Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

Not all elementary functions can be expressed with exp-minus-log

stylewarning.com

Post reply on HN
Print view View original

81–90 of 125 posts

Re: Not all elementary functions can be expressed with exp-minus-log

#81

Post by xigoi »

reikonomusha wrote: post #77
thaumasiotes wrote: post #65

Earlier quoted context omitted.

> This fundamental "cheat" gave rise to some of the most important pure and applied mathematics known. > Can't solve the differential equation y'' = -y? Why not just introduce a function sin(x) as its solution! Problem solved. But that's not how sine was introduced. It's been around since classical geometry. It was always easy to solve the differential equation y'' = -y, because the sine had that property, and we kne…

You can calculate, measure, draw, construct, write a power series for, express as hypergeometric function, etc. the Bring radical too. All of these concepts, from sine to real numbers, Bring radicals to complex exponentials, can all be defined in different, equivalent ways. What is interesting are the properties invariant to these definitions. It still doesn't seem to me that a square root should be any more or less…

For me, what makes the square root more “natural” is that, although it’s usually introduced as an “answer by definition”, it can also be arrived at by wondering what happens if you take something to the halfth power.

Re: Not all elementary functions can be expressed with exp-minus-log

#82

Post by DoctorOetker »

xigoi wrote: post #78
DoctorOetker wrote: post #74

Earlier quoted context omitted.

1) > Related is the paper [What is a closed-form number?], which explores the field E, defined as the smallest subfield of ℂ closed under exp and log. I believe the set of numbers that can be generated using exp-minus-log is a strict subset of this. is that a typo / accidental mis-phrasing? exp-minus-log construction is closed for the operations it supports, and spans both exp and log, so E must be either identical t…

> exp-minus-log construction is closed for the operations it supports, and spans both exp and log, so E must be either identical to or a subset of exp-minus-log; not the other way around. Since E is by definition closed under exp, log and subtraction, it is clearly also closed under EML.

SabrinaJewson claims it is a STRICT subset: EML ⊂ E

I remind the trivial results that both E ⊆ EML and EML ⊆ E and hence EML = E

apart from construction: which is minimal for EML but highly redundant for E.

the EML paper shows that this minimal construction for EML is not unique so other binary operations may be found with perhaps more interesting properties, or admitting shorter binary trees for commonly used functions and values (which may reflect subjective "simplification" of expressions in mathematics.

Re: Not all elementary functions can be expressed with exp-minus-log

#83

Post by arowthway »

cubefox wrote: post #53

Where would EML expressions sit in this fascinating table? https://en.wikipedia.org/wiki/Template:Mathematical_expressi...

Closed-form expressions, I guess?

"For example, if one adds polynomial roots to the basic functions, the functions that have a closed form are called elementary functions."

Re: Not all elementary functions can be expressed with exp-minus-log

#84

Post by cubefox »

arowthway wrote: post #83
cubefox wrote: post #53

Where would EML expressions sit in this fascinating table? https://en.wikipedia.org/wiki/Template:Mathematical_expressi...

Closed-form expressions, I guess? "For example, if one adds polynomial roots to the basic functions, the functions that have a closed form are called elementary functions."

Maybe I'm misunderstanding, but the table says that the "root of a polynomial that is not an algebraic solution" doesn't count as closed-form.

Re: Not all elementary functions can be expressed with exp-minus-log

#85

Post by DoctorOetker »

reikonomusha wrote: post #38
vintermann wrote: post #26

Earlier quoted context omitted.

Yes, this article is kicking in open doors, the original article was quite clear about the scope. The present article could rather have spent time arguing why this isn't like NAND gate functional completeness. I would have thought the differences lie in the other direction: not that trees of EML and 1 can describe too little, but that they can describe too much already. It's decidable whether two NAND circuits implem…

You are correct, it is undecidable by Richardson's theorem [1]. [1] https://en.wikipedia.org/wiki/Richardson%27s_theorem

that result does not apply for EML: EML doesn't have the | . | absolute value function, a prerequisite for Richardson's theorem.

Re: Not all elementary functions can be expressed with exp-minus-log

#86

Post by DoctorOetker »

cubefox wrote: post #54
saithound wrote: post #5

The original article explicitly acknowledged this limitation, that while in "the classical differential-algebraic setting, one often works with a broader notion of elementary function, defined relative to a chosen field of constants and allowing algebraic adjunctions, i.e., adjoining roots of polynomial equations," the author works with the less general definition. Neither the present article, nor the original one ha…

> Odrzywolek's result is immediately obvious Many things that in retrospect seem immediately obvious weren't obvious before, let alone immediately obvious.

and its depressing when the rare actual progress is made, a collection of jealous practitioners comes to party-poop all over the place, for bringing the insights that make the result from then on immediately obvious.

Re: Not all elementary functions can be expressed with exp-minus-log

#88

Post by SideQuark »

rnhmjoj wrote: post #24

> My concern is that the word “elementary” in the title carries a much broader meaning in standard mathematical usage, and in this meaning, the paper’s title does not hold. > Elementary functions typically include arbitrary polynomial roots, and EML terms cannot express them. If you take a real analysis class, the elementary functions will be defined exactly as the author of the EML paper does. I've actually just lea…

> If you take a real analysis class, the elementary functions will be defined exactly as the author of the EML paper does.

I just looked through many of the best known real analysis texts, and not a single one defines them this way. This list included the texts by

Royden, Terence Tao, Rudin, Spivak, Bartle & Sherbert, Pugh, and a few others....

Can you cite a single text book that has this definition you claim is in every real analysis course? I find all evidence points to the opposite.

Re: Not all elementary functions can be expressed with exp-minus-log

#89

Post by derriz »

cyberax wrote: post #41
derriz wrote: post #29

When I first read the exp-minus-log paper, I found it extremely surprising - even shocking that such a function could exist. But the fact that a single function can represent a large number of other functions isn't that surprising at all. It's probably obvious to anyone (it wasn't initially to me), but given enough arguments I can represent any arbitrary set of n+1 functions (they don't even have to be functions on t…

Why would it be surprising? And if you want something truly surprising, Riemann's zeta function can approximate any holomorphic function arbitrarily well on the critical strip. So technically you need only _one_ argument.

> Why would it be surprising?

Because I don’t know as much mathematics as you.

Re: Not all elementary functions can be expressed with exp-minus-log

#90

Post by arowthway »

cubefox wrote: post #84
arowthway wrote: post #83

Earlier quoted context omitted.

Closed-form expressions, I guess? "For example, if one adds polynomial roots to the basic functions, the functions that have a closed form are called elementary functions."

Maybe I'm misunderstanding, but the table says that the "root of a polynomial that is not an algebraic solution" doesn't count as closed-form.

https://en.wikipedia.org/wiki/Closed-form_expression

"Commonly, the basic functions that are allowed in closed forms are nth root, exponential function, logarithm, and trigonometric functions.[a] However, the set of basic functions depends on the context. For example, if one adds polynomial roots to the basic functions, the functions that have a closed form are called elementary functions."

It's only maths, don't expect things to be so black and white.

Post reply on HN