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

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2
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1answer
79 views

Is the function $f(x)=x$ on $\{\pm\frac1n:n\in\Bbb N\}$ differentiable at $0$?

This is really a question of definitions. If a function $f$ is not defined on an open set containing $x$, how do we define the derivative of $f$? Is it sufficient to be locally approximable by linear ...
2
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1answer
33 views

Definition of an integrand

General Question: Say we have integral $$ \int f(z)\ dz $$ Is the integrand in this context (i) $f(z)$ or (ii) $f(z)\ dz$? In any case, is $f(z)\ dz$ a formally defined mathematical object in its ...
2
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1answer
39 views

Why do we define product of morphisms in this way

I'm always forget the definition of product of morphisms in a category, maybe the main reason is because I don't know the motivation beyond the definition: I need help to see this abstract ...
2
votes
1answer
321 views

What is the most accurate definition of the hyperboloid model of hyperbolic geometry?

For simplicity, let's focus on the two-dimensional case (the hyperbolic plane in 3-space). I have seen the hyperboloid model defined as variations on the following i) The positive sheet of a ...
2
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1answer
683 views

About the epsilon definition of a convergent sequence. Is this definition equivalent?

I read that it is appreciated to include the context and motivation of a question. I may have overdone this a little bit in this question. To summarize, my question is: Are the two blockquoted ...
2
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1answer
1k views

Definition of “the surface measure”?

Let $\mu_n$ be the $n$-dimensional Lebesgue measure. Let $||\cdot||$ be a norm on $\mathbb{R}^n$. Define $S^{n-1}=\{x\in\mathbb{R}:||x||=1\}$. I have proven that $\forall A\in\mathscr{B}_{S^{n-1}}, ...
2
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2answers
124 views

definition of primes for higher hyperoperations

I was reading yesterday when I came across the history of counting. There was some evidence of an early understanding of prime numbers. I thought that I would try changing the definition of primality ...
2
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1answer
32 views

Generated equivalence relations in logics

Let $L$ be some logic (FO or stronger which is not important for this purpose). Given a $\tau$-structure $A$ and a formula $\varphi(x_1, \dots x_n) \in L[\tau]$ with free variables $x_1, \dots, x_n$. ...
2
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1answer
84 views

What needs to be linear for the problem to be considered linear?

Harry Altman presented an excellent question in a comment here: What needs to be linear for the problem to be considered linear? So is it enough to a have linear objective function or other ...
2
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1answer
125 views

Sample size from population?

This is probably very rudimentary maths, but given a strict population size ($N = 20$ for example), is the sample size any number $<N$? For use in calculation of confidence intervals using a ...
2
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1answer
45 views

How to define “closer to proportion”

$\def\prop#1#2#3{#1:#2:#3}$Let's say I have a proportion, $\prop 135$ And then a set of other ones, $$ \begin{array}{l} \prop 235\\ \prop 145 \\ \prop 136 \end{array} $$ Which one is closer to ...
2
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1answer
171 views

Help with definition: partition mod ultrafilter.

I do not see how the partition is well-defined. By definition $A\neq\varnothing\mbox{ mod }D\iff A\notin I_{D}$. Since D is a maximal filter $A\notin I_{D}\iff A\in D$ . So ...
2
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1answer
352 views

The definition of $p$ capacity of a set $A\subset\mathbb{R}^n$

I am having a bit of difficult understanding the definition of the $p$-capacity of a set $A\subset\mathbb{R}^n$ and I was wondering if anyone would be able to clarify whether I have the right idea or ...
2
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1answer
3k views

Definition of local maxima, local minima

Wikipedia says that: A real-valued function $f$ defined on a real line is said to have a local (or relative) maximum point at the point $x^*$, if there exists some $\varepsilon > 0$ such that ...
2
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1answer
186 views

how to write the process of decomposition of a graph into shortest closed sub graphs

If I want to decompose a graph in to possible shortest closed cycles (as shown in right side). then how can i describe this process with mathematical notations. to understand please refer below ...
2
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1answer
435 views

What is a tangential gradient?

If $\mathbb{R}_{+}^{n}$ is the half space $\{x\in\mathbb{R}^n\vert\ x_n>0\}$ and $u$ is a twice differentiable function in $\mathbb{R}^{n}$. If we write: \begin{equation} ...
2
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1answer
70 views

Why Can't we define the differentiation of vector fields in the same way as in $\mathbb{R^{n}}$

In $\mathbb{R^{n}}$, if $X$ is a vector field on $\mathbb{R^{n}}$, and $X=$$\sum_{i=1}^{i=n}$ $X^{i}$ $\frac{\partial}{\partial x^{i}}$, $X^{i}$ $\in$$C^{\infty}(p)$. Then 1.The differentiation of ...
2
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1answer
79 views

“[T]ransversely isotropic and mirror-symmetric (space group:$D_{\infty h}$)”, its orbifold notation?

I am trying to understand this frieze pattern $D_{\infty h}$ aka its orbifold notation. This describes spider's silk. The authors call it a space group, some sort of generalization from orbifolds. ...
2
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1answer
62 views

Use of the term “normal section” in a theorem of Maria Lucido.

Prop. 3 in this paper (p.135) states Let $G$ be a solvable group with $\text{diam}\Gamma(G)=4$. Then either $l_F(G)\leq 3$ or $l_F(G)=4$ and $G$ has a normal section isomorphic to $H$. ($H$ is ...
2
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1answer
648 views

What is the intuitive meaning of the adjugate matrix?

The definition of the adjugate matrix is easy to understand, but I have never seen it used for anything. What is the intuitive meaning of this matrix? Are there examples of applications which may ...
2
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1answer
120 views

Algebraic Structures : Compositions and More

I am working through the these notes (pdf) for my enumeration class, and I am having some trouble understanding some of the definitions and theorems. Could someone try and provide me with an ...
2
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1answer
1k views

Functional independence

Definition confusion: I wish to show that $$f(x,y)={-y\over x}$$ and $$g(x,y)=\log |x|$$ are functionally independent on some domain. What does that mean? What do I have to show? And how does one ...
2
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1answer
437 views

What is the meaning of $K/F$ is a cyclic extension?

I have it that $K/F$ is a (finite) field extension, what is the definition of when $K/F$ is called cyclic ? I heard it while I studied Galois theory and it was defined as $K/F$ is called cyclic ...
2
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1answer
90 views

Why is $X^- = -\min\{X, 0\}$?

Why is $X^- = -\min\{X, 0\}$ ? This is how it is defined in a probability book I am self-studying. For the function $X:\Omega \to \mathcal{R}$, The positive part of $X$ is the function $X^+ = \max ...
2
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1answer
663 views

What is a rational character?

Let $G$ be the group of $F$-points of a connected, reductive group over a $p$-adic field $F$. The unramified character of $G$ are $\chi\circ\psi$ where $\chi$ is an unramified character of ...
2
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1answer
319 views

Definition of a universal example

I'm not sure how the term is being used here: Let $R$ be a commutative ring and $X_1,\ldots, X_n$ indeterminates over $R$. Set $P = R[X_1, \ldots, X_n]$. Given a ring homomorphism $\phi: R ...
2
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1answer
217 views

define simultaneous substitution recursively

Can you help me with my approach to the following task: Define simultaneous substitution $\phi[\psi_1,...,\psi_k/p_1,...,p_k]$ recursively. Usually we have recursive definitions about formulas, but ...
2
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1answer
39 views

Terminology clarification: ***exchanges***

I need help with a terminology definition. If we say "R is a reflection that exchanges the sides a and b in some triangle", does it mean sides a and b have the same length and the reflection maps one ...
2
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1answer
49 views

Reducts of categories

There are several ways to reduce a category. The skeleton of a category is the category with isomorphic objects collapsed into one i.e. the only isomorphisms that remain are the identities. What's ...
2
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1answer
292 views

Set-theoretical definitions of the notion of “structure”

What general set-theoretical definitions of the notion of "structure" are there? By general definition of "structure" I mean a formula $\Phi(x)$ in the first-order language of set theory such that ...
2
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1answer
160 views

Free presentations of $\mathbb{Z}G$-modules

Dear All, I have a doubt about a specific definition, but I cannot find any help on the web or on the books that I have. Talking about $\mathbb{Z}G$-modules, what does one intend saying "take a free ...
2
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2answers
257 views

Extending “as x approaches a” to “as g(x) approaches a”

All the definitions I can find of a limit (with functions from R to R) define something like: "as x approaches a, f(x) approaches L" Where x is treated as a variable that is quantified over in the ...
2
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1answer
81 views

a function of a dependent type, a section, a sheaf

I have defined this simple sheaf. Take $E:set, B:set, p:E\to B$. Let $P(B)$ be a set of subsets of $B$. Let $S$ be the set of sections of $p$. Let $F$ be a contravariant functor from $P(B)$ as a poset ...
2
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2answers
185 views

Is everything an expression?

Is everything that you can write in math (that makes mathematical sense) an expression? If not, what would be examples of non-expressions? And would all expressions be composed of expressions ...
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0answers
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 ...
2
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0answers
33 views

Proof $f(x,y)=x_1+e^{x_{2}}$ is strictly convex

I am trying to show that $f(x,y)=x_1+e^{x_2}$ is strictly convex. I can show this using the Hessian Matrix which is positive definite. However for some reason i can not put it together using algebra ...
2
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0answers
70 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 ...
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0answers
35 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
20 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 ...
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0answers
17 views

Measure on $S^1$

My book gives the following definition of Lebesgue measure on $S^1$: For each $B\subseteq [0,1]$ in the Borel sigma-algebra on $\mathbb{R}$, define $\mu(B)=\lambda(B)$. This defines a finite measure ...
2
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1answer
36 views

First version of Strong Law - finite vs bounded

Iirc, bounded functions are finite and not all finite functions are bounded. From Williams' Probability w/ Martingales: Why 'finite 4th moment' ? It's technically right I guess, but why not ...
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0answers
21 views

Definition of Saturating set

Notations:$ \bar{G}(A)$ denotes the subgraph induced by $A$ in the complement of $G$. Definition see definition 2.15: Given a set $s$ of vertices, an edge $e$ of $G(V − s)$ is said to be ...
2
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0answers
25 views

Clarify the definition of a transition matrix

I am reading the book Matrix Variate Distribution by A. K. Gupta and D. K. Nagar. In the first chapter (Definition 1.2.8), they define a matrix $B_{p}$ ($p \in \mathbb{N}^{\ast}$) as follows : ...
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0answers
49 views

Is it possible to define submanifold like this

Wikipedia offers the following definition for an (embedded) submanifold: An embedded submanifold (also called a regular submanifold), is an immersed submanifold for which the inclusion map is a ...
2
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2answers
71 views

What is a primitive root?

So I'm trying to learn about RSA and have come across various subtopics, including the discrete logarithm problem. This mentions primitive roots, which I do not understand. Essentially all I want ...
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0answers
40 views

How Kriging, Bochner theorem and Positive definite (PD) function are related?

This question referes to the link: https://en.wikipedia.org/wiki/Kriging I can understand the relation between Bochner's theorem and PD function. But could not properly understand and connect all ...
2
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0answers
67 views

Are there non-holomorphic or non-analytic polynomials of several complex variables?

Having no prior exposure to several complex variables, I am trying to read some papers involving this subject. I came upon the terms analytic polynomial and holomorphic polynomial. Do they simply ...
2
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3answers
98 views

Injective or one-to-one? What is the difference?

What is the difference between the terms 'injective' and 'one-to-one', 'surjective' and 'onto', and 'bijective' and 'isomorphic'?
2
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0answers
45 views

What vector field property means “is the curl of another vector field?”

I know that a vector field $\mathbf{F}$ is called irrotational if $\nabla \times \mathbf{F} = \mathbf{0}$ and conservative if there exists a function $g$ such that $\nabla g = \mathbf{F}$. Under ...
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0answers
46 views

Quartic operator definition

What is a quartic operator? I googled it but found only some articles which use that term whitout giving a definition (I found that term while studying 2D Ising model, and the use of some ...