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August 25

Ideas for representing higher dimensional spaces

I want to "plot" y = f(x) in a form printable on paper, preferably in black and white, where y is a scalar, and x is a 9D vector. I know this might be hard. What's the best we can do? y vs. a 2D vector? deeptrivia (talk) 02:34, 25 August 2007 (UTC)[reply]

You could do a 3D vector with a movie of a 3D graph, where time is used to represent one of the vector dimensions of X. You could do higher dimensions by having multiple 3D graph movies, but then you would only have a few sample points for each of the additional dimensions or the number of movies would quickly get out of hand. And, if it really needs to be on paper, then the movie would have to be shown as separate frames, too. StuRat 04:06, 25 August 2007 (UTC)[reply]

It may help to tell us why you want to do that, since any stuffing of ten dimensions into two will lose information, and it'd help to know what information can most safely be left out. Here are a few tricks I've heard of:

Graph pairs (or, if you want to get fancy, triples) of variables against each other in several separate graphs. I can think of few situations where it would actually be useful to have every measurement on the same graph.
Depending on the situation you're in, it may be possible to "project" (that is, pull each point in the space onto a single plane) the data in a convenient way. Actually, in a way that's what the first suggestion is, except there are ways of projecting it that might allow you to show interactions of several variables at once in a very fuzzy way.
You can use other things besides position to indicate what variable is being measured. Like StuRat said, you can use time, or you could use other things. Color the points, for instance, in a gradient (for a continuous variable) or in several separate colors (for a discrete variable). Or, if you think you can make it work visually, you could shove multiple variables into the coloring by doing overlapping gradients. One for each primary color, for instance, which would allow them to mix without attacking each other. You can also vary the size and shape of the dots.
If you're able to use transparent pages, you can put layers into the graph. Black Carrot 17:15, 25 August 2007 (UTC)[reply]

Correction: I just noticed the "black and white" restriction. You can still vary the shading, though, or in some cases make the dots big enough that they can be colored in with patterns, like stripes or wavy lines. Black Carrot 20:16, 25 August 2007 (UTC)[reply]

I just happen to have a function that relates 9 independent variables to one dependent variable. I guess I'll just lower my expectation a bit and plot y with two x's at a time. Thanks, deeptrivia (talk) 03:50, 26 August 2007 (UTC)[reply]

The state of the art has progressed, but you might be interested in a decade-old project by Beshers and Feiner at Columbia University. The work of Lloyd Treinish may also be helpful. Cleveland, the author of an excellent but not-so-well-known book on graphing data, has worked on the Trellis Display project, which is worth a look. For serious commercial work, try Tableau Software, a startup backed by some excellent talent (such as Pat Hanrahan). --KSmrq^T 07:13, 26 August 2007 (UTC)[reply]

If not all of the 9 dimensions hold equivalent interest for you, you may want to look at something along the lines of principle component analysis. -- 20:05, 30 August 2007 (UTC)

Critical point theory

In nonlinear functional analysis, what is Critical Point Theory? deeptrivia (talk) 02:55, 25 August 2007 (UTC)[reply]

Critical point theory looks at critical points of functionals (points where the Jacobian vanishes or is singular) on manifolds: that is, we want to classify these points as minima, maxima or saddle points, and generalize techniques from "advanced calculus" like Lagrange multipliers, but also "count" the number of critical points (give lower or upper bounds on the number or the measure of set of critical points). An example of a theorem that is found in many texts that cover "critical point theory" is Sard's lemma. Morse theory is a closely related area. Phils 03:49, 25 August 2007 (UTC)[reply]

Number systems

Today we know a lot about numbers the decimal Binary and more but what I would like to know is what system did the ancient Greeks use? many equations they discovered are still in use. (The so called old British system was far better than the decimal system for working things out in your head. It was used all over Europe it came from the Arabs in 1200 AD) —Preceding unsigned comment added by Zineh (talk • contribs) 04:19, 2007 August 25

I think Greek numerals is the page your looking for. Numeral system may also be worth looking at. --Salix alba (talk) 10:53, 25 August 2007 (UTC)[reply]

I never heard of the "old British" system before; can you tell me more? —Tamfang 03:19, 2 September 2007 (UTC)[reply]

palindromic generation

I would like to know some answers in mathematics:

how to generate palindromic numbers? how to generate palindromic numbers from a given number?Is there any genaeralised method for generating palindromic numbers? are there palindromes in decimals and fractions?
Are mathematical applications used in kitchen? In what ways?

—Preceding unsigned comment added by 82.148.96.68 (talk • contribs) 12:44, 25 August 2007

Please sign your post by typing ~~~~ after it. asyndeton 12:45, 25 August 2007 (UTC)[reply]

To answer the first question, see Palindromic number#Conversion. To answer the second, yes, there is some math involved in the kitchen, such as measuring ingredients and finding the right proportion between them, but I'm not sure whether there's any more complex math used in the kitchen, if that's what you mean. --CrazyLegsKC 17:00, 25 August 2007 (UTC)[reply]

I suppose geometry might be used in the kitchen, say to calculate how much more crust is needed in a 10 inch diameter pie than an 8 inch diameter pie. Cooking times are not straightforward, either, I wonder if there is a formula for them. For example, if something takes an hour to cook at 300, how long will it take at 400 degrees ? (It's not just a straight ratio.) StuRat 18:17, 25 August 2007 (UTC)[reply]

I think the better question is - if something is supposed to be cooked at 300°, will it still be edible if cooked at 400°? -- Meni Rosenfeld (talk) 18:50, 25 August 2007 (UTC)[reply]

I routinely solve partial differential equations whenever I bake a cake. The Heat equation is very useful in this. nadav (talk) 21:32, 25 August 2007 (UTC)[reply]

Modern kitchen work would be absolutely impossible (we'd have to use more primitive standbys) if it wasn't for all the wonderful technology backing it up. Every single kitchen appliance (not to mention the kitchen itself) required math, and a lot of it, to produce. Black Carrot 23:09, 25 August 2007 (UTC)[reply]

You also might have to convert Fahrenheit to Celsius, or vice versa. EamonnPKeane 17:51, 27 August 2007 (UTC)[reply]

Tangency

The tangent line seems to be the standard way of drawing the first derivative of a curve, and the osculating circle seems to be a popular way of drawing the second. Is there any similar visual way of representing direction and curvature for the third derivative or beyond? Black Carrot 16:52, 25 August 2007 (UTC)[reply]

One method I come to think of is using the

n+1

first terms of the Taylor series expansion to illustrate the derivatives up to order

n

. That way, you get the polynomial that has the same derivatives (from zeroth to

n

:th) as your function at the point you are interested in. —Bromskloss 17:27, 25 August 2007 (UTC)[reply]

Another method is to create normal lines, where the length of each represents the magnitude of whatever you want to measure, such as the third derivative, in this case. This method works well for surfaces, too. StuRat 18:00, 25 August 2007 (UTC)[reply]

Marking a quiz

If a quiz involves placing some items in the correct sequence, is the following a 'fair' scoring scheme? (Yes I know that 'fair' is subjective...)

Recieve 1 point for the first item.
For each 'next item' receive 1 point if, in the correct list, it comes after the previous item.

Thus for vowels: AEIOU = 1+1+1+1+1 = 5, whilst AIOEU = 1+1+1+0+1 = 4, and UOIEA = 1+0+0+0+0 = 1.

-- SGBailey 22:16, 25 August 2007 (UTC)[reply]

Maybe, it depends what you're trying to order. For instance, it might be better to base it on the distance of each element from where it should be in the sequence, and add up the absolute errors. As it is, ABCDEF and DEFABC would differ by 1 in score, while BACDFE would lose two points despite being (in a sense) more correct. Or is local order more important than overall order? Another test would be adding up, for each element, how many elements in all are on the wrong side of it. Black Carrot 22:34, 25 August 2007 (UTC)[reply]

Update: Oddly enough, it appears that my two suggestions are equivalent. Whaddayaknow. Black Carrot 07:53, 26 August 2007 (UTC)[reply]

See edit distance.

Class Membership

Do you use the same symbol (the little e thing) to represent class membership as to represent set membership? That is, if I wanted to say a function f(x) is smooth, would I write f(x)eC^oo? Black Carrot 23:06, 25 August 2007 (UTC)[reply]

Well I don't know about any 'little e', but

$x\in A$ might say that x is a member of the set A...

$y\in x$ might say that y belongs to the class of all x's...

The same $\in$ is used for both. Source: Basic Set Theory, Azriel Levy

Micah J. Manary 00:56, 26 August 2007 (UTC)[reply]

That's the symbol I'm talking about, I couldn't find it. So, would

f(x)\in C^{\infty }

be the proper way to write "f is smooth"? I'm not certain f is a C, I thought it was in C. Black Carrot 01:02, 26 August 2007 (UTC)[reply]

That would be theoretically correct. It you really trying to get your idea across,

C^{\infty }

actually says that not only is the function smooth, but its derivative is smooth, and its second derivative is smooth, forever. Just a smooth function is

f(x)\in C^{1}

i.e., the derivative is continuous. Micah J. Manary 02:43, 26 August 2007 (UTC)[reply]

So to answer your original question,

$f(x)\in C^{1}$ implies f is 'smooth' in the loose sense, that it has a continous and existent derivative. $f(x)\in C^{\infty }$ implies f is perfectly smooth, that all derivatices are smooth as well. See smooth function. Micah J. Manary 02:43, 26 August 2007 (UTC)[reply]

In my experience, most texts wouldn't use the (x), since that would refer to the image of some x instead of the function. It would just be $f\in C^{\infty }$ . nadav (talk) 02:48, 26 August 2007 (UTC)[reply]

I'll agree to that, definitely. The function is f, so it would be a member of the class. However, I don't think its wrong to say "a function, f, which maps x, is a member of

\ C^{\infty }

Micah J. Manary 02:53, 26 August 2007 (UTC)[reply]

Also, the class of smooth functions is not a proper class, but just a set. There is no reason to use different notation for it than for other sets. nadav (talk) 03:09, 26 August 2007 (UTC)[reply]

That helps. Thanks. Black Carrot 05:16, 26 August 2007 (UTC)[reply]

In speech, I've always heard "f is C¹", rather than "f is in C¹". As far as I know, the "class" in "f is of class C¹" is the English word, rather than the mathematical term.

It's definitely a set (or class, if you prefer) just like any other set, and it can even be endowed with algebraic and topological structure. nadav (talk) 23:55, 26 August 2007 (UTC)[reply]