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

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1answer
44 views

Want to find the operator norm of a simple matrix, not sure which definition to use

I want to find the operator norm of $A = \begin{bmatrix} 2 & 0 \\ 0 & -3 \end{bmatrix}$ My prof defines the operator norm as $\|A\| = \max_{\|x\| \leq 1} \|Ax\|_2$ In the problem ...
4
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3answers
85 views

Question About Definition of Almost Everywhere

I suppose I'm a bit confused about the definition in the following regard: A property holds a.e. if it holds everywhere except for a set of measure $0$. Now, if the particular property is only ...
0
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2answers
81 views

Arc length definition

I've always felt that the notion of 'area under the curve' is more precise than 'length of a curve', because of the Riemann integral and the reasoning used there: First, we assume the quantity we ...
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0answers
24 views

What is the norm on the space $L^{\infty}([0,T];L^1(\Omega))$

The title is my question, so What is the norm on the space $L^{\infty}([0,T];L^1(\Omega))$
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1answer
56 views

Definition by Recursion - Need for Rigour

Suppose you define the factorial $n!$ by \begin{align} \tag{1}0!&:=1,\\ \tag{2}(n+1)!&:=(n+1)n!. \end{align} Consider the following argument showing that $n!$ is a uniquely defined function. ...
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2answers
34 views

On the alleged circularity of the definition of implication and double implication

Note: I'll use the convention adopted by my professor: "$\rightarrow$" is the symbol for implication and "$\implies$" for deduction. Our professor defined implication as The connective such that ...
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1answer
77 views

What is a norm topology in functional analysis?

I am currently reading up about norm topology, I have a background in functional analysis but I do not know anything about topology, aside from that topology is a collection of open sets with some ...
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2answers
79 views

Upper/lower sum defintion (spivak)

The first (and only) answer to this question: Defintion of Upper & Lower Riemann Sum reminds us of the definition of Upper and lower Riemann sums. My question is about the usage of infimum and ...
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1answer
26 views

Does finite automata take an alphabet or language as input?

I know this is a bit pedantic but I'm curious of the definition of finite automata. In Theory of Computation Sipser states The formal definition says that a finite automaton is a list of those ...
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0answers
40 views

In the definition of an asymptotic function

Recall definition: A series $\sum _{r=0}^{\infty}a_{r}z^{r}$ is said to represent a function $f(z)$ as $z \rightarrow 0$ in a sector $S \subset \mathbb{C}$ around zero if: \begin{eqnarray} z^{-m} ...
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13answers
4k views

What is the definition of a set?

From what I have been told, everything in mathematics has a definition and everything is based on the rules of logic. For example, whether or not $0^0$ is $1$ is a simple matter of definition. My ...
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1answer
23 views

Why tuple in measurable space definition?

Measure Theory for Dummies says First, we need something to measure. So we define a “measurable space.” A measurable space is a collection of events B, and the set of all outcomes Ω, which is ...
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0answers
20 views

Are There General Terms For Operands In An Equation?

I am curious if there are general terms for operands in any equation. For example, these are the terms for the basic operations: Addition: augend + addend = sum ...
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2answers
27 views

Why is $1$ not an irreducible integer?

When defining irreducible integers, we restrict our attention $Z = \{n \in \mathbb{Z} \; | \; |n| > 1\}$. Then we say that for some $p \in Z$, $p$ is irreducible if for some $a,b \in \mathbb{Z}$: ...
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2answers
34 views

Matrix Norm Definition

I don't understand the intuition behind the definition of a matrix norm: $\displaystyle \|A\|_p = \max_{x\neq0}\frac{\|Ax\|_p}{\|x\|}$. Why is the arbitrary vector $\vec{x}$ included in the ...
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1answer
52 views

Katok/ Hasselblatt: Attractor of dynamical system: How to understand this remark?

In "Introduction to Modern Theory of Dynamical System" by Katok and Hasselblatt, the following definition of attractor is given: Definition 3.3.1 A compact set $A\subset X$ is called an ...
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1answer
68 views

Linear span of general set in topological linear spaces

I'm studying Functional Analysis and I'm in doubt with the definition of linear span. The book states that: Let $\mathscr{L}$ be a topological linear space and let $\mathscr{M}$ be a linear ...
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2answers
103 views

Definition extension of $e$ to the reals

I have read proofs that the sequence $a_n=\left( 1+\frac{1}{n}\right)^n,n\in\mathbb{N^*}$ converges and its limit is defined to be $e$. How is this definition of $e$ extended to the real numbers? ...
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1answer
29 views

Continuity Help

If $\lim_{h \to 0}[f(x+h)+f(x-h)-2f(x)] = 0$ for every $x \in \Bbb R$, does it follow that $f$ is continuous? I start by rearranging it to be $\lim_{h \to 0}f(x+h)+f(x-h)=\lim_{h \to 0}2f(x)$ and I ...
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2answers
48 views

Can a sample space with well defined elements be assigned for every probability experiment? [closed]

Can a sample space with well defined elements be assigned for every probability experiment, considering the definition of sample space as-'the set of all distinct elementary events which are all ...
1
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3answers
29 views

Justifying the analytic definition of a line segment

The definition of a line segment $L$ for $x$ and $y$ in a vector space is $$L = \{\lambda x+(1-\lambda)y : \lambda \in [0,1]\}.$$ I had trouble seeing this, so I considered the basic case of ...
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0answers
18 views

Definition of remainder clarification and application to a proof.

I am in a number theory class which does not have a text and the following definition of REM is giving me trouble. That is, "We will denote by $aREMm$ the remainder left when $a$ is divided by $m$ ...
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2answers
62 views

How do you show that the norm for square matrices is submultiplicative?

In these notes: http://www.math.usm.edu/lambers/mat610/sum10/lecture2.pdf It says that square matrices satisfy so called submultiplicative norm $\|AB\| \leq \|A\|\|B\|$. Is it by definition or is ...
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2answers
65 views

What is the precise definition of the ambient space

For instance, what is the ambient space of a singleton $\{x\}$, where $x \in \mathbb{R}$? Can it be the singleton itself? $\mathbb{R}$? $\mathbb{R}^n$? or some arbitary set that happens to contain ...
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1answer
76 views

Can a arithmetic progression have a common difference of zero & a geometric progression have common ratio one?

How many three digit numbers have the property that their digits taken from left to right form an Arithmetic or Geometric Progression? Eg. 123 is in form a AP when the digits are taken from left to ...
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3answers
81 views

Topological definition of continuity (open set characterization)

I want to demonstrate the topologic definition of continuity, using the classical definition with epsilon's and delta's. So we have that: Classical definition - If the function $f:X \rightarrow Y$ ...
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1answer
23 views

Is this backslash a typo here in showing a function?

"Then there exists a homogeneous function $\rho\in C^\infty(R^n \backslash\{0\},R^n)$ with negative degree of homogeneity which specifies the following properties:" - Pablo Monzon (2006), Almost ...
0
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1answer
19 views

What is the motivation of definition of quadratic integer rings for $D=1\mod(4)$

I'm self-studying a ring theory and have a question. Why do we define a quadratic integer ring differently for the case $D=1\mod(4)$? Why don't we just say that $O_{Q[\sqrt{5}]}=\mathbb{Z}[\sqrt{5}]$ ...
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0answers
75 views

What is the meaning of 'que' in math?

As part of a lengthy mathematical proof on density functions, part of the text says: We know that given $\{x_n\}_{n\in N} \subset R $ such that que $ x_n\to z\in R^c$ , we have that the ...
0
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3answers
58 views

Exponentiation with rationals - why algebraic?

Why is exponentiation with rational numbers considered an algebraic operation? I get why exponentiation with integers is since that's just a finite number of applications of multiplication, but this ...
2
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1answer
27 views

two definition in automorphism of group

Let $G$ be finite p-group and $\sigma \in Aut(G)$ (automorphism group). what does below symbols mean 1. $[G,\sigma]$ (commutator) 2. $C_G(\sigma)$ (centralizer)
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1answer
26 views

What does it mean to prescribe a function on a curve?

I have been trying to find the meaning of "prescribing" a function on a curve. This is from some study notes on PDE's: We can then index $Γ$ by a parameter $s: Γ := {(x, y) = (ϕ(s), ψ(s)) : s ∈ I ...
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0answers
26 views

Clarification on the definition of a classical solution of a pde

I just wanted to clarify the definition of a 'classical solution' to a PDE. It seems we want the solution 'of order' $k$ to be $k$ times continuously differentiable. This is what we refer to as a ...
2
votes
1answer
99 views

Alternate limit definition

I came up with an alternate definition of limit, which I would like to verify it is equivalent to the usual one. A sequence $(a_n)$ has a limit $L$ if, for any $a<L<b$, there are only ...
3
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7answers
529 views

Lemma, theorem, corollary… which one is a suitable term for an observation?

I have an observation from say theorem of "the uniquness of interpolating polynomial". And I tried to prove that observation. However, I donot know whether anybody else has made that observation. And ...
0
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1answer
87 views

a very basic definition cannot be found in most abstract algebra books [duplicate]

I am looking for a precise definition of what is meant by an "algebra". The reason is that I want to compare the difference between an "algebra" and other algebraic structures, i.e. a group, ring, ...
0
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1answer
28 views

How do I extend RS-integral to bounded variation parameters?

Definition of Riemann-Stieltjes integration Let $\alpha:[a,b]\rightarrow \mathbb{R}$ be a monotonically increasing function and $f:[a,b]\rightarrow \mathbb{R}$ be a bounded function. ...
2
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2answers
74 views

Is “probability distribution function” a distribution?

I can understand the definition of distribution as written in https://en.wikipedia.org/wiki/Distribution_(mathematics) On the other hand there are three different terms in the definition of ...
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1answer
73 views

The definition of a torus [closed]

Have seen the term torus used almost in any article group-theory/algebraic geometry. However, with no definition (presumably it is well known entity) Nevertheless couldn't find explanation or ...
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4answers
52 views

Clarification on meaning of “Gaussian random variable”

When my lecturer uses the word Gaussian random variable, he always writes the pdf of the Gaussian instead of the random variable itself. For example, given a random variable $X$ Gaussian, $f_X(x) ...
2
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1answer
59 views

Why Fourier transform is tempered distribution?

This question refers to the question Fourier transform in $L^p$ In the answer it is clearly written that "If $s > 2$, this question quickly becomes technical and requires the theory of ...
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11answers
2k views

What is exactly the difference between a definition and an axiom?

I am wondering what the difference between a definition and an axiom. Isn't an axiom something what we define to be true? For example, one of the axioms of Peano Arithmetic states that $\forall ...
0
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1answer
20 views

Definition of even functions for n dimensions

Is there a generalisation of even functions for functions with multiple variables? If so, what are some concrete examples of the use of this definition?
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7answers
3k views

Why does $e$ have multiple definitions?

The number $e$ seems to have multiple definitions: $$\lim \limits_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$$ The unique number $a$ such that $\int_1^a\frac{1}{x} \, dx = 1$ The unique number ...
0
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1answer
38 views

What's the difference between the terms “Numeral System” and “Number System”?

I understand that a "Numeral system" is a way to express numbers. But what is "Number system"? What's the difference between this two terms?
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2answers
57 views

Confused about definition of quotient map

On Wikipedia it says A map $f: X\to Y$ is a quotient map if it is surjective, and a subset $U$ of $Y$ is open if and only if $f^{-1}(U)$ is open. So by definition, if $f^{-1}(U) \subset X$ is ...
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3answers
283 views

Product of two ideals

I am trying to understand what the meaning of product of ideals is. From this site: http://commalg.subwiki.org/wiki/Product_of_ideals I have figured out that it should be: $$ IJ= \sum_{i = 1}^n (a_i ...
2
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1answer
62 views

Closure of a pre compact subspace is compact?

Suppose we have a metric space $(X,d)$ We say that $Y\subset X$ is pre compact if every sequence in $Y$ has a convergent subsequence. Notice that generally the subsequence might not converge in $Y$. ...
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2answers
62 views

Definition of “in terms of” for a constant vs a variable

Say I have a question that says: "answer the question in terms of x" where x is a variable. vs. "answer the questions in terms of n" where n is any constant. What is the difference between ...
3
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2answers
103 views

Strange definition of real differentiable function

I am currently refreshing my complex analysis knowledge (I had a course many years ago, but I've forgotten almost all of it). The German textbook I'm using, which is Funktionentheorie by Fischer & ...