D-branes

In physics, a wormhole is a hypothetical topological feature of spacetime that is essentially a "shortcut" through space and time. A wormhole has at least two mouths which are connected to a single throat. If the wormhole is traversible, matter can 'travel' from one mouth to the other by passing through the throat. The name "wormhole" comes from the following analogy used to explain the phenomenon: imagine that the universe is the skin of an apple, and a worm is traveling over its surface. The distance from one side of t ...

Including:

• Wormhole - Definition
• Wormhole - Wormhole types
• Wormhole - Theoretical basis
• Wormhole - Traversable wormholes
• Wormhole - Wormholes and faster-than-light space travel
• Wormhole - Wormholes and time travel
• Wormhole - Schwarzschild wormholes
• Wormhole - Wormhole Metrics
• Wormhole - Wormholes in fiction

Read more here: » Wormhole: Encyclopedia - Wormhole

The cyclic model is a brane cosmology model of the creation of the universe, derived from the earlier ekpyrotic model. It was proposed in 2001 by Paul Steinhardt and Neil Turok. Cyclic model - The model. In the cyclic model, two parallel orbifold planes or M-branes collide periodically in a higher dimensional space. The visible four-dimensional universe lies on one of these branes. The collisions correspond to a reversal from contraction to expansion, or a big crunch followed immediately by a big ba ...

Including:

• Cyclic model - The model

Read more here: » Cyclic model: Encyclopedia - Cyclic model

The subject takes its name from a particular construction applied by Alexander Grothendieck in his proof of the Grothendieck-Riemann-Roch theorem. In it, a commutative monoid of sheaves of abelian groups under direct sum was converted into a group, by the formal addition of inverses (an explicit way of explaining a left adjoint). This construction was taken up by Michael Atiyah and Friedrich Hirzebruch to def ...

See also:

K-theory, K-theory - Early history, K-theory - K-theory and physics

Read more here: » K-theory: Encyclopedia II - K-theory - Early history

String physics - Closed and open strings. Strings can be either open or closed. A closed string is a string that has no end-points, and therefore is topologically equivalent to a circle. An open string, on the otherhand, has two end-points and is topologically equivalent to a line interval. Not all string theories contain both open and closed strings. However, any theory which contains open strings must also contain closed strings as interactions betwee ...

See also:

String physics, String physics - Types of strings, String physics - Closed and open strings, String physics - Orientation

Read more here: » String physics: Encyclopedia II - String physics - Types of strings

The term 'string theory' properly refers to both the 26-dimensional bosonic string theories and to the 10-dimensional superstring theories created by adding supersymmetry. Nowadays, 'string theory' usually refers to the supersymmetric variant while the earlier is given its full name, 'bosonic string theory'. While understanding the details of string and superstring theories requires considerable mathematical sophistication, some qualitative properties of quantum strings can be understood in a fairly intuitive fashion. For example, qua ...

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String theory, String theory - History, String theory - Basic properties, String theory - Dualities, String theory - Extra dimensions, String theory - Problems, String theory - Popular culture, String theory - References and further reading, String theory - Footnote, String theory - Popular books and articles, String theory - Textbooks, String theory - External links

Read more here: » String theory: Encyclopedia II - String theory - Basic properties

The Jacobi theta function can be thought of as belonging to a representation of the Heisenberg group in quantum mechanics, sometimes called the theta representation. This can be seen by explicitly constructing the group. Let f(z) be a holomorphic function, let a and b be real numbers, and fix a value of τ. Then define the operators Sa and Tb such that (SaSee also:

Theta function, Theta function - Jacobi theta function, Theta function - Auxiliary functions, Theta function - Jacobi identities, Theta function - Product representations, Theta function - Integral representations, Theta function - Relation to the Riemann zeta function, Theta function - Relation to the Weierstrass elliptic function, Theta function - Some relations to modular forms, Theta function - A solution to heat equation, Theta function - Relation to the Heisenberg group, Theta function - Generalizations, Theta function - Ramanujan theta function, Theta function - Riemann theta function, Theta function - Q-theta function

Read more here: » Theta function: Encyclopedia II - Theta function - Relation to the Heisenberg group

The formal definition goes along the same lines as a definition of manifold, but instead of taking domains in Rn as the target spaces of charts one should take domains of finite quotients of Rn. A (topological) orbifold O, is a Hausdorff topological space X with countable base, called the underlying space, with an orbifold structure, w ...

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Orbifold, Orbifold - Formal definition, Orbifold - Orbifolds in string theory, Orbifold - History

Read more here: » Orbifold: Encyclopedia II - Orbifold - Formal definition

Lorentzian traversable wormholes would allow travel from one part of the universe to another part of that same universe very quickly or would allow travel from one universe to another universe. Wormholes connect two points in spacetime, which means that they would allow travel in time as well as in space. Wormhole - Wormholes and faster-than-light space travel. Often there is confusion about the idea that wormholes allow superluminal (faster-than-light) space travel. In fact there is no real superluminal t ...

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Wormhole, Wormhole - Definition, Wormhole - Wormhole types, Wormhole - Theoretical basis, Wormhole - Traversable wormholes, Wormhole - Wormholes and faster-than-light space travel, Wormhole - Wormholes and time travel, Wormhole - Schwarzschild wormholes, Wormhole - Wormhole Metrics, Wormhole - Wormholes in fiction

Read more here: » Wormhole: Encyclopedia II - Wormhole - Traversable wormholes

The term 'string theory' properly refers to both the 26-dimensional bosonic string theories and to the 10-dimensional superstring theories created by adding supersymmetry. Nowadays, 'string theory' usually refers to the supersymmetric variant while the earlier is given its full name, 'bosonic string theory'. While understanding the details of string and superstring theories requires considerable mathematical sophistication, some qualitative properties of quantum strings can be understood in a fairly intuitive fashion. For example, qua ...

See also:

String theory, String theory - History, String theory - Basic properties, String theory - Dualities, String theory - Extra dimensions, String theory - Problems, String theory - References and further reading, String theory - Footnote, String theory - Popular books and articles, String theory - Textbooks, String theory - External links

Read more here: » String theory: Encyclopedia II - String theory - Basic properties

It is convenient to define three auxiliary theta functions, which we may write This notation follows Riemann and Mumford; Jacobi's original formulation was in terms of the nome q = exp(πτ) rather than τ, and theta there is called θ3, with termed θ0, named θ2, and c ...

See also:

Theta function, Theta function - Jacobi theta function, Theta function - Auxiliary functions, Theta function - Jacobi identities, Theta function - Product representations, Theta function - Integral representations, Theta function - Relation to the Riemann zeta function, Theta function - Relation to the Weierstrass elliptic function, Theta function - Some relations to modular forms, Theta function - A solution to heat equation, Theta function - Relation to the Heisenberg group, Theta function - Generalizations, Theta function - Ramanujan theta function, Theta function - Riemann theta function, Theta function - Q-theta function

Read more here: » Theta function: Encyclopedia II - Theta function - Auxiliary functions

The Jacobi theta function is a function defined for two complex variables z and τ, where z can be any complex number and τ is confined to the upper half-plane, which means it has positive imaginary part. It is given by the formula If τ is fixed, this becomes a Fourier series for a periodic entire function of z with period 1; in this case, the theta function satisfies the identity The function also behaves very regularly with respect to addition by τ and satisfies the functional equation ...

See also:

Theta function, Theta function - Jacobi theta function, Theta function - Auxiliary functions, Theta function - Jacobi identities, Theta function - Product representations, Theta function - Integral representations, Theta function - Relation to the Riemann zeta function, Theta function - Relation to the Weierstrass elliptic function, Theta function - Some relations to modular forms, Theta function - A solution to heat equation, Theta function - Relation to the Heisenberg group, Theta function - Generalizations, Theta function - Ramanujan theta function, Theta function - Riemann theta function, Theta function - Q-theta function

Read more here: » Theta function: Encyclopedia II - Theta function - Jacobi theta function

The Jacobi theta function can be expressed as a product, through the Jacobi triple product theorem: The auxiliary functions have the expressions, with q = expiπτ: ...

See also:

Theta function, Theta function - Jacobi theta function, Theta function - Auxiliary functions, Theta function - Jacobi identities, Theta function - Product representations, Theta function - Integral representations, Theta function - Relation to the Riemann zeta function, Theta function - Relation to the Weierstrass elliptic function, Theta function - Some relations to modular forms, Theta function - A solution to heat equation, Theta function - Relation to the Heisenberg group, Theta function - Generalizations, Theta function - Ramanujan theta function, Theta function - Riemann theta function, Theta function - Q-theta function

Read more here: » Theta function: Encyclopedia II - Theta function - Product representations

The Jacobi theta function is the unique solution to the one-dimensional heat equation with periodic boundary conditions at time zero. This is most easily seen by taking z = x to be real, and taking τ = it with t real and positive. Then we can write which solves the heat equation That this solution is unique can be seen by noting that at t = 0, the theta function becomes the Dirac comb: where δ is the Dirac delta function. Thus, general solution can be specified by convolving the (periodi ...

See also:

Theta function, Theta function - Jacobi theta function, Theta function - Auxiliary functions, Theta function - Jacobi identities, Theta function - Product representations, Theta function - Integral representations, Theta function - Relation to the Riemann zeta function, Theta function - Relation to the Weierstrass elliptic function, Theta function - Some relations to modular forms, Theta function - A solution to heat equation, Theta function - Relation to the Heisenberg group, Theta function - Generalizations, Theta function - Ramanujan theta function, Theta function - Riemann theta function, Theta function - Q-theta function

Read more here: » Theta function: Encyclopedia II - Theta function - A solution to heat equation

The theta function was used by Jacobi to construct (in a form adapted to easy calculation) his elliptic functions as the quotients of the above four theta functions, and could have been used by him to construct Weierstrass's elliptic functions also, since where the second derivative is with respect to z and the constant c is defined so that the Laurent expansion of at z = 0 has zero constant term. ...

See also:

Theta function, Theta function - Jacobi theta function, Theta function - Auxiliary functions, Theta function - Jacobi identities, Theta function - Product representations, Theta function - Integral representations, Theta function - Relation to the Riemann zeta function, Theta function - Relation to the Weierstrass elliptic function, Theta function - Some relations to modular forms, Theta function - A solution to heat equation, Theta function - Relation to the Heisenberg group, Theta function - Generalizations, Theta function - Ramanujan theta function, Theta function - Riemann theta function, Theta function - Q-theta function

Read more here: » Theta function: Encyclopedia II - Theta function - Relation to the Weierstrass elliptic function

The relation was used by Riemann to prove the functional equation for Riemann's zeta function, by means of the integral which can be shown to be invariant under substitution of s by 1 − s. The corresponding integral for z not zero is given in the article on the Hurwitz zeta function. ...

See also:

Theta function, Theta function - Jacobi theta function, Theta function - Auxiliary functions, Theta function - Jacobi identities, Theta function - Product representations, Theta function - Integral representations, Theta function - Relation to the Riemann zeta function, Theta function - Relation to the Weierstrass elliptic function, Theta function - Some relations to modular forms, Theta function - A solution to heat equation, Theta function - Relation to the Heisenberg group, Theta function - Generalizations, Theta function - Ramanujan theta function, Theta function - Riemann theta function, Theta function - Q-theta function

Read more here: » Theta function: Encyclopedia II - Theta function - Relation to the Riemann zeta function

The book The Elegant Universe by Brian Greene was adapted into a three-hour documentary for Nova. String theory is also a series of books based in the Star Trek: Voyager universe. ...

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String theory, String theory - History, String theory - Basic properties, String theory - Dualities, String theory - Extra dimensions, String theory - Problems, String theory - Popular culture, String theory - References and further reading, String theory - Footnote, String theory - Popular books and articles, String theory - Textbooks, String theory - External links

Read more here: » String theory: Encyclopedia II - String theory - Popular culture

Jacobi's identities describe how theta functions transform under the modular group. Let Then See also: proof of Jacobi's identity for functions on PlanetMath.. Note that the conventions for z in that reference differ from those here by a factor of π. ...

See also:

Theta function, Theta function - Jacobi theta function, Theta function - Auxiliary functions, Theta function - Jacobi identities, Theta function - Product representations, Theta function - Integral representations, Theta function - Relation to the Riemann zeta function, Theta function - Relation to the Weierstrass elliptic function, Theta function - Some relations to modular forms, Theta function - A solution to heat equation, Theta function - Relation to the Heisenberg group, Theta function - Generalizations, Theta function - Ramanujan theta function, Theta function - Riemann theta function, Theta function - Q-theta function

Read more here: » Theta function: Encyclopedia II - Theta function - Jacobi identities

Theories of wormhole metrics describe the spacetime geometry of a wormhole and serve as theoretical models for time travel. A simple example of a (traversable) wormhole metric is the following: ds2 = − c2dt2 + dl2 + (k2 + l2)(dθ2 + sin2θdφ2) another type of a theory is t ...

See also:

Wormhole, Wormhole - Definition, Wormhole - Wormhole types, Wormhole - Theoretical basis, Wormhole - Traversable wormholes, Wormhole - Wormholes and faster-than-light space travel, Wormhole - Wormholes and time travel, Wormhole - Schwarzschild wormholes, Wormhole - Wormhole Metrics, Wormhole - Wormholes in fiction

Read more here: » Wormhole: Encyclopedia II - Wormhole - Wormhole Metrics

Theories of wormhole metrics describe the spacetime geometry of a wormhole and serve as theoretical models for time travel. A simple example of a (traversable) wormhole metric is the following: ds2 = − c2dt2 + dl2 + (k2 + l2)(dθ2 + sin2θdφ2) another type of equation to descri ...

See also:

Wormhole, Wormhole - Definition, Wormhole - Wormhole types, Wormhole - Theoretical basis, Wormhole - Traversable wormholes, Wormhole - Wormholes and faster-than-light space travel, Wormhole - Wormholes and time travel, Wormhole - Schwarzschild wormholes, Wormhole - Wormhole Metrics, Wormhole - Wormholes in fiction

Read more here: » Wormhole: Encyclopedia II - Wormhole - Wormhole Metrics

Giving a precise definition of a wormhole is slightly tricky. The idea which one wishes to capture is that there is a compact region of spacetime whose boundary is topologically trivial but whose interior is not simply connected. Formalizing this idea leads to definitions such as the following, taken from Matt Visser's Lorentzian Wormholes: If a Lorentzian spacetime contains a compact region Ω, and if the topology of Ω is of the form Ω ~ R x Σ, where Σ is a three-manifold of nontrivial topology, whose boundary has to ...

See also:

Wormhole, Wormhole - Definition, Wormhole - Wormhole types, Wormhole - Theoretical basis, Wormhole - Traversable wormholes, Wormhole - Wormholes and faster-than-light space travel, Wormhole - Wormholes and time travel, Wormhole - Schwarzschild wormholes, Wormhole - Wormhole Metrics, Wormhole - Wormholes in fiction

Read more here: » Wormhole: Encyclopedia II - Wormhole - Definition