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Topic: Discrete space

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Discrete topology

Topological space

Metric space

Topology glossary

Totally disconnected

Compact space

Uniform continuity

Hausdorff space

Pontryagin duality

Morphism

Lipschitz continuity

Argand plane

Cauchy sequence

Real analysis

List of real analysis topics

	PlanetMath: discrete space
		The product of an infinite number of discrete spaces is discrete under the box topology, but if an infinite number of the spaces have more than one element, it is not discrete under the product topology.
		discrete subspace, discrete topology, discrete space, discrete subset
		This is version 14 of discrete space, born on 2002-02-27, modified 2005-06-19.
	planetmath.org /encyclopedia/Discrete.html (231 words)

	Discrete space - Wikipedia, the free encyclopedia
		The underlying uniformity on a discrete metric space is the discrete uniformity, and the underlying topology on a discrete uniform space is the discrete topology.
		Nevertheless, it is discrete as a topological space.
		However, the discrete metric space is free in the category of bounded metric spaces and Lipschitz continuous maps, and it is free in the category of metric spaces bounded by 1 and short maps.
	en.wikipedia.org /wiki/Discrete_space (1106 words)

	Historical Notes: History of discrete space
		In discussing the notion of curved space, Bernhard Riemann remarked in 1854 that it would be easier to give a general mathematical definition of distance if space were discrete.
		The idea that space might be defined by some sort of causal network of discrete elementary quantum events arose in various forms in work by Carl von Weizsäcker (ur-theory), John Wheeler (pregeometry), David Finkelstein (spacetime code), David Bohm (topochronology) and Roger Penrose (spin networks; see page 1059).
		General arguments for discrete space were also sometimes made - notably by Edward Fredkin, Marvin Minsky and to some extent Richard Feynman - on the basis of analogies to computers and in particular the idea that a given region of space should contain only a finite amount of information.
	www.wolframscience.com /reference/notes/1027c (520 words)

	A Discrete and Dynamic Geometry with Characteristics Satisfying Fundamental Requirements of Relativity Theory and ...
		However, if the concept of dimension on the discrete space is first defined, it may be possible to map the discrete space onto a corresponding continuous model and then speak of qualities which are so important to current theory, such as the curvature fundamental to understanding general relativity.
		A discrete space offers the possibility that constants represent a defined quality of that space and have a value which is integral, in fact unity in many cases.
		In a discrete space this action must be described in discrete units as it will be exhibited in terms of length and time, both discrete entities, or in terms of changes in neighborhoods, also discrete.
	www.geocities.com /andybmt/Discrete_Universe.html (8506 words)

	The causality of Quantum Mechanics
		Chapter one that a volume of space is composed or four spatial dimensions and a continuous non-quantized "field" of mass and energy.
		Similarly, space is "made up" of "quantum mass units of space" or discrete resonant "structures" formed by the interactions of a continuous non-quantized field of mass and energy.
		Therefore, these "vibrations" and resulting displacements in a "surface" of a three-dimensional space manifold with respect to a fourth spatial dimension generated by a "standing" matterenergy wave on a "surface" of a three-dimensional space manifold are responsible for the internal energy associated with all quantum particles.
	home.comcast.net /~jeffocal/chapter2.htm (1432 words)

	philosophy \| john bigelow \| work in progress (Site not responding. Last check: 2007-09-17)
		So the kind of "reasoning" which you must engage in concerning a discrete space is - like a bastard child - similar to the "reasoning" concerning an intelligible, continuous space, yet it is conceived and born outside the framework of the laws of geometry.
		For the Greeks who believed in a discrete space, such as the Epicureans, it followed immediately from their metaphysics that there had to be a maximum speed of motion.
		Khora, I say, is a discrete space: yet this is manifestly not beyond the reaches of logic and language, as is made unescapably apparent by a close reading of the text, particularly where it concerns the triangles which are to be imposed on khora.
	www.arts.monash.edu.au /phil/department/bigelow/space.html (12192 words)

	Discrete - Wikipedia, the free encyclopedia
		In topology, a branch of mathematics, a discrete space is a topological space in which all sets are open, and a discrete set is a set of isolated points.
		In discrete mathematics and in theoretical computer science, the abstract world is usually modeled as a discrete space with discrete time.
		In project management, discrete effort is any activity with a pre-defined state of completion; the opposite of discrete effort is continuous, operational or ongoing effort.
	en.wikipedia.org /wiki/Discrete (317 words)

	Ch.4 - Event Symmetric Space-Time
		However, experimental facts are a different matter and the discovery of quantum theory with its discrete energy levels and the Heisenberg uncertainty principle led physicists to speculate that space-time itself may be discrete as early as the 1930s.
		A space-time constructed as a discrete lattice is analogous to a crystal whose atoms are arranged on a regular array.
		These are akin to the discrete quantum numbers of the quantum mechanics of an atom which still also has a continuum description so the answer may be that space and time have a dual discrete and continuous nature.
	www.weburbia.com /press/html/g04.htm (8611 words)

	A New Description of Nature - The Way to Unification
		Space and time are the framework of our view of the world, yet we regard them as "composed" of infinite numbers of zero-size intervals or points.
		The discrete picture, where space and time are no longer continuous but have discrete structure, modifies our space and time symmetries and hence requires modifications of the laws of conservation of momentum, angular momentum and energy.
		Thus, the breakdown of the continuous symmetry of the phase of the de Broglie wave due to the discreteness, as described by the DFSD space model, modifies the law of conservation of electrical charge.
	www.phys.psu.edu /~scalise/misc/crackpot/tousson2.html (1333 words)

	1. Introduction
		Space and time are thought of as a continuum, hence the models potentially specify detail down to arbitrarily fine scales.
		Discrete physical models are an attractive alternative to continuous models such as partial differential equations.
		Likewise, in most discrete models the update rule operates locally in space, but we might ask whether we can devise rules such that the update can happen independently of other sites in time as well, rather than happen as part of a single global update.
	www-swiss.ai.mit.edu /~rauch/dapm/paper/node1.html (802 words)

	Event-Symmetric Space-Time (Site not responding. Last check: 2007-09-17)
		Since the symmetric group acting on space-time can be regarded as a discrete extension of the diffeomorphism group in general relativity, it is worth noting that the diffeomorphism invariance is not all that evident either.
		The analogy is not perfect since it suggests that curved space-time is embedded in some higher dimensional flat space, when in fact, the mathematical formulation of curvature avoids the need for such a thing.
		Ultimately, they argued, space and time would be understood in terms of relationships among physical objects which we observe more directly.
	www.weburbia.demon.co.uk /pg/esst.htm (3248 words)

	Policy 2001-6: Multiple Discrete Networks -- Single Maintainer ID
		Discrete networks must often have separate unique globally routable address space and will often grow at different rates.
		When applying for additional address space from an RIR for new networks or additional space for existing networks the organization must show greater than 50% utilization for the last block granted by the RIR and their allocations as a whole.
		The organization must not allocate additional space to a discrete network unless all the blocks allocated to that network show utilization greater than 80% individually and as a whole.
	www.arin.net /policy/proposals/2001_6.html (621 words)

	Dynamical Systems - Scholarpedia
		A dynamical system consists of an abstract phase space or state space, whose coordinates describe the state at any instant; and a dynamical rule that specifies the immediate future of all state variables, given only the present values of those same state variables.
		A discrete space can also have infinitely many states; for example, a random walk could be restricted to a lattice of points, and the system state is simply which lattice point is currently occupied.
		Other systems that are often modeled with discrete time include population dynamics (the discreteness referring to subsequent generations) and impacting systems like a billiard where only the state at impact is used.
	www.scholarpedia.org /article/Dynamical_Systems (1325 words)

	Digital Fuzzy Sets and Multivalued Logic
		A continuous-universal space membership function has a value defined for each point in universal space and a continuous-universal space system operates on and produces continuous-universal space membership functions.
		If a continuous-universal space membership function A(u) can take on any value (the grade of membership) in the continuous interval [0, 1], then the continuous-universal space membership function A(u) is called an analog fuzzy sets.
		A discrete-universal space membership function has a value (the grade of membership) only at discrete points in universal space and a discrete-universal space system operates on and produces discrete-universal space membership functions.
	documents.wolfram.com /applications/fuzzylogic/DemonstrationNotebooks/7.html (785 words)

	A New Kind of Science: The NKS Forum - I've discovered the TOE. Is it appropriate to post it here? :)
		Discrete space requires special means of the discrete geometry to describe its objects relations.
		It would be useful to note here, that in context of the kekinema and renovation properties, the ether portion shifts in the model should be perceived in discrete space as their disappearance at the beginning of time stroke and subsequent rebirth in adjacent voxel at the end of time stroke.
		This means that in some cases for the discrete space-time model we’ll be unable to distinguish between membership of the element in ensemble of particle, or its attached energy ensemble.
	forum.wolframscience.com /showthread.php?s=&threadid=783 (4609 words)

	Discrete Donut Twisted Chain Model of Space, Matter and Origin of Gravity
		The discreteness and geometry of the donut particle path allow calculation of the gravity to electromagnetic force ratio from basic concepts involving no inputted constants or units.
		Theories that are continuous or lack the exact discrete form of the donut seem incapable of deciphering gravity.
		In discrete donut twisted chain theory (ddtc) the quark is simply the connection between three donut chain segments, the minimum number of segments needed in order to have separate segments.
	www.ncia.com /~rlmarker/ddtcall.htm (6521 words)

	Algebraic Theory of Signal Processing
		The theory is built on top of the concept of a signal model, which is defined as the triple (A, M, Phi), where A is a chosen algebra of filters, M an associated A-module of signals, and Phi a generalization of the z-transform.
		The space shift is symmetric and thus the obtained models are undirected in contrast to the directed time models.
		Associated with the space models are the infinite and finite C-transform, proper notions of filtering or convolution, spectrum, and Fourier transforms.
	www.ece.cmu.edu /~smart/research.html (1518 words)

	A question of discrete space-time, part 2
		discrete space-time (a maximum speed that cannot be exceeded).
		Hilbert space on a discrete lattice, or an infinite dimensional one on a
		>> Hilbert space on a discrete lattice, or an infinite dimensional one on a
	physicsforums.com /showthread.php?t=136111 (3273 words)

	Car following
		While continuous space and continuous time is more realistic, discrete space and time are more natural for a digital computer.
		In consequence, cellular automata (CA) techniques, which are discrete in space and time, plus have a parallel local update, can actually simulate traffic quite well.
		From a practical point of view, traffic models which use discrete time but continuous space are numerically as efficient as the CA models but are much easier to calibrate.
	www.tu-berlin.de /fb10/ISS/FG4/archive/sim-archive/papers/book/html/node166.html (2075 words)

	2.1 Digital Recording Introduction (Site not responding. Last check: 2007-09-17)
		For example, discrete values are needed for both the vertical and horizontal values of the magnetic example (Figure 2.2a) from Los Alamos.
		Discrete amplitude samples and discrete space (or time) samples.
		Figure 2.2d illustrates the way that most geophysical data are digitally recorded, i.e., discrete in both amplitude and space (or time).
	www-rohan.sdsu.edu /~jiracek/digital/digitalrecording/index.html (598 words)

	Is Space-Time Discrete?
		The idea that space-time could be discrete has been a recurring one in the scientific literature of the twentieth century.
		However, such observations and the discovery of quantum theory with its discrete energy levels and the Heisenberg uncertainty principle led physicists to speculate that space-time itself may be discrete as early as the 1930's.
		Hawking makes a valid point but it may be possible to satisfy everyone by invoking a discrete structure of space-time without abandoning the continuum theories if the discrete-continuum duality can be resolved as it was for light and matter.
	www.weburbia.com /pg/discrete.htm (2780 words)

	The real line and configuration space
		Those theories are so dependent on the classical continuum that Einstein recognized any fully discrete theory would imply relativity was only approximately true and would make false predictions at the scale of space-time discreteness (see Section 5.1).
		The nonrelativistic version of quantum mechanics exist not in physical space but in an abstract higher dimensional structure known as configuration space where there is a single time dimension and a separate set of spatial dimensions for every particle.
		The connection between configuration space and physical space is through a probability distribution which gives the probability that a given configuration of particles will be observed.
	www.mtnmath.com /whatth/node55.html (576 words)

	Information Processing: Is Hilbert space discrete?
		Specifically, in a universe with a minimal length (for example, due to quantum gravity), no experiment can exclude the possibility that Hilbert space is discrete.
		So, if the size of the discreteness \epsilon is sufficiently small, it will escape our notice no matter how clever are our experiments (machines will collapse into fl holes before they can detect the smallest \epsilons).
		However, it is possible that discreteness in H might be larger than this threshold value which is concealed by minimum length.
	infoproc.blogspot.com /2005/08/is-hilbert-space-discrete.html (358 words)

	Discrete Constrained Search (DCS)
		Discrete constrained search (DCS) is a software package for finding constrained global minima (CGM) in the discrete variable space of a single-objective discrete constrained nonlinear programming problem (NLP), with no assumptions on the differentiability and continuity of the objective and constraint functions.
		Using the theory, a search for CLM is greatly simplified by looking for points that satisfy the saddle-point condition in discrete space, rather than solving the original constrained optimization problem.
		Each basic solver looks for saddle points in discrete space in the time allocated and differs only in its mechanism of generating and selecting candidate probes.
	manip.crhc.uiuc.edu /CSA (895 words)

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