Approximations of the Sum of States by Laplace's Method for a System of Particles with a Finite Number of Energy Levels and Application to Limit Theorems

We consider a generic system composed of a fixed number of particles distributed over a finite number of energy levels. We make only general assumptions about system's properties and the entropy. System's constraints other than fixed number of particles can be included by appropriate reduction of system's state space. For the entropy we consider three generic cases. It can have a maximum in the interior of system's state space or on the boundary. On the boundary we can have another two cases. There the entropy can increase linearly with increase of the number of particles and in the another case grows slower than linearly. The main results are approximations of system's sum of states using Laplace's method. Estimates of the error terms are also included. As an application, we prove the law of large numbers which yields the most probable state of the system. This state is the one with the maximal entropy. We also find limiting laws for the fluctuations. These laws are different for the considered cases of the entropy. They can be mixtures of Normal, Exponential and Discrete distributions. Explicit rates of convergence are provided for all the theorems.Comment: 23 page

Beyond visual P300 based brain-computer interfacing paradigms

The paper reviews and summarizes recent developments in spatial auditory and tactile brain-computer interfacing neurotechology applications. It serves as the latest developments summary in "non-visual" brain-computer interfacing solutions presented in a tutorial delivered by the author at the IICST 2013 workshop. The novel concepts of unimodal auditory or tactile, as well as a bimodal combined paradigms are described and supported with recent research results from our BCI-lab research group at Life Science Center, University of Tsukuba, Japan. The newly developed experimental paradigms fit perfectly to needs of paralyzed or hearing impaired users, in case of tactile stimulus, as well as for able users who cannot utilize vision in computer or machine interaction (driving or operation of machinery required not disturbed eyesight). We present and review the EEG event related potential responses useful for brain computer interfacing applications beyond state-of-the-art visual paradigms. In conclusion the recent results are discussed and suggestions for further applications are drawn.Comment: 7 pages, 5 figures, Proceedings of the Third Postgraduate Consortium International Workshop on Innovations in Information and Communication Science and Technology, (E. Cooper, G. A. Kobzev, A. F. Uvarov, and V. V. Kryssanov, eds.), (Tomsk, Russia), pp. 277-283, TUSUR and Ritsumeikan, September 2-5, 2013. ISBN 978-5-86889-

Transformation of bound states of relativistic hydrogen-l ike atom into two-component form

A single-step Eriksen transformation of~$1S_{1/2}$,~$2P_{1/2}$ and~$2P_{3/2}$ states of the relativistic hydrogen-like atom is performed exactly by expressing each transformed function (TF) as a linear combination of eigenstates of the Dirac Hamiltonian. The transformed functions, which are four-component spinors with vanishing two lower components, are calculated numerically and have the same symmetries as the initial states. For all nuclear charges~$Z \in [1\ldots 92]$ a contribution of the initial state to TFs exceeds 86\% of the total probability density. Next large contribution to TFs comes from continuum states with negative energies close to~$-m_0c^2-E_b$, where~$E_b$ is the binding energy of initial state. Contribution of other states to TFs is less than~$0.1\%$ of the total probability density. Other components of TFs are nearly zero which confirms both validity of the Eriksen transformation and accuracy of the numerical calculations. The TFs of~$1S_{1/2}$ and~$2P_{1/2}$ states are close to~$1s$ and~$2p$ states of the nonrelativistic hydrogen-like atom, respectively, but the TF of~$2P_{3/2}$ state differs qualitatively from the~$2p$ state. Functions calculated with use of a linearized Eriksen transformation, being equivalent to the second order Foldy-Wouthuysen transformation, are compared with corresponding functions obtained by Eriksen transformation. A very good agreement between both results is obtained.Comment: 13 pages, 6 figure

Law of large numbers unifying Maxwell-Boltzmann, Bose-Einstein and Zipf-Mandelbort distributions, and related fluctuations

We consider a system composed of a fixed number of particles with total energy smaller or equal to some prescribed value. The particles are non-interacting, indistinguishable and distributed over fixed number of energy levels. The energy levels are degenerate and degeneracy is a function of the number of particles. Three cases of the degeneracy function is considered. It can increase with either the same rate as the number of particles or slower, or faster. We find useful properties of the entropy of the system and solve the entropy optimization problem. It turned out, there are several solutions. Depending on the magnitude of the total energy, the maximum of the entropy can be in the interior of system's state space or on the boundary. On the boundary it can have further three cases depending on the degeneracy function. The main result, Law of Large Numbers yields the most probable state of the system, which equals to the point of maximum of the entropy. This point can be either Maxwell-Boltzmann statistics or Bose-Einstein statistics, or Zipf-Mandelbort law. We also find the limiting laws for the fluctuations. These laws are different for different cases of the entropy's maximum. They can be mixture of Normal, Exponential and Discrete distributions. Explicit rate of convergence is provided for all the theorems.Comment: 18 page

Student Teaching and Research Laboratory Focusing on Brain-computer Interface Paradigms - A Creative Environment for Computer Science Students -

This paper presents an applied concept of a brain-computer interface (BCI) student research laboratory (BCI-LAB) at the Life Science Center of TARA, University of Tsukuba, Japan. Several successful case studies of the student projects are reviewed together with the BCI Research Award 2014 winner case. The BCI-LAB design and project-based teaching philosophy is also explained. Future teaching and research directions summarize the review.Comment: 4 pages, 4 figures, accepted for EMBC 2015, IEEE copyrigh

Pullbacks and nontriviality of associated noncommutative vector bundles

Our main theorem is that the pullback of an associated noncommutative vector bundle induced by an equivariant map of quantum principal bundles is a noncommutative vector bundle associated via the same finite-dimensional representation of the structural quantum group. On the level of $K_{0}$-groups, we realize the induced map by the pullback of explicit matrix idempotents. We also show how to extend our result to the case when the quantum-group representation is infinite dimensional, and then apply it to the Ehresmann-Schauenburg quantum groupoid. Finally, using noncommutative Milnor's join construction, we define quantum quaternionic projective spaces together with noncommutative tautological quaternionic line bundles and their duals. As a key application of the main theorem, we show that these bundles are stably non-trivial as noncommutative complex vector bundles.Comment: 18 pages, to appear in Journal of Noncommutative Geometr

Coalgebra Extensions and Algebra Coextensions of Galois Type

The notion of a coalgebra-Galois extension is defined as a natural generalisation of a Hopf-Galois extension. It is shown that any coalgebra-Galois extension induces a unique entwining map $\psi$ compatible with the right coaction. For the dual notion of an algebra-Galois coextension it is also proven that there always exists a unique entwining structure compatible with the right action.Comment: 21 pages, LaTeX, uses amssymb. Major revision. Version to appear in Commun. Algebr

Green functions of electrons in monolayer and bilayer graphene in a magnetic field

Closed expressions for the Green functions of the stationary two-dimensional two-component Schrodinger equation for an electron moving in monolayer and bilayer graphene in the presence of a magnetic field are obtained in terms of the Whittaker functions.Comment: 7 pages, 2 figures, 2 exact results, published verio

Two-step Input Spatial Auditory BCI for Japanese Kana Characters

We present an auditory stimulus optimization and a pilot study of a two-step input speller application combined with a spatial auditory brain-computer interface (saBCI) for paralyzed users. The application has been developed for 45, out of 48 defining the full set, Japanese kana characters in a two-step input procedure setting for an easy-to-use BCI-speller interface. The user first selects the representative letter of a subset, defining the second step. In the second step, the final choice is made. At each interfacing step, the choices are classified based on the P300 event related potential (ERP) responses captured in the EEG, as in the classic oddball paradigm. The BCI online experiment and EEG responses classification results of the pilot study confirm the effectiveness of the proposed spelling method.Comment: 7 pages, 2 figures, accepted for publication in Advances in Cognitive Neurodynamics Volume 5 -- Proceedings of the 5th International Conference on Cognitive Neurodynamics (ICCN 2015

Multi-mode behavior of electron Zitterbewegung induced by an electromagnetic wave in graphene

Electrons in monolayer graphene in the presence of an electromagnetic (or electric) wave are considered theoretically. It is shown that the electron motion is a nonlinear combination of Zitterbewegung (ZB, trembling motion) resulting from the periodic potential of graphene lattice and the driving field of the wave. This complex motion is called "Multi-mode Zitterbewegung". The theory is based on the time-dependent two-band Hamiltonian taking into account the graphene band structure and interaction with the wave. Our theoretical treatment includes the rotating wave approximation and high-driving-frequency approximation for narrow wave packets, as well as numerical calculations for packets of arbitrary widths. Different regimes of electron motion are found, depending on relation between the ZB frequency $\omega_Z$ and the driving frequency $\omega_D$ for different strengths of the electron-wave interaction. Frequencies and intensities of the resulting oscillation modes are calculated. The nonlinearity of the problem results in a pronounced multi-mode behavior. Polarization of the medium is also calculated relating our theoretical results to observable quantities. The presence of driving wave, resulting in frequencies directly related to $\omega_Z$ and increasing the decay time of oscillations, should facilitate observations of the Zitterbewegung phenomenon.Comment: 15 pages, 17 figure