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

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0answers
27 views

What would be the mathematical name for a function like the following?

x is an independent variable, p is a proportionality constant, and k is a critical value for x. Let's say: f(x)=p (a proportionality constant), where x >= k, a critical value. f(x)=0 where x ...
1
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1answer
43 views

One-sided Derivative Question

Let's say we define $$D_{+}f(x):=\lim_{h\to 0^+}\frac{f(x+2h)-f(x+h)}{h}$$ to be the "right-handed" derivative. This way the function does not have to exist (or equal what it 'should') at the point ...
-3
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0answers
73 views

Reflexive. What does it mean? [on hold]

I would like to know the definition for reflexive. I have not found anything on the internet or in my book.
1
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1answer
26 views

What is the name of this measure property?

if we have a function $f \in L^p$ sucht that $||f||_p =1$ and $m$ being a finite measure. Define a new measure $\mu$ by $$\mu(A):=\int_A |f(x)|^p dm(x).$$ Then $\forall \epsilon > 0 \ \ ...
2
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6answers
212 views

Algebra: What does “is defined for” mean?

In algebra what does: "Is defined for" mean? I have a question posted: $\sqrt{a+b}$ is defined for $-b \leq a$. The question posed is: Is this true... My question: WHAT DOES "Is Defined For" ...
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2answers
29 views

Difference between the definition of monoid action and group action?

The question is essentially in the title. From what I read in the wikipedia article about monoids it seems to me that we can define a monoid action in the exact same way we define a group action. Is ...
0
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1answer
25 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
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0answers
21 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 ...
3
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0answers
20 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 ...
3
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1answer
45 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
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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
3
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1answer
83 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
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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
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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 ...
0
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0answers
9 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 ...
12
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2answers
173 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
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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 ...
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0answers
31 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 ...
1
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2answers
69 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
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0answers
126 views

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) ...
1
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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. ...
0
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2answers
45 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 ...
0
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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
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3answers
54 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
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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 ...
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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
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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 ...
0
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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
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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
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0answers
22 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
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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
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1answer
75 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
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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
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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
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1answer
17 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
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5answers
359 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.
0
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2answers
40 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 ...
3
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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
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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. ...
1
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1answer
28 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
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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 ...
17
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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 ...
0
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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
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1answer
31 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 ...
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0answers
22 views

Name of this Formula [Spherical Earth projected to a plane]

I am using a formula to calculate the distance between two coordinates. Basically this is the Pythagorean theorem. I saw this formula on Wikipedia and it works perfectly for my use case. However I ...
2
votes
1answer
64 views

why calling these 'algebra' and 'ring' too?

In measure theory you have 'algebra's' and 'rings' as subsets of the powerset of the underlying set of the measurable space. If I am well informed then you speak of an algebra if it is closed under ...
2
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0answers
41 views

Replacing $q^2$ by $q$

I have a rather strange question. Suppose we are given a formal power series $$S(q^2) = \sum_{n = 0}^\infty a_n q^{2n}.$$ I wish to replace $q^2$ by $q$. This implies that $S(q) = \sum_{n = 0}^\infty ...
2
votes
1answer
24 views

Is there a name for a directed graph in which every vertex has outdegree one?

Per the question title, I'm dealing with a number of directed graphs, all of which are 1-out regular, and figure that there is probably a name for such a thing. Unfortunately, all my search attempts ...
1
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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 ...
0
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1answer
27 views

Solving the equation $z=\sum_i \alpha_i \exp(-\|x_i-z\|^2)x_i$

For $i=1,\dots,M$ vectors $x_i\in\mathbb{R}^N$ and scalars $\alpha_i$, can you find a vector $z$ satisfying the equation $z=\sum_i \alpha_i \exp(-\|x_i-z\|^2)x_i$? Any pointers will also be ...