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Pregeometry (physics)
In physics, a pregeometry is a hypothetical structure from which the geometry of the universe develops. Some cosmological models feature a pregeometric universe before the Big Bang. The term was championed by John Archibald Wheeler in the 1960s and 1970s as a possible route to a theory of quantum gravity. Since quantum mechanics allowed a metric to fluctuate, it was argued that the merging of gravity with quantum mechanics required a set of more fundamental rules regarding Connection (mathematics), connectivity that were independent of topology and dimensionality. Where geometry could describe the properties of a known surface, the physics of a hypothetical region with predefined properties, "pregeometry" might allow one to work with deeper underlying rules of physics that were not so strongly dependent on simplified classical assumptions about the properties of space. No single proposal for pregeometry has gained wide consensus support in the physics community. Some notions related ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." It is one of the most fundamental scientific disciplines. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fock Space
The Fock space is an algebraic construction used in quantum mechanics to construct the quantum states space of a variable or unknown number of identical particles from a single particle Hilbert space . It is named after V. A. Fock who first introduced it in his 1932 paper "Konfigurationsraum und zweite Quantelung" (" Configuration space and second quantization"). M.C. Reed, B. Simon, "Methods of Modern Mathematical Physics, Volume II", Academic Press 1975. Page 328. Informally, a Fock space is the sum of a set of Hilbert spaces representing zero particle states, one particle states, two particle states, and so on. If the identical particles are bosons, the -particle states are vectors in a symmetrized tensor product of single-particle Hilbert spaces . If the identical particles are fermions, the -particle states are vectors in an antisymmetrized tensor product of single-particle Hilbert spaces (see symmetric algebra and exterior algebra respectively). A general state in Fock ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abstract Simplicial Complex
In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely combinatorial description of the geometric notion of a simplicial complex. Lee, John M., Introduction to Topological Manifolds, Springer 2011, , p153 For example, in a 2-dimensional simplicial complex, the sets in the family are the triangles (sets of size 3), their edges (sets of size 2), and their vertices (sets of size 1). In the context of matroids and greedoids, abstract simplicial complexes are also called independence systems. An abstract simplex can be studied algebraically by forming its Stanley–Reisner ring; this sets up a powerful relation between combinatorics and commutative algebra. Definitions A collection of non-empty finite subsets of a set ''S'' is called a set-family. A set-family is called an abstract simplicial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minkowski Space
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of gravitation. It combines inertial space and time manifolds into a four-dimensional model. The model helps show how a spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Mathematician Hermann Minkowski developed it from the work of Hendrik Lorentz, Henri Poincaré, and others said it "was grown on experimental physical grounds". Minkowski space is closely associated with Einstein's theories of special relativity and general relativity and is the most common mathematical structure by which special relativity is formalized. While the individual components in Euclidean space and time might differ due to length contraction and time dilation, in Minkowski spacetime, all frames of reference will agree on the total interval in spacetime between events.This makes spacetime distance an inva ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tree (graph Theory)
In graph theory, a tree is an undirected graph in which any two vertices are connected by path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. A directed tree, oriented tree,See .See . polytree,See . or singly connected networkSee . is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. The various kinds of data structures referred to as trees in computer science have underlying graphs that are trees in graph theory, although such data structures are generally rooted trees. A rooted tree may be directed, called a directed rooted tree, either making all its edges point away from the root—in which case it is called an arborescence or out-tree� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monad (philosophy)
The term ''monad'' () is used in some cosmic philosophy and cosmogony to refer to a most basic or original substance. As originally conceived by the Pythagoreans, the Monad is therefore Supreme Being, divinity, or the totality of all things. According to some philosophers of the early modern period, most notably Gottfried Wilhelm Leibniz, there are infinite monads, which are the basic and immense forces, elementary particles, or simplest units, that make up the universe. Historical background According to Hippolytus, the worldview was inspired by the Pythagoreans, who called the first thing that came into existence the "monad", which begat (bore) the dyad (from the Greek word for two), which begat the numbers, which begat the point, begetting lines or finiteness, etc. It meant divinity, the first being, or the totality of all beings, referring in cosmogony (creation theories) variously to source acting alone and/or an indivisible origin and equivalent comparators. Py ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graph Dynamical System
In mathematics, the concept of graph dynamical systems can be used to capture a wide range of processes taking place on graphs or networks. A major theme in the mathematical and computational analysis of GDSs is to relate their structural properties (e.g. the network connectivity) and the global dynamics that result. The work on GDSs considers finite graphs and finite state spaces. As such, the research typically involves techniques from, e.g., graph theory, combinatorics, algebra, and dynamical systems rather than differential geometry. In principle, one could define and study GDSs over an infinite graph (e.g. cellular automata or probabilistic cellular automata over \mathbb^k or interacting particle systems when some randomness is included), as well as GDSs with infinite state space (e.g. \mathbb as in coupled map lattices); see, for example, Wu. In the following, everything is implicitly assumed to be finite unless stated otherwise. Formal definition A graph dynamical system is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Structure
In mathematics, an ''n''- dimensional differential structure (or differentiable structure) on a set ''M'' makes ''M'' into an ''n''-dimensional differential manifold, which is a topological manifold with some additional structure that allows for differential calculus on the manifold. If ''M'' is already a topological manifold, it is required that the new topology be identical to the existing one. Definition For a natural number ''n'' and some ''k'' which may be a non-negative integer or infinity, an ''n''-dimensional ''C''''k'' differential structure is defined using a ''C''''k''-atlas, which is a set of bijections called charts between subsets of ''M'' (whose union is the whole of ''M'') and open subsets of \mathbb^: :\varphi_:M\supset W_\rightarrow U_\subset\mathbb^ which are ''C''''k''-compatible (in the sense defined below): Each chart allows a subset of the manifold to be viewed as an open subset of \mathbb^, but the usefulness of this depends on how much the charts agree ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Order
In mathematics, especially order theory, a partial order on a set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements needs to be comparable; that is, there may be pairs for which neither element precedes the other. Partial orders thus generalize total orders, in which every pair is comparable. Formally, a partial order is a homogeneous binary relation that is reflexive, antisymmetric, and transitive. A partially ordered set (poset for short) is an ordered pair P=(X,\leq) consisting of a set X (called the ''ground set'' of P) and a partial order \leq on X. When the meaning is clear from context and there is no ambiguity about the partial order, the set X itself is sometimes called a poset. Partial order relations The term ''partial order'' usually refers to the reflexive partial order relations, referred to in this article as ''non-strict'' partial orders. However som ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Set
In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the ''cardinality (or the cardinal number)'' of the set. A set that is not a finite set is called an '' infinite set''. For example, the set of all positive integers is infinite: Finite sets are particularly important in combinatorics, the mathematical study of counting. Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set. Definition and terminology Formally, a set S is called finite if there exists a bijection for some natural number n (natural numbers are defined as sets in Zermelo-Fraenkel set theory). The number n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Causal Sets
The causal sets program is an approach to quantum gravity. Its founding principles are that spacetime is fundamentally discrete (a collection of discrete spacetime points, called the elements of the causal set) and that spacetime events are related by a partial order. This partial order has the physical meaning of the causality relations between spacetime events. History For some decades after the formulation of general relativity, the attitude towards Lorentzian geometry was mostly dedicated to understanding its physical implications and not concerned with theoretical issues. However, early attempts to use causality as a starting point have been provided by Hermann Weyl and Hendrik Lorentz. Alfred Robb in two books in 1914 and 1936 suggested an axiomatic framework where the causal precedence played a critical role. The first explicit proposal of quantising the causal structure of spacetime is attributed by Sumati Surya to E. H. Kronheimer and Roger Penrose, who invented causal s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field (mathematics), field that contains a finite number of Element (mathematics), elements. As with any field, a finite field is a Set (mathematics), set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are the integers mod n, integers mod p when p is a prime number. The ''order'' of a finite field is its number of elements, which is either a prime number or a prime power. For every prime number p and every positive integer k there are fields of order p^k. All finite fields of a given order are isomorphism, isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science, including number theory, algebraic geometry, Galois theory, finite geometry, cryptography and coding theory. Properties A finite field is a finite set that is a fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
