Ancestry DNA and 5 Ways Tech Has Affected Psychological Research

Five Ways That Technology Has Affected Psychological Research

Information technology has played a large role in enhancing research efforts in science of the mind and behavior. Technology has influenced the discipline not only in how research studies are carried out and analyzed, but also in the actual phenomena that psychologists can investigate. Here are five ways that technology has affected psychological research. 

ancestry dna promo codesDNA Tests Make Everyone a Geneticist

While the jury’s out on the efficacy of these DNA tests, the popularity of them means we have a lot more data about people than we used to. Once we figure out how to organize that data to understand people’s behavior better, we may start including DNA tests as part of counselling sessions. Right now, these tests have become very affordable. Coupled with coupon codes that you can find by clicking here, a test can be as cheap as $50.

The people studied 

Psychological research has historically been strongly biased towards the study of psychology students, because they are convenient for research to find and you can basically force them to do it as a compulsory course requirement. Although this bias towards students still exists, it is assuaged somewhat by the ability to use the internet and mobile apps to reach broader samples of people. 

Lower Costs 

The primary research method in psychology is the self-report questionnaire, which typically involves asking participants to rate certain statements on a numerical scale. Although this method has its critics, one advantage is that is can easily be conducted and analyzed online. Where pen and paper surveys once ruled, computers have taken over. This has reduced the cost of surveying dramatically. 

Easier and more complex analysis 

There was a time when psychologists conducted statistical analyses essentially by hand, with the help of whatever rudimentary calculators were available at the time. Software such as SPSS means that people who are “not good with numbers” are no less able to work as psychological researchers, and greatly reduces the time that researchers spend analyzing their data. 

New measurement tools 

Incredibly accurate research on things like reaction times can now be conducted using specialized computer packages. Mobile technology, using a technique called the experience sampling method, allows psychologists to ask people questions as they go about daily lives. But perhaps the largest shift in measurement has come from neuroscience, where brain-imaging technology has drastically enhanced the understanding of the brain. 

Researching technology 

Technology, and people’s relation to it, is a research topic in itself. For example, some studies are looking into how social groups behave online compared to real-life groups. Others are looking into how internet use affects people, such as the link between social media use and anxiety, or how the internet is impacting people’s ability to concentrate. It’s easy to entice people who spend a lot of time online to participate in studies by offering them discount codes for services they tend to use such as, or a product that they might want such as an MP3 player (for top picks you can see the list here).

The Future 

Advances in neuroscience mean that at some point in your lifetime, researchers will be able to monitor people’s thoughts, and display what they are thinking or dreaming on a computer monitor. Other researchers are theorizing that brain implants could be used to treat certain neurological conditions, enhance certain mental functions, or enable several people with these implants to communicate telepathically perhaps for use in the military. 

More and more, technological advancements are being brought into psychological research, and this trend will continue. The future of this marriage may seem like science fiction, but it is coming. Psychology and technology are now locked onto the same path.

What Is Lucid Dreaming? A Guide for Beginners

A Beginner’s Guide to Lucid Dreaming

Lucid dreaming is precisely when a person dreams and can comprehend his/her dream. The act of dreaming in a lucid-state has intrigued people for centuries: but what is it, exactly?   It’s the ability to regulate the dream and make some other reality in the dream state.

There are some ways to make the act of lucid dreaming easier. There’s a very effective lucid dreaming supplement regimen you can take that will enhance your body’s natural REM sleep state (which is thought to be when your mind is dreaming).

LD (Lucid Dreaming) Has a Long History.

A lot of cultures from ancient times took lucid dreaming very seriously. For instance, people living in ancient Egypt thought that dreams opened up the eyes of a person to truths, solutions, and advice that were hidden. In fact, lots of temples were devoted to dream incubation by ancient Egyptians. The practice of learning the numerous lucid dreaming steps, different states of consciousness, and out of body experiences was done by ancient Egyptian so-called masters of secret things. They firmly believed that individuals could get an altered state of consciousness from the lotus flower.

In today’s world, it can be associated with the ‘dark arts’.  And this is just one of the many misconceptions surrounding lucid dreaming. Contrary to what people believe, there are a lot of significant benefits to lucid dreaming. It can have a favorable effect on an individual’s waking life by improving problem-solving abilities, bettering creativity, and bettering a person’s overall self-confidence.

Lucid Dreaming Tips

There are a few techniques you can practice to enhance your chances of having lucid dreams on a regular basis. They involve basic memory exercises and also guided meditation. Training the subconscious mind to follow brand-new rules is the main objective of the different lucid dreaming steps. The very first step when it comes to learning is to cultivate a solid commitment to mastering the skill. The standard learning process takes a little time, research, and mental preparation.

Keep a Dream Journal.

It’s essential to keep a dream journal if you’re new to lucid dreaming. The awareness of the dream state of a person is enhanced by the journal and makes it much easier to get up while dreaming. A person will encounter roughly five dream cycles nightly, which is the average. A person will experience a period of rapid eye movement sleep or REM abbreviation as each sleep cycle concludes. REM sleep lasts longer as it gets nearer to the morning. This gives you the ability to remember dreams quickly after waking up in the morning.

Write In Your Dream Journal Regularly.

When it comes to keeping a dream journal, the steps involved include things such as choosing a journal and keeping it near the bed. This is essential as lots of dreams disappear quickly upon waking. Writing a step by step description of the dream in the present tense and noting the dream’s date is the next step. Next, you should analyze and identify the themes of the dream. After that, give the dream a name.

Reality Checks are Essential.

Not only should the person keep a dream journal, but also novices must do at least 10 reality checks every day. Reality checks bring about more vivid and lucid dreams by building better self-awareness during the morning hours. Reality checks give one the ability to identify the difference between a dream and a waking reality. Utilizing reality checks in the waking reality gives people the power to do them during their dreams.

A Simple Reality-Check Example.

As an example, a fairly easy reality check could be counting the fingers on each hand and will them to be more or less than 10. Obviously, in real life, ten is the total amount of fingers on each hand. However, willing your hand to have more than 10 digits is a great way to know that you are in a dream state. Performing about 20 minutes of meaningful meditation every day is the final method of inducing lucid dreams for novices.

The Next Step in Your Journey.

A few lucid dreaming experts have developed different tips for lucid dreaming that should help you on your journey. One of the most popular methods is the Mnemonic Induction of Lucid Dreams, which incorporates dream recall, lucid affirmations, visualization of the dream, as well as reality checks.
Mastering the method of waking up during a dream isn’t that simple, which is just like many other things worth doing. Time, patience, focus, and practice are the things it requires, which anybody can learn.

Reflection Matrix

It’s been a while since my last math post. It seems that my previous assertion correlating the quality of question with the period of final exams could not have been further from the truth. It now seems quite likely that the occurrence of a few “good” questions on Yahoo Answers in a short period of time a few weeks back was more of a fluke rather than the norm.

Problem: Find the standard matrix for the linear transformation which reflects points in the x-y plane across the line y = \frac{-2x}{3}.

Solution: To find the matrix representing a given linear transformation all we need to do is to figure out where the basis vectors, i.e.

\vec{e_{1}}=\begin{bmatrix} 1\\ 0 \end{bmatrix} and \vec{e_{2}}=\begin{bmatrix} 0\\ 1 \end{bmatrix}

are mapped under the transformation.

This is easy to see from the fact that since \vec{e_{1}} and \vec{e_{2}} form a basis of \mathbb{R}^{2} any element

\vec{u}\ \epsilon \ \mathbb{R}^{2}

can be written as

\vec{u} = a \cdot \vec{e_{1}} + b \cdot \vec{e_{2}}

for some a,b\ \epsilon\ \mathbb{R}.

Now let

A : \mathbb{R}^{2} \to \mathbb{R}^{2}

be a linear transformation. Then, by definition

A(\vec{u}) = A(a \cdot \vec{e_{1}} + b \cdot \vec{e_{2}}) = a \cdot A( \vec{e_{1}} ) + b \cdot A( \vec{e_{2}} )

Thus, the image of any element \vec{u}\ \epsilon \ \mathbb{R}^{2} under a linear transformation is completely determined by the image of the basis of the transformation’s domain.

Let’s find the image of \vec{e_{1}}=\begin{bmatrix} 1\\ 0 \end{bmatrix} under a reflection across the line given by y = \frac{-2x}{3}. First, we need to find a line perpendicular to y = \frac{-2x}{3} that passed through the point p_{1} = (1, 0).

In general, a line perpendicular to y = mx + b is given by y = \frac{-x}{m}, i.e in our case, y = \frac{3x}{2}. To find the one particular line with the same slope, that passes through p_{1} = (1, 0), we need to simply plug in the values of x = 1 and y = 0 into the line equation and add a constant value so that the equality will hold. Then we have

(0) = \frac{3(1)}{2}+b

Thus, the line perpendicular to y = \frac{-2x}{3} that passes through p_{1} = (1, 0) is given by y = \frac{3x}{2} – \frac{3}{2}

reflection matrix chart
The lines given by equations (1) – red and (2) – green

Now let’s find the point q_{1} = (x, y) at which our two lines y = \frac{-2x}{3} and y = \frac{3x}{2} – \frac{3}{2} intersect. This is done easily by solving the two line equation simultaneously:

y = \frac{-2x}{3} \ (1)

y = \frac{3x}{2} – \frac{3}{2} \ (2)

Equating (1) and (2) (and multiplying by 6 for simplicity) gives

-4x = 9x – 9

x = \frac{9}{13}

which then gives

y = \frac{-6}{13}

Thus, our two lines intersect at

q_{1} = \left (\frac{9}{13}, \frac{-6}{13} \right )

Now, let {p_{1}}’ = (x, y) be the reflection of p_{1} = (1, 0) across y = \frac{-2x}{3}. Since {p_{1}}’ and p_{1} are symmetric across q_{1} (because that is how the reflection is actually constructed) it follows that

x = q_{1}x – \left | p_{1}x – q_{1}x \right | = \frac{9}{13} – \left | 1 – \frac{9}{13} \right | = \frac{5}{13}

y = q_{1}y – \left | p_{1}y – q_{1}y \right | = \frac{-6}{13} – \left | 0 – \frac{-6}{13} \right | = \frac{-12}{13}


A( \vec{e_{1}} ) = \begin{bmatrix} \frac{5}{13} \\ \frac{-12}{13} \end{bmatrix}

Similarly as above, we find the image of \vec{e_{2}}=\begin{bmatrix} 0\\ 1 \end{bmatrix} under a reflection across the line given by y = \frac{-2x}{3}. The line perpendicular to y = \frac{-2x}{3} that passed through the point p_{2} = (0, 1) is given by

y = \frac{3x}{2} + 1 \ (3)

The lines given by equations (1) – red, (2) – green, and (3) – blue

Equating (1) and (3) (and multiplying by 6 again) gives

-4x = 9x + 6

x = \frac{-6}{13}

and so

y = \frac{4}{13}

Therefore, (1) and (3) intersect at

q_{2} = \left (\frac{-6}{13}, \frac{4}{13} \right )

and the image {p_{2}}’ = (x, y) of p_{2} = (0, 1) under the reflection across y = \frac{-2x}{3} is given by

x = q_{2}x – \left | p_{2}x – q_{2}x \right | = \frac{-6}{13} – \left | 0 – \frac{-6}{13} \right | = \frac{-12}{13}

y = q_{2}y – \left | p_{2}y – q_{2}y \right | = \frac{4}{13} – \left | 1 – \frac{4}{13} \right | = \frac{-5}{13}

which yields

A( \vec{e_{2}} ) = \begin{bmatrix} \frac{-12}{13} \\ \frac{-5}{13} \end{bmatrix}

Finally, the matrix A : \mathbb{R}^{2} \to \mathbb{R}^{2} for the linear transformation which reflects points in the x-y plane across the line y = \frac{-2x}{3} is given by

A = \begin{bmatrix} \frac{5}{13} & \frac{-12}{13} \\ \frac{5}{13} & \frac{-5}{13} \end{bmatrix}

It turns out (although I’m not going to show to derivation here) that in general, the matrix for a linear transformation which reflects points in the x-y plane across an arbitrary line y = mx + b is given by

A = \begin{bmatrix} cos2\theta & sin2\theta \\ sin2\theta & -cos2\theta \end{bmatrix}

where \theta is the angle that the line makes with the positive x-axis, i.e

\theta = arctan(m) if m \geq 0


\theta = arctan(\left | m \right |) + \frac{\pi }{2} if m < 0

Triton’s Atmosphere More Mysterious Than Previously Thought

The first ever infrared analysis of the atmosphere of Neptune’s moon Triton revealed the presence carbon monoxide and methane. As summer hit the moon’s southern hemisphere, observations made at the Very Large Telescope (VLT) based at the European Southern Observatory (ESO) showed the thin atmosphere to vary with seasons.

triton atmosphere
Artist’s impression of how Triton, Neptune’s largest moon, might look from high above its surface. The distant Sun appears at the upper-left and the blue crescent of Neptune right of center – Credit: ESO/L. Calçada

“We have found real evidence that the Sun still makes its presence felt on Triton, even from so far away. This icy moon actually has seasons just as we do on Earth, but they change far more slowly,” says Emmanuel Lellouch, the lead author of the paper reporting these results in Astronomy & Astrophysics.

On Triton, where the average surface temperature is about minus 235 degrees Celsius, it is currently summer in the southern hemisphere and winter in the northern. As Triton’s southern hemisphere warms up, a thin layer of frozen nitrogen, methane, and carbon monoxide on Triton’s surface sublimates into gas, thickening the icy atmosphere as the season progresses during Neptune’s 165-year orbit around the Sun. A season on Triton lasts a little over 40 years, and Triton passed the southern summer solstice in 2000.

Based on the amount of gas measured, Lellouch and his colleagues estimate that Triton’s atmospheric pressure may have risen by a factor of four compared to the measurements made by Voyager 2 in 1989, when it was still spring on the giant moon.

Carbon monoxide was known to be present as ice on the surface, but Lellouch and his team discovered that Triton’s upper surface layer is enriched with carbon monoxide ice by about a factor of ten compared to the deeper layers, and that it is this upper “film” that feeds the atmosphere. While the majority of Triton’s atmosphere is nitrogen (much like on Earth), the methane in the atmosphere, first detected by Voyager 2, and only now confirmed in this study from Earth, plays an important role as well.

triton moon
Voyager 2 raw image of Neptune’s satellite Triton taken from roughly 500,000 km. Evidence of complex surface features can be seen from this distance – Credit: NASA

Of Neptune’s 13 moons, Triton is by far the largest, and, at 2700 kilometers in diameter (or three quarters the Earth’s Moon), is the seventh largest moon in the whole Solar System. Since its discovery in 1846, Triton has fascinated astronomers thanks to its geologic activity, the many different types of surface ices, such as frozen nitrogen as well as water and dry ice (frozen carbon dioxide), and its unique retrograde motion.

Observing the atmosphere of Triton, which is roughly 30 times further from the Sun than Earth, is not easy. In the 1980s, astronomers theorised that the atmosphere on Neptune’s moon might be as thick as that of Mars (7 millibars). It wasn’t until Voyager 2 passed the planet in 1989 that the atmosphere of nitrogen and methane, at an actual pressure of 14 microbars, 70 000 times less dense than the atmosphere on Earth, was measured. Since then, ground-based observations have been limited. Observations of stellar occultations (a phenomenon that occurs when a Solar System body passes in front of a star and blocks its light) indicated that Triton’s surface pressure was increasing in the 1990′s. It took the development of the Cryogenic High-Resolution Infrared Echelle Spectrograph (CRIRES) at the Very Large Telescope (VLT) to provide the team the chance to perform a far more detailed study of Triton’s atmosphere.

Golden Nature: Closed-Form Formula For Fibonacci Sequence

Almost everyone has heard of this sequence: 1,\ 1,\ 2,\ 3,\ 5,\ 8,\ etc. It is named after Leonardo of Pisa who introduced it to the western world in one of the most influential books ever published in mathematics – Liber Abaci. This book introduced Europe to the Hindu numerals 0 through 9, the word zero, the notion of an algorithm and the subject of algebra.

The beauty of the Fibonacci sequence and the golden ratio (which is intimately connected to it) lies in that they are not just another mathematical construct, but occur throughout the nature.

Have you ever taken a look at a pine cone and noticed that the scales of the cone are in spirals? Have you ever counted the spiral in the clockwise and counterclockwise directions? I would be surprised if you have … but the counts turn out to be 5 and 8 (or 8 and 13 for bigger cones).

fibonacci sequence closed form

How about these? Care to count the petals?

13 petals
fibonacci 21
21 petals

… intriguing stuff indeed.

So, once again, I was reading through problems on Yahoo Answers. Most of the questions “reduce” (lol) to plugging numbers into well known equations, or using a calculator such as TI-89, or Mathematics, Maple, or even Google – basically, what engineers do. Yeah, you “heard” me right! It takes some effort to found a meaningful question that actually requires some knowledge and skills – see the difference between a mathematician and an engineer now? That said, here’s a one I couldn’t resist:

Problem: Find the term F_{386} in the following sequence F_{0}=-4,\ F_{1}=5,\ F_{2}=1,\ F_{3}=6,\ F_{4}=7,\ F_{5}=13

There is a more “elegant” way to tackle this problem and I may write about in the future (maybe quite soon actually), but for now I’ll ignore matrices, diagonalization and eigenspaces (although the reason why the following solution gives the correct result is tightly connected to linear algebra) and focus, instead, on the recursive nature of the Fibonacci sequence.

Solution: To find the term, “all” that is needed is to find the closed form for the n-th terms of the sequence F_{n}. In general, a Fibonacci sequence is given by

F_{n} = F_{n-1} + F_{n-2}\ (1)

and any particular sequence is fully determined by the initial two terms

F_{0} and F_{1}

In our example, we have

F_{0}=-4 and F_{1}=5

To find the closed form, let’s assume

F_{n}=(-4)\times t^{n}

Then using our recursive formula (1) we get

(-4)t^{n+1}=(-4)\left ( t^{n} + t^{n-1}\right )

and so

t^{2}=t + 1\Rightarrow t^{2} – t – 1 = 0\ (2)

Solving the quadratic from (2) gives

t_{1,2}=\frac{1\pm \sqrt{5}}{2}

Using this result, we can write the closed form formula for the n-th term of our sequence as

F_{n}=-4\left ( \ C_{1}\left ( \frac{1+\sqrt{5}}{2} \right )^{n}+C_{2}\left ( \frac{1-\sqrt{5}}{2} \right )^{n}\ \right ) (3)

where C_{1} and C_{2} are constant parameters determined by our initial conditions,

F_{0}=-4 and F_{1}=5

We can calculate these values by solving the following system of equations

F_{0} = -4\times t^{0}=-4 \left ( C_{1}\times t_{1}^{0} + C_{2}\times t_{2}^{0} \right )

F_{1} = -4\times t^{1}=-4 \left ( C_{1}\times t_{1}^{1} + C_{2}\times t_{2}^{1} \right )

Plugging in known values gives us

-4 = -4 \left (C_{1} \left ( \frac{1+\sqrt{5}}{2} \right )^{0} + C_{2} \left ( \frac{1-\sqrt{5}}{2} \right )^{0} \right )

5 = -4 \left (C_{1} \left ( \frac{1+\sqrt{5}}{2} \right )^{1} + C_{2} \left ( \frac{1-\sqrt{5}}{2} \right )^{1} \right )

Simplifying turns the above into

1 = C_{1} \left ( \frac{1+\sqrt{5}}{2} \right )^{0} + C_{2} \left ( \frac{1-\sqrt{5}}{2} \right )^{0} = C_{1} + C_{2} \ (4)

\frac{-5}{4} = C_{1} \left ( \frac{1+\sqrt{5}}{2} \right ) + C_{2} \left ( \frac{1-\sqrt{5}}{2} \right ) \ (5)

Now we need to solve (4) and (5) for C_{1} and C_{2}. Although I said I was going to leave matrices alone, this is a perfect situation to use them. I prefer to use the procedure described below for a system of two equations with coefficients such as these. Substitution or elimination method would, of course, work, but they involves messy arithmetic which can be so easily error prone in situations such as this one.

We have the following system:

\begin{pmatrix} 1 & 1 \\ \frac{1+\sqrt{5}}{2} & \frac{1-\sqrt{5}}{2} \end{pmatrix} \begin{pmatrix}C_{1}\\ C_{2}\end{pmatrix} = \begin{pmatrix}1\\ \frac{-5}{4}\end{pmatrix}\ (6)

In general, to solve the equation


we use


which turns into


Hence, to solve (6) we need to find the inverse of our matrix. This is fairly simple for a 2 by 2 matrix … Let

A=\begin{pmatrix}a & b\\ c & d\end{pmatrix}


A^{-1}=\frac{1}{ad-bc}\begin{pmatrix}d & -b\\ -c & a\end{pmatrix}

and so

\begin{pmatrix} 1 & 1 \\ \frac{1+\sqrt{5}}{2} & \frac{1-\sqrt{5}}{2} \end{pmatrix}^{-1} =\frac{-1}{\sqrt{5}}\begin{pmatrix} \frac{1-\sqrt{5}}{2} & -1 \\ \frac{-1-\sqrt{5}}{2} & 1 \end{pmatrix}

Our equation (6) then becomes

\begin{pmatrix} 1 & 1 \\ \frac{1+\sqrt{5}}{2} & \frac{1-\sqrt{5}}{2} \end{pmatrix}^{-1}\begin{pmatrix} 1 & 1 \\ \frac{1+\sqrt{5}}{2} & \frac{1-\sqrt{5}}{2} \end{pmatrix} \begin{pmatrix}C_{1}\\ C_{2}\end{pmatrix} = \begin{pmatrix} 1 & 1 \\ \frac{1+\sqrt{5}}{2} & \frac{1-\sqrt{5}}{2} \end{pmatrix}^{-1} \begin{pmatrix}1\\ \frac{-5}{4}\end{pmatrix}

\begin{pmatrix}C_{1}\\ C_{2}\end{pmatrix} = \frac{-1}{\sqrt{5}}\begin{pmatrix} \frac{1-\sqrt{5}}{2} & -1 \\ \frac{-1-\sqrt{5}}{2} & 1 \end{pmatrix} \begin{pmatrix}1\\ \frac{-5}{4}\end{pmatrix}

which gives us

C_{1}=\frac{-2(1-\sqrt{5})-5}{4\sqrt{5}}\ (7)

C_{2}=\frac{2(1+\sqrt{5})+5}{4\sqrt{5}}\ (8)

Finally, by plugging (7) and (8) into (3) we get

F_{n}=-4\left (\left ( \frac{-2(1-\sqrt{5})-5}{4\sqrt{5}} \right )\left ( \frac{1+\sqrt{5}}{2} \right )^{n}+\left ( \frac{2(1+\sqrt{5})+5}{4\sqrt{5}} \right )\left ( \frac{1-\sqrt{5}}{2} \right )^{n}\right ) (9)

We can now “easily” compute F_{386} which is given by

F_{386}=-4\left (\left ( \frac{-2(1-\sqrt{5})-5}{4\sqrt{5}} \right )\left ( \frac{1+\sqrt{5}}{2} \right )^{386}+\left ( \frac{2(1+\sqrt{5})+5}{4\sqrt{5}} \right )\left ( \frac{1-\sqrt{5}}{2} \right )^{386}\right )

Google returns F_{386} \cong 5.2783459\times 10^{80} which is just an approximation, but now we have a formula for the precise answer.

If you wish to check for yourself that out formula (9) is indeed correct, click on the terms below to see the computation by Google:







On a side note – Raising a number to the power of 386 may seem like a long computation but it really requires only 10 multiplications! Don’t believe me?

Consider this:

\left ( \frac{1+\sqrt{5}}{2} \right )^{386}=\left ( \frac{1+\sqrt{5}}{2} \right )^{256}\left ( \frac{1+\sqrt{5}}{2} \right )^{128}\left ( \frac{1+\sqrt{5}}{2} \right )^{2}

To calculate

\left ( \frac{1+\sqrt{5}}{2} \right )^{256}

we use the fact that

1.)\ \left ( \frac{1+\sqrt{5}}{2} \right )^{2} = \left ( \frac{1+\sqrt{5}}{2} \right )\left ( \frac{1+\sqrt{5}}{2} \right )

2.)\ \left ( \frac{1+\sqrt{5}}{2} \right )^{4} = \left ( \frac{1+\sqrt{5}}{2} \right )^{2} \left ( \frac{1+\sqrt{5}}{2} \right )^{2}

3.)\ \left ( \frac{1+\sqrt{5}}{2} \right )^{8} = \left ( \frac{1+\sqrt{5}}{2} \right )^{4} \left ( \frac{1+\sqrt{5}}{2} \right )^{4}

8.)\ \left ( \frac{1+\sqrt{5}}{2} \right )^{256} = \left ( \frac{1+\sqrt{5}}{2} \right )^{128} \left ( \frac{1+\sqrt{5}}{2} \right )^{128}

As you can see, in every step (multiplication) we use the result from the previous one and so it takes only 8 steps (multiplications) to calculate

\left ( \frac{1+\sqrt{5}}{2} \right )^{256}

Finally, in the process of calculating

\left ( \frac{1+\sqrt{5}}{2} \right )^{256}

we get

\left ( \frac{1+\sqrt{5}}{2} \right )^{128} and \left ( \frac{1+\sqrt{5}}{2} \right )^{2}

as a bonus, and so to calculate

\left ( \frac{1+\sqrt{5}}{2} \right )^{386}

we now only need to use two more multiplications

\left ( \frac{1+\sqrt{5}}{2} \right )^{386}=\left ( \frac{1+\sqrt{5}}{2} \right )^{256}\left ( \frac{1+\sqrt{5}}{2} \right )^{128}\left ( \frac{1+\sqrt{5}}{2} \right )^{2}

This idea can be generalized to any

x\epsilon \Re

The (maximal) number of multiplications required to compute

x^{n}\ where\ 2^{p}

Star Wars Main Theme Guitar Tab

A Quick Guide to the Star Wars Theme Guitar Tabs:

I’ve played the guitar for some years but only recently I’ve discovered the version of Star Wars Main Theme played by Tomi Paldanius. As a true Star Wars fan i decided to tab it so share with the Star Wars Universe! This version is a bit simplified (only bass, melody and chords without inversions) but it works quite well.

Tomi Paldanius plays Star Wars – Main Theme

Tuning: Standard (EADGBe)

(repeat 2x)




| / slide up
| \ slide down
| h hammer-on
| p pull-off
| ~ vibrato
| + harmonic
| x Mute note

Aliens Found on Mars: A Brief History

A Short and Quick History of NASA Finding Life on the Martian Planet.

The red planet has held a special place in humanity’s collective imagination for centuries. However, since the Viking landers actually travelled to the red planet, the question of whether there is life on Mars has taken several sharp turns. Just in the past few years, there has been some major controversies and developments. We’ll explore the major ones, from most recent all the way back to the ‘Martian Face’ that NASA photographed in the 90’s.

July 2018: NASA Accused of Burning Evidence of Life on Mars

viking lander nasas

This controversy comes from an actual NASA report, but it has been wildly misinterpreted. Here’s how it goes: researchers discovered data from an old test of Martian soil done by the Viking landers back in the 1970’s. The Viking landers found no evidence of organic molecules, which was puzzling since we did find evidence of them when the two Martian rovers sampled Martian soil.
Now, the soil contained a compound called perchlorate, which is very flammable. There’s very little oxygen or heat on Mars, so it normally doesn’t burn. But the NASA report suggests that the Landers accidentally burned evidence of the Martian organic compounds by burning it with the perchlorate.
The controversy is a whole lot of nothing. We know now that organic compounds do exist on Mars, so whether the Viking landers messed up the first tests doesn’t really matter.

Remember that these are organic compounds, which are the building blocks of life, but aren’t evidence of life itself.

June 2018: Organic Matter Found on Mars

NASA announced that they had discovered organic matter preserved in mudstones that are over 3 billion years old (Earth is 4.5 billion years old, for reference). The announcement also mentioned the fact that we see methane being released into the atmosphere of Mars. Methane is evidence of some kind of organic matter being released into the atmosphere. It’s not clear what this organic matter is, however.

Well, at least that’s what UFO spotters looking at this NASA Mars Rover picture saw:

aliens found on mars
What according to some appears to be a skull with eye sockets and a nose – Source: BARCROFT/NASA

I, personally, am more inclined towards a bowling ball, which would definitely indicate the presence of intelligent life on Mars, unlike here, on Earth.

mars face
The “famous” Face on Mars – part of the Cydonia region, taken by the Viking 1 orbiter and released by NASA/JPL on July 25, 1976. – Source: NASA/JPL
female figure mars
A photo of what appears to be a “female” figure taken on the red planet by the Spirit rover. – Source: NASA

Use GoDaddy Coupon Codes To Save Reference Materials You Trust.

Trouble Finding Trustworthy Reference Materials?

Here’s something you may not have considered: create a website and store research papers and other materials that you trust and find valuable. If you’re not trying to profit from a website, then you can host materials there for your own use without any copyright worries.

Use GoDaddy Coupons To Keep the Costs Down.

The costs of setting up a self-hosted website don’t have to be prohibitive either. Use godaddy promo codes and you can have a quick self-hosted website up and running for $1/month, which includes a free domain name. We found valid promo codes at these sites:

Finding Reference Materials You Can Trust

These days it seems that there is more information available than ever before.  Even so, finding information that is both timely and accurate remains as big a challenge as it ever was.  Finding reference materials that can be trusted is often a daunting task, especially if the subject under review is an unusual one.  While online dictionaries and encyclopedias can provide a good starting point, the information provided there is sometimes suspect.  Researchers will often have to turn to other reference materials in order to get the complete story.

Understanding Your Reference Options

It is important for those researching a particular subject to understand and take advantage of all the reference materials at their disposal.  Some good ideas for uncovering the most useful reference materials include:

  • The local library – even in the age of the internet, the local library still has an important role to play when it comes to reference materials.  Libraries often have access to proprietary libraries of reference materials, just waiting to be tapped by eager researchers.
  • Workers in the field – for many subjects there is no better reference point than the people who work in the field every day.  From science to government, tapping the knowledge of the worker in the field is a great way to gather accurate and up to date information.
  • Trade magazines – there are trade magazines and other publications devoted to every imaginable industry, and reading those trade magazines is a great way to get a feel for what is really happening on the ground.  These trade magazines make excellent reference materials for anyone with an interest in a particular industry.

The Importance of In Depth Research

While a quick overview of a given industry or subject may be fine for some uses, it is important for researchers to understand the importance of in depth knowledge.  Having an in depth understanding of the latest trends in a particular field can allow researchers to read between the lines in order to determine what is really happening.  Having a thorough knowledge of a particular industry can also allow those on the outside to understand what is really being said.  This knowledge will be important in many different ways, from financial issues to industry specific problems.

Wax Worm Caterpillars and GoDaddy Renewal Coupon Codes.

For decades the environmental impact of plastics has been a major crisis with no clear solution. Plastics last for thousands of years, and the amount of plastic we make and consume has grown exponentially over the decades. It appears that we may have a solution.

So…the solution to our plastics crisis is….worms? For the past few weeks the internet has been ablaze with reports of wax worms that can break down plastics into organic compounds. No, this isn’t a fake story built out of a little GoDaddy satellite site. This is the real deal. Incidentally if you are looking for savings on building a website with GoDaddy, take a look at that link.

Pollution of any kind is not acceptable to man. Whether it be water, air, or waste pollution, they are all unhealthy to the environment and cause more harm than good to mankind. This explains why every effort is put in to reduce the level of pollution in the world.


The area of plastic waste pollution seems to have gotten a solution in the form of a Wax Worm. The larvae of this Worm type was initially bred to be used as fish bait before a discovery was made of its love for plastic materials by an amateur beekeeper and scientist, Federica Bertocchini. Here are reasons why Scientists think the Wax Worm may be the solution to plastic waste solution.

1. APPETITE FOR PLASTIC: Following Federica’s chance discovery, researchers investigated the report and found that 100 Wax Worms can devour as much as 92 milligrams of polyethylene. The amazing thing is that this quantity would be devoured in the space of 12 hours. Compare this to another scientific fact that about 200 plastic bags, which is the average a human being uses in one year, would take between 100 to 400 years to degrade in landfill sites. With the huge appetite of the Wax Worm for plastic, it makes good sense to use it effectively.

2. POLYETHYLENE BREAKDOWN: The Wax Worm is able to breakdown plastic through the use of an enzyme which they produce. As a solution to all the plastic waste which fills our landfills, oceans, and waterways, Scientists are researching on how to use the enzymes of the Wax Worm in some type of bacteria and marine organisms to degrade plastic wastes in the wild.

3. ENERGY SOURCE: One major factor which would work in the favor of using Wax Worms to fight against plastic waste pollution is to discover why they consume polyethylene. If it is just a passing thing, then, it would only be of benefit to man now. Another explanation may be that it requires it as a source of its energy. If that be the case, it would work perfectly as a solution to plastic waste pollution.

4. BREED LARGE NUMBER OF THEM: Rather than breed only a few of these Wax Worms for fish bait, emphasis may shift to breeding them in large quantities to be released to the landfills, oceans, and waterways to help degrade plastic wastes lined there. This would go a long way in solving plastic waste pollutions.

This chance discovery by Federica may be the solution to plastic waste pollution. Rather than spending long years waiting for polyethylene to degrade in landfills, this solution is both natural and time saving.

Of course we can’t just leave it all up to the worms, we have to do our part to reduce and reuse plastic and other items that typically get cast off as waste. For example instead of throwing out your whole computer, plastic casing and all, when it’s gotten too old consider using cheap computer parts for less to upgrade your machine. It’s the insides that wear out most quickly, and don’t forget you can take those to an electronics recycling depot to keep them out of the landfill.

The Right Way To Lose Weight (According to Science)

Losing weight isn’t a hard process if you know the right way to do it. However, there is a lot of confusion and misinformation out there about what you should do to lose weight. The truth is that sustainable weight loss requires a combination of the right diet and exercise. Supplements can help your weight loss results, but they are not replacements (at least, according to science). We’ll show you how and when you should eat, exercise, and take supplements. As a thank you for reading, we’ll even supply you with some weight loss supplement discount codes to save you money on your weight loss journey. If you find it easier to exercise while listening to music, here are some top-rated picks for headphones.

Let’s take a look at what you’ll have to do to achieve your weight loss goals.


Diet is the most important part of weight loss. It can also be the most difficult aspect of your routine to master. Even professional athletes are constantly refining their diets to find the perfect balance for their body. However, as a beginner, even a small amount of effort will help. Generally speaking, you want to get most of your calories from whole foods. Avoid anything processed if you can. Whole foods tend to give you much more nutrients for fewer calories. You should aim to consume a large amount of protein. Eat about 1 gram of protein per pound that you weigh in order to preserve and build up your lean muscle mass. This lean muscle mass will raise your metabolism, allowing you to burn more calories at rest. It will also make you look much better as the pounds come off. The rest of your calories should come from unsaturated fats and a good amount of complex carbohydrates from sources such as potatoes and brown rice. You will need to eat fewer calories than you are using throughout the day, but you shouldn’t allow your calorie count to get too low. Allow for no more than a 500 Calorie deficit per day. Eating 200 Calories fewer per day should be ideal.


Many people are under the impression that they should avoid lifting weights and only do cardio when trying to shed fat. In fact, the fastest way to lose fat is through a combination of cardio and resistance training. It’s also important to lift heavy when losing weight. Your goal should be to find a weight for each exercise that you can lift no more than eight times per set. Three sets for a particular exercise should suffice. Although eating too few calories may make it so that some people can’t put on muscle, if you are a true beginner, you may find that you can build muscle even when eating in a caloric deficit.

It is also important to do enough cardio when losing weight. You should be able to do some amount of cardio every single day, even if it is a little bit. Do as much as 45 minutes of low-intensity cardio after your lifting session. You should also do some high-intensity interval training on some of the days that you don’t lift. This type of exercise entails performing a brief period of intense cardio followed by a slightly longer period of low-intensity cardio. You then alternate your high-intensity and low-intensity segments a few more times for the rest of the session. Since interval training can be very demanding, you should limit yourself to two HIIT sessions a week.

There are many components to a good weight loss program. Diet, cardio and resistance training are all vital elements that will put you on the fast track to your perfect body. You should also remember that fitness is a lifelong process. Don’t beat yourself up if you miss a workout or have a cheat meal. If you’re putting in the effort, you’re ahead of everyone who’s still sitting on the couch.

Perhaps the most important step of the process is to find a way to keep yourself accountable without using shame as a tactic. There shouldn’t punishment for ‘screwing it up’, you should find ways of positive reinforcement for encouraging the habits and behaviours that you want. One way that’s really helpful is to share with others what you are doing. Perhaps you find a friend to go for walks with, or you set yourself up a blog to chronicle your journey. You will find that other people will be there to cheer you on and you’ll be more motivated to keep going!