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

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1answer
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Definition of “totient”

I had always taken the term "totient" to be defined by saying that the totient of a positive integer $n$ is the number of positive integers less than $n$ that are coprime to $n$. Thus, for example, ...
2
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2answers
119 views

What does it mean to “identify” points of a topological space?

I was recently reading about circle rotations (a basic example in dynamical systems) and got confused by some notation. It said consider the unit circle $S^{1} = [0,1]/{\sim}$, where $\sim$ indicates ...
2
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2answers
135 views

Terminologies related to “compact?”

A set can be either open or closed, and there can either be a finite or infinite number of them. A "compact" set is one where every open cover has finite subcover. Is there such a thing as a set ...
2
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1answer
42 views

What is the names of $A\vec{x}=\vec{b}$ linear equation system components?

Having $A\vec{x}=\vec{b}$ . What is the names of $A\vec{x}=\vec{b}$ linear equation system components?
2
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1answer
188 views

What is the purpose of defining the notion of inflection point?

What is the purpose of defining inflection point? I know that it is defined to be the point where the second derivative is zero and the second derivative sign changes. It has to have some purpose ...
2
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2answers
121 views

Terminology: $H$ and $K$ are subgroups. What is $HK$ called?

Let $H, K\leq G$. I was wondering what you call the "product" $HK$ of $H$ and $K$. I was trying to verbalise the steps of showing $G$ is a semidirect product: Normality of $H$: $H\unlhd G$. Trivial ...
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4answers
501 views

Does the word “integer” only make sense in base 10?

Does the word "integer" only make sense in base 10? I've always wondered this and have never seen it really discussed anywhere. We all understand the typical definition of an irrational number, ...
2
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1answer
114 views

Terminology for a property that holds in the finite but not infinite case?

(I apologize if this is a duplicate, but I don't know what terms to search for. Please feel free to close this if this has already been asked.) There are some properties of finite objects that don't ...
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1answer
1k views

What do [] mean and what does it mean if it is used in an equation?

What do the square bracket symbols mean? Are they what I hear are "sets"? And when it is in an equation, how is it interpreted? Here is an example: $$\dfrac{dy}{dx}[2x2+y(x)2]=50x+2y(dy/dx)=0$$
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2answers
72 views

Shapes bounded only by lines

What is a term for the set of geometric shapes in the plane, that are bounded by one or more continuous closed curves? This set contains simply-connected polygons and circles but also polygons with ...
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2answers
59 views

Physical significance of knot vector in B-spline.

A B-spline blending curve formulation is: $P(u)=\sum_{k=0}^np_k B_{k,d}(u)$ Given $n+1$ control points, B-spline blending functions are polynomials of degree $d-1$, $(1<d<=n+1)$. ...
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1answer
26 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 ...
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1answer
50 views

Correct Terminology in the Context of Rings

Suppose I have a ring $(A, ◦, •)$ where $A$ is a set of elements $\{α, β, γ,\ldots\}$. Can $◦$ and $•$ with which the ring is equipped be properly termed, in English, its "internal laws of ...
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0answers
37 views

A map from one set of words to another that is not a morphism

A word is a concatenation of letters from a non-empty set called an alphabet. For example, if the alphabet is $\{a,b\}$, then $bba$ is a word from that alphabet. Let the set of all finite words made ...
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2answers
61 views

how do you call a function that breaks down on y?

How do you call a (linear) function (or the point), which breaks down to 0 on ordinate (axis y), as soon as you breach a certain x1 value?
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0answers
54 views

With infinite size, we can have $P \cdot M = M \cdot D $ (D diagonal) but where $M^{-1}$ does not exist. Can we say “P is diagonalizable”?

(I had this question in mind for longer time, but it is just triggered now by some comments at that recent question in mse) (Background) I was looking at properties of the Pascal-matrix: ...
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1answer
56 views

Recursive application of a function : Symbol of [duplicate]

I need to apply a function $f(x)$ recursively/repeatedly for n times; how do I express it (mathematically) ? Is their a mathematical symbol which denotes $f(x)$ applied n times ie $g(x,n)$ ...
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1answer
90 views

Difference: Proposition and Observation

As far as I know, a proposition is a statement which might be used to prove a theorem but is also of independent interest. How would you differentiate it from an observation? Would you say it is a ...
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1answer
126 views

Complement and Negation: $P(A)=0\rightarrow P(\neg A)=1$?

My earlier question became too long so succintly: Suppose $P(C)=0.2$. Its complement is 0.8 i.e. $P(C)^C=0.8$ but what does $P(¬C)$ mean? I think I am messing up the term complement and negation? ...
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1answer
82 views

Name of corresponding objects in equivalent categories

This question is only about terminology. Inside a category we have the standard wordings: An arrow $f: X \rightarrow Y$ is an isomorphism if there is another arrow $g: Y \rightarrow X$ such that $g ...
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3answers
719 views

Is there a special name for the operands of a multiplication?

Sometimes operands for a specific operation are given a special name. For example, in division the first operand is a quotient, the second is a divisor. Is there a word that means "one of the operands ...
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1answer
116 views

Sufficient/necessary vs. weaker stronger

I know what sufficient resp. necessary means, but I'm confused, when our professor uses the terminology weaker resp. stronger. I couldn't yet find out, what the translation of these pair of words to ...
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2answers
67 views

Is there a particular notation for a function confined in a set?

For example $f:\mathbb{R}\to\mathbb{R}$ is a function. How to simply express a correspondent function $g:\mathbb{Q}\to\mathbb{R}$ such that $g(x)=f(x)$, $\forall x\in\mathbb{Q}$?
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1answer
115 views

Foam-like graphs

What's the "official" name of a connected planar graph consisting entirely of polygons (cycles), glued together at edges, e.g. - among other things - without "end vertices" (of degree 1) and without ...
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1answer
73 views

Notation and naming for two operations with $p$-form valued $n$-forms

While trying to answer my other question I found I never heard about vector-valued differential forms. I've been searching for them in various mathematical physics books, but didn't get too much. I'm ...
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1answer
91 views

Whether to use 'OR' or 'AND'

My doubt is: while solving equations or inequalities consisting of absolute values when should we use the conjunction 'OR' and when to use 'AND'? whats the difference between them ?
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1answer
254 views

Is there a different name for strongly Darboux functions?

A function $f\colon\mathbb R\to\mathbb R$ is called Darboux function, function with Darboux property or function with intermediate value property for for any $a<b$ and any $z$ between $f(a)$ and ...
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1answer
109 views

Bernoulli Distribution with support different from $\{0,1\}$

Suppose the support of a distribution is $\{12 , 13 \}$ with $P(X = 12) = p$ and $P(X = 13) = 1-p$. Is this still a Bernoulli distribution even if the support is not $\{1, 0 \}$?
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1answer
166 views

What is an honest basis?

In a comment to this question, the commentator stated that "the monomials form an honest basis for your vector space". To be honest, I never heard of that. Is this something elementary?
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1answer
247 views

What does Linear Congruential mean?

How does one interpret the terms "Linear" and "Congruential" as in a "Linear congruential RNG"? I am used to linearity by $f(ax)=af(x)$. This does not seem to me to hold true in this case ($\bmod$). ...
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2answers
875 views

What does a condition being sufficient as well as necessary indicates?

I have a question in a book I am solving(Discrete Structures by Kolman, Busby & Ross). I am unable to make sense from the question. It is stated below, Show that k is odd is a necessary and ...
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3answers
223 views

How to define $-\infty$?

I think I understand the fundamental concept of infinity. Elementary mathematics define $\infty := \frac{x}{0}$, for every $x$. And also $\infty := \frac{-x}{0}$ for every $x$. I know only one ...
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1answer
115 views

Is there a special name for a semigroup whose multiplication is a constant function?

Let $S$ be a (commutative) semigroup with distinguished element 0 such that $ab=0$ for $a,b\in S.$ Of course this is a very simple family of semigroups, defined only by their cardinality. Does it ...
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3answers
152 views

Matrix with exactly one 1 in each row

Is there a name associated to rectangular matrices $M \times N$ that have exactly one entry equal to $1$ in each row and $0$ everywhere else?
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2answers
144 views

What is the name for a function of a matrix that changes the matrix size?

I have a set of functions that map square matrices with $n$ rows and columns to square matrices with $k < n$ rows and columns. Is there a name for this property? I know that 'projection' would be ...
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0answers
32 views

Terms for particular equivalence relation and partition?

Let $T$ be a set of sets. Let $\equiv$ be an equivalence relation on $\bigcup T$ defined by the formula $$a\equiv b \Leftrightarrow \forall X\in T:(a\in X\Leftrightarrow b\in X).$$ Let $S$ be a ...
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2answers
117 views

LambertW: $ x=W(x\cdot e^{x}) $ for $ x \ge -1$ but not for $x \lt-1$. How do I express my formula/my text?

I just found by numerical heuristics for some systematic numbers $q(x)_\text{heuristical}$ depending on $x=1,2,3,4,\ldots$ using WolframAlpha the suggested interpretation in terms of the ...
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1answer
74 views

Writing a chain of implications in English

How to write a theorem of the form $A\Rightarrow B\Rightarrow C\Rightarrow D$ where every $A$, $B$, $C$, $D$ are formulated with words (English) rather than with formulas? One idea: The next item of ...
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1answer
37 views

Notation for permutation corresponding to the action of a group element

Let $G \times X \to X,\ \ (g,x) \mapsto g.x$ be an action of $G$ on $X$, i.e., $e.x = x$ for all $x \in X$; $gh.x = g.(h.x)$ for all $g \in G$, $x \in X$. For a fixed $g \in G$, how should I refer ...
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2answers
155 views

Is an unit-cube polyhedron? What about other platonic solids?

Definitions According to my linear programming course and screenshot here (Finnish), a polyhedron is such that it can be constrained by a finite amount of inequalities such that $$P=\{\bar x\in ...
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1answer
73 views

What does $f$ is measurable as a function $f^{-1}(\mathbb{R})\to\mathbb{R}$ mean?

I saw a question where we have $\overline{\mathbb{R}}:=\mathbb{R}\cup\{\pm\infty\}$ and $(X,S)$ is a measurable space, $f:\, X\to\overline{\mathbb{R}}$ In one part I was told to assume that $f$ is ...
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1answer
133 views

Correct reading of Set builder Notation?

could anyone please let me know the correct reading(sentence form) of set builder notation, confused with different interpretation in different resources. Many Thanks
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1answer
477 views

Elements of order $n$ in a cyclic group of order $N$

The number of elements of order $n$ in a finite cyclic group of order $N$ is $0$ unless $n|N$, in which case it is $N/n$. Is "the number of elements of order $n$" referring to the number of ...
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1answer
76 views

What is the meaning of “countable spread”?

I encountered an example that said: A Tychonoff 2-starcompact space of countable spread which is not $1\frac{1}{2}$-starcompact. My question is this: What's the meaning of "countable spread" ?
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1answer
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A Set is a collection of well defined and distinct objects. What is a collection of well defined objects without being distinct called?

A set is a collection of well defined and distinct objects, considered as an object in its own right. What is the mathematical term for a collection of well-defined objects without distinction ...