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

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2
votes
2answers
107 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
85 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
89 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
116 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
vote
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
vote
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
57 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
219 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
vote
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
272 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
57 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
133 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
votes
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
112 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
2k 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
111 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
134 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
109 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 ...
1
vote
0answers
90 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
135 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
344 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
180 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
78 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
40 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
99 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
90 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
317 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 ...
0
votes
1answer
44 views

Computing $a^b$ as $\lim_{n\to\infty} a^\frac{\lfloor b \cdot 10^n\rfloor}{10^n}$

Let $b \in \mathbb{R}$, then $ \forall n \in \mathbb{N}(\frac{\lfloor b \cdot 10^n\rfloor}{10^n} \in \mathbb{R})$, but does $\frac{\lfloor b \cdot 10^n\rfloor}{10^n}$ have a particular name? And is ...
1
vote
0answers
217 views

Difference between “marginal variance” and “true variance”?

[This question is not about solving a mathematical problem, but about definitions; please tell me if there is a better forum to ask it.] In a paper I am reading, the author writes: ...the sole ...
2
votes
1answer
83 views

What needs to be linear for the problem to be considered linear?

Harry Altman presented an excellent question in a comment here: What needs to be linear for the problem to be considered linear? So is it enough to a have linear objective function or other ...
1
vote
2answers
213 views

What is a convex optimisation problem? Objective function convex, domain convex or codomain convex?

My teacher in the course Mat-2.3139 did not want to answer this question because it would take too much time. So what does a convex optimisation problem actually mean? Convex objective function? ...
1
vote
2answers
286 views

Why nonlinear programming problem (NLO) called “nonlinear”? What does “nonlinearity” actually mean? Is it “not linear” or something different?

My teacher in the course Mat-2.3139 presented the same definition as in Wikipedia for the nonlinear programming problem here but he did not specify what the nonlinearity actually means or what it ...
3
votes
2answers
161 views

What is compactification generally?

In wikipedia, compactification is defined as an topological imbedding $f:X\rightarrow Y$ such that $f(X)$ is dense in $Y$. However, Munkres-Topology requires $Y$ to be Hausdorff to be called a ...
7
votes
4answers
205 views

Is $0$ a composite number and $-1$ a prime number?

If in the set of natural numbers, all prime numbers $p$ have only two divisors, $1$ and $p$, and all composite numbers have at least three divisors, then can we also use these definitions for the set ...
0
votes
0answers
68 views

Definition of restriction of relation

let be $\mathcal{R}$ a binary relation on $X$, and $\mathcal{T}$ a binary relation on $Y$, $\mathcal{T}$ is restriction of $\mathcal{R}$ if: 1) $Y \subseteq X$ 2) $\forall a,b \in Y( a T b ...
1
vote
2answers
112 views

Meaning of $\{ a,b \}$, and comparison with $(a,b)$

What does $\{a,b\}$ mean in real analysis? I'm also little bit confused about set definition Can you tell me the main difference between $(a,b)$ and $\{a,b\}$? Thank you.
7
votes
5answers
230 views

The derivative of $e^x$ using the definition of derivative as a limit and the definition $e^x = \lim_{n\to\infty}(1+x/n)^n$, without L'Hôpital's rule

Let's define $$ e^x := \lim_{n\to\infty}\left(1+\frac{x} {n}\right)^n, \forall x\in\Bbb R $$ and $$ \frac{d} {dx} f(x) := \lim_{\Delta x\to0} \frac{f(x+\Delta x) - f(x)} {\Delta x} $$ Prove that ...
4
votes
5answers
228 views

Trigonometry confusion

I was doing a bit of trigonometry, as I have been for a couple of years and it suddenly dawned on me that I don't really understand the trigonometric functions, at all. You first learn the basic trig ...
0
votes
2answers
55 views

Confusion about the definition of a “loop” in a topological space

A continuous map $f:[0,1] \to X$ is called a path and if $f(0)=f(1)$ then it is called a loop. But any loop looks like a circle which is not a function as it is not well defined. How did it ...
0
votes
1answer
166 views

About definition of “ordered semi-ring”

I need the definition of "ordered semi-ring". Can I use these properties: $a \preceq b \to a + c \preceq b + c$ $0 \preceq a \wedge 0 ≤ b \to 0 \preceq a \cdot b$ (or: $a \preceq b \wedge 0 ...
0
votes
3answers
174 views

Definition of semi-ring homomorphism

I need the definition of semi-ring homomorphism. Thanks in advance!