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

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1answer
13 views

Question about “integrable” random variable

I was reading the definition of Markov's Inequality on Wikipedia and it says If $X$ is any nonnegative integrable random variable and $a > 0$, then $\mathbb{P}(X \geq a) \leq ...
5
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1answer
81 views

Why is $\mathrm{arctan}(0)$ not infinity?

$\arctan x$ is defined as: $$\arctan x = \frac{1}{\tan(x)} = \frac{1}{\frac{\sin(x)}{\cos(x)}}$$ if I now have $x = 0$ I should get: $$\frac{1}{\frac{\sin(0)}{\cos(0)}} = \frac{1}{\frac{0}{1}} = ...
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1answer
27 views

Different definitions of subnet

I encountered two different definitions of subnet. The first is Let $(I, \preceq_I ), (J,\preceq_J )$ be two directed sets and $X$ be the underlying set.$\{ \eta_j \}_{j \in J}$ is a subnet of $\{ ...
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1answer
15 views

Metric spaces - Limit points and Isolated points

I apologise for this, but I have numerous potential misunderstandings. $\Bbb R^2$ is a metric space, since it is a subspace of the Euclidean space, $\Bbb R^n$ I can look at a set $E$ with elements ...
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1answer
24 views

Closed vector space and a subspace of a vector space [duplicate]

What is a closed operation in a vector space? I don't see any difference between a closed operation in some vector Space R$^n$ and the open operation. What I mean by the closed operation is addition ...
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1answer
16 views

Interpretation of the radius of convergence

What interpretation should one give to the radius of convergence of a series $\sum a_nz^n$ ? I do know how it is mathematically defined and what it implies for convergence/divergence, but I'm having ...
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3answers
25 views

E is closed if every limit point of E is a point of E?

E is closed if every limit point of E is a point of E? Should that be "E is closed if every point of E, is a limit point"? I don't understand. Limit points are essentially points that hug other ...
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1answer
56 views

Understand the definition of convex metric spaces

I am trying to understand the following definition: We call a set $E\subset \Bbb R^k$ convex if>$$\lambda x+(1-\lambda)y\in E$$ Whenever $x\in E, y\in E$ and $0\lt \lambda \lt 1$ Clearly ...
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1answer
36 views

Ordinary differential equations of order zero?

Is $x+y+2=0$ a differential equation without derivatives of order $n$, $n>0$? Could it be called a differential equation (for unknown $y(x)$) of order $0$? If not, can we define differential ...
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1answer
18 views

Difference between finite and discrete set of discontinuities

I am reading this paper. In page 395 and 396, theorems $1$ and $2$ use the terms 'finite' and 'discrete' to refer to sets, in this case sets of discontinuities. What I don't understand is: what is the ...
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0answers
32 views

Two definitions for non-singular in codimension 1

I am trying to understand how the following definitions are the same. $\textbf{Shafarevich definition}$ (pg 128) - A variety is $\textit{non-singular in codimension one}$ if the singular locus has ...
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1answer
24 views

Please help with understanding a logic definition: Subformula

Alright, so I am reading "Computability and Logic" by Boolos and Jeffrey, specifically I'm on chapter 9 "A Precis of First-Order Logic: Syntax. There has been more than a handful of definitions in ...
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2answers
24 views

What is a third proportional?

I searched online, couldn't find anything clear. If I had two numbers, $a,b$, what is their third proportional? Apparently it can be either $c$ such that $a/b=b/c$ or $b/a=a/c$, but obviously these ...
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1answer
26 views

Sense of the graph of a function

What makes it necessary to define the graph of a function $f:A\rightarrow B$ as $$\{(x,f(x))\mid x\in A\}$$ which makes it a subset of $A\times B$, when this is equal to the function itself, which is ...
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1answer
13 views

Spectral gap vs. algebraic connectivity

Can someone please clarify how the spectral gap of a graph relates to its algebraic connectivity (aka Fiedler value) and whether these use the adjacency matrix or laplacian matrix?
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1answer
55 views

What's the difference between these two definitions of polynomial function?

Definition 1: Given $a_n,...,a_1,a_0 \in \mathbb{R}$, a polynomial function is a function $p:\mathbb{R} \rightarrow\mathbb{R} $ such that $p(x)=a_nx^n+...+a_1x+a_0$ Definition 2: The function ...
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1answer
19 views

What is the oscillation of a function?

Define the oscillation of a function at a point $x$ to be (for an open interval $I$): $$\omega_f(x)=\inf_{x\in I}\sup_{s,t\in I}|f(t)-f(s)|$$ I am a bit confused about the definition above. How am I ...
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1answer
47 views

Theory of definitions

I am reading "Introduction to Logic" by P Suppes at the moment. In the Chapter 8 - Theory of definitions of it, I 've some confusion, actually about the Conditional Definition. The brief explanation ...
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1answer
43 views

Why do some authors use terms “non-ascending” and “non-descending” instead of ascending and descending?

In my math book, everywhere the author has used "non-ascending" instead of descending and "non-descending" instead of ascending. I was wondering if there is some special meaning or use associated with ...
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0answers
23 views

Derivative of $Ad(c(t))X$

Let $G=SO(3)$ and $V=\{c'(0)|c:(-\epsilon,\epsilon)\to G, c\in C^{\infty} , c(0)=1\}$. For $g\in G$, define $Ad(g): V\to V$ by $Ad(g)(X)=gXg^{-1}$. The book says ...
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1answer
16 views

Definition of monomial

I thought the definition of a monomial is an algebraic term that has no subtraction or addition. I saw on my online college homework that 2/x is not a monomial. Why?
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2answers
17 views

What's the name of a solid that results from extruding an area straight along an axis?

If you have any kind of 2D shape and move it up into the third dimension, what do you call it, because it seems like prism is used only if the base is a polygon. It also seems like extrusion is a ...
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1answer
10 views

What is the connection between slant/oblique asymptote to the polynomial part of the function and polynomial division?

What is the connection between slant/oblique asymptote calculation to the polynomial part of the function and polynomial division? To find the slant asymptote $y=mx+n$ we can can calculate it in two ...
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1answer
61 views

Understanding the condition of $\diamondsuit^+$

I try to understand $\diamondsuit^+$, a variant of the diamond principle. It says that there is a sequence $\langle \mathcal{A}_\alpha\rangle_{\alpha<\omega_1}$ of collection of subsets which ...
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2answers
43 views

Proving rigorously that if $3\mid(a+b)$ then $3\mid(a^3+b^3)$ using divisibility definition

Let $a,b\in \mathbb Z$. Prove rigorously using divisibility definition that if $3\mid(a+b)$ then $3\mid(a^3+b^3)$ After a bit of algebra I get that $$3\overset{?}{\mid}(a+b)^3-3ab(a+b)$$ So now how ...
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0answers
24 views

Probability density function definition

The definition above is given in my lecture notes. However there is no further reference/explanation given for what $o(h)$ represents. Can anyone explain this in this case?
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1answer
20 views

Clarification of some doubts on the definition of submatrix

I don't fully understand how I can choose a submatrix in a matrix. Judging from this definition and picture (http://mathworld.wolfram.com/Submatrix.html), I would assume that you can't pick as a ...
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0answers
44 views

What abstract structures allows us to describe “nets that converge toward each other”?

Topological spaces are equipped with a bare minimum of structure to allow for a formalization of the statement "the net $a$ converges to the point $x$." Actually this isn't strictly true, but its true ...
3
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1answer
30 views

Definition of complex conjugate in complex vector space

I am starting reading about Hodge theory and while reading the definition of abstract Hodge structure a very basic question came to my mind... What is the definition of the conjugate of a subspace of ...
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0answers
24 views

Definition of a vector field

Reading Wikipedia, I see that a vector field is defined as a mapping $X: S \rightarrow \mathbb R^n$ where $S \subseteq R^n$. However, I sometimes see mappings $X: S \subseteq R^m \rightarrow R^n$ ...
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1answer
37 views

Elliptic boundary value problems and elliptic partial differential equations

I am interested in the relation between the definition of an 'elliptic boundary value problem' and an 'elliptic partial differential equation'. From the wiki entries it seems that 'elliptic boundary ...
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1answer
17 views

Is there a term for something equally distributed around zero?

Let's say X is uniformly distributed in [-1 , 1] Then what can we call the distribution of X³ ? It is not uniform, but it "mirrors" around 0 as well. Is there a simple word describing X³ that would ...
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1answer
20 views

Why are the two definitions of covariance equal? [duplicate]

For example in Wolfram mathworld, you get these two definitions of covariance. http://mathworld.wolfram.com/Covariance.html cov(X,Y) = E[ (X - E[X])(Y - E[Y]) ] = E[XY] - E[X]E[Y] ...
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3answers
346 views

How can we think and/or write rigorously about integration by substitution?

Define a function $I:\mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ as follows. $$I(a,b)=\int_a^b \sin t \cos t \,d t$$ Then we can find a more explicit description of $I$ using integration by ...
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0answers
11 views

What is the correct term for the equations which comprise a mathematical model?

I have a mathematical model constructed by myself and my supervisor. In writing my report do I refer to the equations that make up this model as "constitutive equations", or is there another term ...
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0answers
18 views

What is the difference between assuming for $\forall k< n$ and proving for $n$ than assuming for $n$ and proving for $n+1$

What is the difference between assuming for $\forall k< n$ and proving for $n$ than assuming for $n$ and proving for $n+1$ (or $n-1$ and proving for $n$)? both with induction. The first one is ...
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0answers
31 views

The definition of continuously differentiable functions

When we say $f \in C^1$, we mean that f is continuously differentiable. Isn't the continuity a redundant word? I mean, we have a theorem that says if $f$ is differentiable then it is continuous. So ...
6
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1answer
28 views

Definition of gradient?

Reference: A primer of nonlinear analysis - Antonio & Giovanni Let $H$ be a hilbert space over $\mathbb{K}$ and $U$ be open in $H$ and $p\in U$ and $f:U\rightarrow \mathbb{K}$ be a functional ...
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0answers
32 views

What does rigoruous but non-technical mean?

Hi I find the above expression a bit confusing. I am considering buying a book and it says that it's a rigorous but non-technical introduction to optimal stochastic control. Could someone explain ? ...
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1answer
15 views

What is the definition of differentiation in normed space?

I'm trying to generalize implicit&inverse function theorems in Euclidean spaces to the context of Banach spaces. I'm wondering what would be the definition of differentiation in Banach space and ...
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0answers
7 views

$k$-ary labeled trees with distinct labels

Classical definition of $k$-ary labeled trees doesn't restrict somehow the uniqueness of tree labels inside its branches. My question: Is any special definition (name) for such trees? To clarify ...
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0answers
32 views

What is an omega model?

I went to a seminar and a side question was if a theory had an omega model, however from the context I could not deduce the exact meaning. Does an omega model have a general meaning in mathematical ...
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0answers
22 views

Proper usage for term, addend, factor, multiplicand, expression, formula

The definitions and usage of the following words seem to vary, depending on the source text: term addend factor multiplicand expression formula The words are being used in the context of ...
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1answer
18 views

Clarification on definition of a basis

Quick question; lets say that $S$ is a basis of $V$. I understand that this means all vectors in $S$ are linearly independent, and that every vector in $V$ is an element of $\text{span} \ S$. Is it ...
2
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1answer
45 views

Is this relation transitive? $R=\{(1,2),(1,1),(2,1),(2,2)\}$ over $A=\{1,2,3\}$

Is this relation $R$ over $A$ transitive?$$A=\{1,2,3\}$$ $$R=\{(1,2),(1,1),(2,1),(2,2)\}$$ Since from the definition a relation is transitive if $\forall x,y,z\in A (xRy,yRz\to xRz)$, so since $3$ ...
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0answers
83 views

What is the “correct” way of making $\mathcal{P}(X)$ into a topological space?

If $X$ is a topological space, I want to know the "correct" way of making the powerset $\mathcal{P}(X)$ into a topological space. So let $\mathsf{SupLat}$ denote the infinitary Lawvere theory whose ...
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1answer
48 views

Are 'vectors' vectors?

Let us say I have a 'vector' $\vec v$ for which I can do the following operation on $A\vec v$ where $A$ is a matrix. Now most people (i think) would say that $\vec v \in R^n$ however $\vec v$ is not a ...
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1answer
48 views

Given a field extension $K\colon F$, $K$ is an $F$-vector space

I'm having a hard time understanding fields. Could someone help with the following I need to show that if $F$ $\subseteq$ $K$ are both fields and addition and multiplication on F are the ...
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3answers
40 views

If $f\circ g = g \circ f$ does that mean that both functions are to and from the same set and both are bijections? Does it tell us anything else?

If $f\circ g = g \circ f$ does that mean that both functions are to and from the same set and both are bijections? Does it tell us anything else?
0
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1answer
31 views

Question about defintion of inner product space

While practising I came across the following easy question: "Is the space $B(0,1):=\{f:[0,1]\rightarrow\mathbb{R}$ bounded$\}$ an inner product space?" But I'm not quite sure what the correct answer ...