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

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3
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0answers
34 views

How to write this function in a “well-formed” way

Given an input $0 \lt x \lt 1$, find $x$'s Nearest Integer Continued Fraction with structure $$x = a_0 \pm \cfrac{1}{a_1 \pm \cfrac{1}{a_2 \pm \cdots}}.$$ Then $$f(c) = a_0 + 1 \mp \cfrac{1}{a_1 + ...
5
votes
1answer
103 views

Mathematical meaning of “may not”

Does "may not" mean that never allowed or sometimes not allowed? For example: the sequence may not converge. Does this mean that the sequence never converges or that there is no guarantee that the ...
3
votes
1answer
44 views

Difference between “real functions” and “real-valued functions”

According to my textbook: A function which has either $\mathbb R$ or one of its subsets as its range is called a real valued function. Further, if its domain is also either $\mathbb R$ or a ...
1
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3answers
102 views

If function $f$ is differentiable at a point $a$, does it imply that $f'(a)$ exists?

If function $f$ is differentiable at a point $a$, does it imply that $f'(a)$ exists? I want to proof that if function $f$ is differentiable at a point $a$, then the function is continious at this ...
1
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3answers
81 views

What is the difference between an indexed family and a sequence?

For indexed family wikipedia states: Formally, an indexed family is the same thing as a mathematical function; a function with domain J and codomain X is equivalent to a family of elements of X ...
1
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1answer
56 views

Function Definition Issue

I'm currently working through an real analysis text and came across a definition that seemed a little strange to me. When defining a function the text states: For a function, $$f:\,S\rightarrow ...
1
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0answers
27 views

Set of 2-tuples with indices

I want to define the set of two tuples. Actually, I want to describe the degree distribution (or histogram) in terms of ordered tuples. The first item denotes the degree of a node. The second item ...
1
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1answer
32 views

What does it mean to say that a vector is 'coordinate invariant'?

In my lecture notes, it says that a vector is 'coordinate invariant' if it's properties do not depend on the choice of basis used to represent them. I understand that the basis of a vector space is a ...
0
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0answers
29 views

What is the definition of Planar $3$-Sat problem?

I have some steps in my lecture-notes to make an instance of Planar $3$-Sat problem from an instance of $3$-Sat problem. The steps are as follows: Create one vertex for every literal $x_i$ Create ...
0
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1answer
20 views

What is the n-concavification of a Banach space?

I'm reading this paper about polynomials in Banach spaces and the authors use the notion of the n-concavification of a Banach space $X$ It is the first time that I encounter this concept. What is it? ...
4
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1answer
59 views

Is distance between two graphs defined somehow?

If the two graphs are isomorphic, then their distance is zero. And this distance increases, if vertices or edges are added or removed to/from one of the graphs. Does this "distance" have a special ...
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2answers
60 views

The formal definition of an interval

I is A real interval iff ∀ x,y ∈ I the segment [x,y] ⊂ I I can't understand why an interval is defined this way Why it isn't defined the same way segments are? how can the definition of an ...
2
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1answer
49 views

What is the difference between an Ordered Set and a Completely Ordered Set?

When is a set called an Ordered Set and when is it called a Completely Ordered Set?
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3answers
93 views

What am I missing in this argument for $\lim\limits_{x\rightarrow \infty} \ln x = \infty$?

In an appendix of Stewart's Calculus, the logarithmic and exponential functions are built up starting from the defnition $\ln x = \int_1^x \frac{1}{t}\,dt$. Having shown that $\ln(x^n) = n \ln(x)$ ...
0
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0answers
41 views

Understanding manifolds (asking just for confirmation)

In lecture we used the following definition of manifolds: A subset $ M \subset \mathbb{R}^n $ is called a k-dimensional manifold of the class $C^\alpha$, if $ \forall a \in M $ there is an open ...
0
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2answers
96 views

Support of a distribution, what does it mean?

In my course notes the support of a distribution (continous lineair functional) is defined as follows: Definitions First it defines something like open annihilation sets: An open annihilation ...
3
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3answers
92 views

Is there any way to universally define the notion of $\text{Isomorphism}$?

Suppose we want to give a very general definition of the term Isomorphism, first of all, we'll want an isomorphism to be a bijective function. Informally, we want our function to preserve whatever ...
3
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1answer
77 views

Absolutely continuous function on R

What is the definition of absolute continuity in whole $\mathbb{R}$. I know the definition on an interval $[a, b]$. I have a trouble with understanding the definition of absolute continuity in whole ...
2
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1answer
99 views

Lost in terminology: What is the meaning of the words “Constraint” and “Parameter” in a goodness of fit?

This is somewhat related to this previous question of mine. I need a clear distinction and/or definition of the words 'parameter' and 'constraint' in the following context which is the the only source ...
2
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0answers
42 views

Sup-norm of two functions

Suppose $\Omega$ is a bounded set and connected domain in $\mathbb{R}^n$. Consider the operator $L$ in $\Omega$ $L=a_{ij}(x)D_{ij}u+b_{i}(x)D_{i}u+c(x)u$ For $u \in C^{2}(\Omega)\cap ...
0
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0answers
30 views

Inverse Substitution ( Trigonometric Integration Rule Generalized )

I read that we can generalize the kind of substitution that is used in Trigonometric Substitution. The idea is that we replace the old variable with a new one (like we do in U-Substitution) but unlike ...
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2answers
93 views

Is an undefined equality vacuously true?

To quote Wikipedia, [..] equation is an equality containing one or more variables. Solving the equation consists of determining which values of the variables make the equality true. Consider a ...
0
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1answer
50 views

“Countable” and “Sequence” in Toplology

I'm having difficulty with the meanings of these terms in several references. "Countable" may mean either a set with cardinality = N (i.e. countably infinite) , or with cardinality $\le$ N (i.e. ...
4
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3answers
42 views

Alternative definition for Exponential and Logarithmic functions to prove identities (and by extension sin and cos related identities)

Edit : This question is about not about proving identities, but representations that are easier to work with than Taylor series or integral definition for the functions exp or ln functions. Please do ...
0
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1answer
22 views

Defining Compound Function

If $f(x) = (x+x)$ and $g(x) = 2(x-5)$, and we have the compound function $g(f(x))$, how can we "define what the resulting function does"? It's obvious what is happening, but I'm not quite sure how ...
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0answers
30 views

Why was this terminology of holomorphism used here?

page 6 in this notesays: A note on index convention: $i$,$j$, and $\bar{i}$, $\bar{j}$ index over the holomorphic and antiholomorphic components. I never knew there were holomorphic ...
3
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2answers
66 views

How to make sense of the binomial coefficient over $p$-adic integers?

I recently asked this question: Show that $y^2=x^3+1$ has infinitely many solutions over $\mathbb Z_p$. and now I'm trying to make sense of the first answer that was posted. It said that I should show ...
10
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1answer
161 views

Is there a reason for different nomenclature on Calculus of Variations?

While sightseeing aspects of Calculus of Variations, the following fact elludes me: there is a plethora of new definitions which seem redundant to me. This phenomenom happens, of course, with other ...
0
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1answer
51 views

Are math definitions iff statements? [duplicate]

I was wondering if definitions in mathematics are "if and only" statements? I know for sure that theorems are not "iff" statements. Thank you in advance for your help.
0
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1answer
41 views

Mathematical definitions of infill asymptotics

I am writing a paper that uses infill asymptotics and one of my reviewers has asked me to please provide a rigorous mathematical definition of what infill asymptotics is (i.e., with math symbols and ...
1
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1answer
39 views

Compact space - definition

I have a doubt at the definition of compact spaces. So if you have a topological space $X$, then $X$ is compact if every open cover of $X$ has a finite subcover. In other words, if $X$ is the union of ...
0
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2answers
33 views

Definition of linear/ affine/ etc transformation [duplicate]

A linear transformation is a function $f:V\to W$ from vector space $V$ to vector space $W$ such that linear combinations (alternatively vector addition and scalar multiplication) are preserved. An ...
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2answers
53 views

Terminology conventions in mathematics

I am writing a paper where I have more than one lemma (Lemma 1, Lemma 2, and Lemma 3) and when I cite them together I was wondering is it more appropriate to say, for example, because of Lemmas ...
1
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1answer
41 views

Is it ok to make an operation over some definition at very the first time you are defining it?

Suppose we have an expression and want refer to it with some symbol. Usually one can write something like "$X:= \text{Expression}$" to mean the symbol $X$ is defined to be the expression after the ...
2
votes
2answers
48 views

Definition for the complex exponential function

We define the exponential function as $\exp(z)=\sum\limits_{j=0}^\infty= \frac{z^j}{j!}$ for all $z\in \mathbb{C}$. Lets now compute $\exp(0)$, then we would have to calculate $0^0$ which undefined. ...
1
vote
1answer
44 views

Is $H_n(X)$ just a different way of writing$H_n(S_*(X))$?

While studying homology in algebraic topology, I sometimes see the notation $H_n(S_*(X))$, and sometime the notation $H_n(X)$. I think these are supposed to be the same, but I'm not sure. The first ...
0
votes
2answers
61 views

How can some integrals be zero?

This could be a nonsense but I have to try. We are used to know that an integration like $$\int_a^b F(x)\ \text{d}x$$ gives us the area under the curve $F(x)$ from $a$ to $b$. The question is then: ...
2
votes
1answer
102 views

Weak Gâteaux Derivative

Suppose $X$ and $Y$ are Banach spaces. Let $F:X \rightarrow Y$ be a function and $U \subset X$ be an open set. The Gâteaux derivative of $F$ at $u \in U$ in the direction $\phi \in X$ is given by ...
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0answers
27 views

Is this definition correct regarding the logic of recursive sequences?

This is a self study. I come across with a second order sequence of reals defined by: $$φ_{m+2}=φ_{⌊f(φ_{m},φ_{m+1})⌋}..................(*)$$ where $⌊.⌋$ is the integer part function. Here $f$ is a ...
1
vote
2answers
45 views

What does $\mathfrak{m}(\varprojlim R/\mathfrak{m}^i)$ mean?

Let $\mathfrak{m}$ be an ideal of the ring $R$, and consider it as an $R$-module. What does $\mathfrak{m}(\varprojlim R/\mathfrak{m}^i)$ mean? In the limit the morphisms are the natural projections. ...
0
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0answers
20 views

Terminology and Notation: “Sphere” in a Metric Sapce

I see sphere used interchangeably with "ball" in the sense of "open ball", "closed ball", with notation like $B_r(x), B_r[x]$. I also see sphere used to refer to the boundary of a closed ball, e.g. ...
2
votes
2answers
88 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 ...
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1answer
67 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
34 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
vote
1answer
41 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 ...
1
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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
572 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 = ...