Questions tagged [definitions]
The definitions tag has no usage guidance.
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What are the most misleading alternate definitions in taught mathematics?
I suppose this question can be interpreted in two ways. It is often the case that two or more equivalent (but not necessarily semantically equivalent) definitions of the same idea/object are used in ...
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Nonequivalent definitions in Mathematics
I would like to ask if anyone could share any specific experiences of
discovering nonequivalent definitions in their field of mathematical research.
By that I mean discovering that in different ...
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What is the definition of the function T used in Atiyah's attempted proof of the Riemann Hypothesis?
In Michael Atiyah's paper purportedly proving the Riemann hypothesis, he relies heavily on the properties of a certain function $T(s)$, known as the Todd function. My question is, what is the ...
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Can a mathematical definition be wrong?
This question originates from a bit of history. In the first paper on quantum Turing machines, the authors left a key uniformity condition out of their definition. Three mathematicians subsequently ...
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Examples of advance via good definitions
In my research I came across a case where I could derive a known theorem with rather straightforward way by choosing "non-standard" definitions using my knowledge from a related field. This particular ...
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What definitions were crucial to further understanding?
Often the most difficult part of venturing into a field as a researcher is to come up with an appropriate definition. Sometimes definitions suggest themselves very naturally, as when you solve a ...
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Category theoretic interpretation of matroids?
First time poster, long time lurker here. I have a really basic question that has been bugging me for sometime. Specifically, I'm not exactly sure what the 'correct' category theoretic definition of ...
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What is Serre's condition (S_n) for sheaves?
The Serre's condition $(S_n)$, especially $(S_2)$, has been mentioned in a few MO answers: see here and here for example. I am pretty sure I have seen it in other questions as well, but could not ...
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What does the adjective "natural" actually mean?
Terms like "in the natural way" or "the natural X" are used frequently in mathematical writing. While it is certainly clear most of the time what is meant, on occasion, I have been confounded. The ...
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Paradoxical Mathematical Objects Pending for Construction [duplicate]
The possible properties and applications of some mathematical objects have been described far before their rigorous mathematical definition. Some of them even had a seemingly paradoxical description ...
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Shapes for category theory
Most texts on category theory define a (small) diagram in a category $\mathcal{A}$ as a functor $D : \mathcal{I} \to \mathcal{A}$ on a (small) category $\mathcal{I}$, called the shape of the diagram. ...
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Why are there so many fractional derivatives?
I have been interested in fractional calculus for some time now, and I have seen "lots" of definitions of the $\frac {d^\alpha} {dx^\alpha}$ operator.
I started with the book The Fractional Calculus ...
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Main statement as theorem or corollary
In a text there will often be a few important results that are usually called Theorems. Intermediate statements are called Lemmas, and statements that follow immediately from previous results are ...
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When is a classification problem "wild"?
I hope someone can point me to a quick definition of the following terminology.
I keep coming across wild and tame in the context of classification problems, often adorned with quotes, leading me to ...
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How can I improve my formal definitions?
I am a Software Architect and not very familiarized with standard notation in mathematics. Nonetheless, I would like to write a paper explaining a normalization of a computing model for expert systems....
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What are examples of theorems which were once "valid", then became "invalid" as standard definitions shifted?
That is, results established by correct proofs within some framework, yet the manner in which their author or the general mathematical community at the time would describe these results would, in ...
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Why is the standard definition of a $(p, q)$-tensor so bizarre?
At time of writing the first definition of a $ (p, q) $-tensor on the Wikipedia page is as follows.
Definition. A $ (p, q) $-tensor is an assignment of a multidimensional array $$ T^{i_1\dots i_p}_{...
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Why there is a Quot-scheme, not a Sub-scheme?
Let $X$ be a projective variety, and $E$ be a coherent sheaf on $X$. Grothendieck has proven that there is a scheme $\mathrm{Quot}_X(E)$ parametrizing arbitrary quotient sheaves of $E$. It is probably ...
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Surreal exponentiation -- are the varying definitions equivalent? If not, is there agreement on which ones are better?
The surreal numbers are sometimes introduced as a place where crazy expressions like $(\omega^2+5\omega-13)^{1/3-2/\omega}+\pi$ (to use the nLab's example) make sense. The problem is, there seem to ...
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What, precisely, do we mean when we say that a f.d. vector space is canonically isomorphic to its double dual?
I've been reading the Xena Project blog, which has been loads of fun. In the linked post Kevin gives the natural isomorphism $V \to V^{\ast \ast}$ from a f.d. vector space to its dual as an example of ...
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What is a Kelley ring?
I've heard that in some book by someone named Kelley, perhaps an early edition of John L. Kelley's General Topology, the author gave a definition of a ring which turned out to be weaker than the usual ...
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Definition of ind-schemes
What is the correct definition of an ind-scheme?
I ask this because there are (at least) two definitions in the literature, and they really differ.
Definition 1. An ind-scheme is a directed colimit ...
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Definition of area
I am looking for an attractive, but rigorous definition of area;
say in Euclidean plane. Probably there is no short definition. It is OK to make it even longer, but can it be built from useful parts ...
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Understanding the definition of stacks
First of all I should apologies if this question does not count as a research level one. I asked the same question on MathUnderflow and didn't receive any answer. Let me cross post (copy and paste) it ...
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Formal definition of homotopy type theory
The HoTT community is quite friendly, and produces many motivational introductions to HoTT. The blog and the HoTT book are quite helpful. However, I want to get my hands directly onto that, and am ...
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Is there an intrinsic definition of fractal (i.e. not embedded in euclidean space)?
Long ago, manifolds were embedded subsets of euclidean space defined by polynomials. Later, using the gluing of open sets, people realized they could define manifolds intrinsically. And in certain ...
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Is this a vertex algebroid?... What is vertex algebroid?
A couple of day ago, I was lamenting to a friend about the fact that I have no idea what vertex algebroids are.
During our discussion, I came up with a guess of what a vertex algebroid might be.
I'm ...
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Axiomatic approach to means
Recently I have been contemplating on a talk for high school children. One of my favorite topics in high school was the inequality of means. I had a great high school teacher who wrote some very nice ...
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Grothendieck Topologies versus Pretopologies
The wikipedia article(s) as well as the nlab article(s) about Grothendieck topologies and Grothendieck pretopologies are careful to differentiate the two very emphatically and to point out that ...
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The $n$-th derivative has $n$ zeros. Can such a function be unbounded?
I asked this on Math.SE some days ago, but without any success. For some application I need a formal definition of bell-shaped function. So I had the following idea:
Definition. A $C^\infty$-...
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Would it be possible to propose a satisfying categorical definition for the notion of basis?
I once came across a definition for the notion of basis that was independent of the type of the algebraic structure considered (although I cannot find where). Translated into category terminology, it ...
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Does the notion of graphs with vertex multiplicity exist?
I need to use graphs where each vertex gets a natural number, $b(v)$, its multiplicity. These numbers indicate how many 'replications' of the vertex we have.
It is actually a way to write in a ...
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Locally constant functions with compact support = smooth ?
Hello,
I have a trivial question, but I hope that you don't mind helping. I often get confused with basic definitions.
Let F be a p-adic field. Then (from what I understand) $C_c^{\infty}(F)$ is the ...
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Different definitions for integral de Rham cohomology classes
Suppose that $S$ is a compact orientable surface. In this case, the top de Rham cohomology space $H^2(S)\cong \mathbb{R}$, with the isomorphism given by integration on $2$-forms along $S$.
Now, one ...
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how to define the injectivity radius of manifolds with boundary?
For manifolds without boundary one defines the injectivity radius as the maximal radius where the exponential map is a diffeomorphism. One can then show that the injectivity radius is the maximum ...
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Why do mathematicians prefer one definition over the other when they both define the same concept?
Here is a basic, though very important, example:
Hilbert takes as primary the notion of “congruence” (or “equal”) between segments. His first axiom of congruence “requires the possibility of ...
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Is there a term for this graph subset?
Suppose $G$ is a (finite) graph which is $k$-vertex colourable (i.e. $\chi(G)\leqslant k$). Suppose $S$ is a set of vertices of $G$ with the following property:
If $c:V(G)\rightarrow [k]$ is a vertex ...
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Who first introduced the functional definition of symmetry?
Who first introduced the definition of symmetry using functions explicitly? (That is, for instance, a symmetry of a subset $X$ of the plane is a function $F$ from the plane to the plane that preserves ...
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What is the mathematical structure called if we replace commutative group by commutative monoid in the definition of linear space?
Could anyone tell me what the mathematical structure is called if we replace commutative group by commutative monoid in the definition of linear space?
Also, are there any names for "commutative ...
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Geometric picture of invariant differential of an elliptic curve
What is the geometric meaning of $\omega=dx/(2y+a_1x+a_3)$ for an elliptic curve?
This question is an adjunct to MO Q1 on formal laws and L-series. Silverman (Q1) and Darmon (pg. 6) state:
The ...
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What are the relationship between various definitions for quasi coherent sheaves?
It seems that there are many definitions of quasi coherent sheaves(modules). There is a nice page on nLab quasi coherent sheaves
My questions are:
Are there any other definitions of quasi coherent ...
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Definition of forgetful functor
I was wondering if it is possible to make a formal definition of what it means for a functor to be forgetful. That is, using only the terminology of categories. I have seen so many examples of ...
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Unitary groups over number fields
When defining unitary groups over number fields, one usually takes $F$ to be a totally real number field, $E$ a CM quadratic extension of $F$, and $V$ a hermitian space attached to $E/F$. Then $U(V)$ ...
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What makes a distance?
In the answers to my previous question, I learned that there are different concepts of distance, that is of distance-like functions with the usual metric being only the most popular and important one.
...
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A definition of topology using monads (a.k.a. halos)
In nonstandard analysis, there is a way of studying topological spaces known as "monads" (more commonly known as halos, as it turns out). The monad of a point $x$ (written $\mu(x))$ is the set of all ...
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Étale morphisms in SDG
On page 70 here, Kock defines a formally étale morphism $f:M\rightarrow N$ as one for which the following square is a pullback for each $d:\mathbf 1\rightarrow D$
$$\require{AMScd} \begin{CD} M^D @>...
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What is "augmented algebra"?
Really sorry for this question, but googling for some time did not help me. I was trying to understand the meaning of the following phrase:
Let B be an augmented algebra over a semi-simple algebra T.
...
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Measure without measurable sets
This question is a little on the softer and speculative side, so bear with me.
Usually a measurable space is $(\Omega, \Sigma)$, a set $\Omega$ and sigma algebra $\Sigma$ of subsets. A measurable ...
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Analytic/synthetic distinction in mathematics besides geometry?
In a recent answer to an old MO question, I made a distinction between a "definition" of a mathematical object in the sense of axioms that characterize it, and a "definition" that ...
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What are Penrose Tilings, and how do they relate to Quasicrystals?
The question is in the title, but let me elaborate a little.
Background
Penrose Tilings are really pretty and satisfy some remarkable properties. For instance, I believe the following is true: even ...