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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Is every set a subset of a vector space?

I was taught that a functional is a map from a subset (not subspace) of a vector space into the reals, $F: D\subset V \to \mathbb{R}$. I know there are other definitions, but is there any reason to ...
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1answer
60 views

What does $R[[X]]$ and $R(X)$ stands for?

I'm reviewing Linear Algebra these days and I saw these two notations in my notes without definition. Those are, $R[[X]]$ and $R(X)$ where $R$ is a commutative ring with unity. I remember that one ...
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1answer
40 views

How would you describe category $\mathsf{Rel}$?

I encountered two definitions for a category denoted by $\mathsf{Rel}$: Objects are pairs $\left(A,R\right)$ where $A$ is a set and $R$ a relation on $A$. Arrows in ...
288
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18answers
53k views

What are imaginary numbers?

At school I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number which has something to do with the square root of -1. When I ...
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0answers
14 views

What does it mean to say a simplex has all its vertices distinct?

In Hatcher I'm trying to solve the problem: Show that the second barycentric subdivision of a $\Delta-complex$ is a simplicial complex. Namely, show that the first barycentric subdivision produces a ...
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0answers
8 views

What do $A \upharpoonright x$ and $\mu s \ge x$ denote?

I am reading Computability Theory by Cooper and I do not understand the notation in the definition on the page 230: Let $\{A^s\}_{s \ge 0}$ be a $\Delta_2$-approximating sequence for $A \in ...
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1answer
21 views

Maximum/Maximal set

Maximum or maximal set with property $P$ When I was reading some textbooks, I noticed that I do not get the meaning of the following two phrases. ($P1$) $\quad$ maximum set with property $P$ ($P2$) ...
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0answers
27 views

Changing the zero product property and defining division by zero [duplicate]

I know that defining division by zero is not possible because it violates the zero product property we define, that is, $0\times a=0$ for every $a$. I wonder whether we can somewhat circumvent and ...
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19answers
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What is the difference between a point and a vector

I understand that a vector has direction and magnitude whereas a point doesn't. However, the course note that I am using states that a point is the same as a vector. Also, can you do cross product ...
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1answer
151 views

What is the correct statement of the infinitary associativity law?

Let $X$ denote a non-empty set. Write $\mathcal{L}$ for the class of all ordered pairs $(L,f)$ where: $L$ is a linear poset (possibly empty), and $f$ is an arbitrary function $L \rightarrow X.$ ...
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0answers
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+50

Recurrence vs Recursive

Say team 1 is studying the recursive characteristics of a function. Team 2 is studying the recurrent characteristics of the same function. Are the 2 teams studying the same thing? I have found for ...
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4answers
171 views

Why is the expression $\underbrace{n\cdot n\cdot \ldots n}_{k \text{ times}}$ bad?

For writing a (german) article about the power with natural degree I have the following question: In school one defines the power with natural degree via $$n^k = \underbrace{n\cdot n\cdot \ldots ...
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2answers
22 views

Calculating the argument of a complex number… something tends towards infinity?

A simple question, but I like to be clinical with my choice of words: I have a complex number, $z=-i$. If I were to calculate the argument of this complex number, $arg(z) = tan^{-1}( \frac{-1}{0}) ...
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3answers
123 views

Definition of hyperbolic cosine and its relation with exponential function

Ordinary trigonometric functions are defined independently of exponential function, and then shown to be related to it by Euler's formula. Can one define hyperbolic cosine so that the formula ...
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1answer
29 views

Definition of Random Sample in Estimation

In my statistics class, we're just beginning to talk about (point) estimation. I understand the basics for the most part, but I have a small question that might actually be due more to notation than ...
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1answer
17 views

What is the usual definition of a zero divisor?

Let $R$ be a ring. I found there are two distinct definitions: Wikipedia Definition $a\in R$ is a zero divisor iff there exists nonzero $b\in R$ such that $ab=0$ or $ba=0$. Another: ...
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1answer
33 views

Under what assumptions can one compute conditional probability as $p(x)/p(y)$?

Conditional probability is often introduced in the following way: Consider a normal, fair 6-sided die. If you toss it then the probability $p(x=2)=1/6$. Now given that we already observed that the ...
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2answers
45 views

Definition of neighborhood

I am starting to work through Rudin's Principles of Mathematical Analysis. For $(X, d)$ a metric space and $x \in X$, Rudin defines the neighborhood $N_r(x)$ of $x$ to be the set consisting of all ...
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2answers
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Cant understand some definitions of abstract algebra, can you help me please?

I was reading different definitions of vector space but still I have some dark places without meaning... I cant understand exactly what type of relation is defined between the vector space and the ...
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0answers
15 views

Solution to sets of equations

Let ${\bf y}=(y_1,\dots,y_n)\in{\mathbb R}^n$, ${\bf X}$ an $n\times n$ real matrix and $\beta\in{\mathbb R}^n$. Suppose that ${\bf y}= {\bf X}\beta$ has a solution $\beta^*$ and let $A=I_1\times\dots ...
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1answer
22 views

Definition of strong tangent.

Let $\alpha:I\rightarrow \mathbb{R}^3$ a parametrized curve. What is the definition of strong (weak) tangent of $\alpha$ at the point $t_0$? Thanks!
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2answers
60 views

About the definition of homology

can someone explaine me this definition of Homology: "The homology groups of $X$ measure "how-far" the chain complex associated to $X$ is from being exact." I know that homology measure the number ...
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1answer
23 views

Is $\sum c_n z^n$ analytic when $c_n$ is Banach-valued?

I'm trying to define "Analytic function". I want a definition that covers all interesting cases. To be specific, let me explain what exactly I want Here is the definition of analytic function in ...
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2answers
101 views

Why in open balls is radius $r>0$?

the usual definition is the following: Def.1: given $(a,f)$ a metric space, $c \in a$ and $r \in \Bbb{R}_{>0}$, the open ball of radius $r$ about $c$ is the set $$\mathcal{B}_f(c,r)=\{x \in a| ...
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0answers
14 views

Definition of local truncation error

In the first reference of Wikipedia the local truncation error defined as $$ \tau_n = y(t_n)-y(t_{n-1})-hA(t_{n-1},y(t_{n-1}),h,f) $$ But in the second reference mentioned that $$ \frac{\tau_n}{h} ...
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1answer
30 views

Problem on function definition

I'm trying to solve this problem about functions: "Explain why $x^2-4=0$ is not a real function of real variable." I have yet solved many similar problems, but now i have a doubt; is my solution ...
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2answers
66 views

On the meaning of “Class of finite groups”.

What do we mean precisely when we speak about a class of finite groups? Is this simply a collection of some finite groups, maybe collected with a criterion (example: the class of all finite cyclic ...
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3answers
97 views

Notation for Partial Functions

Suppose we have sets $f$, $A$ and $B$ such that $f\subset A\times B$ and $\forall x\in A\space \forall y,z\in B: [(x,y)\in f \land (x,z)\in f \implies y=z]$ i.e. $f$ is a partial function mapping ...
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0answers
19 views

What's the definition of $C_0(\Omega)$?

Here is a definition of $C_0(\Omega)$ in wikipedia: (http://en.m.wikipedia.org/wiki/Vanish_at_infinity) Let $(X,\tau)$ be a locally compact space. Let's call "a function $f:X\rightarrow ...
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1answer
34 views

What is the formal definition of polynomial ring of several variables?

Let's consider a polynomial ring of single variable. One can define them informally by saying $P(X)=\sum_{i=1}^n a_n X^n$ while $X$ is an indeterminate variable. However, since mathematics is based ...
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0answers
49 views

What is the difference between $\propto$ and $\sim$

Suppose I have two physical quantities, lets name them $a$ and $b$. I wonder what the difference exactly is between $$a\propto b,\tag{1}$$ and $$a \sim b.\tag{2}$$ I know for eq. 1, that it means ...
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1answer
43 views

Categorising types of Mathematics

What area of Math do the following fall under? 1) Systems of ODEs and Phase planes 2) Laplace Transforms 3) Fourier series 4) PDEs with grad, div, curl, flux
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2answers
93 views

defining smooth functions on manifolds *without* smooth chart transitions

Let $M$ be a topological manifold, covered by an atlas of charts ${(U,\phi_U)}$ (which are homeomorphisms into Euclidean space), and let $p\in M$. Say a function $f:M\to\mathbb{R}$ is smooth at $p$ if ...
2
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2answers
111 views

What actually constitutes a *definition* for a function?

I'm reading Enderton's text on logic trying to justify ( to myself ) the use of induction on recursively defined sets. This is of course used repeatedly in trying to prove results about well-formed ...
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1answer
35 views

Definition of quotient category

Is there any reason why only gluing of morphisms sharing domain and codomain is usually allowed in the definition of quotient category?
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0answers
16 views

definition of half-integral weight modular forms

i start reading about modular forms of half-integral weight $k/2$ for $\Gamma' \subset \Gamma_0(4)$. As far as i understand these are holomorphic functions $f\colon \mathbb{H} \rightarrow \mathbb{C}$ ...
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1answer
40 views

Definition of a certain matrix

I remember I came across matrix of the form $$\begin{bmatrix} 1 & 0 & 0 & 0\\ 1 & 1 & 0 & 0\\ 0 & 1 & 1 & 0\\ 0 & 0 & 1 & 1\\ \end{bmatrix}$$ There ...
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9answers
662 views

Positive and negative complex numbers?

Can there be such a thing as positive and negative complex numbers? Why or why not? What about positive or negative imaginary numbers? It seems very tempting to say $+5i$ is a positive number ...
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0answers
59 views

Loss of derivatives

In many books on pdes the expression "loss of derivatives" is used when some estimates on solution are proved. Can someone clarify to me (maybe with an example) the meaning of this expression? For ...
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2answers
45 views

Compact Set: Cover by Merely Neighborhoods

Disclaimer: This thread is just a record of thoughts and written in Q&A style. A subset is compact if every open cover admits a finite subcover. What if one replaces open covers with covers by ...
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0answers
61 views

What does it mean to categorify something? [closed]

What does it mean to categorify something? Is there any simple illustrative example of this process? How it is done and what is it useful for?
4
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2answers
98 views

Why is there a 'missing' $1$ in the Euler–Mascheroni constant?

It is easy to show that: $$ \sum_{k=1}^n \frac{1}{k} > \ln(n+1), $$ but the Euler–Mascheroni constant is defined as: $$ \gamma = \lim_{n \to \infty} \left( \sum_{k=1}^n \frac{1}{k} - \ln(n) ...
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2answers
21 views

Discussion on Measures: Sigma-Additivity

Disclaimer: Though this thread is written in a Q&A style any new thoughts are really welcome! What reasons are there to restrict measures to countable additivity rather than uncountable ...
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1answer
25 views

polynomials and minimality

Could someone explain the concept of minimal polynomials? It seems like these are polynomials which cant be reduced further, but at the same time I am confused cause when we consider $\mathbb Z_2[x]$ ...
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2answers
132 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
187 views

Basic question about analyticity vs. differentiability in complex analysis.

In chapter $V$ of Palka, "Consequences of the Local Cauchy Integral Formula," 3.1. If a function $f$ is analytic in an open set $U$, then $f'$ is analytic in $U$. In particular, $f$ belongs to ...
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1answer
66 views

What is $1 / \mathbf{Set}$ if $1$ is a one-element set and $\mathbf{Set}$ a category?

What does $1 / \mathbf{Set}$ denote? A pointed set is a set $X$ equipped with an element (a basepoint) $x \in X$. Let $\mathbf{Set_*}$ be the category of pointed sets and basepoint-preserving ...
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3answers
313 views

Can only one ordered pair be a relation?

I'm sorry, but I really can't find an answer to this no matter how deep I dig. A relation is defined as any set of ordered pairs. But what about a set of only one ordered pair? Is it still a ...
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0answers
48 views

Fountain code and online code

Are fountain code and online code the same? It seems to me they have the same property, which is used in lossy channel and generate unlimited encoded block. If they are the same, then what encoding ...
3
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0answers
41 views

What is the desirable function identification when setting up arrows in the category of types?

My question is which functions can not be allowed in a statically typed programming language, so that the "canonical" category is less coarse than what you get if you define it's arrows to be ...