Questions tagged [classification]

Classification of various mathematical structures. For classification in the sense of statistics / machine learning, use [tag:statistical-classification].

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CFSG-free proof for classifying simple $K_3$-group

Let $G$ be a finite nonabelian simple group. We call $G$ a $K_3$-group if $|G|=p^aq^br^c$ where $p,q,r$ are distinct primes and $a,b,c$ are positive integers. My question is: Is there a CFSG-free ...
user44312's user avatar
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Understanding segments in Bernstein-Zelevinsky Classification

All reps shall be admissible in what follows. Let $k$ be a non-arch. field, $n = a\cdot b$ natural numbers and $P = M \cdot N \subset \mathrm{GL}_n(k)$ the standard parabolic subgroup with $$ M = \...
Maty Mangoo's user avatar
4 votes
1 answer
237 views

Perceptron / logistic regression accuracy on the n-bit parity problem

$\DeclareMathOperator{\sgn}{sign}$The perceptron (similarly, logistic regression) of the form $y=\sgn(w^T \cdot x+b)$ is famously known for its inability to solve the XOR problem, meaning it can get ...
ido4848's user avatar
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17 votes
2 answers
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Counting degrees of freedom in Lie algebra structure constants (aka why are there any nontrivial Lie algebras of dim >5?)

This is a question about the true number of constraints imposed by the Jacobi identity on the structure constants of a Lie algebra. For an $n$-dimensional Lie algebra, there are $\frac{n^2(n-1)}{2}$ ...
Kensmosis's user avatar
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Classification results

A typical classification result for a class $C$ of objects looks like that: Theorem. Each object of $C$ is isomorphic to one object of the following list: [insert list here]. Examples are the ...
user493267's user avatar
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1 answer
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Wild classification problems and Borel reducibility

My question is whether the archetype of 'wild' problems in algebra, namely classifying pairs of square matrices up to similarity, is 'non-smooth' in the sense of Borel reducibility. This was ...
John Baez's user avatar
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Centralizers of automorphisms in finite simple groups (reference request)

I would like to have a precise version of the following statement and, if possible, a reference to such a statement in some standard book. Claim 1: Let $G$ be a finite simple non abelian group with ...
user203598's user avatar
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Twisted root subgroups in twisted Chevalley groups (reference request)

I am trying to find a standard reference for the natural analogue of root subgroups (and their properties) in twisted Chevalley groups. Let me first recall the classical set-up. According to Steinberg'...
user203598's user avatar
3 votes
1 answer
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Is there a classification of the first geodesic nets?

A geodesic net is an embedding of a multigraph $(V,E)$ into a Riemannian manifold $(M,g)$, so that the vertices are mapped to points of $M$ and the edges to geodesics connecting them. Additionally, ...
Leo Moos's user avatar
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Classification of octonionic reflection groups

I know that there exist classification theorems for real, complex, and quaternionic, reflection groups. There are presentations for the real reflection groups, as well as further presentations for the ...
Sean Miller's user avatar
12 votes
1 answer
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Properties of finite dimensional, real division algebras that yield only $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$

It is a classical result by Kervaire and Milnor that every finite-dimensional, real division algebra has dimension 1, 2, 4 or 8, with the most prominent examples being $\mathbb{R}$, $\mathbb{C}$, $\...
Maximilian Keßler's user avatar
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Finite simple groups of automorphisms of finite simple Lie algebras

I begin by briefly recalling some basic facts in order to pose my question in context. According to the classification, the finite simple groups are cyclic of prime order, are alternating on $n \geq 5$...
user203598's user avatar
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Deduce Sheffer's classification of orthogonal polynomials of A-type 0

Theorem 1.9 in Daniel Galiffa and Tanya Riston's paper, An elementary approach to characterizing Sheffer A-type 0 orthogonal polynomial sequences, 2015, presents without proof Isador Sheffer's ...
Andrius Kulikauskas's user avatar
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Is there a known classification of regular multiplicity-free permutation groups?

The question is in the title, but let me clarify the terminology. I consider a permutation group $\Sigma\subseteq\mathrm{Sym}(\Omega)$ on a finite set $\Omega$. $\Sigma$ is regular if it acts ...
M. Winter's user avatar
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Are there any zeta functions with concurrent derivative shifts in multiple variables?

Expressions for rational zeta series have been obtained by considering the Taylor series of zeta functions. For instance, one has \begin{align}\zeta(s,x+y) &= \sum_{k=0}^{\infty} \frac{y^{k}}{k!} \...
Max Muller's user avatar
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What are the known convex polyhedra with congruent faces?

Note: I originally asked this question on math.SE here, where I posted a bounty on the question but received no answers after a week despite apparent interest in the problem. I'm hoping MathOverflow ...
RavenclawPrefect's user avatar
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Can all (inverse) trigonometric functions with periodic iterates be characterized?

I wonder whether all (composites of) trigonometric and inverse trigonometric functions with periodic functional iterations can be found. In order to specify what I mean by that, let's introduce some ...
Max Muller's user avatar
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Difference between semilinear and fully nonlinear

I'm confused why the Hamilton Jacobi Bellman equation: $$\frac{\partial u}{\partial t}(t,x)+\Delta u(t,x) -\lambda||\nabla u(t,x) ||^{2}=0$$ is considered fully nonlinear, but not semilinear. By ...
Ben's user avatar
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1 answer
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On classifying groups of order $p^5$

Can someone suggest me some source where the author has classified all non-isomorphic groups of order $p^5$ ? I need complete classification (not upto isoclinism), and also in finitely presented form ....
Himanshu Setia's user avatar
3 votes
1 answer
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Is there a precise relationship between the goals of moduli theory and the minimal model program?

I want to get into some of the big classification problems in algebraic geometry, but have a very broad question. Ultimately we would like to classify all varieties over some field up to isomorphism, ...
Luke's user avatar
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Comonoids in the category of monoids

Let us give the category of monoids $\mathbf{Mon}$ a monoidal structure with $\otimes = \sqcup$ (coproduct). How can we classify $\mathbf{CoMon}(\mathbf{Mon})$, the category of comonoids of monoids? ...
HeinrichD's user avatar
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3 votes
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Matroids which are transitive on minimal basis exchanges

I am looking for matroids in which all minimal basis exchanges look the same, that is, the matroid is transitive on these. Let me explain what I mean by that. Consider a finite matroid $M$. Define a ...
M. Winter's user avatar
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Why the VC dimension of triangles in 2D space is not greater than 7?

I understand that there are sets of 7 points on a circle that can be fully shattered using triangles.But, it is not clear to me why it cannot shatter 8 points. Is there any intuitive way of arriving ...
wanderer's user avatar
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1 answer
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Name for "partially complete" invariants in classification problems?

For any equivalence $\sim$ on some collection of objects $C$ consider the problem of trying to determine if two arbitrary objects $x$ and $y$ in $C$ are equivalent i.e. if $x\sim y$ now by definition ...
Ethan Splaver's user avatar
9 votes
2 answers
441 views

The "Johnson polychora"

Firstly, a definition: A convex polyhedron, whose faces are regular polygons (2D polytopes). This includes the 92 Johnson solids, 13 Archimedean solids, 5 Platonic solids and two infinite ...
FusRoDah's user avatar
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5 votes
1 answer
452 views

Classification of the quotients of the ring Z/4 [X]

Is it possible to classify all cyclic $\mathbb{Z}/4$-algebras, i.e. the regular quotients of $\mathbb{Z}/4 [X]$? A typical example is $\mathbb{Z}/4 [X] / \langle X^n , 2 X^k \rangle$. For my purposes ...
Martin Brandenburg's user avatar
1 vote
0 answers
103 views

Generalize characterization of upper semicontinous functions

Let $X$ be a metric space and denote $f:X \rightarrow \mathbb{R}.$ It is easy to show that the following two statements are equivalent: $(1)$ For any real number $c$, we have $f^{-1}(-\infty,c)$ is ...
Idonknow's user avatar
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Classification of pointed Hopf algebras up to gauge equivalence

The classification of finite-dimensional pointed Hopf algebras over an algebraically closed field of characteristic zero and whose group of group-like elements is abelian is very much completed. ...
Mathematician 42's user avatar
14 votes
1 answer
627 views

Classification of non-Hausdorff topological vector spaces

It is well-known that up to topological isomorphism there is exactly one Hausdorff topological vector space (say, over $\mathbb{C}$) of a given dimension $n$, namely $\mathbb{C}^n$ with the euclidean ...
HeinrichD's user avatar
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3 votes
0 answers
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Smallest number $n$ for which we don't know the classification of all groups of order $n$

I noticed that in groupprops and wikipedia there are often given tables of classifications of groups of small order. This motivated me to ask, what is the current state of research in classifying all ...
Sidharth Ghoshal's user avatar
2 votes
0 answers
83 views

Regular homotopy of punctured surfaces

A theorem of James and Thomas (Note on the classification of cross-sections, Topology 4) asserts that the space of immersions, up to regular homotopy, from a compact surface $S$ into $\mathbb{R}^3$ ...
Paul-Benjamin's user avatar
27 votes
1 answer
2k views

Has anyone catalogued the "first generation" proof of the classification of finite simple groups?

It has been estimated that the original proof of the CFSG spans around 15,000 journal pages written by hundreds of authors over most of the 20th century. The GLS project attempted to simplify this ...
Mario Carneiro's user avatar
1 vote
0 answers
148 views

Classification of finite subgroup of $PGSp_4(\mathbb{C})$

Is there a classification of the finite subgroups of $PGSp_4(\mathbb{C})$?
Adel BETINA's user avatar
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2 votes
1 answer
540 views

Classification of cubic surfaces in $\mathbb{P}^3$

We know every cubic surface in $\mathbb{P}^3$ is obtained by blowing up $\mathbb{P}^2$ at 6 points in general position. Hence they are all birational to $\mathbb{P}^2$. My question is: Do we have ...
Xuqiang QIN's user avatar
19 votes
4 answers
1k views

Representation theorem for modular lattices?

Birkhoff's representation theorem implies that every distributive lattice embeds into the lattice of subsets of a set. Is there also some representation theorem for modular lattices? For example, I ...
Martin Brandenburg's user avatar
2 votes
2 answers
696 views

Are the closed and unbounded subsets of $\mathbb{R}$ known up to homeomorphism?

I am currently working on a problem for which this knowledge could greatly reduce the number of cases, but I have yet to find anything after searching online. Are the closed unbounded subsets of $\...
AnotherPerson's user avatar
5 votes
0 answers
118 views

Heegaard diagrams of prime 3-manifolds

Are there some known results which give a classification of closed prime 3-manifolds up to their Heegaard diagrams? (That is, providing a collection of Heegaard diagrams which exhausts all prime ...
Pandora's user avatar
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4 votes
1 answer
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Classification of finite abelian hypergroups and table algebras

Update: Originally, I formulated this question for finite abelian hypergroups, but in a discussion with Geoff Robinson below I realized that the abelian hypergroups defined below are equivalent to ...
Juan Bermejo Vega's user avatar
5 votes
1 answer
851 views

Is there a structure theorem or group law for finite groups generated by two elements?

Say that $a, b \in G$ are two elements of a finite group $G$. Is there a structure theorem for the structure of $\langle a,b\rangle$? Is there a way to derive group laws for the group operation in the ...
user avatar
5 votes
0 answers
292 views

A class 3 group of order 243

Let G be a group of order $243=3^5$. We denote by $(G_i)$ its lower central series and assume that $G$ has class $3$ and that $|G:G_2|=|G_3|=9$. We assume moreover that the cubing map factors as a (...
user avatar
4 votes
0 answers
186 views

A relation on the set of isomorphism classes of finitely generated groups

Let $G$ be the set of finitely generated groups up to isomorphism hence its elements will be noted $[B]$ where $B$ is some finitely generated group. On this set we put a relation $\mathcal{ND}$ ("...
Clément Guérin's user avatar
49 votes
2 answers
5k views

How do you *state* the Classification of finite simple groups?

From the point of view of formal math, what would constitute an appropriate statement of the classification of finite simple groups? As I understand it, the classification enumerates 18 infinite ...
Mario Carneiro's user avatar
3 votes
3 answers
586 views

First Explicit Irreducible Representations

Although the classification of simple Lie Algebras and their representations is fully understood, I wonder whether there is some book with exhaustive tables describing explicit irreducible ...
Jjm's user avatar
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3 votes
1 answer
211 views

What is the growth of the rank of a power of a finite simple group?

Which asymptotic bounds (upper and lower) are known for $s_n$ - the minimal number of generators of $S^n$ where $S$ is a nonabelian finite simple group?
Pablo's user avatar
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7 votes
0 answers
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Signatures of latin squares: what about the extremal cases?

For a latin square (LS) of order $n$, we will define a cut (or maybe general transversal, I don't know whether there is an entrenched name for this) as a collection of $n$ cells such that no two share ...
Wolfgang's user avatar
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13 votes
3 answers
1k views

Characterization of Frobenius complements

I have learned that Frobenius complements are characterized (among finite groups) by having a fixed point free complex representation. That is, a finite group $G$ is a Frobenius complement if and only ...
Joonas Ilmavirta's user avatar
18 votes
1 answer
470 views

Lens spaces and generalized Petersen graphs

Recently I came across this mathoverflow question, in which the number of homeomorphism classes of 3-dimensional lens spaces $L(p, q)$ is computed as a function of $p$. Using the OEIS, I found a ...
Fabian Meumertzheim's user avatar
25 votes
2 answers
4k views

In what sense is the classification of all finite groups "impossible"?

I think there is a general belief that the classification of all finite groups is "impossible". I would like to know if this claim can be made more precise in any way. For instance, if there is a ...
Keivan Karai's user avatar
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5 votes
1 answer
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Castelnuovo's rationality criterion on singular surfaces?

Let $S$ be a projective surface over an algebraically closed field. Suppose that $q(S)=h^1(\mathcal O_S)=0$ and $P_2(S)=h^0(\mathcal O_S(2K_S))=0$. If $S$ is smooth, Castelnuovo's rationality ...
Jesus Martinez Garcia's user avatar
8 votes
2 answers
456 views

Uniform-in-p classification* of p-groups of order p^n for each fixed n?

To what extent is there/can there be a description that is uniform in p (for p sufficiently large) of the p-groups of order $p^n$, for each fixed n? Note 1: I used the word "description" rather than ...
Joshua Grochow's user avatar