Tag Archives: space

Isometry

Isometry

From Wikipedia, the free encyclopedia
For the mechanical engineering and architecture usage, see isometric projection. For isometry in differential geometry, seeisometry (Riemannian geometry).
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.
Isometries are often used in constructions where one space is embedded in another space. For instance, the completion of a metric space M involves an isometry from M into M’, a quotient set of the space of Cauchy sequences on M. The original space M is thus isometrically isomorphic to a subspace of a complete metric space, and it is usually identified with this subspace. Other embedding constructions show that every metric space is isometrically isomorphic to a closed subset of some normed vector space and that every complete metric space is isometrically isomorphic to a closed subset of some Banach space.
An isometric surjective linear operator on a Hilbert space is called a unitary operator.

flying objects

Uploaded on Sep 22, 2008

The AirTraffic team presents the global air traffic (simulation over 24 hours).
http://radar.zhaw.ch/


Uploaded on Feb 19, 2010

Courtesy NASA

FACET is a simulation tool for exploring advanced air traffic management concepts.
An efficient and effective air traffic management system is vital to the U.S. transportation infrastructure. Since 1978, when the airline industry was deregulated, the inflation adjusted gross domestic product (GDP) has increased by 62 percent. In this same time period, total output of scheduled passenger air transportation (as measured by Revenue Passenger Miles) has increased by 190 percent and total airfreight ton miles have increased by 289 percent. Since 1997, flight delays have skyrocketed – doubling in only four years. These trends are expected to continue. In 1998, airline delays in the U.S. cost industry and passengers $4.5 billion — the equivalent of a 7 percent tax on every dollar collected by all the domestic airlines combined.


Uploaded on Oct 26, 2010

Simulation of Space Debris orbiting Earth. Created by the Institute of Aerospace Systems of the Technische Universität Braunschweig and shown at the 3rd Braunschweig Lichtparcours from June 19th to September 30th, 2010. Also available as an interactive screen saver for windows and Linux athttp://www.days-in-space.de.
More information about our research at http://www.space-debris.de.
Color Key:
Red: Satellites (operational or defunct)
Yellow: Rocket bodies
Green: Mission Related Objects (bolts, lens caps, etc.)
Blue: Solid rocket motor slag
White: Fragments from explosion events

New Microbe Found in Two Distant Clean Rooms

November 06, 2013

A rare, recently discovered microbe that survives on very little to eat has been found in two places on Earth: spacecraft clean rooms in Florida and South America.

Microbiologists often do thorough surveys of bacteria and other microbes in spacecraft clean rooms. Fewer microbes live there than in almost any other environment on Earth, but the surveys are important for knowing what might hitch a ride into space. If extraterrestrial life is ever found, it would be readily checked against the census of a few hundred types of microbes detected in spacecraft clean rooms.

The work to keep clean rooms extremely clean knocks total microbe numbers way down. It also can select for microbes that withstand stresses such as drying, chemical cleaning, ultraviolet treatments and lack of nutrients. Perversely, microbes that withstand these stressors often also show elevated resistance to spacecraft sterilization methodologies such as heating and peroxide treatment.

“We want to have a better understanding of these bugs, because the capabilities that adapt them for surviving in clean rooms might also let them survive on a spacecraft,” said microbiologist Parag Vaishampayan of NASA’s Jet Propulsion Laboratory, Pasadena, Calif., lead author of the 2013 paper about the microbe. “This particular bug survives with almost no nutrients.”

This population of berry-shaped bacteria is so different from any other known bacteria, it has been classified as not only a new species, but also a new genus, the next level of classifying the diversity of life. Its discoverers named it Tersicoccus phoenicis. Tersi is from Latin for clean, like the room. Coccus, from Greek for berry, describes the bacterium’s shape. The phoenicis part is for NASA’s Phoenix Mars Lander, the spacecraft being prepared for launch in 2007 when the bacterium was first collected by test-swabbing the floor in the Florida clean room.

Some other microbes have been discovered in a spacecraft clean room and found nowhere else, but none previously had been found in two different clean rooms and nowhere else. Home grounds of the new one are about 2,500 miles (4,000 kilometers) apart, in a NASA facility at Kennedy Space Center and a European Space Agency facility in Kourou, French Guiana.

A bacterial DNA database shared by microbiologists worldwide led Vaishampayan to find the match. The South American detection had been listed on the database by a former JPL colleague, Christine Moissl-Eichinger, now with the University of Regensburg in Germany. She is first co-author of the paper published this year in the International Journal of Systematic and Evolutionary Microbiology identifying the new genus.

The same global database showed no other location where this strain of bacteria has been detected. That did not surprise Vaishampayan. He said, “We find a lot of bugs in clean rooms because we are looking so hard to find them there. The same bug might be in the soil outside the clean room but we wouldn’t necessarily identify it there because it would be hidden by the overwhelming numbers of other bugs.”

A teaspoon of typical soil would have thousands more types of microbes and billions more total microbes than an entire cleanroom. More than 99 percent of bacterial strains, as identified from DNA sequences, have never been cultivated in laboratories, a necessary step for the various types of characterization required to identify a strain as a new species.

Microbes that are tolerant of harsh conditions become more evident in clean room environments that remove the rest of the crowd.

“Tersicoccus phoenicis might be found in some natural environment with extremely low nutrient levels, such as a cave or desert,” Vaishampayan speculated. This is the case for another species of bacterium (Paenibacillus phoenicis) identified by JPL researchers and currently found in only two places on Earth: a spacecraft clean room in Florida and a bore hole more than 1.3 miles (2.1 kilometers) deep at a Colorado molybdenum mine.

Ongoing research with Tersicoccus phoenicis is aimed at understanding possible ways to control it in spacecraft clean rooms and fully sequencing its DNA. Students from California State University, Los Angeles, have participated in the research to characterize the newly discovered species.

The California Institute of Technology, Pasadena, operates JPL for NASA.

Guy Webster 818-354-6278
Jet Propulsion Laboratory, Pasadena, Calif.
guy.webster@jpl.nasa.gov

NASA’s Voyager 1 spacecraft at the far reaches of our solar system

PASADENA, Calif. — NASA’s Voyager 1 spacecraft has entered a new region at the far reaches of our solar system that scientists feel is the final area the spacecraft has to cross before reaching interstellar space.

Scientists refer to this new region as a magnetic highway for charged particles because our sun’s magnetic field lines are connected to interstellar magnetic field lines. This connection allows lower-energy charged particles that originate from inside our heliosphere — or the bubble of charged particles the sun blows around itself — to zoom out and allows higher-energy particles from outside to stream in. Before entering this region, the charged particles bounced around in all directions, as if trapped on local roads inside the heliosphere.

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Inequalities

Cauchy–Schwarz inequality

From Wikipedia, the free encyclopedia

In mathematics, the Cauchy–Schwarz inequality (also known as the Bunyakovsky inequality, the Schwarz inequality, or theCauchy–Bunyakovsky–Schwarz inequality), is a useful inequality encountered in many different settings, such as linear algebra,analysis, in probability theory, and other areas. It is a specific case of Hölder’s inequality.

The inequality for sums was published by Augustin-Louis Cauchy (1821), while the corresponding inequality for integrals was first stated by Viktor Bunyakovsky (1859) and rediscovered by Hermann Amandus Schwarz (1888) (often misspelled “Schwartz”).

Statement of the inequality

The Cauchy–Schwarz inequality states that for all vectors x and y of an inner product space,

| \langle x,y\rangle|^2 \leq \langle x,x\rangle \cdot \langle y,y\rangle,

where \langle\cdot,\cdot\rangle is the inner product. Equivalently, by taking the square root of both sides, and referring to the norms of the vectors, the inequality is written as

 |\langle x,y\rangle| \leq \|x\| \cdot \|y\|.\,

Moreover, the two sides are equal if and only if x and y are linearly dependent (or, in a geometrical sense, they are parallel or one of the vectors is equal to zero).

If x_1,\ldots, x_n\in\mathbb C and y_1,\ldots, y_n\in\mathbb C are any complex numbers and the inner product is the standard inner product then the inequality may be restated in a more explicit way as follows:

|x_1 \bar{y_1} + \cdots + x_n \bar{y_n}|^2 \leq (|x_1|^2 + \cdots + |x_n|^2) (|y_1|^2 + \cdots + |y_n|^2).

When viewed in this way the numbers x1, …, xn, and y1, …, yn are the components of x and y with respect to an orthonormal basis of V.

Even more compactly written:

\left|\sum_{i=1}^n x_i \bar{y_i}\right|^2 \leq \sum_{j=1}^n |x_j|^2 \sum_{k=1}^n |y_k|^2 .

Equality holds if and only if x and y are linearly dependent, that is, one is a scalar multiple of the other (which includes the case when one or both are zero).

The finite-dimensional case of this inequality for real vectors was proved by Cauchy in 1821, and in 1859 Cauchy’s studentBunyakovsky noted that by taking limits one can obtain an integral form of Cauchy’s inequality. The general result for an inner product space was obtained by Schwarz in 1885.

Proof

Let uv be arbitrary vectors in a vector space V over F with an inner product, where F is the field of real or complex numbers. We prove the inequality

 \big| \langle u,v \rangle \big| \leq \left\|u\right\| \left\|v\right\|. \,

This inequality is trivial in the case v = 0, so we assume that <vv> is nonzero. Let δ be any number in the field F. Then,

 0 \leq \left\| u-\delta v \right\|^2 = \langle u-\delta v,u-\delta v \rangle = \langle u,u \rangle - \bar{\delta} \langle u,v \rangle - \delta \langle v,u \rangle + |\delta|^2 \langle v,v\rangle. \,

Choose the value of δ that minimizes this quadratic form, namely

 \delta = \langle u,v \rangle \cdot \langle v,v \rangle^{-1}. \,

(A quick way to remember this value of δ is to imagine F to be the reals, so that the quadratic form is a quadratic polynomial in the real variable δ, and the polynomial can easily be minimized by setting its derivative equal to zero.)

We obtain

 0 \leq \langle u,u \rangle - |\langle u,v \rangle|^2 \cdot \langle v,v \rangle^{-1} \,

which is true if and only if

 |\langle u,v \rangle|^2 \leq \langle u,u \rangle \cdot \langle v,v \rangle, \,

or equivalently:

 \big| \langle u,v \rangle \big| \leq \left\|u\right\| \left\|v\right\|, \,

which completes the proof.

Notable special cases

Rn

In Euclidean space Rn with the standard inner product, the Cauchy–Schwarz inequality is

\left(\sum_{i=1}^n x_i y_i\right)^2\leq \left(\sum_{i=1}^n x_i^2\right) \left(\sum_{i=1}^n y_i^2\right).

To prove this form of the inequality, consider the following quadratic polynomial in z.

(x_1 z + y_1)^2 + \cdots + (x_n z + y_n)^2.

Since it is nonnegative it has at most one real root in z, whence its discriminant is less than or equal to zero, that is,

\left(\sum ( x_i \cdot y_i ) \right)^2 - \sum {x_i^2} \cdot \sum {y_i^2} \le 0,

which yields the Cauchy–Schwarz inequality.

An equivalent proof for Rn starts with the summation below.

Expanding the brackets we have:

 \sum_{i=1}^n \sum_{j=1}^n \left( x_i y_j - x_j y_i \right)^2   = \sum_{i=1}^n x_i^2 \sum_{j=1}^n y_j^2 + \sum_{j=1}^n x_j^2 \sum_{i=1}^n y_i^2  - 2 \sum_{i=1}^n x_i y_i \sum_{j=1}^n x_j y_j ,

collecting together identical terms (albeit with different summation indices) we find:

 \frac{1}{2} \sum_{i=1}^n \sum_{j=1}^n \left( x_i y_j - x_j y_i \right)^2   = \sum_{i=1}^n x_i^2 \sum_{i=1}^n y_i^2 - \left( \sum_{i=1}^n x_i y_i \right)^2 .

Because the left-hand side of the equation is a sum of the squares of real numbers it is greater than or equal to zero, thus:

 \sum_{i=1}^n x_i^2 \sum_{i=1}^n y_i^2 - \left( \sum_{i=1}^n x_i y_i \right)^2 \geq 0.

This form is used usually when solving school math problems.

Yet another approach when n ≥ 2 (n = 1 is trivial) is to consider the plane containing x and y. More precisely, recoordinatize Rn with any orthonormal basis whose first two vectors span a subspace containing x and y. In this basis only x_1,~x_2,~y_1 and y_2~ are nonzero, and the inequality reduces to the algebra of dot product in the plane, which is related to the angle between two vectors, from which we obtain the inequality:

|x \cdot y| = \|x\| \|y\| | \cos \theta | \le \|x\| \|y\|.

When n = 3 the Cauchy–Schwarz inequality can also be deduced from Lagrange’s identity, which takes the form

\langle x,x\rangle \cdot \langle y,y\rangle = |\langle x,y\rangle|^2 + |x \times y|^2

from which readily follows the Cauchy–Schwarz inequality.

L2

For the inner product space of square-integrable complex-valued functions, one has

\left|\int f(x) g(x)\,dx\right|^2\leq\int \left|f(x)\right|^2\,dx \cdot \int\left|g(x)\right|^2\,dx.

A generalization of this is the Hölder inequality.

Use

The triangle inequality for the inner product is often shown as a consequence of the Cauchy–Schwarz inequality, as follows: given vectors x and y:

 \begin{align} \|x + y\|^2 & = \langle x + y, x + y \rangle \\ & = \|x\|^2 + \langle x, y \rangle + \langle y, x \rangle + \|y\|^2 \\ & \le \|x\|^2 + 2|\langle x, y \rangle| + \|y\|^2 \\ & \le \|x\|^2 + 2\|x\|\|y\| + \|y\|^2 \\ & = \left(\|x\| + \|y\|\right)^2. \end{align}

Taking square roots gives the triangle inequality.

The Cauchy–Schwarz inequality allows one to extend the notion of “angle between two vectors” to any real inner product space, by defining:

 \cos\theta_{xy}=\frac{\langle x,y\rangle}{\|x\| \|y\|}.

The Cauchy–Schwarz inequality proves that this definition is sensible, by showing that the right hand side lies in the interval [−1, 1], and justifies the notion that (real) Hilbert spaces are simply generalizations of the Euclidean space.

It can also be used to define an angle in complex inner product spaces, by taking the absolute value of the right hand side, as is done when extracting a metric from quantum fidelity.

The Cauchy–Schwarz is used to prove that the inner product is a continuous function with respect to the topology induced by the inner product itself.

The Cauchy–Schwarz inequality is usually used to show Bessel’s inequality.

Other proofs

If either \left|x\right> or \left|y\right> are the zero vector, the statement holds trivially, so assume that both are nonzero.

For any nonzero vector \left|V\right>, \left<V|V\right> > 0. (NOTE: merits own proof)

\displaystyle \left< \alpha X + Y| \alpha X + Y \right> \geq 0

If the inner product is symmetric. Let \alpha be a real scalar.

\displaystyle \alpha^2\left< X |X\right>+\alpha(\left< X |Y\right>+\left< Y |X\right>)+ \left<Y|Y \right> \geq 0

The last expression is a quadratic polynomial that is non-negative for any \alpha.  The quadratic has either two complex roots,or  a single  real root. Intuitively, the polynomial is either ‘floating above’ the horizontal axis, if it has two complex roots, or tangent to it if it has one real root, since it can’t have two real roots because the graph of the function would have to ‘pass under’ the horizontal axis and take some negative values.

The roots are given by the quadratic formula

math

In particular, the term math must either be negative, yielding two complex roots, or zero, yielding a single real root. Thus

math

math
math
math

Substituting the values of mathmath and math into the last of these inequalities, it can be seen that

\displaystyle (\left< X |Y\right>+\left< Y |X\right>)^2 \leq 4\left< X |X\right> \left<Y|Y \right>

If the inner product is symmetric, this proves the inequality.

An alternative proof follows from the expression

\displaystyle  \frac{{(a+b)}^2}{x+y}=\frac{a^2}{x}+\frac{b^2}{y},

valid for a and b real and $x>0$ and $y>0$. This expression is a restatement of (a y - b x)^2 \geq  0. From this one can get a general n-term expression

\displaystyle  \frac{{(a_1+a_2+\cdots+a_n)}^2}{x_1+x_2+\cdots x_n}=\frac{a_1^2}{x_1}+\frac{a_2^2}{x_2}+\cdots +\frac{a_n^2}{x_n}

To get cauchy-Scwarz inequality set a_k=\alpha_k \beta_k and x_k=\beta_k^2.

If the inner product is skew-symmetric, take

\displaystyle \alpha = -\frac{\left<X|Y\right>}{\left<X|X\right>}

\displaystyle    ( -\frac{\left< Y |X\right>}{\left< X |X\right>} \left<X\right| + \left<Y\right|)( -\frac{\left< X |Y\right>}{\left< X |X\right>} \left|X\right> + \left|Y\right> ) \geq 0

\displaystyle    \frac{\left< Y |X\right>\left< X |Y\right>}{\left< X |X\right>} -\frac{\left< Y |X\right>\left< X |Y\right>}{\left< X |X\right>} -\frac{\left< Y |X\right>\left< X |Y\right>}{\left< X |X\right>} + \left<Y|Y\right> \geq 0

\displaystyle    \left< X |X\right>\left<Y|Y\right> \geq \left< Y |X\right>\left< X |Y\right>
QED

The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities

Cauchy-Schwarz Inequality: Yet Another Proof

http://en.wikipedia.org/wiki/Cauchy%E2%80%93Schwarz_inequality

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Isometry

Isometry

From Wikipedia, the free encyclopedia
For the mechanical engineering and architecture usage, see isometric projection. For isometry in differential geometry, seeisometry (Riemannian geometry).
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.
Isometries are often used in constructions where one space is embedded in another space. For instance, the completion of a metric space M involves an isometry from M into M’, a quotient set of the space of Cauchy sequences on M. The original space M is thus isometrically isomorphic to a subspace of a complete metric space, and it is usually identified with this subspace. Other embedding constructions show that every metric space is isometrically isomorphic to a closed subset of some normed vector space and that every complete metric space is isometrically isomorphic to a closed subset of some Banach space.
An isometric surjective linear operator on a Hilbert space is called a unitary operator.