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

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4 views

Characterize in terms of fibre

I am not familiar with the notion "characterize" in the following context. Does this mean to redefine or?.... Any help would be appreciated. Thank you. For a function $f:X\to Y$, and y an element of ...
3
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2answers
55 views

Are all operations functions?

I have looked at Wikipedia(I know it's not completely reliable) but on it an operation is formally defined as: "A function ω is a function of the form $ω : V → Y$, where $V ⊂ X_1 × … × X_k$." and I ...
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2answers
20 views

Fibers of an Element

Prepping myself for a graduate abstract course and we are using Dummit and Foote's Text. We are starting on Chapter 1, so I thought it would be a good idea to go over chapter 0. There is a term that ...
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0answers
5 views

Relation between Gâteaux derivatives and partial derivatives

Definition Let $V_1,...,V_n,W$ be nonzero normed spaces over $\mathbb{K}$ and $E$ be open in $ \prod_{i=1}^n V_i$ and $p\in E$. Define $U_i=\{a\in V_i : ...
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1answer
37 views

Does it matter if you use big $L$ or little $l$ when talking about $L$-norms?

I was reading a post on Quora regarding the application of "$l_1$", "$l_2$" norms for convex linear programming when I became very confused at which $L$-norm the posters are actually referring to. I ...
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1answer
15 views

Question about usage of $\leq$ in definition of Nash equilibrium

Quick definition: Given $g$, a strategy N-tuple $u^* = (u_1^*,...,u^*_N)$ is said to be a Nash equilibrium if: $$J_i(u_i^*, u^*_{-i}) \leq J_i(u_i, u^*_{-i}), i \in N$$ where $J$ is ...
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0answers
25 views

Name for kind of big O notation with leading coefficient

Context: As known the big O notation $O(f(n))$ describes a function $g(n)$ such that there is a constant $C \ge 0$ with $\limsup_{n\to\infty} \left|\frac{g(n)}{f(n)}\right| \le C$ (I assume that ...
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3answers
388 views

Is every axiom in the definition of a vector space necessary?

Definition: A vector space over a field $K$ consists of a set $V$ and two binary operations $+: V \times V \to V$ and $\cdot: K \times V \to V$ satisfying the following axioms: ...
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0answers
31 views

Interval arithmetic with different definitions of intervals

Interval arithmetic normally deals with intervals defined as $[a,b]$ with rules like $$[a,b]\cdot[c,d]=[a+c,b+d]$$ I am interested in interval arithmetic with different interval definitions such as ...
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1answer
35 views

Is there a name for operators of the type $A: M \to M$

In some theorem in functional analysis I have noticed that it is important to assume that an operator $A: M \to M$ where $M$ is some set plus conditions, as opposed to $A: M \to N, M \neq N$ Is ...
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1answer
20 views

Can someone please help me understand what a “player set” is in extensive form game

my text defines player set as: In N-player game $g$, each non-terminating node is partitioned into $N+1$ sets $g^0, ... g^N$. These are player sets. However it makes no attempt to identify ...
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1answer
33 views

Name for some kind of logarithmic norm/error

As known $(\mathbb R, +)$ and $(\mathbb R^{+}, \cdot)$ are isomorphic with $\exp:\mathbb R\to\mathbb R^{+}$ as an isomorphism. When I transfer the absolute value $|\cdot|$ on $(\mathbb R, +)$ via ...
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0answers
20 views

“Second kind” orthogonal polynomials and functions

Recently I've been doing reading in the subject of orthogonal polynomials on the real line (OPRL). Such OPs arise in solving the three-term recurrence relation $$x ...
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1answer
46 views

Where does this definition of convex set come from?

My textbook says the following. We call a set $E\subset R^k$ convex if $$\lambda x+(1-\lambda ) y\in E$$ whenever $x\in E, y\in E, 0<\lambda <1.$ I'm totally at a loss as to what this ...
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1answer
303 views

What is this No thing?

What exactly is this No? Is there any other use of it other than graphs? Thank you so much. I am not trying to cram or anything it's just that I took a course online and a lot of the time it focused ...
3
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1answer
45 views

Why is the solution of an ordinary differential equation required to be defined on an interval?

I am reading A First Course in Differential Equations with Modeling Applications (10th Edition) and here is a definition: Any function $\phi$, defined on an interval $I$ and possessing at least ...
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1answer
38 views

Orthogonality v. Perpendicularity

In Intro to Linear Algebra (my class) two vectors are defined to be orthogonal if their dot product is zero. And the dot product of two $n$-vectors $\vec a\cdot\vec b=0$ means that the two vectors are ...
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3answers
393 views

Definition of homeomorphic?

I am looking up the definition of "homeomorphic" and the source I am looking at says there are two different definitions: Possessing similarity of form, Continuous, one-to-one, in surjection, ...
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9answers
391 views

Motivation for the Definition of Compact Space

A compact topological space is defined as a space, $C$, such that for any set $\mathcal{A}$ of open sets such that $C \subseteq \bigcup_{U\in \mathcal{A}} U$, there is finite set $\mathcal{A'} ...
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1answer
20 views

A conditioned on B is independent of C

Let $A,B,C$ be measurable sets on a probability space. I'm trying to understand the meaning of the sentence: A conditioned on B is independent of C. The conditional probability was defined as: $$ ...
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1answer
19 views

Meaning of 3-disjoint

Definition: Two edges $\{x, y\}$ and $\{w, z\}$ of $G$ are said to be 3-disjoint if the induced subgraph of $G$ on $\{x, y, w, z\}$ consists of exactly two disjoint edges. (See page 5 of this file.) ...
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2answers
66 views

How are asymptotes actually defined in rigorous mathematics?

This question is coming from the analytic geometry viewpoint. Please ignore the viewpoint of algebraic geometry here, unless that viewpoint is somehow able to handle non-algebraic curves like $x ...
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0answers
34 views

Definition of triangle

If a polygon has 3 sides, but one side has zero length (or one angle is zero degree), is it still a triangle by definition of triangle? and how about if it has 2 sides, if not 3 sides are zero?
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1answer
49 views

Can anyone please help to clarify the sentences “ into a fat tail part in L2 plus a fat body part in L1.”

In the link https://en.wikipedia.org/wiki/Fourier_transform#On_Lp_spaces what does this sentences mean: into a fat tail part in L2 plus a fat body part in L1? Would anyone please help?
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0answers
30 views

Notation/definition problem for commutative binary operation

I'm trying to describe/define the commutative binary operation on a three-element set which when the operands are the same, gives the same element and when they are different gives the element which ...
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1answer
39 views

Definition in Operator Theory

I have started learning some Operator Theory. I encountered the following definition. I would like to know why it is that the $f(z)$ in the integrand and the $f(a)$ are both labelled as $f$ where it ...
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3answers
37 views

Axioms for Group Action

Reading the Wikipedia article on group action, I am wondering, why are the axioms stipulating that a group action obey both "compatibility" and "identity"? If a group action is merely a group ...
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2answers
57 views

Why is it called a “multiset”?

According to Wolfram MathWorld, "A set is a finite or infinite collection of objects in which order has no significance, and multiplicity is generally also ignored ..." and A multiset is "A ...
7
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1answer
112 views

What is $\varphi(0)$? [duplicate]

$\varphi$ is Euler's totient function. My question is: When/if $\varphi$ is defined at $0$, what is it usually defined as? Is there a "most natural" or more commonly accepted definition of ...
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0answers
41 views

Can we have different methods to estimate elements from Lp spaces?

Sorry if my question is vague. Consider I have some time samples and it is known to be summation of sinusoidal. Problem is to estimates the frequencies. Generally, Fast Fourier transform (FFT) is the ...
2
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1answer
21 views

question about the definition of Embedding

Suppose that we have two finite geberated vector spaces $V$ and $S$ over a field $\mathbb{K}$. Let $\phi:V \rightarrow S$ a function. What does it mean that $\phi$ is an embedding?
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3answers
68 views

In the real domain, are there any theorems or definitions that state all functions are differentiable? [closed]

I want to ask about basic theory of calculus, say differentiation. We know that not every function can be integrable, but as far as I know all functions are differentiable in the real domain. My ...
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0answers
15 views

Probability Law of Stochastic Process Definition

I am reading Probability and Stochastics by Çınlar, and am confused by the following definition in it: I must be missing something because this definition does not seem correct to me. For ...
2
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2answers
42 views

Mathematical formalism for the “dot product” of three vectors

I know that the dot product of two vectors is the sum of element-wise multiplication. Using pseudo-MATLAB notation: (x,y) = sum(x.*y). I'm interested in ...
0
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1answer
24 views

The definition of a functional structure on a topological space

I am reading Glen E. Bredon's Topology and Geometry and trying to solve the following problem in the book. But I can't understand what the blue g means. Could anyone explain please? Show that a ...
2
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1answer
36 views

Confusion regarding the $\omega$-limit of a set in a flow

In Salamon's Connected Simple Systems, p.8, the author writes that the $\omega$-limit of a set $Y$ inside a flow $\Gamma$ has the two equivalent descriptions $$ \omega(Y) = I(\overline{Y \cdot ...
3
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2answers
59 views

Name of the mathematical term $\frac{a-b}{a+b}$

I calculated a kind of proportion between $a$ and $b$, not $\frac{a}{b}$ but $$\frac{a-b}{a+b}$$ Do this mathematical term has a name? If so, it would help me to explain my calculations...
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0answers
29 views

Trouble With the Definition of $\Delta$-Complex in Hatcher's Book

On Pg. 103 of Hatcher's Algebraic Topology the author has defined what a $\Delta$-complex is: This is what I have gathered from what the author writes: A $\Delta$-complex is a collection of ...
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2answers
15 views

What is the corresponding term for center of mass for a two-dimensional shape?

What is the term for the point on closed surface with no holes which would correspond to the point on that surface directly above the center of mass for a 3-dimensional figure of constant density and ...
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0answers
12 views

Is there relation between vector valued RKHS and interpolation space?

Vector valued RKHS which is covered extensively in the book "Pick Interpolation and Hilbert function spaces" . On the other hand interpolation space which is defined in the wikipedia link: ...
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1answer
120 views

A matrix is positive if and only if it is Hermitian and its eigenvalues are positive [duplicate]

I want to show the equivalence of two definition of positivity. Let $A \in \mathcal{L}(H)$, where $\mathcal{H}$ is the $n-$dimensional Hilbert Space $\mathbb{C}^n$. $A$ is positive if $\langle x,Ax ...
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16answers
5k views

Why do we not have to prove definitions?

I am a beginning level math student and I read recently (in a book written by a Ph. D in Mathematical Education) that mathematical definitions do not get "proven." As in they can't be proven. Why not? ...
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20 views

How is the line integration defined in the most general setting?

A natural generalization of Riemann-Stieltjes integration is Lebesgue integration. Some would say that the use of Lebesgue integration would be overkill when treating differentiable or continuous ...
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0answers
32 views

Open balls in the definition of a Euclidean submanifold

Stroock (Essentials of Integration Theory For Analysis, $\S8.3.4$) defines a $k$-dimensional submanifold of $\mathbb{R}^n$ to be a set $M \subseteq \mathbb{R}^n$ such that for all $x \in M$, there ...
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2answers
46 views

What is meant by “functional analysis is the study of vector spaces endowed with a topology” [closed]

Lecture notes on Functional Analysis by Razvan Gelca open with the definition: Functional analysis is the study of vector spaces endowed with a topology, and of the maps between such spaces. ...
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3answers
216 views

How is the epsilon-delta definition of continuity equivalent to the following statement?

Claim: A function $f: \mathbb{X} \to \mathbb{Y}$ is continuous if given any open set $\mathbb{U} \subseteq \mathbb{Y}$ the inverse image $f^{-1} (\mathbb{U}) \subseteq \mathbb{X}$ is open. How is ...
5
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1answer
47 views

Is a formal deformation of a Lie algebra an example of a formal group law?

I stumbled across the following definition of the formal deformation of a Lie algebra, and it looks like a group object in the category of formal schemes (not necessarily commutative or ...
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1answer
29 views

Trouble Understanding the Statement of a Theorem in Hatcher's Book

Let $X$ and $Y$ be topological spaces and $A$ and $B$ be subspaces of $X$ and $Y$ respectively. We write $f:(X,A)\to (Y, B)$ as a shorthand for writing $f:X\to Y$ and $f(A)\subseteq B$. Now ...
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0answers
73 views

What is an algebra and what is it's representation?

Heyho, i've kind of got an understanding problem what exactely an algebra and especially it's representation is. In my case it is said, that the relation $R_{12}(u-v) (L(u) \otimes \mathrm{I}) \; ...
4
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2answers
152 views

Is $e^x=\exp(x)$ and why?

In the comments to this question a discussion came up wether we have $e^x=\exp(x)$ by definition and what the "correct" definition of $\exp(x)$ is. Building on that, I want to line out the problem ...