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## Results (1-50 of 259 matches)

Label Dim. $$A$$ Field CM RM Traces Fricke sign $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
252.1.h.a $$4$$ $$0.126$$ $$\Q(\zeta_{8})$$ $$\Q(\sqrt{-7})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{8}q^{2}+\zeta_{8}^{2}q^{4}-\zeta_{8}^{2}q^{7}-\zeta_{8}^{3}q^{8}+\cdots$$
252.1.z.a $$2$$ $$0.126$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$1$$ $$q-\zeta_{6}^{2}q^{7}+(\zeta_{6}+\zeta_{6}^{2})q^{13}+(-1-\zeta_{6}+\cdots)q^{19}+\cdots$$
252.2.a.a $$1$$ $$2.012$$ $$\Q$$ None None $$0$$ $$0$$ $$-4$$ $$-1$$ $$+$$ $$q-4q^{5}-q^{7}-2q^{11}-6q^{13}+4q^{17}+\cdots$$
252.2.a.b $$1$$ $$2.012$$ $$\Q$$ None None $$0$$ $$0$$ $$0$$ $$1$$ $$-$$ $$q+q^{7}+6q^{11}+2q^{13}-4q^{19}+6q^{23}+\cdots$$
252.2.b.a $$2$$ $$2.012$$ $$\Q(\sqrt{-7})$$ $$\Q(\sqrt{-7})$$ None $$1$$ $$0$$ $$0$$ $$0$$ $$q+\beta q^{2}+(-2+\beta )q^{4}+(-1+2\beta )q^{7}+\cdots$$
252.2.b.b $$4$$ $$2.012$$ $$\Q(\sqrt{-2}, \sqrt{7})$$ $$\Q(\sqrt{-21})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-2q^{4}-\beta _{2}q^{5}+\beta _{3}q^{7}-2\beta _{1}q^{8}+\cdots$$
252.2.b.c $$4$$ $$2.012$$ $$\Q(i, \sqrt{7})$$ $$\Q(\sqrt{-7})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1+2\beta _{2}+\cdots)q^{7}+\cdots$$
252.2.b.d $$4$$ $$2.012$$ 4.0.2312.1 None None $$1$$ $$0$$ $$0$$ $$-2$$ $$q+\beta _{1}q^{2}+\beta _{2}q^{4}+(-\beta _{1}-\beta _{3})q^{5}+\cdots$$
252.2.b.e $$4$$ $$2.012$$ 4.0.2312.1 None None $$1$$ $$0$$ $$0$$ $$2$$ $$q+\beta _{1}q^{2}+\beta _{2}q^{4}+(\beta _{1}+\beta _{3})q^{5}+(1+\cdots)q^{7}+\cdots$$
252.2.e.a $$12$$ $$2.012$$ 12.0.$$\cdots$$.1 None None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(\beta _{7}-\beta _{9}-\beta _{10}+\cdots)q^{5}+\cdots$$
252.2.f.a $$4$$ $$2.012$$ $$\Q(\sqrt{-2}, \sqrt{3})$$ None None $$0$$ $$0$$ $$0$$ $$-4$$ $$q+\beta _{3}q^{5}+(-1-\beta _{1})q^{7}-\beta _{2}q^{11}+\cdots$$
252.2.i.a $$2$$ $$2.012$$ $$\Q(\sqrt{-3})$$ None None $$0$$ $$-3$$ $$-2$$ $$-5$$ $$q+(-1-\zeta_{6})q^{3}-2\zeta_{6}q^{5}+(-3+\zeta_{6})q^{7}+\cdots$$
252.2.i.b $$14$$ $$2.012$$ $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ None None $$0$$ $$3$$ $$-2$$ $$6$$ $$q+(-\beta _{1}+\beta _{3})q^{3}-\beta _{9}q^{5}+\beta _{10}q^{7}+\cdots$$
252.2.j.a $$6$$ $$2.012$$ 6.0.309123.1 None None $$0$$ $$-2$$ $$-1$$ $$3$$ $$q+(\beta _{2}+\beta _{4})q^{3}+(-\beta _{1}+\beta _{5})q^{5}+(1+\cdots)q^{7}+\cdots$$
252.2.j.b $$6$$ $$2.012$$ 6.0.309123.1 None None $$0$$ $$2$$ $$3$$ $$-3$$ $$q+(1+\beta _{3}+\beta _{4})q^{3}+(-\beta _{1}-\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots$$
252.2.k.a $$2$$ $$2.012$$ $$\Q(\sqrt{-3})$$ None None $$0$$ $$0$$ $$-2$$ $$1$$ $$q+(-2+2\zeta_{6})q^{5}+(-1+3\zeta_{6})q^{7}+\cdots$$
252.2.k.b $$2$$ $$2.012$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$5$$ $$q+(3-\zeta_{6})q^{7}+5q^{13}+(1-\zeta_{6})q^{19}+\cdots$$
252.2.k.c $$2$$ $$2.012$$ $$\Q(\sqrt{-3})$$ None None $$0$$ $$0$$ $$3$$ $$-4$$ $$q+(3-3\zeta_{6})q^{5}+(-1-2\zeta_{6})q^{7}-3\zeta_{6}q^{11}+\cdots$$
252.2.l.a $$2$$ $$2.012$$ $$\Q(\sqrt{-3})$$ None None $$0$$ $$0$$ $$4$$ $$4$$ $$q+(1-2\zeta_{6})q^{3}+2q^{5}+(1+2\zeta_{6})q^{7}+\cdots$$
252.2.l.b $$14$$ $$2.012$$ $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ None None $$0$$ $$0$$ $$4$$ $$-3$$ $$q+\beta _{1}q^{3}+(-\beta _{5}+\beta _{9})q^{5}+\beta _{6}q^{7}+\cdots$$
252.2.n.a $$4$$ $$2.012$$ $$\Q(\zeta_{12})$$ None None $$2$$ $$0$$ $$0$$ $$0$$ $$q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(-2\zeta_{12}+\cdots)q^{3}+\cdots$$
252.2.n.b $$84$$ $$2.012$$ None None $$-1$$ $$0$$ $$0$$ $$0$$
252.2.o.a $$88$$ $$2.012$$ None None $$-3$$ $$0$$ $$0$$ $$0$$
252.2.t.a $$4$$ $$2.012$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None None $$0$$ $$0$$ $$0$$ $$2$$ $$q-\beta _{2}q^{5}+(2-3\beta _{1})q^{7}+(-\beta _{2}-\beta _{3})q^{11}+\cdots$$
252.2.w.a $$16$$ $$2.012$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None None $$0$$ $$0$$ $$0$$ $$-1$$ $$q-\beta _{10}q^{3}+\beta _{7}q^{5}+(\beta _{3}-\beta _{8})q^{7}-\beta _{4}q^{9}+\cdots$$
252.2.x.a $$16$$ $$2.012$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None None $$0$$ $$0$$ $$0$$ $$-1$$ $$q+(-\beta _{1}-\beta _{8})q^{3}+\beta _{15}q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots$$
252.2.ba.a $$72$$ $$2.012$$ None None $$0$$ $$0$$ $$0$$ $$0$$
252.2.bb.a $$88$$ $$2.012$$ None None $$0$$ $$0$$ $$-6$$ $$0$$
252.2.be.a $$32$$ $$2.012$$ None None $$0$$ $$0$$ $$0$$ $$0$$
252.2.bf.a $$4$$ $$2.012$$ $$\Q(\sqrt{-3}, \sqrt{-7})$$ None None $$-1$$ $$0$$ $$6$$ $$0$$ $$q-\beta _{3}q^{2}+(1+\beta _{1}-\beta _{2})q^{4}+(2-\beta _{2}+\cdots)q^{5}+\cdots$$
252.2.bf.b $$4$$ $$2.012$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None None $$0$$ $$0$$ $$0$$ $$-2$$ $$q+\beta _{3}q^{2}-2q^{4}-2\beta _{1}q^{5}+(1-3\beta _{2}+\cdots)q^{7}+\cdots$$
252.2.bf.c $$4$$ $$2.012$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None None $$0$$ $$0$$ $$0$$ $$2$$ $$q+\beta _{1}q^{2}+2\beta _{2}q^{4}+(2\beta _{1}-2\beta _{3})q^{5}+\cdots$$
252.2.bf.d $$4$$ $$2.012$$ $$\Q(\sqrt{-3}, \sqrt{-7})$$ None None $$1$$ $$0$$ $$-6$$ $$0$$ $$q+\beta _{3}q^{2}+(1+\beta _{1}-\beta _{2})q^{4}+(-2+\beta _{2}+\cdots)q^{5}+\cdots$$
252.2.bf.e $$4$$ $$2.012$$ $$\Q(\zeta_{12})$$ None None $$2$$ $$0$$ $$6$$ $$0$$ $$q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+2\zeta_{12}q^{4}+\cdots$$
252.2.bf.f $$8$$ $$2.012$$ 8.0.562828176.1 None None $$-1$$ $$0$$ $$0$$ $$-2$$ $$q+\beta _{5}q^{2}+(-\beta _{4}+\beta _{6}-\beta _{7})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots$$
252.2.bf.g $$8$$ $$2.012$$ 8.0.562828176.1 None None $$-1$$ $$0$$ $$0$$ $$2$$ $$q+(-\beta _{1}-\beta _{5})q^{2}+(-\beta _{2}-\beta _{6}+\beta _{7})q^{4}+\cdots$$
252.2.bi.a $$4$$ $$2.012$$ $$\Q(\zeta_{12})$$ None None $$2$$ $$0$$ $$-6$$ $$0$$ $$q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(2\zeta_{12}+\cdots)q^{3}+\cdots$$
252.2.bi.b $$4$$ $$2.012$$ $$\Q(\zeta_{12})$$ None None $$2$$ $$0$$ $$6$$ $$0$$ $$q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(-2\zeta_{12}+\cdots)q^{3}+\cdots$$
252.2.bi.c $$80$$ $$2.012$$ None None $$-6$$ $$0$$ $$0$$ $$0$$
252.2.bj.a $$4$$ $$2.012$$ $$\Q(\zeta_{12})$$ None None $$-4$$ $$0$$ $$-12$$ $$0$$ $$q+(-1+\zeta_{12}^{3})q^{2}+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+\cdots$$
252.2.bj.b $$84$$ $$2.012$$ None None $$2$$ $$0$$ $$6$$ $$0$$
252.2.bm.a $$16$$ $$2.012$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None None $$0$$ $$0$$ $$0$$ $$-1$$ $$q+(-\beta _{5}-\beta _{10})q^{3}+(\beta _{7}-\beta _{11})q^{5}+\cdots$$
252.3.c.a $$4$$ $$6.867$$ $$\Q(\sqrt{-2}, \sqrt{7})$$ None None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{5}+\beta _{3}q^{7}+(2\beta _{1}+\beta _{2})q^{11}+\cdots$$
252.3.d.a $$2$$ $$6.867$$ $$\Q(\sqrt{-3})$$ None None $$0$$ $$0$$ $$0$$ $$-14$$ $$q-\zeta_{6}q^{5}-7q^{7}-18q^{11}+3\zeta_{6}q^{13}+\cdots$$
252.3.d.b $$2$$ $$6.867$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$-2$$ $$q+(-1-\zeta_{6})q^{7}-2\zeta_{6}q^{13}-4\zeta_{6}q^{19}+\cdots$$
252.3.d.c $$2$$ $$6.867$$ $$\Q(\sqrt{-6})$$ None None $$0$$ $$0$$ $$0$$ $$10$$ $$q+\beta q^{5}+(5-\beta )q^{7}+6q^{11}+\beta q^{13}+\cdots$$
252.3.g.a $$6$$ $$6.867$$ 6.0.1539727.2 None None $$1$$ $$0$$ $$4$$ $$0$$ $$q+\beta _{2}q^{2}+(-\beta _{1}-\beta _{4})q^{4}+(1-\beta _{2}+\cdots)q^{5}+\cdots$$
252.3.g.b $$12$$ $$6.867$$ 12.0.$$\cdots$$.1 None None $$-2$$ $$0$$ $$-8$$ $$0$$ $$q-\beta _{3}q^{2}-\beta _{5}q^{4}+(-\beta _{3}+\beta _{8})q^{5}+\cdots$$
252.3.g.c $$12$$ $$6.867$$ 12.0.$$\cdots$$.1 None None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{2}+(-1-\beta _{6})q^{4}+(-\beta _{2}+\beta _{7}+\cdots)q^{5}+\cdots$$
252.3.h.a $$8$$ $$6.867$$ 8.0.157351936.1 $$\Q(\sqrt{-7})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{7}q^{2}+(\beta _{3}+\beta _{6})q^{4}+7\beta _{3}q^{7}+(\beta _{2}+\cdots)q^{8}+\cdots$$


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# Synthesis of waveguide antenna arrays using the coupling matrix approach

Mahmud, Rashad Hassan (2016). Synthesis of waveguide antenna arrays using the coupling matrix approach. University of Birmingham. Ph.D.

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Mahmud16PhD.pdf
PDF - Accepted Version

## Abstract

With the rapid development in communication systems recently, improvements in components of the systems such as antennas and bandpass filters are continuously required to provide improved performance. High gain, wide bandwidth, and small size are the properties of antennas which are demanded in many modern applications, and achieving these simultaneously is a challenge. This thesis presents a new design approach to address this challenge.

The coupling matrix is an approach used to represent the circuits made of coupled resonators such as filters and multiplexers. The approach has been utilised here to integrate a single resonator-based antenna with an n$$^t$$$$^h$$ order filter. The integrated component is capable of providing a controllable bandwidth and introduces the filtering functionality. The approach is further developed in order to integrate bandpass filters with N×N resonator-based antenna arrays. This is to increase the gain of the array as well. Six novel components have been fabricated for the purpose of validation.

This thesis also looks at a 300 GHz communication system which is proposed at The University of Birmingham with the objective to build a 10 metre indoor communication link. A 300 GHz (8×8) waveguide antenna array has been designed and fabricated for the system.

Type of Work: Thesis (Doctorates > Ph.D.)
Award Type: Doctorates > Ph.D.
Supervisor(s):
Supervisor(s)EmailORCID
Lancaster, M.J.UNSPECIFIEDUNSPECIFIED
Huang, F.UNSPECIFIEDUNSPECIFIED
Licence:
College/Faculty: Colleges (2008 onwards) > College of Engineering & Physical Sciences
School or Department: School of Engineering, Department of Electronic, Electrical and Systems Engineering
Funders: None/not applicable
Subjects: T Technology > TK Electrical engineering. Electronics Nuclear engineering
URI: http://etheses.bham.ac.uk/id/eprint/6564

### Actions

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# Netzschleuder network catalogue, repository and centrifuge

Problems with this dataset? Open an issue.
You may also take a look at the source code.
The network in this dataset can be loaded directly from graph-tool with:
import graph_tool.all as gt
g = gt.collection.ns["sp_kenyan_households"]

# sp_kenyan_households — Kenyan households contacts (2012)

Description

A network of proximity contacts measured between members of 5 households of rural Kenya, between April 24 and May 12, 2012.1

Each file in the downloadable package contains a comma-separated list representing each measured contact between any two household members (member 1 and member 2) over three days of experiment. The first file stores the contacts recorded between members of the same household, the second file stores the contacts between members of different households.

Each line has the form: “h1, m1, h2, m2, age1, age2, sex1, sex2, duration, day, hour”, where:

h1 is the household of member 1; h1=[L, F, E, B, H] m1 is the anonymous ID number of member 1; h2 is the household of member 2; h2=[L, F, E, B, H] m2 is the anonymous ID number of member 2; age1 is the age of member 1; age1 = [0, 1, 2, 3, 4] age2 is the age of member 2; age2 = [0, 1, 2, 3, 4] sex1 is the gender of member 1; sex1 = [F, M] sex2 is the gender of member 2; sex2 = [F, M] duration is the duration of the contact event in seconds; day is the day of experiment; day = [1, 2, 3] hour is the day time of the contact event; hour = [7 – 20]

1. Description obtained from the ICON project.

Tags
Name Nodes Edges $\left<k\right>$ $\sigma_k$ $\lambda_h$ $\tau$ $r$ $c$ $\oslash$ $S$ Kind Mode NPs EPs gt GraphML GML csv


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# What's the probability of getting three matching faces in any one roll of four Fudge dice?

I'm having a little difficulty recalling the procedure for calculating probability of dice results. My question is, using 4 Fudge d6 dice, what is the probability of getting three of any result in one roll? (+++,---,ooo)

• Does "++++" count for this purpose as containing three of any result, or does it have to be exactly and only three of the same? There is an Atomic Robo stunt I wonder if you're referring to, which permits "+++0", but excludes "++++". – doppelgreener Oct 16 '14 at 0:12

There are 81 possible permutations of this roll. Of these, 3 have all 4 results the same, and 24 have 3 results the same. This gives us a probability of 27/81 or 33.33%, for at least 3 results the same, and 24/81 or 29.63% for exactly 3 results the same.

• Could you post a mathematical formula for the percentage chance? – Sandwich Oct 16 '14 at 3:54
• I know the percentage is correct, but for the entire computation, out of curiosity. Normally to determine the percentage of a certain digit coming up on a d6 you use (1/6^n) for the number of dice, Since Fudge dice are technically d3's, You'd use (1/3^n), but that only covers the odds of getting a single result, and it doesn't compute for an additional die, I'd just like to know, for future reference. – Sandwich Oct 16 '14 at 4:07
• "Where did you get the 81 integer?", is what I'm trying to ask. – Sandwich Oct 16 '14 at 4:35
• Ah nevermind, I figured it out, 3 different branches, one for +, One for -, One for o, each containing 9 different results. 9*3*3. 81. I'm dumb. – Sandwich Oct 16 '14 at 4:40
• You have 4 dice, each has 3 possible outcomes (evenly distributed I assume). That makes 3^4 different permutations, or 3*3*3*3 = 81. Note that the result +000 and 0+00 are different permutations, although they show the same totals of pluses, dashes and zeroes. – fgysin Oct 16 '14 at 6:49

## The odds of getting 3 of the same result out of four are 33%.

For each set of Four Fudge dice there are a total of 27 different roll results with 3 alternate outcomes per set of rolls, +, -, and o. For a total of 81 possible results. Those outcomes are below:

++++ ++-+ ++o+
+++- ++-- ++o-
+++o ++-o ++oo

+-++ +--+ +-o+
+-+- +--- +-o-
+-+o +--o +-oo

+o++ +o-+ +oo-
+o+- +o-o +oo+
+o+o +o-- +ooo

This is the set of outcomes for only the "+" set of rolls, There are sets for the "-" and the "o" as well. As you can see, 33% of the outcomes above have three or more of the same result. This includes the rolls beginning in "-" and "o" as well.

• Since there's an Atomic Robo RPG stunt which asks for exactly three, and the question doesn't use "at least" phrasing, I'm not sure this is complete without the probability of a result of exactly three (and without it, this answer appears functionally identical to WaxEagle's). But then, that's why I voted to close the question as unclear. – BESW Oct 16 '14 at 4:53
• Awww. But mine has cool formatting. – Sandwich Oct 16 '14 at 4:55

The probability of 3 of any one result is 33.33%. Here's why:

The basic probability of getting at least 3 of any one outcome of fudge dice is 11.11%. This is 1/9. The fact that you have 3 possibile results from a set where the outcome of any 3 dice is 11% means that the overall probability is 33.33%.

Showing my work. This program utilizes the fact that Fate dice are essentially d3s, and takes into account the fact that you're rolling 4 dice. The "at Least" option from the following command:

 output [count 1 in 4d3]

Gives us the output of 11.11% to show the odds of getting at least 3 of any one number. Since these 3 outcomes are equally likely, and are independent. The probabilities add allowing us to say that on 33.33% of rolls at least 3 dice will show the same face.

With a small number of possibilities like this it is easiest to simply count the number of desirable outcomes in the possible results and use those. However, there are mathematical formulas that allow us to generalize this to any number of results, across any number of rolls with any sized die.

The two numbers that matter here are the number of results that we care about (in this case 9) and the number of total results (in this case 81). The number of total results is easy to derive. That's simply the number of results in the dice (3) raised to the power of the number of times you're rolling in this case 4. (3^4 = 81).

The simple thing for determining the permutations is to think about the number of possible dice rolls that would come up on our target value if we rolled 3d3. 3d3 show the 3 of the same face a single time. So if we use 4d3, we get 3 for the fourth die, and then we also get 2 for each other die since the other 3 dice can show the same face. This gives us 2+2+2+3 = 9 permutations. This obviously doesn't scale and the math for how to scale this is currently escaping me (it's basically the same math I'm showing here, but reducing it to a mathematical form is beyond me right now)

So that means that our formula for three of any 1 die is

 (2+2+2+3)/81 = 9/81 = 11.11%

Multiplying this by 3 gives us 33.33%.

The other thing we need to pay attention to (and since it's not clear from the question which is desired), is that sometimes we may want to know the probability of only getting 3 of a kind. In this case we can take our previous form:

 27/81 (that's 3x9/81)

and subtract 3 permutations from it (one each for ----, ++++ and oooo) Giving us

 24/81 = 29.62%


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# Math Help - A log problem

1. ## A log problem

Can someone please show me step by step how to solve down on this log problem...

8^-2x+4 = 64
log9x = 1/2
log3 (x-1) + log3 (x+1) = 1

Thank you very much.

2. ## Re: A log problem

I assume the first problem is really 8^(-2x+4)=64

take log8 of both sides

log8(8-2x+4)=log8(64)

(-2x+4)log8(8)=2log8(8)

(-2x+4) = 2

x = 1

second problem

I would first rewrite log9x=1/2 as (ln(x))/(ln(9))=1/2

then

ln(x) = (1/2) ln(9)

ln(x) = ln(91/2)

x = 91/2=3

third

first rewrite as

(ln(x-1) + ln(x+1))/(ln3)=1

ln(x-1)+ln(x+1) = ln3
ln(x2-1)=ln3
x2-1=3
x2=4
x=2

x can't be -2 because it would cause you to take the log of a negative number

ln=natural log

3. ## Re: A log problem

Originally Posted by bmoon123
second problem

I would first rewrite log9x=1/2 as (ln(x))/(ln(9))=1/2

then

ln(x) = (1/2) ln(9)

ln(x) = ln(91/2)

x = 91/2=3

third

first rewrite as

(ln(x-1) + ln(x+1))/(ln3)=1

ln(x-1)+ln(x+1) = ln3
ln(x2-1)=ln3
x2-1=3
x2=4
x=2

x can't be -2 because it would cause you to take the log of a negative number

ln=natural log
It's easier to just go

\displaystyle \begin{align*} \log_9{x} &= \frac{1}{2} \\ x &= 9^{\frac{1}{2}} \\ x &= 3 \end{align*}

and

\displaystyle \begin{align*} \log_3{(x - 1)} + \log_3{(x + 1)} &= 1 \\ \log_3{(x - 1)(x + 1)} &= 1 \\ (x - 1)(x + 1) &= 3^1 \\ x^2 - 1 &= 3 \\ x^2 &= 4 \\ x &= \pm 2 \\ x &= 2 \textrm{ due to the fact you can only take the logarithm of a positive number} \end{align*}


text:
# Multiple Gravity Assists Transfer

In this section, the preliminary design of multiple-leg interplanetary transfer trajectories is discussed. This module provides the functionalities for creating transfer trajectories consisting of multiple transfer legs or various types with powered and unpowered gravity assists. This allows high-thrust or low-thrust transfer trajectories with multiple flybys to be designed, as well as a hybrid of low- and high-thrust. For per-function details see the API documentation.

A multiple gravity-assist transfer (MGA) is constituted by a series of nodes and legs. The nodes correspond to the departure, gravity assist, and arrival planets (bodies), and the legs correspond to the trajectories between the nodes. Therefore, there is always one more node than leg. Note that the initial or final node may be departure/capture or a flyby. The legs may be of a number of different types, which is discussed more in-depth later on.

The module is implemented using simplified dynamical models for the nodes and legs. For the unpowered leg trajectory, for example, the transfer is for instance defined under the assumptions of the patched-conics approximation. This module allows for the flexible design of transfers using only analytical and semi-analytical methods, typically without any required numerical integration. This is particularly useful for preliminary mission design, where having a fast method is particularly important.

Note

The MGA model also allows defining transfers with a single leg (without any gravity assist).

## General Model Description

Before evaluating any transfer, it is useful to introduce the concept of nodes and legs more. To assist in this, a schematic representation is given in the figure below. An MGA trajectory is given, with an arbitrary sequence. A number of nodes can be seen that represent the celestial body used as GA body, as well as a number of legs that connect the nodes together. A central body is given, as this is required for the heliocentric evaluation of the legs, but more on that later. A number of different nodes are used and annotated; these are explained below under ‘Nodes and Their parameters’.

It is crucial to understand that both the nodes and legs have an incoming and outgoing velocity vector and that these are determined in different frames. The key difference being that the incoming/outgoing velocities of the legs are evaluated in a heliocentric frame – assuming the Sun is the central body – and the incoming/outoing velocities of the nodes are evaluated in a planetocentric frame – assuming a planet is the GA target. The velocity vectors are converted in to the respective frame to evaluate the unknown parameters. Which parameters are unknown depends on the type of leg and node, which is explained later. For more details on the difference in reference frame and the patching of these trajectories, see section 4.4.2/3 Musegaas (2012)_.

## Supported models

At present, the types of legs are supported (more details can be found below):

• Unpowered legs: A purely ballistic (Keplerian) trajectory between nodes

• Velocity-based deep-space maneuver (DSM) legs: A ballistic trajectory betweene nodes, with the addition of a single impulsive maneuver during the leg (parameterized by its velocity)

• Position-based DSM legs: A ballistic trajectory betweene nodes, with the addition of a single impulsive maneuver during the leg (parameterized by its position)

• Spherical-shaping legs: A shape-based low-thrust trajectory using the spherical shaping method

• Hodographic-shaping legs: A shape-based low-thrust trajectory using the hodographic shaping method

At present, the types of nodes are supported (more details can be found below):

• Departure node: Only available for the first node, to incorporate $$\Delta V$$ required to escape from a parking orbit around the first node

• Swingby node: A node assuming a hyperbolic trajectory w.r.t. the node, with the possibility of performing an impulsive maneuver at periapsis (see Nodes and Their Parameters).

• Arrival node: Only available for the final node, to incorporate $$\Delta V$$ required to enter a closed orbit around the final node

Each leg and node has its own free parameters, which must be provided by the user to evaluate the performance of the overall trajectory (see below).

## General Procedure

To create a transfer trajectory, the user must define settings for the nodes and legs, after which these settings are processed to create the transfer trajectory.

First, the transfer trajectory module can be imported with:

from tudatpy.kernel.trajectory_design import transfer_trajectory

The most commonly-used for procedure for creating an settings of the trajectory is to use factory functions to get the transfer leg has the same type (e.g. all unpowered, all spherical-shaping, etc.). The factory functions to create a set of node and leg settings is:

Manually creating settings for single legs and nodes is described below.

The complete procedure for creating and analyzing an MGA transfer consists of the following. The associated code snippets are taken from an this example application, for an unpowered leg Cassini (EVVEJS) transfer trajectory:

• Define the transfer settings: The transfer leg settings and node settings a are created. These are defined using the body order (bodies through which the spacecraft will pass), the departure and arrival orbit (semi-major axis and eccentricity) and other settings specific to each leg type. Selecting the semi-major axis of the departure/arrival orbit as $$a = \infty$$ corresponds to having the spacecraft depart/arrive from/to the edge of the initial/final body’s sphere of influence (e.g. with zero hyperbolic excess velocity).

# Define central body
central_body = 'Sun'

# Define the order of bodies (nodes) for gravity assists
transfer_body_order = ['Earth', 'Venus', 'Venus', 'Earth',  'Jupiter',  'Saturn']

# Define the departure and insertion orbits
departure_semi_major_axis = np.inf
departure_eccentricity = 0.

arrival_semi_major_axis = 1.0895e8 / 0.02
arrival_eccentricity = 0.98

# Define the trajectory settings for both the legs and at the nodes
transfer_leg_settings, transfer_node_settings = transfer_trajectory.mga_settings_unpowered_unperturbed_legs(
transfer_body_order,
departure_orbit=(departure_semi_major_axis, departure_eccentricity),
arrival_orbit=(arrival_semi_major_axis, arrival_eccentricity))

# Create physical environment
bodies = ...

# Create the transfer calculation object
transfer_trajectory_object = transfer_trajectory.create_transfer_trajectory(
bodies,
transfer_leg_settings,
transfer_node_settings,
transfer_body_order,
central_body)

• Evaluate the transfer: Select the node times, node parameters, and leg parameters, and use them to evaluate the transfer through evaluate(). These parameters are described in the following sections. Note that, in the case of an optimization, this function is called repeatedly to evaluate the transfer trajectory with differeent properties.

# Define free parameters
node_times = ...
leg_free_parameters = ... # (empty)
node_free_parameters = ... # (empty)

# Evaluate the transfer with given parameters
transfer_trajectory_object.evaluate( node_times, leg_free_parameters, node_free_parameters )

• Retrieve the results: Use TransferTrajectory’s properties or functions to retrieve the $$\Delta V$$, time of flight, state history, acceleration history, etc.

# Retrieve total Delta V
total_delta_v = transfer_trajectory_object.delta_v

All available functions and classes are described in detail in the relevant entry of the API reference. For applications see this example and this PyGMO example.

### Manually Creating the Transfer Settings

While in many casses the transfer settings can be created using the factory functions listed in the previous section, there are some cases where the manual creation of these should be preferred. These include transfers with mixed types of legs. The creation of the transfer settings can be divided into two steps: creation of the legs settings and creation of the nodes settings.

The legs settings are a list with the settings of each leg constituting the transfer. The settings of an individual leg can be retrieved using the appropriate factory function:

The nodes settings are a list with the settings of each node constituting the transfer. There are three main types of nodes: departure, swingby, and arrival nodes. The initial node is usually a departure node (although it can also be a swingby node), the intermediate nodes are always swingby nodes (at the time of writing), and the final is usually an arrival node (although it can also be a swingby node). Selecting the initial and final nodes as swingby nodes is useful when individually analyzing different parts of a transfer or when a mission’s objective is to do a swingby of the final body.

The settings of an individual node can be retrieved using the appropriate factory function:

Note

The nodes and legs settings should have the same number of elements as the number of, respectively, nodes and legs in the transfer.

An example of manual creation of nodes and legs settings can be found below. The transfer body order, time of flight values per leg, and number of revolutions per leg are given as input. The definition of the departure and arrival input is necessary as well for the departure and capture nodes, respectively. Subsequently, as shaping functions have to be given for each leg, a loop is performed over the number of legs that add the recommended shaping functions as given by [Gondelach, 2012]. Finally, a node for each planet in the transfer body order must be given: A departure node, swingby nodes equal to the number of GA’s, and a capture node.

Required Show/Hide

from tudatpy.kernel.numerical_simulation import propagation_setup
from tudatpy.kernel.numerical_simulation import environment_setup
from tudatpy.kernel.trajectory_design import shape_based_thrust
from tudatpy.kernel.trajectory_design import transfer_trajectory
import numpy as np

# Define the order of bodies (nodes) for gravity assists
transfer_body_order = ['Earth', 'Venus', 'Venus', 'Earth',  'Jupiter',  'Saturn']

# define ToF values per leg
time_of_flight = np.array([1.97844702e+07, 6.68197546e+07, 1.00154317e+07, 3.2306349e+8, 4.21392438e+8]) #s

# define number of revolutions per leg
number_of_revolutions = np.array([2, 0, 1, 1, 0])

# Define the departure and insertion orbits
departure_semi_major_axis = np.inf
departure_eccentricity = 0.
arrival_semi_major_axis = 1.0895e8 / 0.02
arrival_eccentricity = 0.98

# Determine number of legs and GA's
no_of_legs = len(transfer_body_order) - 1
no_of_gas = len(transfer_body_order) - 2

#Create transfer leg settings
transfer_leg_settings = []
for i in range(no_of_legs):
normal_velocity_functions = shape_based_thrust.recommended_normal_hodograph_functions(time_of_flight)
axial_velocity_functions = shape_based_thrust.recommended_axial_hodograph_functions(time_of_flight,
number_of_revolutions)
transfer_leg_settings.append(transfer_trajectory.hodographic_shaping_leg(

#Create transfer node settings
transfer_node_settings = []
transfer_node_settings.append( transfer_trajectory.departure_node(departure_semi_major_axis, departure_eccentricity) )
for i in range(no_of_gas):
transfer_node_settings.append( transfer_trajectory.swingby_node() )
transfer_node_settings.append( transfer_trajectory.capture_node(arrival_semi_major_axis, arrival_eccentricity) )

Having created the legs and nodes settings, the same procedure described above for creating the transfer trajectory object, evaluating it, and retrieving the computed data can be followed.

To evaluate the transfer one needs to provide a list of transfer parameters. These are:

• Node times

• Leg parameters

• Node parameters

It is possible to a see a list of the parameters required for a given transfer via the print_parameter_definitions()` function.

The node times always need to be specified (regardless of the leg and node times), and correspond to the epoch when the spacecraft reaches each planet/body. The node and leg parameters, which depend on the specific node and leg type, are described next. Note that, for certain leg and node types, there are no free parameters and the trajectory is fully defined by the node times (as is the case for an a series of unpowered legs).

### Legs and Their Parameters

The parameters associated with each type of leg are listed below. For more detailed descriptions, the reader is referred to the literature listed in the API reference for the factory function of each leg type.

• Unpowered legs: Does not have leg parameters.

• Velocity-based DSM legs: 1 leg parameter:

• Fraction of the leg’s time-of-flight at which DSM is applied ($$\in [0,1]$$).

• Position-based DSM legs: 4 leg parameters:

• Fraction of the leg’s time-of-flight at which DSM is applied ($$\in [0,1]$$).

• Position of the DSM in spherical coordinates, with respect to a frame with the x-axis aligned with the position of the departure body, z-axis aligned with the angular momentum of the departure body, y-axis selected to form a right-handed frame. The spherical position is specified as: dimensionless radial position (using as unit of length the radial position of the departure body), in-plane angle, out-of-plane angle.

Note

A leg with position-based DSMs requires more leg parameters than one with velocity-based DSMs. However, a transfer constituted by the former will require less node parameters than one constituted by the latter.

• Spherical-shaping legs: 1 leg parameter:

• Integer number of revolution ($$\geq 0$$).

• Hodographic-shaping legs: at least 1 leg parameter:

• Integer number of revolution ($$\geq 0$$).

• Free coefficients of shaping functions (number of coefficients is greater or equal to zero).

Finally, and before moving on to the description of the node parameters, it is important to analyze the boundary conditions used for the evaluation of each leg of the transfer. These conditions are dealt with internally by the model; hence, the user never has direct contact with them. Nevertheless, understanding them is essential for describing the node parameters.

Four types of boundary conditions are possible: initial position, initial velocity, final position, and final velocity. A given leg will receive some of these as input and produce the others as output. The inputs and outputs associated with each leg type are summarized in the following table. For example, a velocity-based DSM leg receives as input its initial position, initial velocity, and final position, and outputs its final velocity.

Leg initial position

Leg initial velocity

Leg final position

Leg final velocity

Unpowered

Input

Output

Input

Output

DSM-Velocity

Input

Input

Input

Output

DSM-Position

Input

Output

Input

Output

Spherical shaping

Input

Input

Input

Input

Hodographic shaping

Input

Input

Input

Input

### Nodes and Their Parameters

As already mentioned, there are three main types of nodes: departure, swingby, and arrival nodes. These are the only types with which the user has direct contact. However, internally, these are further divided into subtypes. While a user does not need to understand these subtypes for using the MGA model, the user does need to if he/she wishes to understand what a given node is doing behind the scenes and what node parameters are required for a given transfer.

The various node subtypes are described next. The node subtype used in a given instance depends on the types of the legs that precede and follow it; more specifically, it depends on the inputs and outputs of those legs. Using the previous table specifying the inputs and output of each leg and the following description of each node subtype, the user will be able to predict exactly which nodes are used for a given set transfer, as well as the required node parameters.

Note that the incoming velocity of a node corresponds to the final velocity of the previous leg and the outgoing velocity of a node corresponds to the initial velocity of the following leg.

• Departure node: Usually, this is the initial node of the transfer. The outgoing relative velocity at the node (i.e. the excess velocity) is either retrieved from the following leg or specified by the user. The node computes the impulsive $$\Delta V$$ that needs to be applied at the periapsis of the departure elliptic orbit to enter a hyperbolic orbit with the target excess velocity, according to section 4.6 of Musegaas (2012). This node is subdivided into the following types:

• Node with leg-defined outgoing velocity: Does not require node parameters.

• Node with user-defined outgoing velocity: Node parameters:

• Outgoing velocity vector relative to the node, specified with respect to a TNW reference frame defined using the node’s inertial state. The outgoing relative velocity is specified in spherical coordinates: norm of the velocity, in-plane angle ($$\in [0, 2\pi]$$), out-of-plane angle ($$\in [-\pi/2, \pi/2]$$).

• Arrival node: Usually, this is the final node of the transfer. The incoming relative velocity at the node (i.e. the excess velocity) is either retrieved from the previous leg or specified by the user. The node computes the impulsive $$\Delta V$$ that needs to be applied at the periapsis of the hyperbolic orbit with the specified excess velocity to enter the arrival elliptical orbit, according to section 4.6 of Musegaas (2012). This node is subdivided into the following types:

• Node with leg-defined incoming velocity: Does not require node parameters.

• Node with user-defined incoming velocity: Node parameters:

• Incoming velocity vector relative to the node, specified with respect to a TNW reference frame defined using the node’s inertial state. The incoming relative velocity is specified in spherical coordinates: norm of the velocity, in-plane angle ($$\in [0, 2\pi]$$), out-of-plane angle ($$\in [-\pi/2, \pi/2]$$).

• Swingby node: Usually, these are the intermediate nodes of the transfer. The swingby node is subdivided into the following types:

• Node with legs-defined incoming and outgoing velocity: Does not require node parameters. Computes the $$\Delta V \geq 0$$ that needs to be applied during the swingby to patch the incoming and outgoing velocities, according to section 4.5.2 of Musegaas (2012).

• Node with leg-defined incoming velocity, user-defined swingby: Given the known incoming velocity, the node forward propagates the swingby using the user-specified parameters, according to sections 4.4.2/3 of Musegaas (2012). Node parameters:

• Swingby $$\Delta V$$ (applied at the periapsis)

• Outgoing-velocity rotation angle. Defined according to Appendix 7a of “Spacecraft Trajectory Optimization”, Conway (2010). This angle defines the plane in which the swingby occurs (different from the bending angle, which is defined inside that plane). This angle takes values in $$[0, 2\pi]$$.

• Node with user-defined swingby, leg-defined outgoing velocity: Given the known outgoing velocity, the node backward propagates the swingby using the user-specified parameters. Analogous to sections 4.4.2/3 of Musegaas (2012). Node parameters:

• Swingby $$\Delta V$$ (applied at the periapsis)

• Incoming-velocity rotation angle. Defined analogously to the outgoing-velocity rotation angle, which in turn is defined according to Appendix 7a of “Spacecraft Trajectory Optimization”, Conway (2010). This angle defines the plane in which the swingby occurs (different from the bending angle, which is defined inside that plane). This angle takes values in $$[0, 2\pi]$$.

• Node with user-defined incoming, user-defined swingby: Given the known incoming velocity, the node forward propagates the swingby using the user-specified parameters, according to sections 4.4.2/3 of Musegaas (2012). Node parameters:

• Incoming velocity vector relative to the node, specified with respect to a TNW reference frame defined using the node’s inertial state. The incoming relative velocity is specified in spherical coordinates: norm of the velocity, in-plane angle ($$\in [0, 2\pi]$$), out-of-plane angle ($$\in [-\pi/2, \pi/2]$$).

• Swingby $$\Delta V$$ (applied at the periapsis)
• Outgoing-velocity rotation angle. Defined according to Appendix 7a of “Spacecraft Trajectory Optimization”, Conway (2010). This angle defines the plane in which the swingby occurs (different from the bending angle, which is defined inside that plane). This angle takes values in $$[0, 2\pi]$$.


text:
# A particle goes from point A to point B in 1.0 seconds moving in a semicircle of radius 1.0. What is the magnitude of the average velocity?

By, definition, the average velocity on the time interval `Delta t ` is the displacement vector divided by this time interval:

`vec v_(avg) = (Delta vecr)/(Delta t) = (vecr_f - vec r_i)/(Deltat)`

Note that average velocity and the displacement are vector quantities. The displacement is the vector difference between the final and initial position vectors.

As the particles moves from point A to point B in a semicircle, the displacement of the particle is a diameter of the semicircle (please see the attached image for illustration.)

The magnitude of the average velocity is then the length of the diameter of the semicircle divided by the time interval:

`|vec v_(avg)| = (|Delta vecr|)/(Deltat)` , `|Delta vecr| = d` , the length of the diameter of the semicircle. Since the radius of the semicircle is 1, the diameter is double the radius, or 2.

The magnitude of the average velocity is

`|vec v_(avg)| = 2/1 = 2` (units per second, the units in which the radius is measured is not given here.)

The magnitude of the average velocity is 2.

`<br>`

` <br>`

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text:
# Thread: [GRAPH] What are the domain and range of f -1? (domain) (range)

1. ## [GRAPH] What are the domain and range of f -1? (domain) (range)

What are the domain and range of f ^ -1? (It's an inverse right?)

I got
Domain : (-INFINITY, INFINITY)
Range : (-INFINITY, INFINITY)
Is this correct?

2. The domain of $f$ = the range of $f^{-1}$

The range of $f$ = the domain of $f^{-1}$

Your answer could be correct. What is $f$ ?

3. Originally Posted by pickslides
The domain of $f$ = the range of $f^{-1}$

The range of $f$ = the domain of $f^{-1}$

Your answer could be correct. What is $f$ ?
lol i know what, but i want to know what is my boo boo here can u tell? so i can get the answer right? thanks!

4. Originally Posted by crystaltear

What are the domain and range of f ^ -1? (It's an inverse right?)

I got
Domain : (-INFINITY, INFINITY)
Range : (-INFINITY, INFINITY)
Is this correct?
if f is restricted to the values on the graph, then the domain of f is [-3,3] and the range of f would be [-1,3].

... if the function continues outside the visible graph, then there is no way to tell w/o more information about f.

5. Originally Posted by skeeter
if f is restricted to the values on the graph, then the domain of f is [-3,3] and the range of f would be [-1,3].

... if the function continues outside the visible graph, then there is no way to tell w/o more information about f.
thanks i got the first part right but [_,3] domain and [_,3] I didn't get the x for both of them right...

6. Originally Posted by crystaltear
thanks i got the first part right but [_,3] domain and [_,3] I didn't get the x for both of them right...


text:
# If point M is the center of a sphere with the coordinates (-1,3,2) and point T is a point on

###### Question:

If point M is the center of a sphere with the coordinates (-1,3,2) and point T is a point on the surface of the sphere with the coordinates (0,5,1), what is the radius of the sphere?

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# Using Goto and Label in compiled function?

Below I post some rudimentary code to show that there is something wrong using Goto and Label commands in compiled functions (or I believe there is something wrong). It seems that the command Label is not found. Does anybody know what's going on?

toy=Compile[{{x,_Real}},
Label["t"];
y=RandomReal[{-1.,1.}];
If[x*y<-0.,Print[y*x];Goto["t"],Print[y*x," ","done"]]
];

toy[1.]

-

This one is weird, and by my understanding of things, it is a bug. Label and Goto are on the list of compilable functions, and in fact, if you make your function completely compilable (i.e. compilable to code that does not call MainEvaluate), there is no problem. For example, this works without a hitch:

toy = Compile[{{x, _Real}},
c = 0;
Label[t];
c++;
If[x*RandomReal[{-1., 1.}] < -0., Goto[t]];
c]

In your original function, however, the print statements prevented the Goto to be compiled, and it stayed inside a call to MainEvaluate. CompilePrint says your function is compiled into:

2   V17 = MainEvaluate[ Function[{x}, y = RandomReal[{-1., 1.}]][ R0]]
3   R3 = MainEvaluate[ Function[{x}, y][ R0]]
4   R4 = R0 * R3
5   B0 = R4 < R5 (tol R6)
6   R4 = MainEvaluate[ Function[{x}, If[x y < 0., Print[y x]; Goto[t], Print[y x,  , done]]][ R0]]

You see the Goto is still there, inside the MainEvaluate call, but the Label has been compiled away: thus, the warning that the label cannot be found.

If you remove the print statement near the Goto, the If is compiled and MainEvaluate is called only on its second branch:

toy3 = Compile[{{x, _Real}},
Label["t"];
y = RandomReal[{-1., 1.}];
If[x*y < -0., Goto["t"], Print[y*x, " ", "done"]]];

gives

2   V17 = MainEvaluate[ Function[{x}, y = RandomReal[{-1., 1.}]][ R0]]
3   R3 = MainEvaluate[ Function[{x}, y][ R0]]
4   R4 = R0 * R3
5   B0 = R4 < R5 (tol R6)
6   if[ !B0] goto 9
7   goto 2
8   goto 10
9   V17 = MainEvaluate[ Function[{x}, Print[y x,  , done]][ R0]]

which works fine too, because now both the Label and the Goto have been compiled.

The reason I call it a bug is that I don't see this limitation specified anywhere in the documentation. I suggest reporting it to Wolfram support.

-
The label is still there in both cases. The problem is that labels aren't shared between the VM and the main evaluator (which makes sense, as you can't jump into compiled code at an arbitrary point), so Goto[t] does not find its target. That the compiler can be tricked into divorcing a Goto from its corresponding Label probably is a bug, I guess stemming from an attempt to minimize the number of (expensive) calls back into the main evaluator. –  Oleksandr R. Sep 12 '12 at 10:48
I should mention as well that the performance of compiled code like this, peppered as it is with calls out of the VM, will be absolutely horrific. For example, a single Ordering call per iteration in an early version of this consumed about 80% of the runtime--and that's with a fairly substantial loop body. The practical impact of this bug, then, will probably be minimal in most cases (as performance improvement is usually the main reason why one would use Compile in the first place). –  Oleksandr R. Sep 12 '12 at 11:18

There is nothing wrong with Label. The problem is that print can not be compiled. (see the list of compilable functions). Also y must be localized.

So compare

Needs["CompiledFunctionTools"]
CompilePrint[toy]

which returns

        1 argument
2 Boolean registers
9 Real registers
Underflow checking off
Overflow checking off
Integer overflow checking on
RuntimeAttributes -> {}

R0 = A1
R2 = 1.
R6 = 7.
R1 = -1.
R7 = -0.6382060350723182
R5 = 0.
Result = R4

1
2   V17 = MainEvaluate[ Function[{x}, y = RandomReal[{-1., 1.}]][ R0]]
3   R3 = MainEvaluate[ Function[{x}, y][ R0]]
4   R4 = R0 * R3
5   B0 = R4 < R5 (tol R6)
6   R4 = MainEvaluate[ Function[{x}, If[x y < 0., Print[y x]; Goto[t], \
Print[y x,  , done]]][ R0]]
7   Return

(Notice all the call to MainEvaluate)

Now compare to this:

toy1 = Compile[{{x, _Real}}, Block[{y},
Label["t"];
y = RandomReal[{-1., 1.}];
If[x*y < -0., Goto["t"]];
x*y
]]

Needs["CompiledFunctionTools"]
CompilePrint[toy1]

1 argument
1 Boolean register
7 Real registers
Underflow checking off
Overflow checking off
Integer overflow checking on
RuntimeAttributes -> {}

R0 = A1
R2 = 1.
R6 = 7.
R1 = -1.
R5 = 0.
Result = R4

1
2   R3 = RandomReal[ R1, R2]]
3   R4 = R0 * R3
4   B0 = R4 < R5 (tol R6)
5   if[ !B0] goto 8
6   goto 2
7   goto 8
8   R4 = R0 * R3
9   Return

which returns a nicely compiled function without problems.

-
The Print goes to MainEvaluate but other functions are compiled. Compile[{{x, _Real}}, Block[{y}, Label["t"]; y = RandomReal[{-1., 1.}]; Print[x*y]; If[x*y < -0., Goto["t"]]; Print[y*x, " ", "done"]; x*y]] –  PlatoManiac Sep 12 '12 at 10:01
Does CompilationTarget -> "C" works for this example? For me it does not. –  PlatoManiac Sep 12 '12 at 10:05
@PlatoManiac looks like someone forgot to specify the C translation for the label opcode! –  Oleksandr R. Sep 12 '12 at 10:38
Thanks guys! Very helpful. –  Chris Sep 12 '12 at 11:24
@PlatoManiac Doesn't work for me either. Will anybody post a bug report to Wolfram? –  Ajasja Sep 12 '12 at 12:24