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# Topic: Unitary transformation

###### In the News (Fri 2 Dec 16)

 PlanetMath: unitary A unitary transformation is a surjective linear transformation On the other hand, the representing matrix of a unitary transformation relative to an orthonormal basis is, in fact, a unitary matrix. Unitary spaces, transformations, and matrices are of fundamental importance in quantum mechanics. planetmath.org /encyclopedia/UnitaryTransformation.html   (308 words)

 A Critique of Relativity and Localization The transformation equations named for Lorentz are a set of rules to transform observations made relative to one observer into those made relative to another observer. The Lorentz transformations are called point transformations; and it is their point-for-point character which requires the point-by-point definition of the free fields, and leads to their "local" character, their being defined with reference to space and time. In field theory, for example, Haag's theorems [5] and related results establish that there is no acceptable correspondence (unitary transformation) between free-particle fields and the interacting-particle fields when we include interaction in the time-evolution of the system. www.focusing.org /critique_of_relativity.html   (6402 words)

 Unitary state A unitary state is a state or country that is governed constitutionally as one single unit, with one constitutionally created legislature. The United Kingdom is a particularly striking example of a unitary state with a series of parliament-created devolved assemblies, for Scotland, Wales and Northern Ireland, all of which were created in between 1998 and 1999. The high proportion of the world's countries which are unitary states results in large part from the fact that most are insufficiently large enough to produce the complexity necessary to demand the devolution of power on distinct internal territories. www.kiwipedia.com /en/unitary-state.html   (477 words)

 Pure qubit state - Wikipedia The vector space in which all pure qubit states lie is the two-dimensional Hilbert space. Unitary transformations are one kind of basic operation which can be performed on qubits. A two-dimensional unitary transformation transforms a pure qubit state into another. nostalgia.wikipedia.org /wiki/Pure_qubit_state   (213 words)

 Unitary transform methods of identifying defects in imaging devices - US Patent 5325198   (Site not responding. Last check: 2007-11-05) The squared magnitudes of the transform values of the selected regions are summed and the resulting sum is normalized for mask shape and flood image intensity to determine and quantify the presence of specific artifacts. A decision unit compares the normalized transformation sum to a predetermined maximum threshold, and optionally activates a display means, which indicates if the threshold has been exceeded indicating that the imaging means has a significant artifact of the type being tested and has failed the test. A test mask 20 selects regions of the transform field to be tested according to a test mask M and passes the selected regions of transform field T(A) to a deviation calculation means 22 which determines a sum of transformation sum S(A,M) within the selected regions. www.patentstorm.us /patents/5325198.html   (4898 words)

 Unitary transform methods of identifying defects in imaging devices - Patent 5325198 A unitary transform unit receives the normalized intensity values and determines a D-dimensional transform field comprised of transform values T(A).sub.i,j each being an i,jth component of the unitary transform. In the case of the Fourier transform, the component T(A).sub.i,j is the (i,j)th frequency component of the transform. The horizontal and vertical dimensions of the transform fields being the f.sub.x and f.sub.y frequency axes pertain to the vertical x and horizontal y directions of the flood images, respectively. www.freepatentsonline.com /5325198.html   (4864 words)

 Bogoliubov transformation - Wikipedia, the free encyclopedia In theoretical physics, the Bogoliubov transformation is a unitary transformation from a unitary representation of some canonical commutation relation algebra or canonical anticommutation relation algebra into another unitary representation, induced by an isomorphism of the CCR/CAR algebra. and they can be viewed as the Bogoliubov transformations of one another using the operator-state correspondence. In physics, the Bogoliubov transformation is important for understanding of the Unruh effect and Hawking radiation, among many other things. www.wikipedia.org /wiki/Bogoliubov_transformation   (209 words)

 NRaD TD2733 text In general, the linear transformation on the frequency components may be any Green's function transformation from every component in the input segment to every component in the output segment, and it may be as complicated as the transformation one might use in a time domain approach. If we admit arbitrary elementary transformations as building blocks, we face the problem that individual elements of the data sequence may face degenerate or nearly degenerate transformations, so that some of the input data samples may be represented only weakly or not at all in the output. One complication of discrete-time transformational architectures (elementary transformation architectures, as well as frequency analysis/synthesis techniques) is that they tend to produce critically sampled representations of their output signals; hence, accurately smoothing the output samples to a continuous-time signal may not be simple. www.nosc.mil /sti/publications/pubs/td/2733/nradtd2733txt.html   (3001 words)

 Unitary matrix : Unitary transformation   (Site not responding. Last check: 2007-11-05) A unitary matrix is a square matrix U whose entries are complex numbers and whose inverse is equal to its conjugate transpose U*. A unitary matrix in which all entries are real is the same thing as an orthogonal matrix. All unitary matrices are normal, and the spectral theorem therefore applies to them. www.termsdefined.net /un/unitary-transformation.html   (213 words)

 Change of representation A unitary transformation is equivalent to a change of basis. The results of the active and the passive transformation are the same. There are two ways of looking at a unitary transformation. electron6.phys.utk.edu /QM1/modules/m3/change_rep.htm   (568 words)

 Encyclopedia: Unitary transformation   (Site not responding. Last check: 2007-11-05) A unitary transformation is an isomorphism (but not an antiisomorphism; that corresponds to an antiunitary transformation) between two Hilbert spaces or an automorphism of a single Hilbert space. In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich. In functional analysis, a unitary operator is a bounded linear operator U on a Hilbert space satisfying U*U=UU*=I where I is the identity operator. www.nationmaster.com /encyclopedia/Unitary-transformation   (189 words)

 NAG C Library Manual, Mark 7 : Keywords in Context (UNITARY)   (Site not responding. Last check: 2007-11-05) Generate unitary transformation matrix from reduction to tridiagonal form determined by f08fsc Generate unitary transformation matrix from reduction to tridiagonal form determined by f08gsc Unitary reduction of a pair of complex general matrices to generalized upper Hessenberg form www.nag.com /numeric/cl/manual/html/indexes/kwic/unitary.html   (206 words)

 Application of Unitary Transformations in the Study of Phase Diagram of Strongly Correlated Many-Body Systems. (E. ...   (Site not responding. Last check: 2007-11-05) In order to enhance the development in such type of studies we use a procedure that starts from the existence of a given phase (proved with another method, see for example[1]) in a given domain of the parameter space connected to a model Hamiltonian. The procedure is based on the well known fact that an arbitrary unitary transformation does not change the eigenvalues spectrum, so if a state is a ground-state of a given Hamiltonian (H), the transformed state will be the ground-state of the system described by the transformed Hamiltonian (H'). The unitary transformations used by us[1] connect a superconducting state to a ferromagnetic state, a spin-density wavw state to a charge-density wave state, a phase separation in charge to a phase separation in spin. dtp.atomki.hu /HOME-PAGE/prog98/node77.html   (243 words)

 [No title]   (Site not responding. Last check: 2007-11-05) Unitary similarity transformations furnish a powerful vehicle for generating infinite generic classes of signal analysis and processing tools based on concepts different from time, frequency, and scale. Implementation of these new tools involves simply preprocessing the signal by a unitary transformation, performing standard processing techniques on the transformed signal, and then (in some cases) transforming the resulting output. These applications illustrate the utility of the unitary equivalence concept for uniting seemingly disparate approaches proposed in the literature. cmc.rice.edu /docs/docinfo.aspx?doc=Bar1995Oct1UnitaryEqu   (204 words)

 NAG Fortran Library Manual, Mark 21 : Keyword Index (transformation)   (Site not responding. Last check: 2007-11-05) Generate unitary transformation matrix from reduction to tridiagonal form determined by F08FSF Generate unitary transformation matrix from reduction to tridiagonal form determined by F08GSF Generate unitary transformation matrices from reduction to bidiagonal form determined by F08KSF www.nag.co.uk /numeric/fl/manual/html/indexes/kwic/fl_transformation.html   (221 words)

 The analysis of the T=0 phase diagram of many-body systems based on unitary transformations (E. Kovács and Zs. ...   (Site not responding. Last check: 2007-11-05) The analysis of the T=0 phase diagram of many-body systems based on unitary transformations (E. Kovács and Zs. In order to help the further description of the system in these conditions we have applied a method based on unitary transformations. Looking however for unitary transformations that map the Hamiltonian in itself, a relation is obtained between the coupling constants of H' (g' dtp.atomki.hu /HOME-PAGE/progr2000/node37.html   (251 words)

 The Foldy-Wouthuysen Transformation Because of the nature of the FW transformation, each application of the unitary transform results in a residual odd operator which is higher order by one in the fine structure constant each time. The utility of the Foldy-Wouthuysen transformation lies in the fact that the even operator which remains may be utilized as an approximate Hamiltonian which should take account for the preponderance of the effects of relativity. Because of the effort required to develop the proper unitary transformation and subsequently apply it to the total electronic wavefunction, it is not practical to go much beyond separating the odd operator to a greater order than www.uam.es /docencia/quimcursos/Docs/Knowledge/Fundamental_Theory/rltvt/node19.html   (644 words)

 Unitary operators Matrix equations that have this structure are called similarity transformations, and have important properties. On of such properties is that the eigenvalues of a matrix are invariant under a unitary similarity transformation. are different matrices, their eigenvalues are the same since they are related by a unitary similarity transformation. www.physics.unlv.edu /~bernard/phy721_99/tex_notes/node9.html   (218 words)

 pclatrd an unitary similarity transformation Qâ * sub( A ) * Q, and returns the ma...   (Site not responding. Last check: 2007-11-05) an unitary similarity transformation Qâ * sub(A) * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of sub(A) There is currently no version of the SDSM library with support for a default integer size of 8 bytes (64 bits). PCLATRD reduces NB rows and columns of a complex Hermitian distributed matrix sub(A) = A(IA:IA+N-1,JA:JA+N-1) to complex tridiagonal form by an unitary similarity transformation Qâ * sub(A) * Q, and returns the matrices V and W which are needed to apply the transformation to the unreduced part of sub(A). www.uni-kiel.de /rz/nvv/altix-doc/man_html/man3/pclatrd.3s.html   (1001 words)

 The QR Transformation A Unitary Analogue to the LR Transformation--Part 1 -- Francis 4 (3): 265 -- The Computer Journal The QR Transformation A Unitary Analogue to the LR Transformation--Part 1 -- Francis 4 (3): 265 -- The Computer Journal The QR Transformation A Unitary Analogue to the LR Transformation—Part 1 The LR transformation, due to Rutishauser, has proved to be comjnl.oupjournals.org /cgi/content/abstract/4/3/265   (183 words)

 IngentaConnect Unitary transformation approach to the exact solution for the sin...   (Site not responding. Last check: 2007-11-05) IngentaConnect Unitary transformation approach to the exact solution for the sin... Unitary transformation approach to the exact solution for the singular oscillator By performing unitary transformations, the exact solution of the time-dependent singular oscillator is obtained. api.ingentaconnect.com /content/iop/jphysa/1996/00000029/00000011/art00017   (107 words)

 IngentaConnect A unitary transformation approach to the mutual quenching of stru...   (Site not responding. Last check: 2007-11-05) IngentaConnect A unitary transformation approach to the mutual quenching of stru... A new approach based on a unitary transformation, the [iopmath latex="${\bi k}$"] k [/iopmath] -dependent displacement transformation, is proposed for re-examining the properties of the mutual quenching Jahn-Teller system. We show that when a unitary transformation and the perturbation approximation are applied together to this system, the retardation effect must be considered also. api.ingentaconnect.com /content/iop/jphyscm/2001/00000013/00000005/art00321   (220 words)

 The Foldy-Wouthoysen Transformation   (Site not responding. Last check: 2007-11-05) The operators which result from such a transformation can be very complicated, but provide solutions which fit more comfortably into traditional concepts of electronic solutions. The objective of the Foldy-Wouthoysen transformation is to apply a unitary transform to the relativistic, four-component wave equation in order to decouple the large and small components by eradicating the odd portions of the Hamiltonian to some order in an expansion parameter which will go to zero in the non-relativistic limit. This transformed Hamiltonian is identical to the Foldy-Wouthoysen transformed Hamiltonian given in (4.120), with the exception that the rest mass energy of the electron, zopyros.ccqc.uga.edu /%7Ekellogg/docs/rltvt/node18.html   (780 words)

 pzgebrd unitary transformation   (Site not responding. Last check: 2007-11-05) PZGEBRD reduces a complex general M-by-N distributed matrix sub(A) = A(IA:IA+M-1,JA:JA+N-1) to upper or lower bidiagonal form B by an unitary transformation: Qâ * sub(A) * P = B. If M >= N, B is upper bidiagonal; if M < N, B is lower bidiagonal. The scalar factors of the elementary reflectors which represent the unitary matrix Q. TAUQ is tied to the distributed matrix A. See Further Details. The scalar factors of the elementary reflectors which represent the unitary matrix P. TAUP is tied to the distributed matrix A. See Further Details. www.uni-kiel.de /rz/nvv/altix-doc/man_html/man3/pzgebrd.3s.html   (797 words)

 zgebrd - reduce a general complex M-by-N matrix A to upper or lower bidiagonal form B by a unitary transformation zgebrd - reduce a general complex M-by-N matrix A to upper or lower bidiagonal form B by a unitary transformation The scalar factors of the elementary reflectors which represent the unitary matrix Q. See Further Details. The scalar factors of the elementary reflectors which represent the unitary matrix P. See Further Details. docs.sun.com /source/816-2461/zgebrd.html   (577 words)

 APS - DAMOP 1997   (Site not responding. Last check: 2007-11-05) We present a numerical technique, "Unitary Integration," that exactly preserves these invariants to all orders in the time step. This method, which evolves the density matrix via unitary transformations, is the quantum mechanical analog of symplectic integrators commonly used in classical mechanics. Although the evolution is approximate (as in any numerical method), the structure of Liouville's equation is preserved exactly since the time advance map is constructed so as to yield a unitary transformation. hagar.ph.utexas.edu /~shadwick/posters/damop97.html   (191 words)

 A Unitary Transformation in the Contracted Symplectic Model Approach   (Site not responding. Last check: 2007-11-05) In these works a model hamiltonian that takes into account the shell structure, couplings to major shells through a quadrupole-quadrupole interaction, and a residual rotor term were used. In the present contribution a unitary transformation is introduced, which gives rise to a simpler hamiltonian and the matrix elements of its different component terms with respect to the U_b \times U_s(3) basis states are easily calculated. At the same time this unitary transformation in the boson approximation limit yields new insights to the shell model interpretation of the quantum rotor hamiltonian. flux.aps.org /meetings/YR97/BAPSAPR97/abs/S1155015.html   (162 words)

 NAG Fortran Library Manual, Mark 21 : Keyword Index (unitary)   (Site not responding. Last check: 2007-11-05) Form all or part of unitary Q from QR factorization determined by F08ASF or F08BSF Apply unitary transformation determined by F08ASF or F08BSF Apply unitary transformations from reduction to bidiagonal form determined by F08KSF www.nag.co.uk /numeric/fl/manual/html/indexes/kwic/fl_unitary.html   (230 words)

 IngentaConnect Phase diagram regions deduced for strongly correlated systems via...   (Site not responding. Last check: 2007-11-05) From known phase diagram regions of different model Hamiltonians describing strongly correlated systems we deduced new domains of the ground state phase diagram of the same models by a unitary transformation. Different types of extended Hubbard Hamiltonians were used for the starting point and the existence of new stable spin-density waves, charge-density waves, ferromagnetic states and a paramagnetic insulator is demonstrated. The ground state phase diagrams of several strongly correlated systems modelled by extended Hubbard-like Hamiltonians are analysed using unitary transformations. www.ingentaconnect.com /content/tandf/tphb/2001/00000081/00000003/art00008   (179 words)

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