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January 17

Exam on Wednesday! Help me!

Well, math hates me so I don't get any of this! I need to know how to put these into slope intercept form.

The line passes through (8,5) and has a slope of 5/4.

The line passes through the points (15,16) and (13,10).

Please, please, please help me! —Preceding unsigned comment added by Amley2 (talk • contribs) 16:39, 17 January 2009 (UTC)[reply]

Have you tried studying? —Preceding unsigned comment added by 84.220.118.103 (talk) 16:58, 17 January 2009 (UTC)[reply]

Start with the equation y=mx+c, then substitute in all the information you have and solve for m and c. (For the second case, do the substitution separately for each point so you end up with simultaneous equations to solve, for the first case you'll just need one equation). If that doesn't help, go and see your teacher - teaching you this stuff is what they get paid for. --Tango (talk) 17:18, 17 January 2009 (UTC)[reply]

Take a stab at those problems here, and show us your work, and we'll point out any mistakes. StuRat (talk) 17:22, 17 January 2009 (UTC)[reply]

This problem appears incorrect. Maybe I don't understand what the question is (it is not stated clearly) but if the line passes through (15, 16) and (13, 10), it must have slope 6/2 = 3 (the difference of y-coordinates/difference of x-coordinates) so it can't be 5/4. Do you mean the the first line and the second line are different problems? --PS T 23:16, 17 January 2009 (UTC)[reply]

Yes, I'm pretty sure they are different problems. The 3 points clearly aren't collinear. --Tango (talk) 23:19, 17 January 2009 (UTC)[reply]

Answer to first problem: Equation is y = (5/4)*x - 5

Answer to second problem: Equation is y = 3x - 29

Hope this helps. --PS T 17:42, 18 January 2009 (UTC)[reply]

How do you expect that to help? The OP wasn't saying that they weren't sure if they'd got the right answer, they were saying they didn't know how to get an answer. --Tango (talk) 18:46, 18 January 2009 (UTC)[reply]

Well if he/she does attempt the question (after consulting his/her teacher), he can check with these answers. --PS T 20:18, 18 January 2009 (UTC)[reply]

category has fibre products => has equalisers?

The articles on equalisers/fibre products seem to imply that there are categories with equalisers, but no fibre products. (Does the category of schmes have equalisers?) But what is wrong with the following argument: Given $f,g:X\to Y$ , take $\Delta _{Y}:Y\to Y\times Y,(f,g):X\to Y\times Y$ . The fibre product of these arrows should be the equaliser of $f,g$ . —Preceding unsigned comment added by Ringspectrum (talk • contribs) 17:17, 17 January 2009 (UTC)[reply]

Yes, that shows that a category with products and pullbacks has equalizers. It doesn't show that a category with pullbacks has equalizers, nor does it show that a category with equalizers has pullbacks (which of these questions are you asking? you have one in your title, and one in your first sentence). Algebraist 17:32, 17 January 2009 (UTC)[reply]

The question in the title. So if out category has fibre products and a terminal object, then it has equalisers (the product is the fibre product over the terminal object). In particular, the category of schemes has equalisers. I'll add this to the article on schemes (that the category of schemes over S has finite limits), if I haven't made a mistake. Ringspectrum (talk) 17:54, 17 January 2009 (UTC)[reply]

Just out of curiosity, what is more direct to construct in the schemes category, equalizers or fibre products? Because it's also true that if a category has finite products (included the empty one, which is the final object) and equalizers, then it has pullbacks and all finite (projective) limits... --PMajer (talk) 19:05, 17 January 2009 (UTC)[reply]

When you can construct products of schemes, you can easily generalise this to fibre products. I don't know a construction of equalisers in (Sch/S) without fibre products (if the target is separated, the subspace where f,g coincide is closed, and if the source is reduced, you can give it the reduced subscheme structure: this should be the equaliser; but in general I don't know a direct construction). Ringspectrum (talk) 20:46, 17 January 2009 (UTC)[reply]

Spider pursuing a fly on the net of a polyhedron

Resolved

–

\sim

Lenoxus " * " 00:23, 18 January 2009 (UTC)[reply]

Perhaps you've heard of the spider-fly problem in geometry — here's the problem and its solution. The solution seems to depend on finding the right polyhedron net which contains the shortest line. My question, inspired (if I'm not misremembering) by similar musings on the linked pages, is this: is there a straightforward way to find, for any cuboid, the net with the shortest path — or, moving out a bit, can this be done with any polyhedron? Or does solving a problem of this type (the problem of finding the shortest outside-only path between two points on a polyhedron) always mean looking at all the possible nets? $\sim$ Lenoxus " * " 19:21, 17 January 2009 (UTC)[reply]

I strongly suspect that in general you can't do better than checking all geodesics between your two points, in order to find the shortest one. Consider e.g. a polyhedron like this, a bit perturbed, and suppose the two points are nearly antipodal: then there will be a great number of geodesics, approximatively of the same length, and I do not see how to detect the shortest if not measuring all of them. You can of course avoid considering geodesics of higher index (those turning more times around the polyhedron), but that's all probably. And if the polyhedron is not convex it seems even worse. These global optimization problems of combinatorial nature tend to have algorithms that are exponential in time; maybe some expert of complexity theory may confirm. --PMajer (talk) 20:54, 17 January 2009 (UTC)[reply]

Something involving Calculus of variations, possibly? --Tango (talk) 23:21, 17 January 2009 (UTC)[reply]

It can be done in polynomial time but it's complicated. See e.g. DOI:10.1007/3-540-51542-9_23, DOI:10.1137/S0097539793253371, DOI:10.1145/263867.263869, DOI:10.1145/98524.98601, etc. —David Eppstein (talk) 23:39, 17 January 2009 (UTC)[reply]

Okey-dokey, I think that answers my question satisfactorily within the limits of my own understanding. Thanks for your answers, everyone! $\sim$ Lenoxus " * " 00:23, 18 January 2009 (UTC)[reply]

Balls

I try to solve this trick ... will i couldnt ... maybe you can...?

you have a(12)balls ... one ball weights more or less than one kilo and the rist (11) weights one kilogram

how can you ID the different ball ... using a two side scale ... three times only...? —Preceding unsigned comment added by 79.173.197.60 (talk) 21:14, 17 January 2009 (UTC)[reply]

Do you know if it's more or less (rather than just knowing that it's different)? If so, it's really easy (the first idea I thought of worked, I'm sure you can think of it too). If not, I'm not sure how to do it. --Tango (talk) 22:48, 17 January 2009 (UTC)[reply]

Each weighing gives 3 measures: equal weights, left heavy/right light or right heavy/left light. 3 weighings, 3 possible results, 3^3 = 27 states. 12 states for coin N light, 12 for coin N heavy, 3 states left spare... Just assign coins to weighing in the right fashion. For a full answer, see http://www.cut-the-knot.org/blue/weight1.shtml. -- SGBailey (talk) 23:52, 17 January 2009 (UTC)[reply]

(@Tango)I think that the point of the problem is that you only know that one is different. I see a simple way provided one is allowed to change the length of the levers. This allows e.g. to compare twice a ball with other two. Consider this easier version of the problem: only 4 balls, one different; this time you can use the scale only twice, and you have to identify the different ball AND also you have to say if it is either more or less than the other. If you can do this, go back to the 12 balls and group them in 4 sets of three ball each... --PMajer (talk) 00:09, 18 January 2009 (UTC)[reply]

Have a look at the link I supplied, you don't need to change the arm lengths to answer the question. -- SGBailey (talk) 00:13, 18 January 2009 (UTC)[reply]

yes just had a look, very nice--PMajer (talk) 00:17, 18 January 2009 (UTC)[reply]

The solution at [1] (linked from SGBailey's link) is of course much easier to memorize if you were to actually do this live. I don't know how many people would use this as a party trick, but at certain types of parties, it might work. -- Jao (talk) 00:36, 18 January 2009 (UTC)[reply]