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

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0answers
8 views

Space of ordered n-tuples

What is space of ordered n-tuples? What makes it different from $\mathbb{R}^n$? What is the definition of space of ordered n-tuples? I just think that it is space, which elements are n-tuples. But ...
23
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5answers
2k views

Are there dictionaries in math?

Consider the following dictionary in the programming language Python: D = {'A': 1, 'B': 2, 'C': 3} It is saying that the value of A is 1, the value of B is 2, ...
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1answer
18 views

Definition of locally integrable function

I was given two definitions: Let the function $f(x)$ be defined in a interval $[a,\infty)$ we will say that $f$ is locally integrable in $[a,\infty)$ if for all $a<b$ $f$ is integrable in $[...
4
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3answers
113 views

Should we re-define Sine?

Sine is usually defined as the ratio of the opposite side to an angle to the hypotenuse in a right angle triangle. Another common definition is based on the unit circle. However I think these ...
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1answer
32 views

Why do we quotient only by equivalence relations?

In textbooks, I've always seen the notion of quotient set defined with equivalence relations, that is: if $R$ is an equivalence relation on a set $X$, we can define the quotient set $X/R = \{[x]_R \...
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1answer
14 views

Nichtnegativstellensatz the same as Handelman's Theorem?

Wikipedia on "Handelman's theorem: If $K$ is a compact polytope in Euclidean $d$-space, defined by linear inequalities $g_i ≥ 0$, and if $f$ is a polynomial in $d$ variables that is positive on $K$, ...
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0answers
11 views

Perfect powers by Oblath's result [duplicate]

What do you mean by this statement? Obl\'ath proved that the only perfect powers all of whose digits are equal to a fixed one $ a \neq 1$ in decimal representation are 4, 8 and 9. This is equivalent ...
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2answers
74 views

Simple question: Which is the Wikipedia definition of axiom of choice

I looked up Wikipedia, and on the top of the page it says: For every indexed family $(S_i)_{i \in I}$ of nonempty sets, there exists an indexed family $(x_i)_{i \in I}$ of elements such that $...
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1answer
55 views

In which topologies do open sets maintain open under countable or arbitrary intersection?

We know that in the usual topology, countable or arbitrary intersection of open sets can zoom into a singleton, hence is not in the topology. I am curious if there is well known classes of ...
0
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1answer
42 views

What subbase generates metric topology?

Let metric topology be the topology generated by metric balls of a metrizable space $X$ Is there a subbase $S$ that generates the metric topology? I am asking because in most textbooks, it seems ...
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2answers
27 views

How can an axiom ever be dependent

An axiom is defined, officially, as 'a statement or proposition which is regarded as being established, accepted, or self-evidently true.' Yet an Independent axiom is one where it is not derived from ...
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1answer
41 views

Formal definition of “proexample”. [closed]

Where in the literature do we find the preferred formal definition of “proexample” as in: the number zero is a proexample for the existential sentence "some integer is neither positive nor negative"? ...
4
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1answer
55 views

What is an injection in a topos?

I am reading the paper of Andrej Bauer that proves there is a realizability topos in which there is an injection from the internal Baire Space $\mathbb{N}^\mathbb{N}$ to the natural numbers $\mathbb{N}...
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2answers
38 views

Useful analogy to interpret the notion of evolutionary stable strategy (ESS)

I am seeking a good analogy to understand the concept of evolutionary stable strategy (state) Let $\pi$ denote the fitness of a population, $\pi_{ij}$ is the fitness of strategy $i$ against strategy $...
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0answers
16 views

Arithmetic-definability of geometrically-defined arithmetic concepts

Arithmetic-definability of geometrically-defined arithmetic concepts For purposes of discussion take arithmetic to be the study of the [ positive real] numbers, sequences of numbers, etc. and take ...
2
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0answers
64 views

Is there a concept of “Cross determinant”?

Suppose $A = \begin{bmatrix}a & b \\ c & d \\\end{bmatrix}$. The determinant of $A$ is $$\det A = ad - bc.$$ Suppose $B = \begin{bmatrix}e & f \\ g & h \\\end{bmatrix}$. Now one could ...
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0answers
39 views

Advantages and disadvantages of a particular definition of rings and subrings

My professor defines a ring as a triple $(A, +,\ \cdot)$ such that $(A, +, 0)$ is an abelian group, $(A,\ \cdot)$ is a semigroup and $\cdot$ distributes over $+$. Subsequently, $B\subseteq A$ is said ...
3
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1answer
50 views

Is there a meaning to the notation “\arg \sup”?

When $f$ is a function on a set $A$, the notation: $\arg\max_{x\in A} f(x)$ denotes the set of elements of $A$ for which $f$ attains its maximum value. This set may be empty, for example, if $f(x)=x$ ...
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0answers
14 views

Definition of pseudo-differential operator

I'm trying to understand the defintion of pseudo-differential operators ($\psi do$) on manifolds. According to Hörmander, The analysis of Linear Partial Differential Operators, v. III, 18.1.20 (1994), ...
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2answers
37 views

What is the difference between a polynomial and a function or can they be used interchangebly?

I have been wondering over this basic question (seems rather trivial at first sight) for a long time- What is the difference between a polynomial and function? My confusion arises form the ...
2
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1answer
55 views

What does it mean in general to show something is well defined? [duplicate]

There is another post that addresses this but quickly fix the problem to be something in arthmetics, and in turn what it means for that arithematics problem to be well defined. I have never ...
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5answers
910 views

Formal definition of “counterexample”.

What is the preferred formal definition of “counterexample” as in: zero is a counterexample for "every integer is either positive or negative". Where in the literature is the notion of “counterexample”...
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0answers
16 views

set-theoretic definition of sum of sequences, difference of sequences, product of sequences,…

I have $f,g \in \Bbb R^\Bbb N$, I define: $(f+g):=\{(y,z)| z= f(y)+g(y) \}$ $(-f):=\{(y,z)| z=-f(y) \}$ $(f-g):=f+(-g)$ $(f\cdot g):=\{(y,z)| z= f(y)\cdot g(y) \}$ Is it correct? I have problem ...
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1answer
67 views

What is the correct definition for positive operator and positive definite operator?

As far as I know those operators are defined as follows: Positive operator is an operator $L: H\rightarrow H$ such that $\langle L\textbf u|\textbf u\rangle \geq0$ for all $\textbf u \in H$ and the ...
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1answer
39 views

Definition of Limits: x=c [closed]

If $0 < |x-a| < \varepsilon$, and it's possible for $x=a$, then wouldn't the equation become $0 < 0 < \varepsilon$ (which is technically impossible)?
2
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1answer
24 views

What is actually the standard definition for Radon measure?

I see that there are various definitions for Radon measure and they are NOT equivalent, but they are equivalent on locally compact Hausdorff spaces. I think this is the reason why Radon measure has ...
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0answers
15 views

Finitely generated module: terminology.

What's the meaning of the expression: $S$ is a subring of $\mathbb{C}$ finitely generated as $\mathbb{Z}$-module? Maybe that the additive group of the ring $S$ is a finitely generated abelian group? ...
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2answers
30 views

How do you rigorously explain the fact that $u \in L^p$ can be non defined over sets of measure 0?

In all the definitions of $L^p(\Omega)$ spaces I have been given these are defined to be the set of functions $f: \Omega \to \mathbb{R}$ whose norm $||\cdot||_{L^p}$ is finite. We define is as the ...
1
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1answer
24 views

Formal construction of free groups and objections in arguments

For simplicity, consider $X=\{a,b\}$. Let $Y$ be another set in bijection with $X$, and write its elements to be $a^{-1},b^{-1}$. Let $W(X)$ be the collection of all words in $a,b,a^{-1},b^{-1}$, ...
3
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1answer
30 views

Is a relation between A and B the same as a mapping from elements of A to subsets of B?

The way I always saw it was that a relation is a subset of $A \times B$, or a collection of ordered pairs $(a,b)$, where $a \in A$ and $b \in B$. Is there any meaningful distinction between the two ...
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1answer
20 views

What should I change in the definition of a function to make it time dependent $f \to f(t)$?

Suppose I have a function that takes from sets $X \subset \mathbb{R}$, $Y \subset\mathbb{R}$ to $\mathbb{R}$ $$d: X \times Y \to \mathbb{R}$$ The elements are identified by $d(x,y) = xy \in \mathbb{...
0
votes
1answer
26 views

formula for defining terms in a finite set

Suppose there's a finite set, $S$ of terms in $\mathbb{R}$ which have the property $P(x)$. Suppose we know how to define the maximum value of the set by the relation, $max(x)$. We also have the ...
5
votes
1answer
117 views

Relation between open sentences and sets (conceptual question)

Hi I'm a college student getting into the more proof oriented side of math. I was reviewing Mathematical Proofs, A Transition to Advanced Mathematics 2nd edition and after thinking about chapters 1 ...
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2answers
46 views

When 2 functions are equal?

Are 2 functions equal when they have same domain, same codomain and same law ? EXAMPLE 1 $f: \mathbb{R} \to \mathbb{R}$ $x \to x^2$ and $g: \mathbb{R} \to \mathbb{R^+_0}$ (set of positive ...
2
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1answer
44 views

Stabilization of embedding?

In D. Freed's lecture notes he mentions "stabilization of embedding" in theorem 4.48. Does anyone know the definition? I can't find it online.
1
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1answer
25 views

Is there a difference between arc-wise connectivity or path-wise connectivity?

When authors refer to arc-wise connectivity, do they mean path-wise connectivity? I am studying space filling curves and when reading books, I either come across the concept of arc-wise connectivity ...
0
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0answers
20 views

For a graded poset, why do we only consider the characteristic polynomial defined in terms of $\mu$?

When we have a graded poset $P$ with $0$ and $1$ we can define the characteristic polynomial $f_P(t)$ of $P$: $$f_P(t)=\sum_{x\in P}\mu(0,x)t^{r(1)-r(x)}$$ However, given a poset, have two functions, ...
3
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3answers
64 views

Confusion Regarding Munkres's Definition of Basis for a Topology

The definition of Basis for a Topology as given in Munkres's book is as follows, If $X$ is a set, a basis for a topology on $X$ is a collection $\mathcal{B}$ of subsets of $X$ (called basis ...
5
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3answers
130 views

Why there is no value for $x$ if $|x| = -1$? [duplicate]

According to the definition of absolute value negative values are forbidden. But what if I tried to solve a equation and the final result came like this: $|x|=-1$ One can say there is no value for $x$...
0
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2answers
32 views

Clarification about the definition of surjectivity

Related to the question Proof that Laplacian is surjective $\mathcal{P}^n\to\mathcal{P}^{n-2}$, I know in general that the surjectivity is defined to be : $\forall f \in \mathcal{P}^{n-2}$, $\exists \...
0
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0answers
33 views

What does it mean for a tangent bundle to be parallel?

I am looking at page 8 of the paper here, just below definition 2.13. I have never come across a phrase such as '$T\mathfrak{C}\subset{TM}$ is parallel' - what does it mean for a tangent bundle to be ...
0
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1answer
20 views

Diameter of a Topological Manifold

I know that for a Riemannian Manifold is defined the concept of diameter. I wuold know if it's defined a similar concept for a most general Topological Manifold. Thanks in advance.
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2answers
52 views

Is there a way to denote a repeated operation?

For instance, if you have say: 4 + x = 10 We instantly calculate 10 - 4 to derive x, which is 6. So you could say there is only 1 operation, subtraction, where 4 ...
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4answers
533 views

Topology, closure definition - well defined?

I came upon the following definition for closure, Given a subset of a topological space $X$, the closure of $A$ is defined as the intersection of all closed sets containing $A$. How is this ...
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1answer
20 views

What is the connection between positive definite hessian and metric?

In heard from a someone in verbatim that if you take the taylor series of a certain function, if the Hessian is positive definite, then it is a metric. This quote is without context and therefore ...
3
votes
1answer
26 views

Why do we call this transformation non-singular?

In linear algebra books, the authors call the linear transformation $T$ with the property $$T(\alpha)=0\implies \alpha=0$$ non-singular. What's the motivation behind the term "non-singular"?
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0answers
48 views

Definition of a Polynomial Function

How is the constant term of a polynomial function evaluated for every real number $x$? I'm working with the following definition: A polynomial $P$ is a function $P:\mathbb R\rightarrow\mathbb R$ ...
0
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0answers
24 views

What is the meaning of the notation: $dx_\alpha\wedge dy_\alpha$,$dzd\bar{z}$, $dwd\bar{w}=du^2+dv^2$

I am studying some notes on introduction to Riemann Surfaces and the notation $dx_\alpha\wedge dy_\alpha$,$dzdd\bar{z}$, $dwd\bar{w}=du^2+dv^2$ keeps popping up everywhere and I have never seen it and ...
2
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2answers
51 views

A discrete topological space is a space where all singletons are open $\implies$ all sets are clopen? Closed?

I know that a discrete topological space is where all singletons are open. For example, $\mathbb{N}$ with the subspace topology inherited from $(\mathbb{R}, \tau_{usual})$. This is the case because ...
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2answers
51 views

Does topological degree generalize to maps that aren't between closed connected orientable manifolds?

From what I gather, the degree of a map originally arose in the context of studying maps $f:S^n\rightarrow S^n$. Since $H_n(S^n)\cong \mathbb{Z}$, the induced map $f_\star$ has the form $x\mapsto kx$ ...