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Isoperimetric Theorem

Isoperimetric theorem

Isoperimetry literally means "having an equal perimeter". In mathematics, isoperimetry is the general study of geometric figures having equal boundaries.

The isoperimetric problem in the plane

The classical isoperimetric problem dates back to antiquity. The problem can be stated as follows: Among all closed curves in the plane of fixed perimeter, which curve (if any) maximizes the area of its enclosed region? This question can be shown to be equivalent to the following problem: Among all closed curves in the plane enclosing a fixed area, which curve (if any) minimizes the perimeter? This problem is conceptually related to the principle of least action in physics, in that it can restated: what is the principle of action which encloses the greatest area, with the greatest economy of effort? The 15th-century philosopher and scientist, Cardinal Nicholas of Cusa, considered rotational action, the process by which a circle is generated, to be the most direct reflection, in the realm of sensory impressions, of the process by which the universe is created. Johannes Kepler invoked the isoperimetric principle in discussing the morphology of the solar system, in Mysterium Cosmographicum. Although the circle appears to be an obvious solution to the problem, proving this fact is rather difficult. The first progress toward the solution was made by Jakob Steiner in 1838, using a geometric method later named Steiner symmetrisation. Steiner showed that if a solution existed, then it must be the circle. Steiner's proof was completed later by several other mathematicians. Steiner begins with some geometric constructions which are easily understood; for example, it can be shown that any closed curve that is not fully convex, can be modified to enclose more area, by "flipping" the concave areas so that they become convex. It can further be shown that any closed curve which is not fully symmetrical can be "tilted" so that it encloses more area. The one shape that is perfectly convex and symmetrical is the circle, although this, in itself, does not represent a rigorous proof of the isoperimetric theorem (see external links). The theorem is usually stated in the form of an inequality that relates the perimeter and area of a closed curve in the plane. If P is the perimeter of the curve and A is the area of the region enclosed by the curve, then the inequality states that :

4\pi A \le P^2.

For the case of a circle of radius r, we have A = πr² and P = 2πr, and substituting these into the inequality shows that the circle does indeed maximize the area among all curves of fixed perimeter. In fact, the circle is the only curve that maximizes the area. There are dozens of proofs of this classic inequality. Several of these are discussed in the Treiberg paper below. In 1901, Hurwitz gave a purely analytic proof of the classical isoperimetric inequality based on Fourier series and Green's theorem. The isoperimetric theorem generalises to higher dimensional spaces: the domain with volume 1 with the minimal surface area is always a ball.

External links

- [http://www.math.jhu.edu/~js/Math427/coursenotes/node3.html Course notes on The Isoperimetric Inequality]
- [http://www.math.utah.edu/~treiberg/isoperim/isop.pdf Treiberg: Several proofs of the isoperimetric inequality]
- [http://www.cut-the-knot.org/do_you_know/isoperimetric.shtml Isoperimetric Theorem] at cut-the-knot Category:Multivariate calculus Category:Calculus of variations

Perimeter

The perimeter is the distance around a given two-dimensional object. The word perimeter is a Greek root meaning measure around, or literally "around measure." Perimeter can be calculated for any two dimensional object, with the right formula.

Practical Uses

Perimeter and area play a great role in today's world. Perimeter is used in calculating the border of an object such as a yard or flowerbed when a fence or other border is being installed around the edges. Area is used when all the area inside of a perimeter is being covered with something, such as a yard being covered with sod or fertilizer.

Formulas

As a general rule, the perimeter of a polygon can always be calculated by adding all the length of the sides together. Circles: For circles the equation is P= 2 π r, where r is the radius and π is the mathematical constant, or about 3.14. (An equivalent formula is P= π d, where d is the diameter). The equation for area is A= πr². R is the radius and pi is about 3.14. Pi comes from the equation c/d, which is the circumference of a circle divided by the diameter. Triangles: For triangles the equation for perimeter is P=Side1 + Side2 + Side3. The area can be calculated by the equation A=.5bh. B is the length of the base and H is the height of the triangle. Quadrilaterals: For quadrilaterals the equation for perimeter is P=Side1 + Side2 + Side3 + Side4. More area formulas can be found here.

Curve

In mathematics, the concept of a curve tries to capture the intuitive idea of a geometrical one-dimensional and continuous object. Simple examples are the circle or the straight line. A large number of other curves have been studied in geometry. This article is about the general theory. The term curve is also used in ways making it almost synonymous with mathematical function (as in learning curve), or graph of a function (Phillips curve).

Definitions

In mathematics, a (topological) curve is defined as follows. Let

I

be an interval of real numbers (i.e. a non-empty connected subset of

\mathbb

). Then a curve

\!\,\gamma

is a continuous mapping

\,\!\gamma : I \rightarrow X

, where

X

is a topological space. The curve

\!\,\gamma

is said to be simple if it is injective, i.e. if for all

x

y

I

, we have

\,\!\gamma(x) = \gamma(y) \rightarrow x = y

. If

I

is a closed bounded interval

\,\![a, b]

, we also allow the possibility

\,\!\gamma(a) = \gamma(b)

(this convention makes it possible to talk about closed simple curve). If

\gamma(x)=\gamma(y)

for some

x\ne y

(other than the extremities of

I

), then

\gamma(x)

is called a double (or: multiple) point of the curve. A curve

\!\,\gamma

is said to be closed or a loop if

\,\!I = [a, b]

and if

\!\,\gamma(a) = \gamma(b)

. A closed curve is thus a continuous mapping of the circle

S^1

; a simple closed curve is also called a Jordan curve. A plane curve is a curve for which X is the mathematical plane — these are the examples first encountered — or in some cases the projective plane. A space curve is a curve for which X is of three dimensions, usually Euclidean space; a skew curve is a space curve which lies in no plane. These definitions also apply to algebraic curves (see below). This definition of curve captures our intuitive notion of a curve as a connected, continuous geometric figure that is "like" a line, although it also includes figures that can be hardly called curves in common usage. For example, the image of a curve can cover a square in the plane (Peano curve). The image of simple plane curve can have Hausdorff dimension bigger than one (see Koch snowflake) and even positive Lebesgue measure (the last example can be obtained by small variation of the Peano curve construction). The dragon curve is yet another weird example.

Conventions and terminology

The distinction between a curve and its image is important. Two distinct curves may have the same image. For example, a line segment can be traced out at different speeds, or a circle can be traversed a different number of times. Many times, however, we are just interested in the image of the curve. It is important to pay attention to context and convention in reading. Terminology is also not uniform. Often, topologists use the term "path" for what we are calling a curve, and "curve" for what we are calling the image of a curve. The term "curve" is more common in vector calculus and differential geometry.

Length of curves

X

is a metric space with metric

d

, then we can define the length of a curve

\!\,\gamma : [a, b] \rightarrow X

by :

\mbox (\gamma)=\sup \left\

A rectifiable curve is a curve with finite length. A parametrization of

\!\,\gamma

is called natural (or unit speed or parametrised by arc length) if for any

t_1

t_2

[a, b]

, we have :

\mbox (\gamma|_)=|t_2-t_1|

\!\,\gamma

is Lipschitz then it is automatically rectifiable. Moreover, in this case, one can define speed of

\!\,\gamma

t_0

as :

\mbox(t_0)=\limsup_

and then :

\mbox(\gamma)=\int_a^b \mbox(t) \, dt

In particular, if

X = \mathbb^n

is Euclidean space and

\gamma : [a, b] \rightarrow \mathbb^n

is differentiable then :

\mbox(\gamma)=\int_a^b \left| \,  \, \right| \, dt

Differential geometry

Main article: differential geometry of curves While the first examples of curves that are met are mostly plane curves (that is, in everyday words, curved lines in two-dimensional space), there are obvious examples such as the helix which exist naturally in three dimensions. The needs of geometry, and also for example classical mechanics are to have a notion of curve in space of any number of dimensions. In general relativity, a world line is a curve in spacetime. If

X

is a differentiable manifold, then we can define the notion of differentiable curve in

X

. This general idea is enough to cover many of the applications of curves in mathematics. From a local point of view one can take

X

to be Euclidean space. On the other hand it is useful to be more general, in that (for example) it is possible to define the tangent vectors to

X

by means of this notion of curve. If

X

is a smooth manifold, a smooth curve in

X

is a smooth map :

\!\,\gamma : I \rightarrow X.

This is a basic notion. There are less and more restricted ideas, too. If

X

is a

C^k

manifold (i.e., a manifold whose charts are

k

times continuously differentiable), then a

C^k

curve in

X

is such a curve which is only assumed to be

C^k

(i.e.

k

times continuously differentiable). If

X

is an analytic manifold (i.e. infinitely differentiable and charts are expressible as power series), and

\!\,\gamma

is an analytic map, then

\!\,\gamma

is said to be an analytic curve. A differentiable curve is said to be regular if its derivative never vanishes. (In words, a regular curve never slows to a stop or backtracks on itself.) Two

C^k

differentiable curves :

\!\,\gamma_1 :I \rightarrow X

and :

\!\,\gamma_2 : J \rightarrow X

are said to be equivalent if there is a bijective

C^k

map :

\!\,p : J \rightarrow I

such that the inverse map :

\!\,p^ : I \rightarrow J

is also

C^k

, and :

\!\,\gamma_(t) = \gamma_(p(t))

for all

t

. The map

\!\,\gamma_2

is called a reparametrisation of

\!\,\gamma_1

; and this makes an equivalence relation on the set of all

C^k

differentiable curves in

X

. A

C^k

arc is an equivalence class of

C^k

curves under the relation of reparametrisation.

Algebraic curve

Main article: Algebraic curve In the setting of algebraic geometry, a curve is usually defined to be an algebraic curve. These include, for example, elliptic curves, which are studied in number theory and which have important applications to cryptography. Algebraic curves are more akin to surfaces than curves. Non-singular complex projective algebraic curves are in fact compact Riemann surfaces.

History

A curve may be a locus, or a path. That is, it may be a graphical representation of some property of points; or it may be traced out, for example by a stick in the sand on a beach. Of course if one says curved in ordinary language, it means bent (not straight), so refers to a locus. This leads to the general idea of curvature. As we now understand, after Newtonian dynamics, to follow a curved path a body must experience acceleration. Before that, the application of current ideas to (for example) the physics of Aristotle is probably anachronistic. This is important because major examples of curves are the orbits of the planets. One reason for the use of the Ptolemaic system of epicycle and deferent was the special status accorded to the circle as curve. The conic sections had been deeply studied by Apollonius of Perga. They were applied in astronomy by Kepler. The Greek geometers had studied many other kinds of curves. One reason was their interest in geometric constructions, going beyond ruler-and-compass constructions. In that way, the intersection of curves could be used to solve some polynomial equations, such as that involved in trisecting an angle. Newton also worked on an early example in the calculus of variations. Solutions to variational problems, such as the brachistochrone and tautochrone questions, introduced properties of curves in new ways (in this case, the cycloid). The catenary gets its name as the solution to the problem of a hanging chain, the sort of question that became routinely accessible by means of differential calculus. In the eighteenth century came the beginnings of the theory of plane algebraic curves, in general. Newton had studied the cubic curves, in the general description of the real points into 'ovals'. The statement of Bézout's theorem showed a number of aspects which were not directly accessible to the geometry of the time, to do with singular points and complex solutions. From the nineteenth century there is not a separate curve theory, but rather the appearance of curves as the one-dimensional aspect of projective geometry, and differential geometry; and later topology, when for example the Jordan curve theorem was understood to lie quite deep, as well as being required in complex analysis. The era of the space-filling curves finally provoked the modern definitions of curve.

External links

- [http://www-gap.dcs.st-and.ac.uk/~history/Curves/Curves.html List of famous curves] Category:Curves Category:Metric geometry Category:Topology Category:General topology ko:곡선 ja:曲線

Principle of least action

The principle of least action was first formulated by Pierre-Louis Moreau de Maupertuis, who said that "Nature is thrifty in all its actions". See action (physics). Others who developed the idea included Euler and Leibniz. It should be said that, from the point of view of the calculus of variations, a principle of stationary action is a more accurate formulation. Earlier, Pierre de Fermat had introduced the ideas that rays of light, in optical conditions such as refraction and reflection, followed a principle of least time: see Fermat's principle. The principle of least action led to the development of the Lagrangian and Hamiltonian formulations of classical mechanics. Although they are at first more difficult to grasp, they have the advantage that their world-view is more transferable to the frameworks of relativistic and quantum-mechanical physics than that of Newton's laws. This has caused some people to think that this principle is a "deep" principle of physics.

External links

- http://www.eftaylor.com/leastaction.html Category:Calculus of variations ja:最小作用の原理

Rotation

:This article is about rotation as a movement of a physical body. For other meanings, see rotation (disambiguation). rotation (disambiguation) rotation (disambiguation) Rotation of a planar body is the movement when points of the body travel in circular trajectories around a fixed point called the center of rotation. For a three-dimensional body, the rotation is around an axis — it amounts to rotation in each plane perpendicular to the axis around the intersection of the plane and the axis. An example of rotation of a planar figure around a point is the movement of the propeller of an aircraft. A door attached to the wall by two or more hinges rotates around the axis going through the hinges. hinge If the axis of rotation is within the body, the body is said to rotate upon itself, or spin. Among such rotations, the simplest case is that of constant angular frequency (see also below).

Allowed rotations

An object may allow rotation with respect to an attached other object by means of one or more hinges (e.g. a door, scissors, a hinge joint). Gear couplings and universal joints connect two driveshafts at an angle (rotating about their own axes) to transmit torque, by two pairs of hinges allowing rotation about two other axes. A ball-and-socket joint, e.g. in the shoulder, allows rotations about all three axes. See also: rolling, axle, wheel, rolling, gyration, gyroscope, rotations in anatomy

Mathematics

Mathematically, a rotation is a rigid body movement which keeps a point fixed; unlike a translation. This definition is applicable both for rotations in a plane (two dimensions) and in space (three dimensions). It turns out that a rotation in the three-dimensional space keeps fixed not just a single point, but rather an entire line; that is to say, any rotation in the three dimensional space is a rotation around an axis. This is a consequence of Euler's rotation theorem. Any rigid body movement is in fact either a rotation, or a translation, or a combination of the two. If one does a rotation around a point (axis), followed by another rotation around the same point (axis), the total result is yet another rotation. The reverse (inverse) of a rotation is also a rotation. It follows that the rotations around a point or axis form a group. If however one performs rotation around a point (axis) followed by rotation around another point (axis), the overall movement may not be a rotation anymore. group Rotations around the x, y and z axes are called principal rotations. Rotation around any axis can be performed by taking a rotation around the x axis, followed by a rotation around the y axis, and followed by a rotation around the z axis. That is to say, any spatial rotation can be decomposed into a combination of principal rotations. In flight dynamics, the principal rotations are known as pitch, roll and yaw. See also: curl, cyclic permutation, Euler angles, rigid body, rotation around a fixed axis, rotation group, rotation matrix

Astronomy

In astronomy, rotation is a commonly observed phenomenon. Stars, planets and similar bodies all rotate around their axes, while planets also rotate about a star such as the Sun, and moons also rotate about a planet. The motion of the components of galaxies is complex, but it usually includes a rotation component. One consequence of the rotation of a planet is the phenomenon of precession. Precession has the overall effect of introducing a long-term "wobble" in the movement of the axis of a planet. For example, the tilt of the Earth's axis to its orbital plane (obliquity of the ecliptic) is currently 66.5 degrees, but this angle has slowly changed over time due to the action of precession. See also: orbital period, oblate, orbital revolution

Physics

The speed of rotation is given by the angular frequency (rad/s) or frequency (turns/s, turns/min), or period (seconds, days, etc.). The time-rate of change of angular frequency is angular acceleration (rad/s²), This change is caused by torque. The ratio of the two (how heavy is it to start, stop, or otherwise change rotation) is given by the moment of inertia. The angular velocity vector also describes the direction of the axis of rotation. Similarly the torque is a vector. According to the right-hand rule, moving away from the observer is associated with clockwise rotation and moving towards the observer with counterclockwise rotation, like a screw. See also: rotational energy, angular momentum, angular velocity, centrifugal force, centripetal force, circular motion, circular orbit, Coriolis effect, spin, rigid body angular momentum

Amusement rides

Many amusement rides provide rotation. A Ferris wheel and observation wheel have a horizontal central axis, and parallel axes for each gondola, where the rotation is opposite, by gravity or mechanically. As a result at any time the orientation of the gondola is upright (not rotated), just translated. The tip of the translation vector describes a circle. A carousel provides rotation about a vertical axis. Many rides provide a combination of rotations about several axes. In Chair-O-Planes the rotation about the vertical axis is provided mechanically, while the rotation about the horizontal axis is due to the centrifugal force. In roller coaster inversions the rotation about the horizontal axis is one or more full cycles, where the centrifugal force keeps people in their seats.

External links

- [http://www.cut-the-knot.org/Curriculum/Geometry/RotationTransform.shtml Product of Rotations] at cut-the-knot
- [http://www.cut-the-knot.org/Curriculum/Geometry/Connes.shtml When a Triangle is Equilateral] at cut-the-knot Category:Euclidean geometry Category:Celestial mechanics ja:自転

Reflection

The word reflection (also spelt reflexion in British English) can refer to several different concepts:
- In mathematics, reflection is a transformation of a space, a reflection group is a group generated by reflections, and the reflection principle is a theorem of set theory.
- In physics, reflection is a wave phenomenon.
- In computer science, reflection is a programming language feature for metaprogramming, in languages or language engines like Java and .NET.
- In electricity, reflection is reflected voltage in an electrical signal due to an impedance change.
- In music, it may be
- Reflection, a 1998 album by Paradise Lost.
- "Reflection", a song by Tool from the album Lateralus.
- "Reflection", a song by Christina Aguilera.
- In art, "Reflection", also called "Parasight", is an art installation by Shane Cooper.
- Reflection, an online card game developed by Free Fall Associates.
- In psychology, reflection is a process of thoughtful meditation or rumination: see introspection.
- Reflection, one's image as seen in a mirror. See the physical phenomenon above.
- In education, 'reflection' is the art of turning experience into learning.

External links

- [http://rdfintrospector.blogspot.com/2005/01/introspection-warts-and-all.html Blog Entry on the usage of the term Introspection in the news]
- [http://rdfintrospector.blogspot.com/2005/01/reflections-about-usage-of-term.html Blog Entry on the usage of the term Reflection in the news]

Johannes Kepler

Johannes Kepler (December 27, 1571 – November 15, 1630), a key figure in the scientific revolution, was a German mathematician, astronomer and astrologer of famed brilliance. He is best known for his laws of planetary motion, expounded in the two books Astronomia nova and Harmonice Mundi. Kepler was a professor of mathematics at the University of Graz, court mathematician to Emperor Rudolf II, and court astrologer to General Wallenstein. Early in his career, Kepler was an assistant to Tycho Brahe. Kepler's career also coincided with that of Galileo Galilei. He is sometimes referred to as "the first theoretical astrophysicist", although Carl Sagan also referred to him as the last scientific astrologer.

Life

Kepler was born on December 27, 1571 at the Imperial Free City of Weil der Stadt (now part of the Stuttgart Region in the German state of Baden-Württemberg, 30 km west of Stuttgart's center). His grandfather had been Lord Mayor of that town, but by the time Johannes was born, the Kepler family fortunes were in decline. His father earned a precarious living as a mercenary, and abandoned the family when Johannes was 17. His mother, an inn-keeper's daughter, had a reputation for involvement in witchcraft. Born prematurely, Johannes is said to have been a weak and sickly child, but despite his ill health, he was precociously brilliant - he often impressed travelers at the inn [aforementioned] with his phenomenal mathematical faculty as a child. Though he excelled in his schooling, Kepler was frequently bullied, and was plagued by a belief that he was physically repulsive, thoroughly unlikable and, compared to the other pupils, an outsider. This ostracizing probably led him to turn to the world of ideas, as well as an abiding religious conviction, for solace. He was introduced to astronomy/astrology at an early age, and developed a love for that discipline that would span his entire life. At age five, he observed the Comet of 1577, writing that he "...was taken by [his] mother to a high place to look at it." At age nine, he observed another astronomical event, the Lunar eclipse of 1580, recording that he remembered being "called outdoors" to see it and that the moon "appeared quite red." In 1587, Kepler began attending the University of Tübingen, where he proved himself to be a superb mathematician. Upon his graduation from that school in 1591, he went on to pursue study in theology, becoming a part of the Tübingen faculty. However, before he took his final exams he was recommended for the vacant post of teacher of mathematics and astronomy at the Protestant school in Graz, Austria. He accepted the position in April of 1594, at the age of 23. In April 1597, Kepler married Barbara Müller. She died in 1611 and was outlived by two children. In December 1599, Tycho Brahe wrote to Kepler, inviting Kepler to assist him at Benátky nad Jizerou outside Prague. After Tycho's death, Kepler was appointed Imperial Mathematician (from November 1601 to 1630) to the Habsburg Emperors. In October 1604, Kepler observed the supernova which was subsequently named Kepler's Star. In January 1612 the Emperor died, and Kepler took the post of provincial mathematician in Linz. In 1611, Kepler published a monograph on the origins of snowflakes, the first known work on the subject. He correctly theorized that their hexagonal nature was due to cold, but did not ascertain a physical cause for this. The question of snowflakes was not resolved until the 20th century. On March 8, 1618 Kepler discovered the third law of planetary motion: distance cubed over time squared. He initially rejected this idea, but later confirmed it on May 15 of the same year. In August of 1620, Katherine, Kepler's mother, was arrested in Leonberg as a witch; she was imprisoned for 14 months. She was released in October 1621 after attempts to convict her failed. Even though she was subjected to torture, she refused to confess to the charges. However, only the courageous personal intervention of Kepler (despite the risk to be arrested as well) and his reputation as the famous Imperial Mathematician rescued her.

Death

On November 15, 1630 Kepler died of a fever in Regensburg. In 1632, only two years after his death, his grave was demolished by the Swedish army in the Thirty Years' War.

Work

Kepler lived in an era when there was no clear distinction between astronomy and astrology, and no consensus on the scientific method as the correct way to decide what was correct or incorrect in science. His ideas are therefore a fascinating mixture of what would today be considered mathematical physics and nonsensical mysticism. Although the subsections below separate the two, Kepler did not see them as separate. Kepler was a Pythagorean numbers mystic. That is, he considered mathematical relationships to be at the base of all nature. Thus all creation was an integrated whole. This may be contrasted with the Platonic and Aristotelian notion that the Earth was fundamentally different from the rest of the universe, being composed of different substances and with different natural laws applying. Thus Kepler was the first to attempt to discover universal laws. This quest led him to apply terrestrial physics to celestial bodies, which produced the three Laws of Planetary Motion. Conversely, he was convinced that celestial bodies influence terrestrial events. One result of this belief was his correct assessment of the Moon's role in generating the tides, years before Galileo's incorrect formulation. Another was his belief that someday it would be possible to develop a "scientific astrology," while not believing that any astrologic rules currently propounded were at all valid.

Scientific work

Kepler's laws

Kepler inherited from Tycho Brahe a wealth of the most accurate raw data ever collected on the positions of the planets. The difficulty was to make sense of it. The orbital motions of the other planets are viewed from the vantage point of the Earth, which is itself orbiting the sun. As shown in the example below, this can cause the other planets to appear to move in strange loops. Kepler concentrated on the orbit of Mars, but he had to know the orbit of the Earth accurately first. In order to do this, he needed a surveyor's baseline. In a stroke of pure genius, he used Mars and the Sun as his baseline, since without knowing the actual orbit of Mars, he knew that it would be in the same place in its orbit at times separated by its orbital period. Thus the orbital positions of the Earth could be computed, and from them the orbit of Mars. He was able to deduce his planetary laws without knowing the exact distances of the planets from the sun, since his geometrical analysis needed only the ratios of their solar distances. Image:Retrograde-motion-of-mars.png Kepler, unlike Brahe, held to the heliocentric model of the solar system, and starting from that framework, he made twenty years of painstaking trial-and-error attempts at making some sense out of the data. He finally arrived at his three laws of planetary motion: three laws of planetary motion 1. Kepler's elliptical orbit law: The planets orbit the sun in elliptical orbits with the sun at one focus. 2. Kepler's equal-area law: The line connecting a planet to the sun sweeps out equal areas in equal amounts of time. 3. Kepler's law of periods: The time required for a planet to orbit the sun, called its period, is proportional to the long axis of the ellipse raised to the 3/2 power. The constant of proportionality is the same for all the planets. Using these laws, he was the first astronomer to successfully predict a transit of Venus (for the year 1631). Kepler's laws were the first clear evidence in favor of the heliocentric model of the solar system, because they only came out to be so simple under the heliocentric assumption. Kepler, however, never discovered the deeper reasons for the laws, despite many years of what would now be considered non-scientific mystical speculation. Isaac Newton eventually showed that the laws were a consequence of his laws of motion and law of universal gravitation. (From the modern vantage point, the equal-area law is more easily understood as arising from conservation of angular momentum.)

1604 supernova

angular momentum On October 17, 1604, Kepler observed that an exceptionally bright star had suddenly appeared in the constellation Ophiuchus. (It was first observed by several others on October 9.) The appearance of the star, which Kepler described in his book De Stella nova in pede Serpentarii ('On the New Star in Ophiuchus's Foot'), provided further evidence that the cosmos was not changeless; this was to influence Galileo in his argument. It has since been determined that the star was a supernova, the second in a generation, later called Kepler's Star or Supernova 1604. No further supernovae have been observed in the Milky Way, though others outside our galaxy have been seen.

Other scientific and mathematical work

Kepler also made fundamental investigations into combinatorics, geometrical optimization, and natural phenomena such as snowflakes, always with an emphasis on form and design. He was also one of the founders of modern optics, defining e.g. antiprisms and the Kepler telescope (see Kepler's books Astronomiae Pars Optica — i.a. theoretical explanation of the camera obscura — and Dioptrice). In addition, since he was the first to recognize the non-convex regular solids (such as the stellated dodecahedra), they are named Kepler solids in his honor.

Mysticism and astrology

Mysticism

Kepler discovered the laws of planetary motion while trying to achieve the Pythagorean purpose of finding the harmony of the celestial spheres. In his cosmologic vision, it was not a coincidence that the number of perfect polyhedra was one less than the number of known planets. Having embraced the Copernican system, he set out to prove that the distances from the planets to the sun were given by spheres inside perfect polyhedra, all of which were nested inside each other. The smallest orbit, that of Mercury, was the innermost sphere. He thereby identified the five Platonic solids with the five intervals between the six known planets — Mercury, Venus, Earth, Mars, Jupiter, Saturn; and the five classical elements. In 1596 Kepler published Mysterium Cosmographicum, or The Cosmic Mystery. Here is a selection explaining the relation between the planets and the Platonic solids: :… Before the universe was created, there were no numbers except the Trinity, which is God himself… For, the line and the plane imply no numbers: here infinitude itself reigns. Let us consider, therefore, the solids. We must first eliminate the irregular solids, because we are only concerned with orderly creation. There remain six bodies, the sphere and the five regular polyhedra. To the sphere corresponds the heaven. On the other hand, the dynamic world is represented by the flat-faces solids. Of these there are five: when viewed as boundaries, however, these five determine six distinct things: hence the six planets that revolve about the sun. This is also the reason why there are but six planets… sphere from Mysterium Cosmographicum (1596)]] sphere :… I have further shown that the regular solids fall into two groups: three in one, and two in the other. To the larger group belongs, first of all, the Cube, then the Pyramid, and finally the Dodecahedron. To the second group belongs, first, the Octahedron, and second, the Icosahedron. That is why the most important portion of the universe, the Earth—where God's image is reflected in man—separates the two groups. For, as I have proved next, the solids of the first group must lie beyond the earth's orbit, and those of the second group within… Thus I was led to assign the Cube to Saturn, the Tetrahedron to Jupiter, the Dodecahedron to Mars, the Icosahedron to Venus, and the Octahedron to Mercury… To emphasize his theory, Kepler envisaged an impressive model of the universe which shows a cube, inside a sphere, with a tetrahedron inscribed in it; another sphere inside it with a dodecahedron inscribed; a sphere with an icosahedron inscribed inside; and finally a sphere with an octahedron inscribed. Each of these celestial spheres had a planet embedded within them, and thus defined the planet's orbit. In his 1619 book, Harmonice Mundi or Harmony of the Worlds, as well as the aforementioned Mysterium Cosmographicum, he also made an association between the Platonic solids with the classical conception of the elements: the tetrahedron was the form of fire, the octahedron was that of air, the cube was earth, the icosahedron was water, and the dodecahedron was the cosmos as a whole or ether. There is some evidence this association was of ancient origin, as Plato tells of one Timaeus of Locri who thought of the Universe as being enveloped by a gigantic dodecahedron while the other four solids represent the "elements" of fire, air, earth, and water. In 1975, nine years after its founding, the College for Social and Economic Sciences Linz (Austria) was renamed Johannes Kepler University Linz in honor of Johannes Kepler, since he wrote his magnum opus harmonice mundi in Linz. To his disappointment, Kepler's attempts to fix the orbits of the planets within a set of polyhedrons never worked out, but it is a testimony to his integrity as a scientist that when the evidence mounted against the cherished theory he worked so hard to prove, he abandoned it. His most significant achievements came from the realization that the planets moved in elliptical, not circular, orbits. This realization was a direct consequence of his failed attempt to fit the planetary orbits within polyhedra. Kepler's willingness to abandon his most cherished theory in the face of precise observational evidence also indicates that he had a very modern attitude to scientific research. Kepler also made great steps in trying to describe the motion of the planets by appealing to a force which resembled magnetism, which he believed emanated from the sun. Although he did not discover gravity, he seems to have attempted to invoke the first empirical example of a universal law to explain the behaviour of both earthly and heavenly bodies.

Astrology

Kepler disdained astrologers who pandered to the tastes of the common man without knowledge of the abstract and general rules, but he saw compiling prognostications as a justified means of supplementing his meagre income. Yet, it would be a mistake to take Kepler's astrological interests as merely pecuniary. As one historian, John North, put it, 'had he not been an astrologer he would very probably have failed to produce his planetary astronomy in the form we have it.' Kepler believed in astrology in the sense that he was convinced that astrological aspects physically and really affected humans as well as the weather on earth. He strove to unravel how and why that was the case and tried to put astrology on a surer footing, which resulted in the On the more certain foundations of astrology (1601), in which, among other technical innovations, he was the first to propose the quincunx aspect. In The Intervening Third Man, or a warning to theologians, physicians and philosophers (1610), posing as a third man between the two extreme positions for and against astrology, Kepler advocated that a definite relationship between heavenly phenomena and earthly events could be established. At least 800 horoscopes and natal charts drawn up by Kepler are still extant, several of himself and his family, accompanied by some unflattering remarks. As part of his duties as district mathematician to Graz, Kepler issued a prognostication for 1595 in which he forecast a peasant uprising, Turkish invasion and bitter cold, all of which happened and brought him renown. Kepler is known to have compiled prognostications for 1595 to 1606, and from 1617 to 1624. As court mathematician, he explained to Rudolf II the horoscopes of the Emperor Augustus and Muhammad, and gave astrological prognosis for the outcome of a war between the Republic of Venice and Paul V. In the On the new star (1606) Kepler explicated the meaning of the new star of 1604 as the conversion of America, downfall of Islam and return of Christ. The De cometis libelli tres (1619) is also replete with astrological predictions.

Kepler on God

"I was merely thinking God's thoughts after him. Since we astronomers are priests of the highest God in regard to the book of nature," wrote Kepler, "it benefits us to be thoughtful, not of the glory of our minds, but rather, above all else, of the glory of God."

Writings by Kepler

astrological
- Mysterium cosmographicum (The Cosmic Mystery) (1596)
- Astronomiae Pars Optica (The Optical Part of Astronomy) (1604)
- De Stella nova in pede Serpentarii (On the New Star in Ophiuchus's Foot) (1604)
- Astronomia nova (New Astronomy) (1609)
- Dioptrice (Dioptre) (1611)
- Nova stereometria doliorum vinariorum (New Stereometry of wine barrels) (1615)
- Epitome astronomiae Copernicanae (published in three parts from 1618-1621)
- Harmonice Mundi (Harmony of the Worlds) (1619)
- Tabulae Rudolphinae (1627)
- Somnium (The Dream) (1634) - considered the first precursor of science fiction.

Kepler in fiction

- John Banville: Kepler: a novel. London: Secker & Warburg, 1981 ISBN 0-436-03264-3 (and later eds.). Also published: Boston, MA:Godine, 1983 ISBN 0-87923-438-5. Arthur Koestler : The Sleepwalkers; A History of Man's Changing Vision of the Universe. Publihed by Hutchinson. Synopsis; a profound outline of astrology from Pythagoras to Newton with weight given on Kepler.

Machines named in Kepler's honor

Kepler Space Observatory, a solar-orbiting, planet-hunting telescope due to be launched by NASA in 2008.

External links

- [http://posner.library.cmu.edu/Posner/books/annotation.cgi?call=520_K38PN Annotation: Posner Family Collection in Electronic Format] Harmonices mvndi The Harmony of the Worlds in fulltext facsimile in Latin
- Full text of [http://www.gutenberg.net/etext/12406 Kepler] by Walter W. Bryant, from Project Gutenberg
- [http://www.skyscript.co.uk/kepler.html Kepler and the "Music of the Spheres"]
- [http://www.dmoz.org/Science/Astronomy/History/People/Kepler,_Johannes/ Johannes Kepler Directory]
- [http://www.depauw.edu/sfs/backissues/8/christianson8art.htm Gale E. Christianson- Kepler's Somnium: Science Fiction and the Renaissance Scientist]
- Kepler, Johannes Kepler, Johannes Kepler, Johannes Kepler, Johannes Kepler, Johannes Kepler, Johannes Kepler, Johannes Kepler, Johannes Kepler, Johannes als:Johannes Kepler ko:요하네스 케플러 ja:ヨハネス・ケプラー

Mathematician

A mathematician is a person whose area of study and research is mathematics. Today, most mathematicians are professors at a university or other research institution; however, a minority have a non-academic career and are often known as amateur mathematicians. While a number of misinformed people may believe mathematics is fully understood (as it is often presented this way in elementary textbooks), in fact, there is ongoing research into many areas of mathematics. In fact, the publication of new discoveries in mathematics continues at an immense rate in hundreds of scientific journals, many of them devoted to mathematics and many devoted to subjects to which mathematics is applied (such as theoretical computer science and theoretical physics). Unlike the other sciences, research in mathematics generally does not consist of performing experiments. Rather, mathematics is about problem-solving, where truths are deduced from other known truths. Computer experiments and other numerical evidence might be a part of this process, but in the end, mathematics research is about constructing proofs of theorems. In particular, calculation is not a big part of mathematics research, and mathematicians need not have any extraordinary ability in adding or multiplying numbers. See mental calculators to read about prodigies at performing such calculations.

Motivation

Mathematicians are typically interested in finding and describing patterns that may have originally arisen from problems of calculation, but have now been abstracted to become problems of their own. Problems have come from physics, economics, games, generalizations of earlier mathematics, and some problems are simply created for the challenge of solving them. Although much mathematics is not immediately useful, history has shown the eventually applications are found. For example, number theory originally seemed to be without purpose, but after the invention of computers it gained countless applications to algorithms and cryptography.

Differences

Mathematicians differ from philosophers in that the primary questions of mathematics are assumed (for the most part) to transcend the context of the human mind; the idea that "2+2=4 is a true statement" is assumed to exist without requiring a human mind to state the problem. Not all mathematicians would strictly agree with the above; the philosophy of mathematics contains several viewpoints on this question. Mathematicians differ from physical scientists such as physicists or engineers in that they do not typically perform experiments to confirm or deny their conclusions; and whereas every scientific theory is always assumed to be an approximation of truth, mathematical statements are an attempt at capturing truth. If a certain statement is believed to be true by mathematicians (typically as special cases are confirmed to some degree) but has neither been proven nor disproven to logically follow from some set of assumptions, it is called a conjecture, as opposed to the ultimate goal, a theorem that is proven true. Unlike physical theories, which may be expected to change whenever new information about our physical world is discovered, mathematical theories are static. Once a statement is considered a theorem, it remains true forever.

Demographics

As is the case in many scientific disciplines, the field of mathematics has been disproportionately dominated by men. Among the minority of prominent female mathematicians are Emmy Noether (1882 - 1935), Sophie Germain (1776 - 1831), Sofia Kovalevskaya (1850 - 1891), Rózsa Péter (1905 - 1977), Julia Robinson (1919 - 1985), Mary Ellen Rudin, Eva Tardos, Émilie du Châtelet, Mary Cartwright and Marianna Csörnyei.

Quotes

...beware of mathematicians, and all those who make empty prophecies. The danger already exists that the mathematicians have made a covenant with the devil to darken the spirit and to confine man in the bonds of Hell. :-Saint Augustine, De Genesi ad Litteram (actually "mathematicians" in this context refers mainly to astrologers and such) A mathematician is a machine for turning coffee into theorems. :-Paul Erdős Die Mathematiker sind eine Art Franzosen; redet man mit ihnen, so übersetzen sie es in ihre Sprache, und dann ist es alsobald ganz etwas anderes. (Mathematicians are [like] a sort of Frenchmen; if you talk to them, they translate it into their own language, and then it is immediately something quite different.) :-Johann Wolfgang von Goethe Some humans are mathematicians; others aren't. :-Jane Goodall (1971) In the Shadow of Man

Jokes

Several old jokes common amongst the scientific disciplines illustrate the difference between the mathematical mind and that of other disciplines. One goes as follows: :An engineer, a physicist, and a mathematician are all staying at a hotel one night when a fire breaks out. The engineer wakes up and smells the smoke; he quickly grabs a garbage pail to use as a bucket, fills it with water from the bathroom, and puts out the fire in his room. He then refills the pail and douses everything flammable in the room with water. He then returns to sleep. :The physicist wakes up, smells the smoke, jumps out of bed. He picks up a pad and pencil and makes some calculations, glancing frequently at the flames. He then measures exactly 15.6 liters of water into the garbage pail, and throws it on the flames, which are extinguished. Smiling, he returns to sleep. :Finally the mathematician wakes up. He too grabs a pad and begins furiously writing; glancing at the flames; and then writing more. After a while he gets a satisfied look on his face; entering the bathroom, he produces a match, lights it, and then extinguishes it with a bit of running water. "Aha! A solution exists," he murmurs - and returns to his slumbers. Another joke goes thus: :Three men are flying in a hot air balloon and suddenly they realize that they are lost. Luckily they see a man plowing a field and ask, "Where are we?". The man on the ground thinks for a minute and then answers, "You are in a hot air balloon". One of the men in the air then says to his friends, "He was a mathematician - he thought before answering, his answer was totally right and totally useless" And another: :An astrologer, a chemist, and a mathematician are on a bus during their first visit to Scotland. They see a black sheep grazing alone in a pasture as they drive by. The astrologer excitedly exclaims, "Ah, this shows Scottish sheep are black!" The chemist didactically corrects him: "No, no, it just shows some Scottish sheep are black." The mathematician then says, "Actually, we can only be sure there is at least one Scottish sheep of which at least one side is black" And finally: : An experiment is being made. A physicist (or an engineer) and a mathematician are asked to boil hot water, but the kettle is in the living room. The physicist goes to the living room, takes the kettle, returns to the kitchen and puts it on the stove and boils the water. The mathematician does the same. In the second stage, the kettle is in the kitchen and the two are again asked to boil hot water. The physicist simply puts the kettle on the stove and boils the water. However, the mathematician takes the kettle, puts it in the living room and declares: "We have already solved this problem!"

Links and references

References

- A Mathematician's Apology, by G. H. Hardy. Memoir, with foreword by C. P. Snow.
- Reprint edition, Cambridge University Press, 1992; ISBN 0521427061
- First edition, 1940
- Dunham, William. The Mathematical Universe. John Wiley 1994.

External links

- [http://www-history.mcs.st-and.ac.uk/history/index0.html The MacTutor History of Mathematics archive], a very complete list of detailed biographies.
- [http://genealogy.math.ndsu.nodak.edu/ The Mathematics Genealogy Project], which allows to follow the succession of thesis advisors for most mathematicians, living or dead. Category:Mathematical science occupations
- ja:数学者 ko:수학자 th:นักคณิตศาสตร์ __NOTOC__

Inequality

:For the socioeconomic sense, see social inequality. social inequality In mathematics, an inequality is a statement about the relative size or order of two objects. (See also: equality) The notation a b means that a is greater than b. These relations are known as strict inequality; in contrast a ≤ b means that a is less than or equal to b and a ≥ b means that a is greater than or equal to b. If the sense of the inequality is the same for all values of the variables for which its members are defined, then the inequality is called an "absolute" or "unconditonal" inequality. If the sense of an inequality holds only for certain values of the variables involved, but is reversed or destroyed for other values of the variables, it is called a conditional inequality. The sense of an inequality is not changed if both sides are increased or decreased by the same number, or if both sides are multiplied or divided by a positive number; the sense of an inequality is reversed if both members are multiplied or divided by a negative number. Young students sometimes confuse the less-than and greater-than signs, which are mirror images of one another. A commonly taught mnemonic is that the sign represents a hungry alligator that is trying to eat the larger number; thus, it opens towards 8 in both 3 < 8 and 8 > 3.[http://mathforum.org/library/drmath/view/58428.html] The notation a >> b means that a is "much greater than" b. What this means exactly can vary, meaning anything from a factor of 100 difference to a ten order of magnitude difference. It is used in relation to equations in which a much greater value will cause the output of the equation to converge on a certain result.

Properties

Inequalities are governed by the following properties:

Trichotomy

The trichotomy property states:
- For any real numbers, "a" and "b", only one of the following is true:
- a b

Transitivity

The transitivity of inequalities states:
- For any real numbers, "a", "b", "c":
- If a > b and b > c; then a > c
- If a < b and b < c; then a < c

Reversal

The inequality relations are mirror images in the sense that:
- For any real numbers, "a" and "b":
- If a > b then b < a
- If a a

Addition and subtraction

The properties which deal with addition and subtraction states:
- For any real numbers, "a", "b", "c":
- If a > b; then a + c > b + c and a − c > b − c
- If a < b; then a + c < b + c and a − c < b − c

Multiplication and division

The properties which deal with multiplication and division state:
- For any real numbers, "a", "b", and "c":
- If c is positive and a > b; then a × c > b × c and a / c > b / c
- If c is positive and a < b; then a × c b; then a × c b × c and a / c > b / c

Applying a function to both sides

Any strictly monotonically increasing function may be applied to both sides of an inequality and it will still hold.

Chained notation

The notation a < b < c stands for a < b and b < c (from the transitivity above follows also a < c). Obviously, by the above laws, one can add/subtract the same number to all three terms, or multiply/divide all three terms by same nonzero number and reverse all inequalities according to sign. But care must be taken so that you really use the same number in all cases, eg. a < b + e < c is equivalent to a - e < b < c - e. This notation can be generalized to any number of terms, for instance a₁ ≤ a₂ ≤ ... ≤ a_n means that a_i ≤ a_j for any 1 ≤ i ≤ j ≤ n.

Well-known inequalities

See also list of inequalities. Mathematicians often use inequalities to bound quantities for which exact formulas cannot be computed easily. Some inequalities are used so often that they have names:
- Azuma's inequality
- Bernoulli inequality
- Boole's inequality
- Cauchy-Schwarz inequality
- Chebyshev's inequality
- Chernoff's inequality
- Cramér-Rao inequality
- Hoeffding's inequality
- Hölder's inequality
- Inequality of arithmetic and geometric means
- Jensen's inequality
- Markov's inequality
- Minkowski's inequality
- Nesbitt's inequality
- Pedoe's inequality
- Triangle inequality

References

-
-
- Category:Inequalities Category:Elementary algebra ko:부등식 ja:不等式

Analytic proof

In structural proof theory, an analytical proof is a proof whose structure is simple in a special way. The term does not admit an uncontroversial definition, but for several proof calculi there is an accepted notion of analytic proof. For example:
- In Gentzen's natural deduction calculus the analytic proofs are those in normal form; that is, no formula occurrence is both the principal premise of an elimination rule and the conclusion of an introduction rule;
- In Gentzen's sequent calculus the analytic proofs are those that do not use the cut rule. However it is possible to extend both calculi so that there are proofs that satisfy the condition but are not analytic: a particularly tricky example of this is the analytic cut rule: this is a special case of the cut rule where the cut formula is a subformula of side formulae of the cut rule; a proof that contains an analytic cut is by virtue of that rule not analytic.

Fourier series

The Fourier series, named in honor of Joseph Fourier (1768-1830), is an extremely useful mathematical tool. Intuitively, one can use Fourier series to divide certain large problems into more manageable pieces. More precisely, a Fourier series is a representation of a periodic function with period 2π as a sum of periodic functions of the form :

x\mapsto e^,

which are the harmonics of e^{i x}. By Euler's formula, the series can be expressed equivalently in terms of sine and cosine functions. This can be generalized to periodic functions of any positive period. Fourier was the first to study systematically such infinite series, after preliminary investigations by Euler, d'Alembert, and Daniel Bernoulli. He applied these series to the solution of the heat equation, publishing his initial results in 1807 and 1811, and publishing his Théorie analytique de la chaleur in 1822. From a modern point of view, Fourier's results are somewhat informal, due in no small part to the lack of a precise notion of function and integral in the early nineteenth century. Later, Dirichlet and Riemann expressed Fourier's results with greater precision and formality. Many other Fourier-related transforms have since been defined, extending to other applications the initial idea of representing any periodic function as a superposition of harmonics. This general area of inquiry is now sometimes called harmonic analysis.

Definition of Fourier series

Suppose that f(x), a complex-valued function of a real variable, is periodic with period 2π, and is square-integrable over the interval from −π to π. Let :

F_n =\frac\int_^\pi f(x)\,e^\,dx.

Each F_n is called a Fourier coefficient. Then, the Fourier series representation of f(x) is given by :

f(x) = \sum_^ F_n \,e^.

Each term in this sum is called a Fourier mode or a harmonic. In the important special case of a real-valued function f(x), one often uses the equality :

e^=\cos(nx)+i\sin(nx) \,\!

(derived from Euler's formula) to equivalently represent f(x) as an infinite linear combination of functions of the form

\cos(nx) \,\!

and

\sin(nx) \,\!

, that is :

f(x) = \fraca_0 + \sum_^\infty\left[a_n\cos(nx)+b_n\sin(nx)\right]

, where :

a_n = \frac\int_^\pi f(x)\cos(nx)\,dx

and

b_n = \frac\int_^\pi f(x)\sin(nx)\,dx

which corresponds to

F_n = (a_n - i b_n) / 2 \,

and

F_ = F_n^ -

, and therefore

F_ = a_0 / 2.\,

Example

Let f(x) = x be the identity function for x from −π to π. Outside this domain, the Fourier series implicitly requires that we define the function periodically. We will compute the Fourier coefficients for this function. Notice that cos(nx) is an even function, while f and sin(nx) are odd functions. :

a_n=\frac\int_^f(x)\cos(nx)\,dx= \frac\int_^x \cos(nx)\,dx = 0,

b_n= \frac\int_^f(x)\sin(nx)\,dx=\frac\int_^ x \sin(nx)\, dx

=\frac\int_^ x\sin(nx)\, dx= \frac\left(
\left[-\frac\right]_0^+\left[\frac\right]_0^
\right)=(-1)^\frac.

Notice that a_n are 0 because x and x cos(nx) are odd functions. Hence the Fourier series for f(x) = x is: :

f(x)=x=\frac + \sum_^(a_n\cos(nx)+b_n\sin(nx))

=\sum_^(-1)^\frac \sin(nx), \quad \forall x\in (-\pi,\pi).

For an application of this Fourier series, see the value of the Riemann zeta function at s = 2.

Convergence of Fourier series

While the Fourier coefficients a_n and b_n can be formally defined for any function for which the integrals make sense, whether the series so defined actually converges to f(x) depends on the properties of f. The simplest answer is that if f is square-integrable then :

\lim_\int_^\pi\left|f(x)-\sum_^
F_n\,e^\right|^2\,dx=0

(this is convergence in the norm of the space L²). There are also many known tests that ensure that the series converges at a given point x. For example, if the function is differentiable at x. Even a jump discontinuity does not pose a problem: if the function has left and right derivatives at x, then the Fourier series will converge to the average of the left and right limits (but see Gibbs phenomenon). However, a fact that many find surprising, is that the Fourier series of a continuous function need not converge pointwise. A discussion of the counterexample, along with other positive and negative results in the general spirit of "for functions of type X, the Fourier series converges in sense Y" may be found in Convergence of Fourier series.

Orthogonality

The Fourier basis functions are orthogonal in the discrete space :

\sum_^\infty e^e^=2\pi\sum_^\infty
\delta(x-y+2\pi n)=2\pi\,\delta_(x-y)

where δ(x) is the Dirac delta function and δ_T(x) is the Dirac comb function. The Fourier basis functions are orthogonal in the continuous space as well: :

\frac\int_^\pi e^e^\,dx = \delta_

where δ_nm is the Kronecker delta function.

Some positive consequences of the homomorphism properties of exp

Because "basis functions" e^ikx are homomorphisms of the real line (more precisely, of the "circle group") we have some useful identities:

Shifting property

If :

g(x)=f(x-y) \,\!

then (if G is the transform of g) :

G_k = e^F_k. \,\!

Convolution theorems

:Main article: Convolution If h(t) is the cyclic convolution of f(t) and g(t): :

h(t)=\int_^\pi f(t')g(t-t')\,dt'

where g(t) = g(t + 2nπ), then the Fourier series transforms are related by: :

H_n=2\pi\,F_nG_n.\,

Conversely, if H_n = 2πF_nG_n, then h(t) will be the cyclic convolution of f(t) and g(t). In the discrete space, if H_n is the discrete convolution of F_n and G_n: :

H_k=\sum_^\infty F_n G_

then the inverse transforms are related by: :

h(t)=f(t)g(t)\,

and conversely, if h(t) = f(t)g(t), then H_n will be the discrete convolute of F_n and G_n. These theorems may be proven using the orthogonality relationships.

Plancherel's and Parseval's theorem

Another important property of the Fourier series is the Plancherel theorem :

\sum_^\infty F_nG^ - _n = \frac \int_^\pi f(x)g^ - (x)\,dx.

Parseval's theorem, a special case of the Plancherel theorem, states that :

\sum_^\infty |F_n|^2 = \frac \int_^\pi |f(x)|^2 \,dx

which can be restated for the real-valued f(x) case above, :

\frac + \frac \sum_^\infty \left( a_n^2 + b_n^2 \right) = \frac \int_^\pi f(x)^2\, dx.

These theorems may be proven using the orthogonality relationships.

General formulation

The useful properties of Fourier series are largely derived from the orthogonality and homomorphism property of the functions

e^ \,\!

. Other sequences of orthogonal functions have similar properties, although some useful identities concerning e.g. convolutions are no longer true once we lose the homomorphism property. Examples include sequences of Bessel functions and orthogonal polynomials. Such sequences are commonly the solutions of a differential equation; a large class of useful sequences are solutions of the so-called Sturm-Liouville problems.

References

- Yitzhak Katznelson, An introduction to harmonic analysis, Second corrected edition. Dover Publications, Inc., New York, 1976. ISBN 0486633314

External links

- [http://www.falstad.com/fourier/ Java applet] shows Fourier series expansion of an arbitrary function Category:Fourier analysis
- Category:Mathematical series ko:푸리에 급수 th:อนุกรมฟูริเยร์

Space (mathematics)

:This article is about space — the scientific and philosophical concepts. For other uses of space, see space (disambiguation). Attempting to understand the nature of space has always been a prime occupation for philosophers and scientists. Perhaps as a result of this considerable discussion, it is difficult to provide an uncontroversial and clear definition of the nature of space, except its physical definition (see below). This article looks at the way space is dealt with variously by physicists, mathematicians and philosophers, and at the relation between space and the mind.

Physics and Space

Space is one of the few fundamental quantities in physics meaning it can't be defined via other quantities because there is nothing more fundamental known at present. Thus, similar to the definition of other fundamental quantities (like time and mass), space is defined via measurement. Currently, the standard space interval, called a standard meter or simply meter, is defined as the distance traveled by light in a vacuum during a time interval of 1/299 792 458 of a second (exact). In classical physics, space is a three-dimensional Euclidean space where any position can be described using three coordinates. Relativistic physics examines spacetime rather than space; spacetime is modeled as a four-dimensional manifold, and currently, there are theories that can support even eleven-dimensional spaces. Before Einstein's work on relativistic physics, time and space were seen as independent dimensions. Einstein's work unified the two into spacetime. In spacetime, measurements of space and time are held to be relative to velocity.

Measurement

The measurement of physical space has long been important. Geometry, the name given to the branch of mathematics which measures spatial relations, was popularised by the ancient Greeks, although earlier societies had developed measuring systems. The International System of Units, (SI), is now the most common system of units used in the measuring of space, and is almost universally used within science. Geography is the branch of science concerned with identifying and describing the Earth, utilising spatial awareness to try and understand why things exist in specific locations. Cartography is the mapping of spaces to allow better navigation, for visualisation purposes and to act as a locational device. Astronomy is the science involved with the observation, explanation and measuring of objects in outer space.

Astronomy and space

In astronomy, space refers collectively to the relatively empty parts of the universe. Any area outside the atmospheres of any celestial body can be considered 'space'. Although space is certainly spacious, it is now known to be far from empty, and filled with a tenuous plasma. In particular, the boundary between space and Earth's atmosphere is conventionally set at the Karman line.

Mathematics and space

In mathematics, a space is a set, with some particular properties and usually some additional structure. It is not a formally defined concept as such, but a generic name for a number of similar concepts, most of which generalize some abstract properties of the physical concept of space. In particular, a vector space and specifically a Euclidean space can be seen as generalizations of the concept of a Euclidean coordinate system. Important varieties of vector spaces with more imposed structure include Banach space and Hilbert space. Distance measurement is abstracted as the concept of metric space and volume measurement leads to the concept of measure space. As far as the concept of dimension is defined, this need not be 3: it can also be 0 (a point), 1 (a line), 2 (a plane), more than 3, and with some definitions, a non-integer value. Mathematicians often study general structures that hold regardless of the number of dimensions. Kinds of mathematical spaces include:
- Banach space
- Euclidean space
- Hilbert space
- Metric space
- Probability space
- Projective space
- Topological space
- Vector space

The philosophy of space

Space has a range of definitions.
- One view of space is that it is part of the fundamental structure of the universe, a set of dimensions in which objects are separated and located, have size and shape, and through which they can move.
- A contrasting view is that space is part of a fundamental abstract mathematical conceptual framework (together with time and number) within which we compare and quantify the distance between objects, their sizes, their shapes, and their speeds. In this view space does not refer to any kind of entity that is a "container" that objects "move through". These opposing views are relevant also to definitions of time. Space is typically described as having three dimensions, and that three numbers are needed to specify the size of any object and/or its location with respect to another location. Modern physics does not treat space and time as independent dimensions, but treats both as features of spacetime – a conception that challenges intuitive notions of distance and time. An issue of philosophical debate is whether space is an ontological entity itself, or simply a conceptual framework we need to think (and talk) about the world. Another way to frame this is to ask, "Can space itself be measured, or is space part of the measurement system?" The same debate applies also to time, and an important formulation in both areas was given by Immanuel Kant. In his Critique of Pure Reason, Kant described space as an a priori notion that (together with other a priori notions such as time) allows us to comprehend sense experience. With Kant, neither space nor time are conceived as substances, but rather both are elements of a systematic framework we use to structure our experience. Spatial measurements are used to quantify how far apart objects are, and temporal measurements are used to quantify how far apart events occur. Similar philosophical questions concerning space include: Is space absolute or purely relational? Does space have one correct geometry, or is the geometry of space just a convention? Historical positions in these debates have been taken by Isaac Newton (space is absolute), Gottfried Leibniz (space is relational), and Henri Poincaré (spatial geometry is a convention). Two important thought-experiments connected with these questions are: Newton's bucket argument and Poincaré's sphere-world.

The psychology of space

The way in which space is perceived is an area which psychologists first began to study in the middle of the 19th century, and it is now thought by those concerned with such studies to be a distinct branch within psychology. Psychologists analysing the perception of space are concerned with how recognition of an object's physical appearance or its interactions are perceived. Other, more specialised topics studied include amodal perception and object permanence. The perception of surroundings is important due to its necessary relevance to survival, especially with regards to hunting and self preservation. "Veridical perception" is the term used to describe the processing of the information provided by the sensory organs to an extent whereby it allows interaction with the actuality of that perceived. It is worth noting that the way we perceive space may not necessarily be representative of the actuality of space.

Anxiety and space

Space can also cause anxiety in people, with agoraphobia manifesting itself in some people as a fear of open spaces, and claustrophobia being the fear of enclosed spaces. Astrophobia is the fear of celestial space, Kenophobia is the fear of empty spaces and spacephobia is the fear of outer space.

Personal space

The term personal space refers to the amount of space a person likes to maintain between their own person and that of other people.

Use of space

The definition of physical space in relation to ownership, in which space is seen as property, has long been an important issue. Whilst some cultures assert the rights of the individual in terms of ownership, other cultures will identify with a communal approach to land ownership. Spatial planning is a method of regulating the use of space at land-level, with decisions made at regional, national and international levels. Space can also impact on human and cultural behaviour, being an important factor in architecture, where it will impact on the design of buildings and structures, and on farming. Ownership of space is not restricted to land. Ownership of Airspace and of waters is decided internationally. Public space is a term used to define areas of land which are open to all, whilst private property is that area of land owned by an individual or company, for their own use and pleasure.

Reference

[http://search.eb.com/eb/article?tocId=46639 Space perception]. Encyclopædia Britannica from Encyclopædia Britannica Online. Accessed June 12, 2005. Category: Topology Category:Environments ko:공간 ja:空間 simple:Space

Isoperimetric dimension

In mathematics, the isoperimetric dimension of a manifold is a notion of dimension that tries to capture how the large scale behavior of the manifold resembles that of a Euclidean space (unlike the topological dimension or the Hausdorff dimension which compare different local behaviors against those of the Euclidean space). In the Euclidean space, the isoperimetric inequality says that of all bodies with the same volume, the ball has the smallest surface area. In other manifolds it is usually very difficult to find the precise body minimizing the surface area, and this is not what the isoperimetric dimension is about. The question we will ask is, what is approximately the minimal surface area, whatever the body realizing it might be.

Formal definition

We say about a manifold M that it satisfies a d-dimensional isoperimetric inequality if for any open set D in M with a smooth boundary one has :

\mathrm\,(\partial D)\geq C\mathrm\,(D)^

The notations vol and area refer to the regular notions of volume and surface area on the manifold, or more precisely, if the manifold has n topological dimensions then vol refers to n-dimensional volume and area refers to (n - 1)-dimensional volume. C here refers to some constant, which does not depend on D (it may depend on the manifold and on d). The isoperimetric dimension of M is the supremum on all d-s such that M satisfies a d-dimensional isoperimetric inequality.

Examples

A d-dimensional Euclidean space has isoperimetric dimension d. This is the well known isoperimetric problem — as discussed above, for the Euclidean space the constant C is known precisely since the minimum is achieved for the ball. An infinite cylinder (i.e. a product of the circle and the line) has topological dimension 2 but isoperimetric dimension 1. Indeed, multiplying any manifold with a compact manifold does not change the isoperimetric dimension (it only changes the value of the constant C). Any compact manifold has isoperimetric dimension 0. It is also possible for the isoperimetric dimension to be larger than the topological dimension. The simplest example is the infinite jungle gym, which has topological dimension 2 and isoperimetric dimension 3. See [http://www.math.ucla.edu/~bon/jungle.html] for pictures and Mathematica code. The hyperbolic plane has topological dimension 2 and isoperimetric dimension infinity. In fact the hyperbolic plane has positive Cheeger constant. This means that it satisfies the inequality :

\mathrm\,(\partial D)\geq C\mathrm\,(D)

which obviously implies infinite isoperimetric dimension.

Isoperimetric dimension of graphs

The isoperimetric dimension of graphs can be defined in a similar fashion. there is no need to have an area and volume measures. One simply counts points. For every subset A of the graph G one defines

\partial A

as the set of vertices in

G\setminus A

with a neighbor in A. A d-dimensional isoperimetric inequality is now defined by :

|\partial A|\geq C|A|^.

The graph analogs of all the examples above hold. The isoperimetric dimension of any finite graph is 0. The isoperimetric dimension of a d-dimensional grid is d. In general, the isoperimetric dimension is preserved by quasi isometries, both by quasi-isometries between manifolds, between graphs, and even by quasi isometries carrying manifolds to graphs, with the respective definitions. In rough terms, this means that a graph "mimicking" a given manifold (as the grid mimics the Euclidean space) would have the same isoperimetric dimension as the manifold. An infinite complete binary tree has isoperimetric dimension ∞.

Consequences of isoperimetry

A simple integration over r (or sum in the case of graphs) shows that a d-dimensional isoperimetric inequality implies a d-dimensional volume growth, namely :

\mathrm\,B(x,r)\geq Cr^d

where B(x,r) denotes the ball of radius r around the point x in the Riemannian distance or in the graph distance. In general, the opposite is not true, i.e. even uniformly exponential volume growth does not imply any kind of isoperimetric inequality. A simple example can be had by taking the graph Z (i.e. all the integers with edges between n and n + 1) and connecting to the vertex n a complete binary tree of height |n|. Both properties (exponential growth and 0 isoperimetric dimension) are easy to verify. An interesting exception is the case of groups. It turns out that a group with polynomial growth of order d has isoperimetric dimension d. This holds both for the case of Lie groups and for the Cayley graph of a finitely generated group. A theorem of Varopoulos connects the isoperimetric dimension of a graph to the rate of escape of random walk on the graph. The result states Varopoulos' theorem: If G is a graph satisfying a d-dimensional isoperimetric inequality then :

p_n(x,y)\leq Cn^

where $p_n(x,y)$ is the probability that a random walk on G starting from x will be in y after n steps, and C is some constant.

References

- Isaac Chavel, Isoperimetric Inequalities: Differential geomet

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The isoperimetric problem in the plane

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