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# Project based on game 2048

I'm thinking about an assignment based around the game 2048. This would be towards the end of a first programming course at the university level and assumes the students have been introduced to at least two dimensional arrays, loops, conditionals and methods. The purpose is to have the students practice these skills.

I would represent the board as a two dimensional array of int.

I would require a method (e.g. merge(int[] row)) which does the work on an individual row of the puzzle when the move is to slide left. The idea is to encapsulate the work of a slide in a single, testable method. This also emphasizes that a two dimensional array is an array of array, and that one can work on a single row at a time.

What are your opinions on the following ideas:

1. Should a single move in the game be implemented as (move(direction)) or as individual methods (up() down() left() right())?

2. One idea for completing the game involves rotating the board by multiples of 90 degrees. Then a move can be implemented as a three step process:

1. Rotate board to one orientation
2. Slide all rows left (merge(board[row]) for each row
3. Rotate board back to original orientation
3. Another idea is to show them that move right can be implemented by the process

1. reverse a row
2. slide it left (merge(board[row])
3. reverse the row again
4. For columns, copy the values from a column into a one dimensional temp array, then perform the merge(temp) and then copy back to the board. This may also require reverses, depending on the up/down selection.
4. Require a method to produce a list of empty cells for use in picking the location of the next "insertion". This could be a list of int to demonstrate how a single index can map to a row/col using div and mod operations. Alternatively, it could be a list of pairs of row/column.

5. The last wrinkle is to decide what to actually stored in each board location. The simple answer is just the value. However, an interesting variation might be to store the log2 of the value. By doing this, you get an index that can be used to select, for example, the background color of each square of the board. This changes the logic of merge(), so that combining adjacent cells ups the value by 1, rather than adding two values together.

Am I trying to put too many different concepts into this assignment? Your opinions and suggestions are appreciated.

Thanks

• About #5: $0{\text{ }}log_{2} = \infty$, while $0=1{\text{ }}log_{2}$, so there ends up being some kind of special test for the empty square value or the stored value of zero. The conditionals around that problem makes it likely that choosing color based on cell value would be just as fast, and less error-prone. – Gypsy Spellweaver Oct 18 '18 at 1:05
• Yes your are correct, the empty cell being 0 is a special case, but as an index works very nicely. I would hope that the students would see that array[cellValue] is cleaner than a switch on the the cellValue with all the possible cases (2 4 8 15 ...). Perhaps you are correct that the students would be more comfortable with the switch approach. I had thought of this an a way to demonstrate allocating an arrray of constant values. – Fritz Sieker Oct 18 '18 at 18:45

First of all, it seems like rather nice bones of a 2-dimensional array lab. It does not strike me as being too many things at all.

1. The interface is cleaner with move(direction), but the internal logic is cleaner divided into methods. Why not the best of both words? Provide them with starter code that will utilize private methods:

public void move(direction){
switch(direction){
case UP:
slideUp();
case DOWN:
slideDown();
case LEFT:
slideLeft();
case RIGHT:
slideRight();
}
}

2. That is another way, also valid.

3. It seems excessively finicky to require implementation details down to this level.

4. Once again, I would provide the interface, but not these sorts of implementation details. You can provide both ways as suggestions, but leave it up to them. They may also choose a third option that you have not mentioned.

5. This fits very nicely if you are providing some sort of GUI starter code, which would justify an implementation detail like this one.

• I think having 4 public move methods is cleaner than using a single one with a string parameter. At least use an Enum! – Solomon Ucko Oct 24 '18 at 1:53
• @SolomonUcko I don't agree about the cleanliness in the public facing methods, though I am referencing private methods, so we're not far off there. I absolutely agree about the Enum. Thanks for pointing that out! Fixed that, and I also noticed when I went in that my code wasn't formatting. Apparently we now need 8 spaces to start a code block?? In any case, that's fixed now, too. – Ben I. Oct 24 '18 at 12:45
• I'd say that using an Enum parameter is better than using multiple methods, which is better than using a String. Sorry for the tone of my previous comment, I was in a bit of a hurry. – Solomon Ucko Oct 24 '18 at 13:26

This will be mostly just caveats and things to think about.

One problem you will have is that the original code and some other versions are all online as open source. It will be hard to keep your students "in the dark" so that they can grow themselves (mushroom principle of education, I guess).

Second, for a 4 by 4 square array, the standard array operations (scanning, collecting, ...) using loops are a bit inefficient. For example, your rotate method for an array can be done with about 20 assignments, just keeping track of subscripts. That might be a very valuable exercise in itself, of course. But loop overhead in such a small array starts to overwhelm the operations.

Third, I haven't thought in detail about this yet, but you might want to consider a program that uses more space than the minimum, keeping, for example, two versions of the matrix simultaneously, with one rotated 90 degrees from the other. This will achieve adjacency of both rows (in one copy) and columns (in the other).

An object based version might also be possible, so that the "cells" of the matrix can be simultaneously row and column adjacent. This might move some of the logic into those cell objects as well.

• Thanks Buffy, The rotation solution is particularly inefficient. However, in this case the problem size is small and is a don't care because it need only be fast enough for the human player not to be kept waiting. . Premature optimization is rarely a good thing. - Fritz – Fritz Sieker Oct 17 '18 at 19:01
• I actually encourage students to look on line for help and/or solutions. I ask only that they cite the link where they found the code. And, in doing so, that they assume responsibility for the code. Tough luck if it doesn't work. I also tend to introduce some structure not found in the online solutions so that students must, at least, adapt what they find to the requirements. – Fritz Sieker Oct 17 '18 at 21:01
• Actually, I wasn't optimizing. I just think that brute force assignment is the simplest solution here. – Buffy Oct 17 '18 at 22:44


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# Alternating current

Alternating current (green curve). The horizontal axis measures time; the vertical, current or voltage.

{{#invoke: Sidebar | collapsible }} In alternating current (AC), the flow of electric charge periodically reverses direction. In direct current (DC, also dc), the flow of electric charge is only in one direction. The abbreviations AC and DC are often used to mean simply alternating and direct, as when they modify current or voltage.[1][2]

AC is the form in which electric power is delivered to businesses and residences. The usual waveform of an AC power circuit is a sine wave. In certain applications, different waveforms are used, such as triangular or square waves. Audio and radio signals carried on electrical wires are also examples of alternating current. In these applications, an important goal is often the recovery of information encoded (or modulated) onto the AC signal.

## Transmission, distribution, and domestic power supply

{{#invoke:main|main}}

AC voltage may be increased or decreased with a transformer. Use of a higher voltage leads to significantly more efficient transmission of power. The power losses (${\displaystyle P_{\rm {L}}}$) in a conductor are a product of the square of the current (I) and the resistance (R) of the conductor, described by the formula

${\displaystyle P_{\rm {L}}=I^{2}R\,.}$

This means that when transmitting a fixed power on a given wire, if the current is doubled, the power loss will be four times greater.

The power transmitted is equal to the product of the current and the voltage (assuming no phase difference); that is,

${\displaystyle P_{\rm {T}}=IV\,.}$

Thus, the same amount of power can be transmitted with a lower current by increasing the voltage. It is therefore advantageous when transmitting large amounts of power to distribute the power with high voltages (often hundreds of kilovolts).

High voltage transmission lines deliver power from electric generation plants over long distances using alternating current. These lines are located in eastern Utah.

However, high voltages also have disadvantages, the main one being the increased insulation required, and generally increased difficulty in their safe handling. In a power plant, power is generated at a convenient voltage for the design of a generator, and then stepped up to a high voltage for transmission. Near the loads, the transmission voltage is stepped down to the voltages used by equipment. Consumer voltages vary depending on the country and size of load, but generally motors and lighting are built to use up to a few hundred volts between phases.

The utilization voltage delivered to equipment such as lighting and motor loads is standardized, with an allowable range of voltage over which equipment is expected to operate. Standard power utilization voltages and percentage tolerance vary in the different mains power systems found in the world.

Modern high-voltage direct-current (HVDC) electric power transmission systems contrast with the more common alternating-current systems as a means for the efficient bulk transmission of electrical power over long distances. HVDC systems, however, tend to be more expensive and less efficient over shorter distances than transformers. {{ safesubst:#invoke:Unsubst||date=__DATE__ |\$B= {{#invoke:Category handler|main}}{{#invoke:Category handler|main}}[citation needed] }} Transmission with high voltage direct current was not feasible when Edison, Westinghouse and Tesla were designing their power systems, since there was then no way to economically convert AC power to DC and back again at the necessary voltages.

Three-phase electrical generation is very common. The simplest case is three separate coils in the generator stator that are physically offset by an angle of 120° to each other. Three current waveforms are produced that are equal in magnitude and 120° out of phase to each other. If coils are added opposite to these (60° spacing), they generate the same phases with reverse polarity and so can be simply wired together.

In practice, higher "pole orders" are commonly used. For example, a 12-pole machine would have 36 coils (10° spacing). The advantage is that lower speeds can be used. For example, a 2-pole machine running at 3600 rpm and a 12-pole machine running at 600 rpm produce the same frequency. This is much more practical for larger machines.

If the load on a three-phase system is balanced equally among the phases, no current flows through the neutral point. Even in the worst-case unbalanced (linear) load, the neutral current will not exceed the highest of the phase currents. Non-linear loads (e.g., computers) may require an oversized neutral bus and neutral conductor in the upstream distribution panel to handle harmonics. Harmonics can cause neutral conductor current levels to exceed that of one or all phase conductors.

For three-phase at utilization voltages a four-wire system is often used. When stepping down three-phase, a transformer with a Delta (3-wire) primary and a Star (4-wire, center-earthed) secondary is often used so there is no need for a neutral on the supply side.

For smaller customers (just how small varies by country and age of the installation) only a single phase and the neutral or two phases and the neutral are taken to the property. For larger installations all three phases and the neutral are taken to the main distribution panel. From the three-phase main panel, both single and three-phase circuits may lead off.

Three-wire single-phase systems, with a single center-tapped transformer giving two live conductors, is a common distribution scheme for residential and small commercial buildings in North America. This arrangement is sometimes incorrectly referred to as "two phase". A similar method is used for a different reason on construction sites in the UK. Small power tools and lighting are supposed to be supplied by a local center-tapped transformer with a voltage of 55 V between each power conductor and earth. This significantly reduces the risk of electric shock in the event that one of the live conductors becomes exposed through an equipment fault whilst still allowing a reasonable voltage of 110 V between the two conductors for running the tools.

A third wire, called the bond (or earth) wire, is often connected between non-current-carrying metal enclosures and earth ground. This conductor provides protection from electric shock due to accidental contact of circuit conductors with the metal chassis of portable appliances and tools. Bonding all non-current-carrying metal parts into one complete system ensures there is always a low electrical impedance path to ground sufficient to carry any fault current for as long as it takes for the system to clear the fault. This low impedance path allows the maximum amount of fault current, causing the overcurrent protection device (breakers, fuses) to trip or burn out as quickly as possible, bringing the electrical system to a safe state. All bond wires are bonded to ground at the main service panel, as is the Neutral/Identified conductor if present.

## AC power supply frequencies

The frequency of the electrical system varies by country; most electric power is generated at either 50 or 60 hertz. Some countries have a mixture of 50 Hz and 60 Hz supplies, notably electricity power transmission in Japan.

A low frequency eases the design of electric motors, particularly for hoisting, crushing and rolling applications, and commutator-type traction motors for applications such as railways. However, low frequency also causes noticeable flicker in arc lamps and incandescent light bulbs. The use of lower frequencies also provided the advantage of lower impedance losses, which are proportional to frequency. The original Niagara Falls generators were built to produce 25 Hz power, as a compromise between low frequency for traction and heavy induction motors, while still allowing incandescent lighting to operate (although with noticeable flicker). Most of the 25 Hz residential and commercial customers for Niagara Falls power were converted to 60 Hz by the late 1950s, although someTemplate:Which 25 Hz industrial customers still existed as of the start of the 21st century. 16.7 Hz power (formerly 16 2/3 Hz) is still used in some European rail systems, such as in Austria, Germany, Norway, Sweden and Switzerland.

Off-shore, military, textile industry, marine, computer mainframe, aircraft, and spacecraft applications sometimes use 400 Hz, for benefits of reduced weight of apparatus or higher motor speeds.

## Effects at high frequencies

{{#invoke:main|main}} A direct current flows uniformly throughout the cross-section of a uniform wire. An alternating current of any frequency is forced away from the wire's center, toward its outer surface. This is because the acceleration of an electric charge in an alternating current produces waves of electromagnetic radiation that cancel the propagation of electricity toward the center of materials with high conductivity. This phenomenon is called skin effect.

At very high frequencies the current no longer flows in the wire, but effectively flows on the surface of the wire, within a thickness of a few skin depths. The skin depth is the thickness at which the current density is reduced by 63%. Even at relatively low frequencies used for power transmission (50–60 Hz), non-uniform distribution of current still occurs in sufficiently thick conductors. For example, the skin depth of a copper conductor is approximately 8.57 mm at 60 Hz, so high current conductors are usually hollow to reduce their mass and cost.

Since the current tends to flow in the periphery of conductors, the effective cross-section of the conductor is reduced. This increases the effective AC resistance of the conductor, since resistance is inversely proportional to the cross-sectional area. The AC resistance often is many times higher than the DC resistance, causing a much higher energy loss due to ohmic heating (also called I2R loss).

### Techniques for reducing AC resistance

For low to medium frequencies, conductors can be divided into stranded wires, each insulated from one other, and the relative positions of individual strands specially arranged within the conductor bundle. Wire constructed using this technique is called Litz wire. This measure helps to partially mitigate skin effect by forcing more equal current throughout the total cross section of the stranded conductors. Litz wire is used for making high-Q inductors, reducing losses in flexible conductors carrying very high currents at lower frequencies, and in the windings of devices carrying higher radio frequency current (up to hundreds of kilohertz), such as switch-mode power supplies and radio frequency transformers.

#### Techniques for reducing radiation loss

As written above, an alternating current is made of electric charge under periodic acceleration, which causes radiation of electromagnetic waves. Energy that is radiated is lost. Depending on the frequency, different techniques are used to minimize the loss due to radiation.

#### Twisted pairs

At frequencies up to about 1 GHz, pairs of wires are twisted together in a cable, forming a twisted pair. This reduces losses from electromagnetic radiation and inductive coupling. A twisted pair must be used with a balanced signalling system, so that the two wires carry equal but opposite currents. Each wire in a twisted pair radiates a signal, but it is effectively cancelled by radiation from the other wire, resulting in almost no radiation loss.

#### Coaxial cables

Coaxial cables are commonly used at audio frequencies and above for convenience. A coaxial cable has a conductive wire inside a conductive tube, separated by a dielectric layer. The current flowing on the inner conductor is equal and opposite to the current flowing on the inner surface of the tube. The electromagnetic field is thus completely contained within the tube, and (ideally) no energy is lost to radiation or coupling outside the tube. Coaxial cables have acceptably small losses for frequencies up to about 5 GHz. For microwave frequencies greater than 5 GHz, the losses (due mainly to the electrical resistance of the central conductor) become too large, making waveguides a more efficient medium for transmitting energy. Coaxial cables with an air rather than solid dielectric are preferred as they transmit power with lower loss.

#### Waveguides

Waveguides are similar to coax cables, as both consist of tubes, with the biggest difference being that the waveguide has no inner conductor. Waveguides can have any arbitrary cross section, but rectangular cross sections are the most common. Because waveguides do not have an inner conductor to carry a return current, waveguides cannot deliver energy by means of an electric current, but rather by means of a guided electromagnetic field. Although surface currents do flow on the inner walls of the waveguides, those surface currents do not carry power. Power is carried by the guided electromagnetic fields. The surface currents are set up by the guided electromagnetic fields and have the effect of keeping the fields inside the waveguide and preventing leakage of the fields to the space outside the waveguide.

Waveguides have dimensions comparable to the wavelength of the alternating current to be transmitted, so they are only feasible at microwave frequencies. In addition to this mechanical feasibility, electrical resistance of the non-ideal metals forming the walls of the waveguide cause dissipation of power (surface currents flowing on lossy conductors dissipate power). At higher frequencies, the power lost to this dissipation becomes unacceptably large.

#### Fiber optics

At frequencies greater than 200 GHz, waveguide dimensions become impractically small, and the ohmic losses in the waveguide walls become large. Instead, fiber optics, which are a form of dielectric waveguides, can be used. For such frequencies, the concepts of voltages and currents are no longer used.

## Mathematics of AC voltages

A sine wave, over one cycle (360°). The dashed line represents the root mean square (RMS) value at about 0.707

Alternating currents are accompanied (or caused) by alternating voltages. An AC voltage v can be described mathematically as a function of time by the following equation:

${\displaystyle v(t)=V_{\mathrm {peak} }\cdot \sin(\omega t)}$,

where

The peak-to-peak value of an AC voltage is defined as the difference between its positive peak and its negative peak. Since the maximum value of ${\displaystyle \sin(x)}$ is +1 and the minimum value is −1, an AC voltage swings between ${\displaystyle +V_{\rm {peak}}}$ and ${\displaystyle -V_{\rm {peak}}}$. The peak-to-peak voltage, usually written as ${\displaystyle V_{\rm {pp}}}$ or ${\displaystyle V_{\rm {P-P}}}$, is therefore ${\displaystyle V_{\rm {peak}}-(-V_{\rm {peak}})=2V_{\rm {peak}}}$.

### Power

{{#invoke:main|main}}

The relationship between voltage and the power delivered is

${\displaystyle p(t)={\frac {v^{2}(t)}{R}}}$ where ${\displaystyle R}$ represents a load resistance.

Rather than using instantaneous power, ${\displaystyle p(t)}$, it is more practical to use a time averaged power (where the averaging is performed over any integer number of cycles). Therefore, AC voltage is often expressed as a root mean square (RMS) value, written as ${\displaystyle V_{\rm {rms}}}$, because

${\displaystyle P_{\rm {time~averaged}}={\frac {{V^{2}}_{\rm {rms}}}{R}}.}$
Power oscillation
${\displaystyle v(t)=V_{\mathrm {peak} }\sin(\omega t)}$
${\displaystyle i(t)={\frac {v(t)}{R}}={\frac {V_{\mathrm {peak} }}{R}}\sin(\omega t)}$
${\displaystyle P(t)=v(t)\ i(t)={\frac {(V_{\mathrm {peak} })^{2}}{R}}\sin ^{2}(\omega t)}$

By the following trigonometric identity, the power oscillation is double frequency of the voltage.

${\displaystyle \sin ^{2}x={\frac {1-\cos 2x}{2}}}$

### Root mean square

For a sinusoidal voltage:

{\displaystyle {\begin{aligned}V_{\mathrm {rms} }&={\sqrt {{\frac {1}{T}}\int _{0}^{T}[{V_{pk}\sin(\omega t+\phi )]^{2}dt}}}\\&=V_{pk}{\sqrt {{\frac {1}{2T}}\int _{0}^{T}[{1-\cos(2\omega t+2\phi )]dt}}}\\&=V_{pk}{\sqrt {{\frac {1}{2T}}\int _{0}^{T}{dt}}}\\&={\frac {V_{pk}}{\sqrt {2}}}\end{aligned}}}

The factor ${\displaystyle {\sqrt {2}}}$ is called the crest factor, which varies for different waveforms.

${\displaystyle V_{\mathrm {rms} }={\frac {V_{\mathrm {peak} }}{\sqrt {3}}}.}$
${\displaystyle \displaystyle V_{\mathrm {rms} }=V_{\mathrm {peak} }.}$
${\displaystyle V_{\mathrm {rms} }={\sqrt {{\frac {1}{T}}\int _{0}^{T}{[v(t)]^{2}dt}}}.}$

### Example

To illustrate these concepts, consider a 230 V AC mains supply used in many countries around the world. It is so called because its root mean square value is 230 V. This means that the time-averaged power delivered is equivalent to the power delivered by a DC voltage of 230 V. To determine the peak voltage (amplitude), we can rearrange the above equation to:

${\displaystyle V_{\mathrm {peak} }={\sqrt {2}}\ V_{\mathrm {rms} }.}$

For 230 V AC, the peak voltage ${\displaystyle \scriptstyle V_{\mathrm {peak} }}$ is therefore ${\displaystyle \scriptstyle 230V\times {\sqrt {2}}}$, which is about 325 V.

## History

The first alternator to produce alternating current was a dynamo electric generator based on Michael Faraday's principles constructed by the French instrument maker Hippolyte Pixii in 1832.[3] Pixii later added a commutator to his device to produce the (then) more commonly used direct current. The earliest recorded practical application of alternating current is by Guillaume Duchenne, inventor and developer of electrotherapy. In 1855, he announced that AC was superior to direct current for electrotherapeutic triggering of muscle contractions.[4]

Alternating current technology had first developed in Europe due to the work of Guillaume Duchenne (1850s), The Hungarian Ganz Works (1870s), Sebastian Ziani de Ferranti (1880s), Lucien Gaulard, and Galileo Ferraris.

In 1876, Russian engineer Pavel Yablochkov invented a lighting system based on a set of induction coils where the primary windings were connected to a source of AC. The secondary windings could be connected to several 'electric candles' (arc lamps) of his own design.[5][6] The coils Yablochkov employed functioned essentially as transformers.[5]

In 1878, the Ganz factory, Budapest, Hungary, began manufacturing equipment for electric lighting and, by 1883, had installed over fifty systems in Austria-Hungary. Their AC systems used arc and incandescent lamps, generators, and other equipment.[7]

A power transformer developed by Lucien Gaulard and John Dixon Gibbs was demonstrated in London in 1881, and attracted the interest of Westinghouse. They also exhibited the invention in Turin in 1884.

### DC distribution systems

Template:Refimprove section During the initial years of electricity distribution, Edison's direct current was the standard for the United States, and Edison did not want to lose all his patent royalties.[8] Direct current worked well with incandescent lamps, which were the principal load of the day, and with motors. Direct-current systems could be directly used with storage batteries, providing valuable load-leveling and backup power during interruptions of generator operation. Direct-current generators could be easily paralleled, allowing economical operation by using smaller machines during periods of light load and improving reliability. At the introduction of Edison's system, no practical AC motor was available. Edison had invented a meter to allow customers to be billed for energy proportional to consumption, but this meter worked only with direct current.

The principle drawback of direct-current distribution was that customer loads, distribution and generation were all at the same voltage. Generally, it was uneconomical to use a high voltage for transmission and reduce it for customer uses. Even with the Edison 3-wire system (placing two 110-volt customer loads in series on a 220-volt supply), the high cost of conductors required generation to be close to customer loads, otherwise losses made the system uneconomical to operate.

### Transformers

Alternating current systems can use transformers to change voltage from low to high level and back, allowing generation and consumption at low voltages but transmission, possibly over great distances, at high voltage, with savings in the cost of conductors and energy losses.

A bipolar open-core power transformer developed by Lucien Gaulard and John Dixon Gibbs was demonstrated in London in 1881, and attracted the interest of Westinghouse. They also exhibited the invention in Turin in 1884. However these early induction coils with open magnetic circuits are inefficient at transferring power to loads. Until about 1880, the paradigm for AC power transmission from a high voltage supply to a low voltage load was a series circuit. Open-core transformers with a ratio near 1:1 were connected with their primaries in series to allow use of a high voltage for transmission while presenting a low voltage to the lamps. The inherent flaw in this method was that turning off a single lamp (or other electric device) affected the voltage supplied to all others on the same circuit. Many adjustable transformer designs were introduced to compensate for this problematic characteristic of the series circuit, including those employing methods of adjusting the core or bypassing the magnetic flux around part of a coil.[9]
The direct current systems did not have these drawbacks, giving it significant advantages over early AC systems.

### Pioneers

The prototype of ZBD. transformer is on display at the Széchenyi István Memorial Exhibition, Nagycenk, Hungary
The Hungarian "ZBD" Team (Károly Zipernowsky, Ottó Bláthy, Miksa Déri). They were the inventors of the first high efficiency, closed core shunt connection transformer. The three also invented the modern power distribution system: Instead of former series connection they connect transformers that supply the appliances in parallel to the main line.Blathy invented the AC Wattmeter, and they invented the essential Constant Voltage Generator.

In the autumn of 1884, Károly Zipernowsky, Ottó Bláthy and Miksa Déri (ZBD), three engineers associated with the Ganz factory, had determined that open-core devices were impracticable, as they were incapable of reliably regulating voltage.[10] In their joint 1885 patent applications for novel transformers (later called ZBD transformers), they described two designs with closed magnetic circuits where copper windings were either a) wound around iron wire ring core or b) surrounded by iron wire core.[9] In both designs, the magnetic flux linking the primary and secondary windings traveled almost entirely within the confines of the iron core, with no intentional path through air (see Toroidal cores below). The new transformers were 3.4 times more efficient than the open-core bipolar devices of Gaulard and Gibbs.[11]

The Ganz factory in 1884 shipped the world's first five high-efficiency AC transformers.[12] This first unit had been manufactured to the following specifications: 1,400 W, 40 Hz, 120:72 V, 11.6:19.4 A, ratio 1.67:1, one-phase, shell form.[12]

The ZBD patents included two other major interrelated innovations: one concerning the use of parallel connected, instead of series connected, utilization loads, the other concerning the ability to have high turns ratio transformers such that the supply network voltage could be much higher (initially 1,400 to 2,000 V) than the voltage of utilization loads (100 V initially preferred).[13][14] When employed in parallel connected electric distribution systems, closed-core transformers finally made it technically and economically feasible to provide electric power for lighting in homes, businesses and public spaces.[15][16]

The other essential milestone was the introduction of 'voltage source, voltage intensive' (VSVI) systems'[17] by the invention of constant voltage generators in 1885.[18] Ottó Bláthy also invented the first AC electricity meter.[19][20][21][22]

The AC power systems was developed and adopted rapidly after 1886 due to its ability to distribute electricity efficiently over long distances, overcoming the limitations of the direct current system. In 1886, the ZBD engineers designed, and the Ganz factory supplied electrical equipment for, the world's first power station that used AC generators to power a parallel connected common electrical network, the steam-powered Rome-Cerchi power plant.[23] The reliability of the AC technology received impetus after the Ganz Works electrified a large European metropolis: Rome in 1886.[23]

The city lights of Prince George, British Columbia viewed in a motion blurred exposure. The AC blinking causes the lines to be dotted rather than continuous.
Westinghouse Early AC System 1887
(US patent 373035)

In the UK Sebastian de Ferranti, who had been developing AC generators and transformers in London since 1882, redesigned the AC system at the Grosvenor Gallery power station in 1886 for the London Electric Supply Corporation (LESCo) including alternators of his own design and transformer designs similar to Gaulard and Gibbs.[24] In 1890 he designed their power station at Deptford[25] and converted the Grosvenor Gallery station across the Thames into an electrical substation, showing the way to integrate older plants into a universal AC supply system.[26]

In the US William Stanley, Jr. designed one of the first practical devices to transfer AC power efficiently between isolated circuits. Using pairs of coils wound on a common iron core, his design, called an induction coil, was an early (1885) transformer. Stanley also worked on engineering and adapting European designs such as the Gaulard and Gibbs transformer for US entrepreneur George Westinghouse who started building AC systems in 1886. The spread of Westinghouse and other AC systems triggered a push back by in late 1887 by Thomas Edison (a proponent of direct current) who attempted to discredit alternating current as too dangerous in a public campaign called the "War of Currents".

In 1888 alternating current systems gained further viability with introduction of a functional AC motor, something these systems had lacked up till then. The design, an induction motor, was independently invented by Galileo Ferraris and Nikola Tesla (with Tesla's design being licensed by Westinghouse in the US). This design was further developed into the modern practical three-phase form by Mikhail Dolivo-Dobrovolsky and Charles Eugene Lancelot Brown.[27]

The Ames Hydroelectric Generating Plant (spring of 1891) and the original Niagara Falls Adams Power Plant (August 25, 1895) were among the first hydroelectric AC-power plants. The first commercial power plant in the United States using three-phase alternating current was the hydroelectric Mill Creek No. 1 Hydroelectric Plant near Redlands, California, in 1893 designed by Almirian Decker. Decker's design incorporated 10,000-volt three-phase transmission and established the standards for the complete system of generation, transmission and motors used today.

The Jaruga Hydroelectric Power Plant in Croatia was set in operation on 28 August 1895. The two generators (42 Hz, 550 kW each) and the transformers were produced and installed by the Hungarian company Ganz. The transmission line from the power plant to the City of Šibenik was Template:Convert long on wooden towers, and the municipal distribution grid 3000 V/110 V included six transforming stations.

Alternating current circuit theory developed rapidly in the latter part of the 19th and early 20th century. Notable contributors to the theoretical basis of alternating current calculations include Charles Steinmetz, Oliver Heaviside, and many others.[28][29] Calculations in unbalanced three-phase systems were simplified by the symmetrical components methods discussed by Charles Legeyt Fortescue in 1918.

{{#invoke:Portal|portal}}

## References

1. {{#invoke:citation/CS1|citation |CitationClass=book }}
2. {{#invoke:citation/CS1|citation |CitationClass=book }}
3. Pixii Machine invented by Hippolyte Pixii, National High Magnetic Field Laboratory
4. Licht, Sidney Herman., "History of Electrotherapy", in Therapeutic Electricity and Ultraviolet Radiation, 2nd ed., ed. Sidney Licht, New Haven: E. Licht, 1967, Pp. 1-70.
5. Template:Cite web
6. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
7. {{#invoke:citation/CS1|citation |CitationClass=book }}
8. {{#invoke:citation/CS1|citation |CitationClass=book }}
9. {{#invoke:citation/CS1|citation |CitationClass=book }}
10. Hughes, p. 95
11. Template:Cite web
12. {{#invoke:Citation/CS1|citation |CitationClass=journal }}
13. Template:Cite web
14. Template:Cite web
15. Template:Cite web
16. Template:Cite web
17. American Society for Engineering Education. Conference - 1995: Annual Conference Proceedings, Volume 2, (PAGE: 1848)
18. Thomas Parke Hughes: Networks of Power: Electrification in Western Society, 1880-1930 (PAGE: 96)
19. Template:Cite web
20. {{#invoke:Citation/CS1|citation |CitationClass=journal }} Student paper read on January 24, 1896 at the Students' Meeting.
21. The Electrician, Volume 50. 1923
22. Official gazette of the United States Patent Office: Volume 50. (1890)
23. Template:Cite web
24. Hughes, Thomas P. (1993). Networks of Power: Electrification in Western Society, 1880-1930. Baltimore: The Johns Hopkins University Press. p. 98. ISBN 0-8018-2873-2
25. Ferranti TimelineMuseum of Science and Industry (Accessed 22-02-2012)
26. Hughes, Thomas P. (1993). Networks of Power: Electrification in Western Society, 1880-1930. Baltimore: The Johns Hopkins University Press. p. 208. ISBN 0-8018-2873-2
27. Arnold Heertje, Mark Perlman Evolving Technology and Market Structure: Studies in Schumpeterian Economics, page 138
28. I. Grattan-Guinness, History and Philosophy of the Mathematical Sciences - 2003, Page 1229
29. Jeff Suzuki, Mathematics in Historical Context - 2009, page 329

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text: Department of         Mathematical Sciences Events People Colloquia and Seminars Conferences Centers Positions Areas of Research About the Department Alumni Math Colloquium Ibrahim Ekren Univ. of Michigan Title: Multidimensional utility maximization with small nonlinear price impact Abstract: In this talk we discuss the multidimensional utility maximisation problem with small nonlinear price impact. Using homogenisation techniques, we obtain a first order expansion of the utility function for small nonlinear price impact. We give a PDE characterisation of the utility loss for small nonlinear price impact and exhibit a family of asymptotically optimal strategies for the problem. This is a joint work in progress with Erhan Bayraktar and Thomas Caye.Date: Monday, January 22, 2018Time: 4:30 pmLocation: Wean Hall 8220Submitted by:  BohmanNote: Refreshments at 4:00, WeH 6220.


text:
# Question #3e3ea

Nov 29, 2015

It will become isoelectronic to the noble gas neon.

#### Explanation:

The atomic number of fluorine is 9. Therefore fluorine atoms have 9 positively charged protons in their atomic nuclei. Neutral fluorine atoms will also have 9 negatively charged electrons in their electron clouds.

The electron configuration of a neutral fluorine atom is $\text{1s"^2"2s"^2"2p"^5}$.

When a fluorine atom gains one electron, it becomes a fluoride ion with 10 negatively charged electrons and 9 positively charged protons, which gives it a ${1}^{-}$ charge.

The electron configuration of a $\text{F"^(-)}$ ion is $\text{1s"^2"2s"^2"2p"^6}$.

This is the same electron configuration as neon, which has atomic number 10, so it has 10 protons and 10 electrons in its neutral atoms.

The electron configuration of neon is $\text{1s"^2"2s"^2"2p"^6}$.


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# pyrosetta die slow

3 posts / 0 new
pyrosetta die slow
#1

I'm running a python script with minmover function. After pressing Ctrl + C， the program dies in about 30 seconds. I don't know why it takes so long. Is there a way to kill the job immediately ?

Category:
Post Situation:
Sun, 2021-06-13 00:05
Eden

This is a Unix question and not a Pyrosetta question, so maybe StackOverflow can give a better explanation, but Ctr+C sends a SIGINT as in interrupt, while Ctr+Z sends a SIGKILL, which for most code is faster as it is as messy as a segfault.
I use Jupyter notebook and the kernel interrupt (stop button) results in a segfault and is slower (a decent couple of seconds) and than restarting the kernel (instantaneous)...

Mon, 2021-06-14 06:39
matteoferla

Ctrl+Z works.

Thank you for your explaination!

Mon, 2021-06-14 23:10
Eden


text:
Journal article Open Access

# Carbon dioxide in magmas and implications for hydrothermal systems. Miner Deposita

Lowenstern, Jacob B.

### DataCite XML Export

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<identifier identifierType="URL">https://zenodo.org/record/1232605</identifier>
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<creator>
<creatorName>Lowenstern, Jacob B.</creatorName>
<givenName>Jacob B.</givenName>
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<titles>
<title>Carbon dioxide in magmas and implications for hydrothermal systems. Miner Deposita</title>
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<publicationYear>2001</publicationYear>
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<date dateType="Issued">2001-01-08</date>
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<description descriptionType="Abstract">This review focuses on the solubility, origin, abundance, and degassing of carbon dioxide (CO2) in magma-hydrothermal systems, with applications for those workers interested in intrusion-related deposits of gold and other metals. The solubility of CO2 increases with pressure and magma alkalinity. Its solubility is low relative to that of H2O, so that fluids exsolved deep in the crust tend to have high CO2/H2O compared with fluids evolved closer to the surface. Similarly, CO2/H2O will typically decrease during progressive decompression- or crystallization-induced degassing. The temperature dependence of solubility is a function of the speciation of CO2, which dissolves in molecular form in rhyolites (retrograde temperature solubility), but exists as dissolved carbonate groups in basalts (prograde). Magnesite and dolomite are stable under a relatively wide range of mantle conditions, but melt just above the solidus, thereby contributing CO2 to mantle magmas. Graphite, diamond, and a free CO2-bearing fluid may be the primary carbon-bearing phases in other mantle source regions. Growing evidence suggests that most CO2 is contributed to arc magmas via recycling of subducted oceanic crust and its overlying sediment blanket. Additional carbon can be added to magmas during magma-wallrock interactions in the crust. Studies of fluid and melt inclusions from intrusive and extrusive igneous rocks yield ample evidence that many magmas are vapor saturated as deep as the mid crust (10 15 km) and that CO2 is an appreciable part of the exsolved vapor. Such is the case in both basaltic and some silicic magmas. Under most conditions, the presence of a CO2-bearing vapor does not hinder, and in fact may promote, the ascent and eruption of the host magma. Carbonic fluids are poorly miscible with aqueous fluids, particularly at high temperature and low pressure, so that the</description>
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views


text:
## Algebra 2 (1st Edition)

At least $9$ shots consecutively.
Using part a), our equality is: $\frac{60+x}{75+x}=0.82\\60+x=0.82(75+x)\\60+x=61.5+0.82x\\0.18x=1.5\\x=8.33$ Thus she must succeed with at least $9$ shots consecutively.


text:
Location of Repository

## Search for first-generation scalar leptoquarks in $p\bar p$ collisions at $sqrt{s}$ = 1.96 TeV

### Abstract

D0We report on a search for pair production of first-generation scalar leptoquarks (LQ) in ppbar collisions at sqrt(s)=1.96 TeV using an integrated luminosity of 252pb-1 collected at the Fermilab Tevatron collider by the D0 detector. We observe no evidence for LQ production in the topologies arising from LQLQbar->eqeq and LQLQbar->eqnuq, and derive 95% C.L. lower limits on the LQ mass as a function of beta, where beta is the branching fraction for LQ->eq. The limits are 241 and 218 GeV/c2 for beta=1 and 0.5, respectively. These results are combined with those obtained by D0 at sqrt(s)=1.8 TeV, which increases these LQ mass limits to 256 and 234 GeV/c2

Topics: proton-proton inclusive interactions, high-energy elementary particle interactions, semileptonic decays, elementary particle mass, 14.80.-j, 13.85.Rm, [PHYS.HEXP] Physics [physics]/High Energy Physics - Experiment [hep-ex]
Publisher: American Physical Society
Year: 2005
DOI identifier: 10.1103/PhysRevD.71.071104
OAI identifier: oai:HAL:in2p3-00023534v1
Provided by: Hal-Diderot


text:
Under Pressure Crossword Clue, Gray Whale Vs Humpback, Python From Zero To Hero Pdf, Doing The Same Thing And Expecting Different Results Lyrics, Data Governance Architect Job Description, Naples Beach Restaurant, Banana Fish Synopsis, Sacred Worlds Lyrics, Weather In Caribbean In October, " /> Under Pressure Crossword Clue, Gray Whale Vs Humpback, Python From Zero To Hero Pdf, Doing The Same Thing And Expecting Different Results Lyrics, Data Governance Architect Job Description, Naples Beach Restaurant, Banana Fish Synopsis, Sacred Worlds Lyrics, Weather In Caribbean In October, " /> Under Pressure Crossword Clue, Gray Whale Vs Humpback, Python From Zero To Hero Pdf, Doing The Same Thing And Expecting Different Results Lyrics, Data Governance Architect Job Description, Naples Beach Restaurant, Banana Fish Synopsis, Sacred Worlds Lyrics, Weather In Caribbean In October, " />

Ask Question Asked 5 months ago. The Commutator of Covariant Derivatives. $\begingroup$ Partial derivatives are defined w.r.t. We also have the curved-space version of Stokes's theorem using the covariant derivative and finally the exterior derivative and commutator, where Carroll seems to have made a very peculiar typo. In differential geometry, the Lie derivative / ˈ l iː /, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field. 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text:
My watch list
my.chemeurope.com

# Electromagnetic wave equation

The electromagnetic wave equation is a second-order partial differential equation that describes the propagation of electromagnetic waves through a medium or in a vacuum. The homogeneous form of the equation, written in terms of either the electric field E or the magnetic field B, takes the form:

$\left( \nabla^2 - { 1 \over c^2 } {\partial^2 \over \partial t^2} \right) \mathbf{E} \ \ = \ \ 0$
$\left( \nabla^2 - { 1 \over c^2 } {\partial^2 \over \partial t^2} \right) \mathbf{B} \ \ = \ \ 0$

where c is the speed of light in the medium. In a vacuum, c = 299792458 meters per second, which is the speed of light in free space.

The electromagnetic wave equation derives from Maxwell's equations.

It should also be noted that in most older literature, B is called the "magnetic flux density" or "magnetic induction".

## Speed of propagation

### In vacuum

If the wave propagation is in vacuum, then

$c = c_o = { 1 \over \sqrt{ \mu_o \varepsilon_o } } = 2.99792458 \times 10^8$ meters per second

is the speed of light in vacuum. The magnetic constant $\ \mu_0$ and the vacuum permittivity $\ \varepsilon_0$ are important physical constants that play a key role in electromagnetic theory.

Symbol Name Numerical Value SI Unit of Measure Type
$c_0 \$ Speed of light in vacuum $2.99792458 \times 10^{8}$ meters per second defined
$\ \varepsilon_0$ vacuum permittivity $8.85418781787\times 10^{-12}$ Farads per meter derived
$\ \mu_0 \$ magnetic constant $4 \pi \times 10^{-7}$ Henries per meter defined

### In a material medium

The speed of light in a linear, isotropic, and non-dispersive material medium is

$c = { c_0 \over n } = { 1 \over \sqrt{ \mu \varepsilon } }$

where

$n = \sqrt{ \mu \varepsilon \over \mu_0 \varepsilon_0 }$

is the refractive index of the medium, $\ \mu$ is the magnetic permeability of the medium, and $\ \varepsilon$ is the electric permittivity of the medium.

## The origin of the electromagnetic wave equation

### Conservation of charge

Conservation of charge requires that the time rate of change of the total charge enclosed within a volume V must equal the net current flowing into the surface S enclosing the volume:

$\oint \limits_S \mathbf{j} \cdot d \mathbf{A} = - {d \over d t} \int \limits_V \rho \cdot dV$

where j is the current density (in Amperes per square meter) flowing through the surface and ρ is the charge density (in coulombs per cubic meter) at each point in the volume.

From the divergence theorem, this relationship can be converted from integral form to differential form:

$\nabla \cdot \mathbf{j} = - { \partial \rho \over \partial t}$

### Ampère's circuital law prior to Maxwell's correction

In its original form, Ampère's circuital law relates the magnetic field B to the current density j:

$\oint \limits_C \mathbf{B} \cdot d \mathbf{l} = \int \limits_S \mu \mathbf{j} \cdot d \mathbf{A}$

which can be converted to differential form, using Stokes' theorem:

$\nabla \times \mathbf{B} = \mu_0 \mathbf{j}$

### Inconsistency between Ampère's circuital law and the law of conservation of charge

Taking the divergence of both sides of Ampère's circuital law gives:

$\nabla \cdot ( \nabla \times \mathbf{B} ) = \nabla \cdot \mu_0 \mathbf{j}$

The divergence of the curl of any vector field, including the magnetic field B, is always equal to zero:

$\nabla \cdot ( \nabla \times \mathbf{B}) = 0$

Combining these two equations implies that

$\nabla \cdot \mu_0 \mathbf{j} = 0$

Because $\ \mu_0$ is nonzero constant, it follows that

$\nabla \cdot \mathbf{j} = 0$

However, the law of conservation of charge tells that

$\nabla \cdot \mathbf{j} = - { \partial \rho \over \partial t }$

Hence, as in the case of Kirchhoff's circuit laws, Ampère's circuital law would appear only to hold in situations involving constant charge density. This would rule out the situation that occurs in the plates of a charging or a discharging capacitor.

### Maxwell's correction to Ampère's circuital law

Gauss's law in integral form states:

$\oint \limits_S \mathbf{E} \cdot d \mathbf{A} = \frac{1}{\varepsilon_0} \int \limits_V \rho \cdot dV$

which can be converted to differential form using the divergence theorem:

$\nabla \cdot \varepsilon_0 \mathbf{E} = \rho$

Taking the time derivative of both sides and reversing the order of differentiation on the left-hand side gives:

$\nabla \cdot \varepsilon_0 {\partial \mathbf{E} \over \partial t } = { \partial \rho \over \partial t}$

This last result, along with Ampère's circuital law and the conservation of charge equation, suggests that there are actually two origins of the magnetic field: the current density j, as Ampère had already established, and the so-called displacement current:

${\partial \mathbf{D} \over \partial t } = \varepsilon_0 {\partial \mathbf{E} \over \partial t }$

So the corrected form of Ampère's circuital law becomes:

$\nabla \times \mathbf{B} = \mu_0 \mathbf{j} + \mu_0 \varepsilon_0 {\partial \mathbf{E} \over \partial t }$

### Maxwell's hypothesis that light is an electromagnetic wave

In his 1864 paper entitled A Dynamical Theory of the Electromagnetic Field, Maxwell utilized the correction to Ampère's circuital law that he had made in part III of his 1861 paper On Physical Lines of Force. In PART VI of his 1864 paper which is entitled 'ELECTROMAGNETIC THEORY OF LIGHT'. [1] (page 497 of the article and page 9 of the pdf link) Maxwell combined displacement current with some of the other equations of electromagnetism and he obtained a wave equation with a speed equal to the speed of light. He commented:

The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.

(see [2], page 499 of the article and page 1 of the pdf link)

Maxwell's derivation of the electromagnetic wave equation has been replaced in modern physics by a much less cumbersome method involving combining the corrected version of Ampère's circuital law with Faraday's law of induction.

To obtain the electromagnetic wave equation in a vacuum using the modern method, we begin with the modern 'Heaviside' form of Maxwell's equations. In a vacuum, these equations are:

$\nabla \cdot \mathbf{E} = 0$
$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}} {\partial t}$
$\nabla \cdot \mathbf{B} = 0$
$\nabla \times \mathbf{B} =\mu_0 \varepsilon_0 \frac{ \partial \mathbf{E}} {\partial t}$

Taking the curl of the curl equations gives:

$\nabla \times \nabla \times \mathbf{E} = -\frac{\partial } {\partial t} \nabla \times \mathbf{B} = -\mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{E} } {\partial t^2}$
$\nabla \times \nabla \times \mathbf{B} = \mu_0 \varepsilon_0 \frac{\partial } {\partial t} \nabla \times \mathbf{E} = -\mu_o \varepsilon_o \frac{\partial^2 \mathbf{H} } {\partial t^2}$

By using the vector identity

$\nabla \times \left( \nabla \times \mathbf{V} \right) = \nabla \left( \nabla \cdot \mathbf{V} \right) - \nabla^2 \mathbf{V}$

where $\mathbf{V}$ is any vector function of space, it turns into the wave equations:

${\partial^2 \mathbf{E} \over \partial t^2} \ - \ c^2 \cdot \nabla^2 \mathbf{E} \ \ = \ \ 0$
${\partial^2 \mathbf{B} \over \partial t^2} \ - \ c^2 \cdot \nabla^2 \mathbf{B} \ \ = \ \ 0$

where

$c = { 1 \over \sqrt{ \mu_0 \varepsilon_0 } } = 2.99792458 \times 10^8$ meters per second

is the speed of light in free space.

## Covariant form of the homogeneous wave equation

These relativistic equations can be written in covariant form as

$\ \Box^2 A^{\mu} = 0$

where the electromagnetic four-potential is

$A^{\mu}=(\varphi, \mathbf{A} c)$

with the Lorenz gauge condition:

$\partial_{\mu} A^{\mu} = 0\,$.

Here

$\Box^2 = \nabla^2 - { 1 \over c^2} \frac{ \partial^2} { \partial t^2}$ is the d'Alembertian operator. The square box is not a typographical error; it is the correct symbol for this operator.

## Homogeneous wave equation in curved spacetime

Main article: Maxwell's equations in curved spacetime

The electromagnetic wave equation is modified in two ways, the derivative is replaced with the covariant derivative and a new term that depends on the curvature appears.

$- {A^{\alpha ; \beta}}_{; \beta} + {R^{\alpha}}_{\beta} A^{\beta} = 0$

where

${R^{\alpha}}_{\beta}$

is the Ricci curvature tensor and the semicolon indicates covariant differentiation.

The generalization of the Lorenz gauge condition in curved spacetime is assumed:

${A^{\mu}}_{ ; \mu} =0$.

## Inhomogeneous electromagnetic wave equation

Main article: Inhomogeneous electromagnetic wave equation

Localized time-varying charge and current densities can act as sources of electromagnetic waves in a vacuum. Maxwell's equations can be written in the form of a wave equation with sources. The addition of sources to the wave equations makes the partial differential equations inhomogeneous.

## Solutions to the homogeneous electromagnetic wave equation

Main article: Wave equation

The general solution to the electromagnetic wave equation is a linear superposition of waves of the form

$\mathbf{E}( \mathbf{r}, t ) = g(\phi( \mathbf{r}, t )) = g( \omega t - \mathbf{k} \cdot \mathbf{r} )$

and

$\mathbf{B}( \mathbf{r}, t ) = g(\phi( \mathbf{r}, t )) = g( \omega t - \mathbf{k} \cdot \mathbf{r} )$

for virtually any well-behaved function g of dimensionless argument φ, where

$\ \omega$ is the angular frequency (in radians per second), and
$\mathbf{k} = ( k_x, k_y, k_z)$ is the wave vector (in radians per meter).

Although the function g can be and often is a monochromatic sine wave, it does not have to be sinusoidal, or even periodic. In practice, g cannot have infinite periodicity because any real electromagnetic wave must always have a finite extent in time and space. As a result, and based on the theory of Fourier decomposition, a real wave must consist of the superposition of an infinite set of sinusoidal frequencies.

In addition, for a valid solution, the wave vector and the angular frequency are not independent; they must adhere to the dispersion relation:

$k = | \mathbf{k} | = { \omega \over c } = { 2 \pi \over \lambda }$

where k is the wavenumber and λ is the wavelength.

The simplest set of solutions to the wave equation result from assuming sinusoidal waveforms of a single frequency in separable form:

$\mathbf{E} ( \mathbf{r}, t ) = \mathrm {Re} \{ \mathbf{E} (\mathbf{r} ) e^{ j \omega t } \}$

where

• $j = \sqrt{-1} \,$ is the imaginary unit,
• $\omega = 2 \pi f \,$ is the angular frequency in radians per second,
• $f \,$ is the frequency in hertz, and
• $e^{j \omega t} = \cos(\omega t) + j \sin(\omega t) \,$ is Euler's formula.

### Plane wave solutions

Consider a plane defined by a unit normal vector

$\mathbf{n} = { \mathbf{k} \over k }$.

Then planar traveling wave solutions of the wave equations are

$\mathbf{E}(\mathbf{r}) = E_0 e^{-j \mathbf{k} \cdot \mathbf{r} }$

and

$\mathbf{B}(\mathbf{r}) = B_0 e^{-j \mathbf{k} \cdot \mathbf{r} }$

where

$\mathbf{r} = (x, y, z)$ is the position vector (in meters).

These solutions represent planar waves traveling in the direction of the normal vector $\mathbf{n}$. If we define the z direction as the direction of $\mathbf{n}$ and the x direction as the direction of $\mathbf{E}$, then by Faraday's Law the magnetic field lies in the y direction and is related to the electric field by the relation

$c {\partial B \over \partial z} = {\partial E \over \partial t}$.

Because the divergence of the electric and magnetic fields are zero, there are no fields in the direction of propagation.

This solution is the linearly polarized solution of the wave equations. There are also circularly polarized solutions in which the fields rotate about the normal vector.

### Spectral decomposition

Because of the linearity of Maxwell's equations in a vacuum, solutions can be decomposed into a superposition of sinusoids. This is the basis for the Fourier transform method for the solution of differential equations. The sinusoidal solution to the electromagnetic wave equation takes the form

$\mathbf{E} ( \mathbf{r}, t ) = \mathbf{E}_0 \cos( \omega t - \mathbf{k} \cdot \mathbf{r} + \phi_0 )$

and

$\mathbf{B} ( \mathbf{r}, t ) = \mathbf{B}_0 \cos( \omega t - \mathbf{k} \cdot \mathbf{r} + \phi_0 )$

where

$\ t$ is time (in seconds),
$\ \omega$ is the angular frequency (in radians per second),
$\mathbf{k} = ( k_x, k_y, k_z)$ is the wave vector (in radians per meter), and
$\phi_0 \,$ is the phase angle (in radians).

The wave vector is related to the angular frequency by

$k = | \mathbf{k} | = { \omega \over c } = { 2 \pi \over \lambda }$

where k is the wavenumber and λ is the wavelength.

The Electromagnetic spectrum is a plot of the field magnitudes (or energies) as a function of wavelength.

### Other solutions

Spherically symmetric and cylindrically symmetric analytic solutions to the electromagnetic wave equations are also possible.

In cylindrical coordinates the wave equation can be written as follows:

$\mathbf{E} ( \mathbf{r}, t ) = {\mathbf{E}_0 \cos( \omega t - \mathbf{k} \cdot \mathbf{r} + \phi_0 )\over s}$

and

$\mathbf{B} ( \mathbf{r}, t ) = {\mathbf{B}_0 \cos( \omega t - \mathbf{k} \cdot \mathbf{r} + \phi_0 )\over s}$

## References

### Electromagnetism

#### Journal articles

• James Clerk Maxwell, "A Dynamical Theory of the Electromagnetic Field", Philosophical Transactions of the Royal Society of London 155, 459-512 (1865). (This article accompanied a December 8, 1864 presentation by Maxwell to the Royal Society.)

• Griffiths, David J. (1998). Introduction to Electrodynamics (3rd ed.). Prentice Hall. ISBN 0-13-805326-X.
• Tipler, Paul (2004). Physics for Scientists and Engineers: Electricity, Magnetism, Light, and Elementary Modern Physics (5th ed.). W. H. Freeman. ISBN 0-7167-0810-8.
• Edward M. Purcell, Electricity and Magnetism (McGraw-Hill, New York, 1985). ISBN 0-07-004908-4.
• Hermann A. Haus and James R. Melcher, Electromagnetic Fields and Energy (Prentice-Hall, 1989) ISBN 0-13-249020-X.
• Banesh Hoffmann, Relativity and Its Roots (Freeman, New York, 1983). ISBN 0-7167-1478-7.
• David H. Staelin, Ann W. Morgenthaler, and Jin Au Kong, Electromagnetic Waves (Prentice-Hall, 1994) ISBN 0-13-225871-4.
• Charles F. Stevens, The Six Core Theories of Modern Physics, (MIT Press, 1995) ISBN 0-262-69188-4.

• Jackson, John D. (1998). Classical Electrodynamics (3rd ed.). Wiley. ISBN 0-471-30932-X.
• Landau, L. D., The Classical Theory of Fields (Course of Theoretical Physics: Volume 2), (Butterworth-Heinemann: Oxford, 1987). ISBN 0-08-018176-7.
• Maxwell, James C. (1954). A Treatise on Electricity and Magnetism. Dover. ISBN 0-486-60637-6.
• Charles W. Misner, Kip S. Thorne, John Archibald Wheeler, Gravitation, (1970) W.H. Freeman, New York; ISBN 0-7167-0344-0. (Provides a treatment of Maxwell's equations in terms of differential forms.)

### Vector calculus

• H. M. Schey, Div Grad Curl and all that: An informal text on vector calculus, 4th edition (W. W. Norton & Company, 2005) ISBN 0-393-92516-1.

### Biographies

• Andre Marie Ampere
• Albert Einstein


text:
# Is this language regular or non-regular: {ww : w ∈ {a,b}* } [duplicate]

This is a question from a text book that's giving me some trouble. The question is:

Determine whether or not this language is regular. Justify your answer. $$L = \{ww : w \in \{a,b\}^* \}$$

I think this language is not regular because $w$ can be of arbitrary length and adheres to no pattern. So, therefore, it cannot be determined whether $ww$ is part of the language using a finite number of states. Am I correct in this assumption, and does my explanation make sense? Thanks for any help you can give me!

Let $w = ww = (a^p)(b^p)(a^p)(b^p)$. Then consider the Pumping Lemma. Since $|xy| \leq p$ and $|y| \geq 1$, then the $y$ part of the string must be a's. But if we pump up, we'll have more a's in the first part than the second, and $w \neq w \in ww$. Hence, the language can't be regular.

• I think your general idea could work, but you must argue very accurately why it is impossible have a DFA of finite size for the given language. Apart from this, this sounds like you should consider the pumping lemma here. – Rmn Feb 16 '15 at 17:06
• – D.W. Feb 16 '15 at 21:37
• "no pattern" is not a valid reason. Even though $w$ does not have a "pattern", the language $\{w \mid w \in \{a,b\}^*\}$ is regular. So this does not give you anything, and you can not conclude that finitely many states are insufficient. – Raphael Feb 17 '15 at 7:38
• Ah, I see. I'll edit the above post and add my new solution. Thanks. – Victor Brunell Feb 18 '15 at 14:31

Your explanation is correct. To make it more rigorous, assume that L can be recognized by a finite automaton $D$ with $N$ states.
Consider feeding a string $w$ longer than $N$ characters to $D$. Since $D$ has only $N$ states, that means it will go through some state twice. Suppose $w$ can be represented as $w = pqr$, where the state of the DFA is the same before and after $q$.
Then the strings $ww = pqrpqr$ and $pqqq...rpqr$ must be both accepted by $D$, but probably only the first one is in $L$ (some more rigor may be needed to prove that there exists such a string $w$ such that the second string really cannot be in $L$, for some number of repetitions of $q$).


text:
# [Tex/LaTex] Table numbers in beamer

beamertables

I am writing a beamer presentation,but when including tables with \caption then at the left of the title I get "Table: …" but no table number. Is there a way to get tables numbered? Like "Table 1: ….." and then the one on the next slide "Table 2:…".

Here is my MWE:

          \documentclass{beamer}
\usepackage{threeparttable}

\begin{document}
\begin{frame}{Results}
\begin{table}[H]
\centering

\caption{\smallFertility intentions and measures of employment
security and economic uncertainty }

\scalebox{0.8}{
\begin{threeparttable}
\begin{tabular}{lcccc}

\hline
\hline
& (1)   & (2)   & (3)   & (4)  \bigstrut\\
VARIABLES & Baseline 1 & Baseline 2 & Women & Men   &

\hline  \rule{0pt}{3ex}
% &       &       &      & \bigstrut\\
Job keeping & 0.0392 & 0.0300 & 0.0656 & -0.0241\\
& (0.0352) & (0.0362) & (0.0434) & (0.0841) \\
\hline
Job seeking & 0.110*** & 0.0944*** & 0.0754* & 0.165**   \\
& (0.0318) & (0.0321) & (0.0399) & (0.0673)  \\
\hline
Financial security & 0.124*** & 0.0942** & 0.0686 & 0.128  \\
& (0.0438) & (0.0455) & (0.0521) & (0.105)  \\
\hline
Social mobility & 0.0751** & 0.0577 & 0.0387 & 0.0909  \\
& (0.0360) & (0.0358) & (0.0454) & (0.0706)  \\
\hline

\multicolumn{5}{l}{\footnotesize  Significance levels: *** p$<$0.01,
** p$<$0.05, * p$<$0.1.} \\
\end{tabular}
\end{threeparttable}}

\end{table}

\end{frame}
\end{document}

I could not compile your code, but to get numbered caption, you can use

\setbeamertemplate{caption}[numbered]

• floating specifier such as [H] don't make much sense in a documentclass without floats
• please consider using the booktabs package - it gives much nicer spacing within your table


text:
COSAM » Events » 2016 » February » Stochastics Seminar

 Stochastics Seminar Time: Feb 04, 2016 (02:00 PM) Location: Parker Hall 322 Details: Speaker: Dr. Erkan Nane Title: Non-linear noise excitation for some space-time fractional stochastic equations in bounded domains Abstract: In this talk we study non-linear noise excitation for a class of space-time fractional stochastic equations in  bounded domains with Dirichlet  boundary conditions. These equations  were recently  introduced by Mijena and Nane, "Space time fractional stochastic partial differential equations" (Stochastic Process Appl. 125 (2015), no. 9, 3301--3326).   We first study  the existence and uniqueness of the solution of these equations  under suitable conditions on the initial function. We also study the asymptotic behavior of the solution  with respect to a parameter $\lambda$- intermittency. In particular, our results are significant extensions of those  in  M. Foondun, K. Tian and  W. Liu, "On some properties  of a class of fractional stochastic equations" (Preprint available at arxiv.org 1404.6791v1). We will also discuss some open problems on these space-time fractional equations.

Last updated: 02/01/2016


text:
# Setting in memory workspace in ArcPy?

Is it possible to set env.workspace = "in_memory" in ArcPy using both ArcGIS Pro and the ArcGIS 10.2.2 (or 10.3) architectures?

What I am trying to do is get the output of a snap pour point operation written to memory as opposed to disk.

I realize I could write to disk then bring it into memory but this would not help. I am trying to optimize a series of tasks minimizing the use of writing to physical media the output of a process that is merely required as an input to the next process.

I'm going to throw an answer on here because both answers thus far aren't 100% correct.

There are 2 items which can vary from tool to tool.

1. if it honors the workspace environment (this item is always documented on the tool help page)
2. if it can make use of the in_memory workspace (this item may not be explicitly documented. You're more likely to see a note if it DOES NOT support in_memory)

To simply answer the "can you set the environment workspace to in_memory". The answer is YES.

>>> import arcpy
>>> arcpy.env.workspace = r"in_memory"
>>> arcpy.CopyFeatures_management(r"c:\temp\foo.shp", "myinmemoutput")
<Result 'in_memory\\myinmemoutput'>
>>> arcpy.Exists("myinmemoutput")
True

Snap Pour Point does honor the workspace environment per it's documentation and explained Python samples. And a test shows you can write output to in_memory and work with that variable reference...to put into another tool, or save the result

>>> import arcpy
>>> arcpy.env.workspace = r"in_memory"
>>> arcpy.CheckOutExtension("SPATIAL")
u'CheckedOut'
>>> snapOut = arcpy.sa.SnapPourPoint("e:/gpservices101/hydro/US30m/test.gdb/sourcepoint", "e:/gpservices101/hydro/US30m/Region08a/Input/elev_cm", 1,"PourPtID")
>>> snapOut
in_memory\SnapPou_sour1
>>> arcpy.Exists(snapOut)
True
>>> snapOut.save(r"c:\temp\todisk.tif")
>>> arcpy.Exists(r"c:\temp\todisk.tif")
True

Unfortunately, I don't think setting the workspace to the in_memory location like in your question is possible; In such a way that geoprocessing tools will default their output locations to the in_memory workspace.

I just now tried to set the workspace to '\\in_memory' myself, and tested it with a simple CopyFeatures_management() tool run. The string representation of arcpy.env.workspace is indeed set to '\\in_memory', but I'm thinking the CopyFeatures_management() tool is not respecting the workspace environment setting in this case.

It would be awesome though, if we could set in_memory as the default workspace at the map document level. Maybe you could raise this as an ArcGIS Idea?

Best Luck.

EDIT:

Hey I was thinking also since you are using the Snap Pour Point Tool, we may need to approach this a different way. Snap Pour Point appears to act directly on the pour points feature class and does not create an output feature class as part of the tool result. Could we create a temporary, in_memory copy of the pour points and work from there?

EDIT to EDIT:

Oh true, ok you've mentioned you realize this already - sorry.

• I may do that (ArcGIS idea). I wonder why people would vote down this question? I agree with you Jim it would be awesome. Oct 9 '15 at 18:26
• Ah I get the question now. Yes Jim is correct, setting the workspace like that does nothing. You have to specify in_memory for your individual processes. You cannot get it to default with that behavior. Oct 9 '15 at 18:27
• The down vote may have come from the short-length of your question. I see what you mean though - maybe edit your question a bit with some more context and others will vote it back up?
– Jim
Oct 9 '15 at 18:27
• Ok, It is such a direct question I did not see a need to fluff it up. I will do as you say. Oct 9 '15 at 18:29
• reworded question to see if another way exists. Oct 9 '15 at 18:32


text:
NEW
New Website Launch
Experience the best way to solve previous year questions with mock tests (very detailed analysis), bookmark your favourite questions, practice etc...
1

### JEE Main 2015 (Offline)

English
Hindi
If the angles of elevation of the top of a tower from three collinear points $$A, B$$ and $$C,$$ on a line leading to the foot of the tower, are $${30^ \circ }$$, $${45^ \circ }$$ and $${60^ \circ }$$ respectively, then the ratio, $$AB:BC,$$ is :
A
$$1:\sqrt 3$$
B
$$2:3$$
C
$$\sqrt 3 :1$$
D
$$\sqrt 3 :\sqrt 2$$

## Explanation

As $$PB$$ bisects $$\angle APC,$$ therefore $$AB$$ $$:$$ $$BC$$ $$=PA:PC$$

Also in $$\Delta APQ,\sin {30^ \circ } = {h \over {PA}} \Rightarrow PA = 2h$$

and in $$\Delta CPQ,$$ $$\sin {60^ \circ } = {h \over {PC}} \Rightarrow PC = {{2h} \over {\sqrt 3 }}$$

$$\therefore$$ $$AB:BC = 2h:{{2h} \over {\sqrt 3 }} = \sqrt 3 :1$$

तीन संरेख बिंदुओं $$\mathrm{A, B}$$ तथा $$\mathrm{C}$$, एक ऐसी रेखा पर स्थित हैं जो एक मीनार के पाद की दिशा में ले जाती है, से एक मीनार के शिखर के उन्नयन कोण क्रमशः $$30^{\circ}, 45^{\circ}$$ तथा $$60^{\circ}$$ हैं, तो $$\mathrm{AB}: \mathrm{BC}$$ का अनुपात है :

A
$$1:\sqrt 3$$
B
$$2:3$$
C
$$\sqrt 3 :1$$
D
$$\sqrt 3 :\sqrt 2$$
2

### JEE Main 2014 (Offline)

A bird is sitting on the top of a vertical pole $$20$$ m high and its elevation from a point $$O$$ on the ground is $${45^ \circ }$$. It files off horizontally straight away from the point $$O$$. After one second, the elevation of the bird from $$O$$ is reduced to $${30^ \circ }$$. Then the speed (in m/s) of the bird is
A
$$20\sqrt 2$$
B
$$20\left( {\sqrt 3 - 1} \right)$$
C
$$40\left( {\sqrt 2 - 1} \right)$$
D
$$40\left( {\sqrt 3 - \sqrt 2 } \right)$$

## Explanation

Let the speed be $$y$$ $$m/sec$$.

Let $$AC$$ be the vertical pole of height $$20$$ $$m.$$

Let $$O$$ be the point on the ground such that $$\angle AOC = {45^ \circ }$$

Let $$OC = x$$

Time $$t=1$$ $$s$$

From $$\Delta AOC,\,\,\tan {45^ \circ } = {{20} \over x}\,\,\,\,\,\,\,.....\left( i \right)$$

and from $$\Delta BOD,\,\,\tan {30^ \circ } = {{20} \over {x + y}}...\left( {ii} \right)$$

From $$(i)$$ and $$(ii),$$ we have $$x=20$$

and $${1 \over {\sqrt 3 }} = {{20} \over {x + y}}$$

$$\Rightarrow {1 \over {\sqrt 3 }} = {{20} \over {20 + y}}$$

$$\Rightarrow 20 + y = 20\sqrt 3$$

So, $$y = 20\left( {\sqrt 3 - 1} \right)\,\,i.e.,$$

speed $$= 20\left( {\sqrt 3 - 1} \right)m/s$$
3

### AIEEE 2010

For a regular polygon, let $$r$$ and $$R$$ be the radii of the inscribed and the circumscribed circles. A $$false$$ statement among the following is
A
There is a regular polygon with $${r \over R} = {1 \over {\sqrt 2 }}$$
B
There is a regular polygon with $${r \over R} = {2 \over 3}$$
C
There is a regular polygon with $${r \over R} = {{\sqrt 3 } \over 2}$$
D
There is a regular polygon with $${r \over R} = {1 \over 2}$$

## Explanation

If $$O$$ is center of polygon and

$$AB$$ is one of the side, then by figure

$$\cos {\pi \over n} = {r \over R}$$

$$\Rightarrow {r \over R} = {1 \over 2},{1 \over {\sqrt 2 }},{{\sqrt 3 } \over 2}\,\,for$$

$$n = 3,4,6$$ respectively.
4

### AIEEE 2008

$$AB$$ is a vertical pole with $$B$$ at the ground level and $$A$$ at the top. $$A$$ man finds that the angle of elevation of the point $$A$$ from a certain point $$C$$ on the ground is $${60^ \circ }$$. He moves away from the pole along the line $$BC$$ to a point $$D$$ such that $$CD=7$$ m. From $$D$$ the angle of elevation of the point $$A$$ is $${45^ \circ }$$. Then the height of the pole is
A
$${{7\sqrt 3 } \over 2} {1 \over {\sqrt {3 - 1} }}m$$
B
$${{7\sqrt 3 } \over 2}\left( {\sqrt {3 + 1} } \right)m$$
C
$${{7\sqrt 3 } \over 2}\left( {\sqrt {3 - 1} } \right)m$$
D
$${{7\sqrt 3 } \over 2} {1 \over {\sqrt {3 + 1} }}m$$

## Explanation

In $$\Delta ABC$$

$${h \over x} = \tan {60^ \circ } = \sqrt 3$$

$$\Rightarrow x = {h \over {\sqrt 3 }}$$

In $$\Delta ABD{h \over {x + 7}}$$

$$= \tan {45^ \circ } = 1$$

$$\Rightarrow h = x + 7 \Rightarrow h - {h \over {\sqrt 3 }} = 7$$

$$\Rightarrow h = {{7\sqrt 3 } \over {\sqrt 3 - 1}} \times {{\sqrt 3 + 1} \over {\sqrt 3 + 1}}$$

$$\Rightarrow h = {{7\sqrt 3 } \over 2}\left( {\sqrt 3 + 1\,m} \right)$$

### Joint Entrance Examination

JEE Main JEE Advanced WB JEE

### Graduate Aptitude Test in Engineering

GATE CSE GATE ECE GATE EE GATE ME GATE CE GATE PI GATE IN

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Class 12


text:
# Is this a logarithmic spiral?

I'm trying to draw a logarithmic spiral by hand (actually I need to use a plotter to cut a spiral on wood, but that is another story) and I saw this method:

http://www.wikihow.com/Draw-A-Perfect-Spiral

it seems to me like a log spiral, because, if precisely done, the distance to the center is shortened geometrically.

Let me know if I'm correct!.

Thanks!.

• I don't know of an easy mechanical way to draw a logarithmic spiral. I would just calculate and plot points, either on graph paper or in some program. Excel will plot a scatter plot, but it is hard to maintain vertical vs horizontal scale. – Ross Millikan Apr 2 '14 at 22:34

I believe that that is an Archimedian Spiral. Each time it goes around, its shortens by the circumference of the pencil. This arithmetic decreasing is not logarithmic at all.

• The drawings on that website are not to scale. Think about it. Each time the pencil goes around, it shortens the thread to center by the circumference of the pencil. Therefore The distance decreased by equal amounts (the circumference of the pencil) and it is Archimedian – Asimov Apr 2 '14 at 22:27
• Actually it gets shorten by the circumference of the green thumbtack, but, yeah, it is the same. So, then my question is, how much should I shorten the string to achieve a log spiral effect?. – Artemix Apr 2 '14 at 22:39
• a golden spiral aka a logarithmic spiral with growth factor e, is easy enough to draw if you can draw rectangles and do fibonacci numbers. wikihow.com/Draw-the-Golden-Spiral – Asimov Apr 2 '14 at 22:41
• Yeah, the thing is, I need to know, based on the method previously linked (the string, thumbtack and pencil) how much should I shorten the string to achieve a logarithmic effect. This is because I need to plot on wood using a plotter. – Artemix Apr 2 '14 at 22:47
• I don't think you'll have a tidy way to do this "mechanically". A logarithmic (equiangular) spiral is self-similar: for any choice of points one "turn" (2 $\pi$ radians) apart in angle, the arclength must increase by a constant factor. It would be pretty tricky to "pay out" the right amount of string from the pencil everywhere along the path, since the amount of string would have to grow (or decrease, if you're "winding in" toward the limit point) "geometrically". [This is vaguely related to why Jacob Bernoulli got the wrong spiral engraved on his tombstone...] – colormegone Apr 2 '14 at 23:46


text:
1. solving quadratic equation by completing the square

HELP NEEDED

solve the following by completing the square

2x sqaured (dont know to insert power of 2) + 4x - 6

i know i have to divide it by 2 so i get

x sqaured + 2x - 3

but i dont knwo how to work it

also once i have got the answer i have to check it how do i you do this

2. Hi

$\displaystyle 2x^2 + 4x - 6 = 2(x^2 + 2x - 3) = 2(x^2 + 2x + 1 - 4) = 2 \left[(x+1)^2 - 4)\right]$

3. Originally Posted by jm16
HELP NEEDED

solve the following by completing the square

2x sqaured (dont know to insert power of 2) + 4x - 6

i know i have to divide it by 2 so i get

x sqaured + 2x - 3

but i dont knwo how to work it

also once i have got the answer i have to check it how do i you do this

I don't understand what you are asked to solve(?). I guess that you should factor the given term:

$\displaystyle 2x^2 + 4x - 6 = 2(\color{red}x^2+2x\color{black}-3)$

Use the red term to complete the square:

$\displaystyle 2(x^2+2x \bold{{\color{blue}+ 1 - 1}}-3) = 2(\underbrace{(x+1)^2-4}_{\text{difference of squares}})$

A difference of squares can be factored immediately:

$\displaystyle 2((x+1)^2-4) = 2((x+1+2)(x+1-2)) = 2(x+3)(x-1)$

4. Originally Posted by jm16
HELP NEEDED

solve the following by completing the square

2x sqaured (dont know to insert power of 2) + 4x - 6

i know i have to divide it by 2 so i get

x sqaured + 2x - 3

but i dont knwo how to work it

also once i have got the answer i have to check it how do i you do this

unless there's an "= 0" in your problem, you cannot just divide by 2 like that, youwould have to factor out a 2 in that case. the thing to note about complete squares is that the constant term is always $\displaystyle \left( \frac {\text{the coefficient of }x}2 \right)^2$. so we normally add and subtract this term (so that we effectively add 0 and hence do not change the problem) and then simplify to get the complete sqaure.

i will assume we have $\displaystyle 2x^2 + 4x - 6 = 0$

$\displaystyle \Rightarrow x^2 + 2x - 3 = 0$

$\displaystyle \Rightarrow x^2 + 2x {\color{red} + 1^2 - 1^2} - 3 = 0$

(Note that $\displaystyle x^2 + 2x + 1$ is a complete square)

$\displaystyle \Rightarrow (x + 1)^2 - 1 - 3 = 0$

$\displaystyle \Rightarrow (x + 1)^2 - 4 = 0$

can you finish?

EDIT: Pfft! too late!


text:
# Negative value for \$k\$ in binomial theorem

by Victor Su   Last Updated May 15, 2019 16:20 PM - source

Can $$k$$ in the binomial theorem be less than zero so that there is a term that essentially becomes $$x^{n-k}/y^{-k}$$?

Tags :


text:
# Half reaction method justification

I have to say I have never really understood the half reaction method. What bothers me is that it seems to be too artificial. Too many little rules to remember! Why don't we use the mass conservation law to justify the process and use sistematically a set of linear equations to do the balances?

Because the two half-reactions need not necessarily occur at the same place and one half reaction can generally be substituted without affecting the other, it makes sense to think of them separately. This lets us do things like think about different lithium batteries. There are many types and they all use the $\ce{Li -> Li+ + e-}$ oxidation half-reaction, but it's coupled with different reduction half-reactions based on cobalt, iron, etc.


text:
# Global sections of coherent sheaves on determinantal hypersurfaces in $\mathbb{P}^n$

Let us consider the short exact sequence of coherent sheaves on $\mathbb{P}^n$ $$0 \to \mathcal{O}_{\mathbb P^n}(-1)^{r} \stackrel{N}{\longrightarrow} \mathcal{O}_{\mathbb P^n}^{r} \longrightarrow \mathcal F \to 0,$$ where $r \geq 2$ and $N=\{L_{ij}\}$ is a $r \times r$ matrix of linear forms in the variables $x_0, \ldots, x_n$ acting on the left (i.e., $\mathbf{x} \to N \mathbf{x}$).

Then the sheaf $\mathcal{F}$ is supported on the determinantal hypersurface $Y:=\{\det N=0 \} \subset \mathbb{P}^n$, and it is locally free of rank $1$ (i.e., a line bundle) on the open dense locus of $Y$ given by $\textrm{rank } N = r-1$.

Passing to cohomology, we obtain an isomorphism $$\sigma \colon H^0(\mathbb{P}^n, \, \mathcal{O}_{\mathbb P^n}^{r}) \stackrel{\cong}{\longrightarrow} H^0(Y, \, \mathcal F),$$ in particular $h^0(Y, \, \mathcal F)=r.$ In other words, this is a bijective correspondence between $n$-ples $(\alpha_1, \ldots, \alpha_r)$ of complex numbers and global sections of $\mathcal F$.

Question. How can we describe the zero locus in $Y$ of the section $\sigma(\alpha_1, \ldots, \alpha_n)$ in terms of the $\alpha_i$ and the matrix $N$?

Toy model. Consider the case $n=3$, $r=2$ and $$N = \pmatrix{x_0 & x_1 \\ x_2 & x_3}.$$ In this situation, $Y$ is the smooth quadric $\{x_0x_3-x_1x_2=0 \} \subset \mathbb{P}^3$, and $\mathcal{F}$ is a line bundle with $2$ global sections, hence a pencil corrisponding to one of the two rulings of $Y$. I made some (unelegant) local computation and it seems to me that in this case $\sigma(\alpha_1, \, \alpha_2)$ should be something like $\alpha_1 \sigma_{02} + \alpha_2 \sigma_{13}$, where $\sigma_{02}$ is the section whose zero locus is the line $\{x_0=x_2=0 \}$ and $\sigma_{13}$ is the section whose zero locus is the line (in the same ruling) $\{x_1=x_3=0 \}.$ In other words, the global sections of $\mathcal{F}$ are expressed in terms of the columns of $N$.

On the other hand, dualizing our exact sequence and applying Grothendieck duality we get $$0 \to \mathcal{O}_{\mathbb P^3}(-1)^{2} \stackrel{ { }^t N}{\longrightarrow} \mathcal{O}_{\mathbb P^3}^{2} \longrightarrow \mathcal F^*(1) \to 0,$$ where $\mathcal{F}^*:=\mathscr{Hom}(\mathcal{F}, \, \mathcal{O}_{\mathbb P^3})$.

Now the sections of $\mathcal{F}^*(1)$ should correspond the other ruling of the quadric, because $\mathcal{F} \otimes \mathcal{F}^*(1)= \mathcal{O}_Y(1)$. Therefore the global sections of $\mathcal{F}^*(1)$ are given in terms of the rows of $N$, that are the columns of ${}^t N$.

Is this correct? If so, is there any intrinsec or general way to see this?

This is meant to be an integration to Yusuf answer.

Consider a section $$\stackrel{\rightarrow}{\alpha}= \begin{pmatrix} \alpha_1\\ \vdots\\ \alpha_r \end{pmatrix} \in {\mathbb C}^r\cong H^0({\mathbb P}^n,{\mathcal O}^r_{{\mathbb P}^n}).$$ Its image on $H^0(Y,{\mathcal F})$ vanishes in a point $p \in Y$, by definition of quotient, if and only it is in the image of the map induced by $N$ on the stalks over $p$. Then, by the Kronecker-Rouché-Capelli's Theorem, its zero locus equals the set of points where the rank of the matrix $N$ equals the rank of the $r \times (r+1)$ matrix $(N \stackrel{\rightarrow}{\alpha})$.

In your toy model, in all points of $Y$ the rank of $N$ is $1$, so we are looking to the locus where $$rank \begin{pmatrix} x_0&x_1&\alpha_1\\ x_2&x_3&\alpha_2\\ \end{pmatrix}=1$$ that is $\{\alpha_1x_2-\alpha_2x_0=\alpha_1x_3-\alpha_2x_1=0\}$.

This is as in your local computation indeed one of the rulings (but the other one, if my computations are not wrong).

A similar computation in the general case, as pointed out by Yusuf, produces equations involving the $(r-1) \times (r-1)$ minors of $N$.

More precisely, we need the adjoint or cofactor matrix $N^*$ whose $(i,j)$ entry is $(-1)^{i+j}$ times the minor of $N_{i,j}$, the determinant of the $(r-1) \times (r-1)$ matrix obtained by $N$ deleting the $i^{th}$ row and the $j^{th}$ column. In your toy model $$N^*= \begin{pmatrix} x_3&-x_2\\ -x_1&x_0\\ \end{pmatrix}$$

Then the zero locus of the image of $\stackrel{\rightarrow}{\alpha}$ is $$^tN^* \stackrel{\rightarrow}{\alpha}=0$$

The quadric surface example you mention extends to the general case as follows: to each nonzero vector $\vec{\alpha} \in \mathbb{C}^{r} \cong H^{0}(\mathcal{O}_{\mathbb{P}^{n}}^{r})$ we can associate a hyperplane $H_{\vec{\alpha}} \subset H^{0}(\mathcal{O}_{\mathbb{P}^{n}}^{r})^{\ast}$, and we can look at the map from $H_{\vec{\alpha}}$ to $H^{0}(\mathcal{O}_{\mathbb{P}^{n}}(1))^{\oplus r}$ induced by $N^{T}.$ The zero set of the associated section $\sigma(\vec{\alpha}) \in H^{0}(Y,\mathcal{F})$ is the zero set of maximal minors of an $r \times (r-1)$ matrix with coefficients in $H^{0}(\mathcal{O}_{\mathbb{P}^{n}}(1))$ representing this map.