Name: Norman Anthony Aguero
Currently a student at FIU. My major is chemistry and my minor is physics. My goal is to hopefully earn a Ph.D. in physical organic chemistry.
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I don't know why, but this video has always creeped me out.
Microsoft Corporation will be best remembered for its blue screen of death, a randomly occuring, incomprehensible, bright blue message would appear, glibly informing you that your machine has, yet again, crashed and that you've lost all your work and you're going to have to reboot.
Microsoft is the original big company with a staggering cash bank balance, legions of marketing types and a healthy stock price. Microsoft gets rich by selling bad software to everybody, home users, businesses, even computer gamers. Most of the company's influence came about by the so-called "Microsoft Tax", where users purchasing a personal computer would pay Microsoft for use of their OS, whether or not they intended to run it.
Worse than technically inferior software is insecure software. System administrators around the world blindly installed Microsoft web servers and failed to follow up on patches designed to fix security problems. This run-of-the-mill hole allowed attackers to control and cascade viruses, bringing Internet sites to their knees during the "Code Red" virus attack. Microsoft is also to blame for spreading the "Love Bug" virus, a clever bit of code that would allow the remote attacker to execute arbitrary code on one's machine, automatically, thanks to poor choices for default values in Microsoft's e-mail client.
Microsoft effectively owns various markets including desktop operating systems, internet browsers and business productivity software in the form of word processing, spreadsheets and presentation tools. Where Microsoft has had great successes, it has also had a series of failures. Adobe Systems still sells copies of Photoshop in the shadow of Microsoft, as Claris Corporation enjoys a modicum of success with Filemaker Pro. Large companies are slow to adopt Microsoft products as server-side solutions, sticking with industrial-strength UNIX products from Sun, IBM and even free solutions such as Linux. Sure, Oracle runs on Windows NT but would you trust your career with it? Yes, you can get fired for buying Microsoft.
At best, Microsoft is capable of monkey-see, monkey-do, making knockoff workalikes of Apple Computer's desktop software, a poor clone of the ever-present mp3 format and even shamelessly copying from Sun Microsystem's Java programming language.
At the heart of Microsoft is a bowl-haired little geek named William Gates III. Fabulously wealthy and badly deluded into thinking his company has contributed good things to the world, Bill is not much of a programmer and did in fact not create Microsoft's Disk Operating System. But Bill and his flock certainly figured out how to market and bundle software. In 1980, DOS was a pale imitation of UNIX, even more difficult to use, easy to crash and only allowing a single process to run at once. These early, poor engineering decisions still haunt Microsoft to this day.
In 1983, Microsoft released the first version of Windows, what would eventually become the world's most popular operating system. Windows was, of course, simply a large graphic interface placed on top of DOS. From this year on, they figured out a clever scheme: force users to upgrade, i.e., to pay for bugfixes, every year. By 1995, "Windows 95" sold 1 million copies in the first 4 days.
Part of Microsoft's secret was to partner with not only IBM but also with Intel. Without Microsoft's Windows operating system, IBM and Intel have an less compelling product to offer. An Anti-trust action brought against Microsoft in the late 90's accused Microsoft of having used this position to place a stranglehold on the industry and stifle innovation.
In 1995, when Netscape Communications Corporation released their browser, Microsoft noticed. Eventually. It took a few years, the purchase of a smaller browser company and a series of questionable business deals, but they wrestled control of the market from Netscape. The dominating web browser in 1995, Netscape Communicator, had became a footnote in the history of computing within a few years. Netscape didn't do itself any favors by trying to compete with Microsoft, frantically making release after release, each crappier than the next. Even so, Microsoft's swift destruction of Netscape is a harsh lesson for those that want to innovate in Silicon Valley.
| 12 Nov 1975 | Bill Gates and Paul Allen's 7-person software company in Albuquerque, New Mexico adopts the name "Microsoft." | ||||||||
| 27 Apr 2000 | Texas Governor George W. Bush discusses the Microsoft antitrust case with Jim Lehrer.
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| 12 Jun 2000 | Microsoft announces a partnership with DirecTV and Thomson multimedia to develop a TiVo killer it calls UltimateTV. | ||||||||
| 23 Jan 2002 | Microsoft announces it has killed the UltimateTV project, their TiVo killer. |
Explanation: A mere 11 million light-years away, Centaurus A is a giant elliptical galaxy - the closest active galaxy to Earth. This remarkable composite view of the galaxy combines image data from the x-ray ( Chandra), optical(ESO), and radio(VLA) regimes. Centaurus A's central region is a jumble of gas, dust, and stars in optical light, but both radio and x-ray telescopes trace a remarkable jet of high-energy particles streaming from the galaxy's core. The cosmic particle accelerator's power source is a black hole with about 10 million times the mass of the Sun coincident with the x-ray bright spot at the galaxy's center. Blasting out from the active galactic nucleus toward the upper left, the energetic jet extends about 13,000 light-years. A shorter jet extends from the nucleus in the opposite direction. Other x-ray bright spots in the field are binary star systems with neutron stars or stellar mass black holes. Active galaxy Centaurus A is likely the result of a merger with a spiral galaxy some 100 million years ago.
Explanation: These two mighty galaxies are pulling each other apart. Known as " The Mice" because they have such long tails, each spiral galaxy has likely already passed through the other. They will probably collide again and again until they coalesce. The long tails are created by the relative difference between gravitational pulls on the near and far parts of each galaxy. Because the distances are so large, the cosmic interaction takes place in slow motion -- over hundreds of millions of years. NGC 4676 lies about 300 million light-years away toward the constellation of Bernice's Hair (Coma Berenices) and are likely members of the Coma Cluster of Galaxies. The above picture was taken with the Hubble Space Telescope's Advanced Camera for Surveys which is more sensitive and images a larger field than previous Hubble cameras. The camera's increased sensitivity has imaged, serendipitously, galaxies far in the distance scattered about the frame.
By Kristin Elise Phillips, Scienceline
posted: 27 March 2008 ET
Yards and yards of clear plastic sheeting line the cellar floor, dwarfing the corpse: headless, frail, supine. The young bony arms — covered in fine black powder from centuries of immobility in the frozen tundra — are crossed at rest, reminiscent of a ceremonial burial. Camera flashes illuminate the scene. Several dozen scientists stand around the body, murmuring in Russian and English about the find of the day.
How long do you think it was buried? Do you think it’s male or female? How did they get it back to camp? And the pervasive thought: I don’t think we should touch it. He could have died of smallpox.
Smallpox was a vicious disease before its eradication in the 1970s, but the virus is hardy and can survive long-term storage. One such storage unit is the tundra of the high northern latitudes that preserves an unknown number of bodies that could have died from smallpox. Global warming is now rapidly thawing this freezer, increasing the chance that someone could come into contact with a smallpox-infested body, thereby reintroducing the disease.
Smallpox rivals malaria as the most deadly infectious disease ever to affect humans. Throughout history, people looked for ways to combat the disease, priming their immune systems with remedies such as sniffing ground-up scabs or smearing pus into open wounds. The first true vaccine — developed in 1796 by Edward Jenner — was for smallpox.
The variola virus responsible for smallpox, which causes fever, fatigue and pustules that leave deep scars on the skin, decimated the Americas after Columbus landed in the West Indies. The disease similarly ravaged the people of the Arctic, and an estimated 300 million people died from smallpox in the 20th century alone before the World Health Organization’s vaccination campaign was completely effective. The last case from natural exposure was in the late 1970s in Ethiopia.
Today smallpox exists only in highly secure U.S. and Russian laboratories. According to Jonathan Tucker, a senior fellow at the Center for Nonproliferation Studies, “the greatest risk of smallpox infection today is from the continued scientific research with the live virus, as well as from the hypothetical existence of undeclared stocks of the virus that could pose a risk of accidental or deliberate release.” Many scientists agree that an accidental or deliberate release of the virus is a dangerous possibility, especially since vaccinations of the general population ceased in 1972. In response to the attacks on Sept. 11, 2001, the Bush administration ordered the inoculation of U.S. military and health workers so that critical operations would not be affected.
. . .
It was 20 years ago when the headless body was found at a bend in the Kolyma River and brought to camp — at the Northeast Research Station in Cherskii, Siberia. On that day the tundra was changing to red and gold, and longer nights had begun to touch the horizon. Late summer is prosperous near the Arctic Circle: Local fishermen descend on the river to plunder sturgeon, and paleontologists scan the banks by boat and foot for mammoth bones or frozen bodies of ancient musk ox and horses.
Imre Friedmann remembers the day that the body was found. He trudged into the station, finally escaping the plague of swarming mosquitoes, to be told of the body in the cellar. “Everybody avoided handling it,” he recounts in precise, accented English. Friedmann, affiliated with the NASA Ames Research Center, traveled to the Arctic to study the bacteria that thrive in the extreme climate of this region.
Other projects have been similarly affected by the fear of smallpox. Archaeologists halted work in the London crypt Spitalfields in the mid-1980s after finding smallpox scars on a corpse, and a librarian from Santa Fe, N.M., was vaccinated after finding a smallpox scab in a Civil War medical book. In these cases, the virus was no longer viable. But a construction worker in the United Kingdom did contract the disease while demolishing a building that had housed smallpox victims, and researchers in Holland found a live virus in a 13-year-old scab.
Bodies frozen in the north could be even more fertile ground as a reservoir of the virus. Smallpox is resilient when frozen. Louise Parker and James Martel of the Army Corps of Engineers reported that vaccinia, the virus used in the smallpox vaccine, survives short-term freezing and thawing as well as storage at low temperatures. And in the 1950s, U.S. Army scientists found that the variola survived three years of freezing, particularly at very low temperatures.
In the 1980s, a mass grave near Pokhodsk, Siberia, was exposed by a river and local residents demanded testing of the bodies. Researchers took all the necessary precautions of an epidemic: protective gear, antiseptic cleansing and vaccinations. But even though some bodies were well preserved after a hundred years in permafrost, “viable smallpox virus was not detected, but the virus antigen was discovered,” says Sergei Davidov, currently the assistant director of the field station in Cherskii.
Fear of frozen corpses lying beneath the tundra may even be the reason that the United States and Russia maintain stockpiles, according to Donald Henderson. Henderson, an epidemiologist currently at Johns Hopkins University, directed the World Health Organization’s smallpox eradication campaign. After hammering out an agreement between the two countries to reduce smallpox stockpiles, he was “just about ready to take this to the World Heath Assembly when a guy from Britain shows up.” This man was the head of the United Kingdom’s chemical and biological weapons program.
Henderson recalls their conversation.
How could you do that?
How could I do what?
Let me say this: Suppose you have bodies in the tundra? What would we do to protect people — we’ve destroyed the virus.
Henderson explained to the chemist that the possibility of virus frozen in the north has little to do with maintenance of laboratory stockpiles. But the chemist took his concerns to the U.S. Department of Defense, and, according to Henderson, the fear of naturally frozen virus is what led the military to withdraw from the resolution. “I can’t make it up,” he laughs.
Some life does exist in frozen soil and ice. Imre Friedmann, who had been in the research station with the body, points out that “in permafrost we find living bacteria in 3 million-year-old permafrost. So if bacteria survive, I don’t see why viruses don’t survive.” Friedmann is referring to a team from the Russian Academy of Sciences that found bacteria in ancient permafrost. Viruses have also been discovered in old ice cores: Scott Rogers of Bowling Green State University in Ohio found a 140,000-year-old RNA plant pathogen in Greenland.
Taken together — the possibility that viruses survive, the hardiness of smallpox and the expanse of frozen tundra — it seems possible that viable variola could be preserved in permafrost. “If it were going to be anywhere,” Henderson says, “if you were going to find something, [the tundra] would be the likely place.”
Global warming is thawing permafrost. In Siberia, botanists at Tomsk State University estimate that an area twice the size of California has changed from featureless tundra to a lake-dotted, slumped landscape. The decomposition of formerly frozen soil is in turn accelerating global warming because of the release of previously trapped methane gas. The northern Arctic is warming more quickly than other parts of the world, and, according to projections by the National Center for Atmospheric Research in Colorado, the uppermost 10 feet of the Northern Hemisphere’s permafrost may be gone by 2100.
“Obviously the delicate relationship between climate and permafrost is going to have to find new equilibrium,” says Wayne Pollard, a permafrost specialist at McGill University in Montréal.
But what does an accelerated thaw mean for smallpox? Some experts think that climate change reduces the chance of a smallpox reintroduction because the virus cannot survive multiple days unfrozen. To Tucker of the Center for Nonproliferation Studies, “the gradual thawing of the permafrost brought about by global warming [further diminishes] the likelihood of recovering infectious smallpox virus particles from the corpses of victims buried in the Arctic region.”
There is a caveat to this assumption, though. According to Pollard, there are different kinds of permafrost. The ice-rich permafrost is rapidly changing the northern landscape, but dry permafrost, on the other hand, could better preserve a body and the viruses harbored.
“It’s important to say, ‘Never say never,’ with some of these things because it’s like saying life couldn’t have arrived on Earth from an asteroid,” concludes Russell Regnery of the Poxvirus Program at the Centers for Disease Control and Prevention. He thinks that the disease impact from global warming will come from the ooze of tropical diseases such as malaria and leishmaniasis into newly available habitats rather than from the release of pathogens because of permafrost melt.
. . .
The morning after finding the frozen corpse along the Kolyma River, several researchers carried it out of the Cherskii Research Station past a few scraggily evergreens. It was buried that day in 1990, just before the Soviet Union opened. Under normal circumstances, scientists might have examined an old body: one researcher thought the traditional reindeer skin clothing was about 300 years old. But the fear of the unknown — of smallpox — evaporated their intellectual interest.
But fear needs perspective. “These things don’t cough anymore,” says the CDC’s Regnery. Short of people wiping a newly exposed cadaver across their eye, it is hard for him to see how the virus could transfer. Epidemiologist Henderson adds that an outbreak of smallpox would kill people, but it could be contained. Sick people go to bed, and the disease transfers from person to person only when the pustules are obvious. Says Henderson: “There is a lot of docudrama stuff out there that is absolute poppycock.”
By Dave Mosher, Staff Writer
posted: 26 March 2008 03:45 pm ET
This story was updated at 3:30 p.m. ET.
A sniff test of water vapor spewing from Saturn's moon Enceladus shows it is gushing with organic molecules, increasing the possibility of life existing somewhere in the Saturn system.
Scientists have been intrigued by the moon since the fountain of water was first spotted in 2005. Now they've identified a soup of prebiotic material there, similar to what's found in comets, from an analysis of data collected by the Cassini spacecraft.
Nobody really knows how life began, but astrobiologists guess it required chemicals like those tasted by Cassini, a little liquid water and some unknown spark.
Hunter Waite, a Cassini principal investigator at the Southwest Research Institute (SWRI) in San Antonio, said Enceladus' newly understood composition should stir up previous notions of Saturn and its moons.
"These findings will definitely get people to ask new questions about the formation of the Saturn system," Wait told SPACE.com. "The astrobiological potential of the Saturn system just went up a notch or two."
Cassini made its observations during a high-speed pass 30 miles (48 km) above Enceladus on March 12, and recorded the highest temperatures yet detected near tiger stripe-like fissures on the icy moon's southern pole.
Waite and other scientists released their early findings about Enceladus' thermal activity and icy plume composition during a briefing today at NASA headquarters in Washington, D.C.
Big comet?
The new heat maps of Enceladus' surface show temperatures higher than previously observed in the south polar region, with hot tracks running the length of giant fissures.
"They're still awfully cold, but much warmer than background temperatures of the rest of the surface," said John Spencer, a Cassini scientist at SWRI in Boulder, Colo. "This means it has to be even warmer under the surface and raises the possibility of liquid water beneath the [exterior]."
Cassini measured the fissures to be -135 degrees Fahrenheit (-93 degrees Celsius) near their centers. That's about 63 degrees F warmer than previously observed and some 200 degrees F (111 degrees C) warmer than the rest of the moon's surface.
Scientists also say Cassini sampled organic chemicals during its bath in the icy jets and that they are similar to those found in comets.
"A completely unexpected surprise is that the chemistry of Enceladus, what's coming out from inside, resembles that of a comet," Waite said, but noted that Enceladus is also very different from a comet. "Comets have tails and orbit the sun, and Enceladus' activity is powered by internal heat while comet activity is powered by sunlight."
Organic steam bath
In addition to carbon dioxide, carbon monoxide and other compounds, Waite said organic molecules such as methane, propane, acetylene and formaldehyde were sniffed in Enceladus' icy plumes.
"Enceladus' brew is like carbonated water with an essence of natural gas," Waite said of the moon's steamy jets. "Astrobiologically speaking, this moon is one of the most interesting places in the solar system."
Dennis Matson, Cassini project scientist at NASA's Jet Propulsion Laboratory in Pasadena, Calif., said all of the ingredients for life could be present beneath Enceladus' pockmarked surface.
"Enceladus has got warmth, water and organic chemicals," said Dennis Matson. "These three basic ingredients provide a minimum for the origin of life."
But Matson said Cassini will have to gather more data before a key element — liquid water — can be verified to exist on the moon.
"We ... [can't tell] whether the interior of Enceladus contains liquid water and if that water might be a habitat for life," Matson said during today's briefing, "but these are the questions that Cassini will focus on in future flybys."
The spacecraft is set to revisit Enceladus in August and October of this year, followed by five more flybys in the next two years.
Another one, for Roma:
Spring is here.
Well, I've decided to share certain 'paranormal' experiences that I've encountered through my life. I'm sure these moments have led me to pursue the study of sciences. Don't get me wrong, most of my experiences I'm sure have plausible explanations, luckily I wasn't alone when they happened.
1. In 1969, or so, I was into martial arts. I was hit in the solar plexis and rendered unconscious. I went into respiratory and cardiac arrest. I had what is called a 'near death experience.' I saw myself flying through a tunnel, a gray kind of cool one devoid of color. At the end of the tunnel, as I flew towards it, there were 4 or 5 beings that seemed to be conversing, better, communicating with themselves. As I approached them, one seemed to stop its communication with their group and seemed irritated that I was observing them. It looked down on me and put its hand out, hitting me on my head and stopping my travel. I had the feeling it was not my time, so it was kind of pissed at me, like, "what am I doing here?" The beings are now what UFO people call the Grays. They seemed to me entities with large heads, wrap around large eyes, no facial features, and webbed 4 fingered hands. That was many years before all the UFO crap about gray aliens appeared in the media.
2. While married, my ex wife and I saw twice what seemed to me as large clumps of spider webbing, gossamer looking clumps flying through the skies. Angel hair is what ufologist call it.
3. Setting up my 8" Newtonian telescope at home, I saw a satellite crossing the sky from east to west. Then, I saw a red object, intercept the satellite and circle it, at a very high speed, and head of towards the south west. Its speed was mind boggling.
4. Another time, while setting up a telescope, I was with my ex brother in law. Jupiter was very bright but the skies were over cast by cumulus clouds. We saw 8 flying egg shaped objects that were so polished, so mirror like, flying with no sound and reflecting the city lights off their bottoms.
5. I saw about six gold colored objects one night with my friends. That same evening Miami International picked up unidentified objects on their radars. Also, State Troopers saw these objects and reported them.
Well, I've seen more crap. I'm sure there are many explanations. Do I believe?
Yes and no.
Another one for Romma:
190 Nagle Avenue, apt. 5 E.
New York, N.Y. 10034
Tele: LOrainne 7-0586.
That was my address growing up in NYC. Nagle is located on the upper west side of the city, and is intersected by Dyckman Street. Here is a picture of the tennemants I lived in:
The light colored one, on the right in the foreground, is where I lived, on the top floor, the fifth floor. I used to play on the fire escape. I was either Napoleon Solo, or the great 007.
The neighborhood, at the time, was mostly Irish American. It was a very tough place to grow up in. I still have scars from my daily fights with Irish Gangs.
I was one of the very few Hispanics, and that led to the constant brawling. I remember walking down the stairs (no elevator in these tenements) and dodging milk containers filled with blood, spoons and hypodermic needles. The heroin junkies use to get their "fixes" on the stairway.
By that tall, brown high rise in the background, I remember one morning, while walking to school (5th. grade) stepping on something that felt like a "cigar." It turned out to be a human thumb.
These tenement were roach infested. They would crawl on me while sleeping and even use to get into my food. I remember one evening my mother made me a vanilla milkshake and by the time it reached my bedroom, roaches had jumped into it from the ceiling. At first I thought they were raisins, same texture, but when I spit them out, I realized my mistake while vomiting away in the bathroom.
My father was crippled by polio at age four, and twice I remember him falling. Once in the kitchen, and once on the stairs. Everytime he fell, he had this unusual knack for compounded fractures of his bad leg. I'd find him lying in a pool of blood with some part of a bone sticking out of his leg. I must have been between 6 and 10 years of age.
During a coal strike in the mid sixties, it got so cold in that tenement that my pet hamster froze. I buried his little body in between the tenements, where there was a small patch of earth.
What I can say about the neighborhood is that it is absoluetly stunning in both architectural diversity and, located in the Palisades valley cut by the Hudson River, topographically beautiful. Here's more pictures of the neighborhood:
I used to go swimming off the railroad bridge pictured above. This creek is called "Spuyten Duyvil." The creek conects the hudson river with the Harleem River.
The "C" stands for Columbia University.
My goal is to eventually move back here. I still have family that live around this section of NYC. My goal is to live out the rest of my life here in Inwood.
Enough of this crap, back to writing my paper. The title of which is: "Solid Phase Cap and Tag Ogliosaccaride Synthesis."
In mathematics, a line integral (sometimes called a path integral or curve integral) is an integral where the function to be integrated is evaluated along a curve. Various different line integrals are in use. In the case of a closed curve it is also called a contour integral.
The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulas in physics (for example, ) have natural continuous analogs in terms of line integrals (). The line integral finds the work done on an object moving through an electric or gravitational field, for example.
In qualitative terms, a line integral in vector calculus can be thought of as a measure of the total effect of a given field along a given curve.
For some scalar field f : U ⊆ Rn R, the line integral along a curve C ⊂ U is defined as
where r: [a, b] C is an arbitrary bijective parametrization of the curve C such that r(a) and r(b) give the endpoints of C.
The function f is called the integrand, the curve C is the domain of integration, and the symbol ds can be heuristically interpreted as an elementary arc length. Line integrals of scalar fields do not depend on the chosen parametrization r.
For a vector field F : U ⊆ Rn Rn, the line integral along a curve C ⊂ U, in the direction of r, is defined as
where r: [a, b] C is a bijective parametrization of the curve C such that r(a) and r(b) give the endpoints of C.
Line integrals of vector fields are independent of the parametrization r in absolute value, but they do depend on its orientation. Specifically, a reversal in the orientation of the parametrization changes the sign of the line integral.
If a vector field F is the gradient of a scalar field G, that is,
then the derivative of the composition of G and r(t) is
which happens to be the integrand for the line integral of F on r(t). It follows that, given a path C , then
In words, the integral of F over C depends solely on the values of G in the points r(b) and r(a) and is thus independent of the path between them.
For this reason, a line integral of a vector field which is the gradient of a scalar field is called path independent.
The line integral has many uses in physics. For example, the work done on a particle traveling on a curve C inside a force field represented as a vector field F is the line integral of F on C.
Viewing complex numbers as 2D vectors, the line integral in 2D of a vector field corresponds to the real part of the line integral of the conjugate of the corresponding complex function of a complex variable.
Due to the Cauchy-Riemann equations the curl of the vector field corresponding to the conjugate of a holomorphic function is zero. This relates through Stokes' theorem both types of line integral being zero.
The line integral is a fundamental tool in complex analysis. Suppose U is an open subset of C, γ : [a, b] U is a rectifiable curve and f : U C is a function. Then the line integral
may be defined by subdividing the interval [a, b] into a = t0 < t1 < ... < tn = b and considering the expression
The integral is then the limit of this sum, as the lengths of the subdivision intervals approach zero.
If γ is a continuously differentiable curve, the line integral can be evaluated as an integral of a function of a real variable:
When γ is a closed curve, that is, its initial and final points coincide, the notation
is often used for the line integral of f along γ.
The line integrals of complex functions can be evaluated using a number of techniques: the integral may be split in to real and imaginary parts reducing the problem to that of evaluating two real-valued line integrals, the Cauchy integral formula may be used in other circumstances. If the line integral is a closed curve in a region where the function is analytic and containing no singularities, then the value of the integral is simply zero, this is a consequence of the Cauchy integral theorem. Because of the residue theorem, one can often use contour integrals in the complex plane to find integrals of real-valued functions of a real variable (see residue theorem for an example).
Consider the function f(z)=1/z, and let the contour C be the unit circle about 0, which can be parametrized by eit, with t in [0,2π]. Substituting, we find
where we use the fact that any complex number z can be written as reit where r is the modulus of z. On the unit circle this is fixed to 1, so the only variable left is the angle, which is denoted by t. This answer can be also verified by the Cauchy integral formula.
The "path integral formulation" of quantum mechanics actually refers not to path integrals in this sense but to functional integrals, that is, integrals over a space of paths, of a function of a possible path. However, path integrals in the sense of this article are important in quantum mechanics; for example, complex contour integration is often used in evaluating probability amplitudes in quantum scattering theory.
In mathematical optimization problems, the method of Lagrange multipliers, named after Joseph Louis Lagrange, is a method for finding the extrema of a function of several variables subject to one or more constraints; it is the basic tool in nonlinear constrained optimization.
Simply put, the technique is able to determine where on a particular set of points (such as a circle, sphere, or plane) a particular function is the smallest (or largest).
More formally, Lagrange multipliers compute the stationary points of the constrained function. By Fermat's theorem, extrema occur either at these points, or on the boundary, or at points where the function is not differentiable.
It reduces finding stationary points of a constrained function in n variables with k constraints to finding stationary points of an unconstrained function in n+k variables. The method introduces a new unknown scalar variable (called the Lagrange multiplier) for each constraint, and defines a new function (called the Lagrangian) in terms of the original function, the constraints, and the Lagrange multipliers.
Consider a two-dimensional case. Suppose we have a function f(x,y), to maximize, subject to the constraint
where c is a constant. We can visualize contours of f given by
for various values of dn, and the contour of g given by g(x,y) = c.
Suppose we walk along the contour line with g = c. In general the contour lines of f and g may be distinct, so traversing the contour line for g = c could intersect with or cross the contour lines of f. This is equivalent to saying that whilst moving along the contour line for g = c the value of f can vary. Only when the contour line for g = c touches contour lines of f tangentially, we do not increase or decrease the value of f - that is, when the contour lines touch but do not cross.
This occurs exactly when the tangential component of the total derivative vanishes: , which is at the constrained stationary points of f (which include the constrained local extrema, assuming f is differentiable). Computationally, this is when the gradient of f is normal to the constraint(s): when for some scalar λ.
A familiar example can be obtained from weather maps, with their contour lines for temperature and pressure: the constrained extrema will occur where the superposed maps show touching lines (isopleths).
Geometrically we translate the tangency condition to saying that the gradients of f and g are parallel vectors at the maximum, since the gradients are always normal to the contour lines. Thus we want points (x,y) where , and, further, g(x,y) = c. To incorporate both these conditions into one equation, we introduce an unknown scalar, λ, and solve
with
and
As discussed above, we are looking for stationary points of f seen while travelling on the level set g(x,y) = c. This occurs just when the gradient of f has no component tangential to the level sets of g. This condition is equivalent to for some λ. Stationary points (x,y,λ) of F also satisfy g(x,y) = c as can be seen by considering the derivative with respect to λ.
Be aware that the solutions are the stationary points of the Lagrangian F, and are saddle points: they are not necessarily extrema of F. F is unbounded: given a point (x,y) that doesn't lie on the constraint, letting makes F arbitrarily large or small. However, under certain stronger assumptions, as we shall see below, the strong Lagrangian principle holds, which states that the maxima of f maximize the Lagrangian globally.
Denote the objective function by and let the constraints be given by , perhaps by moving constants to the left, as in . The domain of f should be an open set containing all points satisfying the constraints. Furthermore, f and the gk must have continuous first partial derivatives and the gradients of the gk must not be zero on the domain.[1] Now, define the Lagrangian, Λ, as
Observe that both the optimization criteria and constraints gk(x) are compactly encoded as stationary points of the Lagrangian:
and
Collectively, the stationary points of the Lagrangian,
give a number of unique equations totaling the length of plus the length of . This often makes it possible to solve for every x and λk, without inverting the gk.[1] For this reason, the Lagrange multiplier method can be useful in situations where it is easier to find derivatives of the constraint functions than to invert them.
Often the Lagrange multipliers have an interpretation as some salient quantity of interest. To see why this mi
