Discrete stationary classical processes as well as quantum lattice states are asymptotically conf... more Discrete stationary classical processes as well as quantum lattice states are asymptotically confined to their respective typical support, the exponential growth rate of which is given by the (maximal ergodic) entropy. In the iid case the distinguishability of typical supports can be asymptotically specified by means of the relative entropy, according to Sanov’s theorem. We give an extension to the correlated case, referring to the newly introduced class of HP-states. 1
Innerhalb der TUMULT-Veranstaltungen (TUtoriale MULTimediale) zu der Linearen Algebra für Ingenie... more Innerhalb der TUMULT-Veranstaltungen (TUtoriale MULTimediale) zu der Linearen Algebra für Ingenieure an der Technische Universität Berlin wurde ein Konzept von Prelearning eingeführt. Studierende werden dabei dafür "belohnt", sich mit neuen Konzepten vor ihrer Einführung in der Vorlesung zu beschäftigen. In diesem Beitrag werden die Motivation, das Konzept, die Realisierung sowie eine erste Evaluierung des Prelearning in den TUMULT-Veranstaltungen präsentiert.
The workplace of scientists and engineers is about to change: even though numerical software and ... more The workplace of scientists and engineers is about to change: even though numerical software and computer algebra systems remove the burden of routine calculation, it becomes more important than ever to get familiar with new concepts and methods quickly. Given the rapid growth of knowledge in today’s sciences, traditional “learning on supply” (i.e., defining the learning goal as the accumulation of knowledge) is no longer applicable; instead, adequate learning and teaching methods have to be established to guide learners towards efficient self-controlled learning.
It is shown that the integers found by Thouless et al. in the quantized Hall effect are the only ... more It is shown that the integers found by Thouless et al. in the quantized Hall effect are the only quantized quantities associated with the energy bands. It is also proved that if two bands touch and then come apart as a parameter is varied, then their individual integers (conductances) may not be preserved but their sum is preserved.
e describe a precise mathematical theory of the Laughlin argument for the quantization of the Hal... more e describe a precise mathematical theory of the Laughlin argument for the quantization of the Hall conductance for general multiparticle Schrodinger operators with general background potentials. The quantization is a consequence of the geometric content of the conductance, namely, that it can be identified with an integral over the first Chem class. This generalizes ideas of Thouless et al. , for noninteracting Bloch Hamiltonians to general (interacting and nonperiodic) ones.
We discuss and extend an observation of Zinn-Justin that the double-well potential in one dimensi... more We discuss and extend an observation of Zinn-Justin that the double-well potential in one dimension and the anharmonic oscillator in two dimensions have coinciding perturbation expansions.
We investigate Laughlin's fractional quantum Hall effect wave function in the cylinder geometry o... more We investigate Laughlin's fractional quantum Hall effect wave function in the cylinder geometry of Laughlin's integer quantum Hall effect argument, at filling factor 1/3. We show that the plasma analogy leads to a periodic density, and that the wave function admits a representation as a "quantum polymer", reminiscent of the quantum dimer model by Rokhsar and Kivelson. We explain how the representation can be exploited to compute the normalization and one-particle density in the limit of infinitely many particles.
Analysis of the spectrum of a particle on a triangular lattice with two magnetic fluxes by algebraic and numerical methods
Journal of Physics A: Mathematical and General, 1991
ABSTRACT The authors consider an electron on a triangular lattice with two magnetic fluxes. Using... more ABSTRACT The authors consider an electron on a triangular lattice with two magnetic fluxes. Using a C*-algebra formalism a perturbation theory around rational fluxes is developed. This semiclassical type of analysis leads to the Wilkinson-Rammal formula for the energy eigenvalues. The analytic result is verified numerically with high accuracy up to second order. Crossing of eigenvalues near band edges is shown analytically as well as numerically.
The capacity of a quantum channel for transmission of classical information depends in principle ... more The capacity of a quantum channel for transmission of classical information depends in principle on whether product states or entangled states are used at the input, and whether product or entangled measurements are used at the output. We show that when product measurements are used, the capacity of the channel is achieved with product input states, * Partially supported by National Science Foundation Grant DMS-97-05779 and RSDF award from Northeastern University † Partially supported by National Science Foundation Grant DMS-97-06981 and Army Research Office Grant DAAG55-98-1-0374 so that entangled inputs do not increase capacity. We show that this result continues to hold if sequential measurements are allowed, whereby the choice of successive measurements may depend on the results of previous measurements. We also present a new simplified expression which gives an upper bound for the Shannon capacity of a channel, and which bears a striking resemblance to the well-known Holevo bound.
We review the standard JPEG2000 for still image compression and mention some typical applications... more We review the standard JPEG2000 for still image compression and mention some typical applications. Special weight is put onto the core coding system described in Part 1 and the compound image file format for document imaging described in Part 6 including a section on image segmentation.
Die ganze Vielfalt der Dynamik von Flüssigkeiten-sei es das gemächliche Dahinziehen der Wasser-Wi... more Die ganze Vielfalt der Dynamik von Flüssigkeiten-sei es das gemächliche Dahinziehen der Wasser-Wirbel hinter einem Brückenpfeiler, die Wolken-Wirbel auf der Leeseite eines hohen Berges im Pazifik längs einer Kármánschen Wirbelstrasse oder das Fliessen des Blutes in unseren Adern-ist zusammengefasst in einer einzigen nichtlinearen partiellen Differentialgleichung, der Navier-Stokes-Gleichung für das Geschwindigkeitsfeld u(t, x) und den skalaren Druck p(t, x), t ≥ 0, x ∈ G ⊂ R d , d = 2 oder 3 Dimensionen: ρ (u t + (u, ∇)u) = η u − ∇p + f. (1) Dabei bezeichnen η > 0 die Viskosität, ρ > 0 die Dichte und f (t, x) das äussere Kraftfeld. Gesucht sind Lösungen von (1), die der Inkompressibilitätsbedingung (∇, u) = 0 (2) genügen. Der Druck kann aus der Navier-Stokes-Gleichung eliminiert, d.h. durch Geschwindigkeitsfeld und äussere Kraft ausgedrückt werden. Die Gleichungen (1) und (2) beschreiben inkompressible Flüssigkeiten mathematisch vollständig. Deshalb ist die Navier-Stokes-Gleichung die fundamentale Gleichung der
Discrete stationary classical processes as well as quantum lattice states are asymptotically conf... more Discrete stationary classical processes as well as quantum lattice states are asymptotically confined to their respective typical support, the exponential growth rate of which is given by the (maximal ergodic) entropy. In the iid case the distinguishability of typical supports can be asymptotically specified by means of the relative entropy, according to Sanov’s theorem. We give an extension to the correlated case, referring to the newly introduced class of HP-states. 1
Innerhalb der TUMULT-Veranstaltungen (TUtoriale MULTimediale) zu der Linearen Algebra für Ingenie... more Innerhalb der TUMULT-Veranstaltungen (TUtoriale MULTimediale) zu der Linearen Algebra für Ingenieure an der Technische Universität Berlin wurde ein Konzept von Prelearning eingeführt. Studierende werden dabei dafür "belohnt", sich mit neuen Konzepten vor ihrer Einführung in der Vorlesung zu beschäftigen. In diesem Beitrag werden die Motivation, das Konzept, die Realisierung sowie eine erste Evaluierung des Prelearning in den TUMULT-Veranstaltungen präsentiert.
The workplace of scientists and engineers is about to change: even though numerical software and ... more The workplace of scientists and engineers is about to change: even though numerical software and computer algebra systems remove the burden of routine calculation, it becomes more important than ever to get familiar with new concepts and methods quickly. Given the rapid growth of knowledge in today’s sciences, traditional “learning on supply” (i.e., defining the learning goal as the accumulation of knowledge) is no longer applicable; instead, adequate learning and teaching methods have to be established to guide learners towards efficient self-controlled learning.
It is shown that the integers found by Thouless et al. in the quantized Hall effect are the only ... more It is shown that the integers found by Thouless et al. in the quantized Hall effect are the only quantized quantities associated with the energy bands. It is also proved that if two bands touch and then come apart as a parameter is varied, then their individual integers (conductances) may not be preserved but their sum is preserved.
e describe a precise mathematical theory of the Laughlin argument for the quantization of the Hal... more e describe a precise mathematical theory of the Laughlin argument for the quantization of the Hall conductance for general multiparticle Schrodinger operators with general background potentials. The quantization is a consequence of the geometric content of the conductance, namely, that it can be identified with an integral over the first Chem class. This generalizes ideas of Thouless et al. , for noninteracting Bloch Hamiltonians to general (interacting and nonperiodic) ones.
We discuss and extend an observation of Zinn-Justin that the double-well potential in one dimensi... more We discuss and extend an observation of Zinn-Justin that the double-well potential in one dimension and the anharmonic oscillator in two dimensions have coinciding perturbation expansions.
We investigate Laughlin's fractional quantum Hall effect wave function in the cylinder geometry o... more We investigate Laughlin's fractional quantum Hall effect wave function in the cylinder geometry of Laughlin's integer quantum Hall effect argument, at filling factor 1/3. We show that the plasma analogy leads to a periodic density, and that the wave function admits a representation as a "quantum polymer", reminiscent of the quantum dimer model by Rokhsar and Kivelson. We explain how the representation can be exploited to compute the normalization and one-particle density in the limit of infinitely many particles.
Analysis of the spectrum of a particle on a triangular lattice with two magnetic fluxes by algebraic and numerical methods
Journal of Physics A: Mathematical and General, 1991
ABSTRACT The authors consider an electron on a triangular lattice with two magnetic fluxes. Using... more ABSTRACT The authors consider an electron on a triangular lattice with two magnetic fluxes. Using a C*-algebra formalism a perturbation theory around rational fluxes is developed. This semiclassical type of analysis leads to the Wilkinson-Rammal formula for the energy eigenvalues. The analytic result is verified numerically with high accuracy up to second order. Crossing of eigenvalues near band edges is shown analytically as well as numerically.
The capacity of a quantum channel for transmission of classical information depends in principle ... more The capacity of a quantum channel for transmission of classical information depends in principle on whether product states or entangled states are used at the input, and whether product or entangled measurements are used at the output. We show that when product measurements are used, the capacity of the channel is achieved with product input states, * Partially supported by National Science Foundation Grant DMS-97-05779 and RSDF award from Northeastern University † Partially supported by National Science Foundation Grant DMS-97-06981 and Army Research Office Grant DAAG55-98-1-0374 so that entangled inputs do not increase capacity. We show that this result continues to hold if sequential measurements are allowed, whereby the choice of successive measurements may depend on the results of previous measurements. We also present a new simplified expression which gives an upper bound for the Shannon capacity of a channel, and which bears a striking resemblance to the well-known Holevo bound.
We review the standard JPEG2000 for still image compression and mention some typical applications... more We review the standard JPEG2000 for still image compression and mention some typical applications. Special weight is put onto the core coding system described in Part 1 and the compound image file format for document imaging described in Part 6 including a section on image segmentation.
Die ganze Vielfalt der Dynamik von Flüssigkeiten-sei es das gemächliche Dahinziehen der Wasser-Wi... more Die ganze Vielfalt der Dynamik von Flüssigkeiten-sei es das gemächliche Dahinziehen der Wasser-Wirbel hinter einem Brückenpfeiler, die Wolken-Wirbel auf der Leeseite eines hohen Berges im Pazifik längs einer Kármánschen Wirbelstrasse oder das Fliessen des Blutes in unseren Adern-ist zusammengefasst in einer einzigen nichtlinearen partiellen Differentialgleichung, der Navier-Stokes-Gleichung für das Geschwindigkeitsfeld u(t, x) und den skalaren Druck p(t, x), t ≥ 0, x ∈ G ⊂ R d , d = 2 oder 3 Dimensionen: ρ (u t + (u, ∇)u) = η u − ∇p + f. (1) Dabei bezeichnen η > 0 die Viskosität, ρ > 0 die Dichte und f (t, x) das äussere Kraftfeld. Gesucht sind Lösungen von (1), die der Inkompressibilitätsbedingung (∇, u) = 0 (2) genügen. Der Druck kann aus der Navier-Stokes-Gleichung eliminiert, d.h. durch Geschwindigkeitsfeld und äussere Kraft ausgedrückt werden. Die Gleichungen (1) und (2) beschreiben inkompressible Flüssigkeiten mathematisch vollständig. Deshalb ist die Navier-Stokes-Gleichung die fundamentale Gleichung der
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