Finite volume effects in chiral perturbation theory with twisted boundary conditions

We study the effects of a finite cubic volume with twisted boundary conditions on pseudoscalar mesons. We apply Chiral Perturbation Theory in the p-regime and introduce the twist by means of a constant vector field. The corrections of masses, decay constants, pseudoscalar coupling constants and form factors are calculated at next-to-leading order. We detail the derivations and compare with results available in the literature. In some case there is disagreement due to a different treatment of new extra terms generated from the breaking of the cubic invariance. We advocate to treat such terms as renormalization terms of the twisting angles and reabsorb them in the on-shell conditions. We confirm that the corrections of masses, decay constants, pseudoscalar coupling constants are related by means of chiral Ward identities. Furthermore, we show that the matrix elements of the scalar (resp. vector) form factor satisfies the Feynman–Hellman Theorem (resp. the Ward–Takahashi identity). To show the Ward–Takahashi identity we construct an effective field theory for charged pions which is invariant under electromagnetic gauge transformations and which reproduces the results obtained with Chiral Perturbation Theory at a vanishing momentum transfer. This generalizes considerations previously published for periodic boundary conditions to twisted boundary conditions. Another method to estimate the corrections in finite volume are asymptotic formulae. Asymptotic formulae were introduced by Lüscher and relate the corrections of a given physical quantity to an integral of a specific amplitude, evaluated in infinite volume. Here, we revise the original derivation of Lüscher and generalize it to finite volume with twisted boundary conditions. In some cases, the derivation involves complications due to extra terms generated from the breaking of the cubic invariance. We isolate such terms and treat them as renormalization terms just as done before. In that way, we derive asymptotic formulae for masses, decay constants, pseudoscalar coupling constants and scalar form factors. At the same time, we derive also asymptotic formulae for renormalization terms. We apply all these formulae in combination with Chiral Perturbation Theory and estimate the corrections beyond next-to-leading order. We show that asymptotic formulae for masses, decay constants, pseudoscalar coupling constants are related by means of chiral Ward identities. A similar relation connects in an independent way asymptotic formulae for renormalization terms. We check these relations for charged pions through a direct calculation. To conclude, a numerical analysis quantifies the importance of finite volume corrections at next-to-leading order and beyond. We perform a generic Analysis and illustrate two possible applications to real simulations.

Lehrbuch der Arithmetik und der niederen Algebra : für Gymnasien, höhere Bürgerschulen, Kriegsschulen und zum Selbstunterrichte

von Fr. Adrian Köcher

Grundlehren der Algebra : Zum Gebrauch in Schulen und zum Selbstunterricht

Von J. Ineichen, Professor der Physik am Lyceum zu Luzern

An example of Learning based on competences: Use of Maxima in Linear Algebra for Engineers

This paper analyzes the role of Computer Algebra Systems (CAS) in a model of learning based on competences. The proposal is an e-learning model Linear Algebra course for Engineering, which includes the use of a CAS (Maxima) and focuses on problem solving. A reference model has been taken from the Spanish Open University. The proper use of CAS is defined as an indicator of the generic ompetence: Use of Technology. Additionally, we show that using CAS could help to enhance the following generic competences: Self Learning, Planning and Organization, Communication and Writing, Mathematical and Technical Writing, Information Management and Critical Thinking.

Differential algebra techniques for space applications

This project investigates the utility of differential algebra (DA) techniques applied to the problem of orbital dynamics with initial uncertainties in the orbital determination of the involved bodies. The use of DA theory allows the splitting of a common Monte Carlo simulation in two parts: the generation of a Taylor map of the final states with regard to the perturbation in the initial coordinates, and the evaluation of the map for many points. A propagator is implemented exploiting DA techniques, and tested in the field of asteroid impact risk monitoring with the potentially hazardous 2011 AG5 and 2007 VK184 as test cases. Results show that the new method is able to simulate 2.5 million trajectories with a precision good enough for the impact probability to be accurately reproduced, while running much faster than a traditional Monte Carlo approach (in 1 and 2 days, respectively).

Electrooptical behavior of twisted-wedge nematic structures

A new type of domain in twisted-wedge negative nematics is reported. The wedge shape allows the separation of ordinary and extraordinary beams that can be studied separately. In these conditions, at least five different regions can be detected, depending on applied voltages and frequencies. Both rays are shown to yield different diffraction patterns. The relative light intensity of several spots of these paterns is also studied.

Electrohydrodynamic behavior in twisted-wedge nematic structures

A new type of domain for nematic liquid crystals with a twisted-wedge structure is presented. This new type of domain appears from the low frequency range to 10 kHz. This behavior was observed for square and pulsed excitations. The liquid crystal was N-(p-methoxybenzylidene)-p'-butylaniline) (MBBA) used at room temperature. These domains offer a higher degree of complexity than conventional Williams domains. The corresponding stability chart is presented.

Optical memory with twisted nematic liquid crystal (TNLC) devices

A design for obtaining memory in optical bistability with liquid crystals is reported. This design uses optical feedback on a twisted nematie liquid crystal ( TNLC ) through an optoelectronic system. A constant input light is the read-out and its value depends on the desired initial working point, usually at the bottom of the T(V) vs. V curve. Light levels depend on the feedback. An input light pulse change the working point to the top of the transmission curve. When this pulse vanishes, the working point remains at the upper part of the curve. Hence a memory function is obtained. Minimum pulse width needed was 1msec. ON-OPF ratio was 100:3.

DERIVE and Linear Algebra

This work describes an experience with a methodology for learning based on competences in Linear Algebra for engineering students. The experience has been based in autonomous team work of students. DERIVE tutorials for Linear Algebra topics are provided to the students. They have to work with the tutorials as their homework. After, worksheets with exercises have been prepared to be solved by the students organized in teams, using DERIVE function previously defined in the tutorials. The students send to the instructor the solution of the proposed exercises and they fill a survey with their impressions about the following items: ease of use of the files, usefulness of the tutorials for understanding the mathematical topics and the time spent in the experience. As a final work, we have designed an activity directed to the interested students. They have to prepare a project, related with a real problem in Science and Engineering. The students are free to choose the topic and to develop it but they have to use DERIVE in the solution. Obviously they are guided by the instructor. Some examples of activities related with Orthogonal Transformations will be presented.

A toolbox with DERIVE: calculus on several variables, Applications of Computer Algebra

A toolbox is a set of procedures taking advantage of the computing power and graphical capacities of a CAS. With these procedures the students can solve math problems, apply mathematics to engineering or simply reinforce the learning of certain mathematical concepts. From the point of view of their construction, we can consider two types of toolboxes: (i) the closed box, built by the teacher, in which the utility files are provided to the students together with the respective tutorials and several worksheets with proposed exercises and problems,

Libro primero, de Arithmetica Algebratica, enel qual se contiene el arte Mercantiuol, con otras muchas Reglas del arte menor, y la Regla del Algebra, vulgarmente llamada Arte mayor, o Regla de la cosa ... [Texto impreso] ...]

Marca tip. en v. de port. y en v. de última h.

Electron Tomography of Metaphase Nucleolar Organizer Regions: Evidence for a Twisted-Loop Organization

Metaphase nucleolar organizer regions (NORs), one of four types of chromosome bands, are located on human acrocentric chromosomes. They contain r-chromatin, i.e., ribosomal genes complexed with proteins such as upstream binding factor and RNA polymerase I, which are argyrophilic NOR proteins. Immunocytochemical and cytochemical labelings of these proteins were used to reveal r-chromatin in situ and to investigate its spatial organization within NORs by confocal microscopy and by electron tomography. For each labeling, confocal microscopy revealed small and large double-spotted NORs and crescent-shaped NORs. Their internal three-dimensional (3D) organization was studied by using electron tomography on specifically silver-stained NORs. The 3D reconstructions allow us to conclude that the argyrophilic NOR proteins are grouped as a fiber of 60–80 nm in diameter that constitutes either one part of a turn or two or three turns of a helix within small and large double-spotted NORs, respectively. Within crescent-shaped NORs, virtual slices reveal that the fiber constitutes several longitudinally twisted loops, grouped as two helical 250- to 300-nm coils, each centered on a nonargyrophilic axis of condensed chromatin. We propose a model of the 3D organization of r-chromatin within elongated NORs, in which loops are twisted and bent to constitute one basic chromatid coil.

The operator algebra approach to quantum groups

A relatively simple definition of a locally compact quantum group in the C*-algebra setting will be explained as it was recently obtained by the authors. At the same time, we put this definition in the historical and mathematical context of locally compact groups, compact quantum groups, Kac algebras, multiplicative unitaries, and duality theory.