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

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7
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62 views

What's the intuition behind the direct integral of a family of Hilbert spaces?

In order to understand better the mathematically rigorous version of Dirac's formalism in Quantum Mechanics I've been reading about direct integrals of Hilbert spaces, projector-valued measures and so ...
7
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0answers
295 views

Understanding the roots of homomorphism and homeomorphism

I understand the formal (i.e. mathematical) definitions of the terms "homomorphism" and "homeomorphism" as they relate to functions, but I am curious as to the origin of these terms. I don't know ...
5
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0answers
61 views

Looking for functions “similar” to the $ \Gamma $ function

According to wikipedia: https://en.wikipedia.org/wiki/Functional_equation The $\Gamma$ - function is the only solution that suffices the following three equations: $$ f(x)={f(x+1) \over x}\tag{$*$} ...
5
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174 views

From the viewpoint of modern geometry, is there a “best” definition of the term “triangle”?

I can think of at least six different possible definitions of the term "triangle" in Euclidean geometry. a subset of $\mathbb{R}^n$ that can be expressed as the convex hull of three or fewer points. ...
5
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0answers
61 views

Definiton of No Tear and No Paste

Topologists often mention an example beginning by "If there is no tear and no paste, then ...". As a student, I am confused with this "term", and I want to know the exact mean of it. First of all, ...
5
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96 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 ...
5
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0answers
241 views

I think a definition is wrong in “Model Categories” by Hovey.

I am working through the book "Model Categories", by Mark Hovey, and have a doubt about a definition given there. At the beginning of page 50 we read: Define a map $f:X\rightarrow Y$ to be a ...
4
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103 views

What does it mean to categorify something?

What does it mean to categorify something? Is there any simple illustrative example of this process? How it is done and what is it useful for? Here is where I heard it first: "Khovanov homology is ...
4
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111 views

Why not defining a measure as a function on functions?

A measure $\mu$ is a function to $\left[0,\infty\right]$ on the sets belonging to a $\sigma$-algebra. Then for integrable functions $f$ the integral $\int fd\mu$ comes in, having nice properties ...
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95 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 ...
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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 ...
3
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88 views

Definition of $C^{1,2}$?

I just realized that I don't really know what the definition of $C^{1,2}$ (or $C^{m,n}$) means. Two candidates come to mind: 1) For every $y$, the function $x\mapsto f(x,y)$ is $C^1$, and for every ...
3
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29 views

How to write this function in a “well-formed” way

Given an input $0 \lt x \lt 1$, find $x$'s Nearest Integer Continued Fraction with structure $$x = a_0 \pm \cfrac{1}{a_1 \pm \cfrac{1}{a_2 \pm \cdots}}.$$ Then $$f(c) = a_0 + 1 \mp \cfrac{1}{a_1 + ...
3
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0answers
22 views

What is a cusp neighborhood corresponding to a parabolic Mobius transformation in a Riemann surface?

I am referring to this wikipedia entry. So what I understand is that they are defining it using the Fuchsian model. If $\Gamma$ is a Fuchsian group, its parabolic elements correspond to the cusps of ...
3
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37 views

Euclidean Neighbourhoods Retracts and Deformation Retracts

Aim of this question is to clarify the differences between the two concepts written in the title, because it's unclear to me wether one condition implies the other, under which circumstances they are ...
3
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0answers
83 views

What's the derivative of a map defined on manifolds?

I'm going through Warner's book on differentiable manifolds. On page 8 he defines what it means for a map $f: U \subset M \to \mathbb R$ to be differentiable: $f$ is differentiable iff $f \circ ...
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26 views

Characterize in terms of fibre

I am not familiar with the notion "characterize" in the following context. Does this mean to redefine or?.... Any help would be appreciated. Thank you. For a function $f:X\to Y$, and y an element of ...
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135 views

Is There a Common Definition of “Finite Support”?

I thought I understood this term, but when I tried to verify this I found three different and conflicting definitions, none corresponding to mine. Is there a generally agreed definition for this term ...
3
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52 views

Properties of $L^{\infty}$

I'm trying to get a better grasp on the idea of $L^{\infty}$. What are the implications if we are given that $f \in L^{\infty}$? Also, how do we write $\|f\|_{\infty}$ in terms of the inf of a set of ...
3
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57 views

Gradient vector interpretation

I have trouble understanding the gradient vector $\nabla$. For a surface, when we find it's $\nabla$, why is the resultant vector the normal vector when $\nabla$ means gradient vector?? Thanks
3
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71 views

Largest Singular Value / Singular Value

I was wondering, what if the eigenvalues of a matrix A are all negative. So does that simply mean there is no singular value for this particular matrix?, hence I can't calculate the conditional number ...
3
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0answers
39 views

Meaning of the term $X/H$ and orbits

I am trying to find representations of the group $G=GL_2(F(t)/t^2) = (M_2(F_p) , + ) \rtimes GL_n(F)$ So I was trying to do exactly what Serre has explained in this section. I am not quite able to ...
3
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30 views

Definition of Dihedral group via semidirect product

Let $G$ be an abelian group and let $\varphi:Z_2\rightarrow Aut(G)$ be the homomorphism where $\varphi(\overline{1})$ is the inversion automorphism. Define $Dih(G)=G\rtimes_{\varphi} Z_2$. Now set ...
3
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0answers
181 views

What is the “correct” way of making $\mathcal{P}(X)$ into a topological space?

If $X$ is a topological space, I want to know the "correct" way of making the powerset $\mathcal{P}(X)$ into a topological space. So let $\mathsf{SupLat}$ denote the infinitary Lawvere theory whose ...
3
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65 views

Functors that preserve the subset relation on constructs

Is there a name for functors between concrete categories over Set that preserve '$\subset$'? Example, for a functor $F:\mathbf {Top}\to \mathbf {Grp}$, if $Y$ is a subspace of $X$ then $F(Y)$ ...
3
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21 views

Does this matrix have a name: $x=(x_0, \ldots, x_n)$, $S(x)_{i,j} = x_{i+j \mod(n)}$. Can you help me generalise the idea?

Apologies if this question is vague/unclear (especially the title). Given a vector $x=(x_0, \ldots, x_n)$ define the matrix $S(x)_{i,j} = x_{i+j \mod(n)}$. $$ S(x) = \left( \begin{array}{rrrr} ...
3
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0answers
169 views

Anti-Diagonal Matrix

Hi just a quick question. Can a anti-diagonal have zero in its diagonal? I have this definition in my paper. "An anti-diagonal matrix is a square matrix where all the values are zeroes except those ...
3
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0answers
105 views

What is the difference between $\propto$ and $\sim$

Suppose I have two physical quantities, lets name them $a$ and $b$. I wonder what the difference exactly is between $$a\propto b,\tag{1}$$ and $$a \sim b.\tag{2}$$ I know for eq. 1, that it means ...
3
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0answers
79 views

What is the desirable function identification when setting up arrows in the category of types?

My question is which functions can not be allowed in a statically typed programming language, so that the "canonical" category is less coarse than what you get if you define it's arrows to be ...
3
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0answers
88 views

Strong Integral

Is there a strong(!) notion of integral that can face all of those issues: Singularities Decay Modes Oscillations Measure Spaces Locally Convex Spaces For example combining decay modes with ...
3
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0answers
67 views

“Identify” terminology

What does it mean to "identify" two objects in algebra, as distinct from having an isomorphism or automorphism between them? I was led to think about this while reading Stewart's Galois Theory, where ...
3
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80 views

Connected sum of surfaces with boundary

The connected sum of closed surfaces (2-manifolds) is defined by removing a disk from each and gluing the exposed edges together. When defining the connected sum of surfaces with boundary, is the ...
3
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0answers
118 views

Definition of differential forms

I am reading the book Algebraic Geometric Codes: Basic Notions by Tsfasman, Vladut and Nogin. The authors introduce some definitions about differential forms at the beginning of Section 2.2. These ...
3
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0answers
784 views

What is the principal branch of $f(z)=\sqrt{1-z}$, $z\in\mathbb{C}$?

My question is, essencially, about the definition of the principal branch of a function that is not a Logarithm. In this case, $f(z)=\sqrt{1-z}$.
3
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0answers
84 views

Definition of small $o$

In one of my homework assignments (in intro to applied mathematics) there is the following definition: Given two functions $f(\epsilon),g(\epsilon)$ that are defined in $D=(0,\epsilon)$ we say ...
3
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0answers
51 views

Problem of understanding

Please , can someone help me to understand this part of text please What it means "... for some $k\in \mathbb{N} \cup \lbrace 0 \rbrace$ and uniformly fore a.e $t\in [0,2\pi]$" and please how to ...
3
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0answers
200 views

Taylor series of a vector field?

Can someone give me the definition of the Taylor Series of a vector field in $\mathbb{C}^2$? Thanks!
3
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237 views

Lemma of Whitehead

this is the lemma of Whitehead And i really don't understand the proof How to see that $k$ is well defined (i.e how to write an element from $X\cup_{\varphi_i}e^{\lambda} , i=0,1$ ) and how to ...
3
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0answers
210 views

Exercise on measure theory

I have this exercise: Let $\mathscr {S}$ be a collection of subsets of the set $X$. Show that for each $A$ in $\sigma(\mathscr S)$ there is a countable subfamily $\mathscr C_0$ of $\mathscr S$ ...
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0answers
86 views

Define composition of small cyles and making a big graph

I am having following sub graphs and wish to compose all and make a one bigger graph (say G). After that, I want to select the closed path where it is passing along the outer vertices of that ...
3
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0answers
150 views

Steps toward making the Feynman integral rigorous

I would like to know if there has been any progress recently toward giving the Feynman integral a rigorous definition. I heard that Richard Borcherds is working on giving quantum field theory a ...
3
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0answers
51 views

Defining a Certain Class of Plane Graphs

I'm having problems finding the right words to formulate the following class of graphs in a definition. I'm defining a class of plane graphs with the following properties: Removing any vertex of ...
3
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0answers
99 views

Closed / embedded surface

Given a closed surface in $\mathbb R^3$, is it necessarily an "embedded surface"? I think it is true, but that is just because I can't think of a closed surface for which we cannot construct a smooth ...
3
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0answers
232 views

Can GCD be called an operator?

Can $\gcd(a,b)$ be called a binary operator which takes operands $a$ and $b$ and returns their greatest common divisor. And if for some operator —say $\bigotimes$ —$(a_1\bigotimes a_2 ...
2
votes
0answers
34 views

Martingale under conditional prob. measure (definition)

Suppose we are given a probability space $(\Omega, \mathcal{F},P)$ s.t. r.v.s $X$ and $(Y_i)_{i=1}^\infty$ are $\mathcal{F}$-measurable. The relevant filtration is given by ...
2
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0answers
33 views

Proof $f(x,y)=x_1+e^{x_{2}}$ is strictly convex

I am trying to show that $f(x,y)=x_1+e^{x_2}$ is strictly convex. I can show this using the Hessian Matrix which is positive definite. However for some reason i can not put it together using algebra ...
2
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0answers
71 views

The best definition of a knot?

I just started reading a book about knots and links and asked myself what is the best and most precise way to define a knot, just an embedding of $S^{1}$ into $S^{3}$ is not enough, right? Can ...
2
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0answers
35 views

An intuitive interpretation of stopping time

I have the following definition of exercise time. Let $T\in\mathbb{N}$ with $T>0$, let $(\Omega,\mathcal{F})$ be a probability space with the $\sigma$-algebra $\mathcal{F}=2^{\Omega}$ and let ...
2
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0answers
20 views

On karatzas Definition of Kolmogorov-Feller diffusions

we find this definition on Karatzas [Brownian Motion and Stochastic Calculus pg 282] But the definition of Kolmogorov Feller process does not seem to require that $Af$ be continuous in $x$ as is ...
2
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0answers
17 views

Measure on $S^1$

My book gives the following definition of Lebesgue measure on $S^1$: For each $B\subseteq [0,1]$ in the Borel sigma-algebra on $\mathbb{R}$, define $\mu(B)=\lambda(B)$. This defines a finite measure ...