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

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1answer
19 views

What is a differentiable distribution on a manifold

Assume $M$ is a smooth manifold and $d:u\to Q_u<T_uM$ is a distribution on $M$. What does "$d$ is a differentiable distribution" mean? What does it mean for $Q_u$ to depend smoothly on $u$?
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0answers
6 views

meaning of index of a quadratic form for a field of positive characteristic

In "Classification of Algebraic Semisimple Groups" in "Algebraic Groups and Discontinous Subgroups: Procedings of Symposia in Pure Mathematics, Volume IX", Jacques Tits speaks of the index of a ...
3
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2answers
63 views

Why are these representations of e the same? [duplicate]

I heard that $e$ can be defined as the limit as n approaches infinity of $(1 + 1/n)^n$, but I also heard that $e$ is also defined as the sum of the reciprocals of the factorials from $0$ to $\infty$. ...
2
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1answer
33 views

Rouches theorem, $|f(z)|\gt |g(z)|$ at each point on $C$. $\color{red}{\text{On or In?}}$

Rouche's theorem from both of my resources say the following: Let $C$ denote a simple closed contour and suppose that: Two functions $f(z)$ and $g(z)$ are analytic inside and on $C$ $|f(z)|\gt ...
2
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1answer
27 views

Definition of “vector fields never have opposite direction”

Good day! As in my other question I am referring to the book "Differential Equations and Dynamical Systems" by Lawrence Perko, chapter 3.12. I have a question regarding Lemma 2: Lemma 2. If $v$ ...
2
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1answer
54 views

Limit definition of integration

The derivative function has the following definition using the limit: $$f'(x) = \lim_{h\to0} \frac{f(x+h)-f(x)}{h}$$ I was wondering whether I could find a similiar definition for the integral. I ...
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6answers
128 views

Is there proof show that $\log x$ is undefined and make no sense at $ x=0$?

I was asked by someone: why $\log x$ is undefined at $x=0 $? Is there proof show that $\log x$ is undefined at $x=0$? Note(01):: log is the inverse function of the exponential function. note(02): ...
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2answers
120 views

Isn't there a better way to put the canonical smooth structure on the $n$-sphere?

My differential geometry notes put a smooth structure on the $n$-sphere (denoted $S_n$) as follows. Firstly, $S_n$ is taken as a subset of $\mathbb{R}^{n+1}.$ Secondly, we define $U_0 = S_n \setminus ...
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1answer
27 views

Implications of Alternate Definition of the Limit of a Function

In Hubbard's Vector Calculus, I saw an unconventional definition for the limit of a function: Definition: Let $X$ be a subset of $\mathbb{R}^n$. A function ...
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1answer
14 views

Asymmetry of definition of regular value and critical value

Let $f: U \subseteq \mathbb R^n \to \mathbb R^m$ be a smooth map. We say $y \in \mathbb R^m$ is a regular value of $f$ if and only if all points in the set $f^{-1}(y)$ are regular. (see e.g. the ...
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2answers
19 views

Question about notation, subsets of a graph and intersection of vertices

I have the following description of a graph: Let $G$ be a graph such that all of its vertices are subsets with two elements of $\{1,2,...,n\} (n\ge 2)$ where two sets $A,B$ are adjacent iff ...
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1answer
37 views

Set of a matrix

I am working on a homework problem which asks me about the Set of a singular $n\times n$ matrix. specifically whether it is a vector space. I looked in the glossary of the book and searched online and ...
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2answers
35 views

The Standarization of Matrix by Vector Multiplication

I apologize for the trivialness of my question but it has been bugging me as to why the standard for multiplying a matrix by a vector that will give a column matrix mean that the vector has to be a ...
3
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2answers
157 views

Is there a formal definition for antiderivatives?

In the way the derivative can be defined as a limit, specifically $$f'(x):=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$ or any of the other possible variants, is there a way to define the antiderivative, as in ...
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2answers
109 views

How do we know that our definitions/axioms are not contradictory?

Let us assume that we have declared some axioms. Now, we wish to declare a new axiom too. How do we establish that the new axiom is not a consequence of, nor a contradiction to the original set of ...
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3answers
47 views

About $0!=1$ and $a^0=1$ as cases of empty product.

Some useful ''conventions'' as $0!=1$ or $a^0=1$ are particular cases of an empty product, i.e. a product between elements of the empty set. I know that such product is defined as a convention by: $$ ...
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2answers
37 views

When can I not use the chain rule?

If $z=f(x,y)$ where $x=g(r,\theta), y=h(r, \theta)$, then can you give me a good reason why $$\frac{\partial^2 z}{\partial r^2} \neq \frac{\partial}{\partial x}\frac{\partial z}{\partial ...
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1answer
20 views

Question about uncorrelatedness of random variables and distributions

I was wondering, if two random variables are dependent, does that mean that they must be correlated? does one imply on the other or vice versa? Also, if I know that a joint distribution of two ...
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0answers
63 views

Can someone explain presheaf to a calculus student?

From reading some stuff online, there is the claim that continuous functions are presheafs, so that the toy functions I play with in class such as $f(x) = x^2$ can also be thought of as presheafs. ...
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1answer
55 views

Can someone explain to a calculus student what “dual space is the space of linear functions” mean?

I ran across this phrase today in a post and I am slightly confused. From my understanding, the dual space is the space of functions that sends a vector to a real number. There are two confusions: ...
2
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1answer
21 views

What does “points spanned by powers” mean in the Goffinet dragon definition?

The definition of the Goffinet dragon fractal given by Wolfram Mathworld refers to plotting all points spanned by powers of the complex number p=0.65-0.3i What does it mean for points to be ...
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2answers
24 views

What does this definition of $C^k$ surface mean?

Reference: Richard Millman - Elements of differential geometry I'm reading this book and there is a really wierd definition in this book: To descrive the theorem, here are some definitions in this ...
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1answer
77 views

What does it mean when two groups commute?

This is probably an easy question, but I can't find the definition in my book. Let $G$ be a group, and let $H$ and $N$ be subgroups. What does it mean for $H$ and $N$ to commute? I have two ...
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1answer
28 views

Is this task defined mathematically correctly?

I wish to make a math model for predicting users clicking on context advertising. Math definition: Let $X$ be the set of users, and $Y$ be the set of adverts. We make a mapping $F: X×Y→P$ that should ...
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2answers
30 views

Topology definition, finite intersection, infinite union?

In my textbook I find: Definition of a topology $\tau$ as a collection of subsets of $X$ is a topology on $X$ if: $\varnothing, X \in \tau$ If $\mathscr{F}\subseteq \tau$ then ...
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5answers
77 views

Formal definition of the function $Sin$ [closed]

Which is the formal definition of the function $Sin$, starting from axioms of real numbers ? I never found it in any book. Not its Taylor serie, that is based upon the intuitive definition of $Sin$; ...
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6answers
1k views

How to think of a set?

I am doing self study for the last two months on functional analysis. As I get a bit used to the terms like space, topology, manifold, etc, etc, I realized that everything is defined in terms of set. ...
3
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0answers
34 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 ...
4
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1answer
33 views

Definition of a “modular Galois representation”

I am trying to pin down a definition for a $n$-dimensional modular Galois representation $$\rho : \text{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \text{GL}_n(A).$$ I am just looking for ...
0
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1answer
68 views

Product topology - definition

Can someone please give me a detailed explanation of the concept of product topologies? I just can't get it. I have looked in a number of decent textbooks(Munkres, Armstrong, Bredon, Wiki :P, Class ...
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1answer
19 views

Graph: vertex connectivity and edges connectivity

I know that a graph is $k-$ connected if any two vertex can be connected by $k$ independent path. This is what we call the vertex connectivity. But what is the edges connectivity ?
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3answers
73 views

Is the zero vector in the definition of linear dependence arbritary?

The definition of linear dependence according to wikipedia is The vectors in a subset $S=(v1,v2,...,vk)$ of a vector space $V$ are said to be linearly dependent, if there exist a finite number of ...
2
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1answer
44 views

What is a strictly positive probability distribution?

I'm reading about Markov Random Fields. In the wiki page it's written that When the joint probability distribution of the random variables is strictly positive,... I'm so confused! Because a ...
3
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1answer
45 views

Definition of functions in $L^p$ space

I know that if we suppose that $1 \leq p \leq \infty$, and if $f$ is in $L^p$, then this means that $\|f\|_p=[\int_X (f^p) dx]^{\frac{1}{p}}$. But I feel as though I'm missing some important ...
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5answers
845 views

What do people mean by “finite”?

Many arguments about the foundations or philosophy of mathematics centre on the question of whether or not there exist objects or entities (such as certain sets) which are not "finite". (For ...
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1answer
22 views

What does it mean for an orbit to “accumulate”?

For example, from Beardon's "A Primer On Riemann Surfaces", ...Show that g is a homeomorphism of D onto itself, and that no orbit accumulates in D. I'm simply looking for a definition of the ...
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0answers
16 views

Definition of parameterised $n$-manifold

Let $U \subseteq \mathbb R^{n+q}$ and let $W = U \cap \mathbb R^n \times \{0\}$. Let $\phi : U \to \phi (U)$ be a diffeomorphism. Then $M=\phi (W)$ is called a parameterised $n$-manifold in $\mathbb ...
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1answer
23 views

Is this a typo (parameterised $n$-manifold)

In this book here on page 62 parameterised $n$-manifolds are introduced. The example given is that of a regular curve $\gamma (t) = (X(t),Y(t))$ and a parametrisation $\phi (x,y) := (x, y + Y(x))$. ...
4
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1answer
88 views

What is the definition for a fine sheaf/ a partition of the unity on a sheaf?

From Griffiths and Harris, p. 42: A sheaf $\mathcal F$ on $M$ is called fine, if for $U=\bigcup_i U_i\subseteq M$ there is a partition of the unity subordinate to the cover $(U_i\mid i\in I)$. By this ...
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2answers
55 views

What interpretation of the Lie braket is this?

I was reading exercise 1.96 of Gadea's Analysis and Algebra on Differentiable Manifolds. I don't know what definition of Lie braket the author used, but I'm confused, as far as I know ...
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1answer
25 views

Strings in a dictionary. A partial order, strict order, and total order?

Question: The domain is the set of all words in the English language (as defined by, say, Webster's dictionary). Word $x$ is related to word $y$ if $x$ appears as a substring of y. For example, "ion" ...
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1answer
25 views

X - Y in a finite set

Question: The domain is the power set of a finite set. $X$ is related to $Y$ if $X - Y$ is not empty. Is it a partial or strict order? If it is a partial order or strict order, is it also a total ...
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1answer
27 views

Partial order or strict order

Question: The domain is the set of all positive integers. $a$ is related to $b$ if $b = a⋅ 3n$, for some positive integer $n$. Because of the equal sign, isn't this relation symmetric, transitive, ...
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2answers
73 views

How to define a specific ring using a homomorphism

If we have a ring $R$ then I can form a ring of matrices isomorphic to $R$ by setting $r \overset{\phi}{\mapsto} \left( \begin{array}{ccc} r & 0 \\ 0 & 0 \end{array} \right) $ and defining ...
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0answers
12 views

About equal substitutions

When we say that two substitutions, say $\theta,\sigma$ agree on the variables of a term $t$ what exactly do we mean ? Is it that both substitution act on the same variables and substitute them with ...
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0answers
25 views

Transitive relation vs. transitive action

A transitive relation $\circ$ is a relation with the property that: $$ (a\circ b \wedge b \circ c) \Rightarrow a \circ c.$$ A transitive group action is a group action $$\phi : G \times X \rightarrow ...
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1answer
36 views

What is the definition of ``2nd-order quasilinear parabolic'' ? for partial differential systems?

I have to know why the mean curvature flows are 2nd-order quasilinear parabolic. Let $\Omega\subset\mathbb{R}^n$ be a bonded domain (or a smooth manifold of $n$ dimensional) and $N\geq 2$. When the ...
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1answer
25 views

Are these partial order or total orders?

I just want to know if the following are considered partially ordered sets or total set. Here is the definition: $\mathbb{R}^2$ : (a,b) R (c,d) iff $a\leqslant c$ and $b\leqslant d$ $\mathbb{R}^2$ ...
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1answer
9 views

Terminology for probability matrix.

I have two related questions about terminology. If a matrix contains probabilities such that each column (or row or both) sums to $1$ , is this matrix always called a stochastic matrix i.e. even if ...
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1answer
49 views

Is my understanding of a limit correct?

When i first learned about limit i was taught that the limit $\lim \limits_{x \to a}$ f(x) = L exists if f(x) $\rightarrow$ L when x $\rightarrow$ a. Or that f(x) gets arbitrarily close to L when x ...