$. for $A$ a square matrix of size $n$, you define $\det A$ ...
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1answer
174 views

Example of a set that is Dedekind-finite but not Tarski-finite?

Can you give an example of a set that is Dedekind-finite, but not Tarski-finite?
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1answer
119 views

An alternative definition of finite?

Does the following definition adequately characterize the notion of finite? Is it equivalent to, say, Dedekind-finiteness? A set $S$ is finite if and only if for all $x_0\in S$ and all $f:S\to S$, if ...
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3answers
1k views

What is a formal definition of “predicate logic”?

I'm currently trying to get clear about some terms that are often used in computer science (I'm a computer science student), but were never formally introduced. Especially, I would like to know what a ...
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1answer
192 views

Alternative definition of strong/weak operator topology.

Given two normed spaces $(X, ||\cdot||_X)$ and $(Y,||\cdot||_Y)$ the space of bounded linear maps $\mathcal{B}(X,Y)$ can be equipped with the strong operator topology (SOT) as follows: The ...
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1answer
92 views

Definition of “symmetric bilinear (real) form indefinite”

In my studies I use these definition: Def.: $f \in \mathscr{B}_ \Bbb{R}((e \times e), \Bbb{R}) $, $f $ is symmetric bilinear (real) form positive definite if 1) $\forall x \in e(f(x,x)\geq0)$ 2) ...
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1answer
59 views

Equivalent definitions of isometry

Consider a map $T:\mathbb{R}^2\to\mathbb{R}^2$ such that $\lVert T(x)\rVert=\lVert x\rVert$. Is this equivalent to stating that $\langle x, y\rangle=\langle T(x), T(y)\rangle$ for all ...
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2answers
101 views

Meaning of variables and applications in lambda calculus

The wikipedia definition of lambda terms is: The following three rules give an inductive definition that can be applied to build all syntactically valid lambda terms: a variable, $x$, is ...
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1answer
121 views

Why are there so many different definitions for differentiability?

I am studying differentiability for functions of several variables. Here is the first definition of differentiability I came across:$\quad$ A function $f:\Bbb R^n\to\Bbb R^m$ is differentiable at ...
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3answers
46 views

Definition: limit of a sequence

What is the purpose not to choose $|x_n-a|\leq\epsilon$ instead of $|x_n-a|<\epsilon$ in the definition of convergence? Is their a substancial difference (or a practical one)? Thanks in advance.
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1answer
51 views

Question about the definition of a supremum

I want to know if the next definition is correct or I should fix it: Lets say that we have set $K$ and $a\in K$. Then, if $a\sup K$, for every $x\in K\Rightarrow x<a$? Or I shold cahge it to $x\le ...
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2answers
607 views

Rigorous mathematical definition of “much greater than” symbol

What does $f(x) \gg g(x)$ mean mathematically? How can we characterize "much greater than"?
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1answer
68 views

Natural definitions of families of subgraphs

Cluster analysis is a vibrant area of applied mathematics, "used in many fields, including machine learning, pattern recognition, image analysis, information retrieval, and bioinformatics". A subfield ...
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2answers
35 views

What does $\text{mod}\ m$ in $a \equiv b (\text{mod}\ m)$ means

I am trying to do example 3.6 from this http://www.cs.fsu.edu/~lacher/courses/MAD3105/lectures/s1_3equivrel.pdf script, but I am not sure what does $(\text{mod}\ m)$ means. Can somebody explain it to ...
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1answer
286 views

Is a sequence a subsequence of itself?

I know that sets are subsets of themselves, so by that logic is that true for sequences?
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1answer
32 views

About definition of $a^b$ with $a \in \Bbb{R} \wedge b \in \Bbb{N}$

-- let $a,c \in \mathbb{R}$, and $b \in \Bbb{N}$, with $\Bbb{R}$ is a complete ordered field, $c \triangleq a^b$ if $c=\begin{cases} 1, & \mbox{if } a\neq 0 \wedge b=0\\ 0, & \mbox{if } a=0 ...
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1answer
199 views

Define positions of a set of points given (only) the distances between them

I have been thinking about spatial transforms. Given $n$ points, there are $\frac{n!}{(n-2)!2!}$ combinations of selecting two points, so for 64 points in space, there are 2016 single point-to-point ...
2
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1answer
62 views

Why do people not use “partially directed” graphs?

Are there structures in use which are a mix of directed and undirected graphs? I.e. the effective edge-set consists of both directed and undirected vertex pairs. In the case that the graph is simple, ...
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1answer
76 views

Is it correct to say $ x^3+2x+1=y^2 $ is an elliptic curve?

I'm a bit confused about the definition on elliptic curve. For example, can we say that $x^3+2x+1=y^2$ is an elliptic curve? My opinion is that it is not an elliptic curve as the definition given in ...
0
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1answer
64 views

$V, W$ vector fields, $\omega$ one-form, $d\omega(V, W)=?$

$V, W$ vector fields, $\omega$ one-form, what is the definition of $d\omega(V, W)$? I have seen some equalities and I suppose that $d\omega(V,W)=(V^jW^i-V^iW^j) \frac{\partial f_i}{\partial x^j}$ ...
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6answers
256 views

Why is exponentiation defined as $x^y=e^{\ln(x)\cdot y}$?

There are many curves that extend integer exponentiation to larger domains, so why was this one chosen?
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0answers
33 views

Difficulty with terminology/standard definition for Jordan-normal form.

(Perhaps this is something rather for MOverflow, don't know) I've understood the concept of the Jordan-normal-form such that it is similar to the idea of diagonalization of a matrix, but can be ...
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2answers
224 views

Additional ways of defining real powers.

I am familiar with the following 3 way of defining real powers: Given $x,y\in\mathbb{R}$, such that $x\gt0$, we can define $$ x^y = \begin{cases} \sup_{\,q\in\mathbb{Q}}\{x^q :q\le y\} ...
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1answer
56 views

Why the inverse of the matrix in this definition

We have this definition: I didn't understand why in this definition we just define $F^A$ to be $$F^A(X_0:X_1:X_2)=F\bigg((X_0:X_1:X_2)A\bigg)$$ $F^A$ is not the composition $F\circ c$? Why ...
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7answers
846 views

What is the square root of complex number i?

Square root of number -1 defined as i, then what is the square root of complex number i?, I would say it should be j as logic suggests but it's not defined in quaternion theory in that way, am I ...
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1answer
53 views

Can all real numbers be presented via a natural number and a sequence in the following way?

Is there (for each fixed base system with digits $0,1,\dots,m$) and then for each real number $r\in\mathbb R$, an integer $n\in\mathbb Z$ and a sequence $(a_i)_{i\in\mathbb{N}}$ with ...
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1answer
39 views

Is a relation$ R=\{(a,a):a\in I\}$ said to be reflexive, symmetric and transitive?

Consider a relation $R=\{(a,a): a\in A\}$ where $A=\{1,2,3\}$. i.e $R=\{(1,1),(2,2),(3,3)\}$ This clearly reflexive but is it necessary that all such relations are necessary to be symmetric and ...
0
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1answer
88 views

Definition of Limit Superior/Inferior of Real-Valued Functions

Let $f$ be a real valued function and let 1) $$\limsup_{r \to \infty}f(r) = \mu$$ 2) $$\liminf_{r \to \infty}f(r) = \phi$$ Does 1) $\iff$ $\exists\ r_n$ increasing, $r_n \to \infty$ such that for ...
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1answer
49 views

Transforming Nested Fixed-Point Formulas into Infinitary Logic Formulas with Finitely many Variables

There is a definition (actually a description of how it could be defined) of a fixed-point logic formula. The formula is in inflationary fixed point logic (IFP) in this case but it could also be ...