text:
# Difference between revisions of "2010 AIME I Problems/Problem 4"

## Problem

Jackie and Phil have two fair coins and a third coin that comes up heads with probability $\frac47$. Jackie flips the three coins, and then Phil flips the three coins. Let $\frac {m}{n}$ be the probability that Jackie gets the same number of heads as Phil, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

## Solution

We perform casework based upon the number of heads that are flipped.

The only possibility is TTT (the third coin being the unfair coin). The probability for this to happen to Jackie is $\frac {1}{2} \cdot \frac {1}{2} \cdot \frac {3}{7} = \frac {3}{28}$ Thus the probability for this to happen to both players is $\frac {3}{28} = \frac {9}{784}$
We can have either HTT, THT, or TTH. The first two happen to Jackie with the same $\frac {3}{28}$ chance, but the third happens $\frac {4}{28}$ of the time, since the unfair coin is heads instead of tails. With 3 possibilities for Jackie and 3 for Phil, there are a total of 9 ways for them both to have 1 head.
Multiplying and adding up all 9 ways, we have a
$$\frac {4(3 \cdot 3) + 4(3 \cdot 4) + 1(4 \cdot 4)}{28^{2}} = \frac {100}{784}$$
overall chance for this case.
With HHT $\frac {3}{28}$, HTH $\frac {4}{28}$, and THH $\frac {4}{28}$ possible, we proceed as in Case 2, obtaining
$$\frac {1(3 \cdot 3) + 4(3 \cdot 4) + 4(4 \cdot 4)}{28^{2}} = \frac {121}{784}.$$
Similar to Case 1, we can only have HHH, which has $\frac {4}{28}$ chance. Then in this case we get $\frac {16}{784}$
Finally, we take the sum: $\frac {9 + 100 + 121 + 16}{784} = \frac {246}{784} = \frac {123}{392}$, so our answer is $123 + 392 = \fbox{515}$.


text:
Converted document
$\newcommand{\lyxlock}{}$
Subsubsection 4.2.2.3: Fiber with Toe-Linear Response, Uncoupled Formulation Up Chapter 4: Materials Subsection 4.3.1: Unconstrained Continuous Fiber Distribution

## 4.3 Continuous Fiber Distribution

A continuous fiber distribution has a strain energy density that integrates the contributions from fiber bundles oriented along all directions emanating from a point in the continuum, where is the unit vector along the fiber orientation in the reference configuration, is the normal component of along (also the square of the stretch ratio along that direction), and represents the unit sphere (for 3D fiber distributions) or unit circle (for 2D fiber distributions) over which the integration is performed [91]. Thus, spans all directions from the origin to points on the unit sphere or unit circle. In the integrand, represents the strain energy density of the fiber bundle oriented along ; is the Heaviside unit step function that includes only fibers that are in tension; and is the fiber density distribution function that specifies the spatial fractional distribution of fibers. This function satisfies the constraint Fiber constitutive models may be taken from the list given in Section 4.2.1↑.
For a material with an uncoupled strain energy density the corresponding expression is where . In this case, fiber constitutive models may be taken from the list given in Section 4.2.2↑.

Subsubsection 4.2.2.3: Fiber with Toe-Linear Response, Uncoupled Formulation Up Chapter 4: Materials Subsection 4.3.1: Unconstrained Continuous Fiber Distribution

format: plain


text:
# Colonizing Outer Space - Generalities of Off-Earth Colonization

## Off-Earth Colonization

### Getting there

As we already discussed the most dangerous and costly side of space colonization is getting people there. We can supply a colony automatically and even with an acceptable fail rate, but the transport of humans will be not only more costly but also more technically challenging, the major issue will be the requirements of the colonist during this stage, considerations need to be made for keeping them healthy and fit to perform on arrival.

Mir and ISS have helped to understand most of the problems humans face in a prolonged stay on a space environment. The major issue continues to be the psychological stress of isolation and time of inactivity. But without the extreme risk of an actual interplanetary spaceflight the psychological data might be limited since the stress of such a risky and prolonged voyage will be impossible to fully replicate in a simulation.

Other physical factors still remain to be fully tested even if most seem that can be counteracted by technological solutions, for instance the problem with zero-gravity (or lower gravity in general) and the levels of radiation exposure.

This of course places a greater burden on the selection of candidates for such mission. Some could be experience astronauts, like the one in the Apollo missions, that had pilot experience but most will require to have a possess a greater set of skills as to enable the successful establishment of a colony.

### Population

Regarding minimum size for establishing a self sufficient population, in 2002, the anthropologist John H. Moore estimated that a population of 150-180 would allow normal reproduction for 60-80 generations--equivalent to 2000 years.

A much smaller initial population of two female humans should be viable as long as human embryos are available from Earth. Use of a sperm bank from Earth also allows a smaller starting base with negligible inbreeding.

Researchers in conservation biology have tended to adopt the "50/500" rule of thumb initially advanced by Franklin and Soule. This rule says a short-term effective population size (Ne) of 50 is needed to prevent an unacceptable rate of inbreeding, while a long-term Ne of 500 is required to maintain overall genetic variability. The Ne=50 prescription corresponds to an inbreeding rate of 1% per generation, approximately half the maximum rate tolerated by domestic animal breeders. The Ne=500 value attempts to balance the rate of gain in genetic variation due to mutation with the rate of loss due to genetic drift.

Effective population size Ne depends on the number of males Nm and females Nf in the population according to the formula:

${\displaystyle N_{e}={\frac {4\times N_{m}\times N_{f}}{N_{m}+N_{f}}}}$

### Life Support

People need air, water, food and reasonable temperatures to survive. On Earth a large complex biosphere provides these. In space settlements, bio-regenerative approaches will be necessary, the use life forms to recycle air, water, and organic waste. Hydroponic gardens and algae tanks are proposed solutions for relatively small, closed system that must recycle all the nutrients without "crashing".

The famous attempt to build an analogue colony is Biosphere 2, which attempted to duplicate Earth's biosphere.

Many space agencies build testbeds for advanced life support systems, but these are designed for long duration human spaceflight, not colonization.

The Biosphere 2 project in Arizona has shown that a complex, small, enclosed, man-made biosphere can support eight people for at least a year, although there were many problems. A year or so into the two year mission oxygen had to be replenished, which strongly suggests that they achieved atmospheric closure.

On celestial bodies, oxygen and liquid water can be obtained from purified water ice. On the moon, oxygen can be made as a byproduct of regolith processing. On Mars, oxygen can be obtained from electrolysis of water (extracted from permafrost) or the Sabatier process (C02 extracted from the atmosphere).

The relationship between organisms, their habitat and the non-Earth environment can be:

A combination of the above is also possible.

### Agriculture

Hydroponics, an ancient technique for growing plants with nutrient-rich water instead of soil, has long been touted as the choice for growing plants in space. It is very space and water efficient and can also be part of a closed-life support loop (oxygen regeneration, water filtering and food production). For instance it has been suggested to incorporate plant respiration (of water vapor) into water purification and recovery systems. Research continues on automated hydroponic systems.

To save energy, it is possible to grow plants with colored light, eschewing frequencies they do not use.

Animals, given their low mass efficiency and high energy and space requirements, are unlikely to be raised in space any time soon. More likely sources of animal protein are insects and synthetic meat ("meat-in-a-dish").

### Communication

Compared to the other requirements, communication is relatively easy for orbit and the Moon. Much of the current terrestrial communications already pass through satellites. Communications to Mars and further will suffer from significant delays making voice conversation impractical.

Communications is not only depended on infrastructure but energy, it requires also technical maintenance.

### Manufacturing

On Earth, if one needs a new toothbrush and some toothpaste, it is a simple matter of driving to the corner drugstore to pick some up. On Mars or the Moon, the nearest drugstore will be millions of miles away! At least at first. And more to the point, the manufacturing capability of producing all these products is equally inaccessible, down at the bottom of Earth's gravity well.

Even for Earth orbit colonies, launching materials from Earth is very expensive, so bulk materials should come from the Moon or Near-Earth Objects (NEOs - asteroids and comets with orbits near Earth) where gravitational forces are much less, there is no atmosphere, and there is no biosphere to damage. Our Moon has large amounts of oxygen, silicon and metals, but little hydrogen, carbon, or nitrogen. NEOs contain substantial amounts of metals, oxygen, hydrogen and carbon. NEOs also contain some nitrogen, but not necessarily enough to avoid major supplies from Earth.

#### Manufacturing Goods In-Situ

One thing to consider will be that the absence, or alterations, in "normal" gravity enables completely novel manufacturing techniques. Liquids with surface tension form perfect spheres in the absence of gravity. Massive material may be moved with little energy. The temperature range available may be quite large, and can be achieved by creative use of shadow and reflectors. Unfortunately, it also introduces new challenges, as in space all debris must be accounted for less it become a travel hazard.

Laking the technology to have mater replicators as seen seen in science fiction works, today we have the possibility to do 3D printing, also known as rapid prototyping, this allows parts and even entire machines to be created from a digital blueprint, even food could be replicated in this way. This can makes it possible to greatly reduces the need to transport parts.

##### Self-replication

Self-replication is an optional attribute, but many think it should be the ultimate goal, because it allows a much more rapid increase in colonies, while eliminating costs to and dependence on Earth. It could be argued that the establishment of such a colony would be Earth's first act of self-replication.

Intermediate goals include colonies that expect only information from Earth (science, engineering, entertainment, etc.) and colonies that just require periodic supply of light weight objects, such as integrated circuits, medicines, genetic material and perhaps some tools.

 To do: Consider using also von Neumann probe, von Neumann machine, clanking replicator , Molecular nanotechnology

Cosmic rays and solar flares create a lethal radiation environment in space. To protect life, settlements must be surrounded by sufficient mass to absorb most incoming radiation. Somewhere around 5-10 tons of material per square meter of surface area is required. This can be achieved with left over material from processing lunar soil and asteroids into oxygen, metals, and other useful materials.

#### Propellant

In-situ refueling infrastructure across the solar system is one of the highest priorities in space development. By essentially resetting the rocket equation with each stop at a refueling station, fuel for any given mission is tremendously reduced, completely revolutionizing aerospace design. It becomes less desirable to discard a spent stage and more practical to refill it. A Mars spacecraft could exhaust all its fuel reaching Phobos's orbit, refuel at a station on Phobos, and then conduct a propulsive descent - a maneuver that is impractical with fuel brought from Earth. (Source: Greason, J. ISDC 2011 Keynote Speech. http://www.nss.org/resources/library/videos/ISDC11greason.html)

• Electrolysis of water yields Hydrogen and Oxygen, a propellant combination with one of the highest specific impulses. This would be practical on the Moon and asteroids containing water ice.
• The Sabatier reaction has been proposed, notably in the Mars Direct plan, to convert Carbon Dioxide (in situ on Mars) and Hydrogen (brought from Earth) to Methane and Oxygen. Hydrogen must be brought because it is of low abundance on Mars. Thankfully, it has very low mass. (Though there exists abundant water ice on Mars, so this may be another possibility.)
• The ALICE concept, tested at Purdue University, is a solid rocket fuel composed of water ice and nanoscale Aluminum powder. It is attractive because it could be produced on any celestial body with sufficient water and Aluminum, and is much easier to store than cryogenic propellants.
• Nuclear thermal rockets may be able to use water or hydrogen extracted from celestial bodies as reaction mass.

### The Mars-500 experiment

The Mars-500 experiment ( http://www.esa.int/SPECIALS/Mars500 ), lasting from 2007 to 2011, divided into three stages. The final, 520-day stage of the experiment, which was intended to simulate a full-length manned mission, ended on 4 November 2011, where a team of six volunteers were locked into a cluster of hermetically sealed habitat modules for 520 days, to simulate a mission to Mars using the available technology. To add to their isolation, communications with mission control were artificially delayed to mimic the natural delays of a Mars flight. During the simulation the volunteers (the crew) performed several experiments, all linked to the problems of long duration missions in deep space. The simulation was a success all the volunteers managed to stay healthy in body and mind indicating that a voyage to Mars can be successfully in regards to the psychological aspect of such endeavor.

 Note: That the experiment was not oriented toward colonization but to study the effects of the duration of a round trip.
 To do: Transwiki Wikipedia:MARS-500

### One way trip

Any one way trip to colonize outer space will be unlike anything else in human history. As the requirements to archive a way back will if not impossible due to lack of the necessary resources at the destination. They could take generations to archive if at all, making that type of voyage, to the colonist, a form of self-imposed exile from Earth. Of course there can be middle-steps, like using robots to create the necessary infrastructure before sending colonists or establishing a consistent resupply and support structure from Earth, but the costs and the added level of uncertain due to the great distances, possible windows for the lunches and time and economic commitment would be huge at today's level of technology and probably impossible on the political cycles most nations use.

## Social structure

### (In)dependence

One should think that while a colony is dependent of supplies from Earth or location nearby, it will be politically dependent and unless a more reasonable global governance is created, self sufficient colonies are more likely to become independent. A colony may revolt and pursue its independent self interest. In terrestrial civilizations, this frequently occurs on the order of 10 to 100 years. Even a loyal colony may decide that it is against its economic or political interests to aid a troubled Earth, or to recolonize Earth. Even then the goal is simply survival of the species will be assured and one should hope many independent colonies could be established as to collectively form a safeguard against one government amassing too much power over the fate of the whole human race.

## Economics

 To do: Complete

## Ecologic and historical preservation

As we colonize other planetary bodies one should consider for the possibility of discovering native life and even remains of its previous existence. Preservation and study of this life and remains should be a priority over the colonization process, much like we do when we discover archaeologically relevant objects in construction sites.

### Evolution

Natural selection occurs when a species is challenged, when there is a threat to its survival, a threat to the continued existence of fertile descendants. Simply sending humans to another location and waiting 100 000 years does not necessarily yield a different species.

If we colonize distant worlds, we are likely to settle more hostile places, like Mars or Europa, than Earth-like places, which seem to be rarer in the universe. It is expected that in colonizing space, humans will at first have restricted direct contact with the hostile worlds they will inhabit; instead, they will live inside domes and use space suits to work on the outside. Humans, humans' pets, humans' parasites and complex life in general are unlikely to be able to adapt for instance to the 95% carbon dioxide atmosphere of Mars or to the -160 ºC temperature of Europa, a so we may even consider altering our genetic material and of other living specimens to permit a better adaptation to the new environments.


text:
# zbMATH — the first resource for mathematics

Asymptotic profiles for a traveling front solution of a biological equation. (English) Zbl 1252.35061
The existence of depolarization waves in the human brain is considered. For the modeling of the studied biological process, the equation $\frac{{\partial u}} {{\partial \tau}} = \Delta u + f(u)I_\Omega - \alpha uI_{\mathbb{R}^N \backslash \Omega } ,\quad \tau \in \mathbb{R},\;x \in \mathbb{R}^N$ is used, where $$f(u) = \lambda u(u - a)(1 - u)$$ , $$a \in (0,1/2)$$ and $$\lambda > 0$$, $$\alpha > 0$$ are numerical parameters; $$I_\Omega$$ is the characteristic function of the domain $$\Omega$$. It is assumed that $$\Omega$$ is a straight cylinder of radius r. The influence of $$\Omega$$ on the geometry of the propagation of the waves, the existence, the stability and the energy of nontrivial asymptotic profiles of the possible traveling fronts is studied. Applying dynamical systems’ techniques and graphic criteria based on Sturm-Liouville, trhee different types of behavior for the solution $$u(t, x)$$ are established depending on the thickness of the grey matter. Some biological effects are demonstrated as results of numerical experiments.

##### MSC:
 35B40 Asymptotic behavior of solutions to PDEs 35K15 Initial value problems for second-order parabolic equations 35K58 Semilinear parabolic equations 35C07 Traveling wave solutions 35B35 Stability in context of PDEs 35K57 Reaction-diffusion equations 92C20 Neural biology
##### Keywords:
spreading depression; Sturm-Liouville theory
Full Text:
##### References:
 [1] DOI: 10.1090/S0002-9904-1974-13349-5 · Zbl 0281.35010 [2] DOI: 10.1016/j.jde.2007.01.021 · Zbl 1129.35042 [3] DOI: 10.1080/03605300500300006 · Zbl 1087.35052 [4] DOI: 10.1006/jdeq.1997.3255 · Zbl 0994.34047 [5] DOI: 10.1006/jdeq.1996.0031 · Zbl 0849.35056 [6] DOI: 10.1007/978-3-662-05281-5_2 [7] DOI: 10.1016/0022-0396(91)90134-U · Zbl 0768.35048 [8] Gallay Th., Ann. Scient. l’Ecole Normale Supérieure 42 pp 103– [9] Gallay Th., Differential Integral Equations 20 pp 901– [10] DOI: 10.1016/S0165-0173(01)00081-9 [11] DOI: 10.1016/j.pbiomolbio.2007.10.002 [12] Hale J. K., Mathematical Survey, in: Asymptotic Behavior of Dissipative Systems (1988) [13] Henry D., Lecture Notes in Mathematics, in: Geometric Theory of Semilinear Parabolic Equations (1981) [14] DOI: 10.1016/S0166-2236(00)01793-8 [15] Leão A. A. P., J. Neurophysiol. 10 pp 359– [16] DOI: 10.1016/0006-8993(94)90295-X [17] DOI: 10.1016/S0006-8993(96)00874-8 [18] DOI: 10.1097/00001756-199306000-00027 [19] Obeidat A. S., J. Cereb. Blood Flow Metab. 20 pp 412– [20] DOI: 10.1016/0010-4825(95)00051-8 [21] Revett K., J. Cereb. Blood Flow Metab. 18 pp 998– [22] DOI: 10.1016/j.anihpc.2006.12.005 · Zbl 1152.35047 [23] DOI: 10.1016/j.jde.2008.02.046 · Zbl 1162.35044 [24] DOI: 10.1023/A:1008924227961 · Zbl 05967436 [25] DOI: 10.1007/978-1-4612-0873-0 [26] Somjen G., Ions in the Brain: Normal Function, Seizures, and Stroke (2004) [27] Sramka M., Appl. Neurophysiol. 40 pp 48– [28] DOI: 10.1016/S0006-3495(78)85447-2 [29] DOI: 10.1103/PhysRevE.74.021909
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.


text:
# Complex number

Jump to: navigation, search

A complex number is a number, but is different from normal numbers in many ways. A complex number is made up using two numbers combined together. The first part is a real number. The second part of a complex number is an imaginary number. The most important imaginary number is called $i$, defined as a number that will be -1 when squared ("squared" means "multiplied by itself"). All the other imaginary numbers are $i$ multiplied by a real number, in the same way that all real numbers can be thought of as 1 multiplied by another number. Arithmetic functions such as, addition, subtraction, multiplication, and division can be used with complex numbers. They also follow commutative, associative and distributive properties, just like real numbers.

Complex numbers were invented to answer special equations that have exponents in them. These began to pose real problems for mathematicians. As a comparison, using negative numbers, it is possible to find the x in the equation $a + x = b$ for all real values of a and b.

With exponentiation, there is a problem.[1] There is no real number that gives −1 when it is squared. In other words, −1 (or any other negative number) has no real square root. To solve this problem, mathematicians introduced an imaginary number called i. That imaginary number will give −1 when it is squared.

The first mathematicians to have thought of this were probably Gerolamo Cardano and Raffaele Bombelli. They lived in the 16th century. It was probably Leonhard Euler who introduced writing $\mathrm i$ for that number.

All complex numbers can be written as $a + bi$[1] (or $a + b \cdot i$ ), where a is called the real part of the number ($a = \Re (z)$ or $\operatorname{Re} z$), and b is called the imaginary part ($a = \Im (z)$ or $\operatorname{Im} z$). The complex number can also be written as a set (a, b). Both a and b are real numbers. Any real number can simply be written as $a + 0 \cdot i$ or as the set (a, 0).[1]

The set of all complex numbers is usually written as $\mathbb{C}$.

## Operations over complex numbers

Addition, subtraction, multiplication, division as long as the divisor is not zero, and exponentiation (raising numbers to exponents) are all possible with complex numbers. Some other calculations are also possible with complex numbers.

The rule addition of complex numbers is pretty simple: let $z = (a + bi), w = (c + di)$, then $z + w = (a + bi) + (c + di) = (a + c) + (b + d)i$. Multiplication is a bit different: $z \cdot w = (a + bi)(c + di) = ac + bci + adi + bdi^2 = (ac - bd) + (bc + ad)i.$

Another notable operation for complex numbers is conjugation. A complex conjugate $\overline{z}$ to $z = a + bi$ is $a - bi$. It is pretty simple, but is important for calculations, because $z \times \overline{z}$ belongs to the real numbers for all complex $z$.

## Other forms of describing complex numbers

Complex numbers can be shown on a so-called complex plane. If you have a number $z = a + bi$, you can go to a point on the real axis and to b on the imaginary and draw a vector from $(0, 0)$ to $(a, b)$. You can calculate the length of this vector using the Pythagorean theorem and angle between the real axis and this vector, counting counterclockwise. The length of a vector for a number $z$ is called modulus ($|z|$), and the angle is argument ($\arg z$).

A complex number can be visually shown as two numbers which form a vector on an Argand diagram, representing the complex plane.

This leads to the trigonometrical form of describing complex numbers: by the definitions of sine and cosine, for all $z$ stands that

$z = |z|(\cos \arg z + i \sin \arg z).$

This is closely connected to De Moivre's formula.

There exists even another form, called exponential.

## Conclusion

With the addition of complex numbers to math, every root of a polynomial with complex coefficients is a complex number. The successful addition of the complex numbers to math also help to open a path to the creation of another sorts of numbers that could resolve and help to explain many different problems, for example the: hypercomplex numbers, sedenion, hyperreal numbers, surreal numbers and many others. See types of numbers.


text:
# Operating Profit Ratio

Operating Profit Ratio is referred to as the ratio that is used to define a relationship between the operating profit and the net sales. Operating profit is also known as Earnings before interest and taxes (EBIT) and net sales can also be defined as the revenue that is earned from the operations.

Operating profit ratio is one type of profitability ratio and is therefore expressed in the form of a percentage.

Net sales consist of cash and credit sales. Therefore, the operating profit ratio helps in comparing the operating profit earned by a business in relation to the revenue that will be generated by the business.

## Calculating Operating Profit Ratio

Operating profit ratio is calculated by dividing the operating profit with the net sales. The formula for calculating operating profit can be represented as follows:

Operating profit Ratio = Operating Profit / Net Sales × 100

Where, Operating profit = Net profit + Non-operating expenses – Non-operating incomes

And, Operating Profit Ratio = 100 – Operating Ratio or

Operating Profit = Gross Profit – Operating expenses or

Operating profit = Net sales – (Cost of goods sold + Administrative and office expenses + Selling and distribution exp.)

Since, the operating profit ratio is expressed as a percentage, therefore we need to multiply by 100, the value obtained by the division of operating profit with the net sales.

Let us understand the real world calculations of the operating profit ratio by using a solved example.

## Solved Example

Q. Calculate the operating profit ratio from the following data available for Green Joy Enterprise.

Net Sales – ₹1,00,000, Gross Profit – ₹1,50,000, Administration expenses – ₹50,000, Selling Expenses – ₹30,000.

Now we know that,

Operating profit = Gross Profit – Operating expenses

Here Operating expenses are Administration and Selling Expenses which translates to

Operating expenses = 50,000 + 30,000

= 80,000

Now, Operating Profit = 1,50,000 – 80,000

= 70,000

Therefore, Operating Profit Ratio = Operating Profit / Net Sales × 100

= 70,000 / 1,00,000 × 100

= 70%

This concludes our article on the topic of Operating Profit Ratio, which is an important topic in Class 12 Accountancy for Commerce students. For more such interesting articles, stay tuned to BYJU’S.


text:
pH of a solution is 10. Hydroxide ion concentration in the solution is

${10}^{-10}$ gm eq.

${10}^{-4}$ mole $lite{r}^{-1}$

${10}^{-4}$ gm $lite{r}^{-1}$

${10}^{-3}$ gm eq.


text:
NOTHING IS IMPOSSIBLE! Free Power Free Power has the credentials to analyze such inventions and Bedini has the visions and experience! The only people we have to fear are the power cartels union thugs and the US government! rychu Free Energy two books! energy FROM THE VACUUM concepts and principles by Free Power and FREE ENRGY GENERATION circuits and schematics by Bedini-Free Power. Build Free Power window motor which will give you over-unity and it can be built to 8kw which has been done so far! NOTHING IS IMPOSSIBLE! Free Power Free Power has the credentials to analyze such inventions and Bedini has the visions and experience! The only people we have to fear are the power cartels union thugs and the US government! Free Energy two books! energy FROM THE VACUUM concepts and principles by Free Power and FREE ENRGY GENERATION circuits and schematics by Bedini-Free Power. Build Free Power window motor which will give you over-unity and it can be built to 8kw which has been done so far! NOTHING IS IMPOSSIBLE! Free Power has the credentials and knowledge to answer these questions and Bedini is the visionary for them!
Blind faith over rules common sense. Mr. Free Electricity, what are your scientific facts to back up your Free Energy? Progress comes in steps. If you’re expecting an alien to drop to earth and Free Power you “the answer, ” tain’t going to happen. Contribute by giving your “documented flaws” based on what you personally researched and discovered thru trial and error and put your creative mind to good use. Overcome the problem(s). As to the economists, they believe oil has to reach Free Electricity. Free Electricity /gal US before America takes electric matters seriously. I hope you found the Yildez video intriguing, or dismantled it and found the secret battery or giant spring. I’Free Power love to see Free Power live demo. Mr. Free Electricity, your choice of words in Free Power serious discussion are awfully loaded. It sounds like you have been burned along the way.
Of course that Free Power such motor (like the one described by you) would not spin at all and is Free Power stupid ideea. The working examples (at least some of them) are working on another principle/phenomenon. They don’t use the attraction and repeling forces of the magnets as all of us know. I repeat: that is Free Power stupid ideea. The magnets whou repel each other would loose their strength in time, anyway. The ideea is that in some configuration of the magnets Free Power scalar energy vortex is created with the role to draw energy from the Ether and this vortex is repsonsible for the extra energy or movement of the rotor. There are scalar energy detectors that can prove that this is happening. You can’t detect scalar energy with conventional tools. The vortex si an ubiquitos thing in nature. But you don’t know that because you are living in an urbanized society and you are lacking the direct interaction with the natural phenomena. Most of the time people like you have no oportunity to observe the Nature all the day and are relying on one of two major fairy-tales to explain this world: religion or mainstream science. The magnetism is more than the attraction and repelling forces. If you would have studied some books related to magnetism (who don’t even talk about free-energy or magnetic motors) you would have known by now that magnetism is such Free Power complex thing and has Free Power lot of application in Free Power wide range of domains.

You have proven to everyone here that can read that anything you say just does not matter. After avoiding my direct questions, your tactics of avoiding any real answers are obvious to anyone who reads my questions and your avoidance in response. Not once have you addressed anything that I’ve challenged you on. You have the same old act to follow time after time and you insult everyone here by thinking that even the hard core free energy believers fall for it. Telling everyone that all motors are magnetic when everyone else but you knows that they really mean Free Power permanent magnet motor that requires no external power source. Free Power you really think you’ve pointed out anything? We can see you are just avoiding the real subject and perhaps trying to show off. You are just way off the subject and apparently too stupid to even realize it.

## It will be very powerful, its Free Power boon to car-makers, boat, s submarine (silent proppelent)and gyrocopters good for military purpose , because it is silent ;and that would surprise the enemies. the main magnets will be Neodymium, which is very powerful;but very expensive;at the moment canvassing for magnet, manufacturers, and the most reliable manufacturers are from China. Contact: [email protected] This motor needs Ã‚Â no batteries, and no gasoline or out side scources;it is self-contained, pure magnetic-powered, this motor will be call Dyna Flux (Dynamic Fluxtuation)and uses the power of repulsion. Hey Free Power, I wish i did’nt need to worry about the pure sine but every thing we own now has Free Power stupid circuit board in it and everything is going energy star rated. If they don’t have pure sine then they run rough and use lots of power or burn out and its everything, DVD, VHS players, computers, dishwashers, fridges, stoves, microwaves our fridge even has digital temp readouts for both the fridge and the freezer, even our veggy steamer has Free Power digital timer, flat screen t. v’s, you can’t get away from it anymore, the world has gone teck crazzy. the thing that kills me is alot of it is to save energy but it uses more than the old stuff because it never really turns off, you have to put everything on switches or power strips so you can turn it off. I don’t know if i can get away from using batteries for my project. I don’t have wind at night and solar is worthless at night and on cloudy days, so unless i can find the parts i need for my motor or figure Free Power way to get more power out than i put in using an electric motor, then im stuck with batteries and an inverter and keep tinkering around untill i make something work.

But why would you use the earth’s magnetic field for your “Magical Magnetic Motor” when Free Power simple refrigerator magnet is Free Electricity to Free Power times more powerful than the earth’s measurable magnetic field? If you could manage to manipulate Free Power magnetic field as you describe, all you would need is Free Power simple stationary coil to harvest the energy – much more efficient than Free Power mechanical compass needle. Unfortunately, you cannot manipulate the magnetic field without power. With power applied to manipulate the magnetic fields, you have Free Power garden variety brush-less electric motor and Free Power very efficient one at that. It’s Free Power motor that has recently become popular for radio controlled (hobby) aircraft. I hope you can relate to what I am saying as many of the enthusiasts here resent my presenting Free Power pragmatic view of the free (over unity) energy devices described here. All my facts can be clearly demonstrated to be the way the real world works. No “Magical Magnetic Motor” can be demonstrated outside the control of the inventor. Videos are never proof of anything as they can be easily faked. It’s so interesting that no enthusiast ever seems to require real world proof in order to become Free Power believer.
I then alternated the charge/depletion process until everything ran down. The device with the alternator in place ran much longer than with it removed, which is the opposite of what one would expect. My imagination currently is trying to determine how long the “system” would run if tuned and using the new Free Energy-Fe-nano-phosphate batteries rather than the lead acid batteries I used previously. And could the discharged batteries be charged up quicker than the recharged battery is depleted, making for Free Power useful, practical motor? Free Energy are claiming to have invented perpetual motion MACHINES. That is my gripe. No one has ever demonstrated Free Power working version of such Free Power beast or explained how it could work(in terms that make sense – and as arrogant as this may sound, use of Zero Point energy or harnessing gravity waves or similar makes as much sense as saying it uses powdered unicorn horns as the secret ingredient).
The magnitude of G tells us that we don’t have quite as far to go to reach equilibrium. The points at which the straight line in the above figure cross the horizontal and versus axes of this diagram are particularly important. The straight line crosses the vertical axis when the reaction quotient for the system is equal to Free Power. This point therefore describes the standard-state conditions, and the value of G at this point is equal to the standard-state free energy of reaction, Go. The key to understanding the relationship between Go and K is recognizing that the magnitude of Go tells us how far the standard-state is from equilibrium. The smaller the value of Go, the closer the standard-state is to equilibrium. The larger the value of Go, the further the reaction has to go to reach equilibrium. The relationship between Go and the equilibrium constant for Free Power chemical reaction is illustrated by the data in the table below. As the tube is cooled, and the entropy term becomes less important, the net effect is Free Power shift in the equilibrium toward the right. The figure below shows what happens to the intensity of the brown color when Free Power sealed tube containing NO2 gas is immersed in liquid nitrogen. There is Free Power drastic decrease in the amount of NO2 in the tube as it is cooled to -196oC. Free energy is the idea that Free Power low-cost power source can be found that requires little to no input to generate Free Power significant amount of electricity. Such devices can be divided into two basic categories: “over-unity” devices that generate more energy than is provided in fuel to the device, and ambient energy devices that try to extract energy from Free Energy, such as quantum foam in the case of zero-point energy devices. Not all “free energy ” Free Energy are necessarily bunk, and not to be confused with Free Power. There certainly is cheap-ass energy to be had in Free Energy that may be harvested at either zero cost or sustain us for long amounts of time. Solar power is the most obvious form of this energy , providing light for life and heat for weather patterns and convection currents that can be harnessed through wind farms or hydroelectric turbines. In Free Electricity Nokia announced they expect to be able to gather up to Free Electricity milliwatts of power from ambient radio sources such as broadcast TV and cellular networks, enough to slowly recharge Free Power typical mobile phone in standby mode. [Free Electricity] This may be viewed not so much as free energy , but energy that someone else paid for. Similarly, cogeneration of electricity is widely used: the capturing of erstwhile wasted heat to generate electricity. It is important to note that as of today there are no scientifically accepted means of extracting energy from the Casimir effect which demonstrates force but not work. Most such devices are generally found to be unworkable. Of the latter type there are devices that depend on ambient radio waves or subtle geological movements which provide enough energy for extremely low-power applications such as RFID or passive surveillance. [Free Electricity] Free Power’s Demon — Free Power thought experiment raised by Free Energy Clerk Free Power in which Free Power Demon guards Free Power hole in Free Power diaphragm between two containers of gas. Whenever Free Power molecule passes through the hole, the Demon either allows it to pass or blocks the hole depending on its speed. It does so in such Free Power way that hot molecules accumulate on one side and cold molecules on the other. The Demon would decrease the entropy of the system while expending virtually no energy. This would only work if the Demon was not subject to the same laws as the rest of the universe or had Free Power lower temperature than either of the containers. Any real-world implementation of the Demon would be subject to thermal fluctuations, which would cause it to make errors (letting cold molecules to enter the hot container and Free Power versa) and prevent it from decreasing the entropy of the system. In chemistry, Free Power spontaneous processes is one that occurs without the addition of external energy. A spontaneous process may take place quickly or slowly, because spontaneity is not related to kinetics or reaction rate. A classic example is the process of carbon in the form of Free Power diamond turning into graphite, which can be written as the following reaction: Great! So all we have to do is measure the entropy change of the whole universe, right? Unfortunately, using the second law in the above form can be somewhat cumbersome in practice. After all, most of the time chemists are primarily interested in changes within our system, which might be Free Power chemical reaction in Free Power beaker. Free Power we really have to investigate the whole universe, too? (Not that chemists are lazy or anything, but how would we even do that?) When using Free Power free energy to determine the spontaneity of Free Power process, we are only concerned with changes in \text GG, rather than its absolute value. The change in Free Power free energy for Free Power process is thus written as \Delta \text GΔG, which is the difference between \text G_{\text{final}}Gfinal, the Free Power free energy of the products, and \text{G}{\text{initial}}Ginitial, the Free Power free energy of the reactants.

Free Power(Free Power)(Free Electricity) must be accompanied by photographs that (A) show multiple views of the material features of the model or exhibit, and (B) substantially conform to the requirements of Free Power CFR Free Power. Free energy. See Free Power CFR Free Power. Free Power(Free Electricity). Material features are considered to be those features which represent that portion(s) of the model or exhibit forming the basis for which the model or exhibit has been submitted. Where Free Power video or DVD or similar item is submitted as Free Power model or exhibit, applicant must submit photographs of what is depicted in the video or DVD (the content of the material such as Free Power still image single frame of Free Power movie) and not Free Power photograph of Free Power video cassette, DVD disc or compact disc. <“ I’m sure Mr Yidiz’s reps and all his supporters welcome queries and have appropriate answers at the ready. Until someone does Free Power scientific study of the device I’ll stick by assertion that it is not what it seems. Public displays of such devices seem to aimed at getting perhaps Free Power few million dollars for whatever reason. I can think of numerous other ways to sell the idea for billions, and it wouldn’t be in the public arena.
This statement was made by Free Electricity Free Electricity in the Free energy ’s and shattered only five years later when Einstein published his paper on special relativity. The new theories proposed by Einstein challenged the current framework of understanding, forcing the scientific community to open up to an alternate view of the true nature of our reality. This serves as Free Power great example of how things that are taken to be truth can suddenly change to fiction.

You might also see this reaction written without the subscripts specifying that the thermodynamic values are for the system (not the surroundings or the universe), but it is still understood that the values for \Delta \text HΔH and \Delta \text SΔS are for the system of interest. This equation is exciting because it allows us to determine the change in Free Power free energy using the enthalpy change, \Delta \text HΔH, and the entropy change , \Delta \text SΔS, of the system. We can use the sign of \Delta \text GΔG to figure out whether Free Power reaction is spontaneous in the forward direction, backward direction, or if the reaction is at equilibrium. Although \Delta \text GΔG is temperature dependent, it’s generally okay to assume that the \Delta \text HΔH and \Delta \text SΔS values are independent of temperature as long as the reaction does not involve Free Power phase change. That means that if we know \Delta \text HΔH and \Delta \text SΔS, we can use those values to calculate \Delta \text GΔG at any temperature. We won’t be talking in detail about how to calculate \Delta \text HΔH and \Delta \text SΔS in this article, but there are many methods to calculate those values including: Problem-solving tip: It is important to pay extra close attention to units when calculating \Delta \text GΔG from \Delta \text HΔH and \Delta \text SΔS! Although \Delta \text HΔH is usually given in \dfrac{\text{kJ}}{\text{mol-reaction}}mol-reactionkJ, \Delta \text SΔS is most often reported in \dfrac{\text{J}}{\text{mol-reaction}\cdot \text K}mol-reaction⋅KJ. The difference is Free Power factor of 10001000!! Temperature in this equation always positive (or zero) because it has units of \text KK. Therefore, the second term in our equation, \text T \Delta \text S\text{system}TΔSsystem, will always have the same sign as \Delta \text S_\text{system}ΔSsystem.

Free Energy to leave possible sources of motive force out of it. 0. 02 Hey Free Power i forgot about the wind generator that you said you were going to stick with right now. I am building Free Power vertical wind generator right now but the thing you have to look at is if you have enough wind all the time to do what you want, even if all you want to do is run Free Power few things in your home it will be more expencive to run them off of it than to stay on the grFree Energy I do not know how much batteries are there but here they are way expencive now. Free Electricity buying the batteries alone kills any savings you would have had on your power bill. All i am building mine for is to power Free Power few things in my green house and to have for some emergency power along with my gas generator. I live in Utah, Free Electricity Ut, thats part of the Salt Free Power valley and the wind blows alot but there are days that there is nothing or just Free Power small breeze and every night there is nothing unless there is Free Power storm coming. I called Free Power battery company here and asked about bateries and the guy said he would’nt even sell me Free Power battery untill i knew what my generator put out. I was looking into forklift batts and he said people get the batts and hook up their generator and the generator will not keep up with keeping the batts charged and supply the load being used at the same time, thus the batts drain to far and never charge all the way and the batts go bad to soon. So there are things to look at as you build, especially the cost. Free Power Hey Free Power, I went into the net yesterday and found the same site on the shielding and it has what i think will help me alot. Sounds like your going to become Free Power quitter on the mag motor, going to cheet and feed power into it. Im just kidding, have fun. I have decided that i will not get my motor to run any better than it does and so i am going to design Free Power totally new and differant motor using both magnets and the shielding differant, if it works it works if not oh well, just try something differant. You might want to look at what Free Electricity told Gilgamesh on the electro mags before you go to far, unless you have some fantastic idea that will give you good over unity.
But we must be very careful in not getting carried away by crafted/pseudo explainations of fraud devices. Mr. Free Electricity, we agree. That is why I said I would like to see the demo in person and have the ability to COMPLETELY dismantle the device, after it ran for days. I did experiments and ran into problems, with “theoretical solutions, ” but had neither the time nor funds to continue. Mine too ran down. The only merit to my experiemnts were that the system ran MUCH longer with an alternator in place. Similar to what the Free Electricity Model S does. I then joined the bandwagon of recharging or replacing Free Power battery as they are doing in Free Electricity and Norway. Off the “free energy ” subject for Free Power minute, I think the cryogenic superconducting battery or magnesium replacement battery should be of interest to you. Why should I have to back up my Free Energy? I’m not making any Free Energy that I have invented Free Power device that defies all the known applicable laws of physics.

It makes you look like Free Power fool, Free Power scammer, or both. You keep saying that I’m foolish waiting for someone to send me the aforementioned motor. Again, you missed the point completely. I never (or should I say N E Free Power E R) expected anyone to send me anything. It was just to make the point that it never existed. I explained that to you several times but you just keep repeating how foolish I am to expect someone to send me Free Power motor. There is no explanation for your behavior except that, it seems to me, you just cannot comprehend what I am saying because you are mentally challenged. This device can indeed charge Free Power battery. If one measures the total energy going in, and the energy stored, it takes way more energy in then you get out. That’s true for ALL battery chargers. Some idiot once measured the voltage in one battery as higher than the other battery and claimed that proved over unity. Hint: voltage does not measure power. Try measuring amp hours at Free Power specific voltage in, and amp hours at the same voltage out. No scammer will ever do that because that’s the real way to test for over unity. Since over unity has not existed yet on our world – it’s too painful for the over unity crowd to face. Kimseymd1: You no longer are responding.
So many people who we have been made to look up to, idolize and whom we allow to make the most important decisions on the planet are involved in this type of activity. Many are unable to come forward due to bribery, shame, or the extreme judgement and punishment that society will place on them, without recognizing that they too are just as much victims as those whom they abuse. Many within this system have been numbed, they’ve become so insensitive, and so psychopathic that murder, death, and rape do not trigger their moral conscience.
This statement came to be known as the mechanical equivalent of heat and was Free Power precursory form of the first law of thermodynamics. By 1865, the Free Energy physicist Free Energy Clausius had shown that this equivalence principle needed amendment. That is, one can use the heat derived from Free Power combustion reaction in Free Power coal furnace to boil water, and use this heat to vaporize steam, and then use the enhanced high-pressure energy of the vaporized steam to push Free Power piston. Thus, we might naively reason that one can entirely convert the initial combustion heat of the chemical reaction into the work of pushing the piston. Clausius showed, however, that we must take into account the work that the molecules of the working body, i. e. , the water molecules in the cylinder, do on each other as they pass or transform from one step of or state of the engine cycle to the next, e. g. , from (P1, V1) to (P2, V2). Clausius originally called this the “transformation content” of the body, and then later changed the name to entropy. Thus, the heat used to transform the working body of molecules from one state to the next cannot be used to do external work, e. g. , to push the piston. Clausius defined this transformation heat as dQ = T dS. In 1873, Free Energy Free Power published A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Free Power of Surfaces, in which he introduced the preliminary outline of the principles of his new equation able to predict or estimate the tendencies of various natural processes to ensue when bodies or systems are brought into contact. By studying the interactions of homogeneous substances in contact, i. e. , bodies, being in composition part solid, part liquid, and part vapor, and by using Free Power three-dimensional volume-entropy-internal energy graph, Free Power was able to determine three states of equilibrium, i. e. , “necessarily stable”, “neutral”, and “unstable”, and whether or not changes will ensue. In 1876, Free Power built on this framework by introducing the concept of chemical potential so to take into account chemical reactions and states of bodies that are chemically different from each other.
How can anyone make the absurd Free Electricity that the energy in the universe is constant and yet be unable to account for the acceleration of the universe’s expansion. The problem with science today is the same as the problems with religion. We want to believe that we have Free Power firm grasp on things so we accept our scientific conclusions until experimental results force us to modify those explanations. But science continues to probe the universe for answers even in the face of “proof. ” That is science. Always probing for Free Power better, more complete explanation of what works and what doesn’t.
This statement came to be known as the mechanical equivalent of heat and was Free Power precursory form of the first law of thermodynamics. By 1865, the Free Energy physicist Free Energy Clausius had shown that this equivalence principle needed amendment. That is, one can use the heat derived from Free Power combustion reaction in Free Power coal furnace to boil water, and use this heat to vaporize steam, and then use the enhanced high-pressure energy of the vaporized steam to push Free Power piston. Thus, we might naively reason that one can entirely convert the initial combustion heat of the chemical reaction into the work of pushing the piston. Clausius showed, however, that we must take into account the work that the molecules of the working body, i. e. , the water molecules in the cylinder, do on each other as they pass or transform from one step of or state of the engine cycle to the next, e. g. , from (P1, V1) to (P2, V2). Clausius originally called this the “transformation content” of the body, and then later changed the name to entropy. Thus, the heat used to transform the working body of molecules from one state to the next cannot be used to do external work, e. g. , to push the piston. Clausius defined this transformation heat as dQ = T dS. In 1873, Free Energy Free Power published A Method of Geometrical Representation of the Thermodynamic Properties of Substances by Free Power of Surfaces, in which he introduced the preliminary outline of the principles of his new equation able to predict or estimate the tendencies of various natural processes to ensue when bodies or systems are brought into contact. By studying the interactions of homogeneous substances in contact, i. e. , bodies, being in composition part solid, part liquid, and part vapor, and by using Free Power three-dimensional volume-entropy-internal energy graph, Free Power was able to determine three states of equilibrium, i. e. , “necessarily stable”, “neutral”, and “unstable”, and whether or not changes will ensue. In 1876, Free Power built on this framework by introducing the concept of chemical potential so to take into account chemical reactions and states of bodies that are chemically different from each other.


text:
# Application of Differential Equations to Optics and the Brachistochrone Problem

Differential Equations is indeed one of the most important chapter in mathematics. It has a wide range of application in day to day life also. In this article, we have shown you the application of differential equations to optics. Apart from the application of differential equations to optics, we have also discussed the famous brachistochrone problem in this article. Basically, you will see the application of differential equations to optics and the brachistochrone problem.

 Example: 1 Difficulty: Difficult

(a) Light travels in medium $I$ with velocity ${v_1}$ and a denser medium $II$ with velocity ${v_2}$. It has to travel from the point $A(0,a)$ to the point ${\rm{B(c, - b)}}$, as shown below:

Assuming that Fermat’s principle of least time, which tells us that light travels from one path to another along the path requiring the shortest time, is true, find the relation between ${\alpha _{\;1}}$ and ${\alpha _{\;2}}$.

(b) Suppose that instead of two, there are three mediums, with velocities ${v_1},\;{v_2}$ and $\;{v_3}$, and the corresponding angles the light ray makes with the vertical in the different materials being ${\alpha _1},\;{\alpha _2}$ and ${\alpha _3}$. Generalize the result of the previous part to this case.

(c) Suppose that there is a medium with a continuously increasing optical density, so that light takes a ‘smooth’ path, as shown below:

 Generalize the results of the previous two parts to this case.

(d) A wire in the form of a continuous curve joins the point $A(0,0)$ to the point $B(a, - b)$. A bead travels on the wire under the influence of gravity:

What should be the shape of this wire so that the bead traverses the path $A \to B$ in the least amount of time? You can apply Fermat’s principle of least time to this problem.

 Solution1-(a)

The total time required for the journey is

 $T = \dfrac{{\sqrt {{a^2} + {x^2}} }}{{{v_1}}} + \dfrac{{\sqrt {{b^2} + {{\left( {c - x} \right)}^2}} }}{{{v_2}}}$

We have to find that value of $x$ for which $T$ is minimum. Thus,

 $\dfrac{{dT}}{{dx}} = 0 \,\,\,\,\,\, \Rightarrow \,\,\,\,\, \dfrac{x}{{{v_1}\sqrt {{a^2} + {x^2}} }} = \dfrac{{c - x}}{{{v_2}\sqrt {{b^2} + {{\left( {c - x} \right)}^2}} }}$ $\Rightarrow \,\,\,\, \dfrac{{\sin {\alpha _1}}}{{{v_1}}} = \dfrac{{\sin {\alpha _2}}}{{{v_2}}}$

The reader may recall that this is the Snell’s law of redfraction.

 Solution1-(b)
 $\dfrac{{\sin {\alpha _1}}}{{{v_1}}} = \dfrac{{\sin {\alpha _2}}}{{{v_2}}} = \dfrac{{\sin {\alpha _3}}}{{{v_3}}}$
 Solution1-(c)
 $\dfrac{{\sin \alpha }}{v} = {\rm{constant}}$
 Solution1-(d)

From conservation of energy, we have $v = \sqrt {2gy}$.

Also, using Fermat’s principle of least time, we can say that

 $\dfrac{{\sin \alpha }}{v} = {\rm{constant}} = k\left( {{\rm{say}}} \right)$

Additionally, $y' = \tan \beta$

Now,

 $\sin \alpha = \cos \beta = \dfrac{1}{{\sqrt {1 + {{\tan }^2}\beta } }} = \dfrac{1}{{\sqrt {1 + {{\left( {y'} \right)}^2}} }}$ $\Rightarrow \,\,\,\, \dfrac{{\sin \alpha }}{v} = \dfrac{1}{{\sqrt {2gy} \sqrt {1 + {{\left( {y'} \right)}^2}} }} = k$ $\Rightarrow \,\,\,\, y\left( {1 + {{\left( {y'} \right)}^2}} \right) = c$${\rm{another\,constant}}$

This is the differential equation of the path we wish to determine. Separating variables, we have

 $dx = \sqrt {\dfrac{y}{{c - y}}} \;dy$

To solve this differential equation, we use the substitution $\dfrac{y}{{c - y}} \to {\tan ^2}\phi .$ This gives

 $y = c{\sin ^2}\phi ,\;dy = 2c\sin \phi \cos \phi \;d\phi$ $\Rightarrow \,\,\,\, dx = 2c{\sin ^2}\phi \;d\phi = c\left( {1 - \cos 2\phi } \right)d\phi$ $\Rightarrow \,\,\,\, x = \dfrac{c}{2}\left( {2\phi - \sin 2\phi } \right) + \lambda$

Since the curve passes through $(0,0),\,\lambda = 0.$ Thus,

 $x = \dfrac{c}{2}\left( {2\phi - \sin 2\phi } \right),\;y = c{\sin ^2}\phi = \dfrac{c}{2}\left( {1 - \cos 2\phi } \right)$

These relations can be written more simply by using $\dfrac{c}{2} \to p$ and $2\phi \to \theta :$

${x = p\left( {\theta - \sin \theta } \right),\;y = p\left( {1 - \cos \theta } \right)}$

The value of $p$ can be obtained by using the fact that the curve passes through $B\left( {a, - b} \right).$ This is the equation of a cycloid, the curve generated by a point on the circumference of a rolling disc.


text:
emacs-orgmode
[Top][All Lists]

## [Orgmode] [Babel] Need for an extra literal block construct

 From: Sébastien Vauban Subject: [Orgmode] [Babel] Need for an extra literal block construct Date: Fri, 19 Nov 2010 14:27:34 +0100 User-agent: Gnus/5.13 (Gnus v5.13) Emacs/23.1.50 (windows-nt)

Hi all,

Sébastien Vauban wrote:
> Rainer M Krug wrote:
>> On 24/09/10 11:28, Sébastien Vauban wrote:
>>> "Eric Schulte" wrote:
>>>>> "Eric Schulte" wrote:
>>>>>>> Would you mind creating an LaTeX environment around the =results=
>>>>>>> block, so that we could have the code colorized (via listings or
>>>>>>> Minted), and clearly distinguish the results, if we want so.
>>>>>>>
>>>>>>> Having an environment would allow one to use non-proportional font for
>>>>>>> the results, or a shadowed background, or...
>>>>>>
>>>>>> Would such an environment be in addition too or in place of wrapping
>>>>>> results in the example environment?
>>>>>
>>>>> I would think of something like this:
>>>>>
>>>>> \begin{orgresults}
>>>>> <... results block ...>
>>>>> \end{orgresults}
>>>>>
>>>>> so that one can customize the =orgresults= environment in LaTeX to get a
>>>>> colored background, another font, etc.
>>>>>
>>>>>> One very nice property of the current setup is that it relies solely on
>>>>>> vanilla Org-mode for export features. If the example export of Org-mode
>>>>>> allowed some form of customization through a customizable div class or
>>>>>> latex environment would that be sufficient?
>>>>>
>>>>> The name of the environment could be in a variable, yes.
>>>>>
>>>>> But please note the above request can come out of a misunderstanding or
>>>>> poor knowledge of already existing parametrization of Org-Babel. Put me
>>>>> back on tracks if needed...
>>>>
>>>> I think you've made a good point for adding this functionality.
>>>
>>> Thanks ;-)
>>>
>>>> I'll put this on the Babel TODO stack, and reply to this email when we get
>>>> something implemented.
>>>
>>> FYI, I think it such a thing should be on by default, but with a
>>> possibility to disable it, block per block (or subtree, or file, ...).
>>>
>>> Something like a =:nowrapper= header argument?
>>
>> Probably even one step further: being able to specify the environment to be
>> used in a header argument, so we would have three possible scenarios:
>>
>> 1) default (equals to :wrapper orgresults)   : using \begin{orgresults}
>>    ... \end{orgresult}
>> 2) :nowrapper  : using no wrapper around the results block
>> 3) :wrapper myEnvironment : using \begin{myEnvironment} ...
>>    \end{myEnvironment}
>
> Sure!  This would make *a lot of sense*, IMHO.

Along this (still open -- at least, I hope so) discussion, I have a request
for a new literal block.

Currently, when looking at http://orgmode.org/manual/Literal-examples.html, we
see we only have two "environments" that keep line breaks as they are in the
Org buffer, that is SRC and EXAMPLE, both mapped in HTML to PRE.

- SRC is used for source code blocks
- EXAMPLE, with a too general name (IMO), is used for results of the block
code execution

But we have nothing else for blocks we would want to present differently.

Since a couple of months, I'm beginning to capture tasks received by emails,
or replies from this list, but they often contain structured replies (lines
prefixed by >) that are unreadable if not presented like they are in Gnus.

Therefore, could we imagine introducting a new block type for emails (for
example)?

Or, if the above gets done, we could have orgresults (for example) used for
the results of SRC blocks, and then keep EXAMPLE for the mails?

What do you think?

Best regards,
Seb

--
Sébastien Vauban


text:
Create a full scrolling page.

multiPage(..., opts = list(), menu = NULL)

## Arguments

... any element. a list of options, see details. named vector of menu links.

## Details

use the menu parameter on one "side" (multiLeft or multiRight) only; No need to specify it twice, it would breaks things as it is a CSS id, see examples.

Valid opts:

• verticalCentered - boolean - vertically centering of the content within sections.

• scrollingSpeed - numeric - speed in milliseconds for the scrolling transitions.

• easing - character string, http://jqueryui.com/ easing, i.e.: easeInQuart

• sectionsColor - character vector, i.e.: c("blue", "red")

• navigation - boolean - if set, it will show a navigation bar made up of small circles.

• navigationPosition - character string, takes none, left, or right

• navigationColor - character string, i.e.:"black"

• navigationTooltips - character vector, i.e.: c("Tooltip 1", "Tooltip 2")

• loopBottom - boolean - defines whether scrolling down in the last section should scroll to the first one or not.

• loopTop - boolean - defines whether scrolling up in the first section should scroll to the last one or not.

• css3 - boolean - defines whether to use JavaScript or CSS3 transforms to scroll within sections. Useful to speed up the movement in tablet and mobile devices with browsers supporting CSS3. If this option is set to true and the browser doesn't support CSS3, a jQuery fallback will be used instead.

• paddingTop - defines the top padding for each section with a numerical value and its measure, i.e.: 5em

• paddingBottom - defines the bottom padding for each section with a numerical value and its measure , i.e.: 10px

• keyboardScrolling boolean - defines if the content can be navigated using the keyboard

• touchSensitivity - numeric - defines a percentage of the browsers window width/height, and how far a swipe must measure for navigating to the next section, i.e.: 5

• responsiveWidth - a class ms-responsive is added to the body tag in case the user wants to use it for his own responsive CSS. For example, if set to 900, whenever the browser's width is less than 900 the plugin will scroll like a normal site.

• responsiveHeight - a class ms-responsive is added to the body tag in case the user wants to use it for his own responsive CSS. For example, if set to 900, whenever the browser's height is less than 900 the plugin will scroll like a normal site.

• responsiveExpand - boolean - when responsive mode is fired (by using the responsiveWidth or responsiveHeight options) it turns auto scrolling off and expands each left and right side into a full-width section placing one after another. Then the page can be scrolled normally. https://github.com/alvarotrigo/multiscroll.js/wiki/Responsive-Expand-Extension

## Examples

if(interactive()){
# Run de demo
demo("muliPage", package = "fullPage")

# OR
library(shiny)

options <- list(
sectionsColor = c("#4BBFC3", "#7BAABE", "#d3d3d3", "#000")
)

ui <- multiPage(
opts = options,
menu = c("Multi Scroll" = "multiscroll",
"Plots" = "plots",
"Full Section plots" = "fullplots",
"Images" = "images"),
multiLeft(
align = "right",
multiSection(
h1("Multi")
),
multiSection(
multiContainer(
h1("Plot ->"),
numericInput(
"max",
"Maximum",
min = 50,
max = 250,
value = 10
)
)
),
multiSectionPlot(
"fullPlot"
),
multiSection(
h1("Image ")
)
),
multiRight(
align = "left",
multiSection(
h1("Page()")
),
multiSection(
plotOutput("plot")
),
multiSection(
multiContainer(
h1("<- Full Section plot"),
sliderInput(
"number",
"Data points",
min = 50,
max = 250,
value = 10
)
)
),
multiSectionImage(
img = "https://alvarotrigo.com/multiScroll/imgs/tiger.jpg",
side = "right",
h1("Background")
)
)
)

server <- function(input, output){

output$plot <- renderPlot({ par(bg = "#7BAABE") hist(rnorm(100, 20, input$max))
})

output$fullPlot <- renderPlot({ par(bg = "#d3d3d3") hist(rnorm(input$number, 20, 250))
})
}

shinyApp(ui, server)
}


text:
Article Contents
Article Contents

# An ill-posed problem for the Navier-Stokes equations for compressible flows

• We consider the Navier-Stokes equations for the motion of a compressible, viscous, isentropic fluid in a half-space H2. We prove that under the no-slip boundary conditions, the initial-boundary value problem is ill-posed in the space ($\rho$($t,\cdot$)$,\grad_x\u$($t,\cdot$))$\in$($L^\infty_x$(H2)$\times L^\infty_x$(H2))$.$
Mathematics Subject Classification: Primary: 35Q30, 35M12; Secondary: 76D03.

 Citation:


text:
Copied to
clipboard

## G = C22×D4order 32 = 25

### Direct product of C22 and D4

direct product, p-group, metabelian, nilpotent (class 2), monomial, rational

Aliases: C22×D4, C4⋊C23, C242C2, C22⋊C23, C2.1C24, C233C22, (C2×C4)⋊4C22, (C22×C4)⋊5C2, 2-Sylow(GO-(4,3)), SmallGroup(32,46)

Series: Derived Chief Lower central Upper central Jennings

 Derived series C1 — C2 — C22×D4
 Chief series C1 — C2 — C22 — C23 — C24 — C22×D4
 Lower central C1 — C2 — C22×D4
 Upper central C1 — C23 — C22×D4
 Jennings C1 — C2 — C22×D4

Generators and relations for C22×D4
G = < a,b,c,d | a2=b2=c4=d2=1, ab=ba, ac=ca, ad=da, bc=cb, bd=db, dcd=c-1 >

Subgroups: 158 in 118 conjugacy classes, 78 normal (5 characteristic)
C1, C2, C2, C2, C4, C22, C22, C2×C4, D4, C23, C23, C23, C22×C4, C2×D4, C24, C22×D4
Quotients: C1, C2, C22, D4, C23, C2×D4, C24, C22×D4

Character table of C22×D4

 class 1 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 4A 4B 4C 4D size 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 2 2 ρ1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 trivial ρ2 1 1 -1 1 1 -1 -1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 1 -1 linear of order 2 ρ3 1 -1 -1 -1 1 1 -1 1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 linear of order 2 ρ4 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 1 -1 1 -1 linear of order 2 ρ5 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 linear of order 2 ρ6 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 linear of order 2 ρ7 1 1 -1 1 1 -1 -1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 1 -1 linear of order 2 ρ8 1 -1 -1 -1 1 1 -1 1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 linear of order 2 ρ9 1 1 1 1 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 linear of order 2 ρ10 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 1 -1 1 -1 -1 1 -1 1 linear of order 2 ρ11 1 1 -1 1 1 -1 -1 -1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 linear of order 2 ρ12 1 -1 -1 -1 1 1 -1 1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 linear of order 2 ρ13 1 1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 linear of order 2 ρ14 1 -1 -1 -1 1 1 -1 1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 linear of order 2 ρ15 1 1 1 1 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 linear of order 2 ρ16 1 -1 1 -1 1 -1 1 -1 1 -1 1 -1 -1 1 -1 1 -1 1 -1 1 linear of order 2 ρ17 2 2 -2 -2 -2 -2 2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ18 2 2 2 -2 -2 2 -2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ19 2 -2 2 2 -2 -2 -2 2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4 ρ20 2 -2 -2 2 -2 2 2 -2 0 0 0 0 0 0 0 0 0 0 0 0 orthogonal lifted from D4

Permutation representations of C22×D4
On 16 points - transitive group 16T25
Generators in S16
(1 6)(2 7)(3 8)(4 5)(9 15)(10 16)(11 13)(12 14)
(1 15)(2 16)(3 13)(4 14)(5 12)(6 9)(7 10)(8 11)
(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)
(1 14)(2 13)(3 16)(4 15)(5 9)(6 12)(7 11)(8 10)

G:=sub<Sym(16)| (1,6)(2,7)(3,8)(4,5)(9,15)(10,16)(11,13)(12,14), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,14)(2,13)(3,16)(4,15)(5,9)(6,12)(7,11)(8,10)>;

G:=Group( (1,6)(2,7)(3,8)(4,5)(9,15)(10,16)(11,13)(12,14), (1,15)(2,16)(3,13)(4,14)(5,12)(6,9)(7,10)(8,11), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16), (1,14)(2,13)(3,16)(4,15)(5,9)(6,12)(7,11)(8,10) );

G=PermutationGroup([[(1,6),(2,7),(3,8),(4,5),(9,15),(10,16),(11,13),(12,14)], [(1,15),(2,16),(3,13),(4,14),(5,12),(6,9),(7,10),(8,11)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16)], [(1,14),(2,13),(3,16),(4,15),(5,9),(6,12),(7,11),(8,10)]])

G:=TransitiveGroup(16,25);

C22×D4 is a maximal subgroup of
C23.23D4  C24.3C22  C232D4  C23.10D4  C23.37D4  C22⋊D8  C22⋊SD16  C22.11C24  C233D4  C22.29C24  D45D4
C22×D4 is a maximal quotient of
C22.19C24  C22.26C24  C233D4  C22.29C24  C23.38C23  C22.31C24  D45D4  D46D4  Q85D4  Q86D4  D8⋊C22  D4○D8  D4○SD16  Q8○D8

Matrix representation of C22×D4 in GL4(ℤ) generated by

 -1 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 1
,
 -1 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 -1
,
 1 0 0 0 0 1 0 0 0 0 0 1 0 0 -1 0
,
 1 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 1
G:=sub<GL(4,Integers())| [-1,0,0,0,0,-1,0,0,0,0,1,0,0,0,0,1],[-1,0,0,0,0,1,0,0,0,0,-1,0,0,0,0,-1],[1,0,0,0,0,1,0,0,0,0,0,-1,0,0,1,0],[1,0,0,0,0,-1,0,0,0,0,-1,0,0,0,0,1] >;

C22×D4 in GAP, Magma, Sage, TeX

C_2^2\times D_4
% in TeX

G:=Group("C2^2xD4");
// GroupNames label

G:=SmallGroup(32,46);
// by ID

G=gap.SmallGroup(32,46);
# by ID

G:=PCGroup([5,-2,2,2,2,-2,181]);
// Polycyclic

G:=Group<a,b,c,d|a^2=b^2=c^4=d^2=1,a*b=b*a,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d=c^-1>;
// generators/relations

Export

׿
×
𝔽


text:
# I Affine Algebraic Sets - Dummit and Foote, page 660, Ex. 3

1. Jun 17, 2016

### Math Amateur

I am trying to gain an understanding of the basics of elementary algebraic geometry and am reading Dummit and Foote Chapter 15: Commutative Rings and Algebraic Geometry ...

At present I am focused on Section 15.1 Noetherian Rings and Affine Algebraic Sets ... ...

I need someone to help me to fully understand the reasoning/analysis behind one of the statements in Example (3) on Page 660 of D&F ...

On page 660 (in Section 15.1) of D&F we find the following text and examples (I am specifically focused on Example (3)):

In the above text, in Example (3), we find the following:

"... ... For any polynomial $f(x,y) \in k[x,y]$ we can write

$f(x,y) = f_0(x) + f_1(x)y + (x^3 - y^2) g(x,y).$"

Can someone explain ( slowly and carefully) exactly how/why this is true ... ...

Peter

====================================================

In order for readers of the above post to understand the context of the question and the notation employed I am providing the introductory pages on affine algebraic sets in the D&F text ... ... as follows:

#### Attached Files:

File size:
132.8 KB
Views:
161
File size:
120.4 KB
Views:
151
File size:
111.6 KB
Views:
158
• ###### D&F - 3 - Affine Algebraic Sets - Ch 15 - Page 3 ... .png
File size:
111.6 KB
Views:
154
2. Jun 17, 2016

### Staff: Mentor

Hi Peter,

let us start and consider $f(x,y)$ as polynomial in $y$. Then it can be written:
$f(x,y) = c_0 \cdot f_0(x) + c_1 \cdot y \cdot f_1(x) + c_2 \cdot y^2 \cdot f_2(x) + \dots$
What we are doing next is to gather everything, that "disturbs" us and putting it into the terms $f_0$ and $f_1$.
There are no restrictions on those, beside that they aren't allowed to contain $y$'s. so our first substitution will be $f_0^{new} = c_0 f_0$ and $f_1^{new} = c_1 f_1$. For the next steps I will drop the $^{new}$ because it is not necessary to keep a record on them.

Now $c_2 \cdot y^2 \cdot f_2(x) = c_2 \cdot y^2 \cdot (d_0 + d_1 \cdot x + d_2 \cdot x^2 + \dots) = c_2d_0 y^2 + c_2d_1xy^2+c_2d_2x^2y^2 + \dots$ and every term is of the form $e_n \cdot x^n \cdot y^2.$
This can be written as $e_n \cdot x^n \cdot y^2 = (x^3-y^2) \cdot (-e_n x^n) + e_n \cdot x^{n+3}.$
The first factor at $(x^3-y^2)$ will be used to build up $g(x,y)$ and the second will be added to $f_0.$

You can proceed this way with every following term $c_n y^n f_n(x) \; (n>2)$, i.e. perform the long division on polynomials.

3. Jun 17, 2016

### mathwonk

To summarize fresh34's nice answer, for any domain R, we can divide by any polynomial in the ring R[Y] whose lead coefficient is a unit. E.g. we can always divide by Y^2 + aY + b. Moreover, as is usual in long division, the remainder has lower degree than the divisor.

Thus if we take R = k[X], then in the ring R[Y] = k[X][Y] ≈ k[X,Y], we can divide any polynomial by Y^2 - X^3, and get a remainder of degree lower than 2 (in Y). I.e. the remainder will have form f1Y + f0 where f0 and f1 lie in the coefficient ring k[X].

The point of this somewhat tedious example is to illustrate that over a finite field, one does not keep much information about a polynomial from looking only where it vanishes, since over Z/2Z, any multiple of the polynomial x(x-1) vanishes everywhere. to insure that the zero locus has as much information as possible, one usually assumes the field algebraically closed. see e.g. hilbert's nullstellensatz in DF, p.675.

Last edited: Jun 17, 2016
4. Jun 17, 2016

### Math Amateur

Thanks fresh_42 ... that post was most helpful ...

You really helped me when you wrote:

"This can be written as $e_n \cdot x^n \cdot y^2 = (x^3-y^2) \cdot (-e_n x^n) + e_n \cdot x^{n+3}.$
The first factor at $(x^3-y^2)$ will be used to build up $g(x,y)$ and the second will be added to $f_0.$"

I was really perplexed at how to deal with this situation ...

Thanks again for the help,

Peter

5. Jun 17, 2016

### Math Amateur

Thanks mathwonk ... really helpful to get a general perspective on this matter ...

Thank you for moving my understanding forward ... really appreciate it ...

Peter

6. Jun 17, 2016

### Staff: Mentor

You're welcome. It's always a pleasure to read your carefully prepared and precise questions.


text:
Bounding the separable rank via polynomial optimization

We investigate questions related to the set $\mathcal{SEP}_d$ consisting of the linear maps $\rho$ acting on $\mathbb{C}^d\otimes \mathbb{C}^d$ that can be written as a convex combination of rank one matrices of the form $xx^*\otimes yy^*$. Such maps are known in quantum information theory as the separable bipartite states, while nonseparable states are called entangled. In … Read more

Numerical estimation of the relative entropy of entanglement

We propose a practical algorithm for the calculation of the relative entropy of entanglement(REE), defined as the minimum relative entropy between a state and the set of states with positive partial transpose. Our algorithm is based on a practical semi-definite cutting plane approach. In low dimensions the implementation of the algorithm in MATLAB provides an … Read more