In the world of theoretical computer science, P vs. NP is something of a mythical unicorn. It’s become notorious, since it remains an unsolved problem. It basically asks this: If it is easy to check that a solution to a problem is correct, is it also easy to solve that problem? Get back to us when you have an answer. (http://www.claymath.org/millennium-problems)
Why Is This So Hard?
In the P vs. NP problem (http://news.mit.edu/2009/explainer-pnp), the P stands for polynomial and the NP stands for nondeterministic polynomial time. Did we lose you? Let’s back up. In simpler terms, P stands for problems that are easy for computers to solve, and NP stands for problems that are not easy for computers to solve, but are easy for them to check. Here’s when an example might be helpful: “A farmer wants to take 100 watermelons of different masses to the market. She needs to pack the watermelons into boxes. Each box can only hold 20 kilograms without breaking. The farmer needs to know if 10 boxes will be enough for her to carry all 100 watermelons to market.” (https://simple.wikipedia.org/wiki/P_versus_NP)
Behnam Esfahbod / WIkipedia ()
This sample problem is not easy to solve; it requires you to go through every dang possible combination. Checking the final answer, however, is pretty easy. All P problems are also NP problems (if a computer can easily solve it, the computer can also easily check it). The question remains open: Are P problems and NP problems the same type of problem? Or, are there are some problems that are easily verified but not easily solved?
Who Wants To Be A Millionaire—From Math?
You may be wondering who really cares about this sort of thing. Well, if someone could show that P is equal to NP, it would make difficult real-world problems a piece of cake for computers to solve. Oh, and the person who solves this problem would also get $1 million from the Clay Mathematics Institute. The P vs. NP Problem is one of six unsolved Millennium Problems that hold a million-dollar prize for whoever cracks it.
Want to dive even further into the world's toughest math problems? Check out "The Millennium Problems: The Seven Greatest Unsolved Mathematical Puzzles Of Our Time" by Keith J. Devlin. We handpick reading recommendations we think you may like. If you choose to make a purchase, Curiosity will get a share of the sale.
Watch And Learn:
Why is P vs NP Important?
In this video,He explain perhaps the most famous problem in all of Computer Science. Does P = NP? He define the terms and give examples of each. We also programmatically go through the traveling salesman problem. He experiment with a little bit of mixed reality in this video as well.
(note: This animation has no audio track.) – The Open University
Although many moons in the Solar System follow prograde orbits, there are some notable exceptions. The gas giant planets Jupiter, Saturn, Uranus and Neptune have several small outer moons that follow retrograde orbits; this means that they orbit their planet in the opposite direction to the planet’s rotation. In a retrograde orbit, a moon revolves in its orbit in the opposite direction from that in which the planet rotates about its axis.
By Sarah Hansen, OCW Educator Project Manager Assessing students’ learning is one of the most important things we do as educators. It’s also one of the most complicated. There’s a lot to consider: When will assessment happen? (Along the way? At the end of the course?) How will we collect useful information about student learning? […]
Explanation: Last week, a Tesla orbited the Earth. The car, created by humans and robots on the Earth, was launched by the SpaceX Company to demonstrate the ability of its Falcon Heavy Rocket to place spacecraft out in the Solar System. Purposely fashioned to be whimsical, the iconic car was thought a better demonstration object than concrete blocks. A mannequin clad in a spacesuit — dubbed the Starman — sits in the driver’s seat. The featured image is a frame from a video taken by one of three cameras mounted on the car. These cameras, connected to the car’s battery, are now out of power. The car, attached to a second stage booster, soon left Earth orbit and will orbit the Sun between Earth and the asteroid belt indefinitely — perhaps until billions of years from now when our Sun expands into a Red Giant. If ever recovered, what’s left of the car may become a unique window into technologies developed on Earth in the 20th and early 21st centuries.
A mineral is a naturally occurring inorganic crystalline substance, whose compositions are either fixed or vary between certain fixed limits. This excludes, for example, artificial diamonds, coal, volcanic glass. What makes each mineral unique is a combination of its chemical composition and the internal arrangement of its constituent atoms.
The chemical composition may be fixed, as in quartz (SiO2) in which, for every single atom of silicon there are 2 atoms of oxygen. Or it may be variable for example the mineral olivine, which varies between two formulae; Mg2SiO4 and Fe2SiO4. Some minerals have a single chemical composition but a different arrangement of atoms in their three-dimensional structure. This is called “polymorphism” and is well seen in the polymorphs of carbon: graphite and diamond. One is grey/black and very soft, the other is usually colourless and is the hardest natural substance.
There are many different classes of minerals, but only the most common will be examined. These are: silicates (containing the SiO44^- anion), carbonates (with the CO3^2- anion), halides (which contain F^-, Cl), sulphides (with S^2-), sulphates (SO4^2-), oxides and native metals. The following are examples of each of these classes, which will become familiar in this lecture.
The extent to which atomic structure controls the outward shape of a substance is called its crystallinity. For example, quartz can develop in 4 ways. Firstly, as a crystal showing crystal faces, which is termed CRYSTALLISED.
Secondly, as irregular grains of quartz with fully developed internal structure, but not displaying crystal faces, which is termed CRYSTALLINE.
Thirdly, as a finely distributed crystalline aggregate called chalcedony whose grains are only visible under powerful magnification, this is termed CRYPTOCRYSTALLINE.
Fourthly, as a precipitate such as OPAL where there is no regular arrangement of atoms which is termed AMORPHOUS.
3) Physical Properties
What physical properties can be used to identify minerals?
In hand-specimen, minerals can be identified using a combination of physical properties, and it is these that will be studied in this lecture and practical.
a) Crystal Form/Habit
The characteristic shape of an individual crystal of a mineral is called its “habit”. Sometimes it is possible to identify a mineral from its habit alone, for example, quartz often forms a six-sided column with a set of pyramid-like faces at the top. We would describe this as prismatic.There are many common terms to describe the habit of crystals for example:
bladed (elongate crystals flattened in one direction);
botryoidal (rounded masses looking like bunches of grapes);
reniform (kidney like);
fibrous (groups of parallel thread-like crystals);
massive (no regular form);
platy (very flat crystals);
prismatic (elongated crystals with well developed prism faces);
tabular (crystals slightly flattened in one direction).
A fully developed form is referred to as euhedral; the opposite, an
irregular form, is called anhedral.
A diamond is considered to be very hard because no other mineral can scratch it. Quartz can scratch a large number of other minerals. A scale has been devised to describe the hardness of minerals; Mohs’ scale. It varies from 1 (softest) to 10 (hardest). To test the hardness of an unknown mineral a specimen set of minerals called a hardness set is used. Starting with a hard specimen such as corundum the user tries with each in turn to scratch the unknown mineral until one is found that will not scratch the unknown mineral. The hardness of the unknown mineral is between the number of the mineral that will scratch it and the one that will not.
It is useful in the field to know the hardnesses of some everyday objects. For example, a fingernail has a hardness slightly over 2, and can scratch minerals of hardness 2 or less. Teeth have a hardness of around 4 and a copper coin 4.5 to 5; a steel knife blade is 6 and a hard file is around 7. Silicate minerals vary in hardness – the softest is talc (1) the hardest is topaz (8). Oxides, sulphides and many native minerals are very soft (less than 3), but exceptions include corundum Al2O3 which has a hardness of 9.
c) Specific Gravity / Density
Some minerals have an unusually high or low density (mass per unit volume), although most are between 2.5-3.0 g/cm^3. Specific gravity, or relative density, is the ratio of the mass of a mineral to the mass of an equal volume of water. However, it is difficult to estimate in the field except in some cases. Some common ore minerals are particularly heavy, for example, galena PbS (7.5), cassiterite SnO2 (6.9). This is because of the high atomic weight of the elements Pb and Sn. One commonly occurring mineral, barite (barytes) BaSO4, is also unusually heavy (4.5) compared with other superficially similar minerals such as calcite.
Colour is rarely a reliable indicator of the identity of a mineral. This is because in many minerals, impurities elements in trace amounts) can change the colour drastically. An example is a quartz, which is colourless when pure (“rock crystal”) but occurs in the coloured forms of amethyst (purple), rose quartz (pink), smokey quartz (grey) and others. Fluorite (CaF2) also occurs in purple, blue, green and yellow varieties. Some minerals are less variable, and in some cases, the colour is characteristic, for example, malachite (green), pyrite (golden), Galena (silver-grey). Nevertheless, it is better to identify these minerals on other characteristics in addition to the colour, for example, using cleavage or habit.
A slightly more reliable method of determining a mineral’s true colour is to use its streak. This is the name given to the powdered material left behind when a mineral is rubbed on an unglazed porcelain plate. Most pale-coloured minerals have a white streak which is of little use in identification, but the method is very useful with dark coloured opaque minerals such as hematite (Fe2O3) which has a characteristic red/brown streak, or pyrite (FeS2) which has a characteristic black streak.
f) Degree of transparency
Terms such as transparent, translucent and opaque are used to describe the degree to which a mineral can transmit light. However, this often depends on the thickness of a specimen and other factors such as internal inclusions. It is, thus, not a particularly useful guide for mineral identification. However, most ore minerals (pyrite, galena) are opaque, while many silicates are translucent to transparent.
Cleavage in minerals is the splitting or breaking of a crystal along planar surfaces which are determined by the crystal structure. There are often only a small number of possible cleavage planes in a mineral, whereas there can be many possible crystal faces. Cleavage of mica is along sheets, so that parallel smooth flat surfaces can be seen. This is due to weak bonds between the sheets in the structure of mica. As mica can only cleave in this one plane, it is said to have only one cleavage direction. Other minerals such as calcite have three excellent cleavage directions, giving a rhomboid shape. Distinctive patterns of cleavage are good identifying marks for many minerals. Galena and halite both have three good cleavages at 90°, yielding almost perfect cubes.
The universe is the biggest and oldest thing we know. It contains all existing matter and space. And its origin marks the beginning of time as far as we understand it. We don’t know what made the formation of the universe possible, nor why it occurred. The visible universe is currently about 93 billion light years wide.
A light-year is a distance that light travels in a year, which makes the universe about 880 trillion trillion metres wide. The visible universe is, however, still expanding, and we can measure that rate of expansion. Then, working backwards, we can figure out when the universe would have begun. To the best of our knowledge, the universe formed about 13.8 billion years ago in what is commonly referred to as the Big Bang.
This image shows the universe about 370000 years after the Big Bang, which is the oldest light that we’ve been able to record with the greatest precision. The image records ancient light or cosmic microwave background. The colours show tiny temperature fluctuations from an average temperature. These indicate areas of different densities, which became the stars and galaxies of today. Red spots are a bit hotter and blue spots a bit cooler. The image was recorded between 2009 and 2013, during the Planck mission, when the space observatory was operated by the European Space Agency, in conjunction with NASA, the National Aeronautics and Space Administration. Today, the universe is very cold. On average, it is 2.7Kelvin. Kelvin is a measure of temperature with the same magnitude as degrees Celsius. But 0 Kelvin equals minus 273.15 degrees Celsius.
In the universe, the hot parts, such as stars, make up only a tiny fraction. If we wind the clock backwards, the universe gets smaller. And this means the universe was hotter in the past. When matter gets hot, solids melt and liquids boil. The hot matter glows – red at first, but it becomes bluer as the temperature goes up. Eventually, all matter is gas. So we have a bright, glowing blob of gas. Going further back in time, as the gas gets hotter, the electrons are separated from the nuclei and a plasma is made. The temperature at this point is about 3000 to 6000 Kelvin and the glowing blob is white hot. As we go back further in time, the universe gets even smaller and hotter.
The nuclei themselves, containing protons and neutrons, are broken up. The reason for the breakup of nuclei is that the individual particles and the energy of the radiation are so great that the collisions of all this hot stuff are incredibly violent. The light is no longer in the visible spectrum. It is energetic enough to be x-rays and even gamma rays. Between just 10 seconds and 1000 seconds after the Big Bang, subatomic particles, including neutrons and protons, were formed. Neutrons live for just 9 minutes when they are free. Hence only those that stuck to protons during this period survived. All of the ordinary matter present today formed in this short window of time.
At about 1 microsecond after the Big Bang, the universe was very hot, at 10 to the 10 Kelvin, and quarks formed stable particles called hadrons. Before 1 picosecond, or 10 to the minus 12 seconds, the universe was an exotic place. The gas was hotter still and the laws of physics appeared different to how we see them today. The distinction between matter and radiation, such as light, cannot be detected. The forces of electromagnetism and the weak nuclear force also become indistinguishable. At the very earliest times, the universe was so hot and dense that we cannot yet describe them accurately.
Venus and the Triple Ultraviolet Sun
Image Credit: NASA/SDO & the AIA, EVE, and HMI teams; Digital Composition: Peter L. Dove (http://www.flickr.com/photos/pldove/)
Explanation: An unusual type of solar eclipse occurred in 2012. Usually, it is the Earth’s Moon that eclipses the Sun. That year, most unusually, the planet Venus took a turn. Like a solar eclipse by the Moon, the phase of Venus became a continually thinner crescent as Venus became increasingly better aligned with the Sun. Eventually, the alignment became perfect and the phase of Venus dropped to zero. The dark spot of Venus crossed our parent star. The situation could technically be labelled a Venusian annular eclipse with an extraordinarily large ring of fire. Pictured here during the occultation, the Sun was imaged in three colours of ultraviolet light by the Earth-orbiting Solar Dynamics Observatory, with the dark region toward the right corresponding to a coronal hole. Hours later, as Venus continued in its orbit, a slight crescent phase appeared again. The next Venusian transit across the Sun will occur in 2117. </center>
The Earth is an oblate spheroid, being slightly flattened at the
Equatorial radius = 6378 km Polar radius = 6357 km
These measurements are calculated on the assumption that the Earth’s surface is smooth, but this is only an approximation since it disregards mountains and ocean depths. However, the difference between the height of Mount Everest and the depth of the Marianas trench is only about 20 km. Most land is concentrated in seven continents each fringed by shallow seas (flooded continent). Separating these are a number of major oceans including the Pacific, Atlantic and the Indian oceans.
It was Cavendish in 1798 who first calculated the mass of the Earth as 5.977 x 1024kg, and since its volume is known (from 4/3 ∏ r^3 where r is the radius of the Earth), then it can be calculated that the average density is 5.516 g/cm3. However, most rocks exposed at the surface have densities of less than 3g/cc, for example:
Therefore, a material of greater density must exist at deeper levels within the Earth. The Earth has a series of layers or “shells”, but only the outer few km of the Earth can be directly observed; the upper crust, and the deepest boreholes which reach to only about 12.5 kms. Earthquakes provide the key to the structure at depth.
Stresses which develop in the Earth may become great enough to break the rocks, and cause slip along the resulting in fractures (faults). Although the slip distance in a given earthquake may be small (cm to metres), the rock masses involved are large and so the energy released is great. The resulting shock waves, or earthquakes, may cause great damage; greatest near the centre or focus, and less further away. The epicentre is the point on the surface of the Earth vertically above the focus.
Detection of seismic waves.
Earthquake energy is transmitted by several types of waves. Two types will be described:
P waves (primary or compressional) are transmitted by vibrations oscillating in the direction of propagation (push/pull).
S waves (secondary or shear), which vibrate at right angles to the direction of propagation. S waves cannot be transmitted through liquids because liquids have no elastic strength.
The arrival of earthquake waves is recorded by a seismograph. A mass is loosely coupled to the Earth by a spring. A chart is firmly coupled to the Earth. A pen linking them traces the difference in motion between the mass and the Earth’s surface. The arrival of waves from a distant earthquake is recorded as a seismogram on the rotating drum.
Consider what happens to P and S waves as they travel through the Earth.
The most important property of seismic waves is their speed of propagation. The velocity is governed by the physical properties (density, compressibility, rigidity) of the medium through which the wave is travelling.
Earlier in this lecture, it was deduced that the density of the Earth increases with depth. The wave propagation velocity must, therefore, change with depth, and this causes the wave to refract.
If a wave travelling through a medium with a fixed density encounters a new medium with a different density, the wave will change its direction. This “bending” of the wave is called refraction.
Data from seismometers located around the world can record waves from any given earthquake. The differences between recordings at different seismometers reveal properties of the sub-surface and hence the internal structure of the Earth.
For example, it has been discovered that the mantle is solid rock, but the outer core is a liquid. This was discovered, because for any given earthquake:-
Both P and S waves are recorded by seismometers at distances of up to 103o from the epicentre.
At distances greater than 103o, no S waves are recorded. This means that S waves that would have reappeared at > 103o have not propagated. The material at depths travelled by such waves must be liquid and be unable to transmit S waves.
Also, it has been discovered that the outer core must have a lower P wave velocity than the mantle. This is because at distances of 103o to 142o, no strong P waves are recorded. The liquid outer core has a lower P wave velocity, causing the P waves to be refracted to a steeper angle, so they cannot re-emerge between 103o to 142o. They actually re-emerge at angles > 186o. There is one small caveat to this observation. The inner core appears to be solid because some weak P wave arrivals occur between 103o to 142o. This is thought to be due to a slight increase in P wave velocity as waves enter the inner core, causing them to be refracted to a shallower angle, to re-emerge between 103o to 142o. If the inner core is solid, S waves could propa- gate there. The graph shows some calculations of what expected S wave velocities would be, but the inner core structure is still a source of controversy.
In the early 20th century a Yugoslavian seismologist by the name of Mohorovicic was studying seismograms from shallow focus earthquakes (< 40 km) that were nearby <800km. He noticed that there were 2 distinct sets of P waves and S waves involved. He interpreted these waves as a direct set and a refracted set. In the refracted set, waves travel down and are refracted at a boundary by a medium of higher velocity.
This boundary separates the crust with VP of 6-7km/sec from the upper mantle where VP starts at 8km/sec. It is called the Mohorovicic discontinuity but is commonly known as the MOHO.
Today, seismologists use artificial explosions to determine the structure beneath the surface and it is from these data that the depth of the MOHO can be calculated and thus the thickness of the crust. The MOHO is at 5-15 km under ocean crust and 35 km beneath normal thickness continental crust. The MOHO can be as much as 70 km deep beneath mountain belts where converging plates have caused an orogeny or mountain building event.
The Structure of the Earth
Recent advances in seismology now allow tomographic images of the interior of the Earth to be produced from P and S wave velocity data. Just as tomographic images of the interior of human bodies are produced by density contrasts in human tissue and bone subject to wave propagation, density contrasts in the Earth can be mapped by combining wave velocity data from large numbers of earthquakes.
The basic idea is that where the solid mantle is relatively hot, the P and S wave velocities should be anomalously low because the heat will result in a density decrease. One should be able to image hot, ascending plumes of mantle asthenosphere by looking for areas of anomalously low seismic velocity. Conversely, where the solid mantle is relatively cool, the P and S wave velocities should be anomalously fast because the lack of heat will result in a relatively high density.
One should be able to image cool, descending slabs of mantle lithosphere by looking for areas of anomalously high seismic velocity. Such images allow us to study subduction zones and constrain how deep the slabs penetrate. It appears that some slabs do not penetrate beneath 670 km whereas others continue down to the core-mantle boundary. This is an area of controversy in geology.
By Joe Pickett, OCW Publication Director OCW has just published 21G.503 Japanese III, the third in a four-course sequence on Japanese taught at MIT. With relatively few Japanese speakers on the MIT campus, the instructors must make the most of what happens in the classroom and motivate students to work hard outside it. The course […]