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2
votes
0answers
90 views

Does this symmetrization operator have a name? Any theory?

Consider a function $f(x_1,\ldots,x_n)$ of $n$ complex variables. Define $$f_{\mathrm{symm}}(x_1,\ldots,x_n) = 2^{-n}\sum_{\varepsilon_1,\ldots,\varepsilon_n=\pm 1} ...
3
votes
1answer
92 views

Is there a standard notation for off-diagonal transpose?

Given a matrix $A=\begin{pmatrix}a&b\\c&d\end{pmatrix}$, its transpose, obviously, is $A^T=\begin{pmatrix}a&c\\b&d\end{pmatrix}$. But is there a conventional way of notating the matrix ...
13
votes
2answers
384 views

Why is “The Higman Rope Trick” thus named?

I'm studiyng Higman's Embedding Theorem, and a fundamental part of the proof is the following lemma: If R is a benign normal subgroup of finitely generated group F, then F/R can be embedded in a ...
9
votes
0answers
249 views

Who first talked about “holes” in homology?

The question Why do the homology groups capture holes in a space better than the homotopy groups? and many others here use the idea that homology counts the ``holes'' in a space. The comments on this ...
4
votes
1answer
176 views

Continuous-piecewise-linear versus piecewise-linear

Some authors use the term "continuous piecewise-linear" where other authors use the shorter term "piecewise-linear" (with continuity tacit). I'd be interested in people's thoughts about this ...
1
vote
0answers
71 views

Why are they called 'pernicious' numbers?

The definition of a pernicious number: In number theory, a pernicious number is a positive integer where the Hamming weight (or digit sum) of its binary representation is prime. The meaning of ...
5
votes
1answer
346 views

Why is this group called “The Holomorph of a group”

Many years ago I found in google the notation "Holomorph of group". It is the semi direct product of $G$ with $Aut(G)$. Why is the term "Holomorph" used here, while it is usually used for complex ...
4
votes
2answers
273 views

Has this construction, which builds a symmetric multicategory from a commutative monoid, been described or studied anywhere, and if so, where?

Whenever $R$ is a commutative ring, write $R[x^{(n)}]$ for the set of all $p \in R[x]$ such that $p$ is a monic polynomial of degree $n$. Then $R[x^{(n)}]$ is not closed under sums, nor does it ...
5
votes
1answer
196 views

Is there a standard name for this poset

I've run into the following poset and I would expect it has a standard name. Let $n\geq k\geq 0$. Then $P_{n,k}$ consists of all $k$-element subsets of $\{1,\ldots,n\}$ ordered by $X\leq Y$ if ...
2
votes
0answers
98 views

Terminology question for maps between posets

Let $P$ and $Q$ be two poset (partially ordered sets) and $\phi : P \to Q$ an order-preserving function. I would like to know whether there is a name and perhaps a different characterizations of such ...
0
votes
0answers
71 views

What is this graph property: number of vertices it takes to see every vertex?

I am wondering what the name is for the following graph property: given a graph $G$ what is the smallest cardinality of $A\subseteq G$ such that every $v\in G$ is connected to some vertex of $A$? I am ...
1
vote
2answers
184 views

Polar Coordinate Systems on Manifolds [closed]

Is there agreement on how to interpret $r$ and $\varphi$ on a manifold if a reference point and a reference direction are given, or, put differently, does the definition of a reference point and, of a ...
4
votes
5answers
833 views

A generalized diagonal?

A simple question. Let $ f:X\to Y $ be a function and let $ E_f:=\{(x, y): f (x)=f (y)\}\subset X\times X $. What is the name of the set $ E(f) $? It would be nice to have some reference also. It ...
1
vote
0answers
73 views

How would you call a variety that is locally a complete intersection up to defect c?

Let $X$ be an equidimensional variety of dimension $n$ over a field that can be covered by open subvarieties of certain intersections of $N-n$ hypersurfaces in $P^N$ (for a large enough $N$; we ...
-1
votes
1answer
52 views

Integrating factors and integrability of an ODE system

The following argument is from a paper about the Bendixson-Dulac Theorem. Consider a smooth differential equation on the plane $$ x'=g(x,y),\quad y'=h(x,y). $$ Suppose there exists a function ...
0
votes
0answers
82 views

The Euler characteristic of Hilbert series

The Hilbert series of a graded vector space $V=\bigoplus_{n\mathbb Z}V_n$ is the (ordinary) generating function of the dimensions of its homogeneous components, $h_V(t)=\sum_{n\in\mathbb Z}t^n\dim ...
0
votes
1answer
70 views

What is the standard name of an edge-graph

Given a graph $G=(E,V)$, I construct a graph $G'$ where the vertices of $G'$ are given by the edges of $G$ and say that two edges of $G$ are neighbors in $G'$ if they have a common vertex. Is there a ...
1
vote
0answers
87 views

Properties and name of some polynomials

I have encountered in a problem some polynomials given by $P_k(x) = \prod_{j=0}^{k-2} (kx-j)$. I need to understand if these polynomials are known, and if they have certain special properties, as ...
4
votes
1answer
229 views

Intersection of nonzero prime ideals is zero — does it have a name?

The Rabinowitch trick (in Eisenbud's Commutative Algebra with a view toward Algebraic Geometry, page 132) says that $R$ (commutative unital ring) is Jacobson if and only if for every prime ideal $P ...
28
votes
2answers
648 views

Term for “uncheckable constructions”

Is there a term for "uncheckable geometric constructions"? Say, Angle Trisection and Doubling the Cube are checkable; i.e., if the answer is given one can do finite Compass-and-straightedge ...
9
votes
1answer
407 views

Between compact and locally uniform: What is the name of this convergence?

Let $X$ be a topological space, $(Y,d)$ a metric space, $f\in Y^X$, and $(f_n)$ a sequence in $Y^X$ with the following property: For every $x_0\in X$ and every $\varepsilon>0$, there exist a ...
2
votes
1answer
88 views

Standard names and methods for this type of fitting minimization

In material science research, we have come across the following type of problem. Given a m by n matrix A, a m vector b, and error tolerance $\varepsilon$, we want to do this minimization $$\eqalign{ ...
1
vote
0answers
45 views

Name for generalization of bivariate weighted-homogeneous polynomials

A polynomial $f = \sum_j c_j X^{\alpha_j}Y^{\beta_j}\in\mathbb K[X,Y]$ is said weighted-homogeneous if there exist $p$, $q$ and $d$ (where $p$ and $q$ are not both $0$) such that ...
2
votes
0answers
44 views

Combination of convex and multiplicative structures

Combination of linear and multiplicative structures gives an algebra. What if instead of a linear structure one has a convex one? Is there a term for this? A natural example is provided, for ...
5
votes
1answer
168 views

Optimal definition of “paving by affine spaces”?

Cell decompositions have been used in topology for a long time as a tool in computing cohomology, but the notion in algebraic geometry and arithmetic geometry of paving by affine spaces (or "affine ...
2
votes
1answer
129 views

Characteristic Varieties and Associated Varieties

Two notions that occur often in representation theory seem to be that of a "characteristic variety" and that of an "associated variety". The former term seems exclusive to D-module theory while the ...
3
votes
1answer
149 views

Uniformizing a relation on ordered sets

Suppose $A$ and $B$ are (complete) ordered sets. Suppose $R\subseteq A\times B$, and $f(a)=\inf\{b : (a,b)\in R\}$ $g(b)=\inf\{a : (a,b)\in R\}$ then what can we call $f$ and $g$? Perhaps there is ...
5
votes
2answers
287 views

Partial inverse of a matrix - or does it have its own name?

In my calculations I need to use something which is "between" a matrix and its inverse. That is, I invert only some dimensions. I am interested if it has an established name. That is, a matrix (here ...
1
vote
1answer
85 views

Name for (function, set) pairs?

Right now I'm working on a topological graph theory problem. To prove a theorem I introduced some objects. Has anyone heard of something similar before? I would like to call them by the right name. ...
-1
votes
1answer
96 views

terminology: “complex” and “sequence” in homological algebra

It appears that the terms "complex" and "sequence" are used synonymously in homological algebra. But there seem to be collocations (in the linguistic sense) that prefer one of those words. For ...
1
vote
1answer
176 views

Name for series $\sum f_n x^n / (n! (n+k)!)$

Let $(f_n)_{n\ge0}$ be a real sequence. Then $\sum f_n {x^n \over n!}$ is called the exponential generating function of $(f_n)$. Let $k\ge0$ be a nonnegative integer. If we add another factorial ...
2
votes
1answer
247 views

Reference request for generalization of groups with out identity element?

In other words what do we call a magma which is associative and has divisibility property but not existence of identity? Or a groupoid when it loses the identity property? A reference on such ...
3
votes
3answers
203 views

What do you call a fixed point theorem for a mapping from a subset of a space to the whole space?

There are a number of fixed point theorems in which we have a map from some subset of a (metric, topological, ...) space to the whole space. (Usually, there is some condition regarding the behavior ...
3
votes
1answer
248 views

Decomposition vs filtration vs stratification

Are there accepted/standard definitions of "decomposition", "filtration", and "stratification" of a topological space (or of a manifold, or of an algebraic variety) $X$? I tend to understand ...
5
votes
3answers
287 views

Does this property of a first-order structure imply categoricity?

Let $\mathfrak{A}$ be a first-order structure over a relational language and let $\kappa$ be an infinite cardinal. Lets say that $\mathfrak{A}$ has the $\kappa$-property if for every structure ...
5
votes
0answers
108 views

Star shaped sets with a midpoint

Suppose $U$ is an open subset of $\mathbb{R}^n$ which is star shaped with respect to $p\in U$. I'll call $p$ a midpoint of $U$ if for any line $\ell$ through $p$, the point $p$ is the midpoint of the ...
0
votes
0answers
113 views

Name of Property $t=st \text{ and } s=ts$

What is the name of the property shared by a pair of functions $s,t$ with $$t=st \text{ and } s=ts$$ ( Main example: relation-valued domain and range operations on relations, via ...
36
votes
8answers
3k views

What recent programmes to alter highly-entrenched mathematical terminology have succeeded, and under what conditions do they tend to succeed or fail?

I think we all occasionally come across terminology that we'd like to see supplanted (e.g. by something more systematic). What I'd like to know is, under what circumstances is it reasonable to believe ...
2
votes
0answers
50 views

A category of partially-ordered commutative semirings; name/notation?

Organize $\mathbb{N} \cup \{-\infty\}$ as a semiring $S$ in the max-plus fashion. So in particular, we have $1_S = 0$ and $0_S = -\infty$. Then for every commutative semiring $R$, we get a degree ...
3
votes
3answers
385 views

“countable” topology

Given universal set $U$. Is there any name of the collection of subsets of $U$ (call them quasi-open) satisfying the following axioms: i) $\emptyset$ and $U$ are quasi-open; ii) finite intersections ...
32
votes
2answers
4k views

How did “normal” come to mean “perpendicular”?

How and when did the word "normal" acquire this meaning? When I first thought of this, I couldn't really come up with any explanation that wasn't complete speculation -- pretty much all I was able to ...
9
votes
2answers
492 views

Origin of the term “generic” in set theory

In set theory, in particular the context of forcing, if $M$ is a model of $\sf ZFC$ and $P\in M$ is a partial order, we say that $G\subseteq P$ is a generic filter (or $M$-generic or generic over $M$) ...
2
votes
2answers
191 views

What do we call functions that behave like predicate symbols?

Assume a metatheory that supports lambda-abstraction, and an object language that is merely first-order. Now let $\varphi$ denote a formula in the object language with one free variable $x$. Then we ...
3
votes
1answer
198 views

Coxeter groups - Parabolic subgroups

In the theory of Coxeter groups there is the notion of so called parabolic subgroups. I'm wondering is this term just a random name, or there are some historical reasons? Why parabolic? Thanks.
4
votes
1answer
223 views

What is an element of an iterated tangent bundle?

An element of the tangent bundle $T M$ of a manifold is called a "(tangent) vector". An element of its dual $T^* M$ is called a "covector" or a "1-form". An element of the exterior square ...
6
votes
3answers
327 views

Is there a name for a “rigid” sheaf?

Is there a name for the property of a sheaf $\mathcal F$ such that the restriction maps $\mathcal F(V) \to \mathcal F(U)$ are injective when $V$ is connected and $U$ is nonempty? In other words, this ...
5
votes
0answers
242 views

Primes for which 2 is a primitive root

I am writing a paper in which I keep referring to primes p for which 2 is a primitive root mod p and so I want to give a name for these primes. Is there a name for these primes in the literature ...
3
votes
2answers
239 views

Definition of the Moebius Ladder Graph

I found two different definitions of the Moebius Ladder Graph, whose essential difference is, whether the smallest one shall be $K_4$ or $K_{3,3}$. according to Wikipedia ...
4
votes
0answers
134 views

Correspondence between numerical semigroups and polynomials?

A numerical semigroup $A$ is defined as a subsemigroup of the semigroup $(\mathbb{N},+)$ of the positive integers such that the set $\mathbb{N}\setminus A$ is finite. Equivalently (for a subsemigroup) ...
2
votes
1answer
130 views

Terminology for the equation $a=a+b$ in commutative semigroups

Let $(S,+)$ be a commutative semigroup. For $a,b\in S$ consider the equation $a=a+b$. Does such a relation between the given $a$ and $b$ have a name? I am currently using such equations quite often ...