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Topic: Hilbert systems

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Hilbert space

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	Wavefunction - Wikipedia, the free encyclopedia
		In the mathematical formulation of quantum mechanics, the state of any system is represented by an object called a ket, which is an element of an abstract mathematical structure called a Hilbert space.
		Due to the commutation relationship of the position and momentum operators, for a system of spinless particles in Euclidean space the r-space and k-space wavefunctions are Fourier transform pairs.
		If the energy spectrum of a system is (partly) discrete, such as for a particle in an infinite potential well or the bound states of the hydrogen atom, then the position representation is continuous while the momentum representation is partly discrete.
	en.wikipedia.org /wiki/Wavefunction (735 words)

	Proof theory - Wikipedia, the free encyclopedia
		Hilbert's ideas seem to have been based on an analogy, in fact false, with the elimination theory of algebraic geometry familiar to him from his early work in algebra.
		Kurt Gödel's seminal work on proof theory first advanced, then refuted this program: his completeness theorem seemed to bring Hilbert's dream of reducing all mathematics to a small, finitarily meaningful core within reach, then his incompleteness theorems showed that the dream was unattainable.
		The recent discovery of self-verifying theories, systems strong enough to talk about themselves, but too weak to carry out the diagonal argument that is the key to Gödel's unprovability argument.
	en.wikipedia.org /wiki/Proof_theory (1094 words)

	Hilbert
		Hilbert was a member of staff at Königsberg from 1886 to 1895, being a Privatdozent until 1892, then as Extraordinary Professor for one year before being appointed a full professor in 1893.
		Hilbert's eminent position in the world of mathematics after 1900 meant that other institutions would have liked to tempt him to leave Göttingen and, in 1902, the University of Berlin offered Hilbert Fuchs' chair.
		Hilbert himself tried at first to follow Gordan's approach but soon realised that a new line of attack was necessary.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Hilbert.html (1570 words)

	Hilbert's Program
		Hilbert never gave a general account of which operations and methods of proof are acceptable from the finitist standpoint, but only examples of operations and methods of inference in contentual finitary number theory which he accepted as finitary.
		Although Hilbert's first proposals focused exclusively on consistency, there is a noticeable development in Hilbert's thinking in the direction of a general reductivist project of a sort quite common in the philosophy of science at the time (as was pointed out by Giaquinto 1983).
		Hilbert's proof theoretic program can then be seen to be a search for a proof theoretic reduction of all of mathematics to finitary mathematics; in a relativized program one looks for reductions of theories weaker than all of classical mathematics to theories often stronger than finitary mathematics.
	plato.stanford.edu /entries/hilbert-program (7534 words)

	Wavefunction (Site not responding. Last check: 2007-10-14)
		In quantum mechanics the state of a system is represented by an object called a ket vector, which is an element of an abstract mathematical structure called a Hilbert space.
		For isolated systems, the dynamics (or time evolution) of the system can be described by a one-parameter family of unitary operators.
		If the energy spectum of a system is (partly) discrete, such as for a particle in an infinite potential well or the bound states of the hydrogen atom, then the position representation is continuous while the momentum representation is partly discrete.
	www.1-free-software.com /en/wikipedia/w/wa/wavefunction.html (553 words)

	Logic Seminar Abstracts (Site not responding. Last check: 2007-10-14)
		New applicative systems [based on an untyped partial combinatory algebra] are proposed whose provably recursive functions conicide with the functions computable in polynomial time, polynomial space, polynomial time and linear space, as well as linear space.
		New applicative systems [based on an untyped partial combinatory algebra] are proposed whose provably recursive functions coincide with the functions computable in polynomial time, polynomial space, polynomial time and linear space, as well as linear space.
		The main innovation was the invention of the epsilon-calculus, on which Hilbert's axiom systems were based, and the development of the epsilon-substitution method as a basis for consistency proofs.
	www-logic.stanford.edu /Abstracts/Seminar/Winter01.html (924 words)

	Richard Zach Bibliography
		Hilbert was unfortunately not exact in delineating what that viewpoint was, and Hilbert himself changed his usage of the term through the 1920s and 30s.
		The purpose of this paper is to outline what the main problems are in understanding Hilbert and Bernays on this issue, based on some publications by them which have so far received little attention, and on a number of philosophical reconstructions and criticisms of the viewpoint (in particular, by Hand, Kitcher, Parsons, and Tait).
		Hilbert's instrumentalism is usually thought to entail that only the "real," finitistic part of mathematics has meaning and the "ideal" part (the formalized part that involves assumptions of infinity) is a meaningless game of signs.
	sun3.lib.uci.edu /~scctr/philosophy/zach.html (2560 words)

	David Hilbert
		David Hilbert was born in Koenigsberg, East Prussia in 1862 and received his doctorate from his home town university in 1885.
		Hilbert's axioms could be proved as theorems from Moore's and conversely, Moore's axioms could be proved as theorems from Hilbert's.
		Sometimes a mimeographed copy of the the paper would circulate before the actual publication and you would try to get your hands on a copy of it as quickly as possible, especially if you, yourself, had been working on the problem and had been leaning toward a result that was contrary to the current rumor.
	www.sonoma.edu /Math/faculty/falbo/hilbert.html (1070 words)

	A Generalized Hilbert Matrix Problem and Confluent Chebyshev--Vandermonde Systems
		One is for the case where the points involved in the generalized Hilbert matrices satisfy a TH-relation introduced in the present paper, which include equidistant points, clustered points, and Chebyshev points.
		The solution of Chebyshev--Vandermonde systems is also reduced to the generalized Hilbert matrix problem by using J-matches, links of Chebyshev polynomials, and the inversion of a class of generalized Hilbert matrices.
		Third, the results obtained are applied to related problems, for example, confluent Chebyshev--Vandermonde systems for near Chebyshev $\sigma$-points, Hermite interpolation in terms of Chebyshev polynomials, and a class of generalized Hilbert systems.
	epubs.siam.org /sam-bin/dbq/article/30722 (376 words)

	DC MetaData for: Hilbert C-systems for Actions of the Circle Group (Site not responding. Last check: 2007-10-14)*
		Hilbert C*-systems for Actions of the Circle Group
		Hilbert space, $\Gamma$ the conjugation and $P$ a basis projection
		According to a general result for Hilbert systems of this type,
	www.esi.ac.at /Preprint-shadows/esi940.html (141 words)

	[No title]
		This distinction is lifted to type theory, with first class proof objects, through Curry's abstraction algorithm in the Hilbert systems and through Church's typed lambda calculus in the Gentzen systems.
		I would be interested in any opinions on these tradeoffs, further differences between the systems, or recent or historical references that discuss this question.
		Hilbert systems) and lambda calculus (here Gentzen systems) is that the first is "algebraic", whereas the second is not.
	www.tcs.informatik.uni-muenchen.de /~alti/lambda-vl/types-mails.txt (848 words)

	DI & CoS - Modal Logic
		This model theory is generally given using frame semantics, and it is systematic in the sense that for the most important systems we have a clean, exact correspondence between their constitutive axioms as they are usually given in a Hilbert-Lewis style and conditions on the accessibility relation on frames.
		Thus, the formulation of a system in such a framework is an evolutional process and leads to positive proof theoretical results.
		Accordingly, the systems can be divided into two categories: in those which allow semantic-oriented formulae and those which allow formulae in positions not reachable by the usual systems in the sequent calculus.
	alessio.guglielmi.name /res/cos/ML (870 words)

	Brainstorms: A Gödelian Argument Against Darwinism
		Also, Hilbert's goal was, in essence, to create the Ultimate Mathematica, where you would plug in any question, apply rules, and get an answer (at least in theory).
		The systems I am referring to, of course, are axiomatic systems (or formal systems) which are made up of symbols like + and -, and no not all of them are simple.
		But there is now mounting evidence of biological systems for which any slight modification does not merely destroy the existing function but also destroys the possibility of any function of the system whatsoever (see Axe 2000).
	www.iscid.org /boards/ubb-get_topic-f-6-t-000326-p-2.html (4158 words)

	A systematic proof theory for several modal logics (Site not responding. Last check: 2007-10-14)
		The family of normal propositional modal logic systems are given a highly systematic organisation by their model theory.
		This model theory is generally given using Kripkean frame semantics, and it is systematic in the sense that for the most important systems we have a clean, exact correspondence between their constitutive axioms as they are usually given in a Hilbert style and conditions on the accessibility relation on frames.
		Because of this property, it is possible to axiomatise the modal logics in a manner directly analogous to the Hilbert axiomatisation.
	www.linearity.org /cas/papers/sysptf.html (259 words)

	combatindex.com - US Sea Systems: DE 742 (HILBERT)
		Hilbert (DE-742) was laid down 23 March 1943; launched 18 July 1943 by Western Pipe and Steel Co.; sponsored by Mrs.
		Hilbert also played a key role in protecting our oilers which fueled Admiral Marc Mitscher's Fast Carrier Task Force engaged in the Battle of the Philippine Sea - one of the most decisive battles of the war.
		Hilbert, with other units of the 3d Fleet, anchored for the first time in Japanese waters at Sagami Wan 9 September 1945.
	www.combatindex.com /hardware/detail/sea/vintage/destroyers/de/701-800/de742.html (354 words)

	A HILBERT TRANSFORM BASED RECEIVER POST PROCESSOR
		A Hilbert transform technique is used to identify the frequency dependent gain and circularity errors within the receiver.
		The Hilbert transformer is conceptually equivalent to a broadband 90 phase shifter (Slater, 1991, Slater, 1985).
		The I channel signal and Hilbert transform are combined to form an analytic signal which is then used to identify the circularity and gain errors within any receiver configuration.
	www.nearfield.com /amta/amta91ds.htm (2046 words)

	Untitled Document
		A.H. Zemanian, A frequency-domain characterization for the causality of active linear systems.
		A.H. Zemanian, The Hilbert port: An extension of the concept of the n-port.
		L.P. D'Amato and A.H. Zemanian, A boundary-value criterion for semipassive Hilbert ports.
	www.ece.sunysb.edu /~zeman/refereed.html (2483 words)

	Publications in chronological order
		As a consequence, subsystems of a physical system described by a generalized Hilbert space over a division ring K are always described by a generalized Hilbert space over a subdivision ring of K. Aerts, D. Why is it impossible in quantum mechanics to describe two or more separated entities.
		We introduce the formal mathematical structure of a 'state experiment probability system', by using this new type of probability theory, as a general description of a physical entity by means of its states, experiments and probability.
		Abstract: We study a system of two entangled spin 1/2, were the spin's are represented by a sphere model developed within the hidden measurement approach which is a generalization of the Bloch sphere representation, such that also the measurements are represented.
	www.vub.ac.be /CLEA/aerts/publications/chronological.html (13719 words)

	The tensor product procedure
		The structure of isomorphisms between Hilbert space lattices had been investigated by Eugene Wigner in the beginning years of quantum mechanics, and lead to a deep and beautiful theorem (called Wigner's theorem) that shows that each isomorphism can be represented by a unitary or anti-unitary map between the Hilbert spaces.
		Aerts, D. and Daubechies, I. Mathematical condition for a sub-lattice of a propositional system to represent a physical subsystem with a physical interpretation.
		As a consequence, subsystems of a physical system described by a generalized Hilbert space over a division ring K are always described by a generalized Hilbert space over a subdivision ring of K. Aerts, D. Description of compound physical systems and logical interaction of physical systems.
	www.vub.ac.be /CLEA/aerts/publications/tensor_product.html (1104 words)

	Absolute stability of feedback systems in Hilbert spaces (ResearchIndex) (Site not responding. Last check: 2007-10-14)
		Abstract: The problem of absolute stability of a feedback loop of an abstract differential system in Hilbert spaces is considered.
		Applications of Popov's type frequency domain criteria and of the Kalman-Yakubovich Lemma for the construction of Lyapunov functions are illustrated, in two situations pertaining distributed sytems.
		Finally, a new criterion for absolute stability of a class of parabolic systems with boundary feedback is presented.
	citeseer.ist.psu.edu /bucci97absolute.html (476 words)

	Combining Hilbert Style and Semantic Reasoning in a Resolution Framework - Ohlbach (ResearchIndex) (Site not responding. Last check: 2007-10-14)
		Reasoning in Hilbert Systems, however, is extremely inefficient.
		In this paper a combination of Hilbert style and semantic reasoning is proposed.
		It is particularly tailored for cases where either the semantics of some operators is not known, or it is second-order, or it is just too complicated to handle, or...
	citeseer.ist.psu.edu /ohlbach98combining.html (569 words)

	Amazon.de: English Books: Recent Developments in Integrable Systems and Riemann-Hilbert Problems. (Site not responding. Last check: 2007-10-14)
		The goal of the meeting was to foster new research by bringing together experts from different areas.
		Their contributions to the volume provide a useful portrait of the breadth and depth of integrable systems.
		The book is intended for graduate students and researchers interested in integrable systems and its applications.
	www.amazon.de /exec/obidos/ASIN/0821832034/digitalphot00-21 (237 words)

	L-System Based Fractals
		One of the interesting things about this one is that by rescaling the lines at every iteration (by a factor of 3), the overall size remains constant while the length of all the lines added together grows towards infinity.
		The Hilbert Curve, proposed by David Hilbert, is called a space-filling fractal, because as you can see, it continuously attempts to fill in the empty area within.
		Select the Hilbert Curve fractal, reset it, and advance it to the first iteration.
	www.arcytech.org /java/fractals/lsystems.shtml (1020 words)

	SIAM Journal on Control
		691--715 Masao Ikeda and Hajime Maeda and Shinzo Kodama Stabilization of Linear Systems.
		130--147 P. D'Alessandro and A. Isidori and A. Ruberti A new approach to the theory of canonical decomposition of linear dynamical systems.
		653--669 Arthur J. Krener On the equivalence of control systems and linearization of nonlinear systems 670--676 L. Cesari and J. La Palm and D. Sánchez Erratum: ``An existence theorem for Lagrange problems with unbounded controls and a slender set of exceptional points'' (SIAM J. Control \bf 9 (1971), 590--605).
	www.math.utah.edu /ftp/pub/tex/bib/toc/siamjcontrol.html (5768 words)

	Noncommutativity - Geometry and Probability. (Site not responding. Last check: 2007-10-14)
		Tensor product systems of (two-sided) Hilbert modules over a unital C-algebra B arise naturally in the dilation theory of CP-semigroups (semigroups of unital completely positive mappings) on that C-algebra.
		Product systems coming from a CP-semigroup always have a "unit" and the unit allows one to recover the CP-semigroup we started with.
		Existence of a central unit will also help us to show that the product system under consideration is isomorphic to (the product system associated with) a time ordered Fock module - the module analogue of symmetric Fock space.
	www.maths.nott.ac.uk /personal/jmlsjw/NCGP/ABSTRACTS/skeide.html (220 words)

	Hilbert's 16th problem for quadratic systems and cyclicity of elementary graphics (Site not responding. Last check: 2007-10-14)
		Hilbert's 16th problem for quadratic systems and cyclicity of elementary graphics
		In this paper we study the finite cyclicity of several elementary graphics appearing in quadratic systems.
		One originality of the paper is to prove that for certain graphics among quadratic systems some regular transition maps are not tangent to the identity.
	ej.iop.org /EJ/abstract/0951-7715/9/5/008 (302 words)

	The Monoidal Category of Hilbert Spaces
		The simplest case is a so-called `joint system': a system built out of two separate parts.
		This means that from a (pure) state of a joint quantum system we cannot extract (pure) states of its parts.
		For example, if we have a joint quantum system whose two parts evolve in time without interacting, any time evolution operator for the whole system is given by the tensor product of time evolution operators for the two parts.
	math.ucr.edu /home/baez/quantum/node4.html (2679 words)

	Citebase - Hilbert C-systems for actions of the circle group (Site not responding. Last check: 2007-10-14)*
		The paper contains constructions of Hilbert systems for the action of the circle group $T$ using subgroups of implementable Bogoljubov unitaries w.r.t.
		Fock representations of the Fermion algebra for suitable data of the selfdual framework: ${\cal H}$ is the reference Hilbert space, $\Gamma$ the conjugation and $P$ a basis projection on ${\cal H}.$ The group $C({spec} {\cal Z}\to T)$ of $T$-valued functions on ${spec} {\cal Z}$ turns out to be isomorphic to the stabilizer of ${\cal A}$.
		4 (1997) 785 we study Hilbert-C* systems {F,G} where the fixed point algebra A has nontrivial center Z and where A'\cap F=Z is satisfied.
	citebase.eprints.org /cgi-bin/citations?archiveID=oai:arXiv.org:math-ph/0010011 (644 words)

	MATH2071: LAB #7: Solving Linear Systems
		Then we state the linear system problem and consider three methods of solution, using the determinant, the inverse matrix, or Gauss factorization.
		Repeat for a larger N and for the Frank and Hilbert matrices.
		(We already know that the Hilbert matrix was hopeless when we had the exact inverse, so we'll worry now about what happens with the other two.) Presumably, the errors we make in computing the inverse will then be reflected as additional errors in the solution of our linear system.
	www.csit.fsu.edu /~burkardt/math2071/lab_07.html (3170 words)

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