Stack Exchange Network

Stack Exchange network consists of 174 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [definition]

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

1
vote
1answer
19 views

Notion of submodel relation

There is no definition of the essential notion of substructure (=submodel) in Shelah's introduction E56 to AEC, 1st Volume. Could someone please define this for me? I think that $$M \subseteq N$$ ($M$ ...
0
votes
0answers
25 views

What kind of integral or analysis is capable of dealing with cantor's function?

According to Lebesgue’s characterization of Riemann integrable functions (see: http://www.math.ru.nl/~mueger/Lebesgue.pdf) cantor funciton on $[0,1]$ (https://en.wikipedia.org/wiki/Cantor_function) is ...
2
votes
0answers
77 views

Notations in Functional Analysis: $L^p$, $L_p$, $\mathscr{L}^p$, $\mathscr{L}_p$, $\mathcal{L}^p$, and $\mathcal{L}_p$

If my memory doesn't fail me, then to some functional analysts, $L^p$ and $L_p$ spaces are two different things. I understand that many people use $L_p$ to means the space of functions with finite $p$...
0
votes
0answers
36 views

What is the difference between a “ line ” and a “ straight line ”?

Is there actually a difference between a line and a straight line ? Is figure 1 a line . ? . Should I take help from " Euclid "? I believe according to " Euclid " the above figure is a valid line.
0
votes
0answers
30 views

explain why the following definition is not equivalent to the definition of the limit of function f

Let $f:(0,1) \to \mathbb{R}$ be a given function. Explain how the following definition is not equivalent to the definition of the limit $\lim\limits_{x \to x_0} f(x) = L$ of $f$ at $x_0 \in [0,1]$ . ...
0
votes
0answers
11 views

What is a martingale $(X_n)$ index by the finite set $(0,1,…,N)$?

In the book of Yor and Revuz : "Continuous martingale and Brownian motion" third edition, proposition 1.5 page 53 : it's written : If $(X_n)$ is an integrable submartingale indexed by the finite set $(...
1
vote
2answers
41 views

What does the following notation mean: $(L-\lambda I)$

In my tekstbook powervectors are discussed where $(L-\lambda I)^p \textbf{v}=0$ was mentioned. Here is $L$ a linear operator on the vectorspace $V$, $\lambda$ a scalar and $\textbf{v} \in V $. My ...
0
votes
2answers
43 views

Is my conception of limit correct?

$(1)$ Can I define the derivative $\left( \dfrac{dy}{dx}=\lim_{\Delta x \rightarrow 0}\dfrac{\Delta y}{\Delta x} \right)$ as a value which can never be reached when $\Delta x$ approaches zero but ...
1
vote
1answer
42 views

What is uniqueness quantification?

Can someone explain the concept of Uniqueness quantification ∃! in an easily understandable way since I can't understand the definition of it, what's special about it with other logical operators like ...
2
votes
2answers
55 views

If $(U,\varphi)$ is a coordinate chart around $p \in M$, where $M$ smooth manifold, then how does $\varphi$ induce coordinates on $T_p M$?

I am studying differential topology and I have some trouble understanding how coordinates are induced on the tangent space at any point. Let $M$ be an $n$-dimensional smooth manifold, and let $p \in ...
0
votes
1answer
16 views

Definitions of $\epsilon$-regular partition

I am wondering about the equivalence between two definitions of an $\epsilon$-regular partition of a graph. First of all, if $G$ is a graph and $A$ and $B$ are subsets of its vertex set, the density ...
0
votes
0answers
20 views

About the differences between definitions of “Capable Group”

I am looking for the properties of groups having "immediate Descendants", in other therm, "Capable Groups"; The problem that I fond is that "Capable Group" could have many meaning! So, could you ...
0
votes
1answer
41 views

Confusion over the word “ratio” in the definition of $\pi$

According to Wikipedia, pi is "a mathematical constant that was originally defined as the ratio of a circle's circumference to its diameter." However, when I think of the word "ratio", something like ...
-3
votes
2answers
36 views

Self Reference in Arithmetic: Division By Zero [on hold]

The argument is as follows: Given an integer, it is possible to legally define all other integers by fundamental operations of adding or subtracting unit by unit. However for this to happen, we need ...
0
votes
0answers
27 views

Domain of a function given only its formula?

Given an equation $f(x)=\cdots$, is there an accepted convention for defining the domain and range/codomain of $f$ (assuming these are not given). e.g. if $f(x)=x^2$ what is the domain and range of $...
0
votes
0answers
14 views

Generalization of lower semicontinuous functions to Banach space-valued functions

Recall a definition of lower semicontinuous function (taken from Wiki). Suppose $X$ is a topological space, $x_{0}$ is a point in $X$ and $f:X\to \mathbb {R} \cup \{-\infty ,\infty \}$ is an ...
1
vote
3answers
86 views

When I'm given $\sin(x)$, what goes inside of the $x$? [duplicate]

I don't know if I'm just randomly blanking or if I never really knew and have just been going with the flow, but I'm not sure what x represents. In early high school they were degrees, eg. $\sin(30)$ ...
1
vote
0answers
23 views

Why is convex conjugate defined on functions taking values on extended real line?

Recall a definition of convex conjugate (taken from Wiki): Let $X$ be a real topological vector space, and let $X^*$ be the dual space to $X.$ Denote the dual pairing by $$\langle \cdot,\cdot \...
0
votes
1answer
19 views

$G$ is an $(x,y)$-graph if $x>C(G)$ and $y>I(G)$

I dont understand this definition, I'm new in this topic. Definition 1: $I(G)$, the independence number of the graph $G$, is the maximum number of point of $G$ thath can be chosen so that no two are ...
0
votes
1answer
18 views

What exactly does subcritical, critical and supercritical mean in the context of PDEs?

I have seen these terms thrown around a lot in the PDE literature but have struggled to find a definition of what they actually mean. If anything, I get the idea that these notions of sub/super- ...
0
votes
0answers
19 views

Definition of weakly compact operator

My book defines a weakly compact operator $T:X\to Y$ between Banach spaces as a continuous linear operator such that the closure of $T(B)$ is weakly compact for each $B\subset X$ bounded. My question ...
1
vote
0answers
14 views

What is meant by a cubic inflection point on a circle map?

I'm looking at the so-called circle map, defined by the sequence: $$x_{n+1} = f(x_n) = f^n(x_0) = x_n+\Omega - \frac{k}{2\pi}\sin(2\pi x_n) \quad \text{mod }1$$ where $\Omega$ and $k$ are some ...
0
votes
2answers
32 views

A concern on the definition of compactness in a metric space [duplicate]

Let $(X,d)$ be a metric space. This space is compact if any sequence $x_n \subset X$ has a convergent subsequence. This is how I'm given the definition of a compact metric space and it confuses me. ...
-1
votes
0answers
20 views

Difference between compact class and finite intersection property

The following definition of compact class is taken from Bogachev's Measure Theory Volume 1, page $13.$ A family $\mathcal{K}$ of subsets of a set $X$ is called a compact class if for any sequence $...
0
votes
3answers
42 views

Definition of the Lebesgue number of a open cover

Let $\mathcal{U}$ be an oper cover of a topological space $A \subseteq \mathbb{R}^n$. The Lebesgue number of $\mathcal{U}$ is defined as the least upper bound for all numbers $\delta \geq 0$ such that ...
15
votes
4answers
871 views

Help in understanding why the arithmetic derivative is well-defined.

I'm reading into the arithmetic derivative at the moment and I just don't get why it is well-defined. For reference, Ufnarovski tried explaining it here on page 2. I would like to understand the not ...
1
vote
1answer
28 views

Definition of rational numbers

We can uniquely define the set of real numbers as a complete ordered field. But can we do something similar with the set of rational numbers ? I think we need to change the completness axiom with some ...
2
votes
0answers
17 views

In the definition of partial derivative, why the function must be defined on an open set?

On the page Partial derivative on Wikipedia, the following formal definition was found: I am wondering if in this definition, the condition that $U$ being open is always necessary. For example, if in ...
1
vote
0answers
22 views

Am I properly negating the definition of this set?

The statement: Let $A\subset \mathbb{R}$ be defined as follows: $x\in A$ if and only if there exists $c>0$ so that $$ |x-j2^{-k}|\geq c2^{-k} $$ holds for all $j\in \mathbb{Z}$ and integers $k\geq ...
0
votes
2answers
21 views

Complex measure of empty set.

Reading the wikipedia pagina, a complex measure $\mu$ is a function $\mathcal{F} \to \mathbb{C}$ that is $\sigma$-additive. I.e., if $(E_n)_n$ is a set of disjoint sets in the $\sigma$-algebra $\...
0
votes
1answer
22 views

Elementary Substructures vs. Structures

Here is the definition according to Wikipedia of what an elementary substructure is: I am just wondering about the last part, where it says it follows that $N$ is a substructure of $M$. Why? Because ...
0
votes
1answer
19 views

What's the definition of proper subspace of a vector space used in Rudin's Functional analysis

I'm reading through the Rudin's functional analysis, and theorem 3.5 use the term "Proper Subspace", there's a theorem in chapter 2 that uses the same terminology. I'm reading through chapter 1 again,...
1
vote
2answers
20 views

Let $A$ be a Boolean algebra and $F\subseteq A$ be a filter on $A$. Why are the following properties equivalent?

Let $\mathcal{A}$ be a Boolean algebra and $F\subseteq \mathcal{A}$ be a filter on $\mathcal{A}$. Why are the following properties equivalent? $$(1)\,\,\,A\land B\in F\Rightarrow A,B\in F$$ $$(2)\,\,\...
1
vote
1answer
26 views

Let $(G, \cdot)$ be a group, and $H \leqslant G$. Let $x \in H$, what $C_H(x) < C_G(x)$ mean? ($C_A(x)$ notation for “centralizer of $x$ 'in A”)

Let $(G, \cdot)$ be a group. For any $x \in G$, we write: $$ C_G(x) = \{z \in G \mid z \cdot x = x \cdot z\}$$ Let $H \leqslant G$ (subgroup of), and $x \in H$. What does it mean when we write: $$ ...
1
vote
1answer
35 views

Definition of Substructures According to Wikipedia

I have a quick question about the definition for substructures in mathematics from its Wikipedia page: What does it mean by functions and relations are "traces" of the functions and relations of the ...
0
votes
0answers
30 views

Questions about database representaiton

Recently I am reading a paper, and here the author gives the way of database representation. I feel confused. Could someone give me an example of the so-called universe and histograms? My thought is ...
-1
votes
1answer
35 views

P(F > x) : what do you call this? [closed]

What do you call P(F > x) (e.g. P(F > 4.2)? I'm trying to figure out a formula to arrive at the number given after the > sign using just two given degrees of freedom and I figured that if I knew the ...
0
votes
1answer
27 views

Is there a formal name for an equation with multiple solutions?

I saw that there is a related question for an equation with no solution, but I was curious about an equation with more than one solution.
-1
votes
3answers
28 views

is x/x a proper or improper fraction?

If the numerator is greater than the denominator, it's improper. The other way around, it's proper. What if the numerator is the same as the denominator?
0
votes
0answers
8 views

What is an affine transformation whose “centered” part is orthogonal called?

On page 7 of The Meaning of Relativity, Einstein calls transformations of the form $$x^{\overline{i}}=a^{\overline{i}}+e_{\:i}^{\overline{i}}x^{i},$$ or $$\Delta x^{\overline{i}}=e_{\:i}^{\overline{...
3
votes
1answer
46 views

What does it mean for a definition to be covariant or contravariant?

In this answer, Brian M. Scott lists different definitions of continuity, and for some of them mentions whether they are covariant or contravariant. I am familiar with these terms from subtyping in ...
1
vote
0answers
42 views

What does $\sum_{k=1}^{\infty}X_k a_{n,k}$ exists almost surely for each $n$, actually mean?

Prove that $\sum_{k=1}^{\infty}X_k a_{n,k}$ exists almost surely for each $n$. Is it that $P\bigg(\displaystyle\lim_{n\rightarrow\infty}\sum_{k=1}^{\infty}X_ka_{n,k}<\infty\bigg)=1$? In this case ...
1
vote
2answers
45 views

What does this expected value notation mean?

From: Learning from Data, section 2.3.1 - Bias and Variance: Let $f : X \rightarrow Y$, and let $D=\{(x,y=f(x)) : x \in A \subseteq X\}$ where each $x \in A$ is chosen independently with distribution ...
2
votes
0answers
81 views

Definition of manifolds as submanifolds of $\mathbb{R}^m$

I'm having trouble understanding the definition of coordinate charts of manifolds given in the book "Analysis II" by Herbert Amann and Joachim Escher. They define manifolds as submanifolds of $\mathbb{...
-1
votes
0answers
14 views

An example for a probability distribution which is not a multinomial distribution

As far as I know, multinomial distribution is a probability function over a random variable that takes multiple values. My doubt is that are there any probability distributions which are not ...
0
votes
1answer
36 views

If V and W are linear subspaces of $R^n$, then V transversal W means just $V + W = R^n$

If V and W are linear subspaces of $R^n$, then V transversal W means just $V + W = R^n$. Could anyone give me a hint for this exercise please?
0
votes
1answer
18 views

What does it mean for two [multivalued] complex functions to be equal?

For $f:X\subseteq\mathbb{R}\to\mathbb{R}$ and $g:X\subseteq\mathbb{R}\to\mathbb{R}$, we may say that $f$ and $g$ are equivalent if $\forall x\in{X}.f(x)=g(x)$. But for many complex functions $f:Z\...
0
votes
1answer
134 views

Contraries's definition and vacuous truth

Say, $A: \text{Every Americans use English.}$ $B: \text{No American uses English.}$ $A$ and $B$ are said contarary. People say that A and B are contrary when A and B can not be both true ...
0
votes
0answers
88 views

What is the proof of proof by contradiction? [duplicate]

I understand how to use proof by contradiction method. But why is it correct? How do you prove that a false output, say B, achieved by applying logically consistent operations to the input, say A, ...
0
votes
1answer
25 views

Bounds of defining operations for equivalence classes

When operations for number systems are defined in terms of representatives of equivalence classes, can those operations meet the criteria for being well defined if the definition includes specific ...