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## Saturday, December 31, 2005 ... //

### Internet gender gap

First, an off-topic answer. Celal asks me about the leap seconds - why has not the Earth already stopped to rotate if there are so many leap seconds. The answer is that we are now indeed inserting a leap second in most of the years - which means that one year is longer by roughly 1 second than it was back in 1820 when the second was defined accurately enough. More precisely, what I want to say is that one solar day is now longer by roughly 1/365 of a second than it was in the 19th century; what matters is of course that the noon stays at 12 pm.

Although the process of slowing down the Earth's rotation has some irregularities, you can see that you need roughly 200 years to increase the number of the required leap seconds per year by one. In order to halve the angular velocity, you need to increase the number of leap seconds roughly by 30 million (the number of seconds per year), which means that you need 30 million times 200 years which is about 6 billion years. Indeed, at time scales comparable to the lifetime of the solar system, the length of the day may change by as much as 100 percent.

100 percent is a bit of exaggeration because a part of the recent slowing is due to natural periodic fluctuations and aperiodic noise, not a trend. However, coral reefs indeed seem to suggest that there were about 400 days per year 0.4 billion years ago. Don't forget that the slowing down is exponential, I think, and therefore the angular velocity will never quite drop to zero (which has almost happened to our Moon).

BBC informs that

While the same percentage of men and women use the internet, they use it in very different ways & they search for very different things. Women focus on maintaining human contacts by e-mail etc. while men look for new technologies and ways to do new things in novel ways.

• "This moment in internet history will be gone in a blink," said Deborah Fallows, senior research fellow at Pew who wrote the report.

I just can't believe that someone who is doing similar research is simultaneously able to share such feminist misconceptions. The Internet has been around for ten years and there has never been any political or legal pressure for the men and women to do different things - the kind of pressures in the past that is often used to justify similar hypotheses about the social origin of various effects.

People are just doing whatever they find natural to do with the internet, and a result is that there are significant differences between men and women. These differences follow primarily from the fact that we are wired differently. Everyone who is able to look around and evaluate the perceptions knows that.

Growing gender gap in computer science

Meanwhile, the gender gap grows in computer science (CS). Boston Globe describes the computer science department of Tufts University. Twenty years ago, many women were attracted to the department in order for the social engineers to produce "gender equality". The peak of this movement was in 1985 when as many as 37 percent of the computer science Bc. degrees went to women.

You have had everything you need according to the pseudoscientific theories of discrimination: a lot of female role models and good, welcoming atmosphere. Nevertheless, the airplanes don't land. The percentage of women among the new alumni dropped below 20 percent again. Something must be wrong, right? Still, many people are going to continue to propagate their patently false explanations of the observed statistics that have been falsified hundreds of times. For example, the Boston Globe article by Marcella B. wrote

• As the popularity of computer science soared in the first half of the 1980s, many university departments became overburdened and more competitive, some professors argue. Introductory classes were taught in a way that emphasized technical minutiae over a broader sense of what was important and exciting about the field, a style catering to the diehard -- and overwhelmingly male -- techies rather than curious new recruits.

This is a completely unreasonable comment. The reality is that the field of computer science simply is a competitive field. It has been one for quite many years. And the males seem to be doing statistically better both in the technical minutiae of that field as well as in the excitement about its general features. Moreover, both of these things - general excitement as well as technical details - are necessary for the computer science people to do their work competitively. Abandoning "technical minutiae" is nothing else than lowering the standards.

I wonder how many more years, decades or centuries are necessary for the believers in various feminist religions to start to realize that something is wrong with their beliefs.

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# Math Insight

### The dot product

The dot product between two vectors is based on the projection of one vector onto another. Let's imagine we have two vectors $\vc{a}$ and $\vc{b}$, and we want to calculate how much of $\vc{a}$ is pointing in the same direction as the vector $\vc{b}$. We want a quantity that would be positive if the two vectors are pointing in similar directions, zero if they are perpendicular, and negative if the two vectors are pointing in nearly opposite directions. We will define the dot product between the vectors to capture these quantities.

But first, notice that the question “how much of $\vc{a}$ is pointing in the same direction as the vector $\vc{b}$” does not have anything to do with the magnitude (or length) of $\vc{b}$; it is based only on its direction. (Recall that a vector has a magnitude and a direction.) The answer to this question should not depend on the magnitude of $\vc{b}$, only its direction. To sidestep any confusion caused by the magnitude of $\vc{b}$, let's scale the vector so that it has length one. In other words, let's replace $\vc{b}$ with the unit vector that points in the same direction as $\vc{b}$. We'll call this vector $\vc{u}$, which is defined by $$\vc{u} = \frac{\vc{b}}{\|\vc{b}\|}.$$

The dot product of $\vc{a}$ with unit vector $\vc{u}$, denoted $\vc{a} \cdot \vc{u}$, is defined to be the projection of $\vc{a}$ in the direction of $\vc{u}$, or the amount that $\vc{a}$ is pointing in the same direction as unit vector $\vc{u}$. Let's assume for a moment that $\vc{a}$ and $\vc{u}$ are pointing in similar directions. Then, you can imagine $\vc{a} \cdot \vc{u}$ as the length of the shadow of $\vc{a}$ onto $\vc{u}$ if their tails were together and the sun was shining from a direction perpendicular to $\vc{u}$. By forming a right triangle with $\vc{a}$ and this shadow, you can use geometry to calculate that \begin{gather}\vc{a} \cdot \vc{u} = \|\vc{a}\| \cos \theta, \label{dot_product_unit} \end{gather} where $\theta$ is the angle between $\vc{a}$ and $\vc{u}$.

If $\vc{a}$ and $\vc{u}$ were perpendicular, there would be no shadow. That corresponds to the case when $\cos \theta = \cos \pi/2 =0$ and $\vc{a} \cdot \vc{u}=0$. If the angle $\theta$ between $\vc{a}$ and $\vc{u}$ were larger than $\pi/2$, then the shadow wouldn't hit $\vc{u}$. Since in this case $\cos \theta <0$, the dot product $\vc{a} \cdot \vc{u}$ is also negative. You could think of $-\vc{a} \cdot \vc{u}$ (which is positive in this case) as being the length of the shadow of $\vc{a}$ on the vector $-\vc{u}$, which points in the opposite direction of $\vc{u}$.

But we need to get back to the dot product $\vc{a}\cdot \vc{b}$, where $\vc{b}$ may have a magnitude different than one. This dot product $\vc{a}\cdot \vc{b}$ should depend on the magnitude of both vectors, $\|\vc{a}\|$ and $\|\vc{b}\|$, and be symmetric in those vectors. Hence, we don't want to define $\vc{a}\cdot \vc{b}$ to be exactly the projection of $\vc{a}$ on $\vc{b}$; we want it to reduce to this projection for the case when $\vc{b}$ is a unit vector. We can accomplish this very easily: just plug the definition $\vc{u}=\frac{\vc{b}}{\|\vc{b}\|}$ into our dot product definition of equation \eqref{dot_product_unit}. This leads to the definition that the dot product $\vc{a} \cdot \vc{b}$, divided by the magnitude $\|\vc{b}\|$ of $\vc{b}$, is the projection of $\vc{a}$ onto $\vc{b}$. $$\frac{\vc{a}\cdot \vc{b}}{\|\vc{b}\|} = \|\vc{a}\| \cos\theta.$$ Then, if we multiply by through by $\|\vc{b}\|$, we get a nice symmetric definition for the dot product $\vc{a} \cdot \vc{b}$. \begin{gather} \vc{a}\cdot \vc{b} = \|\vc{a}\| \|\vc{b}\|\cos\theta. \label{dot_product_definition}\tag{2} \end{gather}

The picture of the geometric interpretation of $\vc{a} \cdot \vc{b}$ is almost identical to the above picture for $\vc{a} \cdot \vc{u}$. We just have to remember that we have to divide through by $\|\vc{b}\|$ to get the projection of $\vc{a}$ onto $\vc{b}$.

In the following interactive applet, you can explore this geometric intrepretation of the dot product, and observe how it depends on the vectors and the angle between them. (The reported number shouldn't depend on the length of $\|\vc{b}\|$ since we divided by that magnitude.)

The Java applet did not load, and the above is only a static image representing one view of the applet. The applet was created with Geogebra. The applet is not loading because it looks like you do not have Java installed. You can click here to get Java.

The dot product as projection. The dot product of the vectors $\vc{a}$ (in blue) and $\vc{b}$ (in green), when divided by the magnitude of $\vc{b}$, is the projection of $\vc{a}$ onto $\vc{b}$. This projection is illustrated by the red line segment from the tail of $\vc{b}$ to the projection of the head of $\vc{a}$ on $\vc{b}$. You can change the vectors $\vc{a}$ and $\vc{b}$ by dragging the points at their ends or dragging the vectors themselves. Notice how the dot product is positive for acute angles and negative for obtuse angles. The reported number does not depend on $\|\vc{b}\|$ only because we've divided through by that magnitude.

The geometric definition of equation \eqref{dot_product_definition} makes the properties of the dot product clear. One can see immediately from the formula that the dot product $\vc{a}\cdot\vc{b}$ is positive for acute angles and negative for obtuse angles. The formula demonstrates that the dot product grows linearly with the length of both vectors and is commutative, i.e., $\vc{a} \cdot \vc{b} = \vc{b} \cdot \vc{a}$.

However, the geometric formula \eqref{dot_product_definition} is not convenient for calculating the dot product when we are given the vectors $\vc{a}$ and $\vc{b}$ in terms of their components. To facilitate such calculations, we derive a formula for the dot product in terms of vector components. With such formula in hand, we can run through examples of calculating the dot product.

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# Encoders

## What is an Encoder?

Data must be encoded into sparse binary arrays for an HTM system to process it. These binary arrays define an input space for the Spatial Pooling algorithm. SDRs stream across this input space as spatial patterns over time. Spatial patterns must contain semantic meaning within their representations in order for HTM systems to process them.

SDRs are quite different from standard computer representations, such as ASCII for text, in that meaning is encoded directly into the representation. An SDR consists of a large array of bits of which most are zeros and a few are ones. Each bit carries some semantic meaning, so if two SDRs have more than a few overlapping one-bits, then those two SDRs have similar meanings.

Any data that can be converted into an SDR can be used in a wide range of applications using HTM systems. Consequently, the first step of using an HTM system is to convert a data source into an SDR using what we call an encoder. The encoder converts the native format of the data into an SDR that can be fed into an HTM system. The encoder is responsible for determining which output bits should be ones, and which should be zeros, for a given input value in such a way as to capture the important semantic characteristics of the data. Similar input values should produce highly overlapping SDRs.

## The Encoding Process

The encoding process is analogous to the functions of sensory organs of humans and other animals. The cochlea, for instance, is a specialized structure that converts the frequencies and amplitudes of sounds in the environment into a sparse set of active neurons (Webster et al, 1992; Schuknecht, 1974). The basic mechanism for this process (Fig. 1) comprises a set of inner hair cells organized in a row that are sensitive to different frequencies. When an appropriate frequency of sound occurs, the hair cells stimulate neurons that send the signal into the brain. The set of neurons that are triggered in this manner comprise the encoding of the sound as a Sparse Distributed Representation.

One important aspect of the cochlear encoding process is that each hair cell responds to a range of frequencies, and the ranges overlap with other nearby hair cells. This characteristic provides redundancy in case some hair cells are damaged but also means that a given frequency will stimulate multiple cells. And two sounds with similar frequencies will have some overlap in the cells that are stimulated. This overlap between representations is how the semantic similarity of the data is captured in the representation. It also means that the semantic meaning is distributed across a set of active cells, making the representation tolerant to noise or subsampling.

The cochleae for different animals respond to different ranges of frequencies and have different resolutions for which they can distinguish differences in the frequencies. While very high frequency sounds might be important for some animals to hear precisely, they might not be useful to others. Similarly, the design of an encoder is dependent on the type of data. The encoder must capture the semantic characteristics of the data that are important for your application. Many of the encoder implementations in NuPIC take range or resolution parameters that allow them to work for a broad range of applications.

There are a few important aspects that need to be considered when encoding data:

1. Semantically similar data should result in SDRs with overlapping active bits.
2. The same input should always produce the same SDR as output.
3. The output should have the same dimensionality (total number of bits) for all inputs.
4. The output should have similar sparsity for all inputs and have enough one-bits to handle noise and subsampling.

In following sections we will examine each of these characteristics in detail and then describe how you can encode several different types of data. Note that several SDR encoders exist already and most people will not need to create their own. Those who do should carefully consider the above criteria.

#### 1) Semantically similar data should result in SDRs with overlapping active bits

To create an effective encoder, you must understand the aspects of your data that should contribute to similarity. In the cochlea example above, the encoder was designed to make sounds with similar pitch have similar representations but did not take into account how loud the sounds were, which would require a different approach.

The first step to designing an encoder is to determine each of the aspects of the data that you want to capture. For sound, the key features may be pitch and amplitude; for dates, it may be whether or not it is a weekend.

The encoder should create representations that overlap for inputs that are similar in one or more of the characteristics of the data that were chosen. So for weekend encoders, dates that fall on Saturdays and Sundays should overlap with each other, but not as much or at all with dates that fall on weekdays.

#### Preserving Semantics: A Formal Description

Here we formalize the encoding process by defining a set of rules that relate the semantic similarity of two inputs with the number of overlapping one9bits in the corresponding encoded SDRs.

Let $A$ be an arbitrary input space and let $S(n,k)$ be the set of SDRs of length $n$ with $k$ ON bits. An encoder $f$ is simply a function $f$$A \rightarrow S(n,k)$ . A distance score $d_A$ over space $A$ is a function $d_A$$A \times A \rightarrow R$ that satisfies three conditions:

• 1. $\forall x,y \in A,d_A (x,y) ≥ 0$
• 2. $\forall x,y \in A,d_A (x,y) = d_A (y,x)$
• 3. $\forall x \in A,d_A (x,x) = 0$

Equation 1 requires the semantic similarity metric give a distance value of zero or greater. Equation 2 requires the distance metric to be symmetric. And Equation 3 requires that the distance between two identical values be zero.

Given an input space and a distance score, we can evaluate an encoder by comparing the distance scores of pairs of inputs with the overlaps of their encodings. Two inputs that have low distance scores should have SDRs with high overlap, and vice versa. Moreover, if two SDRs have higher overlap than two other SDRs, then the former’s pre-encoding distance score should be lower than the latter’s. We state this formally below.

For SDRs $s$ and $t$ with the same length, let $O(s,t)$ be the number of overlapping bits (i.e. the number of ON bits in $s$ & $t$ ). Then for an encoder $f$$A \rightarrow S(n,k)$ and $\forall w,x,y,z \in A$ ,

• 4. $O(f(w),f(x)) ≥ O(f(y),f(z)) \Leftrightarrow d_A (w,x) ≤ d_A (y,z)$

Equation 4 states that encodings with more overlapping one bits means the values have greater semantic similarity and, inversely, that values with greater semantic similarity will have encodings with more overlapping one bits. It is not always possible to create an encoder that satisfies this, but the equation can be used as a heuristic to evaluate the quality of an encoder.

#### 2: The same input should always produce the same SDR as output

Encoders should be deterministic so that the same input produces the same output every time. Without this property, the sequences learned in an HTM system will become obsolete as the encoded representations for values change. Avoid creating encoders with random or adaptive elements. It can be tempting to create adaptive encoders that adjust representations to handle input data with an unknown range. There is a way to design an encoder to handle this case without changing the representations of inputs that is described below in the section labeled “A more flexible encoder method.” This method allows encoders to handle input with unbounded or unknown ranges.

#### 3: The output should have the same dimensionality (total number of bits) for all inputs

The output of an encoder must always produce the same number of bits for each of its inputs. SDRs are compared and operated on using a bit-by-bit assumption such that a bit with a certain “meaning” is always in the same position. If the encoders produced varying bit lengths for the SDRs, comparisons and other operations would not be possible.

#### 4: The output should have similar sparsity for all inputs and have enough one-bits to handle noise and subsampling

The fraction of total ON bits in an encoder can vary from around 1% to 35%, but the sparsity should be relatively fixed for a given application of an encoder. While keeping the sparsity the same should be the rule, small variations in sparsity will not have a negative effect. Additionally, there must be enough one-bits to handle noise and subsampling. A general rule of thumb is to have at least 20-25 one bits. Encoders that produce representations with fewer than 20 one bits do not work well in HTM systems since they may become extremely susceptible to errors due to small amounts of noise or non-determinism.


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Limits using epsilon delta definition $f(x,y)=xy$ for functions of two variables

Prove: using $\epsilon$-$\delta$ definition, the limit of both $f$ and $g$ as $(x,y)\to (0,0)$ is $0$.

1. $f(x,y)=xy$

2. $g(x,y)=\frac{xy}{x^2 +y^2+1}$

Also, for Q2 can I convert $g(x,y)$ to $m(x,y)/n(x,y)=g(x,y)$ using arithmetic of limits, then prove using $\epsilon$-$\delta$ definition the limit of function $m$ and $n$ separately; then combine the two?

Thanks :)

I wonder if this is correct: $|xy-0|<\epsilon$ given $|x-0|< \delta$ and $|y-0|< \delta$

$|xy-0|< |x-0||y-0|<\delta^2=\epsilon$

therefore: $\delta<\epsilon^{1/2}$

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eddited g(x,y).... – jake Oct 16 '12 at 13:02
why do you care about delta/epsilon? for function of two variable you can do by another way,for first just choose two path ,one for x axis and second on y axis,in both case,if you go on X axis path y is equal to zero,on y axis the same x is equal to zero,so result in both case is zero – dato datuashvili Oct 16 '12 at 13:05
i dont care about epsilon delta :(. my homework question does >< – jake Oct 16 '12 at 13:08
Presumably he cares about delta/epsilon because it is a problem in a chapter on delta/epsilon proofs, and he is meant to apply it. @dato – Thomas Andrews Oct 16 '12 at 13:09
thanks dato! seems i should Google more often :P – jake Oct 16 '12 at 13:28

1 Answer

1. First notice that $$|x|=\sqrt{x^2}\le \sqrt{x^2+y^2}=\|(x,y)\|_2,\ |y|=\sqrt{y^2}\le \sqrt{x^2+y^2}=\|(x,y)\|_2 \quad \forall (x,y) \in \mathbb{R}^2.$$ Given $\varepsilon>0$, let $\delta=\sqrt{\varepsilon}$. We have $$\|(x,y)\|_2\le \delta \Longrightarrow |f(x,y)|=|x||y|\le \|(x,y)\|_2^2 \le \delta^2=\varepsilon$$
2. Observe that $$|g(x,y)|=\frac{|f(x,y)|}{x^2+y^2+1}\le |f(x,y)| \quad \forall (x,y) \in \mathbb{R}^2$$ and use 1.
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What does the subscript "2" mean in the norm notation? – mavavilj Sep 23 '15 at 18:41
It stands for the euclidean norm (as there are many norms on $\mathbb{R}^2$, for instance $\|(x,y)\|_p=(|x|^p+|y|^p)^{1/p}$, where $p$ is a real number with $p\ge 1$). – Mercy King Sep 23 '15 at 22:58


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## JMI2011B-4 Non-existence of elliptic curves with everywhere good reduction over some real quadratic fields (pp.113-117)

Author(s)： Shun'ichi Yokoyama and Yu Shimasaki

J. Math-for-Ind. 3B (2011) 113-117.

Abstract
We prove the non-existence of elliptic curves having good reduction everywhere over some real quadratic fields. These results of computations give best-possible data including structures of Mordell-Weil groups over real quadratic fields $\mathbb{Q}(\sqrt{m})$ up to 100 via two-descent.

Keyword(s).　 Elliptic curves having everywhere good reduction, Mordell-Weil groups, Two-descent


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# Is eqv? the portable way of checking object equality?

I asked here: “Is eqv? the portable way of checking object equality?”.

Will explained:

Grant Rettke wrote:
> From what I have read, eq? is a way of determing whether two objects
> are the same based on whether they point to the same location in
> memory; and generally that eqv? works the same way as eq? but for
> numbers and characters; numbers and characters equality will not be
> determined consistently across different implementations.

There are a few other differences as well, but numbers
and characters are indeed the most commonly encountered.

> I have read that as a result of this one ought to use eqv? as a
> portable means for checking for object equality unless performance is
> such an issue that eq? should be used instead.

That's reasonable, modulo the quibble with "object
equality" below.

> Is eqv? the portable way of checking object equality?

Yes to the first half of that sentence, provided you
replace "portable" by "most portable standard".

As for the second half of the sentence, I have this
quibble:  In general, there is no such thing as *the*
notion of object equality; the appropriate notion of
object equality depends upon the context.  In particular,
it depends on whether you want to compare two objects
with respect to mutability (that is, does a side
effect to one object imply the same side effect to
the other?) or with respect to their current state.

In Scheme (whether IEEE/ANSI/R5RS or R6RS), eqv? is
the most implementation-independent of the pre-defined
equivalence predicates that are defined on all objects
and are guaranteed to distinguish two distinct mutable
objects.  Similarly, equal? is the pre-defined partial
equivalence predicate that distinguishes on the basis
of current state but not on the basis of mutation; the
R6RS equal? is also total.

In Scheme, eq? is basically an efficiency hack that
can be used as an alternative to eqv? when you know
for sure that at least one of the arguments is a
boolean, symbol, empty list, pair, procedure, non-empty
string, non-empty vector, non-empty bytevector (R6RS
only), or record (R6RS only).  The rationale for eq?
is that it is often about ten times as fast as eqv?,
which is enough to matter for some applications, and
the specific list of situations for which eq? is
guaranteed to behave the same as eqv? is about as
inclusive as it can be without sacrificing speed in
typical implementations.

Will

Basically, eq? is pointer equality.


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This is the second post on the Metropolis-Hastings algorithm. While the first post was devoted to explain why the Metropolis algorithm works, here I am concerned with a common mistake that happens (it happened to me at least) when sampling from distributions that are defined on constrained spaces, most notably $\mathbb R_+$ or $[0,1]$.

Suppose we want to sample from the distribution of a positive random variable $X$ with a random walk Metropolis-Hastings algorithm. There is nothing wrong with using a proposal distribution that might result in negative proposals in this situation. The negative proposals are simply rejected and the sampler will continue to sample from the correct target distribution. A shortcoming of this approach, however, is that, sometimes, too many of the proposals will be rejected, rendering the sampler inefficient. Therefore, it is often sensible to specify a proposal that has to be positive as well, such as the Gamma or the Pareto distribution, or to transform the variable into an unconstrained space and use Normally distributed proposals on the transformed space. It is the latter with which I am planning to deal in this post.

The following example might illustrate the point. Say, we want to sample from a $\text{Gamma(3,1)}$ density (where we use the shape-rate parameterization) by using Normally distributed proposals with variance $\delta^2$, i.e., $x^\ast \sim \text{Normal}(x, \delta)$, where $x^\ast$ is the proposed value and $x$ is the current value at each iteration. The acceptance ratio is then

$r = \frac{\text{Gamma}(x^\ast\, \vert \, 3,1) \text{Normal}(x\, \vert \, x^\ast, \delta)}{\text{Gamma}(x \, \vert \, 3,1)\text{Normal}(x^\ast\, \vert \, x, \delta)} = \frac{\text{Gamma}(x^\ast\, \vert \, 3,1) }{\text{Gamma}(x \, \vert \, 3,1)}$

since the Normal distribution is symmetric—i.e., $\text{Normal}(x\, \vert \, x^\ast, \delta) = \text{Normal}(x^\ast\, \vert \, x, \delta)$. We accept the proposal with probability

$\rho = \min\big\{1, r\big\}.$

So, the Metropolis-Hastings algorithm might be coded up as follows. Notice we calculate the acceptace ratio on the log-scale for numerical stability.

The results show that the algorithm successfully approximates the target distribution:

At the same time, however, we see that a lot of propsals near zero are rejected. This is expected: if the current state of the sampler is near zero, the next proposal will have a high probability of being negative, which leads to an automatic rejections.

So, one might wonder why not using a proposal distribution that is strictly positive? One way to do so is to sample on the log-scale. That is, we keep the Normally distributed proposal distribution but sample

$\log(x^\ast) \sim \text{Normal}(\log(x),\delta)$

and then use $x^\ast = \exp(\log(x^\ast))$ as our proposed value. This ensures that all proposals will be strictly positive, preventing the high rate rejections near $x^\ast=0$. Also, as before, $\text{Normal}(\log(x^\ast)\, \vert \, \log(x), \delta) = \text{Normal}(\log(x)\, \vert \, \log(x^\ast),\delta)$, so can use the exact same acceptance ratio.

Yet, as the figure above shows, once we switch to the log-scale the sampler fails to sample from the traget distribution. The reason for its failure lies the fact that we have changed variables and forgot to adjust for this . In order to see where the algorithm went wrong, consider again the expression

$\log(x^\ast) \sim \text{Normal}(\log(x), \delta).$

This is a probabilistic statement about $\log(x^\ast)$ and not $x^\ast$. On the other hand, in calculating the acceptance ratio, $\rho$, we need to account for the probability that $x^\ast$ is proposed from state $x$, and vice versa, not the probability of $\log(x^\ast)$ and $\log(x)$.

To adjust for the change of variables, we can rely on a simple theorem about the distribution of functions of random variables. The general formula can be found in most introductory statistics textbooks and is based on a standard result in calculus. The version below is from Mood et al.’s introductory book with minor changes in notation:

Theorem (Change of Variables, Unidimensional, Continuous)

Suppose $W$ is a continuous random variable with probability density function $f_W$. Set $\mathcal W = \{w: f_W(w) > 0\}$. Assume that

• $g: \mathcal W \rightarrow \mathcal V$ defines a one-to-one transformation of $\mathcal W$ onto $\mathcal V$.
• The derivative of $w = g^{-1}(v)$ with respect to $v$ is continuous and nonzero for $v\in \mathcal V$, where $g^{-1}(v)$ is the inverse function of $g(w)$; that is, $g^{-1} (v)$ is that $w$ for which $g(w) = v$.

Then $V = g(W)$ is a continuous random variable with density

$f_V(v) = \left\vert \frac{\,\text{d}}{\,\text{d} v}g^{-1}(v)\right\vert f_W(g^{-1}(v)) \mathbb I_{\mathcal V} (v).$

The term that appears in the absolute value function is called the Jacobian of the transformation. Intuitively, the Jacobian of a function $f$ evaluated at the point $x$ can be understood as the amount to which a function stretches or contracts volumes near the point $x$. This also gives some intuition about why it appears whenever we want to change variables, either in the univariate context or the multivariate context.

Now, back to the problem. We know that the proposal $\log(x^\ast)$ is Normally distributed with mean $\log(x)$ and standard deviation $\delta$, but we are interested in the distribution of $x^\ast$. To align the notation with that of the theorem, let

$\log(x^\ast) = w\quad \text{ and }\quad x^\ast = v.$

The mapping that transforms $w$ to $v$ is the exponential function and its inverse is the logarithm. Thus, $g(w) = \exp(w)$ and $g^{-1}(v) = \log(v)$. The exponential function, indeed, defines a one-to-one transformation of $\mathcal W = \mathbb R$ onto $\mathcal V = \mathbb R_+$; and $g^{-1}$ is continuously differentiable with respect to $v$ and is non-zero on $\mathbb R_+$. So, all requirements to apply the change of variable formula are satisfied.

Applying the formua, we obtain

\begin{aligned} f_{V}(v) &= \left\vert\frac{\,\text{d}}{\,\text{d} v} g^{-1}(v)\right\vert f_{W}(g^{-1}(v)) \\ &= \left\vert\frac{\,\text{d}}{\,\text{d} v} \log(v)\right\vert f_{W}(\log(v)) \\ &= \left\vert\frac{1}{v}\right\vert\text{Normal}\Big(\log(v)\,\Big\vert\, \log(x), \delta\Big), \end{aligned}

for $v > 0$. So,

$f_{X^\ast}(x^\ast) = \left\vert\frac{1}{x^\ast}\right\vert\text{Normal}\Big(\log(x^\ast)\,\Big\vert\, \log(x), \delta\Big)$

and we see immediatly that the Jacobian is the missing piece we’ve left out.

Lastly, to incorporate the Jacobian, we note that the acceptance ratio is calculated on the log-scale. So, we have to add (not multiply)

$\log\left\vert\frac{\,\text{d}}{\,\text{d} x} \log(x)\right\vert = \log \left\vert \frac{1}{x}\right\vert = -\log(x)$

and $-\log(x^\ast)$, respectively, from the numerator and denominator in the acceptance ratio.

So let’s run the sampler again:

The results look as follows:

Voila! Looks good!


text:
location:  Publications → journals
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Search: MSC category 22D05 ( General properties and structure of locally compact groups )

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1. CMB 2008 (vol 51 pp. 60)

Janzen, David
 F{\o}lner Nets for Semidirect Products of Amenable Groups For unimodular semidirect products of locally compact amenable groups $N$ and $H$, we show that one can always construct a F{\o}lner net of the form $(A_\alpha \times B_\beta)$ for $G$, where $(A_\alpha)$ is a strong form of F{\o}lner net for $N$ and $(B_\beta)$ is any F{\o}lner net for $H$. Applications to the Heisenberg and Euclidean motion groups are provided. Categories:22D05, 43A07, 22D15, 43A20
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text:
# Spacing around longer function argument's parentheses

The line

i(x,y,z,t) = \delta(x-\ell_x(z)) \delta(y-\ell_y(z)) \hat{\imath}(\xi(z),t)

produces (for my eyes, at least) too little space between the successive function calls. It is difficult to visually distinguish the three calls. I guess the underlying reason is that the expressions that serve as function arguments are relatively long. For comparison,

i(x,y,z,t) = \delta(x) \delta(y) \hat{\imath}(\xi(z),t)

already looks a lot better in my opinion.

I know that I could

1. Manually adjust spacing using \, etc., including the option to define a macro that does this automatically.
2. Insert \cdot as a means of adding seperating space.

However, my question is: Is there a canonical way (e.g. a package) of dealing with this problem? I imagine I'm not the first one to encounter this problem... still, I wasn't able to find useful, general solutions to the problem.

A possible solution might be something along the lines of

i(x,y,t) = \delta \funcargs{x-\ell_x(z)} \delta \funcargs{y-\ell_y(z)} \hat{\imath} \funcargs{\xi(z)}{t}

Note that I find the use of \cdot to be unsatisfying since it introduces inconsistency in the document (sometimes having the dot and sometimes not, with the meaning being the same).

• Use \, or \; to add more space. – skpblack Nov 3 '14 at 14:39
• As I said, I know that I can manually adjust spacing. This is not the solution I was asking for. I'll try to further clarify the question. – jhin Nov 3 '14 at 14:47
• How do you feel about wrapping them inside \mathinner{...}s? – morbusg Nov 3 '14 at 14:49
• Because the added space is dependent on what follows (another function[yes] vs. a comma or an additional close paren [no]), making it active would seem to be the only way. And making it active would introduce all sorts of other problems, I would guess. Thus, I agree with @skpblack that adding it manually is the best approach. – Steven B. Segletes Nov 3 '14 at 14:49
• @Steven: With \mathinner, no space would be inserted between it and \mathclose, but \thinmuskip would be inserted between it and \mathpunct. – morbusg Nov 3 '14 at 14:56

First, let me give a paraphrasing from The TeXbook showing the amounts of spaces between different types of math atoms:

where

• Ord includes things like a, \theta, and things inside \mathord or just plain {}'s;
• Op includes things like \sin, \gcd, and things inside \mathop;
• Bin includes things like \pm, \otimes, and things inside \mathbin;
• Rel includes things like \leq, \mid, and things inside \mathrel;
• Open includes things like \langle, (, and things inside \mathopen;
• Close includes things like \rangle, ), and things inside \mathclose;
• Punct includes things like ,, ., and things inside \mathpunct;
• Inner includes things like \cdots and things inside \mathinner.

Here is an example document illustrating a couple of different ways:

\let\funarg\mathinner
\def\funop#1{\mathop{{}#1}}
$$i(x,y,z,t) = \delta(x-\ell_x(z)) \delta(y-\ell_y(z)) \hat\imath(\xi(z),t),$$
$$i\funarg{x,y,z,t} = \delta\funarg{(x-\ell_x(z))} \delta\funarg{(y-\ell_y(z))} \hat\imath\funarg{(\xi(z),t)},$$
$$\funarg{i(x,y,z,t)} = \funarg{\delta(x-\ell_x(z))} \funarg{\delta(y-\ell_y(z))} \funarg{\hat\imath(\xi(z),t)},$$
$$\funop i(x,y,z,t) = \funop\delta(x-\ell_x(z)) \funop\delta(y-\ell_y(z)) \funop{\hat\imath}(\xi(z),t),$$
\bye

So as you can see, you probably want the last option of using a \mathop atom for the function "name".

With LaTeX, this could be defined as follows (the second row below shows the output of your original code):

\documentclass{article}
\newcommand\funop[1]{\mathop{{}#1}}
\begin{document}
$i(x,y,z,t) = \funop\delta(x-\ell_x(z)) \funop\delta(y-\ell_y(z)) \funop{\hat\imath}(\xi(z),t)$

vs.

$i(x,y,z,t) = \delta(x-\ell_x(z)) \delta(y-\ell_y(z)) \hat{\imath}(\xi(z),t)$
\end{document}

(The reason I didn't include the parenthesis in the command is that I felt it would be too much effort to measure the arguments to decide on the size of the fences. I think it's simpler to just manually define the height with \bigl( etc., and also it's nicer to see the parens in the manuscript instead of {}'s)

• Thanks, this is really helpful! What I do right now is \newcommand{\func}[2]{\operatorname{#1} \left( #2 \right)}. I first tried \mathop as it seemed more appropriate than \mathinner but it resulted in weird vertival spacing, so I settled to \operatorname. I have \left and \right fixed as proposed in tex.stackexchange.com/a/2610/64293. The result looks really good in my opinion. – jhin Nov 3 '14 at 17:12
• @jhin: Yes, the first empty group in my plain-tex definition takes care of the weird vertical spacing. I could have sworn there is either a LaTeX or AMSMath command to achieve the same thing, I'm just having trouble finding it. – morbusg Nov 3 '14 at 17:17
• @jhin You can use \mleft and \mright from mleftright package. In any case, I wouldn't use them. I would prefer a wrapper around \DeclarePairedDelimiter\parenthesis{\lparen}{\rparen} and then passing the optional comand like \func[\big]{..}{..} to \parenthesis[\big]{..}. – Manuel Nov 3 '14 at 17:18
• @Mico: Thanks! Though, I was thinking of a \mathop to be the "suggested" math atom because the \mathinner adds a \thinmuskip in case of a following \mathpunct atom. Do you recall if there is a LaTeX or AMSMath command which basically does \mathop{{}#1}? – morbusg Nov 3 '14 at 17:28
• @Manuel - and could you elaborate a bit on why you would prefer such a wrapper over the \mleftrightpackage...? – jhin Nov 3 '14 at 17:42

Inserting a thinspace (\,) between the multiplicative factors should work. Inserting \cdot in these locations should work too. You could also combine the thinspaces with making the outer parentheses a tad bigger than the inner ones (by using \bigl( and \bigr)). The three possibilities are illustrated below:

\documentclass{article}
\begin{document}
$i(x,y,z,t) = \delta(x-\ell_x(z)) \, \delta(y-\ell_y(z)) \, \hat{\imath}(\xi(z),t)$

$i(x,y,z,t) = \delta(x-\ell_x(z)) \cdot \delta(y-\ell_y(z)) \cdot \hat{\imath}(\xi(z),t)$

$i(x,y,z,t) = \delta\bigl(x-\ell_x(z)\bigr) \, \delta\bigl(y-\ell_y(z)\bigr) \, \hat{\imath}\bigl(\xi(z),t)\bigr)$
\end{document}

One can automate the third method by creating a macro such as

\def\myfunc#1{\bigl(#1\bigr)\,}

and then typing

$i(x,y,z,t) = \delta\myfunc{x-\ell_x(z)} \delta\myfunc{y-\ell_y(z)} \hat{\imath}\myfunc{\xi(z),t)}$
• Thanks for the answer. I prefer the solution given by morbusg, though. Note that \bigl etc. introduce additional (undesired) spacing between the function name and the parentheses, see e.g. tex.stackexchange.com/questions/40829/… – jhin Nov 3 '14 at 15:23
• @jhin - You may have misinterpreted that posting: The extra, unwanted space is inserted if one uses \left( and \right). In contrast, no extra space is inserted by \bigl( and \bigr). You can verify this assertion by modifying the third row of the code above to omit the two \, directives : You'll find that there's no absolutely no extra space. – Mico Nov 3 '14 at 15:56
• I wasn't sure whether the two produce the same spacing, so I looked at your attached picture and concluded that yes they do, since in fact there is a larger space between the delta and the parenthesis in the third row than in the two previous rows... might just be the effect of the scaling, though. Thanks for correcting me! – jhin Nov 3 '14 at 16:23
• @jhin - The additional whitespace in the third row comes from the \, macros, not from \bigl( and \bigr). It's exactly the same amount of whitespace as in the first row (which also uses \, macros). – Mico Nov 3 '14 at 17:09
• Sorry, perhaps should have clarified: I'm really referring to the whitespace between the \delta and its successive ( or \bigl, respectively. And that is indeed a tiny little bit larger in the third row than in the two previous rows. – jhin Nov 3 '14 at 17:17


text:
# Fluorescent phosphors and their spectrum

I am doing a spectroscopy demo and I am trying to identify the phosphors used in CFLs (and their spectra). Does anyone have a reputable source for these. The Wikipedia, Sylvania and the DOE websites lack much specific information.


text:
# Safety Margin for a Artificial Orbital Habitat

I'm always interested in the possibilities of human creating an orbiting space habitat like Stanford Torus or O'neill Cylinder. Those habitats were designed to support certain numbers of colonists.

Now for the question: Based on current safety practice (real world),

What should the safety margin of these habitats be? (Ex: If the habitat's intended pop was 10,000 then with a safety margin of 200% they can safely support up to 20,000 people).

Also, how much redundancy should the colony's systems have? (Ex: If the colony life support system was a 4 tiered redundant system then the life support will only failed if all 4 layers failed).

• So... What's your question? I see two question marks, but they're both immediately followed by answers. – Frostfyre Aug 14 '15 at 3:29
• For complex systems, there is a mix a well-understood and not so well understood systems. Well understood systems need a smaller safety margin, cutting edge systems may require very large safety factors. There is also a big difference in how critical systems are. Critical systems require higher safety factor, less critical system lower safety factors. Risk tolerance also varies considerably. A small colony for early adopters will tolerate smaller safety factors than later colonies for the masses. Even cost is a factor, people will be more accepting of risk if costs are high. – Gary Walker Aug 14 '15 at 4:10
• @frostfyre that 2 was the question. The sentence followed the question was a sample answer... since I don't had a clue about how robust the orbital should be designed about the safety margin. Maybe i wrote it on a wrong way.... Siince increasing safety margin would substantially increase the cost, I just want to know what the real world currently adopted. – Satori Wita Aug 14 '15 at 4:41
• @gary walker . Your comment was helpful. Let's just say that I dont really need the detailed system per system information. Just the general limit of how many tines the colony can support people greater than what colony designer intended colony's population. Sorry for the confusing English... – Satori Wita Aug 14 '15 at 4:44
• I made some slight edits in terms of sentence structure, wording, and the examples that will hopefully clarify things. If you see anything changed that doesn't make sense please revert it. – Dan Smolinske Aug 14 '15 at 16:42

I would look at it a different way. There are other questions skirting around the edges of the questions you asked which are more valid for very large population biospheres.

A complex system like an orbital habitat will operate in shades of grey, not black and white. Concepts like "safety margins" and "number of redundant systems" are too gross, and the end result will always be an unsustainable habitat, or a habitat that is 10x too expensive and barely operates.

An excellent example of what an artificial habitat has to deal with is to look at our own habitat. Consider the "safety margin" for number of humans it supports? We think about how hard we are pushing the earth as we climb our way towards 10 billion. But what was the safety margin in the first place. A quick search shows that worrying about population is nothing new. In 2nd century Carthage, Tertuilian lamented that the world population of 190 million was too much and would crush the Earth. By those estimates we're looking at a safety margin of 5200%. A better way of looking at it is that we rarely are anywhere near a "carrying capacity." More accurately there is a fluid quality of life degradation as population increases, and we humans eventually set a bar at a minimum quality of life we will endure (we've also made technological advances to push that quality of life issue out).

Likewise, a focus on number of redundant systems leads astray: most systems are not instant-death if they stop working. There's no system which, upon failure, mindlessly opens all the airlocks and vents the habitat. When a system stops working, a clock starts. As time progresses, quality of life diminishes. What matters is not how redundant the systems are, but how gracefully the systems can be repaired ore replaced after a failure occurs. Consider: you only have 2 corneas, one for each eye, yet they are known for being hardy little structures, because the body is configured to make it rather difficult to damage them beyond what they can repair (including a blink reflex, which helps avoid most major cornea trauma.)

EDIT:

As a closing, working with a complex system is always a balance between two extremes. On one hand, you have to predict ahead of time what the system will do. On the other hand, those predictions will be wrong 100% of the time unless you observe, update, and adapt as you go. When designing large systems, if it feels like you need too much expertise in the design phase, try relying more on listening to the system while it operates and making changes as it goes. If it feels like you can't hear her whispers, then you need to make more predictions. Remember Kaylee from Serenity. She knew her stuff, but what made her special is, in her words, "machines just talk to [her]."

• I see. If that was the case then what kind of approach that we should take when determined how many people that can supported in the colony for short time? Since I think that the colony population would grow (although will be limited) by birth, immigration and added with transient traveler (tourist, businessman, trader, visiting dignitaries, etc). – Satori Wita Aug 14 '15 at 13:58
• The approach I would take is avoiding to need to determine how many people can be supported. I would avoid throwing all my support behind the numbers in the up-front analysis, and instead concentrate on how to perceive how the station adapts to changing loads of people. Sure' I'd make some up front calculations to have a good sense, but I'd concentrate on the station's actual ability to function, rather than how the numbers say the station should function. That also helps in cases where the contractors did not do a perfect job (they were human, after all). – Cort Ammon Aug 14 '15 at 15:48
• For example, one may identify (after a few minor crises) the station's equivalent of a canary in a coal mine, something which is very good at indicating a problem in the near future, but far enough into the future to actually do something about it. That canary indicator should flex with reality, so if 2 oxygen pumps are off, it should react at a lower population. The key is to give smart people enough time to react and solve the problem... they just need to be made aware of it in time. – Cort Ammon Aug 14 '15 at 15:49

Safety factors figure into catastrophic (i.e. unmitigatible) failure scenarios, and cannot reasonably be said to apply to the sort of scenario envisioned. When discussing large and self-repairing systems like this, we talk about the limits in terms of it's ability to function "sustainably", to house people "comfortably", etc.

A space station will possess self-diagnostic tools. Parts of the system will be expected to fail at a certain average rate, and the system will possess a sort of additional capacity in order to make up for that. Even in modern times, a freak unexpected shutdown of vital life-support equipment for several days for a station operating beyond capacity should be relatively easy to manage, provided the persons responsible for logistics were competent. Generally speaking, the crew of the station will always have time to respond to overpopulation pressures long before any negative physical health effects would present themselves.

Provided that the crew expects to be operating beyond the old capacity indefinitely, modifications to the ship can be made to further increase the station's capacity. Provided that they only need to do so for 'a while' (possibly as long as several months) space-efficient but expensive consumable reserves can be delved into.

That said, in general, the limiting life support factor for humans in space currently is actually healthy air, which, because technology, is largely the same as water. The ISS produces via recycling and electrolysis ~80% of its expected oxygen needs, and receives the other 20% via shipments from Earth. It also stores 140+ days worth of backup oxygen for it's expected personnel load. Typically, it gets a resupply ship every 90 days.

Extending the function of the ISS by ratio to our theoretical space station:

The station can run unsupported at  |  For
80%  capacity                       |  practically indefinitely
100% capacity                       |  11.67 duty cycles
133% capacity                       |  3.89 duty cycles
150% capacity                       |  2.92 duty cycles
200% capacity                       |  1.67 duty cycles
250% capacity                       |  1.17 duty cycles
300% capacity                       |  0.90 duty cycles
500% capacity                       |  0.47 duty cycles
750% capacity                       |  0.29 duty cycles
1000%capacity                       |  0.21 duty cycles

Probably oxygen would stop being the limiting factor at some point in this process, but the function described by this data would still be fairly representative of projects like this in general; operating above capacity mostly just decreases the time the crew has to respond to normal problems encountered in the course of operation and makes the success of supply missions more important. At some point (in our ISS-based model, around the 300% capacity mark) a rescue mission or other extreme action is required as the station cannot survive until the next scheduled resupply on its own. Nonetheless, even in the extreme 1000% capacity scenario the crew has a lot of human-experienced-time to decide what to do about the problems (19 days for the ISS itself).

• I see. Your table and explanation help me picturing about the work around mechanism that support the colonist. I can imagine that the colony would had self-diagnostic function for its systems, a dedicated team of technicians, spare-parts cache or tool to make them, and constant modification to the system to accommodate the change in the colonist number. It looks like ~300% figure seems realistic. I need to make similar table.... – Satori Wita Aug 14 '15 at 14:05

What should the safety margin of these habitat? Also how much redundancy the colony system should have? These are excellent questions though except in generalities, we can't answer them here.

## Safety Margin

A simple, naive answer would be a factor of 2. For example, an oxygen system is capable of handling 10,000 m^3 of air per day but the operational requirements are only for 5,000m^3 per day. If you have two such units, your habitat can tolerate the complete failure of one oxygen system and still operate indefinitely. Granted, you want to get that fixed as quickly as possible but you have enough leeway to fix it properly instead of some rushed hack job.

## Redundancy

A simple, naive answer is "Two of everything" and "At least two ways to do everything". You'll see this play out in large computer systems where systems architects are perpetually on the watch for single points of failure and eliminating them as much as possible. Some times, these single points can't be eliminated but in such cases, lots of engineering work goes into making sure the equipment/system is incredibly robust. If the habitat only has a single airlock then that airlock will be incredibly resilient to bad bumps, overspeed docking, torsion moments, vibration, accidental kicks by ingressing astronauts, etc, etc, etc.

## Nature of Failure

As other have pointed out, when a system fails, especially in an environment like space, it should fail-safe, ie, the life support system doesn't vent the remaining atmosphere to space when it fails.

Disasters are rarely the result of single large failures. Most often they are the result of a string of small mistakes and failures that individually are easy to counter but collectively cannot. This has proved true in practically every domain where safety is a concern. Big failures, while an omnipresent possibility, don't happen nearly as often as failure cascades where a small failure of one component leads to another failure which leads to another failure until the entire system is compromised and catastrophically fails. A lot of engineering work on the ground will go into designing systems that preclude failure cascades.

In addition to equipment design, training of astronauts is critical. Resilience research has found that it is often the human element that can accommodate failures in the systems. Proper procedures and training need to be in place to fix problems that can't be anticipated or adding the ability to detect a certain error condition would add an unacceptable cost/weight of a particular system.

I'm going to take a stab at this question, but as James has mentioned, this is up to you. I'll take his suggestion and use benchmarks that you can choose to apply or not.

## Safety Margin

In the oil industry, on-site safety personnel assist in analyzing work environments and designing processes to control, eliminate, and prevent problems. I use this industry, because I'm familiar with it, but also because it is high-risk, expensive, and nobody wants anything bad to happen.

A good rule of thumb is to have 20% contingency on all equipment for safety purposes on rigs that don't have immediate access. For your purposes, I'd make the outer walls 20% thicker than necessary, etc. throughout your design. Again, this is a design solution, depending on your space colony.

For the number of colonists, I would design a capacity for 20% more as well. If you aim to have 1,000 people, design it for 1,200 and ensure that you have a mechanism for removal of babies or other people to keep it around 1,000 or less.

## Redundancy

Redundancy is again your choice. When I was an urban planner in the UAE and they introduced the first Nuclear Power plant, at Baraka, we understood that the rule of thumb for safe nuclear power was "4" for systems such as cooling, etc. I would suggest for your critical items that you have four of everything, because it will take a while for backup supplies.


text:
Open access peer-reviewed chapter

# Modeling Deforestation and CO2 Emissions in Tropical Forests (Western South Amazon)

By Valderli Jorge Piontekowski, Ângela Pereira Bussinguer, Fabiana Piontekowski Ribeiro, Jonas Inkotte, Karla Monique Silva Carneiro Valadão, Marco Bruno Xavier Valadão, Mirella Basileu de Oliveira Lima, Alcides Gatto, Eraldo Aparecido Trondoli Matricardi and Reginaldo Sérgio Pereira

Submitted: April 5th 2017Reviewed: November 29th 2017Published: December 20th 2017

DOI: 10.5772/intechopen.72806

Downloaded: 935

## Abstract

Spatial modeling is a tool to represent deforestation and predict future scenarios according to different landscape change. Establishing 80% Legal Reserve Area (LR) in the Amazon since 90th, the Brazilian forestry code has made clear the biodiversity conservation profile of the largest tropical forest in the world. However, this mechanism did not prevent the advance of deforestation, which in recent years has increased again. This remote tool aims to monitor the deforestation, simulating its possible future trajectories, and thus generate information that can be used to assist in the management of deforestation reduction. The spatial modeling in the prediction of different deforestation scenarios based on public policies and their changes to the state of Acre (north of Brazil). Using the methodological processes of the Dinamica EGO software, three scenarios were projected up to the year 2050: (1) deforestation “Business as usual”, (2) deforestation with 50% LR and (3) deforestation with 80% LR provided by law. Based on these results it was evident that maintaining and respect 80% LR, it’s possible reduce the CO2 emissions more than 76%, avoiding around 119,534,836 t of CO2 and influences positively on reducing deforestation. Dinamica EGO proved to be an effective to represent the deforestation.

### Keywords

• legal reserve area
• Dinamica EGO
• state of Acre
• Brazilian Amazon

## 1. Introduction

Tropical forests represent the largest terrestrial sink of anthropic emissions of carbon dioxide (CO2) and are globally responsible for significant carbon stocks (C). The path for the global reduction of terrestrial CO2 is controlled by the equilibrium between C fixation via accumulation of plant biomass and emission by burning [1, 2].

Modeling deforestation represents a powerful tool in different type of natural forest worldwide. The magnitude of the impact is been attested by predict changes in futures scenarios. In Brazil, modeling mechanisms is use to gauge the efficiency of the environmental policy established in Brazilian forestry code. Brazilian Amazon, since 1996 each property located in the must maintain a Legal Reserve Area (LR) of 80% where the vegetation cover consists of forest phytophysiognomies.

After more than 20 years and recent amnesty for properties the change landscape until July 22nd 2008, recently, in 2016 Legal Amazon lost 786,900 km2, corresponding to 15.7% of the entire region in 2016 [3]. Precisely, the most common illegal practice is deforestation beyond the limit defined by law of LR [4].

Many landowners in the Amazon region claim to be unable to comply with the Forestry Code. Thus, the dominant debate in the region has always been favorable to noncompliance with legislation and not to its improvement [5]. Such positions may contribute to the loss of forest cover and may also cause direct environmental impacts, such as increased CO2 emissions into the atmosphere.

The ways to understand the impacts of different methods of using land can be better understood from the construction of scenarios, which may represent the spatial design of future deforestation and its effects on global CO2 emissions. In view of this, tools for remote deforestation monitoring, spatial dynamics modeling and assessments of the use and occupation of deforested areas can be used to increase the effectiveness of the implementation of environmental policies. The Satellite Monitoring Project of the Amazon Forest (PRODES) and the Amazon Near Real Time Deforestation Detection System (DETER) developed by the National Institute of Space Research (INPE) are good examples of the use of these tools.

PRODES is a project that has been monitoring deforestation in the Amazônia Legal since 1988, estimating annual deforestation areas and rates [3]. DETER is a warning system of changes in forest cover, which has the capacity to detect altered areas larger than 25 ha, thus making public policies to combat and control deforestation more effective [6]. The modeling of landscape dynamics allows the prediction of disturbances and monitoring the dynamics of vegetation, as these have the capacity to explore the role of deforestation, fire, climate, and weather [7].

The modeling process is a useful and multi-application tool. The results of modeling facilitate the study and understanding of deforestation and CO2 emission issues, allowing simple representation of the complex deforestation processes by simulating the possible future trajectories according to the scenarios to be elaborated.

Thus, for the present case study, a model of landscape simulation based on cellular automata was developed and applied to the state of Acre, located in the Western South Amazon, northern region of Brazil, using the software platform Dinamica EGO [8]. Scenarios were projected for the entire territory of the state of Acre, represented by analysis applied to the simulation process, in order to assess the impacts of deforestation and CO2 emission, in space and time.

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## 2. Overview of spatial and temporal modeling for environmental services

In order to carry out effective measures of forest management and protection, a basic demand is the construction of methods for the collection, processing, analysis and publishing of spatial data regarding vegetation cover of these areas [9]. It is also important to evaluate the use of soil and changes in ecosystem dynamics, as well as observe advances in legislation, and other factors assessed by simulation models.

The main programs used for spatial image processing in remote sensing worldwide are QGIS, ArcGIS, Udig and Spring in Brazil. FIA Map Maker is used in the making of tables and maps in the USDA Forest Service’s Forest Inventory and Analysis Data Base (FIADB), and the ACORn (A Comprehensive Ozark Regeneration simulator) program is used to predict the number of tree species and their potential individuals [10, 11].

Currently, there are versions of mapping programs in mobile devices, such as ArcGIS mobile, ArcPad and Avenza Maps, which replaces PDF Maps. They have been used successfully in assessing forest inventories, ecology studies, and conifer restoration projects in the United States [12, 13].

Many databases used in these programs may be compatible with statistical programs such as R. In addition, most of these programs have tools and interfaces that can be accessed by a variety of programming languages such as C++, Java, Python, Perl, C# and Ruby.

The Dinamica EGO is a spatial simulator of landscape dynamics widely used in Brazil. This simulator allows the construction of explicit scenarios that simulate the dynamics of an environmental system from transition probabilities, reproducing the way its spatial patterns evolve and allowing the projection of the probable ecological and socioeconomic consequences of the dynamics of the systems [8, 14].

Based on previous linear regression analysis, another model used in South America and that presents satisfactory results is the Land Change Modeler. It was used in a study in Peru with the purpose of simulating the loss of primary forest cover and changes in the characteristics of the landscapes from 2015 to 2025 [15].

MELA is a widely used simulator in Finland developed by the Finnish Forest Research Institute. It aims to model predictions for forest indicators (biomass, C sequestration) and economic indicators (yield and wood supply) [16].

In Africa, the MODIS (Moderate Resolution Imaging Spectroradiometer) platform is used in conjunction with the Land Degradation Surveillance framework (LDSF) and is implemented as part of the Africa Soil Information Service (AfSIS) in a soil-quality monitoring project. It has been employed in several tropical landscapes around the globe and also by NASA observatories [17].

The LDSF model stands out for its ability to perform joint map evaluations of several variables such as soil organic matter (SOM), pH, sand, sum of exchangeable bases, quantitative occurrence and degree of root depth restriction in the first soil layers, water infiltration capacity, soil erosion levels, and vegetation cover data, as well as information on structure, distribution, diversity and even species richness. From the joint analysis of these variables, the identification of soil and ecosystem quality can be improved, as well as the contrasts between land uses over time and also the possibility of verifying management interventions at various scales, helping to identify the adoption of these techniques [18].

The LANDIS-II model has been used to simulate the potential of C sequestration of vegetation under different fire regimes and prediction of biomass. The LANDIS-Pro model performs simulations of biomass, forest succession, seed dispersal, species establishments and disturbances in general. The PHENIPS model can be used in phenological prediction in tropical forests, in addition to the productivity and cycling of C and nitrogen in coniferous forests as a result of multiple validated models [19].

The SIMILE and TROLL models have been used in the simulation of plant succession processes [20]. However, such models have a limitation in the validation of their predictions, since most of the studies were conducted in specific times and areas (short-term predictions), which is not compatible with the simulations carried out in the large temporal space used for the most part of these simulations. Such short-term predictions can be more realistic by calibrating the models employed with the use of field data in time series, such as forest inventory data, on an appropriate timeline scale, providing greater predictability credibility [19].

Finally, the DSS-Wuk model, widely used in Germany, follows the tendency of the models to gather several variables (complex dynamic simulation) [21]. Recently, the model’s ability to project forest development in the period from 2011 to 2100, compared to the reference period (1971–2000) [22], was evaluated in the country based on two climatic scenarios.

The DSS-Wuk combines several submodels (regionalization of climate, allometric and phenological, nutrient rates, water in the soil, mortality due to drought stress, wind and insect damage, site index, and others) for the purpose of better describing the impacts of climate change, besides including a sub-model of economic evaluation. It is recognized as a very realistic and very promising approach in the modeling area [23].

In the modeling carried out in Geographic Information System (GIS) programs, the analysis known as Anselin Local Moran’s I can also be used, a type of Cluster analysis that verifies outliers and which, combined with geostatistical and remote sensing data, is able to evaluate patterns of environmental variables related to forest ecosystems. This was done recently in the simulation of future C sequestration potential in soils in China from 2000 to 2300 [24], and in the analysis and simulation of forest deforestation in Iran from 1972 to 2010 due to residential growth [25].

According to the rapid changes in the landscape observed globally, it is perceived that the systems of earth science have been improved by the use of remote sensing data and models based on the conversion of land cover, especially in a temporal space resolution, starting from the promising use of several submodels and the union of GIS data with data obtained in the field.

Advances in the field of forest modeling present the capacity to improve the predictive ability of different scenarios and the possibility of understanding ecological processes. The results of these models enable the adoption of better practices for the conservation of natural resources, along with the economic aspects of their exploration, as well as predicting the potential of forests, especially tropical forests, to mitigate GHG emissions.

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## 3. Deforestation scenarios for the western south Amazon (case study)

The state of Acre is located in the northern region of Brazil, in the South Western Amazon, covering an area of 164,215.98 km2 [26]. It borders Bolivia and Peru, and the states of Rondônia and Amazonas in Brazil. About 50% of its territory is composed of protected areas, distributed and classified according to the Brazilian system named “System of Units of Conservation” (SNUC), Law 9985/2000 in 18 Units of Conservation of Sustainable Use, 3 Integral Protection Units and 36 Indigenous Lands (Figure 1).

The natural forest types that occurs in the study area are: Open Broadleaf Forest with Palm trees, Open Broadleaf Forest with Dominant Bamboo, Open Broadleaf Forest with Bamboo, Dense Rainforest, Open Alluvial Rainforest with Palm Tree, Open Alluvial Rainforest with Bamboo, Open Alluvial Rainforest with Palms + Secondary Vegetation, Open Alluvial Forest with Palms + Pioneering vegetation.

### 3.1. Database

The data sources used for the modeling were: Acre data––Economic Ecological Zoning of the state of Acre [26]; PRODES data––Project of Estimation of Deforestation in the Brazilian Amazon, carried out by the INPE [3]; and the National Water Agency (ANA1) data. The software used to manipulate and generate the results were ArcGIS®10.x and Dinamica EGO, version 1.8.9.

### 3.2. Data processing

The entry maps of the deforestation model were obtained by PRODES (land use and land cover map for the years 2006 and 2009) and converted to the matrix format with spatial resolution of 100 × 100 m. Then, the maps were reclassified to the following classes: forest, non-forest and deforestation, as hydrography was defined as null. The 2006 map represented the initial input landscape of the model and the 2009 map represented the final input landscape model.

The static variables used in the deforestation model are subdivided into categorical, continuous and dynamic. The set of categorical variables is composed of maps of vegetation, soils and protected areas. In the set of continuous variables are the maps of altitude, slope, distance to main rivers, urban attraction distance, distances to population nuclei, distance to all roads, and distance from main roads (Figure 2). The deforested area distance was used as a dynamic variable.

For the analysis of future deforestation, the behavior of subregions was considered, for which a micro-basin map (OTTO basins level 12) was used. The regionalization method allows defining sequences of specific operations for each region (microbasin), modeling as a result the local context influenced by particular phenomena [27].

With the input data of the model, maps of the initial and final landscape plus the explanatory variables to deforestation, the calibration, validation and future projection of deforestation were carried out. The procedures for developing the model using the software Dinamica EGO can be visualized in Figure 3.

Then, the model was performed generating a simulated map for the year 2009 and a probability map (which indicated the areas most susceptible to deforestation) based on the weights of evidence from the variable maps. From the simulated map, the validation process was carried out to verify the similarity between the simulated map and a reference map—land use and land cover map for year 2009 of the PRODES classification. For the validation, the model that uses the method of the constant decay function was applied, with windows of variable sizes from 1 × 1 to 13 × 13 pixels. The method is termed as “Fuzzy Similarity” in a familiar local context established by Hagen [29].

### 3.3. Generation of future scenarios

The study of future deforestation in the state of Acre was evaluated under the following scenarios: Scenario 1––deforestation “Business as usual”; Scenario 2—deforestation with LR of 50%; Scenario 3—deforestation with LR of 80% as provided by law. To evaluate the influence of LR size in scenarios 2 and 3, the useful areas of each basin (OTTO basins level 12) were used, with an average size of 4800 ha. In all scenarios, deforestation was simulated by the year 2050.

For all scenarios, the total deforestation in hectares and the amount of CO2 emitted in tons were calculated. For scenarios 2 and 3, the amount of deforestation and CO2 emissions that would be reduced under the following circumstances was calculated: (a) if there was a containment of the increase of liabilities in the micro-watershed already with liabilities up to the year 2010, and (b) if there was no generation of liabilities in micro-basins that, until the year 2010, were with forest assets. Environmental liabilities are deforested Legal Reserve Areas and environmental assets are units of areas that exceed LR size.

### 3.4. Model for accounting for CO2 emissions

In order to account for CO2 emissions, an annual biomass loss map was generated from areas that would be deforested annually, overlapping with a forest biomass map.

The biomass map used in the analysis was made by Saatchi et al. [30], where it was considered that 50% of forest biomass corresponds to C [31]. The CO2 values resented were obtained by multiplying C by the ratio between the molecular weights of CO2 and C, which is 44/12 (3.67 t CO2 equivalent (CO2 eq)).

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## 4. Future scenarios modeled

In scenario 1, with the “Business as usual” pattern, from 2011 to 2050 more than 550,400 ha will be deforested, representing 42,324,000 t of C originally contained in these deforested forests and released into the atmosphere in the form of CO2 of 155,329,080 t. The amount deforested by clear-cutting in the Western South Amazon (Acre State) will exceed 20.5% in 2010 [3] to more than 27% in the year 2050.

The areas of liabilities and assets already consolidated up to the year 2010 in the micro-basins and how these areas would remain until the year 2050 are presented in Table 1. It also shows the difference of the liabilities that would increase and the assets that would be reduced, based on the scenario of land cover change, considering the “Business as usual” prospects of deforestation with LR of 50 and 80%.

Forest assets and liabilities (ha)
Until 2010Projection until 2050Difference 2010–2050
LiabilitiesAssetsLiabilitiesAssets+ Liabilities− Assets
LR 80%988.700900.1001.422.200783.200433.500116.900
LR 50%312.8002.750.900545.5002.433.300232.800317.600

### Table 1.

Table of liabilities and forest assets (LR of 50% and 80%) considering “business as usual” deforestation until 2010 and with the projection of deforestation up to 2050.

Considering the hypothesis that the legislation is not fulfilled, containing deforestation as described in the Law, deforestation trends in hectares and CO2 emissions in tons would behave as shown in Figure 4. Deforestation, maintaining an LR of 80% (Scenario 3), would show a reduction of around 79%, with some 433,500 ha being deforested in relation to “Business as usual” deforestation patterns (Scenario 1). However, if LR changes back to 50% (Scenario 2), there will be an increase in deforestation by more than 170% when compared to an LR of 80%, with about 200,700 hectares being deforested (Figure 4).

One of the consequences of deforestation is the emission of CO2, and the modeling shows that CO2 emissions, with an LR of 80%, would decrease by more than 76%, avoiding around 119,534,836 t of CO2 to be released into the atmosphere when compared to “Business as usual” deforestation patterns. However, when considering a LR of 50%, the CO2 emissions would increase by more than 169% compared to the LR of 80% (Figure 5).

In view of the projections made, although the state of Acre (Western South Amazon) presents an even greater part of its territory in forest, it is evident that maintaining and respecting an LR of 80% would be of great importance for maintaining the environmental quality and, consequently, avoiding the reduction of biodiversity.

The set of variables used in this study presented a peculiar performance in the representation of deforestation, showing that it is mainly concentrated near the areas already deforested and the areas of influence of the current road network of the State. Because it is precisely in areas that are affected by the implantation of agricultural activities and logging, the latter has great importance in the expansion of deforestation in Acre [26]. In this sense, it is worth noting that the scenario presented for the state of Acre may take on another configuration as new roads and branches are deployed in the region. According to Nogueira [32], the influence of static variables on deforestation does not result from the individual action of each one, but from a complex relation of different factors of several causal variables, which has its origin in complex social, political and economic conditions.

The scenario “Business as usual” (Scenario 1) projects the historical evolution of deforestation in the region. The set of policies prohibiting land acquisition and implementation of rural settlement projects in areas of primary forest cover, together with the commitment of land use with adequate techniques and the prevention of clear cutting of vegetation [26] function as a hypothesis that reflects its effects on deforestation rates from 2011 to 2050.

The rates of deforestation on this period are smaller when compared to the deforestation rates of the 1980s, 1990s and the beginning of the twenty-first century when the state’s growth rate was still under strong effects of the policy of incentive to occupy and deforest for production. However, deforestation reduction is already a reality throughout Amazonia [3, 26], even if due to the smaller amount of forest available for this activity.

As expected, the 80% LR size positively influenced the decrease in deforestation over a 50% LR, presenting approximately 63% reduction. The difference between the scenarios in maintaining an LR of 80% and reducing it to 50% implies, at the end of 41 years, the emission of around 60,610,417 t of more CO2 into the atmosphere. This is equal to nearly 7 years of CO2 emissions from the solid waste and domestic effluent sector in the state of São Paulo in 2008 [33].

The commercialization of CO2 emission certificates avoided by deforestation is still an uncertain market. However, if it was to be regularized and implemented de facto, the Clean Development Mechanism (CDM) market, established in Article 12 of the Kyoto Protocol, the emission difference with an LR of 50% in relation to that of 80% may represent a loss of about $606.00 million if we consider an average value of$10.00 a ton of CO2. And the prospect can be even worse when considering the 80% LR difference with the value obtained from the “Business as usual” scenario, reaching a loss of up to \$1.19 billion.

The CO2 considered in these accounts is only that contained in the C above the ground. Thus, this loss would only be related to the emission of CO2, without taking into account other benefits offered by forests, such as the conservation of scenic beauty, socio-biodiversity, water and its services, climate regulation and soil conservation.

In the areas of the Brazilian Amazon, one of the elements that facilitate the dynamics of deforestation is the implantation of roads, since this practice allows easy access to forest remnants areas with greater potential for deforestation [34, 35]. In addition to the roads, rivers are also a source of easy access that enables deforestation, without taking into account that their use is often cheaper than transport by road. Ferreira et al. [36] argue that the process of deforestation usually begins with the official or clandestine opening of roads that allow the irregular occupation of land to the predatory exploitation of timber. Normally, after the exploitation of the forest, this area is converted to agriculture or pasture for the extensive creation of livestock, especially in large estates, being responsible for about 80% of deforestation in the legal Amazon.

Some researchers claim that the causes of deforestation in the Amazon are complex and often interrelated. Among them, there is the absent role of states, which in many situations do not restrict illegal deforestation [14, 37]. Added to this, there is still the ambiguity of public policies that stimulate, on one hand, the deforestation by great landowners, who thus assure the land ownership in the face of the “threat” of agrarian reform [37].

The state of Acre follows the model described, especially in the southeast region of the state. In this region, there is the highest concentration of deforestation, as is the case for the municipalities of Senador Guiomar, Acrelândia, Plácido de Castro, Rio Branco, Xapuri, Brasiléia, Epitaciolândia, Bujari and Porto Acre. However, it is in these municipalities that most of the population of Acre is found, along with most of the agricultural activity of the state and the greater demand for water resources. The region has hundreds of properties with 50–100% of its areas deforested [38]. This is a situation that may qualify these properties as illegal when facing the law, regardless of the size of the LR kept inside the property.

The other regions of the state still maintain most of their green areas preserved or, at least, conserved [26, 39]. Law No. 4771 of 1965 which refers to the Forest Code determines, among other obligations, the protection of sensitive areas and that landowners must maintain a portion of the native vegetation inside the rural properties, as is the case of the LR and Permanent Preservation Areas (APPs2). In the latter case, in general, forest management is prohibited, except in specific cases. Thus, one of the well-debated points is the size of the portion of the LR area to be maintained within each rural property, since from the implementation of Provisional Measure No. 1511 of 1996 it was determined that the LR in the Amazon region would increase from 50 to 80% of each rural property [5].

The changes in legislation aimed to restrict the rampant devastation that was taking place within the Amazon from 1994 to 1995, time and place in which the highest deforestation rate in history was recorded, with more than 29,000 km2 of the forest devastated [40]. The size of the LR of a property functioned as a legislative component, which has its direct implications under the prospect of reducing deforestation.

Modification of the property use limit has curtailed deforestation for some years. However, deforestation rates returned to exponential growth starting in 1998 and had its peak in 2004, when, through public policies and more efficient geospatial monitoring, the deforestation scenario in the Amazon was again reduced.

Forests are of fundamental importance for balancing CO2 emissions globally. When these are suppressed, most of the C stored in them is released into the atmosphere by burning or, more slowly, by decomposition [41].

CO2 emissions due to changes in soil cover from land use have become a major component in the last decades, already considered as “dangerous interferences” in the global climate system. Approximately 0.8 ± 0.2–2.2 ± 0.8 billion tons of CO2 from tropical forest deforestation is released annually, accounting for about 10–35% of global CO2 emissions [5].

Deforestation in Brazil is the process that contributes most to CO2 emissions, accounting for 70% of national emissions [27]. The stock of C in the state of Acre present in the aerial and underground plant biomass and the C contained in the soil varies from 75 to 270 Mg ha−1, according to data published by the Environment State Secretariat (SEMA3) [42].

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## 5. Final considerations

From a modeling point of view, the methodology and concepts used in the Dinamica EGO software were effective in representing deforestation under the three projections for the Western South Amazon. The variables selected and used in the scenarios clearly showed their influences both in the sense of restricting, as well as in potentializing the conversion of forest areas.

About the results of the deforestation of the projected scenario up to the year 2050, and considering the prospects of maintaining the current “business as usual” deforestation patterns, a severe intensification of deforestation is predicted over the years. As a consequence, the South Western Amazon is expected to reach very high levels of degradation, increasing CO2 emissions to 155 million tons.

In the simulation of scenario 2, a reduction of CO2 emissions of approximately 59 million tons would occur, when compared to the permanency of the current scenario. In this case, the reduction of the Legal Reserve (LR) to 50% must be considered, as it would lead to a loss, within the legality, of approximately 200,700 ha of native forests in relation to an LR of 80%. In addition, CO2 emissions would increase by about 609,119,217 t. Based on the results of this study, maintaining the size of LR in 80% of rural properties would prevent the emission of millions of tons of CO2 into the atmosphere.

Thus, modeling tools are important mechanisms for forecasting and decision-making that can be used in order to assist in environmental conservation practices.

## Notes

• Agência Nacional de Águas.
• Áreas de Preservação Permanente.
• Secretaria de Estado do Meio Ambiente.

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Valderli Jorge Piontekowski, Ângela Pereira Bussinguer, Fabiana Piontekowski Ribeiro, Jonas Inkotte, Karla Monique Silva Carneiro Valadão, Marco Bruno Xavier Valadão, Mirella Basileu de Oliveira Lima, Alcides Gatto, Eraldo Aparecido Trondoli Matricardi and Reginaldo Sérgio Pereira (December 20th 2017). Modeling Deforestation and CO2 Emissions in Tropical Forests (Western South Amazon), New Perspectives in Forest Science, Helder Filipe dos Santos Viana and Francisco Antonio García Morote, IntechOpen, DOI: 10.5772/intechopen.72806. Available from:

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text:
# Test Function Convexity

I'm trying to test if $f : \mathbb{R}^n\rightarrow \mathbb{R}$ given by :

$$f(x) = \ln \left(\sum\limits_{i=1}^n e^{x_i}\right)$$ is convex.

I tried computing positivity of the Hessian $H$ matrix by definition, multiplying by $a \in \mathbb{R}^n$. I got the expression

$$a^T H a = \displaystyle\sum_{i=1}^n a_i^2 e^{x_i} -(a_1 + ... + a_n)\frac{a_1 e^{x_1} + ... + a_n e^{x_n}}{e^{x_1} + ... + e^{x_n} }$$

Where $a_i$ is the $i$-th coordinate of $a$.

Is there a way to check if this expression is higher than zero? I can't see if an averages inequality would help.

Let $\lambda, \mu \in \langle 0,1\rangle$ be such that $\lambda + \mu = 1$ and $x = (x_1, \ldots, x_n), y= (y_1, \ldots, y_n) \in \mathbb{R}^n$. We wish to show that $$f(\lambda x + \mu y) \le \lambda f(x) + \mu f(y)$$
We shall use Hölder's inequality for conjugate exponents $\frac1{\lambda}$ and $\frac{1}\mu$. We have:
Hence, $f$ is convex on $\mathbb{R}^2$.


text:
# Universal generation of the cylinder homomorphism of cubic hypersurfaces

Research paper by Renjie Lyu

Indexed on: 29 Oct '18Published on: 29 Oct '18Published in: arXiv - Mathematics - Algebraic Geometry

#### Abstract

In this article, we prove that the Chow group of algebraic cycles of a smooth cubic hypersurface $X$ over an arbitrary field $k$ is generated, via the natural cylinder homomorphism, by the algebraic cycles of its Fano variety of lines $F(X)$, under an assumption on the $1$-cycles of $X/k$. As an application, if $X/\mathbb{C}$ is a smooth complex cubic fourfold, our result provides a proof of the integral Hodge conjecture for curve classes on the polarized hyper-K\"ahler variety $F(X)$. In addition, when $k$ is finitely generated over its prime subfield with $\text{char}(k)\neq 2, 3$, we use our conclusion to prove the integral analog of the Tate conjecture for $1$-cycles on $F(X)$ for a smooth cubic fourfold $X/k$.


text:
# Tag Info

1

I just found the answer myself. In this paper: Lyle Ramshaw, Robert E. Tarjan (2012). "On minimum-cost assignments in unbalanced bipartite graphs". Technical reports, HP research labs. in section 5, the authors show that the Hopcroft-Karp algorithm in fact solves the following problem: given an integer $s$, find matchings with $1,\ldots,s$ edges. The ...

4

The problem can be solved in time $\tilde{O}(n^{4/3})$, using several algorithms: Agarwal, Partitioning arrangements of lines II: Applications. Chazelle, Cutting hyperplanes for divide-and-conquer. Matoušek, Range searching with efficient hierarchical cuttings. There is an essentially matching lower bound due to Erickson, New lower bounds for Hopcroft's ...

1

Your problem is known as maximum coverage.

2

It seems you are confusing the terms minimal and minimum. The Hitting set problem is to find a hitting set of minimum cardinality, not a minimal set. A set $S$ is called minimal with respect to some property (in this case, being a hitting set) if there exists no strict subset $T$ of $S$ (i.e. $T\subsetneq S$) that also satisfies that property. On the ...

1

There is no heuristic that is "universal" in practice. Which heuristic works best in practice often depends on the specific problem you're dealing with. There's no one "silver bullet" heuristic that works great on every optimization problem.

0

Your question is not quite clear, but I believe you are asking how to show that 1in3SAT is NP-complete via a reduction from the known NP-complete problem 3SAT. If this is your question, then consider the following set up, using your notation: Let an instance of 3SAT be defined by a set of $n$, Boolean variables $V = \{x_1, ..., x_n\}$ and a Boolean equation ...

0

I believe the best algorithm known is Hopcroft and Karp, "An $n^{5/2}$ Algorithm for Maximum Matchings in Bipartite Graphs", SIAM Journal of Computing 2:4 (1973), pp 225-231.

0

During the forward pass, you compute F such that F[i] represents the maximum possible profit when restricting yourself to the elements with indices 0,1..,i. During the backward pass, you compute B such that B[i] represents the maximum possible profit when restricting yourself to the elements with indices i,i+1,..,n-1 (where n is the size of your list). The ...

1

This statement seems quite vague and confused. NP-hardness is precisely defined, and you shouldn't loosely say "optimization TSP problems" because it doesn't specify the problem(s). In fact, it is not uncommon that seemingly small changes in a problem lead to different complexities. Further, because NP-hardness is precisely defined, a problem is not NP-...

0

The question is whether the distances between cities fulfil the triangular equation: For any three A, B and C, is distance (A, B) ≤ distance (A, C) + distance (C, B)? In your example, where distance (A, B) = 10, distance (C, B) = 1,000,000,000 and distance (X, B) = infinite (or very, very large) for all other X: Yes, it seems that going A->B->A avoids the ...

Top 50 recent answers are included


text:
# Why do TTL pulse generators generate an asymmetrical square wave (unlike CMOS)?

I've read that the asymmetrical (about 1/3 mark-to-space) output waveform is due to input gate characteristics of a TTL inverter, but I want to know more about which characteristics of the TTL input gate causes this asymmetry. Thanks!

• For one, the switching threshold for TTL is about 0.8 V while for CMOS it is Vdd/2. – The Photon Sep 29 '17 at 16:03
• ... and the TTL inputs source some significant current. – Transistor Sep 29 '17 at 16:22

Two reasons. First, a TTL input is actually an output. It sources a current that must be shunted to ground to be seen as a logic 0. This current affects the charging rate of the timing capacitor, adding to the current from the output through the timing resistor. Above 2.4 V the input no longer sources current out the input pin. When the output goes low, the timing resistor discharges the timing capacitor, but when the capacitor voltage gets low enough the input starts sourcing current into it, changing the discharge rate.

Separate from that, as above, the two input transition levels (there are two because of hysteresis) are not centered about a voltage equal to half way in between the two extremes of the output voltage. A typical TTL high output is around 3.5 V, so a "CMOS equivalent" set of input transition levels would be centered around 1.7 V. But the two TTL transition levels are centered around 1.3 V (original 7414), so the charging and discharging currents through the timing resistor are not equal.

https://en.wikipedia.org/wiki/Transistor%E2%80%93transistor_logic

https://en.wikipedia.org/wiki/Schmitt_trigger

CMOS Schmitt triggers have the two threshold voltages at $\frac 1 3 V_{DD}$ and $\frac 2 3 V_{DD}$. Furthemore $V_{OH}=V_{DD}$ and $V_{OL}=0$.

Since this is symmetric with respect to $V_{DD}/2$, it follows that the timings $T_H$ and $T_L$ are approximately equal, from the equations:

$$V(t=T_H)=V_{t+}=\frac {2}{3}V_{DD}=V_{DD}+(\frac {V_{DD}} 3 -V_{DD})e^{-\frac {T_H} \tau}$$ $$V(t=T_L)=V_{t-}=\frac {1}{3}V_{DD}=(\frac {2} 3 V_{DD})e^{-\frac {T_L} \tau}$$

(The two equations give the same $T_L$ and $T_H$).

This of course holds true if R is much higher than the inverter's pMOSFET and nMOSFET on-state resistances, and if the thresholds are exactly as shown. Minor variations will give you a non 50% duty cycle.

TTL devices, not only have asymmetrical thresholds (0.9 V and 1.7 V), but also the no-load logic levels are approximately $V_{OH}=3.5\ V$ and $V_{OL}=0.2\ V$. The two thresholds are not "centered" in this range. Furthermore they have a very non negligible input current (especially for large values of the resistor), which changes the equivalent charging/discharging resistance seen by the capacitor.

EDIT:

If you need a 50% duty cycle with the TTL (but also on CMOS), then you can for instance use half the capacitance (the frequency doubles) and cascade a T-flip flop (the frequency what you originally wanted, and the duty cycle is 50%).

Your example circuit shows a Schmitt-trigger inverter. Look at the threshold voltages in the example table...their average is less than half-Vcc:
The capacitor charges toward Vcc (or Vdd) faster than it discharges toward ground. This voltage threshold asymmetry is a major contributor to pulse-width asymmetry.

If Schmitt thresholds were symmetrical about half-Vcc, output pulses should be close to 50% high, 50% low. I am assuming that the gates' output stage pulls high (toward Vcc) in a similar way as it pulls low (toward ground). Asymmetry in a gate's output stage is another source of asymmetry, if current is flowing into a load.

• Those do not look like TTL transitions levels. – AnalogKid Sep 29 '17 at 19:04
• @AnalogKid Certainly, the table is for HCMOS schmitt...(TTL is only spec'd for Vcc=5V). For the TTL schmitt (7414), average threshold is worse, about 1.2V, causing even worse pulse skew. And TTL output stage asymmetry causes even more pulse skew in the OP's oscillator. HCMOS is preferable, but not perfect. – glen_geek Sep 29 '17 at 22:42

The character of CMOS thresholds allows a symmetric placement of the positive and negative trip points of the Schmitt circuit, and of the logic output drive voltages relative to those trip points. So, the voltage across the feedback resistor may be 5V - 1.66V when the CMOS first goes HIGH, and 0V - 3.33V when it first goes LOW. Those equal (though opposite) voltages make equal currents and take equal times to charge that capacitor, so you get a 50% duty cycle. Charge and discharge currents are in the ratio 1:1.

In TTL, however, the thresholds may be around 1.4V (like, 1.2V low and 1.6V high), while the output drive levels are 3.2V high and 0.5V low. So, voltage drops across the feedback resistor are 3.2V-1.2V in one phase, and 0.5V-1.6V in the other. That means the currents that charge and discharge the capacitor are in the ratio of 2.1:1.1, and that is NOT how you get a 50% duty cycle.

The important character of TTL is that input thresholds are better controlled than CMOS (have a small uncertainty) and the output voltages have less symmetry than CMOS. After that, it's just figuring of details.

Do not assume you have 50% duty cycle with CMOS. Process variations: doping, implant ion beam energy, annealing temperatures and times, etc contrive to implement functioning circuit that may be far from nominal performance.