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

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3
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2answers
198 views

Math and Origami

I am working on a project for class about the mathematics behind origami and write now I am looking into what is and is not constructible. I've gotten to the definition of origami constructible points ...
2
votes
1answer
218 views

Can we say that each monotonic sequence is a bitonic?

Is it mathematically correct to state that every monotonic is actually a bitonic with 0 number of elements in other half? I.e., if it is increasing then we can say it has 0 elements in decreasing ...
1
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2answers
60 views

A single word to represent a sequence of repeating numbers

I asked this question on english.stackexchange too. What is a noun to represent a sequence of repeating numbers? For example: 777777777
2
votes
1answer
32 views

Generated equivalence relations in logics

Let $L$ be some logic (FO or stronger which is not important for this purpose). Given a $\tau$-structure $A$ and a formula $\varphi(x_1, \dots x_n) \in L[\tau]$ with free variables $x_1, \dots, x_n$. ...
1
vote
1answer
46 views

Notation of double-sided infinite sum

The notation $\sum_{k=1}^\infty a_k$ always means $$\lim_{n\rightarrow\infty}\sum_{k=1}^n a_k.$$ What about $\sum_{k=-\infty}^\infty a_k$, such as in the Laurent series? Does it always means ...
0
votes
1answer
120 views

Compact Operators and Complete Metrics Spaces

I have a couple of questions about compact operators and compactness in complete metric spaces: 1.I have the following implications: Let $Y$ be a metric space with $A$ a subset of $Y$. $A$ is ...
0
votes
1answer
91 views

Definition of continuity without limit

Without limits, I define $f(x)$ is continuous at $x=a$, when it: $f(a)$ exists; For every $d>0$, in the close interval $[a-d,a+d]$, there exist a maximum $M$ and a minimum $m$; For every ...
6
votes
2answers
195 views

Definition of $H_\lambda$ (hereditary cardinality)

It seems to me that the definition of $H_\lambda$ (the set of sets of hereditary cardinality less than $\lambda$) on the web page at Cantor's Attic is not quite correct. From the page: ...
6
votes
1answer
2k views

Difference between Analytic and Holomorphic function

A function $f : \mathbb{C} \rightarrow \mathbb{C}$ is said to be analytic in an open set $A \subset \mathbb{C}$ if it is differentiable at each point of the set $A$. The function $f : \mathbb{C} ...
3
votes
2answers
137 views

Existence of differential form on a manifold

I have a fundamental question about the existence of differential forms on manifolds. A $k$-form on a manifold in local coordinates looks like $f(x_1,...,x_n)dx_{i_1}...dx_{i_k}$, where $f$ is a ...
1
vote
0answers
71 views

Understanding a.. weird definition

I came across the following definitons: Let $n$ be a positive integer and $\zeta_n$ be a primitive $n^{th}$ root of unity. For any prime $p\nmid n$, let the degree of the extension ...
1
vote
2answers
177 views

Definition of a $n$ - form on a manifold

I am confused about the definition of a differential form on a manifold. The definition I have comes from Bott and Tu and is as follows: A differential form, $\omega$, on a manifold $M$ is a ...
2
votes
1answer
39 views

definition clarification in graph theory

I was studying about Almost Self-Centered Graphs (ASC). ASC graphs are introduced as the graphs with exactly two non-central vertices. Of course, the remaining two vertices are diametrical. My doubt ...
4
votes
1answer
79 views

Definitions of adjoints (functional analysis vs category thy)

If I have a linear operator $f$ on a Hilbert space, then I define the adjoint of $f$ to be $f^*$ where, $(fx,y)=(x,f^*y)$ for all $x,y$. I am confused because this definitions is very different to ...
2
votes
2answers
105 views

Clarification on limit definition.

I know that in the definition of limit they say that if $\ \ \forall \varepsilon > 0\ \ \exists \ \delta > 0 \ \ $But why not $\ \ \forall \delta> 0\ \exists \ \varepsilon > 0$? Thanks ...
4
votes
1answer
84 views

Has anyone succeeded in formalizing the notion of a complete vector space? (Not using topological ideas).

In order theory, we have the concept of a lattice, which is defined as consisting of an underlying set $L$ together with two binary operations $\wedge$ and $\vee$. Now when $L$ is finite, the concept ...
1
vote
2answers
88 views

Proof that the derivative of a linear function is $0$.

We have a function $f(x)=x$ which is defined and is continuous on the set $S$ of all real numbers. The derivative at point $x$, $$f'(x)=\lim_{h\to 0} \frac{f(x+h)-f(x)}h$$ Using the theorems of ...
1
vote
1answer
114 views

How to tell if relation on set is a partial order when relation is defined as a set of ordered pairs?

Determine whether $R$ is a partial order on set the $S$ and justify your answer. $S=\{1,2,3\}$ and $R=\{(1,1),(2,3),(1,3)\}$. So the job is to consider if $R$ is reflexive, antisymmetric and ...
1
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2answers
27 views

A confusion regarding the definition of continuous functions.

Let us suppose $f:X→Y$ is a continuous, non-surjective mapping. Also assume that there is an open set $A\subset Y$ which contains points which are not in $f(X)$. What would $f^{−1}(A)$ be? Would it ...
1
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0answers
19 views

How do I state a reduction in cost?

I have developed an algorithm and am having a hard time stating its benefit versus a baseline. Suppose that the baseline cost of solving the problem is 1000 seconds. Now suppose that my algorithm ...
1
vote
0answers
91 views

$ |\sin (x) | \leq 1$ by it's definition

I was told that it can be shown that $|\sin(x)| \leq 1$ by its definition $\sin (z):= \frac{1}{2i} \bigl( (\exp(iz)-\exp(-iz)\bigr) $ I am aware that as soon as I choose $x \in \mathbb{R}$ and ...
1
vote
1answer
55 views

How does $\exp(0)=1$ follow from the definition $\exp(z):= \sum_{n=0}^\infty \frac{1}{n!} z^n$

We introduced the Exponential function as follows: $$ \exp: \begin{cases} \mathbb{C} & \longrightarrow \mathbb{C} \\z & \longmapsto \displaystyle \sum_{n=0}^{\infty} \frac{1}{n!}z^n ...
1
vote
2answers
212 views

How to define the Nabla-Operator

As I began to teach myself in differential geometry, I finally used to use the Nabla-Operator. I know and understand its usage as in $$ \nabla f := \left( \begin{matrix} \frac{∂f}{∂x_1} & ...
0
votes
1answer
53 views

What is the name of this theorem

Let $f: G\to K$ a morphism of groups. If $H\subset \ker f$, then there existe a unique morphism of groups $g : G/H\to K$ such that $f=gs$. Moreover, $g$ is surjective if $f$ is surjective ; $g$ ...
1
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0answers
46 views

Subgraph without “holes”

does everyone know, if there already exists a definition of subgraphs, which do not contain a "hole"? EDITED: That means: I presuppose a planar embedding of a graph G and I want to find a connected ...
62
votes
14answers
4k views

Are “if” and “iff” interchangeable in definitions?

In some books the word "if" is used in definitions and it is not clear if they actually mean "iff" (i.e "if and only if"). I'd like to know if in mathematical literature in general "if" in definitions ...
4
votes
0answers
86 views

Should $0$ be considered a prime? [duplicate]

Typically, a prime is defined as follows: $p$ is prime iff $(p \mid xy \implies p \mid x$ or $p \mid y)$ and $p$ is not a unit or zero. But for ideals, we say the zero ideal is prime. There is a ...
0
votes
3answers
243 views

Solution of a differential equation having a singularity (not everywhere defined) [closed]

Remind me about ordinary differential equations, whose solutions are not everywhere defined (have a singularity). I want to remember the exact definition of a solution with singularity, which I ...
1
vote
2answers
55 views

Matrix with Functions as Entries

What do we call a matrix with functions as entries? $$\textbf{f(x)}=\begin{bmatrix} f_{11}(x) & f_{12}(x) \\ f_{21}(x) & f_{22}(x) \end{bmatrix} $$
0
votes
1answer
127 views

Difference between a holomorphic and diffeomorphic function

A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain. What I interpret from this is ...
0
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0answers
55 views

definition of derived algebra $[L,L]$ of a Lie algebra $L$

Definition of derived algebra of a Lie algebra $L$ is given by linear span of commutators $[x,y]$ for $x,y \in L$. but here why do we take linear span and why cant we just consider collection of all ...
0
votes
1answer
34 views

Does the sum-of-divisors contain each divisor only once?

To calculate perfect numbers, is each divisor added to the sum-of-divisors only once or as often as it appears in the factorisation of the number?
-1
votes
1answer
110 views

Use the definition of functional limit to prove…

I'm kinda stuck on this problem: Use the definition of functional limit to prove that the limit as $x\to 0$ of $\frac{x}{|x|}$ does not equal $-1$. Here is the definition of a functional limit: ...
3
votes
4answers
1k views

What precisely is the difference between Euclidean Geometry, and non-Euclidean Geometry?

I was wondering, what it is precisely which defines the difference between Euclidean and non-Euclidean Geometry, in a few words/equations/diagrams? Would I be correct in understanding that ...
1
vote
1answer
105 views

Forward and backward composition in relational algebra

Forward composition: $p\,;q$ $$\forall p,q\cdot p\in S\leftrightarrow T\land q\in T\leftrightarrow U\implies\\p\,;q=\{x\mapsto y\mid(\exists z\cdot x\mapsto z\in p\land z\mapsto y\in q)\}$$ ...
0
votes
2answers
127 views

definition of separation axioms in topology

I am learning the Separation Axioms and came across the definition of regular space. In the definition they say, "Suppose the one point sets are closed in $X$" My question is: how can one point sets ...
1
vote
3answers
29 views

Prove that as $ \lim_{x \to a} f(x) = L, \,\text{then}\, \lim_{x \to a} |f(x)| =|L|$

Basically so far I have managed to break it down to the following : $|f(x)-l|< \varepsilon$ then $||f(x)|-|L||\leq|f(x)-L|< \varepsilon$ then squeeze $||f(x)|-|L||< \varepsilon$ I'm not ...
2
votes
1answer
99 views

Adjoint of a Matrix Definition

Tom M. Apostol in his book "calculus Vol. 2" page 122 (see image below) defines adjoint of a matrix as the transpose of the conjugate of the matrix. Is this definition always correct ? Does it agree ...
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0answers
89 views

$f: E \to \mathbb{R}^m$ is not continuous at $x_0$, show that $g:E \cup \lbrace x_0 \rbrace \to \mathbb{R}^m$ is continuous at $x_0$

Let $E \subset \mathbb{R}^k, f: E \to \mathbb{R^m} $ be a function. Let $x_0 \in \mathbb{R}^k$ such that $x_0 \notin E$ but note that $ \displaystyle \lim_{x \to x_0}_{x \in E}=y_0$ does exist. ...
1
vote
1answer
131 views

What sets are Lebesgue measurable?

I cannot detect the fallacy in the set of the following statements in my inconsistent notes: A sigma algebra is a set of the sets in the generating set closed under the set operations countable ...
5
votes
2answers
340 views

What are some physical, geometric, or otherwise useful interpretations of divergent sums?

Is there any practical application to discussing the 'sum' of sequences that are not convergent under Cesàro summation? Why would we want to assign a value to an otherwise divergent sequence and ...
0
votes
2answers
40 views

What is the intersection between the set of all expressions, of all equations and of all functions?

I am studying the definition of mathematical expression, of equation and of function and I want to draw a venn diagram with the intersection between the set of these objects. Some people say every ...
0
votes
1answer
70 views

Definition for $\lim(s_n)$ and $\limsup(s_n)$

Can someone provide me the definition of a (finite ) $\lim (s_n)$ and how it correlates to the definition of $\limsup(s_n)$? $\lim(s_n)=+\infty$ if $\forall M>0, \exists N=N(M)\in \Re$ ...
0
votes
1answer
176 views

Riemann Integral (Rudin)

I was reading Rudin's, "Principles of Mathematical Analysis", specifically the section about the Riemann Integral and I've ran into some "shaky" notation. Can someone just explain to me geometrically ...
1
vote
2answers
77 views

How do these two definitions of uniformly elliptic fit together?

Consider a bounded domain $\Omega\subset\mathbb{R}^2$. We defined that a semi-linear PDE of degree 2 (whereat $A=(a_{ij})$ is the coefficient matrix of the main part of the PDE and ...
1
vote
1answer
39 views

Is $S^0$ a manifold?

Consider a singleton space $\{x\}$, it is a manifold and it is locally euclidean as there is a homeomorphism to $\mathbb{R}^0$. However, consider $S^0=\{-1,1\}$ with the discrete topology, there does ...
0
votes
1answer
36 views

Holder Space series term

Holder space $C^{k,\gamma}(\bar{U})$ has the norm $||u||_{C^{k,\gamma}(\bar{U})} := \sum_{|\gamma | \leq k}||D^{\alpha}u||_{C(\bar{U})} + \sum_{|\gamma|=k}[D^{\alpha}u]_{C^{0,\gamma}(\bar{U})}$, where ...
0
votes
2answers
96 views

Different ways of defining Absolute Value

Calculus I presents this definition of absolute value: $$f(x)=y=|x|=\left\{\begin{array}{}\;\;\;x&\text{ if}\,\,\,x\geq 0\\-x&\text{ if}\,\,\,x<0\end{array}\right.$$ But you can also ...
2
votes
1answer
89 views

Acceptable Definition for $\sqrt{a}$?

Is this an acceptable definition for $\sqrt{a}$, where $a \in \mathbb{R}$? If $a\geq 0, \sqrt{a} = b \in \mathbb{R}$ s.t. $b\geq 0, b^2 = a$. I'm proving some theorems involving $\sqrt{a}$, in the ...
0
votes
1answer
296 views

Limit of Identity Function vs. limit of Squaring Function

$$\lim_{x\rightarrow a} x = a$$ and $$\lim_{x\rightarrow a} x^2 = a^2$$ $f(x)=x^2=x \times x$, i.e.: two identity functions. I'm a bit confused on how $x^2$ can be interpreted as being ...