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Questions tagged [definition]

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

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1answer
14 views

How to understand a probability space in dicrete time

It is common in probability to define a prob. space as : $$(\Omega,\mathscr{F},P)$$ This can be understood as sample space, events, and probabilities for each event. However I don't know how to to ...
0
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1answer
15 views

Definition of subpartition

I stumbled on the expression subpartition of a set: in my context $V$ is a set (of nodes) and "$A, B$ is a subpartition of $V$". What does this phrase means exactly and how does it differ, if it does,...
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3answers
52 views

Circular Definition of Experiment in probability

I was trying to understand what an experiment was in the theory of probability. I found several definitions. Definition by Wikipedia Any procedure that can be infinitely repeated and whose ...
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2answers
43 views

Is “alignment” (or something else) a better word than “direction” for a senseless direction?

When I encountered the concept of direction being prior to the ordering of points on a line representing (parallel to) the direction, I thought it was a valuable distinction. I resolved to keep this ...
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0answers
19 views

Word for cyclic, non-periodic function

The additive decomposition of a time series can be written as $$ Y_t = T_t + C_t + S_t + I_t $$ where $T_t$ is the trend component, $C_t$ is the cyclic component, $S_t$ is the seasonal component, ...
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30 views

Under the subgroups of the group of all affine transformations what can or cannot be measured?

Edit: It was pointed out in the comments, sheers are transformations which are volume preserving, and not orthogonal. That was sloppy of me. Consider that part of my question answered. I believe ...
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4answers
938 views

Is it possible to define “Straight-line” logically? If it is possible, How you will define it? [duplicate]

Recently I am studying the "ELEMENTS" of Euclid. It is a translation of SIR THOMAS L. HEATH. In the definition part of the first book, the second definition is, "A line is Breathless length". My ...
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2answers
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What is limit point?

I'm new in calculus and can't understand what the limit point is.... Here, the definition from textbook that I've a question If E=(1,2) 1.It' say that 1 and 2 is also limit point,I can't understand ...
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1answer
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Understanding Taylor Approximations

I am curious about what quantity a Taylor approximation actually optimizes, when it produces, as they say, the "best" possible nth-degree approximation of a function around the given x-value. ...
2
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1answer
46 views

Definition of a mathematical interval, why not defined this easier way?

According to Wikipedia, "In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set" If we ...
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0answers
11 views

Most natural choice for the sign in the definition of the resolvent

When $T$ is a densely defined unbounded operator on a complex Hilbert space $H$, depending on the author its resolvent is defined as $$R_\lambda = (T - \lambda)^{-1} \quad \text{or} \quad R_\lambda = ...
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Is there a term denoting the index list of a multiply-inexed object, such as a tensor component?

Suppose we have a tensor expressed in component form as $T_{ij\dots{k}}$. Is there a name for the construct ${ij\dots{k}}$? I might call it a multi-index, but prefer not to invent newfangled ...
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2answers
36 views

Are charts for smooth manifolds homeomorphisms or diffeomorphisms?

I will link the following lecture notes, because it makes no sense to keep pasting from them. When reading them, there are two things I do not understand. The author introduces smooth manifolds by ...
2
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1answer
81 views

What is the right definition of the limit of a function?

So I came to this question while trying to answer simpler one: "Is square root function continuous at $0$ or only right continuous?" If you look at the wikipedia page about $\varepsilon$-$\delta$ ...
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2answers
38 views

What does “principal diagonal of an n-cube” mean to you? [on hold]

I want to use the term principal diagonal of an n-cube. Or, given some n-dimensional rectangular parallelpiped $\mathcal{P};$ the principal diagonal of $\mathcal{P}$. It has an obvious meaning to me....
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3answers
155 views

Doubts regarding $\epsilon$-$\delta$ definition of limits

I am learning $\epsilon$-$\delta$ definition of limits. I was confused on a few points and read some of the related answers on this and other sites. But I couldn't find discussion on any of these ...
3
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2answers
66 views

Can a function with infinitely many holes in the domain still be continuous?

If we have a function $f:A\rightarrow \mathbb{R}$ where $A$ has infinitely many holes and is subset of $\mathbb{R}$( i.e. $\mathbb{Q} \subset \mathbb{R}$). Then can $f$ be a continuous function still?...
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2answers
65 views

How do you define a mathematical definition/formula? [closed]

I am unable to find any resources on Google for this (I may just be entering the wrong search terms, in which case I can delete this post if needed); but my question is straight-forward. How do I ...
2
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1answer
66 views

What is it called when a decimal value has a pattern while infinite? [closed]

I recently made an edit to this post concerning $\pi$ and it containing all possible combinations of numerical values; and this answer to it brought forward an interesting number: 0....
1
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1answer
44 views

A weird definition of regular function

Consider the following definition, where immersions are defined: Later the author states: The problem: In the definition, a regular function is defined to have constant rank in particular. But later ...
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0answers
34 views

More general definition of holomorphic functions

I just began my course in complex analysis and have a few questions. I know this definition of when a function $f:U \to \mathbb{C} $ is holomorphic where $U \subseteq \mathbb{C} $ is an open set, ...
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$z^c = e^{c \log z}$ vs. $z^c = e^{\log z^c}$ and the domain of equality

Definition of $x^r$ for $x\in \mathbb{R}$ and $r\in \mathbb{Q^c}$ makes sense if we define it as a limit of $x^{r_n}$ for a sequence ${\{r_n}\} \subset \mathbb{Q}$ for $\lim_{\infty} r_n=r$. And the ...
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1answer
54 views

About the definition of a discrete valuation ring

In the script of my professor he introduces the valuation as follows Let $R$ be a Dedekind domain. If $I$ is a nonzero ideal of $R$, then for any nonzero prime ideal $\mathfrak{p}$ of $R$ we define ...
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1answer
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definition of complete vector lattice

Suppose $M$ is a von-Neumann algebra, $L=M\cap M'$ is the centre of $M$. The last line on page 29, C*-algebras and their automorphism groups, states that the self-adjoint part $L_{sa}$ of $L$ is a ...
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3answers
33 views

Probability problem with or/and (meaning of “neither”). [closed]

In a certain Algebra 2 class of 28 students, 5 of them play basketball and 21 of them play baseball. There are 5 students who play neither sport. What is the probability that a student chosen randomly ...
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1answer
175 views

Royal Road to Free Groups and Free Products

This question is more about strategy, which can be used when developing group theory, then about a particular proofs. $ \newcommand{GRP}{\mathsf{GRP}} \newcommand{SET}{\mathsf{SET}} $ One way to ...
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2answers
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Confusion between function and multivalued function.

"What is a function?" can be answered as "Single-valued relations are called functions". But how can "What are the multi-valued function?" be answered? Will someone clarify my doubt why multi-valued ...
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0answers
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How to formally define what a reading comprehension question answering problem is?

I'm trying to formally define what Intelligent Agents with Reading Comprehension Question Answering agents are in mathematical terms for a dissertation. To my mind we can say we have on the one hand ...
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0answers
30 views

Defining tensors as multilinear maps, without defining the dual space first

A $(p,q)$-tensor can be defined as a multilinear function in several ways, which are mostly equivalent: $$T:(V^*)^p\times V^q\to\mathbb R$$ or $$T:V^q\to V^p$$ or $$T:V^q\to \mathbb R\times V^p$$ ...
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1answer
34 views

Meaning of surface measure

While studying PDE, I came across this trace operator which talks about the class $L^{p}(\partial \Omega)$ for $\Omega \subset \mathbb{R}^{n}$ is open with $C^{1}$ boundary. I don't understand what ...
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Transformation on surface patches that preserves length of curves.

suppose $\mathcal{S} \subset \mathbb{R}^3$ is a surface patch, and let $\mathcal{C}$ a simple curve on $\mathcal{S}$, suppose $f$ is a transformation/function/mapping such that $f(\mathcal{S}) = \...
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4answers
80 views

Are trig functions only defined for unit circles?

In my textbooks the trig functions are defined with the help of a unit circle. So does it always have to be a circle with radius $1$ unit? Can't we define trig functions with the help of a circle with ...
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0answers
60 views

Is there a generalization of the concept of variance for a collection of probability distributions?

If I have a collection of numbers, I can obtain a measure of how much they're "spread" by computing the sample variance of them, i.e. $$\frac{1}{n}\sum_{i=1}^n(x_i-\mu)^2~,$$ where $\mu$ is the sample ...
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Definition of Local Functor

I have seen local functors being mentioned while constructing products of varieties and schemes but I couldn’t find the definition of a local functor. Any comments/reference will be appreciated.
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1answer
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what does it mean that something fibers?

For example, in an article I have found that "compact abelian group which fibers over the circle $S^1$ [...]" and surely I have heard that phrase before. What does it mean?
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2answers
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How are the properties satisfied in this induction proof?

I have some notes on the topic of the Principle of Induction (POI) from the perspective of the Well-Ordering Principle (WOP). The following claim has just been proved: Claim: (Principle of Induction) ...
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1answer
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Equality or not in definition of limits and infinity and sequences?

See in the first case, there is an equality also, between $n$ And $N$, but in the second case, there is a strict inequality between $x$ and $N$. So what I want to ask is that if we remove the ...
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42 views

Is this statement redundant?

Given any monotone set function $\mu$ (i.e. $\mu$ satisfies $A\subseteq B\implies \mu(A)\subseteq \mu(B)$) one can define a partial order $\preceq $ by setting $X\preceq Y\iff [X\subseteq Y]\land [\mu(...
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0answers
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Alternative definition of “sheaf”

Let $(X,\tau)$ denote a topological space and $\mathcal{O}$ denote a presheaf on this space with codomain $\mathbf{Set}$. We can take the category of elements of $\mathcal{O}$, which consists of a ...
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3answers
56 views

Definition of dependence in probability

Here is classical definition and example of dependent events. "When two events are said to be dependent, the probability of one event occurring influences the likelihood of the other event. For ...
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2answers
253 views

What is the difference between vector space and dual space?

I read that in Dirac notation, kets are elements of a vector space and bras are elements of the dual space. My question is, what is the difference between vector space and dual space, and why are bras ...
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4answers
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Is there a formal term for the placement of two symbols next to each other to imply an operation?

An example of what I am talking about is indicating multiplication by writing $$ab\equiv{a}\times{b},$$ in traditional real number algebra. I was writing some notes involving matrix multiplication. ...
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0answers
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Definition of zerovalent vertex in a tree

In the paper "Recurrence relations for the number of labeled structures on a finite set" by Blatter and Specker the authors speak of univalent, zerovalent and multivalent points of a tree. It seems ...
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1answer
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Definition of “definition”: use iff or if? [duplicate]

There are topics with the same name but my question is not as abstract as in those. My question is as follows: taken a generic definition like $x\;\mathbf{ is\; something}$ if $y$ it could be written ...
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2answers
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Are my definitions of limits accurate?

I wrote the definitions below. Are they accurate? If not, what correction(s) should be made? You may think based on an apparent lower bound for my level of mathematical maturity that I could answer ...
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3answers
454 views

Rolle's theorem: what's the right statement of the theorem?

In the fourth edition of "Introduction to Real Analysis" by Bartle and Sherbert, theorem 6.2.3 (Rolle's theorem) states, Suppose that f is continuous on a closed interval $I := [a, b]$, that the ...
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1answer
64 views

Understanding the definition of norm of tensors on a Riemannian manifold

I am teaching myself Riemannian Geometry in order to studying Mean Curvature flow. I was reading Lecture Notes on Mean Curvature Flow by Carlo Mantegazza and I'm trying understand the following ...
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2answers
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Representation of elements of direct sum of groups

I am confused about direct sum of groups, which is something that I thought I understood for a long time. By definition, the direct sum $\oplus_{\alpha}G_{\alpha}$ of groups $G_\alpha$ has elements ...
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6answers
143 views

What is a number in math? [closed]

Before I begin, let me give you so background. I previously asked a question on "How to prove that −x is not equal to x just because they yield the same result when in $x^2$". This got me thinking. ...
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Definition of Especial matrix in Linear Algebra

Consider $n \times n$ matrices ${\bf A}_i$, $1\leq i \leq n$, over a field. Let an $n \times n$ matrix $\bf B$ is obtained as follows: $$ {\bf B}=\prod_{i=1}^n \, {\bf A}_i \times \prod_{j=n}^{2n-1} \,...