Tagged Questions

Definitions are the core of mathematical precision; they come to answer "what is X" in mathematics. Into this category fit questions regarding equivalence of definitions; clarifications regarding complicated definitions; as well questions with purposed definitions for mathematical notions with ...

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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} ...
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8 views

What is the definition of “sheet”?

Here is a definition for evenly covered set. Definition Let $C,X$ be topological spaces and $p:C\rightarrow X$ is a continuous function. Let $U$ be an open subset of $X$. Then $U$ ...
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0answers
14 views

Composite residuosity statement.

Consider the following definition. A number $z$ is said to be $n$-th residue modulo $n^2$ , if there exists a number $y \in \mathbb{Z}_{n^2}^*$ such that $$z\equiv y^n \mod n^2$$ Let us take $n=6$ ...
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1answer
39 views

Why $R^{op}$ used in the definition of “category of $R$-modules”?

Earlier today, I was thinking: "Oh, an $R$-module is just an additive functor $R \rightarrow \mathbf{Ab}.$" Anyway, I had a bit of a read over at nLab, and it says: For any small ...
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0answers
22 views

different representations of strong induction

I've seen 2 forms of strong induction; just wondering how one follows from the other. $1) f(n_0)\wedge f(n_1)\wedge\cdots \wedge f(n_{k-1})\wedge f(n_k)\wedge \forall_n[f(n-k)\wedge ...
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0answers
11 views

Functorial cofibrant replacement does not have to be fibration?

I'm new to model category theory, and I find myself confused about the different meanings of cofibrant replacement in literature. The usual definition is that we assign to every object $X$ in our ...
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1answer
24 views

Caratheodory: Alternative Definition

Idea My idea is to facilitate Caratheodory's construction by composing it with Hahn-Kolmogorov. Problem Given a premeasure on a ring $\mu:\mathcal{R}\to\overline{\mathbb{R}}_+$. Do the ...
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2answers
51 views

What does it mean for a function to be holomorphic?

I am trying to wrap my head around the definition of holomorphic. Wikipedia tells me that: A holomorphic function is a complex-valued function of one or more complex variables that is complex ...
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1answer
27 views

Why Not Define Connectedness to Mean Path Connected?

All spaces I have seen which are connected are also path connected (apart from examples to show that the two are not equivalent). Is there a reason for using the weaker definition of connectedness ...
2
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1answer
42 views

What is a linear equation?

How do we define the linear equation? I mean, it looks like a polynomials with degree one but I'm not sure if $ax+by+c=0$ is a linear equation if $a=b=0$?
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3answers
39 views

Is modular arithmetic defined for all rational numbers (with denominators coprime to modulus)?

In the expression $\frac{1}{b}\pmod m$, where $(b,m)=1$, is $\frac{1}{b}$: a) a rational number (and so rational numbers are defined in modulo arithmetic using multiplicative inverses)? b) just ...
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0answers
33 views

What is the formal definition of a structure?

I can't seem to find one anywhere, and I've looked in several books and pages. I don't even know how to tag this.
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1answer
17 views

Cauchy - Sequences. Different definitons

Is $$\forall \epsilon > 0 \exists N \in \mathbb{N}: \forall j \ge N |a_j - a_N| < \epsilon $$ equivalent to $$\forall \epsilon > 0 \exists N \in \mathbb{N}: \forall m,n \ge N |a_m - a_n| ...
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0answers
10 views

anisotropic vector valued function definition

Does anyone have insight into the significance of the following statement which defines an anistropic function $$a:\Omega \times \mathbb{R} \times \mathbb{R}^{N} \rightarrow \mathbb{R}^{N}$$ The ...
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0answers
12 views

Help with understanding why a particular set was chosen in theorem and corollary involving the limit of a function

Here's a theorem from Ross' Elementary Analysis, he gives the limit definition of a function in terms of $\delta$-$\epsilon$ And the corollary that follows: Why is it that in the corollary, the ...
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2answers
27 views

What is effectively continuous?

In Soare's book Recursively Enumerable Sets and Degrees I saw a sentence: $\Phi_e$ is an effectively continuous functional from the Cantor space $2^\omega$ to itself. What does it mean for a ...
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0answers
17 views

Clarification of Definition: Free Algebra

I need some clarification on the definition of free algebra. Here is an extract from Lie Algebras and Lie Groups by Jean-Pierre Serre: I am somewhat confused about the definition of free algebra. ...
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15 views

trail vs path in Graph Theory v/s Graphical Models

In my course on probabilistic graphical models, I learnt (quoting from page 36 of the book Probabilistic Graphical Models: Principles and Techniques by the same author) Path: We say that X1 , . . . ...
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5answers
327 views

Seeking elegant proof why 0 divided by 0 does not equal 1

Several years ago I was bored and so for amusement I wrote out a proof that $\dfrac00$ does not equal $1$. I began by assuming that $\dfrac00$ does equal $1$ and then was eventually able to deduce ...
2
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0answers
31 views

Synonyms for “Theorem”

Some mathematical results, despite being formally proven, are not actually called "theorem". Examples include: Bertrand's postulate Pigeonhole principle Law of large numbers Do these names imply ...
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0answers
17 views

Why is an open interval needed in this definition? (definition of a limit of a function)

Here's a part of the definition Ross' Elementary Analysis states for limits of a function: In both parts of the definition, why are open intervals needed? Would it fail if it were a closed interval ...
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1answer
19 views

On the definition of transition maps

When defining a manifold the domain and codomain of the transition maps is usually denote like this: $$\varphi_\eta \circ \varphi_\lambda^{-1}: \varphi_\lambda(U_\lambda \cap U_\eta) \to ...
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0answers
21 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 ...
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1answer
66 views

What is Bourbaki's definition of subfield? or categorical definition of subfield?

Let $F$ be a field. Let $K$ be a subset of $F$ which is closed under two binary operations $+,\cdot$ Assume $(K,+,\cdot)$ is a field. Is $K$ called a subfield of $F$ in Bourbaki's definition? Or, ...
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1answer
24 views

What is a sparse subset?

In a work about fully homomorphic encryption I found usage of the expression: "sparse subset", as in: Our hint will consist of a set of vectors that has a (secret) sparse subset of vectors whose ...
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1answer
36 views

Examples of Free Lie Algebra

In wikipedia, free Lie algebras are defined using the universal property. Can anyone give some concrete examples of free Lie algebras? "In mathematics, a free Lie algebra, over a given field $K$, is ...
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8answers
956 views

Why is the absolute sign needed in the definition of a bounded function

A function $f$ is bounded if there exists a real number $M$ such that $|f(x)| \le M$ for all $x \in \operatorname{dom}(f)$. Why is the absolute sign needed?
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0answers
39 views

Analytic Functions: Notation? [duplicate]

Analytic functions are usually denoted by $\mathcal{C}^\omega$. What does the $\omega$ stand for? (The infinity symbols of a colleague of mine really look like omegas...)
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0answers
30 views

Definition of orientation preserving linear map: is this welldefined?

I am reading a definition in this document here about what it means for a linear map to be orientation preserving: It seems to me though that this is not well-defined: take the unit $x$-vector ...
2
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1answer
51 views

On the definitions of $n$-manifold etc.

I'm doing 3rd year undergraduate geometry (an introductory subject), and we've been given formal definitions of terms like "$n$-manifold" and "smooth $n$-manifold." However, I tend to think about ...
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3answers
53 views

When finding the derivative using its definition, why do we plug $0$ for $h$?

If $\lim h\to 0$, when finding the derivative of the function, why do you plug in the limit that is being approached. Like why would you plug in $0$ in the function $4x+2h$ (which is the derivative of ...
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2answers
40 views

What is the polarization identity?

Hi I am studying stochastic calculus and my professor often mentions "Polarization Identity" but I do not know how it is defined. I tried googling it but could not find the right definition and ...
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0answers
17 views

Two definitions of “affine stratification”

I see two different definitions of "affine stratification" in the literature: A stratification where each stratum is isomorphic to $\mathbb{A}^n$ for some $n$. A stratification where each stratum is ...
2
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2answers
39 views

Is the angle between a vector and a line defined?

Is the angle between a vector and a line defined? The angle between two lines $a,b$ is defined as the smallest of the angles created. The angle between two vectors $\vec{a},\vec{b}$ is the ...
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0answers
37 views

What is the definition of boundary-parallel Dehn twist?

I have not been able to find a working definition for the term: "boundary-parallel Dehn twist ". I know what a boundary-parallel surface is, and what a parallel surface is, but I have not been able to ...
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0answers
22 views

Differences between the definition for direct and inverse images.

I am reading a book and it introduces direct and inverse images with the following definitions: Given a function $f:X \to Y$ and subsets $A \subseteq X$ and $B \subseteq Y$, the direct image of $A$ ...
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2answers
35 views

Concept: The graph component

I have the following definition for a Component of a graph: A subgraph $H$ of a graph $G$ is a component of $G$ if $H$ is a maximal connected subgraph of $G$, ...
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1answer
35 views

Characteristics of $f(x,y)=\sqrt{xy}$

$f(x,y)=\sqrt{xy}$ (i) Determine the maximal domain of $f$, thus that the range is real. (ii) Is $f$ continuous its domain? (iii) Determine and describe all Level sets of $f$ To (i) $D:=\{(x,y)\in ...
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1answer
26 views

Examples of the Geometric Realization of a Semi-Simplicial Complex

I am reading The Geometric Realization of a Semi-Simplicial Complex by J. Milnor and here are the definitions: I find it difficult to visualize without specific examples. Can anyone help to provide ...
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3answers
118 views

What is in clear mathematical terms the definition for a sequence of integers, to be called *random*?

Sequences of integers might be ordered, totally ordered,... For all such attributes we find definitions in clear mathematical terms. (1) But what is in clear mathematical terms the established and ...
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2answers
601 views

Is the Euler characteristc defined wrong? If not, why not?

Ever since learning that $$\chi(S_0\# S_1) = \chi(S_0)+\chi(S_1)-2$$ (where $\chi$ denote the Euler characteristic), I've wondered whether $\chi$ isn't "defined wrong." If we let $\chi' = 2-\chi,$ ...
3
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1answer
51 views

On the Definition of multiplication in an abelian group

In class we had the following Definition: Let $(A,+)$ be an abelian Group with $a \in G$. We define: $$na:= \begin{cases}na, \ \forall n \in \mathbb{N} \\ |n|(-a), \ \forall n \in ...
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0answers
35 views

What does it mean if $P^n$ is irreducible for every $n\in\mathbb{N}$?

If $P$ is the transition matrix belonging to a markov chain, then what does it mean that $P^n$ is irreducible for every $n\in\mathbb{N}$? For $n=1$ it means that all states communicate with each ...
3
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0answers
42 views

Conceptual question on showing properties of the absolute value function on $\mathbb{Q}$

I have a rather conceptual question about showing certain small lemmas regarding the absolute value function on $\mathbb{Q}$. I want to only give one example: Let $a,b \in \mathbb{Q}$ and $|.|$ ...
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3answers
45 views

Contrapositive of a Definition

I have a problem (in real analysis class) that states "What is the contrapositive of the definition of "closed"?" The definition in our class of closed is: "a set E is closed iff the set contains all ...
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3answers
4k views

Do we have negative prime numbers?

Do we have negative prime numbers? $..., -7, -5, -3, -2, ...$
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0answers
11 views

What is “atomistic distribution”?

Could anyone give me some definition or explanation of "atomistic distribution"? I found a definition of "atomless distribution" by googling. It says: "Atomless distribution is any distribution ...
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0answers
15 views

Set definition with tuple [closed]

I am defining a set of tuples as "" I would like to mention "c" is constant and all tuples' "d" are between "d1" and "dN". Could somebody help me how to write it in set theory definition ?
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2answers
28 views

X and Y have the same cardinality if and only if bijection from X to Y? [duplicate]

My textbook says "Let X and Y be sets. We say X and Y have the same cardinality if there is a bijection f: X --> Y." I was wondering why the text does not say "if and only if." A bijection implies ...
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1answer
14 views

Codomain confusion

I'm confused about the codomain of a linear transformation. If we have a linear transformation which maps from $\mathbb{R}^n$ to $\mathbb{R}^m$ and the range of the linear transformation is only the ...