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

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How to formulate this property in a precise way?

Recently I've came across the following situation when studying Quantum Mechanics: suppose we have two operators $A,B$ on the space of functions $L^2(\mathbb{R}^3)$ and suppose they have the following ...
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20 views

Quotient topology from Delta complex,

A $\Delta$-complex structure on a space X is a collection of maps $\sigma_{\alpha} : \Delta^n \rightarrow X$ with n depending on index $\alpha$ such that 1)The restriction ...
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16 views

Can we attach a space with discrete signal?

This question refers to the link https://en.wikipedia.org/wiki/Space_(mathematics) and https://en.wikipedia.org/wiki/Discrete-time_signal. My question is how can we associate a discrete signal with a ...
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1answer
17 views

Given a basis $U$, what conditions are needed for an orthogonal basis for it?

Given a basis $U$, what conditions are needed for an orthogonal basis for it? For example, in the following vector space $U$, if $U =sp\{(1,1,1),(1,3,7)\}$ then what conditions are needed for an ...
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4answers
80 views

Codomains and the definition of a function

A function $f$ is defined as a set of ordered pairs $(x, y)$ such that $(x,b), (x,c) \in f \Longrightarrow b=c$. Since $y$ is determined uniquely by $x$, it is customarily denoted $f(x)$. One ...
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2answers
25 views

Alternative version of definition of convergence

Knowing the definition of convergence of a sequence in $\Bbb R$: ($x_n$) converges to x iff $\forall \epsilon \gt 0$ $\exists N$ such that for every $n\gt N$, $d(x_n,x)\lt \epsilon$. Consider a new ...
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1answer
16 views

Heine definition of limit of a function at infinity using sequences

I couldn't find the answer neither on Google, nor this website, so decided to ask. The Heine definition of limit: from Wikipedia $\lim_{x\to a}f(x)=L$ if and only if for all sequences $x_n$ (with ...
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46 views

If the sample space is an Euclidean Space, we can use a different type of PDF

The title resume all the point I'll try to make now. Reading this post, I realize that is possible to have another type of PDF (probability density function). Usually, we have a probability space ...
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1answer
38 views

Is this a valid notation in set theory?

I have three sets, $A:=\{a_1,\ldots,a_n\}$, $B:=\{b_1,\ldots,b_n\}$ and $C:=\{0\}$. Let $D:=A\times B \cup C$. I do not know if this is a valid notation? For example, Is $(0,b_2)\in D$? Or, is ...
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25 views

About the definition of $L^{\infty}$ norm

Let $\Omega$ a limited domain in $\mathbb{R}^{n}$, the space $L^{\infty}(\Omega)=\{f: \Omega\to\mathbb{R} $ measurable $; ||f||_{L^{\infty}(\Omega)}<\infty\}$. Then if a function $f \in ...
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3answers
65 views

Is there anything wrong with this definition of discontinuity?

Is there anything wrong with this definition of discontinuity for a function y = f(x)? $\forall \delta>0\, \exists \varepsilon>0$ such that $\vert x-c\vert < \delta$, but $\vert f(x) - ...
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3answers
139 views

In the definition of a limit, why do we care about all $\epsilon > 0$?

Definition of $\lim_{x \to a} f(x) = L$: $\forall \epsilon > 0, \exists \delta > 0 s.t. |f(x) - L| < \epsilon$ $ if \ 0 < |x-a| < \delta$ Why can't we weaken the assumption ...
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3answers
121 views

Book on foundational reasoning of standard arithmetics “curriculum”

I am interested in a book that is about arithmetics but the presentation is not just the known to all formulas but the foundational logic behind it. The closest example I can think about is the way ...
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90 views

Why every fundamental group isn't a trivial fundamental group?

I don't understand exactly the definition of the Fundamental Group. Munkres defined Fundamental Group in your book 'Topology' like "Let $X$ be a space; let $x_0$ be a point of $X$. A path in $X$ that ...
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1answer
79 views

How does Hartshorne's definition of group schemes encode the law for the neutral element?

Hartshorne's Algebraic Geometry says A scheme $X$ with a morphism to another scheme $S$ is a group scheme over $S$ if there is a section $e\colon\;S\to X$ (the identity) and a morphism ...
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88 views

Definition of $C^{1,2}$?

I just realized that I don't really know what the definition of $C^{1,2}$ (or $C^{m,n}$) means. Two candidates come to mind: 1) For every $y$, the function $x\mapsto f(x,y)$ is $C^1$, and for every ...
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2answers
42 views

What is the official definition of $S^1$ [closed]

I need an official definition of $S^1$ that is better than $\{circle\}$. The reason is because I am interesting in defining a function $f: \mathbb{R} \to S^1$ where $\mathbb{R}$ is the interval $[0, ...
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0answers
16 views

Does every collection of edge vectors of a cone span a face of the cone?

Definition 1 : A polyhedral cone is a subset of a real vector space which is the intersection of finitely many closed half spaces. (The defining planes of these half spaces must pass through $0$.) A ...
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34 views

Martingale under conditional prob. measure (definition)

Suppose we are given a probability space $(\Omega, \mathcal{F},P)$ s.t. r.v.s $X$ and $(Y_i)_{i=1}^\infty$ are $\mathcal{F}$-measurable. The relevant filtration is given by ...
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2answers
48 views

Mathematical usage of “$\dots$” during enumeration, is it ok to be imprecise?

I am guilty of writing things like the following in proofs: so by lemma 1.2 we have that for $k<n$ all of the integers $k,k+1,k+2,k+3,\dots ,n$ are pompous. I really like how this looks, the ...
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18 views

Local maxima and minima of functions on a discrete domain

Could we say every point of functions on a discrete domain is local maximum and local minimum? (as long as we are not concerned with strict maximum and minimum) For example, if we we define a ...
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2answers
27 views

Finding the probability space of the given experiment.

Specify the probability space completely for the following experiment: tossing a fair coin till we see the first heads. Here is what I have done so far: The sample space is simply $T^n H$ where $n$ ...
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1answer
24 views

What does “modulo equivalence relationship” mean?

I am reading something on completion of metric spaces and it says: Let $\hat S$ be $\mathcal{C}$ modulo equivalence relationship of co-Cauchy sequences. Where $\mathcal{C}$ is the set is all ...
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28 views

Definition of mathematical expression

According to wikipedia: "In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical ...
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1answer
22 views

Understanding the definition of an Interface

Im starting to learn about modeling of moving interfaces and am feeling daft about the basic definition itself: Given an $n$ dimensional space $\Omega$ an interface $\Gamma$ is a co-dimension 1 ...
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2answers
16 views

A small confusing part in the definition of initial value problem

Assume $U\subset\mathbb{R}\times\mathbb{R}^{n}=\mathbb{R}^{n+1}$, $U$ is open and $(t_0, \bf{x}$$_0)\in U$. Assume ${\bf f} (= {\bf f}(t,{\bf x})) : U \to \mathbb{R}$ is continuous. Then the following ...
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17 views

“Equidecomposable”: informal meaning

I am having trouble understanding the definition of the term "equidecomposable". Is it like two sets are split into many sets and then these many sets can be joined together to make either of the two ...
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1answer
135 views

Do mathematical objects have underlying types?

I came upon this issue while I was trying to think about what "type" of object is $\mathbb R^n$ - is it a set, a vector space, an inner product space, an affine space, a metric space? Perhaps the ...
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2answers
38 views

Definition of $L^p$ norm of a vector-valued function

If $u$ is a vector the definition of the discrete norm will be $$\|u\|_{l^p}=(\sum |u_i|^p)^{1/p},$$ If $u$ is a function, $$\|u\|_{L^p}=\left(\int|u|^p\right)^{1/p}$$ But when $u$ is a vector-valued ...
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1answer
26 views

Difference finitely many and arbitrarily many

Is there a difference between "finitely many" and "arbitrarily many"? Some notes I am reading are making a point of distinguishing between the two and I thought they meant the same thing.
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60 views

Is there an open / established ontology for mathematics?

I am currently thinking about how to bundle some of the efforts of students to get / create good educational material. One idea of this little project (wiki-ed - still in the very early phase) is to ...
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2answers
26 views

Is there a concrete definition/formula for finding the leading coefficient of any polynomial?

Is there a concrete definition that tells one the leading coefficient of any polynomial? Using logic, I derived this formula: $$ a=\frac{\frac{d^p}{dx^p}f(x)}{p!}$$ where $f(x)$ is a polynomial, $p$ ...
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1answer
29 views

Some clarification on this textbook's definition of linear ODEs?

The textbook I am reading (Zill's "A First Course in Differential Equations with Modeling Applications) describes classifying ODEs as linear vs. nonlinear with the following statement: An nth-order ...
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2answers
88 views

How to formally write down mapping to a category with math notation?

I'm writing a paper and would like to describe my approach with formulas too. However, I have a problem with writing down the following mapping step (just a tiny step of my algorithm). Imagine, that ...
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0answers
19 views

stable cup-i product

I am confused on the definition of the cup-i product in page 15 Mosher and Tangora's cohomology operations and applications in homotopy theory. Let $X$ be a simplicial complex and $S^\infty$ be the ...
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1answer
34 views

Alternate definitions of width (of a partial order) without Choice?

Say an antichain of a poset $P$ is a set of pairwise incomparable elements of $P.$ Typically, the width of a partial order is defined to be the supremum of the cardinalities of antichains of $P.$ When ...
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1answer
23 views

Definition of limit of sequences in text (Taylor's Foundation of Analysis)

Definiton. A sequence ${a_{n}}$ of real numbers is said to convergence to the number $a$, or have limit equal to $a$, if, for each $\varepsilon >0$, there is a 'real number' $N$ such that $\mid ...
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2answers
58 views

Defining vertical tangent lines

In looking at the definition of vertical tangent lines in some popular calculus texts, I noticed that there are a few different definitions for this term, including the following: A function $f$ ...
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3answers
80 views

Why the terminology “global fields” and “local fields”

Let $p$ be a prime number. A global field is defined as a finite extension of $\mathbb Q$ or $\mathbb F_p(t)$. On the other hand one can show that a local field (which is by definition a complete ...
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1answer
37 views

What is the precise relationship between vector space and vecctor field?

I have looked up precise definition of a vector field and a vector space but I could not understand the relationship between them. On wikipedia A vector field is: Given a subset $S$ in $R^n$, a ...
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1answer
47 views

Does the sum of exterior angles of a simple, convex polygon truly = 360°?

The question being asked of me is the following: What is the sum of a polygon's exterior angles? Assuming, again, that the polygon is simple and convex, the answer I see repeatedly given is ...
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3answers
84 views

Is there a relationship between isometry as defined on metric spaces and those on vector spaces?

I am taking a course on linear algebra and another on real analysis. In linear algebra we defined that two vector spaces are isomorphic if there existed a bijective and linear map between ...
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46 views

Why does the dissimilar looking operation on two different sets of numbers have the same name of multiplication?

The operation of incremented addition i.e. $2 \times 3$ is $2 + 2 + 2$ or $3+3$, is termed multiplication. The following operation on rational numbers $\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$ ...
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26 views

Common distribution function of continuous and discrete random variables

Given two random variables, one discrete (X) and the other one continuous (Y), and given $P_X$ and $f_Y$, how would you refer to their common distribution function? As $P_{X,Y}$ or $f_{X,Y}$?
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1answer
45 views

How many dots do I have to write?

This seems very odd and silly. But I do not know where else to ask. This question occurs to me whenever I write an infinite sequence, sum or decimal points etc. Ex: $ 1.2 + 2.3 + 3.4 + ……………$ Ex: ...
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1answer
58 views

Evolution of Definitions

I try to understand how the definitions of mathematics have evolved (or formulated)... I'll use the epsilon-delta continuity definition as an example to ask my question... It may seem trivial, but ...
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2answers
45 views

Are the limit points of a sequence and of a set defined differently?

Are limit points of a sequence and an interval differently defined? I ask because I've been given the following definition of a limit point: A limit point is one such that every of its ...
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0answers
105 views

What is forcing isomorphism? [closed]

This question is from Kunen's set theory book. My questions are: What is the definition of isomorphism between forcing notions? When do we say that two forcing notions $\mathbb{P},\mathbb{Q}$ ...
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3answers
24 views

How is countably infinite addition defined

In the axiom of additivity of probability theory, the concept of a countably infinite sum, i. e. the sum of countably infinitely many real numbers, is used. Could someone please tell me how that kind ...
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1answer
21 views

What does “proportional” mean?

I used to think of $\propto$ as indicating the one quantity is proportional to the other, with possibly an additive constant involved, i.e. $f(x) \propto g(x)$ if $f(x) = ag(x) + b$. Is that ...