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

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1answer
6 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
10 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 ...
2
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2answers
64 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
36 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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1answer
35 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 ...
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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 ...
4
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1answer
481 views

Tangent space to a surface at boundary points

Let $M$ be a $2$-dimensional compact oriented surface in $\mathbb R^3$ with boundary $\partial M$. For any $p\in M \setminus \partial M$ tangent vectors are defined as speed vectors of smooth curves $\...
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2answers
35 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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1answer
39 views

Formal definition of “proexample”. [on hold]

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
54 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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7answers
2k views

What is the square root of complex number i?

Square root of number -1 defined as i, then what is the square root of complex number i?, I would say it should be j as logic suggests but it's not defined in quaternion theory in that way, am I wrong?...
3
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1answer
48 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
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 ...
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5answers
904 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”...
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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3answers
488 views

Alternative definition of hyperbolic cosine without relying on exponential function

Ordinary trigonometric functions are defined independently of exponential function, and then shown to be related to it by Euler's formula. Can one define hyperbolic cosine so that the formula $$\...
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0answers
36 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 ...
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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 ...
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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), ...
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 ...
25
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6answers
3k views

Chain Rule Intuition

We know that the chain rule is used to differentiate a composite function ,say $$f(x) = h(g(x))$$ It's defined as the derivative of the outside function times the derivative of the inner function or ...
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1answer
66 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 ...
388
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21answers
66k views

What are imaginary numbers?

At school I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number which has something to do with the square root of $-1$. When I ...
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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
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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4answers
667 views

What is the “Principle of permanence”?

While reading the book "The Number-System of Algebra (2nd edition)." term "Principle of permanence" occurred to me. I remember I had read this in the book "Beginning algebra for college students.". I ...
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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 ...
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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 $a,b,a^{-1},b^{-1}$, ...
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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.
3
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1answer
27 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 ...
0
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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
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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
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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 ...
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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 ...
3
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3answers
113 views

Strange question about free abelian group

I was given the following question for homework, but it makes no sense to me. Let $F$ be a free abelian group over a set $S$ with respect to the function $\varphi \colon S \to F$. Identify the set ...
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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, ...
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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 ...
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3answers
129 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 $x$...
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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 \...
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4answers
530 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
30 views

Equivalent definition of Lebesgue measurability for sets?

When introducing measurability, we noted that we wanted the following property to hold for disjoint $A, B \in \mathcal{P}(\mathbb{R})$ $m(A \cup B) = m(A)+m(B)$ (additivity) We then defined a set A ...
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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 ...
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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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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
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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"?
4
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1answer
463 views

Question about Definition of Boundary in Stokes' Theorem

I was wondering if what my teacher said was correct and complete in that in Stokes' Theorem the "boundary curve" of a surface can be defined as the mapping along the boundary of the two-dimensional ...