While reading a paper (citation omitted to protect the "guilty"), I came across a reference to an "n-dimensional tetrahedron", meaning the subset of Rn given by
x1, ..., xn ≥ 0 and x1 w1 + ... xn wn ≤ τ
for positive constants w1,..., wn and τ.
Of course this is an n-simplex. But calling it a "tetrahedron" is etymologically incorrect -- that means "four faces", while an n-simplex has n+1 faces. This probably occurs because most of us tend to visualize in three dimensions, not in arbitrary high-dimensional spaces.
I'm not saying that "tetrahedron" shouldn't be used here -- I'm just pointing out an interesting linguistic phenomenon.
Showing posts with label nomenclature. Show all posts
Showing posts with label nomenclature. Show all posts
30 December 2008
06 July 2008
Nomenclature clash
Prime Numbers for June 29 to July 5, from today's New York Times. (I don't know if this is a weekly thing; it could be but I don't recall seeing it before.)
The numbers are 46, 62000, 30, 18%, and 30000; each is important to some news story from this week. (If you want to get technical, 62000 and 30000 are approximations.)
Presumably they mean "prime" in the sense of "important". Or in the sense of "composite", but that would be a bit perverse.
The numbers are 46, 62000, 30, 18%, and 30000; each is important to some news story from this week. (If you want to get technical, 62000 and 30000 are approximations.)
Presumably they mean "prime" in the sense of "important". Or in the sense of "composite", but that would be a bit perverse.
19 November 2007
The complexity zoo
From bit-player (Brian Hayes): Until NEXPTIME, on the proliferation of complexity classes.
The trivial chemical nomenclature does, though. ("Trivial" in chemistry doesn't have the mathematician's meaning; in chemistry it means the way in which names are assigned in a one-to-one, ad hoc manner to chemical structures.) Perhaps the Right Thing to do is not to try to regularize the current nomenclature for complexity classes, but to come up with a new systematic naming system that could overlay the current one. But that might require knowing more about complexity theory than we currently do.
There's an inclusion diagram for complexity classes at the Complexity Zoo (which also exists in a wikified version here which appears to be more current.)
Have you ever tried to explain to your grandmother why NP is named NP? Does she get it when you say that problems labeled NP-complete are the hardest problems in NP, but NP-hard problems might be harder, and not in NP?Yes, complexity classes are confusing. As Hayes points out, "There are hints of structure in the naming scheme". I'm not sure if that makes it better or worse. My question is: do there exist complexity classes A, B, C, and D such that the pairs of names (A, B) and (C, D) are related in the same way but the actual classes are related in different ways? In other words, does this nomenclatures have the potential for false generalizations? Hayes points out that the chemists have come up with a good systematic nomenclature for chemical compounds, of which there are many more than there are complexity classes. It's even useful if you're not a chemist; I've caught myself on occasion using the chemist's nomenclature for alkanes to describe trees. (I studied chemistry and math in college.) In particular, if I remember correctly the systematic chemical nomenclature doesn't allow false generalizations.
The trivial chemical nomenclature does, though. ("Trivial" in chemistry doesn't have the mathematician's meaning; in chemistry it means the way in which names are assigned in a one-to-one, ad hoc manner to chemical structures.) Perhaps the Right Thing to do is not to try to regularize the current nomenclature for complexity classes, but to come up with a new systematic naming system that could overlay the current one. But that might require knowing more about complexity theory than we currently do.
There's an inclusion diagram for complexity classes at the Complexity Zoo (which also exists in a wikified version here which appears to be more current.)
06 October 2007
Dual numbers
You know that trick where you invent some number ε such that ε2 = 0 and use it to, basically, take derivatives?
For example, (x+ε)2 = x2 + 2xε, so if we change x by some small amount ε then we change x2 by 2x times that amount. Thus the derivative of x2 must be 2x.
It turns out that trick has a name; it's calculation in the algebra of dual numbers, which I discovered by randomly poking around Wikipedia, and it's apparently used in at least some computer algebra systems to do differentiation. I didn't know that.
Edit (Monday, 4:36 pm): Charles of Rigorous Trivialities has pointed out that dual numbers are used extensively in deformation theory.
For example, (x+ε)2 = x2 + 2xε, so if we change x by some small amount ε then we change x2 by 2x times that amount. Thus the derivative of x2 must be 2x.
It turns out that trick has a name; it's calculation in the algebra of dual numbers, which I discovered by randomly poking around Wikipedia, and it's apparently used in at least some computer algebra systems to do differentiation. I didn't know that.
Edit (Monday, 4:36 pm): Charles of Rigorous Trivialities has pointed out that dual numbers are used extensively in deformation theory.
28 September 2007
Nested radicals, smoothness, and simplification
I saw an expression involving a nested radical, namely
.
(Write φ = (1 + φ)1/2 and solve for φ.) The Wikipedia article on nested radicals led me to Simplifying Square Roots of Square Roots by Denesting. The authors tell us that:
This reminds me of a couple things that happened in my class yesterday. First, I was defining what it means for a curve to be smooth; our definition was that the curve given by the vector function r(t) is smooth if r'(t) is continuous and never zero, except perhaps at the endpoints of the interval over which it's defined. (This makes smoothness a property of a parametrization, which is a bit counterintuitive. I suppose that one could define a curve -- as an abstract set of points -- to be smooth if it has a smooth parametrization. Although I haven't worked it out, I assume that if a curve has a smooth parametrization, the arc-length parametrization is smooth.) One of the students said "but the professor said 'smooth' means something else!" I'm not sure if the professor actually said "smooth means X" or if he said "some people think smooth means X", but it's a good point. (In particular, "smooth" often seems to mean that a function has infinitely many continuous derivatives.)
Second, the article is about using computer algebra systems to simplify expressions like
where the left-hand side is "simpler"; sometimes my students worry that they are not presenting their answer in the simplest form. While I'll accept any reasonably simple answer (unless the problem statement specifies a particular form), it is remarkably difficult to define what "simple" means.
One rule I have figured out, though, is that 4x - 4z - 8 = 0 should be simplified to x - z - 2 = 0 by dividing out the common factor. In general, given a polynomial with rational coefficients, one probably wants to multiply to clear out the denominators and then divide by any common integer factor of the new coefficients, so the resulting coefficients are relatively prime integers. The article addresses this sort of "canonicalization" in the context of nested radicals. I keep telling my students that they should keep that sort of thing in mind, especially since our tests will be mostly multiple-choice.
(Sometimes I'm tempted to define "simplest" as "requires the fewest symbols"... but how does one prove that some 100-character expression one has written can't be written in 99 characters? And how do you count something like "f(x, y)= (x+y)1/2 - (x-y)1/2, where x = foo and y = bar?" ("foo" and "bar" are supposed to be very complicated expressions.) Do you plug foo and bar into the original equation and then count the characters, or do you count the actual characters that are between the quotation marks?)
(Write φ = (1 + φ)1/2 and solve for φ.) The Wikipedia article on nested radicals led me to Simplifying Square Roots of Square Roots by Denesting. The authors tell us that:
The term surd is used by TeX as the name for the symbol √ Maple has a function called surd that is similar to the nth root defined here; like all good mathematical terms, the precise definition depends upon the context. In general, a mathematical term that does not have several conflicting definitions is not important enough to be worth learning.
This reminds me of a couple things that happened in my class yesterday. First, I was defining what it means for a curve to be smooth; our definition was that the curve given by the vector function r(t) is smooth if r'(t) is continuous and never zero, except perhaps at the endpoints of the interval over which it's defined. (This makes smoothness a property of a parametrization, which is a bit counterintuitive. I suppose that one could define a curve -- as an abstract set of points -- to be smooth if it has a smooth parametrization. Although I haven't worked it out, I assume that if a curve has a smooth parametrization, the arc-length parametrization is smooth.) One of the students said "but the professor said 'smooth' means something else!" I'm not sure if the professor actually said "smooth means X" or if he said "some people think smooth means X", but it's a good point. (In particular, "smooth" often seems to mean that a function has infinitely many continuous derivatives.)
Second, the article is about using computer algebra systems to simplify expressions like
where the left-hand side is "simpler"; sometimes my students worry that they are not presenting their answer in the simplest form. While I'll accept any reasonably simple answer (unless the problem statement specifies a particular form), it is remarkably difficult to define what "simple" means.
One rule I have figured out, though, is that 4x - 4z - 8 = 0 should be simplified to x - z - 2 = 0 by dividing out the common factor. In general, given a polynomial with rational coefficients, one probably wants to multiply to clear out the denominators and then divide by any common integer factor of the new coefficients, so the resulting coefficients are relatively prime integers. The article addresses this sort of "canonicalization" in the context of nested radicals. I keep telling my students that they should keep that sort of thing in mind, especially since our tests will be mostly multiple-choice.
(Sometimes I'm tempted to define "simplest" as "requires the fewest symbols"... but how does one prove that some 100-character expression one has written can't be written in 99 characters? And how do you count something like "f(x, y)= (x+y)1/2 - (x-y)1/2, where x = foo and y = bar?" ("foo" and "bar" are supposed to be very complicated expressions.) Do you plug foo and bar into the original equation and then count the characters, or do you count the actual characters that are between the quotation marks?)
18 September 2007
A convention for quadrant/octant/orthant numbering
In two-dimensional Cartesian geometry, it's conventional to refer to the area where x and y are both positive as the "first quadrant", to that where x is negative and y is positive as the "second quadrant", to that where y is negative and x is negative as the "third quadrant", and to that where y is negative and x is positive as the "fourth quadrant". In tabular form, we have
There seems to be no conventional way to extend this to the eight octants in three-dimensional Cartesian geometry. The only thing that people seem to do consistently is to call the octant where x, y, and z are all positive the "first octant", or sometimes the "positive octant". And I've noticed that a few of my students want to call the octant where x, y, and z are all negative the "eighth octant"; I actually want to do the same. It's far from the first octant; it should have a number that is far from 1. But we don't call the quadrant where x and y are negative the "fourth quadrant".
Actually, the "next step" in this recursion would lead to a scheme in which the (-, -, -) octant is called the fifth octant, and in general the octant obtained by reflecting the nth octant over the xy-plane "should be" (9-n)th octant. This preserves the property that the kth octant and the (k+1)st octant are adjacent; there's no a priori reason why this is a good condition to have, but it's the same recurstion that says that the 3rd and 4th quadrants are obtained by reflecting the 2nd and 1st quadrants, respectively, over the x-axis.
More generally, in d+1 dimensions, the point (ε1, ε2, ..., εd, 1), where each ε is ±1, is in the (d+1)-dimensional orthant with the same number as the d-dimensional orthant containing (ε1, ε2, ..., εd); call this number k. The point (ε1, ε2, ..., εd, -1) is in the orthant with number (2d+1+1-k).
This means that if we traverse the octants (or, more generally, the "orthants" in d-dimensional space) in numerical order, we get a Hamiltonian tour of the vertices of the d-dimensional cube, which is kind of cute.
This page from Math Forum seems to advocate calling the all-negative octant the "seventh octant", which comes from a hybrid of my convention and the convention that octants which border each other over the plane z=0 have numbers differeing by 4. I suppose it's okay in three dimensions, but my sense of aesthetics demands that we have a definition which recurses to higher dimensions.
Still, there's really no reason to be able to number the orthants other than the all-positive one (often called the "first orthant"), and anyone who needs to refer to a particular orthant really ought to just specify it in the form x1 * 0, x2 * 0, ... where each * is < or > in order to avoid confusion.
| x < 0 | x > 0 | |
| y > 0 | 2nd | 1st |
| y < 0 | 3rd | 4th |
There seems to be no conventional way to extend this to the eight octants in three-dimensional Cartesian geometry. The only thing that people seem to do consistently is to call the octant where x, y, and z are all positive the "first octant", or sometimes the "positive octant". And I've noticed that a few of my students want to call the octant where x, y, and z are all negative the "eighth octant"; I actually want to do the same. It's far from the first octant; it should have a number that is far from 1. But we don't call the quadrant where x and y are negative the "fourth quadrant".
Actually, the "next step" in this recursion would lead to a scheme in which the (-, -, -) octant is called the fifth octant, and in general the octant obtained by reflecting the nth octant over the xy-plane "should be" (9-n)th octant. This preserves the property that the kth octant and the (k+1)st octant are adjacent; there's no a priori reason why this is a good condition to have, but it's the same recurstion that says that the 3rd and 4th quadrants are obtained by reflecting the 2nd and 1st quadrants, respectively, over the x-axis.
More generally, in d+1 dimensions, the point (ε1, ε2, ..., εd, 1), where each ε is ±1, is in the (d+1)-dimensional orthant with the same number as the d-dimensional orthant containing (ε1, ε2, ..., εd); call this number k. The point (ε1, ε2, ..., εd, -1) is in the orthant with number (2d+1+1-k).
This means that if we traverse the octants (or, more generally, the "orthants" in d-dimensional space) in numerical order, we get a Hamiltonian tour of the vertices of the d-dimensional cube, which is kind of cute.
This page from Math Forum seems to advocate calling the all-negative octant the "seventh octant", which comes from a hybrid of my convention and the convention that octants which border each other over the plane z=0 have numbers differeing by 4. I suppose it's okay in three dimensions, but my sense of aesthetics demands that we have a definition which recurses to higher dimensions.
Still, there's really no reason to be able to number the orthants other than the all-positive one (often called the "first orthant"), and anyone who needs to refer to a particular orthant really ought to just specify it in the form x1 * 0, x2 * 0, ... where each * is < or > in order to avoid confusion.
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