# 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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### Classify all open affine subschemes of a projective variety

Currently I am pondering a question from algebraic geometry that can be stated in very simple terms: Let $X \subseteq P^n_k$ be a projective variety, subscheme of projective $n$-space over an ...

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### Can one formalize the prevalence of the Big Five systems of reverse math?

Simpson's systems of second order arithmetic turn out to be five in
number; to simplify notation let's denote them A, B, C, D, E. What
seems to be an empirical observation is that most theorems in
...

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### Bayes classifiers with cost of misclassification

A minimum ECM classifier disciminate the features $\underline{x}$ to belong to class $t$ ($\delta(\underline{x}) = t$) if $\forall j \ne t$:
$$\sum_{k\ne t} c(t|k) f_k(\underline{x})p_k \le \sum_{k\ne ...

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### Classification of all connected simple real Lie groups?

Is there an explicit(!) classification of all(!) connected real simple Lie groups up to isomorphism? Not just simply connected or adjoint, but all of them?

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### Almost simple groups and their involutions without CFSG

Suppose $A$ is a finite almost simple group (meaning, by definition, that there exists a finite simple group $P$ such that $P \leq A \leq \mathrm{Aut}(P)$). Suppose furthermore that $A$ acts $2$-...

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### Finding inverses of certain elements in the set of normal invariants of a smooth manifold

Let, $V$ denote the Stiefel manifold of 2-frames $V_{10,2}$ . Consider the the map $S_\text{diff} (V) \xrightarrow{\eta} N_\text{diff} (V) $ in the surgery exact sequence of a smooth manifold. . ...

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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 ...

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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 = \...

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### 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 ...

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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 $\frac{n^2(n-1)}{2}$ ...

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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 ...

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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 ...

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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 ...

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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'...

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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, ...

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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 ...

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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 $\mathbb{R}$, $\mathbb{C}$, $\...

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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 $n \geq 5$...

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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 ...

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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 ...

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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!} \...

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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 ...

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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 ...

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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 ...

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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 ....

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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, ...

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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?
...

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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 ...

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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 ...

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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 ...

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### 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 ...

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### 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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### 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 ...

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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. ...

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### 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 ...

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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 ...

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### 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 $\mathbb{R}^3$ ...

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

Is there a classification of the finite subgroups of $PGSp_4(\mathbb{C})$?

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### 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 ...

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### 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 ...

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### 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 $\...

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### 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 ...

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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 ...

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### 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 ...

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### 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 (...

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### 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 $\mathcal{ND}$ ("...

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### 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 ...

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### 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 ...

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### 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?