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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How to make precise the notion of “the multiset of roots of a polynomial function”?

A (real) polynomial function can be defined as a function $f : \mathbb{R} \rightarrow \mathbb{R}$ such that there exists a sequence $a : \mathbb{N} \rightarrow \mathbb{R}$ such that the terms of $a$ ...
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2answers
143 views

Is an unit-cube polyhedron? What about other platonic solids?

Definitions According to my linear programming course and screenshot here (Finnish), a polyhedron is such that it can be constrained by a finite amount of inequalities such that $$P=\{\bar x\in ...
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3answers
161 views

If there is only one one-sided limit, the limit exists?

Question on theory: If $\lim_{x\to a^+}F(x)=L$ and $\lim_{x\to a^-}F(x)$ doesn't exist, then does the $\lim_{x\to a}F(x)$ exists and is $L$? Thanks.
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1answer
180 views

a question about definition of regular surface

While I am reading Do Carmo's differential geometry,I have several questions about the definition of regular surface. From condition 2,the author said : "...... $x^{-1}:V \cap S \rightarrow U$ ...
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5answers
2k views

What does it mean when a function is finite?

When someone says a real valued function $f(x)$ on $\mathbb{R}$ is finite, does it mean that $|f(x)| \leq M$ for all $x \in \mathbb{R}$ with some $M$ independent of $x$?
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real numbers and number line

While reading some articles, I got a bit confused by the definitions of numbers. Specifically, Can the number line contain decimal values? I read that Real numbers = All numbers on the number ...
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2answers
864 views

What is which distinguishes Pure Mathematics from Applied Mathematics? Please explain [closed]

I like to know in depth what really differs Pure Maths from Applied Maths. What are their respective applications? Also when this distinction was made in the history of Mathematics and why?
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1answer
25 views

Contractible objects in model categories [closed]

Please, what is the definition of contractible object in a (closed) model category (if it exists)?
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1answer
10 views

Forms - unitary group?

If $Id$ is the $n$ by $n$ identity matrix and $J$ is the $2n$ by $2n$ matrix with $Id$ in the upper right corner and $-Id$ in the lower left corner, then $Sp_{2n} = \{G\in Gl_{2n} : G^{tr}JG = J\}$ is ...
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1answer
37 views

On the definition of commutators

We know that given a group $G$, for every $x,y\in G$ we define $[x,y]:=x^{-1}y^{-1}xy$, the commutator of $x$ and $y$. I saw something more general, commutators involving more than two elements, like ...
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1answer
51 views

Has this topology a name?

let $(X,\tau )$ be the topological space where $\tau =\{\emptyset, X, \{x\}, X-\{x\}\}$ , $ x \in X$. Does this topology have a name? Thanks in advance!
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1answer
31 views

Does a bijection between two sets $A$ and $B$ implies $P(a \in C ) = P(b \in C)$, if $A,B \subset C$?

I'm just thinking about it. For example, a bijection between $\mathbb{Z^*_+}$ and $\mathbb{Q}$ implies that the probability of a random real number being rational or positive integer is the same (in ...
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1answer
26 views

Convergence of vector spaces

I'm considering vector spaces over $\mathbb{R}$ or $\mathbb{C}$ : $(E_n)_n$ is a sequence of vector space (all included in $\mathbb{R}^p$ for example, with $p$ fixed). What meaning(s) do we usually ...
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1answer
19 views

Formal notation for finite intersection

How to state following sentence formally? Let $\tau$ be family of sets (possibly infinite). Any finite intersection of sets in $\tau$ is in $\tau$. My attempt is: $\tau$ is family of sets and ...
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2answers
22 views

What is the meaning of an “objective function” ?

I have studied Mathematics in french. Right now I am in front of this expression "objective function" and "objective functionals". I don't know its meaning, some one can help please ?
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1answer
25 views

Simple closed curve definition of genus

The genus of a connected surface can be defined as the maximum number of disjoint simple closed curves that can be removed from it without disconnecting it. Why must the simple closed curves be ...
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1answer
67 views

What are some practical uses of functions? [closed]

Functions are basically formal equations that relate a set of inputs to output. What are some practical uses for functions and inverse functions?
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1answer
42 views

Lipschitz continuity of $f(x,y)=4x^2+xy-\frac{1}{y-1}$ on an open set $U \subset \mathbb{R} \times (\mathbb{R} \setminus \lbrace 1 \rbrace)$

Problem: Find an open set $U \subset \mathbb{R} \times (\mathbb{R} \setminus \lbrace 1 \rbrace )$ which includes the points $(0, 1/2$) and $(0,3/2)$ such that the function ...
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2answers
55 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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2answers
65 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 ...
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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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4answers
56 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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1answer
89 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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1answer
83 views

Multiplication of positive fractional numbers.

I am reading this answer. We know that multiplication of two positive rational number $\dfrac{a}{b}$ and $\dfrac{r}{s}$ respectivly is defined as follows: $\dfrac{a}{b}\times\dfrac{r}{s} = ...
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1answer
45 views

Equivalent definitions of isometry

Consider a map $T:\mathbb{R}^2\to\mathbb{R}^2$ such that $\lVert T(x)\rVert=\lVert x\rVert$. Is this equivalent to stating that $\langle x, y\rangle=\langle T(x), T(y)\rangle$ for all ...
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2answers
120 views

Rigorous mathematical definition of “much greater than” symbol

What does $f(x) \gg g(x)$ mean mathematically? How can we characterize "much greater than"?
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1answer
62 views

$V, W$ vector fields, $\omega$ one-form, $d\omega(V, W)=?$

$V, W$ vector fields, $\omega$ one-form, what is the definition of $d\omega(V, W)$? I have seen some equalities and I suppose that $d\omega(V,W)=(V^jW^i-V^iW^j) \frac{\partial f_i}{\partial x^j}$ ...
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1answer
47 views

Why the inverse of the matrix in this definition

We have this definition: I didn't understand why in this definition we just define $F^A$ to be $$F^A(X_0:X_1:X_2)=F\bigg((X_0:X_1:X_2)A\bigg)$$ $F^A$ is not the composition $F\circ c$? Why ...
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1answer
32 views

Is a relation$ R=\{(a,a):a\in I\}$ said to be reflexive, symmetric and transitive?

Consider a relation $R=\{(a,a): a\in A\}$ where $A=\{1,2,3\}$. i.e $R=\{(1,1),(2,2),(3,3)\}$ This clearly reflexive but is it necessary that all such relations are necessary to be symmetric and ...
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1answer
26 views

A notation question on how to properly denote a function that takes inputs only of a certain form.

Suppose I have a set $B = \{n^2 + n + 1 : n \in \mathbf{N}\}$ and I want to define a function $g: B \rightarrow \mathbf{N}$ that only accepts as it's arguments numbers of the form $n^2 + n + 1$ for $n ...
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2answers
42 views

In a commutative algebraic theory, do all constant symbols necessarily represent the same value?

Let $T$ denote a commutative algebraic theory with two nullary function symbols $a$ and $b$ (i.e. constants). Is it an automatic consequence of the definitions that $a=b$ is a theorem of $T$? My ...
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2answers
47 views

Laws of indices

Well, given any real number $x$ and any positive integer $n$, the number $x^{n}$ is defined to be the product $x.x.x. ... x.x $ ($n$ times). But, how do we define $x^{r}$ when $r$ is a negative ...
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1answer
156 views

For all but finitely many $n \in \mathbb N$

In my book I have the following theorem: A sequence $\langle a_n \rangle$ converges to a real number $A$ if and only if every neighborhood of $A$ contains $a_n$ for all but finitely many $n \in ...
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2answers
42 views

Question about $(A - \lambda I_A)\vec{x} = 0$.

Finding a solution to $C\vec{x} = (A - \lambda I_A)\vec{x} = 0$ is the equivalent of considering the determinant of $C$ when it is zero. This means the matrix is linearly dependent and has infinite ...
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1answer
44 views

Definition for the action of a category on a set.

I'm trying to understand the definition of the action of a category on a set which is given in nLab, more particularly the first one. If one has a functor $\rho: C \to Set$, one takes the set S as the ...
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1answer
41 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 ...
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2answers
179 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 ...
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1answer
84 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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2answers
85 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\{ ...
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1answer
225 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 ...
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1answer
66 views

What is a “lemma”? [duplicate]

As per title, what is a "lemma"? How is it different from "theorem"? ASAIK, I have to prove a self-proposed theorem in my paper. Do I also have to provide the proof for a self-proposed lemma?
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1answer
57 views

$K$-Category $M(0, 0) = M(A, 0) = M(0, A)$ using definition from Swan's 'Sheaf Theory'

I'm using the following definition: A category $\mathcal C$ is given by the following: A collection of objects $A$ A set $M(A, B)$ for any two objects $A, B \in \mathcal C$. A function $M(B, C) ...
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2answers
111 views

Complex Analysis Question About Analytic Functions

I have some questions about knowing where and where not functions are analytic. Here's a function, f(z)= $\frac{Log(z+4)}{z^2+i}$ -I know that this function is not defined for ...
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1answer
50 views

Does this definition say what I want it to?

I don't have a real background in math but I should still be able to define stuff in my MSc thesis, although the thesis does not involve a lot of math. I want to define an object $o$'s ...
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1answer
15 views

Subseqeunce convergence definition

Definition: A subsequence $(a_{n_k})$ of $(a_n)$ is convergent if given any $\epsilon >0$, there is an $N$ such that $\forall k\geq N \implies\vert a_{n_k} - \ell\vert < \epsilon$ Why do we ...
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1answer
78 views

Definition of initial segment

the definition of initial segment is correct: "let $\preceq$ be a linear ordering of a set $A$, and $B \subsetneqq A$, $B$ is initial segment of $A$ under $\preceq$ if $\forall a \in A, \forall b \in ...
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1answer
152 views

What does “topological dual of a Banach space” mean?

I am not sure what does the "topological" imply. Thanks.
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3answers
239 views

What is the epsilon-delta definition of limits, exactly?

I am a bit confused with infinitesimals, and want to know why they were discarded and the epsilon-delta definition is being used? What is the epsilon-delta definition of limit? What is the intuition ...
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1answer
95 views

question about Graph Theory notation

I'm just starting to learn graph theory. I have two questions about notation: 1). For a graph $G$ we denote the vertex set $V$ and the edge set $E$ by $G=(V,E)$. So we have a graph $G=$ ...
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1answer
32 views

How could I define this $\mathrm{nw}(X)$ by using only one sentence?

A family $\mathcal N$ of subsets of a topological space $X$ is a network for $X$ if for every point $x\in X$ and any neighbourhood $U$ of $x$ there exists an $M \in \mathcal N$ such that $x\in M ...