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

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3
votes
1answer
69 views

Is a homomorphism expected to be a (structure-preserving) map?

Is a homomorphism a special type of morphism, namely a structure-preserving map? For a morphism (of a category), it is clear that we can't always expect that a morphism is necessarily a ...
4
votes
3answers
354 views

What is the difference between being unique, unique up to isomorphism and unique up to unique isomorphism?

Could anyone explain the difference between the above mentioned terms "unique", "unique up to isomorphism" and "unique up to unique isomorphism", preferably with an example? Are there other ...
0
votes
1answer
20 views

What's maximal clique?

I'm unable to understand what maximal clique is. I mean how a clique can't be extended by a node and remain a clique? If I add a node and then I connect this node to every other nodes in the clique, ...
0
votes
0answers
16 views

mean number of links in adjacency matrix

I have converted from an individual-level adjacency matrix to one for clusters and I am trying to show mathematically how I programmed up determining the mean number of inter-cluster links. I am not ...
13
votes
7answers
4k views

Difference between “Show” and “Proof”

In most of mathematics problems you see "prove that..." or "show that..." statement. My question is what's the difference between these two words? I mean is "showing" something different from ...
3
votes
0answers
17 views

Is there a name for this partial order between metrics?

Suppose we have a set $X$ and two metrics $d_1,d_2$ on it (which may or may not attain $\infty$). Assume furthermore that $d_1,d_2$ have the same metric components (where a metric comoponent is a ...
9
votes
2answers
386 views

Etymology of the word “normal” (perpendicular)

While the word "normal" is one of the most overloaded mathematical terms, in linear algebra, it is usually associated with the notion of being perpendicular to something, as in "normal vector" or ...
3
votes
1answer
42 views

Is there a name for the trivial probability distribution P(X=x) = 1 for a unique x?

Is there a name for the trivial probability distribution given by $P(X=x) = 1$ for a unique $x$ and $P(X=y) = 0$ for all $y \ne x$? I know it is very trivial, but since it is the distribution that ...
0
votes
1answer
11 views

What is the different between generating expression and infinite expansion?

What is the different between generating expression and infinite expansion? I can't figure it on my own
12
votes
2answers
160 views

Is there any distinction between these products: scalar, dot, inner?

I hope you will forgive a math question that comes up in physics contexts where language is loose. This question migrated from Physics SE. I'm finding that I sometimes don't know what kind of product ...
0
votes
0answers
57 views
+50

Differend kind of recursion.

How are called these special case of definition by recursion? given increasing $f,g: \Bbb N \rightarrow \Bbb N$ and $b,b_1,b_2 \in \Bbb N$ 1 (a special case of primitive recursion) ...
0
votes
1answer
24 views

Semantic question about chirality

Is it enough to generally say that an object is (or is not) chiral in some space/some number of dimensions according to some convention, or is some sort of structure or description of how it is chiral ...
4
votes
0answers
25 views

Subsets of cyclic group with distinct pairwise differences

Given a number $m\in\mathbb N$, let $\mathbb Z_m=\{0,1,\dots,m-1\}$ denote the ring of integers modulo $m$ (although we won't need multiplication, so any cyclic group of order $m$ will do). Given a ...
0
votes
0answers
8 views

Is there a name for a 'weak' sublattice?

For a subset of a lattice to be a sublattice, we need that it is closed under the meet and the join operations. However there are other subsets which are not sublattices, but such that the poset on ...
1
vote
2answers
57 views

Confusion related to definitions involving free groups

From Wikipedia ...the free group $F_{S}$ over a given set $S$ consists of all expressions (a.k.a. words, or terms) that can be built from members of $S$, considering two expressions different ...
0
votes
0answers
36 views

Is there a traditional name for the “eigenspace” function?

Let $A$ denote a field, $X$ denote an $A$-vector spaces, and suppose $\varphi : X \rightarrow X$ is a linear transformation. Is there a traditional name for the corresponding "eigenspace" function? By ...
0
votes
0answers
30 views

The term $rank$ in methematics

Reading wikipedia's disambiguation page about the "rank" word I see many concept of rank of many different matematical object. I only know about the rank of a graded poset and the rank of a set that ...
2
votes
2answers
259 views

probability terminology for parameter in a Markov process

Suppose $$P(\text{feature present at time} \ t \ \text{and} \ t+\Delta t) = \beta^{2}+\beta(1-\beta) \exp(\Delta t/\tau)$$ where $\tau = 1/(\pi_{01}+\pi_{10})$. What is $\tau$?
1
vote
0answers
18 views

Terminology: Contraction *of* normed spaces? *Between* normed spaces? *On* normed spaces?

I have a terminological question. Suppose $X$ and $Y$ are normed spaces, and let $f$ be a contraction $X \to Y$. Which of the following expressions is correct? $f$ is a contraction of normed spaces. ...
1
vote
0answers
23 views

Is there a name for this type of tensor rank?

Let $A\in\mathbb{R}^{n_1\times n_2\times n_3 \times n_4}$ be a tensor. Suppose that $k$ is the minimum integer there exist matrices $X_1,\ldots,X_j\in\mathbb{R}^{i_1\times i_2}$ and ...
2
votes
4answers
82 views

Values of square roots

Good-morning Math Exchange (and good evening to some!) I have a very basic question that is confusing me. At school I was told that $\sqrt {a^2} = \pm a$ However, does this mean that $\sqrt {a^2} ...
0
votes
2answers
44 views

What subject in mathematics investigates the type of problems that constitute the LSAT “logic games” (example given)?

For my own curiosity, I read part of an LSAT study guide yesterday. The "logic games" section comprised questions like, An advertising executive must schedule the advertising during a particular ...
10
votes
3answers
166 views

What do we call entities (like $\sum$) that bind variables?

In logic, we refer to entities like $\forall$ and $\exists$ as quantifiers, because they bind variables. However, variable-binding doesn't just occur at quantifiers. For example, the symbol $i$ ...
0
votes
1answer
17 views

Limiting probability of Markov chain(Terminology)

If I am asked to find the limiting probability of a Markov chain, what does this pertain to? $\lim \limits_{n \to \infty} P^n$? Where $P$ is the stepping matrix and $n$ is the number of steps. "What ...
2
votes
3answers
51 views

What's the name of the quantity $\mathbb{P}(A\cap B)/(\mathbb{P}(A)\mathbb{P}(B))\;$?

In a physics book, I've come across the quantity $$ \frac{\def\P{\mathbb{P}}\P(A\cap B)}{\P(A)\P(B)}\,, $$ where $A$ and $B$ are events. The author calls this quantity the correlation of $A$ and ...
5
votes
1answer
113 views

What exactly is a dimension?

Maybe this is too broad a question, maybe I need to be more specific. I am just clearing my head here, feel free to ignore at your pleasure. In Linear Algebra, we learned that the dimension of a ...
54
votes
14answers
3k views

Are “if” and “iff” interchangeable in definitions?

In some books the word "if" is used in definitions and it is not clear if they actually mean "iff" (i.e "if and only if"). I'd like to know if in mathematical literature in general "if" in definitions ...
0
votes
0answers
12 views

Is it appropriate to define $f_j=f(x_j)$?

I am reading a textbook that introduce the Fourier transofrm, yet the question is not confined to this subject. The author defines that $f_j=f(x_j)=f(\frac{2j\pi}{n})$ From my point of view, ...
0
votes
1answer
34 views

2D is to face as 3D is to?

Essentially, if a point is a zero-dimensional component of an object, a line is a one-dimensional component, and a face is a two-dimensional component, what is a three-dimensional component? If there ...
15
votes
5answers
2k views

Is 'no solution' the same as 'undefined'?

Today in class my teacher wrote something along the lines of: $6^x = 0$ And proceed to heed a response from the class. A few people shouted undefined. So the teacher then writes: no solution ...
0
votes
1answer
8 views

An indexed family of filters and their elements

Let $X$ is an indexed (by some set $n$) family of filters (on some poset $\mathfrak{A}$). Is there any standard notation/terminology for the set $\{ y\in \mathfrak{A}^n \,|\, \forall i\in n:y_i\in ...
0
votes
1answer
61 views

Is there a name for the one-point compactification of $\mathbb{C}$?

Let $\hat{\mathbb{C}}$ be the one-point compactification of $\mathbb{C}$. This space $\hat{\mathbb{C}}$ is called the Riemann sphere. If I want to designate the topology $\tau$ on ...
1
vote
0answers
21 views

I need help understanding what r-th and s-th rows are.

Let E be the matrix obtained from the unit $n \times n$ matrix by multiplying the $r$-th row with a number $c$ and adding it to the $s$-th row, $r \neq s$. Let $A$ be an $n \neq n$ matrix. Then ...
1
vote
1answer
38 views

Confused by a step in a proof that $a^x - b^y = c$ has at most two solutions in positive integers $x,y$

The theorem is Theorem 1.1 from Michael A. Bennett in his "On Some Exponential Equations of S.S. Pillai". Here is the statement of the theorem: Theorem 1.1. If $a,b,c$ are nonzero integers with $a,b ...
2
votes
1answer
70 views

Definition of the $\sec$ function

I am a postgraduate student of mathematics from Slovenia (central Europe) with quite some experience in mathematics. While answering questions on this site, I often encounter the function $\sec(x)$ ...
0
votes
0answers
26 views

3D - surface area, 2D - perimeter, 1D - ?? (how do you call the equivalent term)

The other day I was thinking (while grinding some sugar to make it easier to dissolve in water): When you 'cut' a 3D object in multiple 3D objects: the total volume remains the same but the total ...
0
votes
0answers
9 views

Terminology for 'clusters' in a discrete probability distribution?

I have attached an image of a probability distribution. As you can see their are peaks, and in my opinion thee 'clusters' in this distribution. There is the cluster that spans the origin and goes out ...
0
votes
1answer
16 views

Need help in understanding $ord_p{a}$ as used in Theorem 1.1 from “On Some Exponential Equations Of S. S. Pillai”

I have a question about very early argument in the proof of Thereom 1.1. Theorem 1.1 of On Some Exponential Equations of S.S. Pillai states that if $a,b,c$ are nonzero integers with $a,b \ge 2$, then ...
3
votes
5answers
358 views

Injection vs. Surjection: Mnemonic to remember which is which

What are some mnemonics to help one remember that Injection = One-to-one and Surjection = Onto? The only thing I can think of is 1njection = 1-1.
1
vote
1answer
80 views

Terminology: Sheaves with surjective structure maps?

Is there established terminology for sheaves (on a topological space), the structure maps of which are all surjective? I have come across some "cosheaves" with injective structure maps and would like ...
6
votes
3answers
983 views

If and only if, which direction is which?

I can never figure out (because the English language is imprecise) which part of "if and only if" means which implication. (A if and only if B) = (A <=> B), but is the following correct: (A only ...
0
votes
2answers
38 views

About matrix $R$, what is this called: $R^TR$? What is it for?

I am doing singular value decomposition on a matrix $R$. The first step is to compute such a matrix $R^TR$. What is this matrix? A reference told me this is cross product of matrix R. I use a ...
16
votes
4answers
2k views

Is there a way to denote the calculation $1+2+3+\dots+n$? [duplicate]

Since $n!$ represents $$1\cdot2\cdot3\cdots n,$$ I am wondering if there is a way to represent $$1+2+3+\dots+n?$$ What are some usual notations for the computation of some common sequences? Any other ...
10
votes
4answers
2k views

What is the term for a factorial type operation, but with summation instead of products?

(Pardon if this seems a bit beginner, this is my first post in math - trying to improve my knowledge while tackling Project Euler problems) I'm aware of Sigma notation, but is there a function/name ...
3
votes
1answer
71 views

Which one is correct? $(n/2)$nd or $(n/2)$th? [closed]

I am reading a textbook in which I find a writing problem: Squaring them produces the $(n/2)$nd roots of unity. My question: Which one is correct? $(n/2)$nd or $(n/2)$th?
1
vote
1answer
63 views

What is this metric called?

Ahlfors -complex analysis p.20 Consider a stereographic projection between the 2-sphere and $\overline{\mathbb{C}}$ (i.e. one-point compactification of $\mathbb{C}$) Let $z,w$ be complex numbers. ...
0
votes
0answers
25 views

Mapping a set of sets to a partitioning.

I've been experimenting with the following idea, and I wondered if there's a name for it: Suppose $S_0, S_1, ... S_{n-1}$ is an array of $n$ sets of elements in $U$. Now for any element $e \in U$ we ...
1
vote
1answer
26 views

Why are isotropy groups named as such?

Why are isotropy groups, also known as stabilizers, named as such? In physics, the word isotropy means having the same property in all directions. Can one draw an analogy from this to interpret the ...
0
votes
0answers
39 views

Ring $R$ as $R[x]$-module

My professor mentioned some interesting examples of modules, giving as an example the following two: $R$ as an $R[x]$-module, in which multiplication by $x$ was taken to be evaluation under a fixed ...
2
votes
1answer
30 views

Embedding and monomorphism

What is the difference between an embedding and a monomorphism? As far as I can see, most introductory abstract algebra texts treat them as if they are the same, i.e. an injective function from one ...