Questions on the usage and meaning of words in mathematics, the names for mathematical entities, and other such questions.

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A Terminology in Group Theory

The term soluble (or solvable) group came from the works of Galois, Abel, Lagrange, relating to solvability of equations by radicals. One may ask then, how the terminology nilpotent group came? The ...
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0answers
30 views

How to read notation about little o notation

On page 2 of this pdf the author introduces this equivalence relation: $$f(x) \sim_{x_0} g(x) \iff f(x) - g(x) = o(x-x_0)\quad \text{as } x\to x_0$$ Assuming this is a standard notation, how does ...
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0answers
47 views

The simplest way to present a Lie algebra to a wide audience?

I would like to get suggestions from you as to the best way to present the idea and contents of Lie algebras to a wide public of people with no detailed background in maths. What wlould you explain to ...
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0answers
21 views

In mathematical writing, what does the phase “Class of nonlinear system” mean?

In many engineering and applied mathematics resources, author specify his work or theory for "a class of Y , Y={Nonlinear systems, Hybrid systems, discontinuous systems .. etc}" what does this mean? ...
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1answer
57 views

fractal curve and fractal set

Would it be correct to say that all fractal curves are fractal sets, but not all fractal sets are fractal curves? If that is correct, what would be an example of a fractal set that is not a fractal ...
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1answer
51 views

Is distance between two graphs defined somehow?

If the two graphs are isomorphic, then their distance is zero. And this distance increases, if vertices or edges are added or removed to/from one of the graphs. Does this "distance" have a special ...
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1answer
35 views

Are axiom schema of specification and axiom of specification the same terminology?

I sometimes see books just having "axiom of specification" rather than lengthy "axiom schema of specification". Are these two the same thing?
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5answers
123 views

What are the English names for the rules for expanding a sum or a difference squared?

$(a+b)^2=a^2+2ab+b^2$ $(a-b)^2=a^2-2ab+b^2$ What are these two called in English? Are they called anything at all? I looked around on some math websites and Wikipedia, but I didn't find these rules. ...
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1answer
32 views

What is the name of this construction of an induced order?

Let $<_1$ be a total order on a non-empty set $A$. Consider the set $\mathscr{J}(A)$ given by $$\mathscr{J}(A) = \bigsqcup_{n=1}^\infty A^n$$ ...where $\sqcup$ denotes "disjoint union" (or ...
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3answers
88 views

Is there any way to universally define the notion of $\text{Isomorphism}$?

Suppose we want to give a very general definition of the term Isomorphism, first of all, we'll want an isomorphism to be a bijective function. Informally, we want our function to preserve whatever ...
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2answers
60 views

Is there accepted terminology for multisets of vectors that are “as independent as possible”?

Let $V$ denote a finite-dimensional real vector space. Suppose $A$ is a multiset of elements of $V$. Then if $\mathrm{card}(A) > \mathrm{dim}(V)$, it follows that $A$ cannot be linearly ...
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0answers
68 views

Differences between solenoidal and rotational vector fields

In classification of vector fields, one of the 4 different type vector fields is "solenoidal and irrotational vector field" (both divergence-free and curl-free). If solenoidal and rotational vector ...
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3answers
604 views

Why is the word associative used to represent the concept of the associative property?

For the commutative property ... According to wikipedia: The word "commutative" is a combination of the French word commuter meaning "to substitute or switch" and the suffix -ative meaning ...
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2answers
75 views

Is there a name for a binomial expansion without coefficients?

I am investigating a problem from George E. Andrews Number Theory (Dover, 1971), discussed previously here: Induction Proof that $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1})$ I was led ...
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1answer
99 views

Lost in terminology: What is the meaning of the words “Constraint” and “Parameter” in a goodness of fit?

This is somewhat related to this previous question of mine. I need a clear distinction and/or definition of the words 'parameter' and 'constraint' in the following context which is the the only source ...
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1answer
34 views

Is there a standard for math terminology?

I searched for the definition of "quotient" and I felt there were many answers. Is there a place that defines math terms? Something like a standard?
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0answers
31 views

difference between covariant derivative and connection

Is there any difference between covariant derivative and connection, or it is just a case of using two expressions to describe the same thing?
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1answer
32 views

How is this called? (“Syncing sequences”)

I recently needed something like the following: Let $(a_0, a_1, \dots, a_m)$ and $(b_0, b_1, \dots, b_n)$ be two finite sequences of integers, not necessarily of the same length, with the same ...
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2answers
89 views

Is there standard terminology to describe the not-quite-a-limit behavior of ${\tan( \log x) \over x}$ as x approaches infinity?

Suppose I want to describe the long term behavior of ${\tan(\log x) \over x}$ as x increases towards positive real infinity. Now, $$\lim_{x \rightarrow \infty}{\tan(\log x) \over x}$$ obviously ...
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1answer
11 views

Standard deviation without square

For $(x_i)_{1 \leq i \leq N}$, the standard deviation is of course $$\sigma = \sqrt{\frac{1}{N} \sum_i (x_i - m)^2},$$ where $m$ is the mean value of the $(x_i)$. We can say that the variance ...
2
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1answer
22 views

Is there a term in mathematics for Metcalfe's Law?

Metcalfe's Law states that the value of a network is proportionate to the square of the number of users. This comes from the idea that there are $N*(N-1)/2$ pairs in a network of size $N$. Does this ...
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1answer
35 views

Actual convention for the term Sinusoidal Phase “Shift” or “Offset”

I have seen the convention for a sinusoid appear as: $x(\theta) = A \cdot \sin( B \cdot (\theta - \phi) \ ) + D$ $y(\theta) = A \cdot \sin( B \cdot (\theta + \phi) \ ) + D$ Is "offset": $\phi$ ...
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1answer
30 views

Is there a name for the farthest/shortest component of a vector?

Say I have a diagonal line segment/vector such that the horizontal component is longer than the vertical component (or vice versa). Is there a common name / term for each of the components? I'm ...
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1answer
38 views

Probability distribution that always yields the same number

What would be the correct name for a probability distribution which has a 100% chance of yielding a specific number? I'm tempted to call it "expected value distribution" or some such, but I'm ...
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1answer
73 views

What does it mean if a free algebra has an unsolvable word problem?

I wonder how hard identity testing (similar to polynomial identity testing) can be for a free algebra. I thought that in a certain sense, the problem should always be semi-decidable, because the free ...
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1answer
41 views

Remove Ambiguity From Meaning Prime Element, Prime Ideal, Prime for $x^2+2$ in $\mathbb Z_4[x]$

Remove ambiguity from prime element, prime ideal, prime. Here is the context. In $\Bbb Z_4[x]$, given the polynomial $x^2+2$ and asked show it is irreducible, but not prime. Not sure if in this ...
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2answers
38 views

What does it mean for a matrix to be “non-negative definite”?

In some old course notes I'm reading to touch up on statistical forecasting methods, the book often makes reference to "non-negative definite" matrices. I know what a semi-positive definite, positive ...
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4answers
974 views

A name for this property?

Let $*$ be an operation such that $(xy)^* = y^*x^*$, e.g. if $x,y$ are $2\times2$ matrices and $*$ is "take the inverse" or if $x,y$ are operators and if $*$ is the adjoint. Is there a name for such ...
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4answers
72 views

Origin of the phrase 'in polynomial time'

What is the origin and context of the phrase 'solvable in polynomial time' in computer science? Are they related to the notion of 'polynomials' in mathematics?
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1answer
23 views

General Notation for a Reductive Operation, such as Sum (Σ) or Product (Π)

In functional programming, people often use operations like "fold" or "reduce", to convert from a collection to a single object using a binary operation. This is analogous to the sum and product ...
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0answers
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Why was this terminology of holomorphism used here?

page 6 in this notesays: A note on index convention: $i$,$j$, and $\bar{i}$, $\bar{j}$ index over the holomorphic and antiholomorphic components. I never knew there were holomorphic ...
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1answer
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Is there a name for the linear maps $u_i: E_i \to \prod_k E_k$ defined by $u_i(t) = (0,…,0,t,0,…,0)$?

Let $E_1,...,E_n$ be vector spaces. We know that a function $p_i: \prod_k E_k \to E_i$, $p_i(x_1,...,x_n) = x_i$ is called a projection function. I often have to use the function $u_i: E_i \to ...
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1answer
36 views

What is the name of this shape? (two arcs and two tangents)

Think the shape of a two-pulley belt or a fully tightened bicycle chain: If it was symmetrical, it could be called rounded rectangle, but not when the radii are not equal.
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1answer
58 views

Name for sets $X$ equipped with a bijection $X \times X \cong X$

Is there a common name for pairs $(X,\alpha)$, where $X$ is a set and $\alpha : X \times X \to X$ is a bijection? Once I have heard "heap" for this, but this already has a different meaning. Notice: ...
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11answers
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Does “Doing a thing to both sides of an equation” have a name?

A two part question. 1 True or False: when working with an equation or inequality, everything that you do is either: a substitution, or an operation performed on each side Note that algebraic or ...
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1answer
46 views

Are math definitions iff statements? [duplicate]

I was wondering if definitions in mathematics are "if and only" statements? I know for sure that theorems are not "iff" statements. Thank you in advance for your help.
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What is a “standard symbol” in analysis?

I was reading some Wikipedia pages about analysis when I came across this strange "standard symbol" terminology in the "Fourier integral operator" page. It seems to be a function (or a distribution) ...
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57 views

On the History of the Concept of Module

I am interested in knowing a little bit more about the history of the concept of module. As far as I know, there are two primary meanings of the word in mathematics, namely, modules as derived from ...
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5answers
229 views

Must perpendicular (resp. orthogonal) lines meet?

In geometry books in French, there has traditionally been a very clear distinction between 'orthogonal' lines and 'perpendicular' lines in space. Two lines are called perpendicular if they meet at a ...
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2answers
19 views

What object does the term “probability distribution” typically refer to, for both discrete and continuous spaces, in different fields of study?

In the discrete case, is it taken to refer to a probability mass function (PMF) or a cumulative distribution function (CDF)? Similarly, on continuous probability spaces, is it generally taken to refer ...
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1answer
54 views

What does it mean to say that “formula is exact for all polynomials of degree less $n$”? [closed]

What does it mean that "formula that is exact for all polynomials of degree less than or equal to 2?'
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2answers
29 views

Projective Linear Group and Projective Space

my aim : Why $GL(V)/Z(GL(V))$ is termed as projective general linear group? The reason seen in books says This group acts on the projective set/space faithfully. But, there can be many groups acting ...
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2answers
50 views

Terminology conventions in mathematics

I am writing a paper where I have more than one lemma (Lemma 1, Lemma 2, and Lemma 3) and when I cite them together I was wondering is it more appropriate to say, for example, because of Lemmas ...
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1answer
28 views

Terminology request: operations on graph adjacency matrices

I have a physics/computer science background. I'm writing up some research, and I keep hesitating and second-guessing myself when it comes to the formal definitions. I would like to settle my mind and ...
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1answer
44 views

Is a single point a closed interval?

For example, is {0} considered a closed interval? Why or why not? Doesn't it contain all (it's only) limit point of 0?
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179 views

What is the name for this operation?

What is the name for this operation? Effectively, take a range and adjust its center point to 0 on a number line. In the case of the example above, I'm specifically looking for the name or ...
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1answer
99 views

What does algebra $A \rtimes G$ stand for?

Let $k$ be an algebraically closed field, $A$ a finite dimensional (unital associative) $k$-algebra of finite dimension, and $G$ a torus over $k$ acting on $A$. What does $k$-algebra $A \rtimes G$ ...
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0answers
30 views

Terminology of 'impossible' and 'indeterminate' results

In logic (whether boolean logic or real-valued logic), what are the proper terminology for impossible and indeterminate values? I define impossible as the outcome of a statement that cannot exist for ...
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1answer
19 views

On non-modular lattices and orto-modularity

I would like to have a definition for non-modular lattices which clearly sets them appart from their modular counterparts, thereby focusing on their main distinctive feature. Besides, I would be very ...
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34 views

Is there a name for this property (on sequences)

Suppose we have a sequence $(x_n)_{n=1}^\infty\subseteq X$ with $(X,d)$ a metric space, and we have the following property: $$\exists x\in X\forall\epsilon>0[|B_\epsilon(x)\cap ...