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

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2answers
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Closed Subspaces of Vector Spaces

Question: In Functional Analysis we can note things like: every closed subspace of a Banach space is Banach. In this case, what does "closed subspace" mean? Does this mean closed under the norm ...
11
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1answer
5k views

Difference between a theorem and a law

There are plenty of theorems out there as well as laws within mathematics. For example, in Boolean algebra: Theorems Idempotent Involution Theorem of Complementarity Laws Commutative ...
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1answer
2k views

Relationship between tuples, vectors and column/row matrices

I am taking a course in linear algebra at the moment, and the book I have uses $1\times n$ matrices, $n\times 1$ matrices and $n$-tuples to represent vectors. In condition I have been taught that ...
11
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3answers
685 views

Differentiable at a point

My roommates and I have an argument you guys can help to settle (peace is at stake, don't let us down!) In undergrad calculus courses, one usually explains what it means for a function to be ...
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8answers
415 views

Is the 'variable' in 'let $y=f(x)$' free, bound, or neither?

Consider the string 'Let $y = f(x)$." Suppose that it occurs in some elementary context, such as when graphing the function $f$ using $x$/$y$ coordinates. How is this to be understood in predicate ...
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1answer
124 views

does the definition of model depend on a theory or just a signature?

Σ:signature T:Σ-theory For M:Σ-model which satisfies T, is it proper to name it “Σ,T-model” or “Σ-model satisfying T”? In comparison, in the context of Lawvere theories, any Lawvere theory L ...
12
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2answers
658 views

Formalizing Those Readings of Leibniz Notation that Don't Appeal to Infinitesimals/Differentials

[disclaimer: I've studied a lot of logic but never been good at analysis, so that's the angle I'm coming from below] in my attempt to find a precise version of the 'definitions' usually given when ...
4
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1answer
349 views

Definition of simplicial approximation

I have been given the following definition of simplicial approximation in lectures: Let $K, L$ be simplicial complexes and $f : |K| \to |L|$ be a continuous map of their polyhedra. A simplicial ...
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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 ...
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2answers
294 views

Almost surely in definition of Martingale

$(X_n), n \in \mathbb{N}$ is a stochastic process. I saw in one definition of Martingale that $$E [X_{n+1} |X_0 , X_1 , . . . , X_n ] = X_n \quad a.s., \forall n \geq 0.$$ I understand what "almost ...
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5answers
3k views

Difference between kernel and function?

I have been looking around for this question, but all results I found only describe the definition and not the answer I seek. Is "kernel" basically a synonym of "function"? When should be the time we ...
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2answers
660 views

Are negative or noninteger powers still power series?

I saw definitions and theorem about power series are in the form of $\sum_{k=0}^n a_k (x-x_0)^k$. And it definitely doesn't include negative or noninteger powers. Nevertheless, I saw the theorems like ...
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1answer
4k views

What is a standard curve?

I know you may find definitions out there for standard curve. But, how can we define it in a way that makes it more understandable and clear of what it does?
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2answers
164 views

The Polynomial concept needs to include both variables and contants?

As in Wikipedia: In mathematics, a polynomial is an expression of finite length constructed from variables (also known as indeterminates) and constants. So, it's only considered a polynomial if ...
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2answers
495 views

Proving that a point is a Stationary point with the definition of the derivative

How do you demonstrate that a local minimum of a function has its derivative equal to zero with the definition of the derivative.
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2answers
182 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 ...
4
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2answers
451 views

Describability of a set

As Wikipedia says, there are two ways to describe a particular set: intensional definition and extensional definition. For a set able to be described by extensional definition, is it necessary and ...
15
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3answers
586 views

Generalization of a ring?

I've just started learning about rings. Rings are one additive abelian group strung together, through the associative law, with another structured operation. Couldn't we continue stringing together ...
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2answers
542 views

Definition of Symplectic Matrix

In Wikipedia and MathPlanet an equivalent definition of a symplectic matrix is given: $$\left( \begin{array}{ccc} A & B \\ C & D \end{array} \right)$$ is symplectic if and only if: ...
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3answers
2k views

What is a special function?

When I read some issues here I see from time to time incorrect references to the field special functions, it might e.g. be a discussion around Dirac's $\delta$-function which is tagged ...
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2answers
412 views

What is a simple loop?

I'm asking here because no textbook or website that I know of gives a definition of the above mentioned term. Since there's no obvious way (that I can think of) to define a normal subloop, I don't see ...
1
vote
1answer
133 views

Field of sets versus a field as an algebraic structure

During my studies of real analysis I've come across the definition of a field (or algebra) of sets. My question is: What is the connection between this structure and the structure of a field or ...
4
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1answer
424 views

Definition of direct image?

I read a little remark in a topology text that the direct image of an open set under a continuous mapping is not necessarily open. What is the definition of direct image in this case? I tried ...
4
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1answer
545 views

Definitions for limsup and liminf

I was wondering what are the general spaces that the concepts limsup and liminf can apply to? Is complete lattice one of them? Also How about metric space? What are limsup and liminf specified with ...
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3answers
2k views

What is the proper geometric description of a the oval used for a horse racetrack?

I'm talking about the shape made up of a rectangle with a semi-circle at each end. Does it have a particular name?
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4answers
1k views

Alternate definition of prime number

I know the definition of prime number when dealing with integers, but I can't understand why the following definition also works: A prime is a quantity $p$ such that whenever $p$ is a factor of ...
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7answers
2k views

Why do we require a topological space to be closed under finite intersection?

In the definition of topological space, we require the intersection of a finite number of open sets to be open while we require the arbitrary union of open sets to be open. why is this? I'm assuming ...
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3answers
2k views

Why doesn't 0 being a prime ideal in Z imply that 0 is a prime number?

I know that 1 is not a prime number because $1\cdot\mathbb Z=\mathbb Z$ is, by convention, not a prime ideal in the ring $\mathbb Z$. However, since $\mathbb Z$ is a domain, $0\cdot\mathbb Z=\{0\}$ ...
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6answers
5k views

What is the Direction of a Zero (Null) Vector?

To be more precise, I am interested in knowing if the intuition that a Euclidean zero vector does not have a particular direction is actually correct, and if there is a rigorous formulation that would ...
8
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7answers
2k views

Infinite limits

Does a limit that has the value of infinite exist or not? I've recently come across certain sources that say that if the value of a limit is infinite, then that limit does not exist. This contradicts ...
28
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11answers
4k views

How do you define functions for non-mathematicians?

I'm teaching a College Algebra class in the upcoming semester, and only a small portion of the students will be moving on to further mathematics. The class is built around functions, so I need to ...
8
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2answers
468 views

Definition of a set

What is a set? I know that results such as Russell's paradox mean that the definition isn't as straight forward as one might expect.
4
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2answers
11k views

Approximation symbol: Is π ≈ 3.14.. equal to π ≒ 3.14..?

This could be a trivial question, but what is exactly the difference of between these two expressions? Am I correct to state the both interchangeably whenever I need to express the approximation of ...
9
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5answers
766 views

Are $x \cdot 0 = 0$, $x \cdot 1 = x$, and $-(-x) = x$ axioms?

Context: Rings. Are $x \cdot 0 = 0$ and $x \cdot 1 = x$ and $-(-x) = x$ axioms? Arguably three questions in one, but since they all are properties of the multiplication, I'll try my luck...
3
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2answers
570 views

Unital homomorphism

What is a unital homomorphism? Why are they important?
21
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9answers
1k views

Why is $x^0 = 1$ except when $x = 0$?

Why is any number (other than zero) to the power of zero equal to one? Please include in your answer an explanation of why $0^0$ should be undefined.
6
votes
10answers
2k views

What is a real number (also rational, decimal, integer, natural, cardinal, ordinal…)?

In mathematics there seem to be a lot of different types of numbers. What exactly are: Real numbers Integers Rationals numbers Decimals Complex numbers Natural numbers Cardinals Ordinals And as ...
7
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12answers
1k views

What is a limit?

This limit thing keeps coming up in my calculus textbook. What is it?