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

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2
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2answers
84 views

What is the agreed upon definition of a “positive definite matrix”?

In here: http://ocw.mit.edu/courses/mathematics/18-06sc-linear-algebra-fall-2011/positive-definite-matrices-and-applications/symmetric-matrices-and-positive-definiteness/MIT18_06SCF11_Ses3.1sum.pdf ...
1
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1answer
64 views

Stationary accumulation points

I have been reading about trying to prove global convergence of general optimization alrgoithms and am have come across the term "stationary accumulation point" and am trying to decipher exactly what ...
0
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1answer
32 views

Global convergence versus convergence to a global

I have been reading many optimization papers and wanted to know what the difference between global convergence and convergence to a global is. Sounds like the same thing to me.
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4answers
1k views

Is this alternative notion of continuity in metric spaces weaker than, or equivalent to the usual one?

I will try to be as clear as possible. For simplicity I will assume that the function $f$ for which we define continuity at some point is real function of a real variable $f: \mathbb R \to \mathbb ...
0
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0answers
16 views

Definition of the inverse image functor

Suppose we have a continuous map of topological spaces $f:X\to Y$ and a sheaf $\mathcal B$ over Y. We can define a presheaf $\mathcal A$ over X by setting $\mathcal A(U) = \mbox{lim}\: \mathcal B(V)$ ...
1
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1answer
37 views

Definition of finite-sheeted covering

What is the definition of a finite-sheeted covering $q: E \to X$? Does it mean that every open $V \subseteq X$ has a pre-image $q^{-1}(V)$ that is the disjoint union of a finite number of sheets? Or ...
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0answers
14 views

Does the usual law for image also hold for relations?

Let $R \subseteq U \times V$ be a relation, and $S_0, \dots, S_{m-1} \subseteq U$ Then does the following hold? $$R\left(\bigcap_{j=0}^{m-1} S_j \right) \subseteq \bigcap_{j=0}^{m-1} R(S_j)$$ It ...
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1answer
42 views

Why do all the quantities define $\|A\|$?

Let $A$ be a $n \times n$ matrix ($A \in \mathbb{R}^{n \times n}$). Then for each $x \in \mathbb{R}^n$ the vector $Ax$ is defined and so we can see the matrix $A$ as a function $A: \mathbb{R}^n \to ...
11
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3answers
566 views

Why is the matrix multiplication defined as it is? [duplicate]

Matrix multiplication is defined as: Let $A$ be a $n \times m$ matrix and $B$ a $m\times p$ matrix, the product $AB$ is defined as a matrix of size $n\times p$ such that $(AB)_i,_j = ...
2
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1answer
37 views

What is the definition of $\ell^2(G)$ where $G$ is a group?

First I'll give some context for my question. I'm learning about crossed products of dynamical systems involving $C^*$-algebras and I've just seen the definition of a covariant representation. I have ...
2
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2answers
71 views

Examples of little Lipschitz functions

Suppose that $(X,d)$ is a metric space. A function $f:X \rightarrow \mathbb{R}$ is called little Lipschitz if for all $\epsilon>0$, there exists a $\delta>0$ such that for all $x,y \in X$, ...
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0answers
17 views

Definition of a vector of rational coordinates

What is a vector of rational coordinates? Consider a random vector $X$ of dimension $k\times 1$ taking values in $\mathbb{R}^k$. Then take $\mathbb{Q}_k:=(q_1,q_2,...)$ as the vectors with rational ...
0
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3answers
49 views

Definition of the Isomorphism theorem

I have a question about the definition of the Isomorphism theorem given my notes. The Isomorphism Theorem: Let $\phi : G \rightarrow G'$ be a group homomorphism with kernel $K$. Then $\phi$ induces ...
0
votes
1answer
27 views

Why does the composition of functions $g(f(x))$ require the codomain of $f$, and not its image, be the same as the domain of $g$?

As the question title states, for a composition of functions $f(g(x))$ I was taught that the domain of $g$ has to be the same as the codomain of $f$. Why? Can't you say that the image of $f$ and the ...
1
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1answer
93 views

Why $0$ (zero) is a purely imaginary number? [duplicate]

I was reading this Wikipedia article and found that $0$ is a purely imaginary number. Why? Is it because $i0=0$? So zero is the only number which is real as well as purely imaginary? Any explanations ...
1
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1answer
30 views

What, in set-theoretical terms, is a satisfactory definition of the complex logarithm?

A bijection $\log^+:\Bbb R_{>0}\times\Bbb R\to \Bbb C:(r,\theta)\mapsto\ln r+\mathrm i\theta$ can be defined, along with a family of smooth bijections $\mathrm c_{\alpha}:\Bbb R_{>0}\times ...
0
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0answers
19 views

Contraction of a vector form

I'm trying to make sense of this definition, but I cannot see why the resulting map is in a space of dimension $k-1$, surely as it is comprised of k vectors this maps a k-form to a (k+1)-form? I ...
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1answer
37 views

How would you write formal definition for indeterminate limit?

The original one, I believe, should be that for $$\lim_{{x}\to{\infty}}f(x)=L$$ $\forall\epsilon>0, \exists M \in ℝ$ such that $x>M \Rightarrow |f(x)-L|<\epsilon$ But what if it is that x ...
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0answers
26 views

Definition of a local cross section

I came across the following definition of a cross section and local section in Wikipedia - Let $\pi : E\rightarrow B$ be a fibre bundle. A cross section of this bundle is defined as a continuous map ...
2
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1answer
40 views

Confused about multinomials. Can we write $\binom{n}{a,b,c}=\binom{n}{a}\binom{n-a}{b}\binom{n-a-b}{c}$ if $a+b+c \le n$?

Can we write $\binom{n}{a,b,c}=\binom{n}{a}\binom{n-a}{b}\binom{n-a-b}{c}$ if $a+b+c \le n$? The definition for multinomial says $a+b+c=n$ must hold or else $\binom{n}{a,b,c}=0$. I found that if ...
0
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1answer
24 views

Why are $\lambda$-systems defined with the stronger condition $A\subset B\implies B\setminus A\in\Lambda$

A class $\Lambda$ over subsets of $E$ is called a $\lambda$-system if among other conditions the following is fullfilled: $$(A,B\in\Lambda) \wedge (A\subset B) \implies B\setminus A\in\Lambda$$ My ...
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0answers
33 views

What is a cusp neighborhood corresponding to a parabolic Möbius transformation in a Riemann surface?

I am referring to this wikipedia entry. So what I understand is that they are defining it using the Fuchsian model. If $\Gamma$ is a Fuchsian group, its parabolic elements correspond to the cusps of ...
0
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1answer
34 views

On the mere definition of divergence

If we have $$\lim_{x\to\infty}f(x)=\infty$$ we say that as x tends towards infinity $f(x)$ diverges. For what reason do we not say as x tends towards infinity $f(x)$ converges to $\infty$?
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1answer
74 views

Confusion over definition of wedge sum in topology

I'll ask my question using the specific example of the wedge sum of unit intervals (pointed at 1), motivated by the actual exercise I'm trying to answer. So we have a countable collection of pointed ...
0
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1answer
57 views

Difference between trivial component and isolated vertex

I have following definitions in my Graph Theory lecture notes: The components of a graph G are its maximal connected subgraphs A component (or graph) is trivial if it has no edges An isolated vertex ...
0
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1answer
37 views

Mean curvature is the divergence of the normal

As a definition, I was told that for a surface in 3D, $ 2H = -\nabla \cdot \nu$ where $H$ is the mean curvature and $\nu$ is the normal unit vector. In some results that I am studying, the factor 2 ...
2
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0answers
71 views

The best definition of a knot?

I just started reading a book about knots and links and asked myself what is the best and most precise way to define a knot, just an embedding of $S^{1}$ into $S^{3}$ is not enough, right? Can ...
3
votes
2answers
89 views

The $n$-cells of a modular lattice are antichains

The terminology "$n$-cell" is made up by me, and I'd love to hear if this has an official name. Given a poset $(X,\le)$, define the set $A_n$ of $n$-cells of $X$ recursively as follows: An ...
2
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0answers
44 views

An intuitive interpretation of stopping time

I have the following definition of exercise time. Let $T\in\mathbb{N}$ with $T>0$, let $(\Omega,\mathcal{F})$ be a probability space with the $\sigma$-algebra $\mathcal{F}=2^{\Omega}$ and let ...
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0answers
29 views

Definition of derivative as best affine approximation

I'd like to try to redefine the derivative (in a way equivalent to the usual definition) of a function $f: U \subseteq R \to R$ to make it clear that the derivative $Df(a)$ is the linear part of the ...
4
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1answer
83 views

Construction of a relative cup product $H^p(X, A) \otimes H^q(X, B) \to H^{p + q}(X, A \cup B)$

Let $A$ and $B$ be subspaces of a space $X$. What is the construction of a relative cup product$$H^p(X, A) \otimes H^q(X, B) \to H^{p + q}(X, A \cup B)?$$Here, we take cohomology with coefficients in ...
0
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2answers
18 views

Can someone clarify in simple terms what it means to “apply an inequality to a measure”?

I am reading wikipedia's entry on Holder's inequality https://en.wikipedia.org/wiki/H%C3%B6lder%27s_inequality#Counting_measure There are quite a few variations of Holder's inequality "applied to ...
2
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0answers
21 views

On karatzas Definition of Kolmogorov-Feller diffusions

we find this definition on Karatzas [Brownian Motion and Stochastic Calculus pg 282] But the definition of Kolmogorov Feller process does not seem to require that $Af$ be continuous in $x$ as is ...
2
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2answers
39 views

Different definitions for the complex inner product.

I have asked this question on P.S.E. and have gotten some nice answers, but I felt I might get even more satisfactory answers if I post it here. "I have just now noticed that Griffiths (in his book ...
0
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1answer
84 views

What are the difference among entire, smooth, analytic and holomorphic in complex function?

In complex analysis, I know that they have similar meaning. However what are the difference among them? Especially for entire function, does entire function mean analytic function?
2
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1answer
32 views

Understanding Homotopy Definition

I'm having trouble with part of the definition of homotopy, but I may just have a misunderstanding about continuous maps from product spaces. This is the definition my book uses: Two continuous ...
0
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0answers
36 views

What is the definition of a $p$-element in the context of finite groups

I suspect that a $p$-element in a finite group $G$ is an element whose order is a power of $p$, $p$ being a prime number but I am not sure. I would appreciate any feedback. Thank you.
0
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1answer
93 views

Derivative definition vs its requirements for existence

In the case of an extremely disconnected function such as ${\left(-2\right)}^{x}$. One definition requires that a derivative must be continous. $(-2)^x$ has two paths that could make it discontinuous ...
0
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1answer
21 views

Wikipedia definition of graph homomorphism: is this correct?

Consider the definition of a graph homomorphism given here: A graph homomorphism $f$ from a graph $G=(V,E)$ to a graph $G'=(V',E')$, written $f:G \rightarrow G'$, is a mapping $f:V \rightarrow V' $ ...
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1answer
20 views

Family of decompositions of a probability space and sigma algebra generated by a discrete random variable.

While reading the textbook "Martingale Methods in Financial Modelling" by Musiela and Rutkowski I am puzzled with a new definition (i.e. "the family of decompositions") that I never encountered and ...
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0answers
24 views

Barycenter of a measure

Suppose that $X$ is a Banach space and $\mathcal{F}(X) = \overline{span}(\{ \delta_x : x \in X \})$, where $\delta_X : Lip(X) \rightarrow \mathbb{R}$ and $Lip(X)$ is the set of real-valued Lipschitz ...
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0answers
30 views

Formal definition of an array or vector

First of all, excuse me if I compare arrays and vectors erroneously, I'm not mathematician. I need to know how to define formally an array of length n and composed by ones and zeros depending on ...
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2answers
46 views

Minimum vs lowerbound

What is the difference between the minimum value and the lower bound of a function? To me, it seems that they are the same.
3
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3answers
76 views

How to abbreviate $a_0=a_1=\cdots=a_n$?

We write $\sum_{j=0}^n a_j =a_0+a_1+\cdots+a_n$ or $\prod_{j=0}^n a_j=a_0\cdot a_1 \cdots a_n$. How to abbreviate $a_0=a_1=\cdots=a_n$? Maybe ${\Large =}| _{j=0}^n a_j$? Or $\{\forall j,k| ...
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1answer
41 views

Definition of a pushout of a short exact sequence

What is the definition of pushout of a short exact sequence? In this paper1, page $126$, under the proof of Proposition $2.8$, I don't understand how the author justify the existence of the operator ...
5
votes
1answer
59 views

Is the width of a poset well-defined?

According to the Wikipedia definition (current revision), the width of a poset is the cardinality of any maximum antichain, where "maximum antichain" here means an antichain of maximal cardinality. ...
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0answers
62 views

What's the meaning or definition of $\frac{\bigtriangleup {x}}{x}$

My teacher circled $\frac{\bigtriangleup{x}}{x}$ today on my answer and wrote "you should know that". I'm trying to figure out what it could mean but I can't come up with anything. ...
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1answer
30 views

Definition of space $L_f^2$ where $f$ is a function?

http://it.tinypic.com/r/2iqjvbl/9 Hi guys! I'm writing my thesis for my degree and it's about Sturm-Liouville theory applications. I'm using the book "Al-Gwaiz M.A. Sturm-Liouville Theory and Its ...
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3answers
2k views

Is there an example for an undefinable number?

This question is motivated by a comment of Robert on the question Can any Real number be typed in a computer? : Can you "think of" an undefinable number? – Robert Israel I would like to ...
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0answers
88 views

Definition: is a graph allowed to have a “dangling” edge without a vertex at its end(s)?

My textbook gives the following definition "a graph $G=(V,E)$ consisting of $V$, a nonempty set of vertices and $E$, a set of edges. Each edge has either one or two vertices associated with it." Now ...