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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26 views

Approximation Theory: Iterative Methods

Can someone explain to me the general idea of what's going on? I don't get how the functions are being formed, nor do I know why we redefine $h(x)$ to ensure convergence after we defined $g$ ...
5
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2answers
118 views

If a finite poset has greatest and least elements, is it a lattice?

Let $P$ be a finite partially ordered set with elements $0$ and $1$ such that $0 \le x \le 1$ for any $x \in P$. Does it follow that $P$ is a lattice? If not, what is a counterexample? I believe this ...
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2answers
59 views

What is the definition for holomorphic functions on the Riemann sphere?

I'm trying to study complex analysis in general setting, but i have troubles with defining things in this general setting. I have skimmed definitions in Ahlfors, Conway and wikipedia, but these all ...
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3answers
420 views

Definition of set.

A set is defined as a collection of distinct objects. Why have we defined a set to contain only distinct objects? Why is a collection of objects which may have identical objects not called a set? ...
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1answer
38 views

Equivalent definitions for strictly positive elements

We have two usual definitions for strictly positive elements in C*-algebras: Let $A$ be a C*-algebra Definition (a) [MURPHY, C$^*$-algebras and Operator Theory] An element $a\in A_+$ is said to be ...
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424 views

What is the definition of a positive integer?

I am reading the book "The Number-System of Algebra (2nd edition)". At the starting of page-4 the author writes: A positive integer is a symbol for the number of things in a group of distinct ...
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1answer
67 views

What is this metric called?

Ahlfors -complex analysis p.20 Consider a stereographic projection between the 2-sphere and $\overline{\mathbb{C}}$ (i.e. one-point compactification of $\mathbb{C}$) Let $z,w$ be complex numbers. ...
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1answer
42 views

A question on Holder spaces

A function $f$ is said to belong to the Holder space if Holder condition is satisfied, i.e. $\exists \beta,L\geq0$ such that $$|f(x)-f(x')|\leq L|x-x'|^\beta$$ for all $x,x'$ in the domain of $f$. ...
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1answer
24 views

Why do we define product of morphisms in this way

I'm always forget the definition of product of morphisms in a category, maybe the main reason is because I don't know the motivation beyond the definition: I need help to see this abstract ...
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3answers
29 views

Can someone explain me how to read members of a set to prove uncountability?

I have trouble in understanding what the elements in the following set are: $$ V = \{f:\mathbb{N}\to \mathbb{N} \mid \text{there is $N_f \in \mathbb{N}$ so that $f(x) \le N_f$ for all ...
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0answers
60 views

what is the argument of 0?

When $z\neq 0$, $\arg z$ is defined to be the set $\{\theta \in \mathbb{R} : z=|z|e^{i\theta}\}$. What if $z=0$? Usually does one leave the argument of $0$ undefined? Or is $\arg 0 = \mathbb{R}$?
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1answer
52 views

Embedding and monomorphism

What is the difference between an embedding and a monomorphism? As far as I can see, most introductory abstract algebra texts treat them as if they are the same, i.e. an injective function from one ...
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1answer
47 views

How to understand structure groups?

I'm studying fiber bundles and I'm somewhat confused on how Structure Groups appears. The definition of fiber bundle I have is the following: A bundle is a tuple $(E,B,\pi)$ where $E,B$ are ...
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2answers
70 views

Limits to infinity?

As a part of homework, I was asked What does $\lim_{x\to a} f(x)=\infty$ mean? In an earlier calculus class I was taught that in order for $L=\lim_{x\to a}f(x)$ to exist, we need that ...
5
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3answers
171 views

What precisely is a vacuous truth?

Is there a proper and precise definition that goes something like this? Definition. A statement $S$ is a vacuous truth if ... ...
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1answer
37 views

$\epsilon$ rules for uniform and pointwise convergence.

Could someone please provide with the $\epsilon$ definitions of uniform and pointwise convergence. I'm trying to really get my head around the differences between them (I do know the differences, but ...
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3answers
94 views

Can I create my own function like Trigonometric or Exponential

When I want to solve mathematical problems, most of the time I meet the following functions Algebraic like polynomials. Trigonometric like sin(), cos(), tan(), cot(). Logarithmic like log(). ...
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2answers
100 views

How does the epsilon-delta definition define a limit?

I understand what the epsilon-delta definition is saying in regards to the distance from a point c and the distance from your limit, but I don't understand how this defines a limit. Any help is ...
3
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1answer
29 views

Linear function definition

I'm trying to figure out what is linearity and what is a linear function. But the wikipedia page confused me. Firstly it defines as polynomial : $f(x) = ax +c$ Than it defines as linear map: ...
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4answers
82 views

Vertical bar notation: $\frac{d}{dt}|_{t=0}f(a+tv)=$?

The definition of the directional derivative is $D_vf(a)$=$\left.\dfrac{d}{dt}\right|_{t=0}f(a+tv)=\lim \limits_{t\to0}\dfrac{f(a+tv)-f(a)}{t}$. However, I do not understand how this bar notation is ...
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4answers
264 views

What is the *exact* definition of a bounded subset in a metric space (in relation with the Heine-Borel Theorem)?

I see quite a lot of different definitions of a bounded space. For instance, from nLab: Let $E$ be a metric space. A subset $B⊆E$ is bounded if there is some real number $r$ such that ...
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1answer
63 views

A set with a supremum and an infinum inside

What is the name of a set $A$ that has its infimum $i \in A$ and a supremum $s \in A$? Examples: $[0,1]$ has a supremum $1$ and an infimum $0$ which are inside $[0,1]$. $\{0,1,2,3\}$ has a supremum ...
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4answers
94 views

How is the word “contains” defined in set theory? (In relation with neighborhoods in topology).

From Wiki: Some basic sets of central importance are the empty set (the unique set containing no elements) Thus, this make me think that "contained" is equivalent to the $\in$, as in: if $a$ is ...
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1answer
47 views

Clear definition of degeneracy of a graph.

There are at least two questions on this topic but the answers are not clear to me and WiKi link didn't make it any clear either. Could someone please clarify is the degeneracy of a graph $G$ ...
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1answer
70 views

What is the definition of “Winning Strategy” in an Ehrenfeucht-Fraïssé game?

I've read many descriptions and applications of a Winning Strategy, as much as many for a Strategy tout court, but when a formal, algebraic definition is called upon, I've found close to no input. I ...
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2answers
36 views

Confused about class definition

I find this in my Set theory material: [0] = {x:0==x(mod2)} = {x:2|0-x} , where I'm replacing equivalence sign ("=" with extra horizontal line) with double ...
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0answers
71 views

Are the words “function”, “map”, and “mapping” synonymous?

Is it correct to say that "A function or a map or a mapping is a binary relation such that ..."
3
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2answers
63 views

Should “together with” be taken as slang for an n-tuple?

When an algebraic structure is defined, it is often defined as a set $S$ "along with"/"together with"/"having" operations $\circ_1, \circ_2, \ldots, \circ_n$, and "denoted" by $(S, \circ_1, \circ_2, ...
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1answer
35 views

What can you tell me about integrable functions and riemann integrals?

Define the concepts of integrable function and Riemann integral for functions of two variables (across a rectangle and over an arbitrary area). I know how to define for a rectangle but not an ...
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0answers
37 views

About definition of “extended absolute value” (in $\overline{\Bbb{R}}=\Bbb{R}\cup\{+\infty,-\infty\}$)

Is correct following definition? Def.: Let be $a \in \overline{\Bbb{R}}=\Bbb{R}\cup\{+\infty,-\infty\}$, $$|a|=\begin{cases} |a|& \text{ if } a\in \Bbb{R} \\ +\infty & \text{ if } a \in ...
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1answer
29 views

Analytic, never zero on domain D graph and its local max. min

If $f$ is analytic and never zero on domain D, then $|f(z)|$ has no local minimum or maximum. Now, If I use the fact that $|f(z)|\leq$ max $|f(z+re^{it})|$ ($0\leq t\leq 2\pi$), then I can show ...
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2answers
53 views

Definition of subalgebra

Is it generally implicitly assumed that if $B$ is a subalgebra of a unital Banach algebra $A$ then $1 \in B$? I tried to find a definition of subalgebra but the only definition I found was in Murphy ...
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1answer
18 views

'Union' of maps

Let $f : A \to Y$, $g : B \to Y$. Suppose that $f(x) = g(x)$ whenever $x \in A \cap B$. Define $$ h : A \cup B \to Y, \\ h(x) = \begin{cases} f(x) & \text{ if $x \in A$} \\ g(x) & \text{ if ...
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4answers
65 views

Definition of Normalized Number

Which is correct? Are they both correct? Definition 1 A floating point number is called normalized if the leading digit of the fraction is nonzero. for example $(0.10101)_{2}\times 2^{3}$ is ...
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2answers
47 views

What is the easiest way to prove by induction?

Is there any easy way to do this? I get the basic step.. where you prove it for some number.. but I don't get the induction step. Do you literally take the given equation that you just proved with ...
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3answers
480 views

What is category theory?

I have read the page about category theory in wikipedia carefully, but i don't really get what this theory is. Is category theory a content in ZFC-set theory? (Just like measure theory, group theory ...
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2answers
43 views

Limit Point of a Set

Definition. A point $x$ is a limit point of a set A if every $\varepsilon$-neighborhood of $x$ intersects the set A in some point other than $x$. I understand the definition in that $x$ is our limit ...
0
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1answer
92 views

Is the value of the sum $1-1+1-1+1-1+\cdots$ does not exist? [duplicate]

Let $a_k:=(-1)^k$ where $k\in\mathbb{N}$. $\mathbb{N}$ is the set of all non-negative integer. And we define the partial sum $S_n:=\sum \limits_{k=0}^{n}a_k$. Notice that the sequence $\{S_k\}$ ...
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2answers
40 views

Help with understanding the definition of operation

I'm having trouble understanding this excerpt from Wikipedia, which defines an operation: Mainly, I don't understand what is meant by $V \subset X_1 \times...\times X_k$. Why does an operation ...
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1answer
38 views

Definition of an $n$-tuple agreeing with the Kuratowski's definition of an ordered pair

Is there a nice and elegant definition of an $n$-tuple ($n$ is a nonnegative integer) in ZFC, which will at the same time agree with the Kuratowski's definition of an ordered pair, i.e. $\left ( a,b ...
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0answers
31 views

Depth in acyclic graphs

I am struggling to understand a definition in a paper: Given a acyclic (directed!) graph $D=(V,E)$ we define a sequence $Q_i \subset V(D)$ of sets: $$Q_0 = \emptyset,$$ $$ Q_i \textrm{ is ...
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0answers
36 views

Solvable Group, which Quotients need to be Abelian?

In Wikipedia it says a group $G$is solvable if it has a subnormal series $\{e\}=G_1\lhd G_2,\dots \lhd G_n=G$ where $G_i$ is a normal subgroup of $G_{i+1}$ and all the factor groups are abelian. My ...
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2answers
55 views

How do i define 'complex rational function'?

http://en.wikipedia.org/wiki/Rational_function I don't get the definition in wikipedia. It would be great to define "complex rational function" with the domain $\overline{\mathbb{C}}$, namely ...
3
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1answer
26 views

which is the usual definition of Argument?

Let $z\in\mathbb{C}\setminus\{0\}$ Then, there exists a unique $\theta \in [0,2\pi)$ and $\phi\in(-\pi,\pi]$ such that $z=|z|e^{i\theta}=|z|e^{i\phi}$. Between $\phi$ and $\theta$, which is the ...
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In the definition of NP, is it required to have polynomially bounded length of certificate?

So given the definition in our lectures, we were told that NP is defined as the set of languages $L$ s.t. there exist a polynomial time bounded Turing-acceptor M s.t. $L ={w: M accepts(w#c) for some ...
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0answers
42 views

Particular “upper bound” and “lower bound” of sequence

Is it possible to define the following? Let $a:\Bbb{N} \to \Bbb{R}$, and $M \in \Bbb{R}$, $M$ is ? if $$\exists n \in \Bbb{N}(\forall p \in \Bbb{N}(p\geq n \to a(p)\leq M))$$ Let $a:\Bbb{N} \to ...
2
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2answers
28 views

Does the definition of “mean”/“average” require the result to be in the domain set?

If I have a function that calculates the mean value of a set of elements that is an arbitrary subset of some set $X$, does the mean, by definition, have to also be in $X$? (In other words, if the mean ...
0
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1answer
47 views

Fiding a derivative

I need to find the derivative of $\sqrt{x^2+3x}$ using the definition of derivative. e.g. $\frac{f(x)-f(a)}{x-a}$ as x->a. Normally I get these but the $x^2$ is messing me up. I am at $$\lim ...
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2answers
87 views

Total/Gaussian curvature is intrinsic, yet mean curvature is extrinsic, why?

What characteristics define the total/mean curvature to be intrinsic/extrinsic accordingly? What is different geometrically about these curvatures that cause them to be defined as this?
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1answer
24 views

Question about measure on set that is not in $\sigma$-algebra

I think I have problem with badly written book, or I just can't understand statement. Let $(X,\mathcal{A},\mu)$ be any measure space and $\mu^*$ be outer and $\mu_*$ inner measure inner measure ...