New answers tagged

0

The most general concept/word would be singularity. In this case, I might call it a "corner", a concept which can be made formal with the concept of a "smooth manifold with corners" as mentioned in this MSE answer, for instance. Depending on the context/exact situation, something like this may be called a cusp or crunode.


0

As you say, "partition" doesn't really fit here since that (almost always) would require that none of the sets are empty. I would use the word "decomposition" instead. One example of this use of "decomposition" is in the Hahn decomposition theorem, where either one of the positive and negative sets may be empty.


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I like to propose a notation that just crossed my mind. To me, zero is no number I would call natural. It is finite, though. So I propose to use $$\mathbb N = \{1, 2, 3, …\}\quad\text{and}\quad\mathbb F = \{0, 1, 2, 3, …\},$$ where “$\mathbb F$” stands for finite numbers or rather finite ordinals or finite cardinals. This fits nicely with the usual practice ...


1

As others have said, there's no consensus on this. However, if you need unambiguous notation, you can use: $\mathbb{Z}_{\geq 0}, \mathbb{Z}_{\geq 1}.$ This is a good option if you're writing something short and sweet, e.g. for posts to this website. In something longer, like an article or PhD, you may wish to spend a sentence or two establishing a ...


3

These lecture notes from a combinatorics course given for many years by N.G. de Bruijn suggest a helpful alternative: Due to the confusion caused by N. Bourbaki about the natural numbers, we feel obliged to define: $$\begin{align}\Bbb N_0 & = \{0,1,2,\ldots\}\quad \text{ and } \\ \Bbb N_1 & = \{1,2,3,\ldots\}. \end{align}$$ (Page 4)


1

Or does everyone really just refer to points and vectors interchangeably in the Euclidean space $\Bbb R^n$? Yes. Everyone does. Okay, hyperbolic statements like that one are almost guaranteed to not be true. But still, it is close to true. Practically no one balks at the concept. However, it is also not completely true that they are used ...


0

The most common term I've seen for $n$ in the expression $\sqrt[n]{x}$ is the index, or root index. Wikipedia's article on $n$th roots mentions this, but only twice (for the first time in the third paragraph of the page).


1

Let $\mathcal K$ be a fixed closed collection. For all $x\in X$ define $R(x)\subset Y$ as $\{f(x)|f\in\mathcal K\}$ and set $$\mathcal L=\{g:X\to Y|\forall x\in X,g(x)\in R(x)\}.$$ Obviously $\mathcal L$ is closed and $\mathcal K\subset\mathcal L.$ We claim that actually $\mathcal K=\mathcal L.$ For if $g\in\mathcal L$ then we have a family of functions ...


2

Using the participle form "conjugated" is not necessary, I would avoid it. The proposition "with" doesn't really add anything, I would stick with "to" (or stick to "to"; prepositions are such fuzzy things). The general stylistic principle here is: longer words that add neither meaning nor intuition should be avoided. That leaves 1, 5, and 7, all of which ...


0

I think this might help you: https://proofwiki.org/wiki/Definition:Conjugate_(Group_Theory) Using the example you've written, $a$ and $b$ are conjugate with respect to $c$ so all of your statements are equivalent except for number $6$ I would say which doesn't really make sense


0

Let me give a simple example that I used last week in my lecture to pre-service teachers. We define $\pi$ to be the ratio of the circumference and the diameter of a circle. But how do we know that this does not depend on our choice of circle? In fact, Euclid proves that given two circles, this ratio is the same. Therefore this definition is well-defined, ...


0

$l_p$ metric on $\mathbb R^m$: $$ d_p(x-y) = \left(\sum_{i=1}^m \big|x_i-y_i\big|^p\right)^{1/p} $$ $1\le p < \infty$. I will let you do the $l_\infty$ metric.


1

(some changes in order to improve precision) About your last interrogation. Any normed vector space defines naturally a metric space by the relationship $d(x,y)=\|x-y\|_p$. Thus, indeed, any $\|\cdot\|_p$ norm induces naturally an $\ell_p$ "metrics" (synonym : "$\ell_p$ distance"). A point of vocabulary about the words "metrics" vs. "distance". "Metrics" ...


1

Let $(x_1,x_2,\ldots, x_n)$ be the given list of numbers, and let $s:=\max_{1\leq k\leq n} x_k$. Then I'd call $d_k:=s-x_k\geq0$ the defect of $x_k$, so that $(d_1,\ldots,d_n)$ would be the list of defects of the given data. (You can replace the word "list" by "set" for a less stringent denomination of terms.)


1

An element $x$ such that $x+x=0$ is called an involution. See here under the "group theory" heading: https://en.m.wikipedia.org/wiki/Involution_(mathematics) This is the opposite of the property you describe, and seems to be the rarer case. An operation under which no element has order 2 could perhaps be called "involution-free"


3

You might want to say that $(\mathbb{S}, +)$ does not have $2$-torsion, or perhaps that it is $2$-torsion-free.


1

You may also encounter well-definedness in such context: There are situations when we are more interested in object's properties then actual form. In such cases we say that we define an object axiomatically or by properties. After stating this kind of definition we have to be sure that there exist an object with such properties and that the object is unique ...


2

An example that I like is when one tries to define an application on a domain that is a "structure" described by "generators" by assigning a value to the generators and extending to the whole structure. Sometimes, because there are relationships between generators, the function is ill-defined (the opposite of well-defined). For a concrete example, the ...


5

In simplest terms, $f:A \to B$ is well-defined if $x = y$ implies $f(x) = f(y)$. At first glance, this looks kind of rediculous because we think of $=$ as meaning exactly the same thing but that is not really how it is used. For example we know that $\dfrac 13 = \dfrac 26.$ The function $f:\mathbb Q \to \mathbb Z$ defined by $f\left(\dfrac xy ...


18

The term well-defined (as oppsed to simply defined) is typically used when a definition seemingly depends on a choice, but in the end does not. In most (but not all) cases, this applies to the definition of a function $f\colon A\to B$ in terms of two given functions $g\colon C\to A$ and $h\colon C\to B$: For $a\in A$ we want to define $f(a)$ by first ...


1

It is widely used in constructions with equivalence classes and partitions.For example when H is a normal subgroup of the group G, we define multiplication on G/H by aH.bH=abH and say that it is well-defined to mean that if xH=aH and yH=bH then abH=xyH.


13

There is an additional, very useful notion of well-definedness, that was not written (so far) in the other answers, and it is the notion of well-definedness in an equivalence class/quotient space. Take an equivalence relation $E$ on a set $X$. We can then form the quotient $X/E$ (set of all equivalence classes). Take another set $Y$, and a function $f:X\to ...


9

The statement '' well defined'' is used in many different contexts and, generally, it means that something is defined in a way that correspond to some given ''definition'' in the specific context. As a simple example: if I say: given the function $f(x)=\sqrt{x}=y$ such that $y^2=x$ this function is not well defined. A function is well defined only ...


2

It might differ depending on the context, but I suppose it's in a context that you say something about the set, function or whatever and say that it's well defined. For example: Let $f(x)$ be a function defined on $\mathbb R^+$ such that $f(x)>0$ and $(f(x))^2=x$, then $f$ is well defined. This means that the statement about $f$ can be taken as a ...


3

"Is standard" modifies "automorphism." "With $\frac{1}{2}$" means that $2$ is invertible in the ring. That is, the sentence should be parsed We prove that every automorphism [of a Chevalley group of type $B_{\ell}, \ell \ge 2$, [over a commutative local ring with $\frac{1}{2}$]] is standard, i.e....


2

The whole class of such functions used in computer science are activation functions, and to be more descriptive you might say that it is shifted sigmoid function, but I don't know if there's a specific term for this. It is also scaled, but nobody really mentions scaling as it is usually irrelevant. Neither do people really care about the restriction to ...


0

The way I'd do it is as the $\sigma$-algebra generated by the sets of the form $\Delta(a_{1}, \ldots, a_{n}) = \{ (s_{k})_{k \in \mathbb{N}} \in \{T, H\}^{\mathbb{N}} : s_{k} = a_{k}, k = 1, \ldots, n\}$, that is, the set of sequences of tails and heads for which the first $n$ flips will agree with some prescribed sequence of $n$ outcomes, i.e. ...


2

You have the right sample space, and you are right that if the $\sigma$-algebra were always the power set of the sample space, then there wouldn't be a point in having such a concept. This is a discrete sample space. "Discrete" means the sum of the probabilities assigned to subsets of the sample space having just one member is $1$. In other words, all of ...


4

This means that $\hat{S}$ is the set of all equivalence classes of $\mathcal{C}$. It assumes that we have in mind some specific equivalence relation $\sim$ on $\mathcal{C}$. In other words, every element $s$ of $\hat{S}$ is a nonempty subset of $\mathcal{C}$. The following two conditions hold: If $x \in s$ then we have $y \in s$ iff $x \sim y$ For every ...


0

If $G$ is a group acting on a set $X$, one notation you can use for the set of fixed points of the action is $X^G$. And if $f$ is a function acting on a set $X$, one notation you can use for the set of fixed points of $f$ is $\text{Fix}(f)$. So I think either $A^{\sigma}$ or $\text{Fix}(\sigma)$ is fine, depending on whether you want to emphasize $A$ or ...


3

The notation $A\sqcup B$ (and phrase "disjoint union") has (at least) two different meanings. The first is the meaning you suggest: a union that happens to be disjoint. That is, $A\sqcup B$ is identical to $A\cup B$, but you're only allowed to write $A\sqcup B$ if $A$ and $B$ are disjoint. The second meaning is that $A\sqcup B$ is a union of sets that ...


1

Not sure why this was a comment rather than an answer so... "I believe the standard notation in the book is $Sym(A,\sigma)$. You can call them symmetric elements, or simply elements fixed by $\sigma$. – @dbluesk"


2

Recall the definitions: A space is connected if it cannot be written as union of two non-empty disjoint open subsets. A space is disconnected if it is not connected. A space is totally disconnected if it has no non-trivial connected subsets. What about the empty space? It is connected, in fact vacuously so as it lacks non-empty subsets in the ...


0

These deviations are usually called the errors or residuals. (The distinction between these two terms becomes important if your sequence is a sampling from a probability distribution, in which case error is measured with respect to the true mean instead of the sample mean.)


3

From Online Etymology Dictionary: echelon (n.) 1796, echellon, "step-like arrangement of troops," from French échelon "level, echelon," literally "rung of a ladder," from Old French eschelon, from eschiele "ladder," from Late Latin scala "stair, slope," from Latin scalae (plural) "ladder, steps," from PIE $\ast$skand- "to spring, leap" (see scan ...


1

It is called an oblique cone. Its vertex isn't on the top of the base, making it slanted.


-1

"scale of a number": Scale is the number of digits to the right of the decimal point in a number. For example, the number 123.45 has a precision of 5 and a scale of 2. other similar question and some oracle db documentation


7

As has already been suggested, it has its origins in military vocabulary. The Wiki article on echelon formation contains many pictures of stuff in echelon formation, and you can immediately see why one might say that the rows of a row-echelon matrix are in "echelon formation." I would have liked to include the wiki images, but for some reason they would not ...


3

It simply means the rows are ordered in a unique hierarchy by the positions of their leading ones (rows of all zeros being lumped at the bottom).


1

Also, be careful when you take an inverse function of $W$ since $W$ has two branches. Probably, you meant the principal branch $W_0$ of it that assumes values from the range $[-1/e,\infty]$. Then, the corresponding inverse function of $W_0$ will be $f(x) = xe^x$ on a domain $x \in [-1, \infty]$.


0

A simple mathematical way of describing this is by using the log function. A number between 0 and 1 has a negative log and a number larger than 1 has a positive log... So "negative log" and "positive log" could be a way of referring to this. Note that with this notation, your kills/deaths ration becomes $$ \log (\frac{kills}{deaths})=\log(kills)- \log ...


0

I'm not sure that I would use a term from mathematics. You might consider using a term like "subpar", where "par" would be $1:1$, as in "One kill to four deaths is a subpar kill/death ratio."


3

Your friends aren't wrong for describing a 1kill, 4 deaths score as negative. They are simply using a different metric, kills-deaths, instead of what you're using, kills/deaths. Both are useful in different scenarios and you should aim to go positive with an improper fraction for a kdr.


3

Since you seem to be primarily interested in rational numbers, a good candidate is proper fraction.


3

You may refer to them as "$k$-vectorspace homomorphisms" if you wish. In a little more detail: if $k$ is a field, then a $k$-vectorspace can be thought of as a set $X$ equipped with, for each sequence $a \in k^n$, a corresponding "linear combination mapping" $X \leftarrow X^n$. By convention, the structure-preserving mappings between $k$-vectorspaces are ...


4

I picture a (countable) sequence of shrinking neighborhoods around a point in $\mathbb{R}^2$. (Specifically, the balls $B_{1/n} (x)$.) If I draw any blob around this point, I just have to wait a bit and my neighborhoods will shrink enough so that they are eventually contained in it. If something is first countable, then you can check if it is second ...


4

Imagine a point and around it a sequence of ever smaller disks with center that point. Whatever form you draw, if that point is in its interior there will be a disk small enough to be contained in that form, too. If you need help to keep apart "first" and "second" countable, you could recall that the definition of 1st depends on what happens around 1 ...


1

Linearity has different (yet equivalent) definitions. One is what you described: preservation of linear combinations (linear combination of inputs yields linear combination of outputs), which is a godsend when solving equations - solve each part separately, and you have a general solution. In physics, this is called the principle of superposition. Secondly, ...


1

"Arbitrarily many" can be infinite where "finitely many" cannot. A specific example is in topology, where the intersection of finitely many open sets is still open, while the union of arbitrarily many open sets is open. An example to show the infinite intersection of open sets may not be open comes from $\Bbb R$ with the usual topology. All the sets of ...



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