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

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1answer
51 views

$xRx'$ and $yRy'$ implies $f(x,y)Rf(x',y')$

Let $R$ be a binary relation. Is there a name for the following property? $f(x,y)Rf(x',y')\quad$ if $xRx'$ and $yRy'$ Note:$f$ is a function.
2
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1answer
200 views

Bounded open interval

This is just a quick terminology question. My textbook is talking about the continuity of $ f $ over a bounded open interval $ (a,b) $. Am I right in assuming that this means $a$ and $b$ are finite? ...
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2answers
339 views

What is an indeterminate in a polynomial ring?

I am currently studying polynomial ring. I have a basic doubt. What is $x$ in a polynomial? A polynomial is an expression $\sum_{k = 0}^n a_k x^k$, where $n,k \in \mathbb{N}$ and $a_k \in R$ where ...
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1answer
48 views

Optimum equals extremum?

Is there any difference between optimum and extremum? It seems to me that they are the same. Am I right?
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2answers
129 views

Alternatives to percentage for measuring relative difference?

If the value of something changes from $\;a\;$ to $\;b\;$, their relative difference can be expressed as a percentage: $$ \newcommand{\upto}{\mathop\nearrow} (D0) \;\;\; a \upto b \;=\; (b-a)/a ...
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1answer
64 views

If the solution dosen't exist, does it mean that problem is not well-posed?

A well-posed problem is defined as here. If that is so, then does it mean that the solution/s of any over-determined or under-determined system is NOT well-posed? Similarly, if the solution doesn't ...
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7answers
592 views

What is a number?

A dictionary I consulted said a 'number' is a 'quantity', so I looked up what quantity means and the same dictionary said it is an amount or number of some material or thing. Since quantity and ...
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1answer
96 views

What does it mean when a set is said to be a “finitely generated vector space”?

I've somehow managed to go through 3 years of my Maths degree without truly understanding what the term "finitely generated vector space" means. Generated by what? Generating what? And for what? I ...
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3answers
54 views

Is there a name for this particular linear fractional transformation?

Is there a conventional name for this function? $$ \begin{align} g(t) & = \frac{1+it}{1-it} \\[15pt] & = \frac{1-t^2}{1+t^2} + i\frac{2t}{1+t^2}. \end{align} $$ This function comes up from ...
3
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0answers
36 views

Origin of the terminology “connected algebra”

I was wondering what is the origin of the word "connected" for a connected algebra ? To be more precise, why is a graded $R$-algebra $A_{\ast}$ with an augmentation $A_{\ast} \to R$ that restricts to ...
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1answer
49 views

Why are “restricted Lie algebras” called restricted?

Restricted Lie algebras are Lie algebras of characteristic $p$ with an additional unary operation which is like raising to $p$th power. I didn't find any motivation for this strange choice of the name ...
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1answer
179 views

What is an “essentially sharp” estimate?

I frequently encountered theorems here and there, which claim to establish essentially sharp upper bounds for certain quantities. However, I am confused about what “essentially sharp” exactly means, ...
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2answers
145 views

What is meant by canonical?

So I came across the term canonical multiple times by now, and still dont have a very good idea of what it means. So e.g. a matrix $M$ w.r.t. a canonical basis $B$. What is makes a basis canonical? ...
4
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1answer
80 views

Is there a name for this “co-dual” vector space?

Let $V$ be a $K$-vector space, then the linear transformations $T:K\rightarrow V$ (where $K$ is considered a 1-dimensional vector space over itself) form a vector space $$ A = ...
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0answers
320 views

Support of a vector

What is the support of a signed vector? By signed vector, I mean a vector which is determined by considering the signs of the coefficients of the entries of another vector.
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1answer
31 views

Realizability in mathematics

What does it mean in mathematics when someone says "it is realizable"? If someone could give me a general and intuitive explanation I would appreciate it. Thank you!
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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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1answer
63 views

Canonical direct product (in a category)

In some categories there are more than one (isomorphic) direct products: For example in Set there are $A\times B$ and $B\times A$ products (as well as many others). But only one of these products ...
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2answers
112 views

The word “and”.

If I say the difference between $x$ and $y$ Does that mean $x - y$ or $y - x$ or $|x - y|$?
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1answer
85 views

name of the unit of adjunction between $-\times C$ and $\cdot^C$

Answers to a earlier question about the categorical interpretation of first-order quantification led me to learn more about adjoints. Now, I understand that a category $\mathscr{C}$ with products has ...
1
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1answer
46 views

The “standard” definition of discrete subspace

Let $X$ be a topological space and $D\subset X$ a generic subspace. In literature I have found the following different definitions: $D$ is discrete in $X$ if $X$ doesn't contain limit points of $D$. ...
2
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0answers
142 views

Difference between soundness and correctness

Is there any actual semantic difference between soundness and correctness? Can I use these words interchangeably when talking about formal reasoning, proof, logics, etc.? Otherwise, is there a ...
11
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7answers
646 views

How to say $(a,b]$ and $[a,b)$ in English?

I read on Mathworld they are called half-closed interval; however, it doesn't tell me how to say it in English. Also, how does one denote which one of the two options that is the one the one ...
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3answers
121 views

What is the difference between commutatitivity and distributivity?

Inspired by linearity property, I see that operator A distributes over sum: $A(\sum {f_i}) = \sum {(A(f_i))}\;\,$. What do I have between A and ∑: commutativity or ...
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0answers
96 views

Analogue of the term 'summand' for unions and intersections.

If we have a sum $\sum\limits_{i=1}^na_i$, we call the terms $a_i$ summands. In fact, in the cases of addition, subtraction, multiplication, and division, we have a large vocabulary to describe the ...
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4answers
121 views

Are there two conventional definitions of “holomorphic”?

In Walter Rudin's Real and Complex Analysis, second edition, on page 213, two definitions are stated. One of them says the derivative of $f$ at $z_0$ is $$f'(z_0)=\lim_{z\to ...
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0answers
123 views

Symbol for the incidence relation between vertices and edges.

Q: Suppose $G$ is a graph whose vertices are $V$ and edges are $E$. Is there a standard symbol for the relation $R$ on $V\times E$ such that $vRe \iff $ v is a vertex of $e$? ...
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1answer
40 views

Terminology for a weaked vector space

Let $S$ be a semigroup on which acts $\mathbb{R}_{\geq0}$. Does this structure has a name? For example $S$ can be the set of convex bodies in $\mathbb{R}^n$ with the Minkowsky sum.
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6answers
503 views

Difference in terminology between Let and Assume?

I was writing an solution to a problem in a textbook about how to factor a quadratic equation. I was told that my use of assume was incorrect and it should have used let; however, my teacher ...
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2answers
113 views

What is the $uv$ pair, or $uv$-plane, exactly?

Maybe the answer to this question is easier than computing $1+1$, but I often find this $uv$ pair on pretty much all the parametric equations that have something to do with 3D geometry and all the ...
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2answers
104 views

“Collection”: What does it mean?

I've seen a lot of question of same ilk as the request I'm about to pose, but what I'd like to know is what does "any collection" mean in the following request: Prove that the intersection of any ...
1
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1answer
40 views

Terminology for a generalization of multilinearity to any algebraic structure.

Let $S$ and $T$ be sets with the same algebraic structure. Let $\Phi:S^n\to T$ such that for any $i\leq n$, and $s_1,\ldots,s_n\in S$, the aplication $s\in S\mapsto ...
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2answers
55 views

Notation for “incommensurate” elements?

Say that $x\wedge y\not=x,y$, that is neither $x\leq y$ nor $y\leq x$. Is there a way to denote this? I've been saying $x<>y$ but that's completely made up.
2
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1answer
193 views

please help me grasp the literal meaning of residue

In complex analysis we study a term RESIDUE of a function given by some formulas. While going through its meaning I found that it means left out term or remainder kind of thing. so I was wondering why ...
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0answers
51 views

Metalanguage on mathematics

I've heard that in the first class of my degree are teaching concepts like metalanguage. What does it mean. Could you give some examples? I've searched on google that metalanguage means representing ...
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2answers
115 views

Why is this convolution true?

I am a little puzzled by how the following summation has been written as a convolution, with one of the inputs reversed in time. Consider the following sum on the LHS, and the convolution on the RHS. ...
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2answers
158 views

The bijective property on relations vs. on functions

I recently encountered what seems to me like an inconsistency in the usage of the term bijective. This is pretty basic stuff, but it somehow never occurred to me before. I'd like to make sure I'm ...
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0answers
39 views

Question on change of variables during convolution/correlation

I am trying to understand how the following two statements are equivalent: $$ \sum_{l=-\infty}^{\infty} h^*[l] \ R_{xx}[m+l] = \sum_{i=-\infty}^{\infty} h^*[i-m] \ R_{xx}[i] $$ I get that we made ...
2
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1answer
52 views

characterization of certain idealizers of submodules

Consider two commutative rings with identity, $R$ and $S$ where $R$ is a principal ideal domain, $S$ is free and finitely-generated as an $R$-module and suppose there is a $R$-module homomorphism ...
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2answers
550 views

What does “characteristic” mean in mathematics?

In Germany, I have heard "ABC is 'characteristic' for XYZ" sometimes from math students. It was used like "if you know ABC, then you know you're talking about XYZ" or "ABC describes XYZ completely". ...
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1answer
53 views

What is the name of the below method

What is the name of the below method? : 100 = 10 10 = x x = 10 * 10 / 100 = 1 Any ideas?
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6answers
350 views

Solve the equation: $x²-5x-500=0$ Take the positive value of $ x$.

Solve the equation: $x²-5x-500=0$. Take the positive value of $ x$. I am having a really bad time with this one.. Can someone give me the exact answer I posted above? I need the answer then the ...
2
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0answers
84 views

What type of graph is this. I shows kind of cause and effect.

I am trying to create a chart like this that shows item 1 effects items 2 and 4 type chart. I didn't really know where to post this so I thought mathematics would be the right place because of the ...
2
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1answer
34 views

A name for a special class of complete lattices

Is there a name for sets $S\in\mathfrak{A}$ for a complete lattice $\mathfrak{A}$ such that for all $T\subseteq \mathfrak A$, $$\bigvee T\in S \iff S\cap T\ne\varnothing?$$ Here $\bigvee$ is the join ...
3
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1answer
94 views

One-to-one mapping vs one-to-one correspondence

Does the phrase "one-to-one mapping" mean the same thing as "one-to-one correspondence?" I know that the latter refers to a bijection. Does the former refer to an injection (i.e. it is the same as ...
11
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1answer
259 views

What is “inner” about the inner product?

The inner product I am asking about is the one that generalizes the dot product for an arbitrary inner product space. Why is it called an "inner" product? Is there an outer product? Who named it ...
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3answers
166 views

Are statements like “Every time I've done X, Y has happened” (vacuously) true if I've never done X?

I've recently been wondering about vacuous truths. I know a statement like "I've never been beaten in a race" is true if I've never been in a race, but what I'm wondering is if the following ...
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2answers
121 views

Can we refer to the standard form of a quadratic equation as the general form as well?

I would like to know if we can refer to $$ax^2+bx+c=0$$ as the "general form" of a quadratic equation, or is it only called the standard form?
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2answers
461 views

Step Function and Simple Functions

Definitions: Simple Function: Any functions that can written in the form:$$s(x)=\sum_{k=1}^na_n\chi_{A_n}(x).$$ Note the finite terms here. It should follow that neither all simple functions are ...
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1answer
54 views

Rowwise matrix multiplication, what is the name of this?

Let $A=\begin{pmatrix} 1 & 2 & 3 \\ 1 & 1 & 1 \end{pmatrix}$ and $\operatorname{SomeOperation}(A)=\begin{pmatrix}1*2*3 \\ 1*1*1\end{pmatrix} =\begin{pmatrix}6 \\ 1\end{pmatrix}$. What ...