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

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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
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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
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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 ...
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1answer
42 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
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0answers
212 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
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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 ...
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2answers
201 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? ...
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2answers
277 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
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2answers
160 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 ...
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4answers
195 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 ...
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0answers
63 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 ...
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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
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5answers
226 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
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5answers
221 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
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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
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1answer
160 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 ...
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3answers
166 views

Definition of semi-ring homomorphism

I need the definition of semi-ring homomorphism. Thanks in advance!
2
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1answer
58 views

Distinguishing characteristic and spurious properties

When working in set theory with definitions of somehow primitive notions like ordered pairs (Kuratowski) or ordinal numbers (von Neumann) one is supposed to use only the characteristic properties of ...
2
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0answers
72 views

What are defintions of ellipticity?

Ellipticity seems to have many definitions. So far, I'm aware of three. Are there other definitions of ellipticity that you know of, and if so, where did you encounter them? The three definitions ...
0
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1answer
136 views

What does it mean to be an “instance of a rewriting rule”?

What is the definition of the following statement? The rewriting rule $l_{1}\rightarrow r_{1}$ is an instance of another rule $l_{2}\rightarrow r_{2}$. PS:This statement comes from the paper of ...
2
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0answers
86 views

Clarification of Hölder norm in terms of oscillation

Let $\Omega\subset\mathbb{R}^2$ be an open bounded set, $B(x_0, \rho)=\{x\in\mathbb{R}^2\ |\ |x-x_0|\leq \rho\}$, $\Omega(x_0, \rho)\equiv B(x_0, \rho)\cap \Omega$, $u\in L^{\infty}\big(\Omega(x, ...
2
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1answer
110 views

Sample size from population?

This is probably very rudimentary maths, but given a strict population size ($N = 20$ for example), is the sample size any number $<N$? For use in calculation of confidence intervals using a ...
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1answer
1k views

Definition: finite type vs finitely generated

The mathematical term "finite type" appears more and more in the modern articles nowadays. But it is still hard to be found in the standard textbooks. I learned the definition of it from Stacks ...
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2answers
123 views

Lebesgue Measure Definition

Given a subset $A \subset \mathbb{R}$ with the length of an open interval $\mu_L(I_k) = b_k -a_k : I \doteq [a_k,b_k]$ The lebesgue measure is defined as $$ \lambda^{\ast} (A) \doteq \inf \Big\{ ...
3
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2answers
69 views

degree of commutativity

What is the exact definition of the degree of commutativity of a $p$-group? When we use notations $d(G)$ and $c(G)$ for other concepts, what is the best notation for degree of commutativity of $G$?
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0answers
84 views

About definition of topology

let be $m$ a function "$m: X \to \mathcal{P}(\mathcal{P}(X))$" (and I denote: $m(x) := m_x, \forall x \in X $), $m$ is topology on $X$ if: 2)$ \forall x \in X (\forall t \in m_x(x \in t)) $ 3)$ ...
3
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2answers
101 views

Terms of graph theory in english

Can anyone please tell me how are these graphs called in english? If we can divide a set of graph vertices in two disjoint sub sets, such as all edges connect vertices only inside these sub sets? ...
0
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2answers
697 views

What is a Tensor Product?

If you were to explain the concept of a tensor product to an undergraduate(post linear algebra), how would you do so? I would like to hear your definition, your take, on the definition of a tensor ...
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2answers
76 views

“Preimage” of a binary relation

Consider the binary relation $R \subseteq X \times Y$. Is there a standard name and notation for the set $X' = \{x\ |\ (x, y) \in R\}$? ProofWiki calls $X'$ the preimage of $R$, denoted as ...
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2answers
237 views

Can someone explain definition of number 1?

I've found the next piece of text: As an example of the second failing, Poincaré recalled the definition of the number 1 offered by another of the logicists, Burali-Forti: $$1 = ...
1
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2answers
97 views

What *really* are the local maxima and local minima

In math is the local max and local min just any peak ... point where slope of the function changes from positive to negative or vice-versa... Or are the LOCAL max and min just the highest point of the ...
4
votes
1answer
422 views

Set-builder notation function definition

I know that a function is a subset $f \subseteq X \times Y$ such that \begin{eqnarray} \forall x \in X, \exists ! y \in Y | (x,y) \in f \end{eqnarray} First, is it possible to express what a ...
0
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1answer
115 views

Ambiguity of Defining Subsequences

Let $(x_n)_{n\in\mathbb{Z}_+}$ be a real sequence such that $x_n=1$ for all $n\in\mathbb{Z}_+$. Consider the sequence $(x_2,x_1,x_3,x_4,x_5,\ldots)$. Argument 1: $(x_2,x_1,x_3,x_4,x_5,\ldots)$ is a ...
1
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1answer
98 views

Definition of a group

What defines a group mathematically, please explain both in Mathematical language and in English if possible. My current understanding: Four things are required to define a group: Closure - Any ...
2
votes
1answer
227 views

Radian, an arbitrary unit too?

Why is the radian defined as the angle subtended at the centre of a circle when the arc length equals the radius? Why not the angle subtended when the arc length is twice as long as the radius , or ...
2
votes
2answers
65 views

Definition verification from two different books?

In Kaplansky's Set Theory And Metric Spaces, he mentions a useful example of a neighborhood of $x$ is a closed ball with center $x$. However, one of the theorems in baby Rudin is "Every neighborhood ...
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2answers
302 views

Rudin's definition of continuity in terms of pre-images (inverse images). Is this simple function continuous or not?

I am reading W. Rudins book ``Principles of Mathematical Analysis''. I find it hard to exactly understand the definition of continuity in terms of pre-images. Rudins definition of a continuous ...
1
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1answer
44 views

What is the lower bound of the subset $2^n,\; n\in\mathbb{N}$

Let: $$ A = \{2^n,\; n\in\mathbb{N}\},\quad A\subset \mathbb{R} $$ Is the lower bound: $(-\infty,0]$ $(-\infty,1]$ $(-\infty,1)$ ? I think it can be the first because ...
2
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1answer
86 views

Is the definition of linearity redundant?

For some two functions f(x) and g(y) and for the transformation T, T is linear if: ...
3
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1answer
158 views

$C^{2, \alpha}$ regularity for elliptic equations with Neumann boundary conditons

Say $\Omega\subseteq \mathbb{R}^n$ is a bounded open set and $0<\alpha<1$. I need some $C^{2, \alpha}(\overline\Omega)$ regularity result for elliptic equations with Neumann boundary conditions ...
0
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2answers
167 views

Meaning of NOT all but finitely often

Can someone clarify for me the meaning of the statement "NOT all but finitely often"? It's driving me crazy. I'm not able to break it up. Thanks.
0
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3answers
557 views

Quotient spaces in linear algebra

There's a statement in some notes I'm reading that goes like this: "...$V/U$ is a 'simplified version' of $V$ where the elements of $U$ are ignored" ($V$ and $U$ are vector spaces). I'm still ...
2
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2answers
271 views

Problem with definition of regular surface in classical differential geometry

I am reading Do Carmo's differential geometry book and the definition of a regular surface in the second chapter is given to be this: I have few doubts about this definition: 1) Why we need to find ...
1
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1answer
454 views

Is this definition of limit at infinity of complex functions correct?

In my book (Churchill), a limit of a function at infinity is defined as: $$ \lim\limits_{z \to \infty}f(z) \equiv \lim\limits_{z \to 0}f(\frac{1}{z}) $$ But why can't you define the point at infinty ...
0
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1answer
25 views

Why isn't $r^{\frac{n}{2}}$ classified as an involution?

Suppose you have a $(2n)$-gon for some $n \in \mathbb{N} : n > 1$. Then the rotation $r^{\frac{n}{2}}$ where $n$ is the number of vertices imposed on the $(2n)$-gon is the same as an involution. ...
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1answer
65 views

How to see a 2-group as a 2-category with only one object?

We'll take the following definition of a 2-group: A 2-group $\mathsf{G}$ is a category internal to $\mathsf{Grp}$ Namely, it is a group $\mathsf{G_0}$ of objects, a group $\mathsf{G_1}$ of ...
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3answers
71 views

Definition of Linear Differential Equation

I am a 13 year old self teaching myself Differential Equations from a website and a book, I came across the definition of a Linear Differential Equation but I didn't understand the definition, I ...
2
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2answers
98 views

Similar linear operators and change of coordinates

Let $S, T$ be operators in $\mathcal{L}(V)$, the space of all linear maps from $V$ to itself. In my lecture notes, I have the definition of similar: "We say that operators $S,T \in \mathcal{L}(V)$ ...
3
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3answers
1k views

Why do functions with compact support include those that vanish at infinity?

The support of a function is defined in Wikipedia as "the set of points where the function is not zero-valued, or the closure of that set". Functions with compact support in $X$ are defined in ...
2
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1answer
118 views

What is a good definition of Hilbert space?

Motivation of my question: in my opinion, in view of the common definition, the statement "$\ell_p$ is a Hilbert space if and only if $p=2$" makes no sense because there is no inner product in the ...