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naughtDE
156 Posts
On August 04 2020 23:14 Ciaus_Dronu wrote: From what I've heard, James R Munkres 'Topology' is a great textbook. I haven't read it, but my experience of his writing in 'Analysis on Manifolds' has been wonderful, he's very clear and insightful. God that was fast. Thanks man, gonna check out it! | ||
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Acrofales
Spain15328 Posts
On August 04 2020 23:14 Ciaus_Dronu wrote: From what I've heard, James R Munkres 'Topology' is a great textbook. I haven't read it, but my experience of his writing in 'Analysis on Manifolds' has been wonderful, he's very clear and insightful. That was the textbook for my uni's course in topology. It is a good book, but you need a pretty solid basis, mainly in algebra, before tackling this, as it is definitely not an easy topic, and he assumes quite a lot of prior knowledge. | ||
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CoughingHydra
172 Posts
On August 04 2020 23:34 naughtDE wrote: God that was fast. Thanks man, gonna check out it! I learned first from Munkres' book and I find it is good for general topology, i.e., if you do not have a specific direction in mind (geometry, functional analysis, etc.). But if you would like to go in a specific direction, for example for geometry, then I would recommend reading Lee's Introduction to topological manifolds - I find that this book gives you a much better understanding of the relationship between topological definitions and geometrical intuition, and as Munkres has also a lot of exercises to chew through. | ||
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naughtDE
156 Posts
On August 05 2020 00:06 CoughingHydra wrote: I learned first from Munkres' book and I find it is good for general topology, i.e., if you do not have a specific direction in mind (geometry, functional analysis, etc.). But if you would like to go in a specific direction, for example for geometry, then I would recommend reading Lee's Introduction to topological manifolds - I find that this book gives you a much better understanding of the relationship between topological definitions and geometrical intuition, and as Munkres has also a lot of exercises to chew through. Seems like Lee has a nice appendix ^^. | ||
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CoughingHydra
172 Posts
Well, it's mostly a review of definitions and basic results. Just to reiterate: with Munkres you would definitely obtain a more comprehensive knowledge in topology - Lee concentrates on the geometric aspects (e.g. he has a nice chapter on the classification of 2-dim. compact manifolds), but barely touches other typical stuff one does in general topology - conditions when a topological space is metrizable (interesting in itself), Tychonoff theorem, Stone-Čech compactification, and Baire theory (all three important in functional analysis). | ||
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Joni_
Germany137 Posts
On August 04 2020 22:53 naughtDE wrote: Any book recommendations for thoroughly self-studying topology? If the DE in your name means german, then you could check out the book "Allgemeine Topologie" by Rene Bartsch, simply because it is written in an enticing, non-dry way in its prose, introductions and motivations but is still a proper maths book. Rene used to give a Topology lecture here at TU Darmstadt that followed it (for obvious reasons), although I'm not sure if the lecture came before the book or vice versa. I mainly remember the book because the better students in the lecture enjoyed his lecture and the book so much, quite a bunch of them bought the book and proof-read it thoroughly, some accumulating a decent amount of corrections that made it into the errata (available on Rene's webpage) and because the small part of the book I read (Uniform Spaces) was a great read, at least for me. | ||
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Melliflue
United Kingdom1289 Posts
The direct method is more abstract, but quicker. It is probably better for students who prefer set theory or logic. This is the approach taken by Munkres. The geometric approach is more common because most students find it more intuitive. This approach defines metric spaces first and gives properties, theorems, etc about them. Then the definition of a topological space is given, followed by the same/analogous properties, theorems, etc. For books that take this approach you can try Mendelson's 'Introduction to Topology' or Sutherland's 'Introduction to Metric and Topological Spaces'. There are also many notes freely available online; lots of professors put up lecture notes. | ||
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naughtDE
156 Posts
On August 05 2020 04:54 Joni_ wrote: If the DE in your name means german, then you could check out the book "Allgemeine Topologie" by Rene Bartsch, simply because it is written in an enticing, non-dry way in its prose, introductions and motivations but is still a proper maths book. Rene used to give a Topology lecture here at TU Darmstadt that followed it (for obvious reasons), although I'm not sure if the lecture came before the book or vice versa. I mainly remember the book because the better students in the lecture enjoyed his lecture and the book so much, quite a bunch of them bought the book and proof-read it thoroughly, some accumulating a decent amount of corrections that made it into the errata (available on Rene's webpage) and because the small part of the book I read (Uniform Spaces) was a great read, at least for me. It does not, but it is one of the languages I speak. I tried to read some of the first semester stuff from Siegfried Bosch years ago, but basically came running back to English, as German when used in a scientific context can become the most obfuscated language. I just had a look inside "Allgemeine Topologie" by Rene Bartsch and he surely puts things very clearly, maybe I give German another go. @Melliflue That's a +1 for munkres then! | ||
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greys
14 Posts
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Simberto
Germany8436 Posts
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JimmyJRaynor
Canada14383 Posts
On October 08 2020 21:12 greys wrote: I study math in college right now, it is probably the hardest subject for me the hardest math course i ever took was MAT 344: Applied Combinatorics. It had two midterms worth 25% and a final exam worth 50%. The first midterm occurred just before you were still eligible to drop the course without incurring an academic penalty. There were 30 people in the class before the first midterm. There were 14 remaining after that first midterm exam. More than half the class bailed.  | ||
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JimmyJRaynor
Canada14383 Posts
On October 08 2020 21:14 Simberto wrote: Very reasonable to bump a thread which has had no activity for three months with that information. Are you an adbot? i do not think he is an adbot. to wit... https://tl.net/forum/entertainment/87868-movie-discussion?page=458#9142 | ||
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maybenexttime
Poland4281 Posts
I'm working on several different materials. Each of them consists of 3-4 phases. Their phase composition overlaps to some extent. Each phase in a given material has a characteristic volume fraction (F). The phase properties I'd be looking at are elastic modulus (E), hardness (H) and plastic yield strength (Y). They may take somewhat different values between nominally the same phases in different materials. Some phases can be inherently brittle and not deform plastically, which means they don't have Y that could be measured. Each material has an overall wear resistance (R). Below is an illustration of what I mean: material #1: phase A (F, E, H, Y) + phase B (F, E, H, Y) + phase C (F, E, H) -> R material #2: phase A (F, E, H, Y) + phase C (F, E, H) + phase D (F, E, H) -> R material #3: phase B (F, E, H, Y) + phase Z (F, E, H, Y) -> R A phase not being present in a given alloy is equal to it having a fraction equal zero. F is a weighting coefficient in a way. The relationship between phase properties or phase fraction and R is not necessarily linear. We'd like to be able to predict R based on F, E, H and Y of the constituent phases (if there is indeed a link), as well as determine which variables may be superfluous. | ||
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Acrofales
Spain15328 Posts
On December 04 2020 20:29 maybenexttime wrote: I seem to have exhausted my other options, so I thought maybe I'd ask here. Is there an analytical method that would be suitable for the problem I've described below? I'm looking for a way to correlate one response variable with multiple independent variables. I'm working on several different materials. Each of them consists of 3-4 phases. Their phase composition overlaps to some extent. Each phase in a given material has a characteristic volume fraction (F). The phase properties I'd be looking at are elastic modulus (E), hardness (H) and plastic yield strength (Y). They may take somewhat different values between nominally the same phases in different materials. Some phases can be inherently brittle and not deform plastically, which means they don't have Y that could be measured. Each material has an overall wear resistance (R). Below is an illustration of what I mean: material #1: phase A (F, E, H, Y) + phase B (F, E, H, Y) + phase C (F, E, H) -> R material #2: phase A (F, E, H, Y) + phase C (F, E, H) + phase D (F, E, H) -> R material #3: phase B (F, E, H, Y) + phase Z (F, E, H, Y) -> R A phase not being present in a given alloy is equal to it having a fraction equal zero. F is a weighting coefficient in a way. The relationship between phase properties or phase fraction and R is not necessarily linear. We'd like to be able to predict R based on F, E, H and Y of the constituent phases (if there is indeed a link), as well as determine which variables may be superfluous. Sounds like a regression problem. If you're explicitly interested in finding your superfluous variables, lasso regression is good, and in general any linear regression model is a good place to start. If your problem is very non-linear and not easily linearized, you can use a decision tree or forest regression model, but it won't give you a neat formula that looks like the ones you described. | ||
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maybenexttime
Poland4281 Posts
On December 07 2020 17:29 Acrofales wrote: Sounds like a regression problem. If you're explicitly interested in finding your superfluous variables, lasso regression is good, and in general any linear regression model is a good place to start. If your problem is very non-linear and not easily linearized, you can use a decision tree or forest regression model, but it won't give you a neat formula that looks like the ones you described. Thanks for the suggestion. I will have a look. I considered multiple linear regression at the very start, but wasn't sure whether it's suitable for this kind of problem. The variables are not truly independent here. E.g. (1) the phase fraction must somehow affect how each phase's properties contribute to the net R value. If the fraction of a phase is at 0.05 you'd expect its impact to be low regardless of its properties. (2) H is a function of and E and Y, but not in a straightforward way - different combinations of E and Y can give the same H, but H generally increases with increasing E or Y. I would've skipped H altogether if it weren't for the fact that some phases don't have Y at all. (3) E, H and Y are a function of the phases' chemical composition and crystal structure. I'm not sure how to treat F. Should it be a variable alongside E, H and Y or rather another coefficient modulating them? It's hard to say how non-linear the problem is. Studies on simpler materials (single phase, with maybe some fraction of a strengthening phase) revealed some general trends in terms of Y and H/E ratio, but not a clear function. Most likely due to other confounding factors, such as the crystal structure or oxide type. | ||
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Acrofales
Spain15328 Posts
On December 07 2020 20:19 maybenexttime wrote: Thanks for the suggestion. I will have a look. I considered multiple linear regression at the very start, but wasn't sure whether it's suitable for this kind of problem. The variables are not truly independent here. E.g. (1) the phase fraction must somehow affect how each phase's properties contribute to the net R value. If the fraction of a phase is at 0.05 you'd expect its impact to be low regardless of its properties. (2) H is a function of and E and Y, but not in a straightforward way - different combinations of E and Y can give the same H, but H generally increases with increasing E or Y. I would've skipped H altogether if it weren't for the fact that some phases don't have Y at all. (3) E, H and Y are a function of the phases' chemical composition and crystal structure. I'm not sure how to treat F. Should it be a variable alongside E, H and Y or rather another coefficient modulating them? It's hard to say how non-linear the problem is. Studies on simpler materials (single phase, with maybe some fraction of a strengthening phase) revealed some general trends in terms of Y and H/E ratio, but not a clear function. Most likely due to other confounding factors, such as the crystal structure or oxide type. Couple of approaches here. The first is to just try it and see. This is almost always the first approach as it is simple. In particular the tree/forest models deal fairly well with covariant variables, but even a simple linear regression model with l1 and/or l2 regularization can do well. It all depends on the data and the problem. If these don't work well at all, modelling it is going to be tough and you may need some feature engineering to get more information out of your data. However, you can still use some other tricks to tease out some of the covariance. You can use LDA to transform your data to an orthogonal base, and the underlying coefficients will tell you which variables were particularly informative. PCA does something similar, but doesn't take your dependent variable into account specifically. Either can help you weed out some of the covariance. | ||
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maybenexttime
Poland4281 Posts
Could you tell me if those methods are suitable for problems which do not feature large datasets? I have a total of 7 materials (M1-M7), composed of 5 different phases (A, B, X, C, D). Each of them consists of 2-3 phases (I decided to ignore some minor phases to makes things simpler) and has a net R value. Each of those phases is characterized by 3-4 parameters (F, E, H and possibly Y). R is a function of the properties of the phases that comprise the material (how do I add lower indices?): R = f(F_A, E_A, H_A, Y_A, F_B, E_B, H_B, Y_B, F_X, E_X, H_X, F_C, E_C, H_C, F_D, E_D, H_D). Since the materials don't contain all the phases, the fractions of the non-present phases effectively equal zero while the other phase parameters do not exist (which I suppose is different from being equal to zero?). M1: R = f(F_A, E_A, H_A, Y_A, F_X, E_X, H_X, F_D, E_D, H_D) M2: R = f(F_A, E_A, H_A, Y_A, F_X, E_X, H_X, F_D, E_D, H_D) M3: R = f(F_A, E_A, H_A, Y_A, F_X, E_X, H_X, F_C, E_C, H_C) M4: R = f(F_A, E_A, H_A, Y_A, F_D, E_D, H_D) M5: R = f(F_A, E_A, H_A, Y_A, F_B, E_B, H_B, Y_B, F_C, E_C, H_C) M6: R = f(F_B, E_B, H_B, Y_B, F_X, E_X, H_X, F_D, E_D, H_D) M7: R = f(F_A, E_A, H_A, Y_A, F_C, E_C, H_C) We only get one set of values for each material because R and F, E, H and Y are measured is different experiments, using different samples. We can't measure R twice for a given material, then measure F, E, H and Y twice, and say this set of F, E, H and Y corresponds to this R and that set to that R. The number of data points is equal to the number of materials studied, and it's much lower than the number of variables. Is this a problem? Edit: According to Wiki, lasso does perform variable selection, which is somewhat reassuring. :-) Feature selection techniques are often used in domains where there are many features and comparatively few samples (or data points). Archetypal cases for the application of feature selection include the analysis of written texts and DNA microarray data, where there are many thousands of features, and a few tens to hundreds of samples. | ||
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Acrofales
Spain15328 Posts
On December 10 2020 22:55 maybenexttime wrote: Thanks a bunch! Many of those terms sound completely foreign to me. I will have to do some reading. :-) I've been exploring the possibility of using dimensional analysis (suggested by one professor), but the problem is so different from the examples shown in the literature, that I got lost. Could you tell me if those methods are suitable for problems which do not feature large datasets? I have a total of 7 materials (M1-M7), composed of 5 different phases (A, B, X, C, D). Each of them consists of 2-3 phases (I decided to ignore some minor phases to makes things simpler) and has a net R value. Each of those phases is characterized by 3-4 parameters (F, E, H and possibly Y). R is a function of the properties of the phases that comprise the material (how do I add lower indices?): R = f(F_A, E_A, H_A, Y_A, F_B, E_B, H_B, Y_B, F_X, E_X, H_X, F_C, E_C, H_C, F_D, E_D, H_D). Since the materials don't contain all the phases, the fractions of the non-present phases effectively equal zero while the other phase parameters do not exist (which I suppose is different from being equal to zero?). M1: R = f(F_A, E_A, H_A, Y_A, F_X, E_X, H_X, F_D, E_D, H_D) M2: R = f(F_A, E_A, H_A, Y_A, F_X, E_X, H_X, F_D, E_D, H_D) M3: R = f(F_A, E_A, H_A, Y_A, F_X, E_X, H_X, F_C, E_C, H_C) M4: R = f(F_A, E_A, H_A, Y_A, F_D, E_D, H_D) M5: R = f(F_A, E_A, H_A, Y_A, F_B, E_B, H_B, Y_B, F_C, E_C, H_C) M6: R = f(F_B, E_B, H_B, Y_B, F_X, E_X, H_X, F_D, E_D, H_D) M7: R = f(F_A, E_A, H_A, Y_A, F_C, E_C, H_C) We only get one set of values for each material because R and F, E, H and Y are measured is different experiments, using different samples. We can't measure R twice for a given material, then measure F, E, H and Y twice, and say this set of F, E, H and Y corresponds to this R and that set to that R. The number of data points is equal to the number of materials studied, and it's much lower than the number of variables. Is this a problem? Edit: According to Wiki, lasso does perform variable selection, which is somewhat reassuring. :-) Basically the more variables you have and the fewer samples, the greater the chance for overfitting, so generally you'll want to use the methods to prevent overfitting more aggressively. Regularization and feature selection are techniques that do exactly that. Dimensionality reduction (with LDA, PCA or a few more complex methods) are another (alternative/complementary) way of doing that. But you need to just dive in and start experimenting with the various techniques, as there is no theoretical best way to do this (well, I guess in theory, there is, we just don't know what it is, and it is different for every problem). One word of caution: with so few samples, and if running more experiments is hard and expensive, then be extra careful to set aside a few samples for your test set and *do not ever use them*. You can then use various ways for splitting your train and validation data, and once you think you have a method that works well, then use your test data to make sure that is indeed a good method. If you use your test data too much to guide the rest of the process, you can still be overfitting and will need new different data to test whether your method actually works! | ||
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maybenexttime
Poland4281 Posts
On December 17 2020 20:03 Acrofales wrote:Basically the more variables you have and the fewer samples, the greater the chance for overfitting, so generally you'll want to use the methods to prevent overfitting more aggressively. Regularization and feature selection are techniques that do exactly that. Dimensionality reduction (with LDA, PCA or a few more complex methods) are another (alternative/complementary) way of doing that. But you need to just dive in and start experimenting with the various techniques, as there is no theoretical best way to do this (well, I guess in theory, there is, we just don't know what it is, and it is different for every problem). I've been learning about those methods from StatQuest. Your recommendations seem well-suited for the problem I'm working on. Very solid advice. You've been really helpful. :-) I'm not sure if it's possible to solve it this way, but it's my best bet. Even if it doesn't work out, I'll still have plenty of data to analyze qualitatively. One word of caution: with so few samples, and if running more experiments is hard and expensive, then be extra careful to set aside a few samples for your test set and *do not ever use them*. You can then use various ways for splitting your train and validation data, and once you think you have a method that works well, then use your test data to make sure that is indeed a good method. If you use your test data too much to guide the rest of the process, you can still be overfitting and will need new different data to test whether your method actually works! Thanks for the warning. I'm aware of that. Having this few data points will make splitting my data tricky. Unfortunately, adding more data points would require making new materials and testing them. That would probably cost several thousand pounds per data point. :-P | ||
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