47

You ask a very insightful question that I wish were emphasized more often. EDIT: It appears you are seeking reputable sources to justify the above. Sources and relevant quotes have been provided. Here's how I would explain this: In probability, the emphasis is on population models. You have assumptions that are built-in for random variables, and can do ...


41

A general 2nd order linear PDE in two variables is written $$Au_{xx} + 2Bu_{xy} + Cu_{yy} + Du_x + Eu_y + F = 0$$ and $A,B,C,D,E,F$ can be functions depending on $x$ and $y$. We say a PDE is elliptic, hyperbolic or parabolic if \begin{align} B^2 - AC &= 0, &\text{parabolic} \\ B^2 - AC &>0, &\text{hyperbolic} \\ B^2 - AC &<0, &...


29

Disclaimer: I'm basically just summarizing what I found on the English Wikipedia page for "Euclidean" through the lens of my experience, biases, and understanding of the math. It seems there are three main categories of usage of the word "Euclidean" meaning "related to the ancient Greek mathematician Euclid of Alexandria". ...


20

All quadratic curves can be studied using the equation $Ax^2+2Bxy+Cy^2 + Dx + Ey + F=0$ the discriminant of which is $B^2-AC$ and the solution curve will be a ellipse, hyperbola, or parabola depending on whether the discriminant is positive, negative, or zero. Partial second-order differential equations take a much similar form with $Au_{xx}+2Bu_{xy}+Cu_{yy}...


19

I interpret that to mean the constant is less than one, and also the integral of the function from zero to one is zero. There should be a comma there, that would help.


18

It makes for easier formulation of some theorems or definitions: a space can be called locally compact if it has a (base of) compact neighbourhoods, or locally connected if it has a local base of connected neighbourhoods (regardless of openness). The formulation of local continuity is also easy: $f$ is continuous at $x$ if $f^{-1}[N]$ is a neighbourhood of ...


18

I'd like to address this comment of the OP to one of the answers: But I think the solutions of the corresponding differential equations have nothing to do with their name? Or do they have?. Actually, we can relate the names to the corresponding geometrical curves. To do this, consider how the following homogeneous PDE will look in the Fourier-transformed ...


17

They share lots in common! The product of two even functions is even, the product of an even function and an odd function is odd, and the product of an odd and an even function is odd. This is exactly how even and odd numbers add, and is a nice motivating example of an isomorphism. The Taylor series (at $x = 0$) of an even function has only even terms $x^{...


15

The first definitions you gave are correct and standard, and statisticians and data scientists will agree with this. (These definitions are given in statistics textbooks.) The second set of quantities you described are called the "sample mean" and the "sample variance", not mean and variance. Given a random sample from a random variable $...


14

It is true that they are different. For instance, when learning to solve quadratic equations, you encounter $$ ax^2+bx+c=0 $$ In this case, the letters $a,b,c$ ("blank boxes" for you to fill in whatever quadratic equation you happen to come across) logically mean something distinctly different from the letter $x$ (the unknown you want to solve for)....


14

Whatever you prove after "Let $x$ be [blah]" is a theorem under the assumption that $x$ is a blah. Notice that it's not a theorem under the assumption that there exists a blah, but that $x$ is such a blah. In particular it is a result with a free variable, but it doesn't say anything about whether such a blah exists. "Let $x$ be a real number ...


14

This is just for clarity. One of the common stumbling blocks in interpreting a piece of mathematics is the issue of type: keeping track e.g. of what's a set of points versus what's a set of sets of points, and so on. It seems to be the case that using varied terminology often makes things a bit easier to read. As to formal content, there is none: there are ...


11

This is just another way of saying that for all but finitely many primes $p$, we have $\alpha(p) = 0$. In particular, the "largeness of the prime" depends upon the $a$ you are given to start with. Specifically how this relates to your case. You are given $a \geq 1$, then, for the most unrefined bound, $p > a$ can never be factors of $a$.


10

Here is one example when it is convenient not to require neighborhoods to be open. The following is either a lemma or a definition: A map $f: X\to Y$ of two topological spaces is continuous at a point $x\in X$ if and only if for every neighborhood $V$ of $f(x)$, $f^{-1}(V)$ is a neighborhood of $x$. Note similarity with the definition of a continuous ...


10

either I have discovered an error in the foundations of general topology that somehow eluded mathematicians for ~100 years, or the McTextbook explanation of second-countable spaces uses the words "is," "same," "the," and "base" incorrectly. Or, your argument is flawed. Your claim beginning "Therefore" is incorrect: in general, just because $\mathbb{A}\not=\...


10

Manifold-with-boundary is not (unless the boundary is empty) a manifold, a persistent source of confusion. Also: "delta function." Sigh. Others please feel free to add your contributions.


9

Basically, Euclidean means "globally flat". In the 1800s, non-Euclidean geometries were discovered. In these geometries, there is implicit curvature in the space (so they're non-flat). Of course, by this time $R^n$ had been invented, and it was realized that these curvature calculations ended up with zero curvature on $R^n$ (so they're flat). It ...


9

The "fix" means "consider some particular $J$" or "for each $J$ define $S$ to be ..." . Your description of $S$ in terms of $J$ is correct. No need to overthink this.


9

Six pages earlier, at the beginning of 9.6.1, the first subsection on Islamic trigonometry, a parenthetical remark notes: The Islamic sine of an arc, like that of the Hindus, was the length of a particular line in a circle of given radius $R$. We will keep to our notation of "Sine" to designate the Islamic sine function Additionally, in 8.7.1 on ...


8

My guess is that it's because they can be defined by a pretty natural operations from inside the group. The most basic operation we have is "multiply" (although "invert" is arguably 'more basic', but let me go on...). So we can look at a map like $$ f_a(x) = a \cdot x $$ which takes $G$ to $G$ (where $a$ is some fixed element of $G$). This isn't an ...


8

(Transferring from a comment.) You'll find in mathematics that terminology can vary from author to author. (We can't even agree on whether the natural numbers include zero!) Context is key. If someone shows a semicircular region and asks its perimeter, then the diameter would certainly need to be included. On the other hand, if someone is discussing a ...


8

It's possible to make sense of this question as follows. Let $G$ be a group together with a distinguished central element of order $2$ which we call $-1$; we'll write the product $(-1)a$ as $-a$. Say that two elements $a, b \in G$ anticommute if $ab = -ba$. Then we have, more or less by definition: Claim: Every pair of elements in $G$ either commutes or ...


8

Actually, it is quite related to commutativity. $\require{AMScd}$ \begin{CD} A@>{f}>> A\\ @V{g}VV @VV{g}V\\ A @>{f}>> A \end{CD} Let $f,g$ be endomorphisms of some algebraic structure $A$. Then $f\circ g=g\circ f$ if and only if the diagram above commutes. So, in some cases commutativity of these operators is equivalent to commutativity of ...


8

Because the notation $\frac1a$ suggests that this is the number that you get when you divide $1$ by $a$. And, yes, it is! But one should use that notation only after it was proved that $a^{-1}{}{}{}{}{}{}$ is indeed what you get when you divide $1$ by $a$.


8

The term cofinal is a compound of the prefix co- and the adjective final. Here the meaning of the prefix is ‘together; mutually; jointly’: if $B$ is cofinal in $A$, the two sets reach the ‘end’ together, so to speak. The prefix is from Latin co-, a variant of con-, and indeed one occasionally sees confinal in place of cofinal. The Latin prefix con- in turn ...


7

No. Only if $P$, then $Q$ means that $Q$ cannot happen without $P$: if $Q$ happens, $P$ must have happened. It does not say that if $P$ happens, then $Q$ must also happen. It’s exactly equivalent to $Q$ implies $P$. Note that this is a true statement if $P$ occurs and $Q$ does not. If and only if $P$, then $Q$ says that if $Q$ occurs, then $P$ must have ...


7

As John Douma pointed out in a comment, a clue can be found in note c of the English Wikipedia page for the film: Much, and even most ( if not all, ) of the mathematical imagery consists of graphical matter to be found in "Jahnke and Emde." That is the Dover Edition of Tables of Functions by Eugene Jahnke and Fritz Emde. A version of that famous ...


7

It abbreviates the word respectively. In this context it means that two results are being stated simultaneously: if $A,B$ are two $f$-sets, and $A\cap B\ne\varnothing$, then $A\cup B$ is an $f$-set; and if $A,B$ are two $i$-sets, and $A\cap B\ne\varnothing$, then $A\cup B$ is an $i$-set. In other words, read it first without the parenthetical items, and ...


7

A Hilbert-basis is not a basis.


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