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 ...

learn more… | top users | synonyms

1
vote
1answer
25 views

Exact definition of a circular helix

I know that a cylindrical helix is a curve $\alpha: I \to \Bbb R^3$, such that exists a constant vector $\bf u$ which makes a constant angle with the tangent vector $\bf T$ to the curve, at every ...
0
votes
0answers
32 views

Definition question.

Suppose we have $x \in \mathbb{R} / [0,1)$. We call $\lfloor x \rfloor$ the integer part of $x$. What do we call $\bar{x}=\frac{x}{\lfloor x \rfloor}$??? I would call it fractional part but it ...
0
votes
1answer
20 views

Confusion about the associative property and the mechanics of Parenthesis

This is a follow up question on my earlier post (Updated): Showing that a set $M$ with two elements classifies as a field. I feel this post is necessary because I realize that what confuses me is how ...
0
votes
2answers
23 views

Give the definition of S $\subseteq$ T for general sets S and T.

The answer I can come up with is; S is a subset of T, denoted by S $\subseteq$ T, or equivalently, T is a superset of S, denoted by T $\supseteq$ S. Can someone correct me if I am wrong, or provide ...
0
votes
0answers
36 views

Definition of a summation from minus infinity

Formally, an infinite series is defined as the limit of its partial sums: $$\sum_{n = 0}^\infty a_n \equiv \lim_{n \to N} \sum_{n = 0}^N a_n$$ However, how does this work for summations such as the ...
-4
votes
2answers
39 views

What do you mean by translation invariance of Lebesgue integral? [on hold]

What do you mean by translation invariance of Lebesgue integral?
1
vote
1answer
38 views

Why is the structure group for lengths $\mathbb{R}^+$ and not automorphisms in $\mathbb{R}^+$?

While reading what is a gauge to understand what a gauge is, I got stuck at a point where Terry Tao wrote: This isomorphism group is called the structure group (or gauge group) of the class of ...
2
votes
3answers
95 views

Understanding of the formal and intuitive definition of a limit

The intuitive definition for $\lim\limits_{x \to a} f(x) = L$ is the value of $f ( x )$ can be made arbitrarily close to $L$ by making $x$ sufficiently close, but not equal to, $a$ . I can easily ...
2
votes
1answer
26 views

Concise description of Lebesgue measure in $\mathbb{R}^{n}$

I would like to confirm that the following is an acceptable description of the Lebesgue measure in $\mathbb{R}^{n}$. The outer Lebesgue measure $E \subset \mathbb{R}^{n}$: $$\lambda^{*}(E) = ...
0
votes
2answers
27 views

Regarding retractions of $X$ onto subspaces

Let $A \subset X$ be a subspace of $X$. Recall that a retraction of $X$ onto $A$ is a continuous map $r: X \to A$ such that $r(a) = a$ for every $a \in A$. Let $X = \bf R$ endowed with the standard ...
-3
votes
1answer
34 views

using the definition to find derivatives [closed]

Calculate the derivate of the given function directly from the definition of derivative, and express the result using differentials when $f(x) = \sqrt{x^2 + 3}.$
0
votes
2answers
29 views

Product over a vector space

When looking at the definition of a vector space, I see that it's basically a set with two operations and a set of 8 axioms. However, none of those axioms talk about the product of two vectors. Is ...
0
votes
1answer
21 views

The definition of a submanifold

I am wondering why it is insufficient to define a submanifold of a manifold $M$ as a subset $S\subset M$ such that $S$ itself is a manifold. Why do we need the notions of embedded submanifolds or ...
2
votes
1answer
33 views

What is the intuition behind homeomorphism, especially behind the geometrical notion of “gluing together”?

Intuitively, a homeomorphism is a way of mapping two spaces without any tearing or gluing together. Thus, I would expect the formal definition of homeomorphism in terms of continuous functions to be ...
0
votes
2answers
34 views

Why is that for any trigonometric function $f, f(2\pi + \theta )=f(\theta )$ for any value of $\theta$ [proof reading]

Here was the question asked to me :: Why is that for any trigonometric function $f, f(2\pi + \theta )=f(\theta )$ for any value of $\theta$ I spontaneously said that it was because of their very ...
2
votes
0answers
46 views

Commutative diagrams with antiparallel arrows

A diagram in category theory is said to commute when for all objects $A$ and $B$ in it, every the composite morphism resulting from a possible path from $A$ to $B$ are the identical. Does that mean ...
2
votes
0answers
22 views

Different definitions of an affine algebraic set

Hartshorne assumes we are working over an algebraically closed field $k$ and we simply define algebraic sets to be any set of the form $V(I)\subset k^n$, where $I\subset k[X_1,\ldots,X_n]$ is an ...
0
votes
2answers
32 views

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 ...
1
vote
1answer
63 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 ...
0
votes
0answers
15 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 ...
0
votes
0answers
10 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 ...
0
votes
1answer
23 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 $P$ ($P2$) ...
0
votes
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 ...
1
vote
1answer
45 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 ...
0
votes
2answers
23 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}) ...
1
vote
4answers
182 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 ...
0
votes
1answer
18 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: ...
0
votes
1answer
31 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 ...
2
votes
1answer
68 views

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 ...
0
votes
1answer
34 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 ...
3
votes
2answers
47 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 ...
-1
votes
2answers
59 views

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 ...
0
votes
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 ...
0
votes
1answer
30 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!
1
vote
2answers
67 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 ...
0
votes
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 ...
0
votes
0answers
19 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} ...
6
votes
1answer
154 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 $L \rightarrow X.$ ...
1
vote
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 ...
3
votes
2answers
68 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 ...
0
votes
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 ...
0
votes
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 ...
3
votes
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 ...
0
votes
1answer
45 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
1
vote
0answers
17 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 $f\colon \mathbb{H} \rightarrow \mathbb{C}$ ...
2
votes
2answers
96 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
votes
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 ...
2
votes
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 ...
2
votes
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 ...
0
votes
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 ...