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

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1answer
13 views

Different percent-valued Majority gates

It has been defined that a Majority gate follows this formula concerning its behavior about outputs: $$\operatorname{Maj} \left ( p_1,\dots,p_n \right ) = \left \lfloor \frac{1}{2} + ...
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0answers
20 views

Definition of a topology on a set $X$: may $I$ contain uncountable many elements or is it restricted to be finite or countable?

I've the following definition of a topology $\mathcal I$ on a set $X$: (T1) $U_a \in \mathcal I, a \in I \Rightarrow \cup_{a \in I} U_a \in \mathcal I$ (T2) $U_1, U_2 \in \mathcal I \Rightarrow U_1 ...
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1answer
31 views

About the definition of universal covering space

There are some references (for instance in Greenberg & Harper) that consider the universal covering space to be not only simply connected but also locally path connected. This definition seems to ...
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0answers
22 views

Why do we define the ordered pair in this way? [duplicate]

When we define an ordered pair (x,y) in a set, why, in many textbook, do we define it as {x,{x,y}} or {{x},{x,y}} instead of {x,{y}} or {{x},{y}} which obviously makes more intuitive sense if we ...
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0answers
17 views

What is a “nonparametric function”?

I am looking for a formal definition of the term "nonparametric function." I understand the term and I use nonparametric regressions http://en.wikipedia.org/wiki/Nonparametric_regression but I ...
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2answers
68 views

Logical consequence in all structures in Kripke semantics

I read* the following definition of logical consequence in all structures within Kripke semantics:$$X\models A\iff\text{ for every } (W,R),\text{ if }(W,R)\models X,\text{ then }(W,R)\models A$$ ...
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3answers
91 views

Field axioms: Why do we have $ 1 \neq 0$?

In the definitions of a field, we have $ 1 \neq 0$. I know that in regular multiplication $0 \times 1=0$ but for reciprocal we don't have inverse of $0$. But all the spaces and different definitions ...
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1answer
77 views

If a series converges does its sequence of partial sums converge?

By Definition, a series $\displaystyle \sum_{n=1}^\infty a_n$ converges if it's sequence of partial sums $\displaystyle S_n = \sum_{k=1}^n a_k$ converges. My question is, is the converse true? If ...
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0answers
13 views

definition of limit at a point and relationship with gradient

Why is $\lim\limits_{t\to\infty}\frac{f(x^*+td)-f(x^*)}{t}=\nabla f(x^*)^Td$ for $t>0$ and sufficiently small?
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2answers
54 views

Proper divisors of 1?

What are the proper divisors of 1? I understand proper divisors do not include the number itself, but is 1 an exception or does it have no proper divisors?
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0answers
21 views

What the meaning 'directed'

What's the meaning of 'directed' and 'cofinal'.It's about a partially ordered set. Please give me an example? A preordered set (I, ≤) is directed if every finite subset F of I has an upper bound. A ...
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2answers
57 views

Zero: What is it and can I define it? [closed]

Struggling here... So I understand that zero is utterly and completely necessary. I've been reading a lot about this and it seems like some people get heated. I'm not a math guy, so if I offend you ...
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0answers
59 views

Is it correct to say that if $\lim\limits_{x \to a}f(x) = 0$ it is an Infinitesimal?

I think I'm misuderstanding something here, because to my understanding the definition of infinitesimal given in my textbook does not convey the same thing as in other sources. I've read the ...
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1answer
80 views

Difference between transform and transformation.

I was told that there is a difference between a transform and a transformation. Can anyone point out clearly. For example : Is Laplace Transform not a transformation ?
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0answers
37 views

why does function relates input to only one output? [duplicate]

I'm trying to understand the rationale behind definition of function. Why is function defined as a relation which relates input to only 1 output? What would happen if we allow function to relate ...
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1answer
40 views

What is an omega model?

I went to a seminar and a side question was if a theory had an omega model, however from the context I could not deduce the exact meaning. Does an omega model have a general meaning in mathematical ...
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2answers
55 views

Trigonometry is to triangle as _____ is to circle.

What is the most suitable word to put in that gap? Something that corresponds to the study of circles.
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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.
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2answers
28 views

Group action and Right action

Sorry if this may seem trivial - I just started studying Group Theory. This is the problem: Prove that $(g,h) \rightarrow hg$ does not define a group action with $g$ acting on $h$. Prove instead ...
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1answer
33 views

What's the difference between list and sequence (as mathematical concepts not programming point of view)?

What distinguishes between set, multiset and list is whether the order is important or not, and whether repetitions of elements is allowed: List: order is important and repetition is allowed ...
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1answer
17 views

Definition of an induced matrix norm.

Could someone explain the second equality in the definition of a induced matrix norm to me? Let $\| \cdot \|$ be a norm for $\cdot \in \mathbb{R}^n$, then the induced matrixnorm for $A\in ...
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0answers
40 views

Uniform Boundedness: Am I right or my TA?

I am a student, and I disagree with the solutions our TA has prepared. I am seeking verification that I am correct or explanation as to why I am wrong. It seems to be a disagreement or ...
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0answers
13 views

Definition of argument-based $\min$ and $\max$

When I say "argument-based", I mean $\min$ and $\max$ functions that take arguments, as opposed to the min and max of a function. For example: $\min(3,4,-2)=-2$, $\min(100,3)=3$, $\max(3,4,-2)=4$, ...
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2answers
50 views

What is $\mathbb{Z}[a]$?

Let $\alpha$ be algebraic over a field $F$. Then, $F(\alpha)$ denotes the subfield of $F$ generated by $\alpha$. This is the standard definition of $F(\alpha)$. Under this definition, for example, ...
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1answer
23 views

What is the justification for calling a hereditary system an independent system?

I was learning about set systems and hereditary systems and I noticed that they also call a hereditary system a independence system and that didn't quite make sense to me intuitively. First recall ...
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2answers
219 views

Definition of a bounded sequence

My professor gave the following definition: A sequence $\{x_n \}$ is said to be bounded if $\exists M > 0$ such that $|x_n| \le M$ for all $n \in \mathbb N^+.$ But then what about the sequence ...
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1answer
69 views

Left Hand Derivative Definition

What is the actual definition of Left Hand Derivative ? I bumbed into this site and the second white box on their site gives the definition . Is that wrong ? What is the correct one then ?
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1answer
43 views

Precise definition of distinct vectors

What is the precise definition of 'distinct vectors'? In particular, are the vectors (2, 1) and (4, 2) distinct, seeing as they are multiples of each other?
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1answer
21 views

Definition of algebraic structure

Is there a definition of an algebraic structure? Wikipedia says: "a set (called carrier set or underlying set) with one or more finitary operations defined on it." In particular, what is the ...
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0answers
10 views

Complex conjugate of operator on differential forms

In a Lecture note on Kahler manifolds, the author writes the following two identities are equivalent as they are conjugates of each other : $[\Lambda, \bar{\partial}]=-i\partial^{*} $ and ...
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5answers
665 views

What's the difference between tuples and sequences?

Both are ordered collections that can have repeated elements. Is there a difference? Are there other terms that are used for similar concepts, and how are these terms different?
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0answers
47 views

How to say this in math lang?

Well, I have a paper to write in which I have to formalize the following definitions: A transaction is composed of actions (and cascading actions), which affect tables, which are made of records. ...
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1answer
36 views

Is it legal to define a function to be undefined for some x?

Is the following definition legal: f(x) = 1 for 0<x<1, and undefined otherwise. Such functions clearly exist, but the question is if it is legal to just ...
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1answer
37 views

Definition of Euclidean domain

Today we learned about Euclidean domains in class but I don't understand why we need one of the conditions stated in the definition . We called an integral domain R a Euclidean domain if there exists ...
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0answers
19 views

Is $\int^{\infty}_{-\infty}\delta(x-x_0)f(x) \, dx = f(x_0)$ sufficient to define delta distributions?

Most of the sources start introductory section of the delta distributions by defining \begin{eqnarray} \delta(x-x_0)&=&\begin{cases} \infty, & \text{if $x=x_0$}.\\ 0, & ...
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0answers
173 views

What is the “correct” way of making $\mathcal{P}(X)$ into a topological space?

If $X$ is a topological space, I want to know the "correct" way of making the powerset $\mathcal{P}(X)$ into a topological space. So let $\mathsf{SupLat}$ denote the infinitary Lawvere theory whose ...
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1answer
60 views

A strange definition for a strange set.

Given a ring $A$ we know that its center $C=\{x : yx= xy \;, \forall y \in A\}$ is a well defined subset of $A$. Now I want define a set that, intuitively, is '' The set of all elements that commute ...
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2answers
62 views

Is definable $\emptyset$-definable?

A set $A$ is $X$-definable in an $L$-structure $\mathcal{M}$ with a domain $M$ iff there are an $L$ formula $\phi(a,\bar x)$ and $\bar x \in X^n$ such that $A=\{a \in M^m | \mathcal{M} \models ...
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1answer
56 views

What is $g^1_3$?

I'm trying to find the definition of $g^1_3$ in algebraic geometry Hartshorne's book, anyone who is used with this book could help me to find this definition? Thanks Remark: this extract is from ...
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1answer
34 views

Prove a limit using the formal definition of the limit

So I have a sequence {a_n} = π/2^n where n=1,2,3,4.... And I need to prove that its limit is 0. Here is what have done, can someone check and tell me if this is correct.? Definition: A sequence ...
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1answer
97 views

Why is it difficult to define n-category?

Forgive me for the vagueness in the following paragraph, but I don't know how to communicate what I am thinking more formally. If we have a definition for 1-categories (category) and a definition for ...
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3answers
92 views

why does we need to satisfy $d(x,y)\le d(x,z)+d(z,y)$ in order to show metric space?

In metric space some axioms must be satisfied . I wonder why we need to satisfy $d(x,y)\le d(x,z)+d(z,y)$ in order to be metric space. If this axiom is not satisfied, does any problems occur? ...
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1answer
56 views

Caratheodory: Construction

Given a semiring. Consider a premeasure. Regard the following constructions: $$\inf_{A\subseteq S_1\sqcup\ldots\sqcup ...
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0answers
39 views

Definition: Mathematical way to define the “left” and the “right”

How do we define the left and the right, (such as left/right handness) from a mathematical way of definition? How can we explain this definition to some inhabitant of a distant stellar system who has ...
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1answer
20 views

Question about function composition

All textbooks and websites I've consulted define function composition thus: Let $f: A \rightarrow B$ and $g: B\rightarrow C$ be functions. The composite of $f$ and $g$ is the function $f \circ g: ...
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2answers
61 views

What is manifold in Geometry?

What is manifold in geometry? WE always use this word like non-manifold geometry but I was wondering what is manifold in the first place. I got some definition online but couldn't understand. A ...
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3answers
312 views

The “Empty Tuple” or “0-Tuple”: Its Definition and Properties

(I would like to link to a previous discussion on the subject: What is A Set Raised to the 0 Power? (In Relation to the Definition of a Nullary Operation)) In axiomatic (ZFC) set theory, we define ...
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1answer
14 views

Definition of open relative, can't understand explanation

I can't figure out this paragraph: $E$ is open relative to $Y$ if to each $p\in E$ there is associated an $r\gt 0$ such that $q\in E$ whenever $d(p,q)\lt r$ and $q\in Y$ Does this look like ...
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1answer
30 views

“Length” of an element in a free group

Is there any universally agreed definition of "length" (or "width", or whatever term) of an element in a free group $F_n(x_1,\cdots,x_n)$? Intuitively, I would like the length of $1$ to be $0$; the ...
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2answers
113 views

$(+\infty,+\infty,\cdots,+\infty)$ exists in $\mathbb{R^{n}}$?

$(\mathbb{R}^{n},d)$ is a metric space and $d$ is the standard metric on $\mathbb{R^{n}}.$ Let $(\mathbb{R^{n},\tau_{d}})$ is the topology space induced by metric space $(\mathbb{R}^{n},d)$ .We can ...