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Welcome to ZOJ
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ZOJ Problem Set - 2607
Laboratory

Time Limit: 10 Seconds      Memory Limit: 32768 KB      Special Judge

John is working in a secret laboratory. Now he is preparing a special experiment involving four dangerous liquid substances with code names 'A', 'B', 'C', and 'D'.

A special container is used to keep the substances. The container is made of a special strong glass. It consists of four sections: 'A', 'B', 'C', and 'D' as shown on the picture.

Each substance is to be kept in its own part, marked with the letter that coincides with the code name of the substance. Since the container must be transported using standard carts, the height of the container is fixed and must be equal to one inch.

John knows, that he needs a, b, c, and d cubic inches of substances 'A', 'B', 'C', and 'D' respectively. Now he wonders what can be the minimal possible total square of the container so that the required volume of each substance would fit its section. Help him to find it out.

Input

Each test case contains four positive integral numbers a, b, c, and d, not exceeding 106. Proceed until the end of file.

Output

For each case, output the answers in two consecutive lines. On the first line print S - the minimal possible square of the container. On the next line print four real numbers p1, p2, q1, and q2 - the dimensions of the container sections as shown on the picture. Print at least six digits after the decimal point. Do not print any blank lines.

Sample Input

5 6 8 7
2 1 1 1

Sample Output

27.8571428571
1.6641005886 1.9414506867 4.1206300292 3.6055512754
5.8284271247
1.41421356237309505000 1.00000000000000000000 1.41421356237309505000 1.00000000000000000000

Author: Andrew Stankevich
Source: Andrew Stankevich's Contest #7
Submit    Status

format: plain


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# Homotopy

A homotopy of a coffee cup into a donut (torus).

Homotopies are studied in a branch of mathematics known as Algebraic Topology. A homotopy is a deformation (change of form) of one thing into another without cutting it. For example, if we imagine a stretchy object, then all the shapes we can stretch or twist it into are "homotopy equivalent". One famous example is the homotopy equivalence of a coffee cup and a donut. To a topologist, these two shapes are the same.

Homotopy between two functions in the plane.

The first example of a homotopy that a topology student is introduced to is a homotopy between functions. A homotopy between two functions ''f'' and ''g'' provides a collection of functions that start at ''f'' and eventually become ''g''. What this means is that this homotopy takes a parameter and gives a function, and there is a possible parameter that gives ''f'' and one that gives ''g''. Typically, we choose the interval ''[0, 1]'' because this has a starting and ending point, and numbers are allowed to vary smoothly within ''[0, 1]''. To output a function, this means that the homotopy takes a second parameter, which is the argument of the output function. That is, if ''f'' takes a number in the interval and outputs pairs of real numbers, the homotopy takes a number in the interval as the parameter and another number in the interval for ''f''. The homotopy then outputs the interpolated function evaluated at the given pair. That is, if ''H'' is a homotopy from ''f'' to ''g'', then we know ${\displaystyle H:[0,1]\times [0,1]\to \mathbb {R} ^{2}}$where ${\displaystyle H(p,0)=f(p)}$and ${\displaystyle H(p,1)=g(p)}$.


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# How many tosses are necessary that $n$ players produce pairwise different numbers of tosses with the result “heads”?

$n$ players toss a fair coin. The number of tosses with the result "heads" is recognized for each player.

The game stops if the numbers are pairwise different.

Let $X$ be the number of tosses for each player until the game is over.

• What is $E(X)$ ?

Simulation gives the following table

      n               E(X)        Var(X)       number of games

2                 2             6           100 000 000
3                 5.22         38.277       100 000 000
4                10.262       143.219       100 000 000
5                17.568       407.487        10 000 000
6                27.480       971.944        10 000 000
7                40.338      2051.034         1 000 000
8                56.529      3960.319         1 000 000
9                76.313      7110.368         1 000 000
10               100.355     12154.328         1 000 000
11               128.885     19864.376           100 000
12               162.443     31217.448           100 000

• For $n=2$, the answer is $E(X)=2$ because the game stops as soon as the players produce a different result. But already for $n=3$, I have no idea how to calculate $E(X)$. – Peter May 18 '15 at 8:48
• I'm not sure of something; With result $\{H,T\},\{H,T\},\{H,H\}$, the game ends? Or the numbers have to be pairwise different, for all pairs? – Masclins May 18 '15 at 9:48
• As I mentioned, the numbers have to be pairwise different. So, the game would not stop in your example. – Peter May 18 '15 at 9:49

While I cannot really trust the higher numbers (since E(n) is of the same order of magnitude of the number of games I ran), I do believe the trend is consistent, at the beginning I thought it looks like a 2nd degree polynomial in $N$, but there are too many deviations from this model.
The only other thing I managed to do is to calculate the probability of the game ending after the minimal number of steps (being equal to the number of coins, $N$) which is given by
$P_N(n=N)=N!\prod_{j=0}^N{\binom{N}{j}}\frac{1}{2^N}$.
• 1) $1000$ games are not enough to have a satisfiable estimate. 2) I do not think that a polynomial of degree $2$ will be a good approximation 3) I would like to have exact values or at least values very close to the true ones. 4) The minimum length is $n-1$, if $n$ players participate. – Peter May 19 '15 at 11:45


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# 12

Frontpage

A dozen years ago, Eliezer Yudkowsky asked us which was less (wrong?) bad:

• 3^^^3 people each getting a dust speck in their eyes,

or

• 1 person getting horribly tortured continually for 50 years.

He cheekily ended the post with "I think the answer is obvious.  How about you?"

To this day, I do not have a clue which answer he thinks is obviously true, but I strongly believe that the dust specks are preferable. Namely, I don't think there's any number we can increase 3^^^3 to that would change the answer, because I think the disutility of dust specks is fundamentally incomparable with that of torture. If you disagree with me, I don't care--that's not actually the point of this post. Below, I'll introduce a generalized type of utility function which allows for making this kind of statement with mathematical rigor.

## Linearity and Archimedes

A utility function is called linear if it respects scaling--5 people getting dust specked is 5 times as bad as a single dust specker. This is a natural and common assumption for what utility functions should look like.

An ordered algebraic structure (number system) is called Archimedean if for any two positive values , , there exists natural numbers , such that and . In other words, adding either value to itself a finite number of times will eventually outweigh the other. Any pair , for which this property holds of and is called comparable. Note that we're taking the absolute value because we care about magnitude and not sign here: of course any positive number of people enjoying a sunset is better than any positive number of people being tortured, but the question remains whether there is some number of sunset-views which would outweigh a year of torture (in the sense that a world with additional sunset watchers and 1 additional year of torture would be preferable to a world with no more of either).

Another way to think about comparability is that both values are relevant to an analysis of world state preference: a duster does not need to know how many dust specks are in world 1 or world 2 if they know that world 2 has more torture than world 1, because the preference is already decided by the torture differential.

Convicted dusters now face a dilemma: a linear, Archimedean utility function necessarily yields some number of dust specks to which torture is preferable. One resolution is to reject linearity: very smart people have discussed this at length. There is another option though which, as far as the author can tell, hardly ever gets proper attention: rejecting Archimedean utilities. This is what I'll explore in the post.

## Comparability

Comparability defines an equivalence class on utilities of world states (and their utilities).

• Any world state is comparable to itself (reflexivity),
• if state 1 is comparable to state 2, then state 2 is comparable to state 1 (symmetry), and
• if state 1 is comparable to state 2 and state 2 is comparable to state 3, then state 1 is comparable to state 3 (transitivity).

These properties follow immediately from the definition of comparability provided above. Now we have a partition of world states into comparability classes. Note that the comparability classes do not form a vector space: 1 torture + 1 dust speck is comparable to one torture, but their difference, 1 dust speck, is not comparable to either.

As I said before, I don't care if you disagree about dust specks. If you think there are any pair of utilities which are in some way qualitatively distinct or otherwise incomparable (take your pick from: universe destruction, death, having corn stuck between one's teeth, the simple joy of seeing a friend, etc--normative claims about which utilities are comparable is outside the scope of this post), then there is more than one comparability class and the utility function needs a non-Archimedean co-domain to reflect this.

If you staunchly believe that every pair of utilities is comparable, that's fine--there will be only one comparability class containing all world states, and the construction below will boringly return your existing Archimedean utility function.

We can assign a total ordering (severity) to our collection of comparability classes: means that, for any world states in classes , respectively, for all . It is clear from the construction of comparability classes that this does not depend on which representatives are chosen. Note that severity does not distinguish between severe positive and severe negative utilities: the classes are defined by magnitude and not sign. Thus, the framework is agnostic on such matters as negative utilitarianism: if one believes that reducing disutility of various forms takes priority over expanding positive utility, then that will be reflected in the severity levels of these comparability classes--perhaps a large portion of the most severe classes will be disutility-reduction based, and only after that do we encounter classes prioritizing positive utility world states.

Then we can think of utilities as being vectors, where each component records the amount of utility of a given severity level. We'll write the components with most severe first (since those are the most significant ones for comparing world states). Thus, if an agent believes that there are exactly two comparability classes:

• a severe class which contains e.g. 50 years of torture, and
• a milder class containing e.g. being hit with a dust speck,

then world A with 1 person being tortured and 100 dust-speckers would have utility . Choosing between this and world B with no torture and 3^^^3 dust speckers is now easy: world B has utility , and since the first component is greater in world B, it is preferable no matter what the second component is. If this seems unreasonable to the reader, it is not a fault of the mathematics but the normative assumption that dust specks are incomparable to torture--if comparability classes are constructed correctly, then this sort of lexicographic ordering on utilities necessarily follows.

This is what we refer to as a pseudograded vector space: the utility space consists of -dimensional vectors, which are I-pseudograded by severity, meaning that for any vector we can identify its severity as the comparability class of the first non-zero component. Equivalently, utility values are functions from I to , and thus utility functions have type (where is the world state space) (this interpretation becomes important if we would like to consider the possibility of infinitely many comparability classes).

Formally, the utility vector space admits a grading function which gives the index of the first (most severe) non-zero coordinate of the input vector. In our example, and . In general, if we're comparing two world states with different severity gradings, we need only check which world has greater severity, and then consult the sign of (that's a function evaluation, not a product!) to determine if it's preferable to the alternative (the sign will be non-zero by definition of ). More generally, we compare worlds by checking the sign of . Note that two utility vectors satisfy exactly when , so the lexicographic ordering is determined fully by which utilities are preferable to 0, the utility of an empty world. We will refer to such utility vectors as positive, keeping in mind that this only means that the most severe non-zero coefficient is positive and not that the whole vector is.

Expected utility computations work exactly as in Archimedean utility functions: averages are performed component-wise. Thus, existing work with utilities and decision theory should import easily to this generalization.

Observe that satisfies ultrametric-like conditions (recalling those found in -adic norms):

• For any non-zero and utility vector ,
• For any pair of utility vectors , we have with equality if

One may reasonably be concerned about the well-definedness of if they believe : isn't it possible that I is not well-ordered under severity, meaning that some vectors may not have a first non-zero element? [Technically we are considering the well-orderedness of the reverse ordering on , since we want to find the most severe non-zero utility and not the least severe].

Consider for instance the function defined on the positive reals: for all its smoothness, it has no least input with a non-zero output, and what's worse the output oscillates signs infinitely many times approaching such that we aren't prepared to weigh a world state with such a utility against an empty world with utility 0. Luckily, our setting is actually a special case of the Hahn Embedding Theorem for Abelian Ordered Groups, which guarantees that any utility vector can only have non-zero entries on a well-ordered set of indices (thanks to Sam Eisenstat for pointing this out to me), and thus is necessarily well-defined. We'll refer to this vector space as : the space of all functions which each only have non-zero values on a well-ordered subset of , and thus in particular all have a first non-zero value.

## Basis Independence

Working with vector spaces, we often think of the vectors as lists of (or functions from the index set to) numbers, but this is a convenient lie--in doing so, we are fixing a basis with which the vector space was not intrinsically equipped. Abstractly, a vector space is coordinate-free, and we merely pick bases for ease of visualization and computation. However, it is important for our purposes here that the lexicographic ordering on utility functions is basis-independent", for a suitable notion of basis (otherwise, our world state preference would depend on arbitrary choices made in representations of the utility space). In fact, classical graded vector spaces have distinguished homogeneous elements of grading i (e.g. polynomials which only have degree i monomials, rather than mixed polynomials with max degree i, in the graded vector space of polynomials), but this is too much structure and would be unnatural to apply to our setting, hence pseudograded vector spaces.

The standard definition of basis (a linearly independent set of vectors which spans the space) isn't sufficient here, since we have extra structure we'd like to encode in our bases--namely, the severity grading. Instead, we'll define a graded basis to be a choice of one positive utility vector for each grading (i.e., a map such that --in categorical terms, is a section of ).

Then suppose Alice and Bob agree that there are two severity levels, and even agree on which world states fall into each, but have picked different graded bases. Letting X=50 years of torture, and Y=1 dust speck in someone's eye, perhaps Alice's basis has and , but for some reason Bob thinks the sensible choice of representatives is and --he agrees with Alice that , but factors the world state differently (while Bob may seem slightly ridiculous here, there are certainly world states with multiple natural factorizations).

Then Alice's utility for world A in her basis is and her , while Bob writes (since ) and . Note that they will both prefer world B, since the change in graded basis did not affect the lexicographic ordering on utilities. Specifically, the transformation from Alice's coordinates to Bob's is induced by left multiplication by the matrix

We observe two key facts about :

• is lower triangular (all entries above the main diagonal are 0), and
• has positive diagonal entries.

These will be true of any transformations between graded bases of the same utility space: lower triangularity follows from the ultrametric property of the grading , and positive diagonals follow from our insistence that the graded basis vectors all have positive utilities. In general (when ), it won't make sense to view as a matrix, but we will have a linear transformation operator with the same properties.

Recalling that the ordering on utilities is determined by the positive cone of utilities (since iff ), we simply observe that lower triangular linear transformations with positive diagonal entries will preserve the positive cone of : if the first non-zero entry of is positive, then will also have 0 for all entries before index (since lower triangularity means changes only propagate downstream) and will scale by the positive diagonal element, returning a positive utility. Thus, our ordering is independent of graded basis and depends only upon the severity classes and comparisons within them.

## What's the point?

I think many LW folk (and almost all non-philosophers) are firm dusters and have felt dismayed at their inability to justify this within Archimedean utility theory, or have thrown up their hands and given up linearity believing it to be the only option. While we will never encounter a world with people in it to be dusted, it's important to understand the subtleties of how our utility functions actually work, especially if we would like to use them to align AI who will have vastly more ability than us to consider many small utilities and aggregate them in ways that we don't necessarily believe they should.

Moreover, the framework is very practical in terms of computational efficiency. It means that choosing the best course of action or comparing world states requires one to only have strong information on the most severe utility levels at stake, rather than losing their mind (or overclocking their CPU) trying to consider all the tiny utilities which might add up. Indeed, this is how most people operate in daily life; while some may counter that this is a failure of rationality, perhaps it is in fact because people understand that there are some tiny impacts which can never accumulate to more importance than the big stuff, and it can't all be chalked up to scope neglect.

Finally, completely aside from all ethical questions, I think there's some interesting math going on and I wanted people to look at it. Thanks for looking at it!

This is my first blog post and I look forward to feedback and discussion (I'm sure there are many issues here, and I hope that I will not be alone in trying to solve them). Thanks to everyone at MSFP with whom I talked about this--even if you don't think you gave me any ideas, being organic rubber ducks was very productive for my ability to formulate these thoughts in semi-coherent ways.

Frontpage

# 12

New Comment

Apologies if this is not the discussion you wanted, but it's hard to engage with comparability classes without a framework for how their boundaries are even minimally plausible.

Would you say that all types of discomfort are comparable with higher quantities of themselves? Is there always a marginally worse type of discomfort for any given negative experience? So long as both of these are true (and I struggle to deny them) then transitivity seems to connect the entire spectrum of negative experience. Do you think there is a way to remove the transitivity of comparability and still have a coherent system? This, to me, would be the core requirement for making dust specks and torture incomparable.

I agree that delineating the precise boundaries of comparability classes is a uniquely challenging task. Nonetheless, it does not mean they don't exist--to me your claim feels along the same lines as classical induction "paradoxes" involving classifying sand heaps. While it's difficult to define exactly what a sand heap is, we can look at many objects and say with certainty whether or not they are sand heaps, and that's what matters for living in the world and making empirical claims (or building sandcastles anyway).

I suspect it's quite likely that experiences you may be referring to as "higher quantities of themselves" within a single person are in fact qualitatively different and no longer comparable utilities in many cases. Consider the dust specks: they are assumed to be minimally annoying and almost indetectable to the bespeckèd. However, if we even slightly upgrade them so as to cause a noticeable sting in their targeted eye, they appear to reach a whole different level. I'd rather spend my life plagued by barely noticeable specks (assuming they have no interactions) than have one slightly burn my eyeball.

Theron Pummer has written about this precise thing in his paper on Spectrum Arguments, where he touches on this argument for "transitivity=>comparability" (here notably used as an argument against transitivity rather than an argument for comparability) and its relation to 'Sorites arguments' such as the one about sand heaps.

Personally I think the spectrum arguments are fairly convincing for making me believe in comparability, but I think there's a wide range of possible positions here and it's not entirely obvious which are actually inconsistent. Pummer even seemed to think rejecting transitivity and comparability could be a plausible position and that the math could work out in nice ways still.

• The thing you called "pseudograding" is normally called "filtration".

• In practice, because of the complexity of the world, and especially because of the presence of probabilistic uncertainty, an agent following a non-Archimedean utility function will always consider only the component corresponding to the absolute maximum of , since there will never be a choice between A and B such that these components just happen to be exactly equal. So it will be equivalent to an Archimedean agent whose utility is this worst component. (You can have an without an absolute maximum but I don't think it's possible to define any reasonable utility function like that, where by "reasonable" I roughly mean that, it's possible to build some good theory of reinforcement learning out of it.)

The thing you called "pseudograding" is normally called "filtration".

Ah, thanks! I knew there had to be something for that, just couldn't remember what it was. I was embarrassed posting with a made-up word, but I really did look (and ask around) and couldn't find what I needed.

...Although, reading the definition, I'm not sure it's exactly the same...the severity classes aren't nested, and I think this is probably an important distinction to the conceptual framing, even if the math is equivalent. If I start with a filtration proper, I need to extract the severity classes in a way that seems slightly more convoluted than what I did.

In practice, because of the complexity of the world, and especially because of the presence of probabilistic uncertainty, an agent following a non-Archimedean utility function will always consider only the component corresponding to the absolute maximum of I, since there will never be a choice between A and B such that these components just happen to be exactly equal. So it will be equivalent to an Archimedean agent whose utility is this worst component.

See my response to Dacyn.

If I understand what you do correctly, the severity classes are just the set differences , where is the filtration. I think that you also assume that the quotient is one-dimensional and equipped with a choice of "positive" direction.

Yes! This is all true. I thought set differences of infinite unions and quotients would only make the post less accessible for non-mathematicians though. I also don't see a natural way to define the filtration without already having defined the severity classes.

I've tried intuitive approaches to thinking along these lines which failed so it's really nice to see a serious approach. I see this as key anti-moloch tech and want to use it to think about rivalrous and non-rivalrous goods.

Firstly I will focus on the most wrong part. The claim that non archimedian utilities are more efficient. In the real world there aren't 3^^^3 little impacts to add up. If the number of little impacts is a few hundred, and they are a trillion times smaller, then the little impacts make up less than a billionth of your utility. Usually you should be using less than a billionth of your compute to deal with them. For agents without vast amounts of compute, this means forgetting them altogether. This can be understood as an approximation strategy to maximize a normal archimedian utility.

There is also the question of different severity classes. If we can construct a sliding scale between specks and torture then we find the need for a weird cut off point, like a broken arm being in a different severity class than a broken toe.

Intuitively speaking broken arm and broken toe are comparable. Broken arm is worse, broken toe is still bad. I'd rather get a broken arm than torture for 50 years, or even torture for 1 day.

For sliding scale of severities: there's a very difficult to compute but intuitively satisfying emphasis that can be imposed so the scale can't slide. It's the idea of "bouncing back". If you can't bounce back from an action that imparts negative utility, it forms a distinct class of utilities. Compare broken arm with torn-off toe. Compare both of those to 50 years of torture.

P.S: If you're familiar with Taleb's idea of "antifragility", that's the notion I'm basing these on.

The idea is that we can take a finite list of items like this

Torture for 50 years

Torture for 40 years

...

Torture for 1 day

...

Broken arm

Broken toe

...

Papercut

Sneeze

Dust Speck

Presented with such a list you must insist that two items on this list are incomparable. In fact you must claim that some item is incomparably worse than the next item. I don't think that any number of broken toes is better than a broken arm. A million broken toes is clearly worse. Follow this chain of reasoning for each pair of items on the list. Claiming incomparably is a claim that no matter how much I try to subdivide my list, one item will still be infinitely worse than the next.

The idea of bouncing back is also not useful. Firstly it isn't a sharp boundary, you can mostly recover but still be somewhat scarred. Secondly you can replace an injury with something that takes twice as long to bounce back from, and they still seem comparable. Something that takes most of a lifetime to bounce back from is comparable to something that you don't bounce back from. This breaks if you assume immortality, or that bouncing back 5 seconds before you drop dead is of morally overwhelming significance, such that doing so is incomparable to not doing so.

Broken arms vs toes: I agree that any number of broken toes wouldn't be better than a broken arm. But that's the point, these are _comparable_.

Incomparable breaks occur where you put the ellipses in your list. Torture for 40-50 years vs torture for 1 day is qualitatively distinct. I imagine a human being can bounce back from torture for 1 day, have scars but manage to prosper. That would be hellishly more difficult with torture for 40 years. We could count torture by day, 1-(365*40) and there would be a point of no return there. A duration of torture a person can't bounce back. It would depend on the person, what happens during and after etc, which is why it's not possible to compute that day. That doesn't mean we should ignore how humans work.

Here's the main beef I have with Dust Specks vs Torture: Statements like "1 million broken toes" or "3^^^3 dust specks" disregard human experience. That many dust specks on one person is torture. One on each is _practically nothing_. I'm simulating people experiencing these, and the result I arrive at is this; choose best outcome from (0 utils * 3^^^3) vs (-3^^^3 utils). This is easy to answer.

You may say "but 1 dust speck on a person isn't 0 utils, it's a very small negative utility" and yes, technically you're correct. But before doing the sum over people, take a look at the people. *Distribution matters.*

Humans don't work like linear sensory devices. Utility can't work linearly as well.

What if I make each time period in the "..." one nanosecond shorter than the previous.

You must believe that there is some length of time, t>most of a day, such that everyone in the world being tortured for t-1 nanosecond is better than one person being tortured for t.

Suppose there was a strong clustering effect in human psychology, such that less than a week of torture left peoples minds in one state, and more than a week left them broken. I would still expect the possibility of some intermediate cases on the borderlines. Things as messy as human psychology, I would expect there to not be a perfectly sharp black and white cutoff. If we zoom in enough, we find that the space of possible quantum wavefunctions is continuous.

There is a sense in which specs and torture feel incomparable, but I don't think this is your sense of incomparability, to me it feels like moral uncertainty about which huge number of specs to pick. I would also say that "Don't torture anyone" and "don't commit attrocities based on convoluted arguments" a good ethical injunction. If you think that your own reasoning processes are not very reliable, and you think philosophical thought experiments rarely happen in real life, then implementing the general rule "If I think I should torture someone, go to nearest psych ward" is a good idea. However I would want a perfectly rational AI which never made mistakes to choose torture.

we find the need for a weird cut off point, like a broken arm

For the cut-off point on a broken arm, I recommend the elbow [not a doctor].

Suppose there was a strong clustering effect in human psychology, such that less than a week of torture left peoples minds in one state, and more than a week left them broken. I would still expect the possibility of some intermediate cases on the borderlines. Things as messy as human psychology, I would expect there to not be a perfectly sharp black and white cutoff. If we zoom in enough, we find that the space of possible quantum wavefunctions is continuous.

I agree! You've made my point for me: it is precisely this messiness which grants us continuity on average. Some people will take longer than others to have qualitatively incomparably damaging effects from torture, and as such the expected impact of any significant torture will have a component on the severity level of 50 years torture. Hence, comparable (on expectation).


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# Problem for bold symbol in math mode with iota [duplicate]

I am using \mathbf{} for include bold symbols in my equation. It works fine with all symbols with the exception of \iota. I tried to use equation environment and to exclude subindex. The following is my writing:

\begin{align}
\mathbf{Y}=\rho \mathbf{WY}+\alpha \mathbf{\iota}_{N}+\mathbf{X\beta}+\mathbf{WX\theta}+\mathbf{u},\label{eqgeneral1}\\
\mathbf{u}=\lambda\mathbf{Wu}+\mathbf{\varepsilon}  \label{eqgeneral2}
\end{align}

I obtain this:

I want this:

¿How can I solve this?

## marked as duplicate by Au101, Stefan Pinnow, Sebastiano, user36296, TeXnicianAug 30 '18 at 20:19

• It's not just iota though? It's all of the Greek letters right, and there's a very good reason for that. \mathbf{} and the standard LaTeX font applies to Roman letters only, not Greek letters, not symbols like + - =, nothing like that. Try: tex.stackexchange.com/questions/595/… – Au101 Aug 30 '18 at 19:29
• You can use \bm or \boldsymbol. – Bernard Aug 30 '18 at 19:44
• For new documents, I would recommend loading unicode-math instead of legacy font packages. Any Unicode math font is going to come with bold math symbols. You can use them with \symbf{\iota} or \mathbf{\iota}. It’s also possible to do more complex things, such as using a different font for your Greek alphabet. – Davislor Aug 31 '18 at 2:30
• If you’re forced to use pdflatex, you might try the isomath package, which supports a large number of Greek math alphabets. You can also use a LGR-encoded legacy Greek font with mathastext. – Davislor Aug 31 '18 at 2:31

Thanks to user Au101. I used \boldsymbol and it works for me:
\begin{align}


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A- A+
Alt. Display

# An Octave/MATLAB® Interface for Rapid Processing of SMOS L1C Full Polarization Brightness Temperature

## Abstract

A tool to process the SMOS microwave radiometer level 1C polarized brightness temperatures data product has been developed. The SMOS L1C science product contains the dual and full (Stokes vector) polarization brightness temperatures at L-band for multiple incidence angles. In order to use the L1C product, the measurements are processed by a number of procedures including radio frequency interference (RFI) filters, conversion of the polarization plane from the antenna (X- & Y-pol) to the Earth’s surface frame (H- & V-pol), and averaging to fixed classes of incidence angles. The software allows for the processing of data for the entire daily half-orbit product, or for specific regions of interest, and can be adapted as a bash-job to process a large number of data files e.g. for time series analysis. This paper describes the tool which was developed in GNU C++ with the capability to be compiled as MEX function to work with Octave or MATLAB® without any source code adjustment.

Funding statement: Deutsche Forschungsgemeinschaft DFG under grant number SI 606/24-1.

Keywords:
How to Cite: Saavedra, P. and Simmer, C., 2018. An Octave/MATLAB® Interface for Rapid Processing of SMOS L1C Full Polarization Brightness Temperature. Journal of Open Research Software, 6(1), p.2. DOI: http://doi.org/10.5334/jors.165
Published on 08 Jan 2018
Accepted on 03 Nov 2017            Submitted on 16 Feb 2017

## (1) Overview

### Introduction

The Soil Moisture Ocean Salinity (SMOS) mission is a European Space Agency (ESA) earth observation satellite carrying the Microwave Imaging Radiometer using Aperture Synthetic (MIRAS) instrument. MIRAS is a passive 2-D interferometric radiometer operating at a frequency of 1.413 GHz (21 cm, L-band) and is used to measure the microwave emission from Earth’s surface as brightness temperature (TB), from which global maps of upper-level soil moisture and ocean salinity can be estimated [10].

The SMOS satellite follows a sun synchronous orbit (dawn/dusk at 6:00/18:00), and measurements are performed for a range of incidence angles between 0° and 60° across a swath of approximately 1000 km width with a spatial resolution ranging from 35 to 50 km depending on incidence angle. The MIRAS interferometer measures brightness temperatures in dual and full polarization (4 Stokes vector) mode.

The instrument data product comprises three level-1 sub-products:

• [L1A] calibrated visibilities between individual antenna receivers (i.e. the interferometric measurements from the sensor including the redundant visibilities), combined per integration time of 1.2 s (snapshot) in full polarization and prior the application of image reconstruction, provided in pole-to-pole (half orbit) time-based segments,
• [L1B] brightness temperature product as output of the image reconstruction of the SMOS observations, which consist of the Fourier components of brightness temperatures in the antenna polarization reference frame (XY polarization, hereafter),
• [L1C] multi-incidence angle brightness temperatures at the top of the atmosphere (XY polarization), geo-located in an equal-area grid. The L1C product comprises separate datasets available for ocean and land pixels with information available per pixel and per snapshot. An addition a so-called browse product contains the brightness temperatures averaged for a fixed incidence angle of 42.5°.

This document describes the analysis and processing of the L1C scientific data product, which consists of half-orbit daily archives with two separated physical files for each half-orbit: the product header file (HDR) and the data block binary file (DBL).

The Earth surface is viewed by SMOS in a snapshot-like configuration, which is restructured in the L1C product on an equal area Digital Global Grid (DGG) [13] with pixels based on the Icosahedral Snyder Equal Area (ISEA) projection at a spatial separation of about 15 km [11].

The SMOS L1C data product arranges the brightness temperatures sequentially for the real components of the XX, XY or YY polarizations, along with the same polarization sequence for the imaginary polarization components. The imaginary part is always zero for the single polarization components (e.g. XX or YY polarization). Figure 1 illustrates typical SMOS L1C half-orbit measurements stored in a DBL data file.

Figure 1

SMOS unprocessed brightness temperatures at X polarization from a morning overpass half-orbit L1C data file for 2 July, 2015. The color scale has been limited to 330 K, but the data contains values as high as 900 K indicating pixels with RFI.

In order to use SMOS measurements in further studies for soil moisture or ocean salinity retrievals and for data assimilation, the L1C brightness temperature needs to be transformed from the antenna reference frame (XY) into the Earth’s surface reference frame (HV polarization, hereafter). This in done by considering the amount of rotation suffered by the polarization plane due to geometric and Faraday rotation.

Although the SMOS radiometer operates at L-band, which is a protected frequency band, observations are partly affected by Radio Frequency Interference (RFI) from various sources like radars and TV transmitters. Therefore RFI detection and mitigation for L1C brightness temperatures are important procedures, which need to be applied before brightness temperatures are used for retrievals of soil moisture and ocean salinity. The magnitude at which RFI affect the data, depends on the orbit (ascending or descending), polarization, and observation incidence angle. E.g., at higher incidence angles the pixel footprint is larger, which increases the impact of RFI compared to observations at nadir.

Some tools for polarization reference frame transformation and RFI mitigation have already been developed (e.g. SMOSview and SMOS-Box).

SMOSview is a JAVA based Graphical User Interface (GUI) software, that decodes SMOS products, displays their content graphically or numerically as tables and exports the data to alternative formats. SMOSview is not intended for a detailed analysis, visualization or processing of SMOS data but only for quick and easy look at the raw data products.

Similarly SMOS-Box is a module for the BEAM GUI software dedicated to the decoding, visualization and basic analysis of all operational data products (Levels 1 and 2) from the SMOS mission (see SMOS/Toolbox+BEAM website). SMOS-Box imports SMOS L1C and L2 science data products into the BEAM software and use the existing BEAM functionality to decode, display, manipulate, analyze and export SMOS data products to alternative formats in an interactive way or as batch processing. The BEAM software development has, however, been discontinued.

The Sentinel Application Platform (SNAP), the successor of BEAM, is a software developed as a common architecture for all Sentinel toolboxes (i.e. Sentinel-1 to -4 and SMOS, see http://step.esa.int/main/toolboxes/snap/). SNAP utilizes the SMOS-Box toolbox and allows to convert any SMOS data levels to NetCDF using its graph processing command line tool or a graphic user interface, also for specific regions of interest. SNAP also provides the Python module snappy, which allows the access to its SNAP Java API from Python; hence it can also work in batch processing mode.

The Centre Aval de Traitement des Donnes provides a Level 3 (L3) brightness temperature product in daily archives, which comprise daily global-coverage products that include all brightness temperatures in full polarization and transformed from the antenna frame (L1B) to the Earth’s surface polarization reference frame. The brightness temperatures (TBs) are binned and averaged for fixed angle classes. L3 TBs are already re-sampled on an Equal-Area Scalable Earth-grid (EASE-grid) [1] with 25 km resolution on a global scale; no detailed processing can be done in regions of interests at higher resolution on the L1C data product on the 15 km ISEA-grid.

### Motivation

The discussed software tools mostly work with SMOS data in an interactive way with single data files at a time or via batch jobs. They load the whole data file and the user manually interacts with the software to glance through specific snapshots or regions for a given date. These tools are very helpful for a case-by-case analysis, but when a specific regions of interest need to be analyzed for a longer time period and large amounts of files must be processed limitation in term of reading and processing time becomes obvious. Since BEAM and SNAP are Java based they need to be interpreted by a Java Virtual Machine, which slows down its execution. The snappy module is a wrapper, which calls SNAP from Python and its performance, in terms of speed, might be even lower as expected from a scripting language. Therefore when many SMOS data files need to be fed to observation operators, e.g. for inversion algorithms or for data assimilation, speed-ups for reading and processing are paramount.

We developed the “Process_SMOSxL1C” mex-function as an alternative tool for rapid reading and processing of SMOS L1C data products with Matlab® [9] or GNU/Octave [4] programming languages. A mex-function is a C/C++ source code compiled in order to run within the Matlab workspace as a native Matlab function at a high execution speed. The “Process_SMOSxL1C” mex-function gives the user the ability to either work with a single file and exploit the powerful features of Matlab and Octave for analysis and visualization, or to run batch-jobs for many files and perform the same processing for a selected region and have the results as well as the original data automatically stored as MAT-files for future analysis. In addition it is possible to easily port the function to high performance computer (HPC) systems since most of them have C/C++ as default programming languages.

### Implementation and architecture

The “Process_SMOSxL1C” function is implemented in a sequential structure. Since the SMOS L1C datablock (DBL) is a binary file, the data needs to be loaded sequentially and sorted to different variable types according to their specification [11]. The scientific L1C datablock consists of two datasets: one contains a list of snapshots and associated information in the swath (hereafter Swath Snapshot Information or SSI) and a second one contains the list of brightness temperature (TB) samples and associated geophysical information for each grid point sensed by the instrument (hereafter TSF for Temperatures Swath Full). In order to process the TBs, information from both datasets are needed, therefore SSI and TSF needs to be loaded sequentially before a region of interest is selected.

Every piece of information contained in a L1C datablock is sorted as a Matlab structure in their corresponding dataset of origin, either in SSI or TSF. Then the “Process_SMOSxL1C” mex-function returns the TSF structure with the original data contained in the DBL file and also gives the possibility to optionally return the SSI structure (in case of further re-processing).

The processing procedure follows five steps:

1. data filtering according to the RFI flags and other criteria,
2. interpolation of TBXY values to obtain the four components of the Stokes vector at every time step,
3. transformation from the antenna reference frame (X- & Y-polarization) to the Earth’s surface reference frame (H- & V-polarization),
4. for every grid point the TBs are averaged for equally-spaced classes of incidence angles into 1° bins from 0 to 60°,
5. results are returned either to the Matlab workspace or saved as a MAT-file containing only the selected region of interest.

Step 1 uses the RFI flags contained in the TSF datablock for every grid point [6]. Additionally, other standard RFI detection criteria have been implemented as described by [6] and [15], namely:

(1)
$S{T}_{4}<50K$
(2)
$\left({\mathit{\text{TB}}}_{S1}-\overline{{\mathit{\text{TB}}}_{S1}}\right)>5+4{\mathit{\text{DTB}}}_{X}$

with ST4 the 4th Stokes parameter, and $\overline{{\mathit{\text{TB}}}_{S1}}=0.5\left({\mathit{\text{TB}}}_{X}+{\mathit{\text{TB}}}_{Y}\right)$ the mean value of the halved 1st Stokes parameter for all observations in one pixel [6], and DTBX the radiometric uncertainty for TBX. Additional criteria (not strictly related to RFI) constrain out-of-range TB values by the requirement $50K<\sqrt{{\mathit{\text{TB}}}_{X}{}^{2}+{\mathit{\text{TB}}}_{Y}{}^{2}}<500K$. Here the upper extreme may be related to RFI.

For the step 2, the TB values come in sequence for X-pol, Y-pol or XY, YX-pol at every snapshot time step. In order to have TB values with four Stokes parameters, an interpolation is needed in such a way that at every time step the TBs contain the [X, Y, Re{XY}, Im{XY}] Stokes parameters.

Step 3 transforms the polarization reference frame via a transformation matrix given by [6] as:

(3)
$\left[\begin{array}{c}{A}_{1}\\ {A}_{2}\\ {A}_{3}\\ {A}_{4}\end{array}\right]=\left[\begin{array}{cccc}{\mathit{\text{cos}}}^{2}\left(\alpha \right)& {\mathit{\text{sin}}}^{2}\left(\alpha \right)& -\mathit{\text{cos}}\left(\alpha \right)\mathit{\text{sin}}\left(\alpha \right)& 0\\ {\mathit{\text{sin}}}^{2}\left(\alpha \right)& {\mathit{\text{cos}}}^{2}\left(\alpha \right)& \mathit{\text{cos}}\left(\alpha \right)\mathit{\text{sin}}\left(\alpha \right)& 0\\ \mathit{\text{sin}}\left(2\alpha \right)& -\mathit{\text{sin}}\left(2\alpha \right)& \mathit{\text{cos}}\left(2\alpha \right)& 0\\ 0& 0& 0& 1\end{array}\right]×\left[\begin{array}{c}{\mathit{\text{TB}}}_{H}\\ {\mathit{\text{TB}}}_{V}\\ {\mathit{\text{ST}}}_{3}\\ {\mathit{\text{ST}}}_{4}\end{array}\right]$

where the leftmost vector is A = [Re{TBX}, Re{TBY}, 2Re{TBXY}, –2Im{TBXY}]T for full polarization and A = [TBX, TBY]T for dual-polarization. The angle α = ωFa + αge is the sum of ωFa, the Faraday angle, and αge, the geometric rotation angle. Every matrix and vector element depends on the incidence angle of observation.

In order to obtain the Earth’s polarization reference frame vector, the above matrix system is solved for [TBH, TBV, ST3, ST4]T at every incidence angle and for every pixel-point by using the GNU Scientific Library [7].

Step 4 re-samples the vector [TBH, TBV, ST3, ST4]T at fixed incidence angle classes of 1° between 0 and 60° in the current version. This can be changed to other ranges and bin sizes, but requires re-compilation of the source code. The new re-sampled Stokes (or dual-pol) TBs are sorted in the following elements of the TSF structure:

> TSF.TB_Fixed_IncAngle,
> TSF.Fixed_IncAngle

where the first is an 3D array with the number of DGG pixel points in the first dimension (depending on the geographic region selected), the number of incidence angles (61 elements) in the second dimension and the polarization (either 4 or 2 elements) in the third dimension.

All information contained in the L1C datablock DBL is kept inside the structures TSF and SSI, with TSF having additional information related to the specific processing; therefore any step can be repeated or re-calculated within Matlab/Octave workspace. For a detailed description of the complete list of elements in the TSF and SSI structures, see the mex-function documentation at https://github.com/pablosaa/Process_SMOS_L1C/wiki.

The general usage of the “Process_SMOSxL1C” mex-function for a desired geographic region needs to be called from the Matlab or Octave workspace as:

> [TSF, SSI] = Process_SMOSxL1C
(‘DBL_filename’,…
[lat_min, lon_min, lat_max, lon_max],…
‘PATH_TO_OUTPUT’);

with ‘DBL_filename’ a string with the absolute location of the L1C DBL data file, the second argument the limits of a latitude and longitude box for a geographic region of interest, and the last argument an optional string which indicates where the MAT-file containing the TSF and SSI variables will be saved. At least one output argument is mandatory, i.e. TSF, while SSI is optional.

Figure 2 shows the TBH extracted and processed from the raw data file shown in Figure 1 for the Neckar catchment in south-west Germany, with the limits of the geographic region given by [47.5, 7.5, 50, 10] and assigned to the second argument of “Process_SMOSxL1C”.

Figure 2

Brightness Temperature extracted for the Neckar catchment after precessing for three incidence angles 35° (left), 40° (middle) and 45° (right).

See Table 1 for a reference benchmark regarding the processing time for a full archive with half-orbits as in Figure 1 and for a selected region as in Figure 2.

Table 1

Benchmarks for processing time with Matlab and Octave for two case studies: Half-orbit and a region (Neckar catchment). Linux 64bit PC with 4 GB RAM and CPU 2534 MHz Inter Centrino 2 core. The “m-script” column represents only loading the raw data, without any processing.

Processing Time in Seconds
Case Study No. DGGs Matlab m-script Matlab MEX Octave MEX

Half-orbit 114369 2957.16 84.01 82.73
Sub-sector 265 6.12 6.02
Storage as output MAT-file +3% +25% +33%

### Quality control

The “Process_SMOSxL1C.cpp” code has been successfully compiled with GNU C++ version 4.7.1 in a Linux 64bit system. The compiled mex-function has been tested in three different Matlab versions: v7.8.0.347 (R2009a), v7.11.0 (R2010b), and v9.1.0.441655 (R2016b). Similarly, the code has been compiled and tested for Octave version 3.6.2. and 4.0.3.

The processing speed is listed in Table 1 for two case studies: a whole SMOS L1C data product containing one half-orbit (see Figure 1), and for a region of interest limited by its bottom-left corner [47.5°N, 7.5°E] and upper-right corner [50°N, 10°E] latitude and longitude, respectively (see Figure 2). This region contains the Neckar river catchment and is used for data assimilation studies by [12, 14] using of the “Process_SMOSxL1C” mex-function tool.

Table 1 includes in its 2nd column the number of Digital Global Grid pixel points contained. The 3rd column shows the result of only loading the raw L1C data product into the workspace by a Matlab m-script without using the mex-function, and the 4th and 5th columns indicate the total processing time by using the “Process_SMOSxL1C” mex-function. The last row in Table 1 indicates the percentage of minimum time needed in addition, when the option to save the processing outputs as a MAT-file is used.

Table 1 clearly shows the efficiency in terms of processing time by the mex-function, with Octave being slightly faster than Matlab. When the processed data is saved as MAT-file, Octave requires more time because all variables need to be translated to the MATIO library variable types [8].

For validation purposes the processed SMOS observations are “compared” against measurements by NASA’s Soil Moisture Active Passive (SMAP) mission launched in March 2015. SMAP carries a radiometer sensing microwave radiation at 1.4 GHz at one fixed incidence angle of 40° [5, 2]. SMAP has an antenna configuration different from SMOS, and SMAP’s L1C data products (36 km resolution) are delivered with the Earth’s polarization reference frame, i.e. no further processing is needed.

For this validation exercise the region corresponding to the Neckar catchment has been selected. Observations by SMOS and SMAP from April to December 2015 have been taken into account when both satellites have a morning overpass at the same day. Figure 3 shows the statistics of TBs at H-, and V-polarization after SMOS TB has been processed with the “Process_SMOSxL1C” mex-function. Note the good agreement despite the fact that the data come from two completely different instruments and methodologies.

Figure 3

Statistics for SMOS vs SMAP brightness temperatures at H-polarization (top panel) and V-polarization (bottom panel) for April to December 2015. SMOS is the result of the “Process_SMOSxL1C” processing software. The dots represent the median TBs within the selected region and the bars are the 1st and 3rd quartiles.

Figure 4 shows the excellent agreement in dynamic range and daily variability between SMOS and SMAP TBs after processing the SMOS L1C product, which lends strong confidence to the capabilities of the software to adequately yield results free of artifacts which could jeopardize further applications of the data.

Figure 4

Time series of SMAP brightness temperature measurements and SMOS L1C product processed with “Process_SMOSxL1C” mex-function (top). The bottom panel shows the residuals of the median within the distribution.

### Compilation and testing

The best way to ensure the that “Process_SMOSxL1C” works for a particular system is by compiling the C++ source code on the used system. To that goal, the user needs first to assure, that Matlab’s mex compiler is properly configured and tested. Compilation procedures are indicated in the https://github.com/pablosaa/Process_SMOS_L1C/wiki. The user can also try the pre-compiled version provided and test whether or not the Process_SMOSxL1C mex-function is working. The easiest way to test is by trying to process a single file interactively from the Matlab or Octave workspace, following the simple command:

> TSF = Process_SMOSxL1C;

Immediately, a file browser pops-up for navigating thought directories to select a SMOS L1C DBL file. Once selected another dialogue-box pops-up (for the Matlab version) for entering the latitude and longitude limits for the processing region (these are done provided as command line inputs for the Octave version). If the Process_SMOSxL1C mex-function shows any message or warnings, it could mean that the input arguments are wrong, but indicates already that the mex-function is working in the system. When no warning appears the process should be working and the user can look at the content of the TSF structure.

Note that the “Process_SMOSxL1C” mex-function has as file extension either “.mex” or “.mexa64” for Octave and Matlab, respectively. “Process_SMOSxL1C” requires that the corresponding L1C DBL and HDR files are located in the same directory. Further description and examples on how to run the function and plot results can be found at https://github.com/pablosaa/Process_SMOS_L1C/wiki.

## (2) Availability

### Operating system

GNU/Linux kernel 3.4.63-2.44, tested on OpenSuse version 12.2 or higher.

### Programming language

Source code is written in GNU C++ language.

Minimum Memory RAM: 2 GB, disk space depending on number of half orbit L1C binary data files (each file is ~435 MB).

### Dependencies

To compile the software the MEX library is needed, with two possibilities for compilation:

• as a MATLAB® function via MEX compiler, revision: 1.1.6.3 [9],
• as a GNU/Octave [4] mex function via MKOCTFILE compiler version 3.6.2 or higher.

The program requires:

• The GNU Scientific Library (GSL) version 2.2.1 for C and C++ language,
• The GNU C GSL Basic Linear Algebra Subprograms (GSLCBLAS) version 3.6.0. [7],
• MATIO library version 1.5.9 [8] is required for Octave to support MAT-file archiving.

### List of contributors

Pablo Saavedra Garfias: project scientist, Meteorological Institute, University of Bonn.

Clemens Simmer: project Principal Investigator (PI), Meteorological Institute, University of Bonn.

### Software location

#### Archive

Name: Process_SMOS_L1C-v1.0.zip

Persistent identifier:https://doi.org/10.5281/zenodo.1036856

Licence: GNU General Public License version 3

Publisher: Pablo Saavedra Garfias

Version published: v1.0

Date published: 26/10/2017

#### Code repository

Name: GitHub

Persistent identifier:https://github.com/pablosaa/Process_SMOS_L1C

Licence: GNU General Public License version 3

Date published: 13/02/2017

English

## (3) Reuse potential

SMOS datasets are widely used for research and application in studies regarding ocean salinity, soil moisture, root zone soil moisture, ice-thickness, landslides monitoring, and precipitation estimation, among others. The “Process_SMOSxL1C” mex-function can be used by researchers, who need to work directly with brightness temperatures rather than with other geophysical parameter retrievals provided in L2 or L3 products. It can also be used to develop or test other retrieval approaches. With the SMOS satellite in operation since 2009, time series are becoming rich in terms of information of the Earth’s surface and ocean sensed by the SMOS instrument. A rapid tool to process and re-analyze large numbers of data files is becoming a necessity and “Process_SMOSxL1C” can be used to that goal.

The “Process_SMOSxL1C” mex-function is being used for calibration and comparisons with synthetic observations obtained by satellite simulators alike SMOS or SMAP by [12]. The calibrated synthetic brightness temperatures are then utilized as state variables for data assimilation experiments applied to land-surface-atmosphere coupled models as shown by [14]. For instance, Figure 5 depicts the agreement of simulations by the Community Microwave Emission Model (CMEM) [3] after being calibrated against SMOS multi-incidence angle brightness temperatures processed by “Process_SMOSxL1C” mex-function. The box-plots are composed by the median (read dot), the inter-quantile region (blue box), the extremes (whiskers), and outliers as green-dots.

Figure 5

Simulation of brightness temperatures at 1.4 GHz and three incidence angles: 20° (top), 30° (center) and 40° (bottom). The black-line is the median of the TB distribution within the Neckar catchment, the gray area is the inter-quantile region and the lighter gray area indicate the 5% to 95% quantiles. The Box-plots depict the statistics of the distributions of the real measurements by SMOS and processed by “Process_SMOSxL1C” mex-function.

The “Process_SMOSxL1C” mex-function is documented online at https://github.com/pablosaa/Process_SMOS_L1C/wiki.

So far the “Process_SMOSxL1C” mex-function is able to work only with L1C scientific data files (SC), however SMOS also provides the L1C browse product (BW) which only contains data for the incidence angle of 42.5°. “Process_SMOSxL1C” is not yet adjusted to support this browse product, but any contribution is welcome by making a pull request to the Process_SMOS_L1C GitHub repository. Any support mechanisms is based on a voluntary basis via the repository.

## Acknowledgements

The authors thank to the German Science Foundation (Deutsche Forschungsgemeinschaft DFG) for funding the research unit FOR2131 and its sub-project P2 on “Scale-Problems in Assimilation of Passive Microwave Observation into Coupled Models”. Special thanks to the research group FOR2131, in particular to Dr. Bernd Schalge, for providing the Community Land Model (CLM) outputs to perform the SMOS simulations in Figure 5.

## Competing Interests

The authors have no competing interests to declare.

## References

1. Brodzik, M J and Knowles, K W 2002 EASE-Grid: A Versatile Set of Equal-Area Projections and Grids. In: Goodchild, M (ed.), Discrete Global Grids. Santa Barbara, California USA: National Center for Geographic Information & Analysis.

2. Chan, S, Njoku, E and Colliander, A 2014 Soil Moisture Active Passive (SMAP), Algorithm Theoretical Basis Document Level 1C Radiometer Data Product. Jet Propulsion Laboratory.

3. de Rosnay, P, Drusch, M, Boone, A, Balsamo, G, Decharme, B, Harris, P, Kerr, Y, Pellarin, T, Polcher, J and Wigneron, J P 2009 Amma land surface model intercomparison experiment coupled to the comunity microwave emission model: Almip-mem. J. Geophys. Res., 114(D05): 108.

4. Eaton, J W, Bateman, D, Hauberg, S and Wehbring, R 2014 GNU Octave version 3.8.1 manual: a high-level interactive language for numerical computations. CreateSpace Independent Publishing Platform. ISBN: 1441413006.

5. Entekhabi, D and Coauthors 2010 The soil moisture active and passive (SMAP) mission. Proc. IEEE, 98: 704–716. DOI: https://doi.org/10.1109/JPROC.2010.2043918

6. ESA 2011 Algorithm theoretical basis document (ATBD) for the SMOS L2 soil moisture processor development continuation project. Technical report, European Space Agency. ESA No.: SO-TN-ARR-L2PP-0037.

7. Galassi, M, Davies, J, Theiler, J, Gough, B, Jungman, G, Alken, P, Booth, M, Rossi, F and Ulerich, R 2015 GNU Scientific Library Reference Manual. http://www.gnu.org/software/gsl/, 3rd edition. ISBN: 0954612078.

8. Hulbert, C C 2016 MATIO User manual for version 1.5.9. https://matio.sourceforge.io/.

9. MathWokrs Inc 2010 MATLAB version 7.10.0 (R2010b). The MathWorks Inc., Natick, Massachusetts.

10. Kerr, Y H, Waldteufel, P, Wigneron, J P, Front, J and Berger, M 2001 Soil moisture retrieval from space: The soil moisture and ocean salinity (SMOS) mission. IEEE Trans. Geosci. Re. Sens., 39(8): 1729–1735. DOI: https://doi.org/10.1109/36.942551

11. Indra, S A 2015 SMOS DPGS: SMOS level 1 and auxiliary data products specifications. Technical report, INDRA sistemas S.A., Ref. SO-TN-IDR-GS-0005.

12. Saavedra, P, Simmer, C and Schalge, B 2016 Evaluation of modeled high resolution virtual brightness temperatures compared to space-borne observations for the neckar catchment. In: 14th Specialist Meeting on Microwave Radiometry and Remote Sensing of the Environment, 85–90. IEEE. DOI: https://doi.org/10.1109/MICRORAD.2016.7530510

13. Sahr, K, White, D and Kimerling, A J 2003 Geodesic discrete global grid systems. Cartography and Geographic Information Science, 30(2): 121–134. DOI: https://doi.org/10.1559/152304003100011090

14. Schalge, B, Rihani, J, Baroni, G, Erdal, D, Geppert, G, Haefliger, V, Haese, B, Saavedra, P, Neuweiler, I, Hendricks, H J, Ament, F, Attinger, S, Cirpka, O A, Kollet, S, Kunstmann, H, Vereecken, H and Simmer, C 2017 High-resolution virtual catchment simulations of the subsurface-land surface-atmosphere system. Discussion paper. In preparation.

15. Schlenz, F, dall Amico, J T, Mauser, W and Loew, A 2012 Analysis of smos brightness temperature and vegetation optical depth data with coupled land surface and radiative transfer models in southern germany. Hydrology and Earth System Sciences, 16: 3517–3533. DOI: https://doi.org/10.5194/hess-16-3517-2012


text:
# One side of a rectangle is twice the length of the other. The area is 100. What are the dimensions?

Jun 10, 2016

$\textcolor{b l u e}{\implies L = 5 \sqrt{2}} \text{ "larr" shorter sides}$

$\textcolor{b l u e}{\implies 2 L = 10 \sqrt{2}} \text{ "larr" longer side}$

#### Explanation:

Let the shorter side length be $L$

Then the longer side length is $2 L$

Thus given area $= 100 = \left(2 L\right) \times \left(L\right)$

$\implies 2 {L}^{2} = 100$

Divide both sides by 2 giving

$\frac{2}{2} {L}^{2} = \frac{100}{2}$

But $\frac{2}{2} = 1 \text{ and } \frac{100}{2} = 50$

${L}^{2} = 50$

Square root both sides

$\sqrt{{L}^{2}} = \sqrt{50}$

But $\text{ "50" " =" " 10xx5" " =" " 2xx5xx5" " =" } 2 \times {5}^{2}$

$\implies L = \sqrt{2 \times {5}^{2}}$

$\textcolor{b l u e}{\implies L = 5 \sqrt{2}} \text{ "larr" shorter sides}$

$\textcolor{b l u e}{\implies 2 L = 10 \sqrt{2}} \text{ "larr" longer side}$


text:
0 like 0 dislike
I need help asap, if u can help me with other questions after this I’ll give u 50+ points

0 like 0 dislike
C. 31 $$\frac{1}{6}$$

Step-by-step explanation:

1. Convert mixed numbers into improper fractions:

$$\frac{29}{4}$$ · 2² + ( $$\frac{17}{2}$$ - 2) ÷ 3

2. Calculate within the parenthesis:

$$\frac{29}{4}$$ · 2² + $$\frac{13}{2}$$ ÷ 3

3. Calculate the exponent:

$$\frac{29}{4}$$ · 4 + $$\frac{13}{2}$$ ÷ 3

4. Multiply and divide (left to right):

29 + $$\frac{13}{2}$$ ÷ 3

29 + $$\frac{13}{6}$$

$$\frac{187}{6}$$
= 31 $$\frac{1}{6}$$


text:
# 1 Introduction

## 1.1 Goals

• Gain experience with Observium

• Commands preceded with "$" imply that you should execute the command as a general user - not as root. • Commands preceded with "#" imply that you should be working as root. • Commands with more specific command lines (e.g. "rtrX>" or "mysql>") imply that you are executing commands on remote equipment, or within another program. # 2 Observium lab All components for Observium should already be installed - it will be now up to you to configure the service, add hosts, etc... If this is not the case, a different lab (observium-install.txt) is available. Follow the instructions in this lab to complete the initial Observium installation. Connect to your virtual machine as the sysadm user, then: # 3 1. Access Observium! A user has already been created for you. This is the user 'admin', and the password is the standard class password: http://observiumX.ws.nsrc.org/  User: admin Password: the class password You should now be logged in to Observium. There should not be any devices configured. Take a look at the interface for a few minutes. # 4 2. Edit the configuration file for Observium, to tell it about our network $ sudo editor /opt/observium/config.php

Now make the following changes:

Find the line:

$config['snmp']['community'] = array("public"); ... and change 'public' to 'NetManage' (NOT "netmanage" or "NETMANAGE"), so it looks like this: $config['snmp']['community'] = array("NetManage");

Also, add the following line below:

$sudo ./poller.php -h all ... Of course, we don't want to do this manually! We will automate this in the next steps. # 6 4. Tell Observium to run a discovery (scan) of the network, and start collecting data $ cd /opt/observium
$sudo ./discovery.php -h all Note that a LOT of information will appear! We'll run a data collection again manually: $ sudo ./poller.php -h all

# 7 5. Go back to the web interface

http://observiumX.ws.nsrc.org/

What do you notice ? How do you think Observium found that there were other devices on the network, and how did it know how to contact them ?

# 8 6. Enable cron jobs

It's time to make the data collection happen automatically.

Create the file /etc/cron.d/observium:

$sudo editor /etc/cron.d/observium ... and paste the following lines: 33 */6 * * * root /opt/observium/discovery.php -h all >> /dev/null 2>&1 */5 * * * * root /opt/observium/discovery.php -h new >> /dev/null 2>&1 */5 * * * * root /opt/observium/poller-wrapper.py 1 >> /dev/null 2>&1 It will take a little while for data to appear in the graphs. # 9 7. While we wait for the traffic to appear, we can navigate the interface Go to http://observiumX.ws.nsrc.org/ Click on "Devices" in menu at the top. Find the router for your group in the list, and click on its name. You will land on the Overview page for your device. You will notice that Observium has automatically detected many, many things about your router! At the top, under your router's hostname, you will see a list of tabs, all of them presenting information about your device: Overview | Graphs | Health | Ports | Routing | Inventory | Logs | Alerts ## 9.1 Graphs Under "Graphs", you will see all information which can be presented in the form of a graph: Network IO, Disk IO, Memory and CPU usage, etc... ## 9.2 Health You will also see the "Health" tab, which offers various metrics on how the device is doing hardware wise - if this information is available - such as temperature, voltage, fan speed, etc... Notice that some of this information is already shown in the overview page for the device (which you get when you click on the name of the device). ## 9.3 Ports Next we'll take a look at the Ports menu tab. Under "Ports" you will find a summary of traffic for all ports, including bits per second and packets per second, port speed, and the type of connection (Ethernet or other). Note that all elements can be clicked on, including the small graphs, and you will be taken to the corresponding page for the data source. ## 9.4 Routing The Routing tab will present you with an overview of enabled routing protocols. If you have enabled OSPF or BGP on your routers, you will get information about the active sessions, neighbors or peers, and other protocol specific information. ## 9.5 Inventory Inventory contains a full list of hardware modules and serial numbers for equipment installed on your router. Note that this will not work on all vendors. ## 9.6 Logs Under Logs you will see a list of events for your router: changes that have taken place in the configuration of the device, or in the state of interfaces, services, etc... ## 9.7 Alerts Alerts is currently empty for your router - we will see more about this later. ## 9.8 Topology Map Also, check out the "Map" function, available under "Ports". The Map function is an automatically generated diagram of the topology of your network, as seen from the point of view of your router. This will only function for CDP/LLDP enabled devices (Cisco, IOS, ...) You can enable CDP/LLDP on your Linux devices by installing 'ladvd' (sudo apt-get install ladvd). Try adding your Linux devices this way. It will only be available if you have enabled 'autodiscovery' 'xdp' as detailed in step 4. # 10 8. Add a location and contact for your router If you navigate back to the Overview page for your router (Go to Devices, click on your router's name), you will see that Observium has picked up the Hardware platform, the Operating System and the Uptime for your device. Now, log into your router with SSH and add the following information to your SNMP configuration: Set the location (make it a City, Country format, so that your host will appear on the geo map overview page) Set the contact (email address or name) To do this: rtr8> enable Password: rtr8# conf terminal Enter configuration commands, one per line. End with CNTL/Z. rtr8(config)# snmp-server contact user@email.address rtr8(config)# snmp-server location City, Country rtr8(config)# exit rtr8# write memory You want to replace "City, Country" with the city and country (or state) we are currently in. For example: Bloomington, Indiana Thimphu, Bhutan etc... If you wait a few minutes (maximum 5), you should see this information appear automatically on the Overview tab for your router in Observium. For fun, talk to your colleagues in the classroom, and ask them to pick a different City and Country for each of the routers! Then wait until Observium has re-scanned the devices, and look at the World map view on the main Observium page (http://observiumX.ws.nsrc.org/) # 11 9. Adding an interface description While you are still logged in to your router, add a pseudo interface to your router, to see if Observium will pick it up: rtr8(config)# interface loopback123 rtr8(config-if)# description A useless interface rtr8(config-if)# exit rtr8(config)# exit rtr8# write memory This should also show up in Observium after a while - try and find the description under "Ports" # 12 10. IP address lookup From Observium's main page, you will find, under the Overview menu at the TOP of the page), a submenu "Search" with 5 options: • IPv4 search -> http://observiumX.ws.nsrc.org/search/search=ipv4/ • IPv6 search -> http://observiumX.ws.nsrc.org/search/search=ipv6/ • MAC search -> http://observiumX.ws.nsrc.org/search/search=mac/ • ARP/NDP tables -> http://observiumX.ws.nsrc.org/search/search=arp/ • FDB tables -> http://observiumX.ws.nsrc.org/search/search=fdb/ Using the IPv4 and ARP table searches, try and search IPs of equipment in the class: • IP of the GW (10.10.0.254) • IPs of the routers (10.10.X.254) • IPs of the PCs (10.10.1.1, 10.10.5.17, etc...) Try and search IPs for your OWN laptop! Find the IP of you own laptop, and see if you can search it in Observium. Can you ? Why ? # 13 11. Add the switch From the TOP menu "Devices", choose "Add device". Fill out the fields: Hostname: sw.ws.nsrc.org Community: NetManage Click on "Add Host". After a few seconds, it should have added the switch. After a few minutes, data should appear in Observium for the switch. Notice the "Ports" line at the TOP right of the Observium interface. Does it say that ports are down ? Which ones ? Try and click on the 'X down' ports message to see which ports are down on which devices. # 14 12. Let's disable ports that are unused! Find the Overview page for "sw.ws.nsrc.org". From there, you can configure the device (the "key" icon on the top right). You are then presented with a configuration page for the device. Click on Ports, then you will get an overview of the Port status on the switch. Look at the ones that are "down". For those that are "down", check the "Ignore" box, then "Save" under "Index". If you call back the main Observium page: http://observiumX.ws.nsrc.org/ ... Observium should no longer complain about these ports! # 15 15. Have you tried adding your PC yet? Other PCs in the class? More fun things: • From the top menu, choose Devices -> All devices • Find your router, and click on it on the list. • Below the graph summary, you'll see a list of the interfaces on your router:  Fa0/0, Fa0/1, Null0 • Click on Fa0/0 • Click on "Real Time" # 16 16. Region settings for the map Optionally, you can change the part of the world the map will show when you log into observium. Look at the parameters for this listed at http://www.observium.org/wiki/Configuration_Options#Map_overview_settings In particular,$config['frontpage']['map']['region'] can be set to a particular country or region. There is more information at https://developers.google.com/chart/interactive/docs/gallery/geochart

In reference to the configuration settings in step 4.

The parameter \$config['frontpage']['map']['region'] can take multiple values. From:

region: The area to display on the map. (Surrounding areas will be displayed
as well.) Can be one of the following:

* 'world' - A map of the entire world.
* A continent or a sub-continent, specified by its 3-digit
code, e.g., '011' for Western Africa.
* A country, specified by its ISO 3166-1 alpha-2 code, e.g.,
'AU' for Australia.
* A state in the United States, specified by its ISO 3166-2:US
code, e.g., 'US-AL' for Alabama. Note that the resolution
option must be set to either 'provinces' or 'metros'.

You can test these options by changing the parameter and reloading the overview page of Observium.


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1

# What is the Best 5 HTP Supplement for Anxiety

Are you suffering from fatigue? Do you feel hyper-vigilant or have racing thoughts? If these symptoms keep you up at night, you may be suffering from anxiety.

As a leading mental health advocate, I know that anxiety disorders are the most common mental illness in the U.S., as reported by the National Institute of Mental Health. I also know that symptoms of anxiety, like panic attacks, phobias, and social anxiety can be paralyzing.

Anxiety disorders are remarkably treatable, but only about a third of anxiety sufferers seek treatment. Let’s look at how 5-HTP works for anxiety and panic attacks.

## How Do I Know If My Anxiety is Severe?

To understand how severe anxiety symptoms can become, take a look at this list of symptoms as provided by Mayo Clinic:

• Sudden or overwhelming fear
• Palpitations
• Sweating
• Trembling
• Shortness of breath
• Sense of choking
• Chest pain
• Nausea
• Dizziness
• A feeling of being detached from the world
• Fear of dying
• Numbness or tingling in the limbs or entire body
• Chills or hot flushes

## Does 5 HTP Reduce Anxiety?

For most people, anxiety serves a valuable purpose. When you face a situation that makes you apprehensive, the rates of your breathing and heart increase. This causes more blood to flow to your brain and signals your brain to release serotonin. These physiological changes allow you to think and respond under pressure.

Check out this video by Dr. John Gray, author of “Men Are from Mars, Women are from Venus,” where he explains how serotonin helps us recover from the symptoms of anxiety. We consume Tryptophan in our diet, which is then converted to 5 htp in our brains. 5 htp helps release serotonin, which has a calming effect on the limbic system to control anxiety.

According to WebMD, 5 htp is proven to be effective for anxiety and other afflictions such as:

• Anxiety
• Sleep disorders
• Depression
• Migraines
• Fibromyalgia
• Obesity
• Premenstrual syndrome
• Premenstrual dysphoric disorder
• Seizure disorders
• Parkinson’s disease

## 5 HTP for Anxiety Gets Strong Reviews

5 htp (5-Hydroxytryptophan) is produced naturally within our bodies as a by-product of the protein building block L-tryptophan. Manufacturers are able to reproduce it from the seeds of Griffonia Simplicifolia, which is a plant that is grown in Africa. See this video by Dr. Robin Murphy, who presented on Griffonia Simplicifolia as a forgotten natural remedy, at a seminar of his peers.

The National Center for Biotechnology Information reports that studies in rodents prove the effectiveness of 5 htp. You can get an idea about the effectiveness of 5 htp by reading the strong user reviews on Amazon.com.

## What is the 5-HTP Recommended Dosage for Anxiety?

In this video, Dr. Ray Sahelian, he discusses his recommendations for 5 htp anxiety dosage. Dr. Sahelian recommends starting with a dosage of 50mg. to reduce some of the negative side-effects, such as nausea or dizziness.

He notes that 5 htp lingers in the system, so it may take a few days before feeling the positive effects. The dosage can be increased to 100mg of 5 htp for anxiety if symptoms persist.

Progressive Health reports that numerous studies have proven effectiveness in treating children for depression, anxiety, night terrors, and sleep disorders.

The recommended dosage of 5 htp is lower for children than for adults, due to the potential for serotonin toxicity in the brain. Ask your child’s pediatrician for the proper dosage of 5 htp for your child.

Children that have liver disease should not take 5 htp because of the risks of liver impairment, which reduces the effects of serotonin over their lifespan. 5 htp is also reported to cause seizures in children with Down syndrome.

## Where Do I Get 5 HTP for Anxiety?

Absorb Health has one of the best 5htp supplements for anxiety. The formula comes in 100mg. capsules. Another good place to find 5 htp for anxiety is at Amazon.com, where you will find many other brands. Consumers give top ratings to Irwin Naturals, NOW Foods, and Best Naturals.

Irwin Naturals includes a blend of herbals and botanicals. Reviewers who took the brand from NOW Foods reported good results when taking it either with green tea or after eating blueberries. The Best Naturals brand comes in a lower dose of 50 mg., which might be a good brand to start with for anyone that tends to have a sensitive stomach.

5 htp is affordable with most prices around $10-$15 per bottle. It’s a great value when you consider it’s an investment in feeling better.

If you’ve suffered from sleeplessness, anxiety, or other similar symptoms, you know that it leaves you feeling weak and down. There’s no reason to feel worn out when 5thp for anxiety can help. Any side effects are usually mild and dissipate over time. Doctors recommend starting with a small dose of 50mg. and increasing it to 100mg. as needed.

Try one of these products or one of the other 5htp products for anxiety at Amazon.com:

Taking 5 htp helps relieve anxiety and here are a few other things you can do that will help:

• Eat healthily and don’t skip meals
• Try a relaxing activity like yoga, meditation, or massage
• Limit alcohol and caffeine intake
• Get exercise daily
• Accept that you can’t control everything


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# 3.7: Recursive Definitions

Recursion occurs in programming when a subroutine is defined—or at least partially defined—in terms of itself. But recursion also occurs outside of programming. A recurs- ive definition is a definition that includes a reference to the term that is being defined. A recursive definition defines something at least partially in terms of itself. As in the case of recursive subroutines, mathematical induction can often be used to prove facts about things that are defined recursively.

As I already noted, there is a recursive definition for n!, for in N, and we can use this definition to prove facts about the factorials. We can define 0! = 1 and n! = · (− 1)!for > 0. Do you see how the base case and the inductive case in an inductive proof can correspond to the two parts of the recursive definition?
Other sequences of numbers can also be defined recursively. For example, the famous Fibonacci sequence is the sequence of numbers $$f_{0}, f_{1}, f_{2}, \ldots,$$ defined recursively by

$$f_{0}=0$$

$$f_{1}=1$$

$$f_{n}=f_{n-1}+f_{n-2} \qquad$$ for $$n>1$$

Using this definition, we compute that

$$f_{2}=f_{1}+f_{0}=0+1=1$$
$$f_{3}=f_{2}+f_{1}=1+1=2$$
$$f_{4}=f_{3}+f_{2}=2+1=3$$
$$f_{5}=f_{4}+f_{3}=3+2=5$$
$$f_{6}=f_{5}+f_{4}=5+3=8$$
$$f_{7}=f_{6}+f_{5}=8+5=13$$

and so on. Based on this definition, we can use induction to prove facts about the Fibonacci sequence. We can prove, for example, that $$f_{n}$$ grows exponentially with $$n,$$ even without finding an exact formula for $$f_{n} :$$

Theorem 3.14.

The Fibonaccisequence, $$f_{0}, f_{1}, f_{2}, \ldots,$$ satisfies $$f_{n}>\left(\frac{3}{2}\right)^{n-1},$$ for $$n \geq 6$$

Proof. We prove this by induction on $$n .$$ For $$n=6,$$ we have that $$f_{n}=8$$ while $$1.5^{n-1}=$$ $$1.5^{5},$$ which is about $$7.6 .$$ So $$f_{n}>1.5^{n-1}$$ for $$n=6 .$$ Similarly, for $$n=7,$$ we have $$f_{n}=13$$and $$1.5^{n-1}=1.5^{6},$$ which is about $$11.4 .$$ So $$f_{n}>1.5^{n-1}$$ for $$n=7$$

Now suppose that is an arbitrary integer with > 7. Suppose that we already know that $$f_{n}>1.5^{n-1}$$ for $$n=k-1$$ and for $$n=k-2 .$$ We want to show that the inequality then holds for as well. But

$$f_{k}=f_{k-1}+f_{k-2}$$

$$>1.5^{(k-1)-1}+1.5^{(k-2)-1}$$    (by the induction hypothesis)

$$=1.5^{k-2}+1.5^{k-3}$$

$$=(1.5) \cdot\left(1.5^{k-3}\right)+\left(1.5^{k-3}\right)$$

$$=(2.5) \cdot\left(1.5^{k-3}\right)$$

$$>\left(1.5^{2}\right) \cdot\left(1.5^{k-3}\right)$$    $$\left(\text { since } 1.5^{2}=2.25\right)$$

$$=1.5^{k-1}$$

This string of equalities and inequalities shows that $$f_{k}>1.5^{k-1} .$$ This completes the induction and proves the theorem.

Exercises

1. Prove that the Fibonacci sequence, $$f_{0}, f_{1}, f_{2}, \dots,$$ satisfies $$f_{n}<2^{n}$$ for all natural numbers $$n \geq 1$$

2. Suppose that $$a_{1}, a_{2}, a_{3}, \ldots,$$ is a sequence of numbers which is defined recursively by $$a_{1}=1$$ and $$a_{n}=2 a_{n-1}+2^{n-1}$$ for $$n>1 .$$ Prove that $$a_{n}=n 2^{n-1}$$ for every positive integer $$n$$


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## power series of inverse trig functions

How do you find the power series for inverse trig functions? Can I find the power series for arcsin by manipulating the power series for sin?

Thanks!
 PhysOrg.com science news on PhysOrg.com >> Galaxies fed by funnels of fuel>> The better to see you with: Scientists build record-setting metamaterial flat lens>> Google eyes emerging markets networks
 Blog Entries: 5 Recognitions: Homework Help Science Advisor I don't think so. Even for the simplest of functions you already run into trouble, consider for example y = x2n (for n = 1, 2, ...). The power series for x1/(2n) is already non-trivial.
 You can, it is called "reversion" of a series. But the formulas get more and more complicated as you proceed. For arcsin, a better way to find the series is to start with the binomial series for $(1-x^2)^{-1/2}$ and integrate term-by-term.

## power series of inverse trig functions

Identify the inverse trig function as a hypergeometric function, and manipulate the series expansion of the hypergeometric function. Any book on hypergeometric functions will give the necessary formulae.


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# Problem: Find ΔE for the freezing of water at -9.50˚C. The specific heat capacity of ice is 2.04 J/g•˚C and its heat of fusion is -332 J/g.

###### FREE Expert Solution

$\overline{)\mathbf{∆}\mathbf{H}\mathbf{=}\mathbf{∆}\mathbf{E}\mathbf{+}\mathbf{P}\mathbf{∆}\mathbf{V}}$

$\mathbf{∆}\mathbf{E}\mathbf{=}\mathbf{∆}\mathbf{H}\mathbf{-}\mathbf{P}\mathbf{∆}\mathbf{V}$

ΔHfreezing, 9.50°C = -311.67 J/g

q = -311.67 J/g

85% (34 ratings)
###### Problem Details

Find ΔE for the freezing of water at -9.50˚C. The specific heat capacity of ice is 2.04 J/g•˚C and its heat of fusion is -332 J/g.


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# Math Help - Need some help on a couple integrations

1. ## Need some help on a couple integrations

You dont need to do them all but one would be nice. I want to see how to do it rather than just get answers. Thanks.

Integral of [(e^x + cos(x)]^2 dx
I tried to expand it and then got confused with integration by parts but I think you use that

Integral of e^2x ln (1+e^x) dx
I always hated ln

Integral of (12-4x-x^2)^.5 dx
I tried to factor it and got it to [-(x+6)(x-2)]^.5

Integral of cos^4 (x) sin^3 (x) dx
Should i change the cos^4 (x) to like (1-sin(x))^2 or something?

Integral of [(4-x^2)^.5]/ x dx

2. Originally Posted by DaAzNJRiCh

Integral of e^2x ln (1+e^x) dx
I always hated ln
I assume,
$\int e^{2x}\ln (1+e^x)dx$
You can write it as,
$\int [(1+e^x)-1] \ln (1+e^x) e^x dx$
Then we can express this as,
$(f\circ g) \cdot g'$
Where,
$g=1+e^x$
$f=(x-1)\ln x$
Thus we need to find,
$F=\int (x-1)\ln x dx$
Which you can do with integration by parts.
$F\circ g+C$

3. Originally Posted by DaAzNJRiCh
You dont need to do them all but one would be nice. I want to see how to do it rather than just get answers. Thanks.

Integral of [(e^x + cos(x)]^2 dx
I tried to expand it and then got confused with integration by parts but I think you use that
$\int [e^x+\cos(x)]^2\ dx=\int e^{2x}+2e^x \cos(x)+(\cos(x))^2\ dx=$ $
\int e^{2x}\ dx +2\int e^x \cos(x)\ dx+\int (\cos(x))^2\ dx$

The first and last integrals on the right are standard integrals and should pose
no problem. Now consider the middle integral on the right:

$
2\int e^x \cos(x)\ dx=2\ Re \left[ \int e^x e^{ix}\ dx \right]=2\ Re \left[ \int e^{x(1+i)}\ dx \right]
$

Which is fairly simple to deal with.

RonL

4. Originally Posted by DaAzNJRiCh

Integral of (12-4x-x^2)^.5 dx
I tried to factor it and got it to [-(x+6)(x-2)]^.5
Okay you have,
$\int \sqrt{12-4x-x^2} dx$
You factored, but you should have completed the square.
$\int \sqrt{-(x^2+4x-12)}dx$
Thus,
$\int \sqrt{-(x^2+4x+4-12-4)}dx$
Thus,
$\int \sqrt{16-(x+2)^2}dx$
Use a linear subsitution,
$u=x+2$
Thus,
$\int \sqrt{4^2-u^2} du$
This is a standard inverse sine function substitution.

5. Originally Posted by DaAzNJRiCh

Integral of cos^4 (x) sin^3 (x) dx
Should i change the cos^4 (x) to like (1-sin(x))^2 or something?
Almost!
$\int \cos^4 x\sin^2 x\sin x dx$
$\int \cos^4 x(1-\cos^2 x)\sin xdx$
Let,
$u=\sin x$
Thus,
$\int u^4(1-u^2)du$
The rest is trivial.


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0 like 0 dislike
A transportation worker takes a simple random sample of 105 toll booth receipts for one day and finds that the mean is $9.45 with a standard deviation of$2.41. what is the 99% confidence interval for the mean amount of all the receipts that day?


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# Why isn't CO2 flammable? [closed]

Given that $\ce{CO2}$ is comprised of carbon and oxygen, why isn't it a flammable gas?

• What has burned can never burn. – Ivan Neretin Nov 18 '16 at 5:33
• fully oxidized. – gannex Nov 18 '16 at 7:33


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Q

# A hemispherical bowl made of brass has inner diameter 10.5 cm. Find the cost of tin-plating it on the inside at the rate of Rs 16 per 100 cm^2.

Q : 5    A hemispherical bowl made of brass has inner diameter $\small 10.5\hspace{1mm}cm$. Find the cost of tin-plating it on the inside at the rate of  Rs 16 per $\small 100\hspace{1mm}cm^2$

Views

Given,

The inner radius of the hemispherical bowl = $r = \frac{10.5}{2}\ cm$

We know,

The curved surface area of a hemisphere = $2\pi r^2$

$\therefore$ The surface area of the hemispherical bowl = $2\times\frac{22}{7}\times\left (\frac{10.5}{2} \right )^2$

$=11\times1.5\times10.5$

$= 173.25 \ cm^2$

Now,

Cost of tin-plating $\small 100\hspace{1mm}cm^2$ =  Rs 16

$\therefore$ Cost of tin-plating $\small 33\hspace{1mm}cm^2$ =  $\small \\ Rs. \left (\frac{16}{100}\times173.25 \right )$

$\small = Rs. 27.72$

Therefore, the cost of tin-plating it on the inside is $\small Rs. 27.72$

Exams
Articles
Questions


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L'Hopital's rule

Tom92

Hi,
Why can't i use L'Hopital's rule in this limit:

lim (2x+sin2x+1)/(2x+sin2x)(sinx+3)^2 as x->infinity

Thanks...

Prove It

MHF Helper
Hi,
Why can't i use L'Hopital's rule in this limit:

lim (2x+sin2x+1)/(2x+sin2x)(sinx+3)^2 as x->infinity

Thanks...
I assume this is

$$\displaystyle \lim_{x \to \infty}\frac{2x + \sin{2x} + 1}{(2x + \sin{2x})(\sin{x} + 3)^2}$$.

You can only use L'Hospital's Rule if you get $$\displaystyle \frac{0}{0}$$ or $$\displaystyle \frac{\infty}{\infty}$$ from direct substitution.

But in this case, you'll get $$\displaystyle \frac{1}{0}$$.

Tom92

Sudharaka

I assume this is

$$\displaystyle \lim_{x \to \infty}\frac{2x + \sin{2x} + 1}{(2x + \sin{2x})(\sin{x} + 3)^2}$$.

You can only use L'Hospital's Rule if you get $$\displaystyle \frac{0}{0}$$ or $$\displaystyle \frac{\infty}{\infty}$$ from direct substitution.

But in this case, you'll get $$\displaystyle \frac{1}{0}$$.
Dear Prove It,

I think you have made mistake in this case. Note that the limit goes to "infinity". Hence the denominator and numerator both tends to "infinity."

Prove It

MHF Helper
Dear Prove It,

I think you have made mistake in this case. Note that the limit goes to "infinity". Hence the denominator and numerator both tends to "infinity."
Oh, oops. Hahaha, I subsituted $$\displaystyle x = 0$$ instead.

Then I don't see why you can't use L'Hospital's Rule in this case...

Sudharaka

Oh, oops. Hahaha, I subsituted $$\displaystyle x = 0$$ instead.

Then I don't see why you can't use L'Hospital's Rule in this case...
Dear Prove It,

But notice that,

$$\displaystyle \lim_{x\rightarrow{\infty}}\frac{2x+\sin{2x}+1}{(2x+\sin{2x})(\sin{x}+3)^2}$$

Dividing the numerator and denominator by "2x";

$$\displaystyle \lim_{x\rightarrow{\infty}}\frac{1+\frac{\sin{2x}}{2x}+\frac{1}{2x}}{(1+\frac{\sin{2x}}{2x})(sinx+3)^2}$$

Therefore the limit does not exist.