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

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1answer
672 views

Norm of a vector-valued function?

When studying commutator estimates, I have encountered the following problem. Consider $f\in C^1(\mathbb{R}^d,\mathbb{R})$ with $\nabla f\in L^p$. So $\nabla f(x)\in\mathbb{R}^d$. My question is ...
2
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1answer
97 views

What is $\lim_{x=0\rightarrow1}x^\infty$?

I know it's all about definition... But still I want to know whether the answer is $0$, $1$, impossible to say or something else, like that the mathematical statement is wrong. However, to clarify, ...
2
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2answers
168 views

A definition of algebraic expression

Definition of algebraic expression An algebraic expression is a collection of symbols; it may consist of one or more than one terms separated by either a $+$ or $-$ sign. If by symbol we only mean ...
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1answer
81 views

Triangle of Multinomial Coefficients

What is the "Triangle Of Multinomial Coefficients" seen here: http://oeis.org/A036038 (OEIS: A036038) I can see that the diagonals of this triangle are just factorials... for example the last number ...
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3answers
142 views

Calculating derivative by definition vs not by definition

I'm not entirely sure I understand when I need to calculate a derivative using the definition and when I can do it normally. The following two examples confused me: $$ g(x) = \begin{cases} x^2\cdot ...
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1answer
290 views

What is the definition of “formal identity”?

In Ahlfors' Complex Analysis he remarks that harmonic $u(x,y)$ can be expressed as $$ u(x,y) = \frac{1}{2}[f(x + i y) + \overline{f}(x - i y)] $$ when $x$ and $y$ are real. He then writes "It is ...
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1answer
357 views

Perpendicular Symbol as Matrix Superscript

If $A$ is a matrix that is not (necessarily) square, then what is $A^\perp$? What I do know is: $A^\perp$ is a matrix, not the orthogonal complement It is related to the QR Decomposition. And ...
3
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1answer
307 views

Lie derivative along time-dependent vector fields

In "Lectures on Symplectic Geometry" by A. C. da Silva (http://www.math.ist.utl.pt/~acannas/Books/lsg.pdf) the author gives the following definition: $$ \mathcal{L}_{v_t} := \frac{\mathrm d }{\mathrm ...
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4answers
4k views

What is a fraction in which the greatest common factor of the numerator and the denominator is 1?

What is this fraction: A fraction in which the greatest common factor of the numerator and the denominator is 1?
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1answer
104 views

Why this functional isn't differentiable?

I saw in the article Alt, H. M. and Caffarelli, L. A. Existence and regularity for a minimum problem with free boundary. J. Reine Angew. Math., 325, (1981), 105–144. That the functional ...
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1answer
123 views

I don't understand the “rational” arguments for trig functions

For example look at $\cos(x)$. I imagine that $x$ can sometimes be rational (or integer) or irrational, and sometimes $\cos(x)$ can be rational (or integer), or irrational. Under what circumstances ...
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2answers
240 views

Why is a section of a sheaf over closed set defined this way?

Why is a section of a sheaf $F$ over closed set $S \subset X$ is defined as inductive limit $$ \varinjlim_{S\subset U} F(U)\; ?$$ From my point of view, we should define it as a function, which each ...
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1answer
121 views

Definition of the higher dimensional mapping tori

This is proving harder to search for than I imagined. The usual definition of a mapping torus $\mathcal{M}_h$ associated to a homeomorphism $h\colon X\rightarrow X$ on a topological space $X$ is the ...
3
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1answer
77 views

definition of the Fourier transform of function on the sphere

Let $f: S^{n-1}\longrightarrow R^n$ be even continuous function. What is the Fourier transform of $f$?
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1answer
161 views

What is a finite partial function from $\mathfrak c$ to $D=\{0,1\}$

While I reading a paper, there is a notation is called finite partial function. I searched by google, I cannot find its definition. So I post it here as a question: What is a finite partial ...
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1answer
159 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 ...
3
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1answer
53 views

What is the meaning of $f(x) \rightarrow a$ as $g(x) \rightarrow b$?

The motivating example was the case: $$f(x, y)\rightarrow0\mathrm{\ \ as\ \ }\sqrt{x^2+y^2}\rightarrow\infty$$ What exactly does this mean? I might define it as: Any sequence $x_n$ with ...
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6answers
1k views

What does “homomorphism” require that “morphism” doesn't?

I'm starting to learn category theory, but there's one thing I don't get: all morphisms seem to be homomorphisms; the definition seems to be the same. What's the difference between these two? Can you ...
3
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3answers
236 views

what are first and second order logics? [duplicate]

The only knowledge I have on logic is due to a book I read a couple of years ago called Introduction to logic: and to the methodology of deductive sciences by Alfred Tarski. And in it he talks about ...
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1answer
66 views

Definition question of convex orbit of finite group action

Assume that a finite group or discrete group $G$ acts on a manifold $M$. Here what does it mean that orbit $G\cdot x$ is convex ? Thank you in advance.
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1answer
106 views

What is the difference between a reflexive relation and an identitive relation

Given a set $X$ and a relation $R$ over $X$, we say that $R$ is reflexive if \begin{equation} xRx\ \forall\ x\in X. \end{equation} What does 'identitive' mean? Is it the same as antisymmetry? Seen ...
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1answer
70 views

Over-constrained general solution to wave equation

d'Alembert's formula states that the general solution to the one-dimensional wave equation is $$ u(x,t) = f(x+ct) + g(x-ct).$$ for any well-behaved functions $f$ and $g$. This is a well-known and ...
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2answers
128 views

Name for grid system

Is there a name for a type of grid you might find in Battleship? Where coordinates don't relate to points on a grid but rather the squares themselves?
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1answer
90 views

What is metastable range?

Wikipedia states that In 1953 George W. Whitehead showed that there is a metastable range for the homotopy groups of spheres. What is the metastable range? I was unable to find the ...
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2answers
127 views

Why is this a sentence with a quantifier an open sentence?

From these notes on Relational Logic, the following two sentences are given as examples of 1) open and 2) closed sentences. The definition of an open sentence is one with at least free variables. ...
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1answer
212 views

Elliptic Operators

I'm studying Elliptics Operators like this: $$Lu=a_{ij}(x)D_{ij}u+b_i(x)D_iu+c(x)u$$ for $u\in C^2(\Omega)\cap C(\overline{\Omega})$. I want to know what the difference when: $L$ is elliptic in ...
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2answers
73 views

“uniquely written” definition

I'm having troubles with this definition: My problem is with the uniquely part, for example the zero element: $0=0+0$, but $0=0+0+0$ or $0=0+0+0+0+0+0$. Another example, if $m \in ...
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1answer
130 views

question about Graph Theory notation

I'm just starting to learn graph theory. I have two questions about notation: 1). For a graph $G$ we denote the vertex set $V$ and the edge set $E$ by $G=(V,E)$. So we have a graph $G=$ ...
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1answer
283 views

Definition of Hölder space

I am wondering what is the definition for Hölder space $C^\gamma$ when $\gamma\in \mathbb{N}$. Let's take the underlying field $\mathbb{R}^d$. Is it $$ C^\gamma = \{f:\ f\in C^{\gamma-1}\}\cap \{f: ...
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3answers
59 views

Vector bundle definition

Is the condition $$ \pi \circ \varphi (x,v) = x $$ in the definition of a vector bundle needed? In Milnor/Stasheff "Characteristic classes" the definition is given without it.
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1answer
35 views

Help with this definition of $(G:_M I)$

I didn't understand why in this definition $I$ has to be an ideal to make sense. REMARK This is from Steps in Commutative Algebra, page 107. Thanks a lot
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3answers
272 views

Non-symmetric $A^T=A$

Wikipedia says that symmetric matrices are square ones, which have the property $A^T=A$. This assumes that one can have non-square $A^T=A$ and, because it does not satisfy the first property of ...
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1answer
152 views

$\Sigma_k^\text{P}$−SAT definition is not clear to me

I don't understand if by saying there are $k$ alternating quantifiers on the variables $x_1$,$x_2$...$x_k$, It means we quantify ALL variables (there are only $k$ variables in the SAT formula) or just ...
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1answer
44 views

Coordinate-free definition of pseudotensors

How to define pseudotensors (particularly, pseudovectors) in a coordinate-free form? Can it be defined on a manifold (like a tensor field)? Or may be the objects that physicists model via ...
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1answer
39 views

How could I define this $\mathrm{nw}(X)$ by using only one sentence? [closed]

A family $\mathcal N$ of subsets of a topological space $X$ is a network for $X$ if for every point $x\in X$ and any neighbourhood $U$ of $x$ there exists an $M \in \mathcal N$ such that $x\in M ...
2
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1answer
111 views

Understanding equivalent definitions of left cosets

I understand the standard definition of left coset, but the one I do not understand (or see why it is advantageous) is the definition that follows: Let $H\leq G$. Then a left coset of $H$ is a ...
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1answer
180 views

Definition of compact set/subset

I found one exercise in my book Let $X$ be a compact subset of a metric space $M$. Prove that $X$ is closed. In the definitions, the book only mentions compact space and never compact set. ...
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3answers
5k views

What is limit superior and limit inferior?

I've looked at the Wikipedia article, but it seems like gibberish. The only thing I was able to pick out of it was the concept of infimum (greatest lower bound) and supremum (least upper bound), as I ...
3
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1answer
282 views

What's the difference between Abstract Algebra and Group Theory?

I'm slowly beginning a student of certain higher mathematics. I'm trying to see if I would prefer to study Group Theory or Abstract Algebra. I know that Abstract Algebra seems to "come before" ...
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2answers
213 views

Defining infinitesimals

Can such definition of infinitesimals hold? $$\mathrm{d} x :=a:(a>0 \;\And\; \forall b \in \mathbb{R}^+\backslash \{ a \}\;(a<b))$$ And, if the above definiton works, then obviously ...
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1answer
94 views

question about epsilon, delta limit definition

Sometimes, when describing the closeness of $x$ to $a$ as being less than $\delta$, it's stated as $|x-a|<\delta$ and sometimes it's stated as $0<|x-a|<\delta$. What is the " $0<$ " part ...
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3answers
114 views

What does boson-type realization mean?

I have seen several different contexts the expression "boson-type realization", for instance in the study of algebras growth and realization of affine algebras. To be or not be a boson-type ...
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2answers
1k views

Definition of a monoid: clarification needed

I'm only in high school, so excuse my lack of familiarity with most of these terms! A monoid is defined as "an algebraic structure with a single associative binary operation and identity element." ...
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2answers
2k views

What is the difference between isomorphism and homeomorphism?

I have some questions understanding isomorphism. Wikipedia said that isomorphism is bijective homeomorphism I kown that $F$ is a homeomorphism if $F$ and $F^{-1}$ are continuous. So my question ...
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2answers
88 views

Conventions on definitional if(f)

When defining a term it seems common to use 'if' when the stronger 'iff' is also true. For instance: Definition 1: A set $A$ is open in $(X,d)$ if $\forall x \in A$, $\exists \epsilon \gt 0$ such ...
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0answers
149 views

Understanding Gray-Level Hit-or-Miss-Transform Definition

From a paper I am using for my Bachelor thesis: The HMT transform by the pair $(A,B)$ associates to a binary image $X$ the set $X\otimes(A,B)$ of positions where the translate of $A$ fits inside ...
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2answers
103 views

Hartshorne Lemma(II 6.1)

Hi I am wondering what Hartshorne means by "an open affine subset on which $f$ is regular". This is part of the first sentence in the proof of lemma 6.1 in chapter two of the book "Algebraic Geometry ...
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3answers
78 views

Idea of a Random Variable

Two weeks into my class, I am still struggling with the idea of a random variable. I can see why random variables make sense when the outcomes are numbers, like monetary gains and losses. But if ...
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1answer
76 views

Is this definition of diagonal matrix correct?

I need to know if the following definition: Let $A:=\|a_{i,j}\|_{\substack{i=1,...,m \\ j=1,...,m}}$ be a square matrix. $A$ is diagonal matrix if $$i\neq j \implies a_{ij}=0, \quad\forall i,j \in ...
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4answers
465 views

Confusion about the usage of points vs. vectors

As far as definitions go, understand the difference between a vector and a point. A vector can be translated and still be the same vector, whereas a point is fixed. But I would like some clarification ...