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

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3
votes
6answers
164 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): ...
17
votes
8answers
37k views

What is the difference between equation and formula?

Sometimes equation and formula are used interchangeably, but I was wondering if there is a difference. ...
2
votes
2answers
159 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 \...
1
vote
1answer
44 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 $\mathbf{f}:X\rightarrow\mathbb{R}^m$...
0
votes
1answer
46 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 ...
1
vote
0answers
79 views

With infinite size, we can have $P \cdot M = M \cdot D $ (D diagonal) but where $M^{-1}$ does not exist. Can we say “P is diagonalizable”?

(I had this question in mind for longer time, but it is just triggered now by some comments at that recent question in mse) (Background) I was looking at properties of the Pascal-matrix: ...
1
vote
1answer
80 views

What do you call a set of numbers with the property that each subset sum is unique?

I require a set of numerical elements on which the sum of some of these elements is unique to the set, that it's to say, no other combination in the sum of elements will result the same outcome. ...
1
vote
1answer
40 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 ...
0
votes
2answers
34 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 $A\...
2
votes
2answers
65 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
votes
2answers
233 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 ...
1
vote
3answers
57 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: $$ \...
1
vote
2answers
47 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 r}\frac{\...
1
vote
1answer
30 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 ...
4
votes
1answer
80 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: ...
3
votes
1answer
30 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 "...
4
votes
0answers
107 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 ...
1
vote
2answers
40 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 ...
22
votes
3answers
2k views

Understanding induced representations

Let $G$ be a group and $H$ be a subgroup. Let $\phi:H\rightarrow GL(V)$ be a representation of $H$. There are three constructions in Wikipedia, but I am not really convinced by these. My question is: ...
0
votes
5answers
110 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 $Sin$; ...
-1
votes
1answer
35 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 ...
2
votes
2answers
180 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 $\...
3
votes
0answers
72 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 ...
1
vote
3answers
1k 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 ...
4
votes
1answer
48 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
votes
1answer
94 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 ...
0
votes
1answer
96 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 ?
4
votes
1answer
233 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 ...
3
votes
1answer
336 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
votes
1answer
49 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 ...
7
votes
5answers
1k 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 instance,...
0
votes
1answer
35 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 ...
1
vote
1answer
27 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 $\phi (x,y) := (x, y + Y(x))$. ...
2
votes
1answer
76 views

Characterizing affine subspaces order-theoretically

Let $V$ denote a real vectorspace and $\mathrm{Con}(V)$ denote the poset of convex subsets of $V$. The goal is to identify those elements of $\mathrm{Con}(V)$ that happen to be affine subspaces of $V$ ...
1
vote
2answers
67 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 $[\frac{\partial}...
3
votes
1answer
47 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" ...
2
votes
1answer
43 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 ...
0
votes
1answer
33 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, ...
2
votes
2answers
80 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 ...
0
votes
0answers
14 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 ...
0
votes
1answer
28 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 $b\leqslant d$ $\mathbb{R}^2$ ...
0
votes
1answer
495 views

The Definition of Consistency and Compactness in FOL

First order logic: "consistency," "compactness"? Consistency: A set $\Sigma\subseteq\text{WFF}$ is consistent iff there is no $\varphi\in\text{WFF}$ such that $\Sigma\vdash\varphi$ and $\Sigma\vdash\...
2
votes
1answer
205 views

How the branch cut make a multi-valued function several branches of single valued function?

In the wikipedia article, it describe the branch points and branch cuts: A branch cut is a curve in the complex plane such that it is possible to define a single analytic branch of a multi-valued ...
0
votes
1answer
20 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 ...
1
vote
1answer
53 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 ...
2
votes
1answer
57 views

Rephrasing the definition of a limit

Is $\large{\lim_{x \to a}f(x)=L}$ The same as saying "one can get $\large{f(x)}$ as close as imaginable to $\large{L}$, by setting $\large{x}$ close enough to $\large{a}$" and vice versa "one can ...
1
vote
0answers
34 views

What is a projective rational function in Fulton's book?

I'm reading Fulton's algebraic curves and I don't know if I understood correctly this definition on the page 46: What does the author mean by $f,g\in \Gamma_h(V)$ being of the same degree? Since $f$...
1
vote
5answers
68 views

How is “smaller than” defined on $\mathbb{R}$?

According to http://en.wikipedia.org/wiki/Binary_relation Binary relations are used in many branches of mathematics to model concepts like "is greater than", "is equal to", and "divides" in ...
0
votes
1answer
158 views

What is the definition of a binary relation?

According to http://en.wikipedia.org/wiki/Binary_relation it is first defined as "a collection of ordered pairs of elements of A" and then as "an ordered triple (X, Y, G) where X and Y are arbitrary ...
2
votes
1answer
69 views

Epsilon Delta Limit Intuition

I am having a hard time grasping the definition of a limit. I initially learned this loose definition of a limit: $\lim\limits_{x \to a}f(x)=L$ iff $f(x)$ approaches L when $x$ approaches $a$ from ...