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Questions tagged [classification]

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1answer
87 views

Minimum number of support vectors? [closed]

I'm learning SVM and its written everywhere that the minimal number of support vectors is 2? But I couldn't find any formal proofs of that. Why cant there be less than 2 support vectors? Can somebody ...
2
votes
1answer
233 views

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 ...
0
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0answers
116 views

How does one obtain a classification of reducible holonomy groups from Berger's classification?

How does one obtain a classification of reducible holonomy groups from Berger's classification with de Rham's decomposition theorem? It was suggested to me that I should requiring that each factor in ...
2
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1answer
58 views

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 ...
6
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1answer
154 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 ...
5
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1answer
372 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 ...
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0answers
100 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 ...
5
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1answer
101 views

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. ...
12
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1answer
299 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 ...
3
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0answers
167 views

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 ...
2
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0answers
71 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 $\mathbb{R}^3$ ...
23
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1answer
1k 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 ...
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0answers
138 views

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

Is there a classification of the finite subgroups of $PGSp_4(\mathbb{C})$?
2
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1answer
266 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 ...
16
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4answers
701 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 ...
2
votes
2answers
367 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 $\...
5
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0answers
105 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 ...
4
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1answer
237 views

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 ...
5
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1answer
657 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 ...
5
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0answers
249 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 (...
4
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0answers
148 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 $\mathcal{ND}$ ("...
37
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2answers
3k 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 ...
3
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3answers
543 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 ...
3
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1answer
193 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?
7
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0answers
168 views

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 ...
9
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2answers
533 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 ...
17
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1answer
399 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 ...
15
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2answers
1k 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 ...
5
votes
1answer
293 views

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 ...
8
votes
2answers
344 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 ...
2
votes
2answers
255 views

How much information does the multiplicative semigroup of an algebra contain?

How much do we know about an given algebra when we only know its semigroup strucure under the product law? How far can two algebras be distinguished by knowing only their semigroup strucure? The ...
4
votes
1answer
306 views

Are all symmetric idempotent Latin squares known?

Are all symmetric idempotent Latin squares known? There is such a square of order $n$ if and only if $n$ is odd. However, is there a classification of all of them? (The motivation for the question ...
7
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1answer
401 views

Taking direct sums in $K$-theory in Kirchberg-Phillips classification

A theorem by Kirchberg and Phillips states that two unital separable nuclear simple purely infinite $C^*$-algebras (so called Kirchberg algebras) satisfying the Universal Coefficient Theorem are ...
8
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2answers
936 views

basics of classification of trilinear forms (when is it non-discrete)

Consider tri-linear forms, $\{A_{ijk}\}$ where $i=1,..,n_1$, $j=1,..,n_2$, $k=1,..n_3$, over a field of zero characteristic, up to the equivalence $A\to (U_1,U_2,U_3)(A)$, by three matrices. What is ...
2
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1answer
495 views

The PDE $u_t=u_{xx}-u_{yy}$: The simplest linear second-order PDE that isn't elliptic, parabolic, or hyperbolic.

I know that there have been several questions on here and stackexchange about linear PDE's which don't fall into the standard classification, but I had a more focused question which I haven't seen ...
1
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2answers
598 views

Finding Decision Boundary from empirical distribution

Based on measuring a certain characteristic, we want to classify measurements as coming from either of two populations. The true population distributions are unknown (and we don't want to take any ...
23
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1answer
1k views

Rock-paper-scissors…

A directed graph whose underlying undirected graph is complete is called a tournament. Let us call a (finite) directed graph balanced if every vertex has as many incoming as outgoing edges. The ...
2
votes
0answers
444 views

bielliptic surfaces

Definition: A surface is called $S$ bielliptic if $S \cong (E \times F) /G$, where $E,F$ are elliptic curves and $G$ is a finite group of translations of $E$ acting on $F$ such that $F /G \cong \...
5
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1answer
606 views

Del pezzo surfaces in positive characteristic

For me a Del Pezzo surface $X$ over an algebraically closed field of characteristic $p$ is an algebraic surface where the anticanonical bundle $\omega^{-1}_X$ or $-K_X$ is ample. (I prefer the second ...
10
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0answers
254 views

“Locally Cartesian” varieties

Differentiable manifolds can be described in terms of local charts to open subsets of $\mathbb{R}^n$ and transition functions that are diffeomorphisms. Trying to put $\mathbb{A}^n$ (over an ...
18
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6answers
3k views

Is there a classification of open subsets of euclidean space up to homeomorphism?

I hope this question is reasonable enough to have a well known answer. i.e either there is a simple invariant (like the homotopy groups) that characterizes the homeomorphism type of such set among ...
7
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7answers
766 views

Classifications of finite simple objects

I'm curious to know if other classifications are known of "finite simple structures" in the same spirit of the monumental classification of finite simple groups. Here I mean "classification" in the ...