• Mathematician Chat v. 3.999...
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[QUOTE=Ausare;51680384]How would you solve this equation? [IMG]http://i.imgur.com/kRbVXRq.png[/IMG] This is my attempt at a solution which is wrong. [IMG]http://i.imgur.com/l7U3iBs.png[/IMG][/QUOTE] Go back to your third line and let c = n(n+1)/2. Then you have c + nx = ax + b. From here it's pretty straight-forward: c - b = ax - nx (c - b) / (a - n) = x
[QUOTE=Wunce;51681400]Go back to your third line and let c = n(n+1)/2. Then you have c + nx = ax + b. From here it's pretty straight-forward: c - b = ax - nx (c - b) / (a - n) = x[/QUOTE] Thanks. Okay, is -4501495/2976 a solution to this? [img]http://i.imgur.com/vnVTbJu.png[/img] Sorry if this is a bother to anyone. Thanks again.
sum3000(i + x) = sum3000(i) + sum3000(x) = 24x + 5 sum3000(i) - 5 = 24x - sum3000(x) 3000*3001/2 - 5 = (24-3000)x 4501495 = -2976x x = -4501495/2976
[url]http://www.math.ru.nl/topology/Notes%20on%20Homological%20Algebra.pdf[/url] These notes are pretty dope, I'm using them to review for my algebraic topology class which starts tomorrow. My professor was a student of Moerdijk's too. :v:
[QUOTE=Fourier;51709161]sum3000(i + x) = sum3000(i) + sum3000(x) = 24x + 5 sum3000(i) - 5 = 24x - sum3000(x) 3000*3001/2 - 5 = (24-3000)x 4501495 = -2976x x = -4501495/2976[/QUOTE] Thanks. I was either being trolled or confused on what was being asked because when I gave that exact answer I was told, "no."
Anyone got a good source on higher order (>2) cartesian tensor eigenvalues? (if that even is a thing).
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[QUOTE=Number-41;51718681]Anyone got a good source on higher order (>2) cartesian tensor eigenvalues? (if that even is a thing).[/QUOTE] I am curious, why do you to know this :D? [editline]24th January 2017[/editline] [QUOTE=SenhorCreeper;51718747]Hey Whats. Up It's. Hot-ThotFucker94 Here. I've Come Into. This Thread To Ask. Which Mathematical Equation Is. Most Likely To Make. Me Score A. Hot Flaming Pussy. I Tried Wolfram Alpha's. Scientific Super-Computer And. It's Only Making Me Feel. Stupid. Damn. [IMG]http://i.imgur.com/o0IXsg0.png[/IMG][/QUOTE] Just go to math college, there is a lot of girls there.. =D
[QUOTE=Fourier;51719029]I am curious, why do you to know this :D? [/QUOTE] I started my PhD in Diffusion Tensor Imaging a few months ago (yeah it's funny where you can end up with a physics degree :v:). In non-isotropic media (like white-matter tracts in the brain), water diffusion will have preferential directions. You can measure this with an MRI scanner, and the signal can be modelled with a tensor equation that assumes some diffusivity model. For a crash course, see [URL="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3163395/"]this source[/URL]. The simplest model uses a second order cartesian tensor. It has 3 eigenvalues and -vectors which you can map to an ellipsoid. So a long cigar-like ellipsoid would correspond with diffusion tensor with a single large eigenvalue about e.g. the z-axis and two equal eigenvalues along the x- and y-directions. This is what you could see as a single white-matter bundle. Now this is only a basic model. It assumes Gaussian diffusion, which is a distribution with infinite support. In a capillary, you would not expect such a distribution. A solution for this is to use higher-order tensor models (which also correspond with non-Gaussian distributions, a.k.a. kurtosis models). Also, fibers can cross, which messes up a whole lot of other things. Now, for some tensor, you want to simplify your model with certain symmetry assumptions. So a WM-bundle will be cylindrically symmetric, so your 2nd order diffusion tensor has 2 equal eigenvalues (x and y) and a third one. I already found an analytical method to find higher order tensors with some spatial symmetry (thank you General Relativity :v:) but I do not know how this relates to its eigenvalues (if it is meaningful to talk about eigenvalues- and vectors for higher order tensors at all). I think material deformation stuff also deals with cartesian tensors with spatial symmetries, but they always relate it to stress and strain and whatnot. I need a more general treatment. The benefits of knowing what a tensor looks like with spatial symmetries, is that it contains fewer independent parameters, and thus measuring this tensor will require less measurements, leading to faster scanning times. Considering a scan is abour $400/hour, you really want this...
Interesting thing :). You might be able to help yourself with Kernel-SVD with this sort of stuff. [IMG]https://i.stack.imgur.com/D7vyt.png[/IMG] SVD also finds orthogonal ellipse (rotation and scale), but if you apply non-linear functions to vectors/data before applying SVD, you can find non-linear (higher order) relationships. Though SVD might be quite slow compared to single eigenvalue algorithm. (this is what popped to my mind though might be completely unrelated to your problem, in this case sorry) How do you calculate first order tensor from MRI image though? With derivatives (filter approximations) yes? Or how. I also wanted to do some arhihectural stability thingy with tensors but I don't understand them good enough yet :/. But visualizing 3D orthogonal tensor isn't hard. (it's a 3x3 matrix after all heh) [editline]24th January 2017[/editline] [QUOTE=Number-41;51719225] The benefits of knowing what a tensor looks like with spatial symmetries, is that it contains fewer independent parameters, and thus measuring this tensor will require less measurements, leading to faster scanning times. Considering a scan is abour $400/hour, you really want this...[/QUOTE] Is that because MRI machine or because processing is so intense?
[QUOTE=Fourier;51719545] How do you calculate first order tensor from MRI image though? With derivatives (filter approximations) yes? Or how. [/QUOTE] Your signal is S=S_0 * exp(-bg^T D g) with D your tensor, g the vector corresponding with the gradient direction and magnitude, so you just take the log and you get a relation a la B=Ax, which is easy to solve. [QUOTE=Fourier;51719545] Is that because MRI machine or because processing is so intense?[/QUOTE] You might have interpreted me wrongly, the scan time should be as short as possible because operating an MRI costs roughly $400/hour, because they are expensive as hell and the MRI room is expensive as hell and the maintenance is expensive as hell and the coils, and the radiologist, etc. Processing can also take very long but we have access to super computers so that's not really an issue.
[QUOTE=Number-41;51719743] You might have interpreted me wrongly, the scan time should be as short as possible because operating an MRI costs roughly $400/hour, because they are expensive as hell and the MRI room is expensive as hell and the maintenance is expensive as hell and the coils, and the radiologist, etc. Processing can also take very long but we have access to super computers so that's not really an issue.[/QUOTE] I though it was MRI the expensive one, because processing power today is not so expensive, just needed to ask to be sure. So, if I understand you correctly, you control the machine (scanning parameters) real-time to gather the data/signal? What about sparse scanning, is that not an option.. like do general scan first then scan areas that are more dense, to structure octree tensor field or some other (insert 3d spatial struct here) tensor field? Though in this case (because MRI is cylindrical scanner) would be better to make cylindrical data structure. P.S. Still don't know answer to original question about eigenvalues D:
For clinical applications you still want to avoid having to wait a week for the results of some scan, that's why processing time should also be minimised. You also do not control the machine in real-time, you just tell it to scan in some specific way and then it does that. A single run might take 10 seconds to 10 minutes, so real-time changing of scanning parameters does not really apply here. Also, because you sample (heh) Fourier space, you cannot just adjust your sampling rate locally for ROI's. Before scanning, the radiologist would already determine which area to scan anyway. After that, it is assumed that you want maximum detail in each region that you scanned. Also, the field of view of a scan is typically much smaller than the machine bore, so there's no need to adjust your data structure (also I don't see how that would matter anyway as you typically vectorize your data first) Your SVD suggestion is a bit general and vague :P I don't know how you'd apply it.
Math is supreme, unfortunately I don't get much math in my Computer Science bachelor. Which makes me want to switch to Mechanical Engineering or something. Shit.
[QUOTE=Number-41;51723970]For clinical applications you still want to avoid having to wait a week for the results of some scan, that's why processing time should also be minimised. You also do not control the machine in real-time, you just tell it to scan in some specific way and then it does that. A single run might take 10 seconds to 10 minutes, so real-time changing of scanning parameters does not really apply here. Also, because you sample (heh) Fourier space, you cannot just adjust your sampling rate locally for ROI's. Before scanning, the radiologist would already determine which area to scan anyway. After that, it is assumed that you want maximum detail in each region that you scanned. Also, the field of view of a scan is typically much smaller than the machine bore, so there's no need to adjust your data structure. Your SVD suggestion is a bit general and vague :P I don't know how you'd apply it.[/QUOTE] Ah, so you deal with output signal data from MRI and don't really deal with scanning. I thought you were also dealing with scanning algorithm. For SVD, well here: I am not very knowledgeable with tensors as you have notices, and even less with this tensor imaging technique. I did peek at link you provided and well, I saw those tensor field images many times. So I have to ask, what are you doing here, do you process tensor field or you process signal, and need faster way to convert signal to tensor field? Because there exists way to many ways you can use SVD. Though it's processing intense so I doubt it might very useful. (time consuming)
You infer the tensor field from the signal (see eq. 1 in the source I linked) using some tensor model. My question was if anyone knew a good general source on tensor eigenvalues.
[QUOTE=ljh;51724074]Math is supreme, unfortunately I don't get much math in my Computer Science bachelor. Which makes me want to switch to Mechanical Engineering or something. Shit.[/QUOTE] Add a math minor. Math is a big benefit to a computer scientist.
[QUOTE=Number-41;51724540]You infer the tensor field from the signal (see eq. 1 in the source I linked) using some tensor model. My question was if anyone knew a good general source on tensor eigenvalues.[/QUOTE] I know what was original question :v:, but you mentioned you needed to extract higher order features because normal Gaussian was not ok. Sorry about that. Don't know reply to original question, well if you have 3x3x3x.. tensors that is. If you have 3x3 tensors (basically, a matrix?), though I doubt you would ask those questions anyway.
[QUOTE=JohnnyMo1;51724760]Add a math minor. Math is a big benefit to a computer scientist.[/QUOTE] Well, actually my Bachelor is in Software Engineering, though the two disciplines sort of overlap if you ask me. Unfortunately the curriculum isn't that much engineering and my local university is kind of a mess. At least the Comp. Sci. and Information Technology department. I almost have completed my first year. I get my results somewhere in the coming 2 weeks. But man I am thinking of picking a more "hands dirty" type of major.
Hello guys, I know this isn't math per se, but I know a lot of you are in electrical engineering and kinda know of this stuff. I haven't touched this subject for more than a couple of years so I'm kinda in need of a helping hand determining a diferential expression for a differential circuit in orther to determine the Transfer Function later on. Transfer Function being: H(x)=Output(x)/Input(x). The circuit's the following: [IMG]http://i.imgur.com/zNlCbbG.png[/IMG] And I need to determine the differential equation that defines the circuit's behavior. So far this is all I got (sorry for the low quality): [IMG]http://i.imgur.com/i1W18tJ.jpg[/IMG] I can't seem to find a good expression for substituting iL since I want a differential equation. It's been a while since I did circuit analysis so I know I'm missing something. Can anyone help me out please? Thank you.
Is there any neat way to "visualize" things of group theory? I had a little introduction to it in university last year, and I'm reading the infite napkin (one of the books that were recommended above), and I can't seem to get a solid grasp on it, and I think visualizarion can go a long way,. I mean the kind of visualization that allows us to see eigenvectors and the kernel of a matrix with linear transformations on R3, for example.
[QUOTE=Cosa8888;51726606]Is there any neat way to "visualize" things of group theory? I had a little introduction to it in university last year, and I'm reading the infite napkin (one of the books that were recommended above), and I can't seem to get a solid grasp on it, and I think visualizarion can go a long way,. I mean the kind of visualization that allows us to see eigenvectors and the kernel of a matrix with linear transformations on R3, for example.[/QUOTE] Some groups have obvious visualisations (e.g. the operations on a Rubiks Cube, matress flipping) but in general there is no easy way. Memorising the axioms and basic theorems of groups and doing lots of exercises will give you an intuitive sense of "structure" regarding groups. In a sense you're training a new, abstract eye. For example, if the kernel of a group homomorphism contains only the identity then the homomorphism is injective. Due to the structure of groups, this "feels" right.
[QUOTE=Cosa8888;51726606]Is there any neat way to "visualize" things of group theory? I had a little introduction to it in university last year, and I'm reading the infite napkin (one of the books that were recommended above), and I can't seem to get a solid grasp on it, and I think visualizarion can go a long way,. I mean the kind of visualization that allows us to see eigenvectors and the kernel of a matrix with linear transformations on R3, for example.[/QUOTE] I think often I view groups as sets, sometimes with points arranged such that connections given by the generators are obvious (like the dihedral group). I think in general though, the structure of groups are too weak for "visualizations," but there are "ways of thinking" about what group theoretic constructions are doing. For instance, I hate the coset picture of quotient groups, but I find other conceptualizations of them very intuitive. [editline]25th January 2017[/editline] Some groups, like free groups, do admit nice pictorial representations. [editline]25th January 2017[/editline] Mostly unrelated: my algebraic topology professor is an ∞-category theorist, and he has ∞-categorical notation creeping into his lectures that include only regular old 1-categories. Makes it really hard to figure out what the hell is going on sometimes.
You can imagine whole group as a static structure, and number is just a point lying on this structure.
[QUOTE=Cosa8888;51726606]Is there any neat way to "visualize" things of group theory? I had a little introduction to it in university last year, and I'm reading the infite napkin (one of the books that were recommended above), and I can't seem to get a solid grasp on it, and I think visualizarion can go a long way,. I mean the kind of visualization that allows us to see eigenvectors and the kernel of a matrix with linear transformations on R3, for example.[/QUOTE] It can range from really simple (e.g. Z/6) to more geometrical (symmetries of platonic solids and polyhedrons) [URL="https://en.wikipedia.org/wiki/Monster_group"]to don't even bother[/URL]. See: [url]https://en.wikipedia.org/wiki/Group_(mathematics)#Examples_and_applications[/url] Personally I've always struggled with cosets and stuff like that...
Do you guys have any Mathematics books recommendations for first year university engineering students? My basics are pretty solid, but can use refreshment.
I want to use 2-fold cross-validation to determine which degree of a polynomial fits a certain data set the best. How do I iterate this process? Average the residual from each iteration?
[QUOTE=Number-41;51728928]It can range from really simple (e.g. Z/6) to more geometrical (symmetries of platonic solids and polyhedrons) [URL="https://en.wikipedia.org/wiki/Monster_group"]to don't even bother[/URL]. See: [url]https://en.wikipedia.org/wiki/Group_(mathematics)#Examples_and_applications[/url][/QUOTE] Then you get group objects in categories: topological groups, Lie groups, group varieties, group schemes... Groups everywhere.
They look very fun, I'm confident I'll eventually get a solid grasp on the subject. I have to ask, what are some applications of group theory? I know the question "what is that for?" tends to bother some people, but I really like it when there's a solid answer to the question. Plus, it seems to have a lot of potential.
[QUOTE=Cosa8888;51738982]They look very fun, I'm confident I'll eventually get a solid grasp on the subject. I have to ask, what are some applications of group theory? I know the question "what is that for?" tends to bother some people, but I really like it when there's a solid answer to the question. Plus, it seems to have a lot of potential.[/QUOTE] Well, a basic one is practically all the number systems you're likely to have seen are groups in some way! The natural numbers, integers, real numbers, and complex numbers under addition are all groups, so understanding groups allows you to understand features of all these number systems at once. Probably the most famous application is that they're used to describe symmetries via actions. For instance, the symmetries of a regular polygon are given by the [I]dihedral group[/I]. A triangle in the plane can be rotated 1/3 of a full rotation and look the same as it did when you started, or 2/3 of a full rotation. A full rotation takes everything to where it started, so it corresponds to the identity of the group. Then you have reflections across an axis. You'll likely cover it in class, but the dihedral group is basically built out of two distinguished elements: one to describe rotations, and another to describe a flip. All the others can be built out of these. Other sorts of symmetries can be described in this way. For instance, complete rotational symmetry of some system about an axis corresponds to the existence of an "SO(2) (or U(1)) action" on the system. It turns out that this is quite general. In category theory, which is quite an abstract setting, the set of automorphisms of any object of a category naturally has the structure of a group. So if you have some sort of mathematical thing, and you consider transformations from that thing to itself which preserve some structure of interest and are reversible, you've got a naturally defined group regardless of what the thing you're studying is. It could be sets, or groups themselves, or manifolds, or schemes, or almost any kind of thing that mathematicians regularly are interested in. [editline]28th January 2017[/editline] One place that this pops up in the real world is particle physics. Gauge theory is an extremely important tool of particle physics, and groups are inextricably linked with gauge theory, like the fact that electric charge exists because of a U(1) gauge symmetry of the theory of electrodynamics.
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