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

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1answer
73 views

Positive definite kernel vs. positive definite function

What is the difference between positive definite kernels and positive definite functions? As I understand it, a positive definite kernel is a positive definite function if it is translation ...
5
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1answer
111 views

Is there a rigorous definition of the term “coordinate system”?

You hear the term coordinate system thrown around a lot, and we all know the usual examples (polar coordinates in $\mathbb{R}^2$, spherical coordinates in $\mathbb{R}^3$, etc.), but in truth I have no ...
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1answer
120 views

Is this a correct definition of the natural numbers in ZF?

Set $s$ is a natural number if $s$ is transitive and for every $x$, $y$ and $z$ $y\in{s}\rightarrow(y$ is transitive$)$, and if $x\in{P}s\wedge(x$ is transitive$)\wedge{z}\in{P}x\wedge(z$ is ...
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0answers
121 views

How is this the definition of equi-integrable?

Let $Q=(0,T)\times\Omega.$ I am completely lost with this: No definition of equi-integrability I have seen looks anything like this. Can someone help me please? Presumably it is a fact that a ...
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0answers
97 views

Defining “Penon Infinitesimals”.

In this lecture (which is accompanied by these slides), right near the end (so page 9 in the pdf of the slides; I don't think you have to watch the lecture), P. Johnstone refers to the "Penon ...
2
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1answer
371 views

What is the most accurate definition of the hyperboloid model of hyperbolic geometry?

For simplicity, let's focus on the two-dimensional case (the hyperbolic plane in 3-space). I have seen the hyperboloid model defined as variations on the following i) The positive sheet of a ...
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2answers
732 views

Closed immersion definition

Hartshorne defines a closed immersion as a morphism $f:Y\longrightarrow X$ of schemes such that a) $f$ induces a homeomorphism of $sp(Y)$ onto a closed subset of $sp(X)$, and furthermore b) the ...
2
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1answer
950 views

About the epsilon definition of a convergent sequence. Is this definition equivalent?

I read that it is appreciated to include the context and motivation of a question. I may have overdone this a little bit in this question. To summarize, my question is: Are the two blockquoted ...
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2answers
40 views

Let $x\in \mathbb{R}$ and $z\in\mathbb{C}$. And define equality ($x=z$) iff $x=(0,x)$. Is this equality well defined?

Let $x\in \mathbb{R}$ and $z\in\mathbb{C}$. And define equality ($x=z$) iff $x=(0,x)$. Is this equality well defined ? Okay. It is easily shown that something goes wrong if you define equality in ...
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4answers
719 views

Definition of General Associativity for binary operations

Let's say we are talking about addition defined in the real numbers. Then, by induction we define $\sum_{i=0}^{0}a_i=a_0$ and $\sum_{i=0}^{n}a_i=\sum_{i=0}^{n-1}a_i+a_n$ for $n> 1.\:$ Now, how do ...
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1answer
73 views

About definition of series associated to sequence!

let $f:\Bbb{N}\to \Bbb{R}$ a sequence, $n \in \Bbb{N}$ and $S \in \Bbb{R}$, $S$ is $n$th-partial sum of $f$ if $$S=\sum_{i=0}^nf(i)$$ let $g:\Bbb{N} \to \Bbb{R}$ and $h:\Bbb{N} \to \Bbb{R}$ two ...
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3answers
104 views

What is the usual definition of “ordinal number”?

My definition for the ordinal number is "von-Neumann ordinal". I thought this is the only definition for the ordinal, but i found some other definitions in wikipedia. What is the usual definition of ...
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1answer
327 views

What does “weakly compact” mean when applied to subsets $X \subset Y$?

Let $X$ be a subset of a Banach space $Y$. Please can you give me a definition of what "$X$ is weakly compact" means? I want one which is in terms of sequences and boundedness, as opposed to one with ...
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1answer
118 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} = ...
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3answers
364 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 $3$ ...
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3answers
314 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
1k 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
134 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
541 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 ...
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1answer
95 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 ...
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2answers
800 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
61 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
1k 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
52 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
479 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
55 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
60 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
450 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
56 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
249 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
482 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
65 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 ...
3
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1answer
33 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
121 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
69 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
272 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?
5
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1answer
132 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
3k 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 ...
3
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1answer
265 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
130 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
78 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
150 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
130 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
48 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
54 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
985 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
79 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
39 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
658 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 ...