
#127
Feb2312, 11:55 PM

P: 6,863

Again, this is not esoteric math. 



#128
Feb2312, 11:56 PM

P: 6,863





#129
Feb2312, 11:57 PM

P: 6,863





#130
Feb2412, 12:03 AM

P: 6,863

Also the rules of vector spaces are that the mathematical operations are *defined*, but necessarily that they always lead to physically possible results. I can count the number of banana trees with integers. The fact that it's possible to talk about a trillion banana trees when in fact there are not that many trees in the world doesn't invalidate the use of numbers to describe banana trees. Also, I haven't seen a "negative banana tree" but it's possible to define an additive inverse anyway.
Also for election results, it's possible to do factor analysis and all sorts of pretty complicated linear algebra with those results. The fact that it is *possible* to do those mathematical operations is what renders it a vector space. What's really cool is that once you've defined some basic operations, then you end up getting the mathematics of vector spaces. And people *do* use these things in "real life". You assign each election district a number indicated for example "ruralness" and do coorelations, and that tells you how to plan out your next campaign. 



#131
Feb2412, 06:59 AM

Mentor
P: 16,481

I am not sure about modulo, but I doubt that a modulo arithmetic results in vectors which satisfy all of the axioms of a vector space. I would have to see a proof. In any case, as you reach further and further to get valid vectors from election results you also get further and further from a mathematical structure that has any natural relationship to election results. The reason that e.g. momentum is a vector is that the properties of momentum have a natural relationship with the properties of vectors. I.e. the result of adding two objects' momentum vectors corresponds naturally to the momentum of the system of the two objects. What election result does the sum of two election results modulo 1 represent? Does such a result have any natural relationship to the two original election results? 



#132
Feb2412, 08:01 AM

PF Gold
P: 706

You can ask questions like, What's the total number of votes in those three counties? Or How many more votes did the candidates receive in this county than that county? What do you have then? 



#133
Feb2412, 08:15 AM

PF Gold
P: 706

So let's say I'm in the ocean, outside of the sight of land. What kinds of questions would you ask me to determine my latitude and longitude? You can ask me what species of fish are in the water You can ask me, when I call out on the radio, what language is being answered in... And lets say by asking some general questions like these, you manage to figure out that I must be in the Pacific Ocean, somewhere southwest of Hawaii. The thing is, we set up the rules so that you can't ask about any specific landmarks, but the whole goal of the game is to circumvent those rules, and to find a specific landmark, which I am next to. If you have figured out that there is a unique location on Earth where you have trout and starfish and they speak Hawaiian on the radio, etc, then that unique intersection of properties actually is, in itself, a landmark. 



#134
Feb2412, 12:33 PM

P: 177

If one tries a similar construction for the integers or nonprime mods, then one obtains an Rmodule (wikipedia Module, where R denotes a ring), which is not a vector space. 



#135
Feb2412, 05:31 PM

Mentor
P: 16,481

I am tired of this game. It does not seem to me that "election results are a vector" is a very natural concept. You can continue to do little tweaks and may eventually come up with something that is mathematically a vector, but as you do it seems that you are getting further away from a useful representation of elections. In general, to say of some realworld concept that "X are vectors" requires the following: 1) There needs to be a bijection of the different X to different mathematical objects 2) There needs to be a correspondence between realworld operations on X and operations on the mathematical objects 3) Those mathematical objects and operations need to satisfy all of the axioms of vector spaces (http://en.wikipedia.org/wiki/Vector_space) I don't see a way of doing that for election results. Even step 1) seems questionable to me. If you really want to do it then please go ahead, but just make sure that you are careful because it is a really unclear fit. 



#136
Feb2512, 03:48 AM

P: 97

To answer the original question, we have to agree on what is measurable.
To me a measure is some set of elements each of which is invariant with respect to everything. In this sense they are "scalars". Strictly speaking, only scalars are physical quantities because results of experiments are not arbitrary: as someone said before, the result of a measure of a given quantity done with a specific instrument in a specific condition cannot possibly change. A couple examples.  Let us take one man jumping off a cliff. The acceleration acting on his center of mass is not measurable, because of the fact we can perform a change of coordinates such that it goes to zero (free fall frame). Nonetheless in every possible frame that man is going to die in a horrible way. What is measurable is the set of components his acceleration has with respect to the ground in a given frame in some chosen units. This numbers are not going to change with the frame, because they are defined in a specific coordinate system (like the mass, wich is the zeroth component of the fourmomentum in the rest frame). So those numbers are "scalars". Similarly the colour of a laser is not a physical quantity, but colour of a laser in a given frame it is.  Try now to do something similar with components of the electromagnetic fourpotential. This time it's components in a given frame are no well defined because of the gauge invariance. To perform a measure of the fourpotential we have to choose not only the frame and the units but also the gauge. What we usually call physical quantities are simply the set of all possible measure we can get of a object (note that this set is, by definition, invariant). The fourpotential is a Lorents vector: if we know the result of a measure of its component in a given frame, gauge and units, then we can calculate every possible result of any other experiment in another frame, gauge and units by performing a Lorentz, gauge and units transformation. Please tell me if you agree ^^ Ilm P.s. sorry for my bad english 



#137
Feb2512, 09:59 AM

PF Gold
P: 706

If you want to end the game, answer these questions: Are there any examples of nondirectional vector spaces? If so, give one. If not, why not? 



#138
Feb2512, 08:34 PM

Mentor
P: 16,481





#139
Feb2612, 02:40 PM

PF Gold
P: 706

I'm still a little confused about how an election, as we've set up above, with each voter being allowed to vote using a real number between 1 and 1, for each of, say, 12 candidates, doesn't also create a 12 dimensional, nondirectional vector space, though. The 12th order polynomial functions are essentially a list of 12 real coefficients associated with 12 powers of x, whereas the election results are 12 real numbers associated with 12 votingresults. 



#140
Feb2612, 04:34 PM

P: 669





#141
Feb2612, 05:57 PM

P: 177





#142
Feb2612, 06:42 PM

Mentor
P: 16,481





#143
Feb2612, 07:04 PM

Mentor
P: 16,481

You can consider many different sets of basis vectors for this space, the most obvious being [itex](x^{12},x^{11},x^{10},...)[/itex], and in terms of that basis you can write the vector as (5,4,8,...) for notational convenience, but the 13 numbers are no longer just an arbitrary list of numbers, but are instead coordinates in a specified basis. You could consider different basis vectors, such as the Legendre polynomials. Then you could write the same vector as a different list of numbers corresponding to coordinates in a different basis. Although the list of numbers would be different, the vector is the same, since the vector is the polynomial and not the list of numbers. 1) Define the bijection between different election results and different vectors 2) Demonstrate the correspondence between realworld operations on elections and operations on the vectors 3) Prove that the vectors and operations satisfy all of the axioms of vector spaces (http://en.wikipedia.org/wiki/Vector_space) 


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