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Viruses in wine? Identification of mid-80s time travel tv show I am studying homology groups and I am looking to try and develop, if possible, a little more intuition about what they actually mean. I've only been studying homology for a short while, so if possible I would prefer it if this could be kept relatively simple, but I imagine it is entirely possible there is no real answer to my query anyway.   As I said above, I want to gain a little deeper understanding of what the n-th homology group actually means: I can happily calculate away using Mayer-Vietoris but it doesn't really give me a great deal of intuition about what the n-th homology group actually means. For example, with homotopy groups, the fundamental group is in some sense a description of how loops behave on the object in question, and it is obvious to me why that is what it is for say, the torus or the circle. However, I have no idea what, if anything, I am actually saying about a triangulable object when I talk about it having 0-th homology group this or 1st homology group that.  The best I have been able to find online or in my limited book selection is the brief description "intuitively, the zeroth homology group counts how many disjoint pieces make up the shape and gives that many copies of $\Bbb Z$, while the other homology groups count different types of holes". What 'different types of holes' are there, roughly speaking? I appreciate that it may often be completely non-obvious what the low-order homology groups are for some complicated construction, but perhaps in simpler examples it might be more explicable. Are there (simple) cases where I could say, just from looking something like e.g. the torus, what its zero-th or first or second etc. homology group was based on the nature of the object? I guess in the zero-th case it is, as my source () above says, related to the number of disjoint pieces. Can we delve deeper than this for the other homology groups?   Any book/website suggestions would be welcomed (preferably websites as I am nowhere near a library!) - I have Hatcher but not a great deal else, and I haven't gleaned as much as I wish to from that alone. Of course I know that there is a great deal we don't know about homology groups even today, so I don't expect some magical all-encompassing answer, but any thoughts you could provide would be appreciated. I hope this question is appropriate for SE Mathematics, apologies if not! -M I've been riding a mountain bike for years, but I prefer to go longer distances on paved ground, so I think a road bike would be better. When I ask what is a good bike, I don't need brands or models - more like what components of the bike should I look for, and what should I avoid? Additionally, is it necessary to "go big or don't go at all" as far as spending - getting a real good quality road bike for $500 to a few thousand, or should I cheap out on a $100-200 road bike, as it is my first, then get a better one when I'm more experienced? Help me understand adjusted odds ratio in logistic regression I've been having a hard time trying to understand the use of logistic regression in a paper. The paper available  uses logistic regression to predict probability of complications during cataract surgery.   What is confusing me is that the paper presents a model that assigns odds ratio of 1 to baseline described as follows:     A patient whose risk profile was in the reference group for all risk indicators (ie adjusted OR=1.00 for all in Table 1) may be regarded as having a ‘baseline risk profile’, and the logistic regression model indicates a ‘baseline predicted probability’ for PCR or VL or both=0.736%.   So the probability of 0.00736 is presented with odds ratio of 1. Based on the transformation from probabilities to odds ratios: $o=\frac{p}{1-p}$, this can't be equal to 1:  $0.00741=\frac{0.00736}{1-0.00736}$.  It gets even more confusing. The composite odds ratios that represents multiple covariates having values different than the baseline is used to calculate predicted risk.      ...the composite OR from Table 1 would be 1.28 X 1.58 X 2.99 X 2.46 X   1.45 X 1.60 = 34.5, and from the graph in Figure 1, we see that this OR corresponds with a predicted probability of PCR or VL or both of around 20%   The only way to arrive at the values the paper is giving as examples is to multiply the baseline probability with composite odds like this: $0.2025=\frac{(34.50\ \times\ 0.00736)}{1\ +\ (34.50\ \times\ 0.00736)}$.  So what is going on here? What is the logic for assigning odds ratio 1 to a baseline probability that is not 0.5? The update formula I've came up with above comes up with the right probabilities for examples in the paper but this is not the direct multiplication of odds ratio I'd expect. What is it then? Is there a difference between "Alpha Helix" and "Alpha subunit"? Galois extension of intersection of fields I have finite Galois extensions: E/K and E/L. $$M:=K \cap L$$ I am trying to prove that if the extension E/M is finite then it is also Galois. Any suggestions? Thanks I want to be able to login to a remote via ssh without having to enter the password all the time.   How do I set it up? Is a different command required to execute a password-less session? What can cause "System.CalloutException: You have uncommitted work pending. Please commit or rollback before calling out"? What else, other than insert/update/upsert/delete, can cause "System.CalloutException: You have uncommitted work pending. Please commit or rollback before calling out"?  I've search the debug logs multiple times, and there absolutely is not a DML statement executed between the startTest and the failing callout.  Are there any other commands, or perhaps another callout (even though it's only an inquiry), that can cause this?  Does anyone know how to get around this problem? Make Google stop sending 'new sign in from ...' emails I get at least two emails from Google every time I sign in to my account and I want it to stop.  I tried changing the notification settings in Gmail, but they're still coming and they're useless.  How can I stop this? I'm a bit overwhelmed by the available options in regards to the three major photo editing apps from Adobe.  Does anyone know of an article or site that lists the features of Lightroom, PS Elements, and PS, side by side?  I know that Lightroom revolves around edits and workflow, and that PS is the big kahuna of image editing.  Where does Elements fit in to this understanding? Why is the "user was removed" message so vague? I've sectioned of this area in the sea near my base, and I need to get rid of all the water from it.  From what I've tried the only way I can see to clear it is fully cover one layer, delete the blocks, then go down to the next layer. Is that the only way I can clear the area or is there a better approach?  Ideally I'd like a survival method (if there is a better one), but any commands what could be used I'd like to hear just so I can learn more commands :) In the construction of types of numbers, we have the following sequence:  $$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S}$$  or:  $$2^0 \mathrm{-ions} \subset 2^1 \mathrm{-ions} \subset 2^2 \mathrm{-ions} \subset 2^3 \mathrm{-ions} \subset 2^4 \mathrm{-ions} $$  or:  "Reals" $\subset$ "Complex" $\subset$ "Quaternions" $\subset$ "Octonions" $\subset$ "Sedenions"  With the following "properties":   From $\mathbb{R}$ to $\mathbb{C}$ you gain "algebraic-closure"-ness (but you throw away ordering). From $\mathbb{C}$ to $\mathbb{H}$ we throw away commutativity. From $\mathbb{H}$ to $\mathbb{O}$ we throw away associativity. From $\mathbb{O}$ to $\mathbb{S}$ we throw away multiplicative normedness.   The question is, what lies on the right side of $\mathbb{S}$, and what do you lose when you go from $\mathbb{S}$ to one of these objects ? Combine and Minify Multiple CSS / JS Files What program can be used to package software into one executable? I need to package an application into a single executable file (so that I don't have to install it, so that it will be portable).  Does anybody know what I can use to make a program portable? Firstly, I apologise if my thinking is completely incorrect - I am by no means a physicist (yet!), I have included layman illustrations to hopefully assist in explaining my thinking.  I am imagining the big bang to be, for lack of a better word, an explosion, thus the universe to be progressively expanding:    Therefore, to me it seems logical to suggest that there is an edge.  If this is the case, then is it reasonable to suggest that explosion + edge = 'bright edge'? Where light projects in all directions at the edge, not just outwards into the 'nothingness'.     So, this brings me to my question: why, when someone looks deeply enough 'at the edge' do they see an absence of light?   Again, apologies if I'm jumping to irrational conclusions &amp; am way off. I know how to solve "simple" recurrence relations. For instance, say you have:  $$c_0 = 20$$ $$c_1 = 30$$ $$c_n = 3 c_{n-1} - 2 c_{n-2}$$  We can write the characteristic equation as:  $$3x^{n-1} - 2x^{n-2} = x^n$$  Solving this with $n=2$, we get $x = 1$ or $x = 2$. This lets us write the relation $c_n = \alpha_1 1^n + \alpha_2 2^n$, and we can solve for $\alpha_1$ and $\alpha_2$ with the initial states $c_0$ and $c_1$.  However, this depends on the fact that $3x^{n-1} - 2x^{n-2} = x^n$ has two roots.  Now, I'm stuck on another problem where the characteristic equation has fewer roots than terms.  Say I have this recurrence relation instead:  $$a_0 = 0$$ $$a_1 = 2$$ $$a_2 = −1$$ $$a_n = 9a_{n-1} - 15a_{n-2} - 25a_{n-3}$$  The characteristic equation would be:  $$9x^{n-1} - 15x^{n-2} - 25x^{n-3} = x^n$$  However, solving with $n=3$, we only get two roots: $x=-1$ or $x=5$. There are not enough roots to write a relation in the form of $a_n = \alpha_1 r_1^n + \alpha_2r_2^n + \alpha_3r_3^n$. How do I proceed? How to represent it with UNIQUE indexes and foreign keys? "Group" is a group of pupils. "Topic" is a topic of lessons.  To every group corresponds exactly one topic. (So there may be multiple groups per topic.)  To a pupil corresponds multiple groups, but maximum one group per topic.  How to implement this in terms of UNIQUE indexes and foreign keys in MySQL? Distinct Sylow P subgroups