The Relativistic Levinson Theorem in Two Dimensions

In the light of the generalized Sturm-Liouville theorem, the Levinson theorem for the Dirac equation in two dimensions is established as a relation between the total number $n_{j}$ of the bound states and the sum of the phase shifts $\eta_{j}(\pm M)$ of the scattering states with the angular momentum $j$: $$\eta_{j}(M)+\eta_{j}(-M)~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~$$ $$~~~=\left\{\begin{array}{ll} (n_{j}+1)\pi &{\rm when~a~half~bound~state~occurs~at}~E=M ~~{\rm and}~~ j=3/2~{\rm or}~-1/2\\ (n_{j}+1)\pi &{\rm when~a~half~bound~state~occurs~at}~E=-M~~{\rm and}~~ j=1/2~{\rm or}~-3/2\\ n_{j}\pi~&{\rm the~rest~cases} . \end{array} \right.$$ \noindent The critical case, where the Dirac equation has a finite zero-momentum solution, is analyzed in detail. A zero-momentum solution is called a half bound state if its wave function is finite but does not decay fast enough at infinity to be square integrable.Comment: Latex 14 pages, no figure, submitted to Phys.Rev.A; Email: [email protected], [email protected]

Finite difference approximations for a size-structured population model with distributed states in the recruitment

In this paper we consider a size-structured population model where individuals may be recruited into the population at different sizes. First and second order finite difference schemes are developed to approximate the solution of the mathematical model. The convergence of the approximations to a unique weak solution with bounded total variation is proved. We then show that as the distribution of the new recruits become concentrated at the smallest size, the weak solution of the distributed states-at-birth model converges to the weak solution of the classical Gurtin-McCamy-type size-structured model in the weak$^*$ topology. Numerical simulations are provided to demonstrate the achievement of the desired accuracy of the two methods for smooth solutions as well as the superior performance of the second-order method in resolving solution-discontinuities. Finally we provide an example where supercritical Hopf-bifurcation occurs in the limiting single state-at-birth model and we apply the second-order numerical scheme to show that such bifurcation occurs in the distributed model as well

Isovector Giant Dipole Resonance of Stable Nuclei in a Consistent Relativistic Random Phase Approximation

A fully consistent relativistic random phase approximation is applied to study the systematic behavior of the isovector giant dipole resonance of nuclei along the $\beta$-stability line in order to test the effective Lagrangians recently developed. The centroid energies of response functions of the isovector giant dipole resonance for stable nuclei are compared with the corresponding experimental data and the good agreement is obtained. It is found that the effective Lagrangian with an appropriate nuclear symmetry energy, which can well describe the ground state properties of nuclei, could also reproduce the isovector giant dipole resonance of nuclei along the $\beta$-stability line.Comment: 4 pages, 1 Postscript figure, to be submitted to Chin.Phys.Let

A refined invariant subspace method and applications to evolution equations

The invariant subspace method is refined to present more unity and more diversity of exact solutions to evolution equations. The key idea is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces that evolution equations admit. A two-component nonlinear system of dissipative equations was analyzed to shed light on the resulting theory, and two concrete examples are given to find invariant subspaces associated with 2nd-order and 3rd-order linear ordinary differential equations and their corresponding exact solutions with generalized separated variables.Comment: 16 page

Petri net controllers for Generalized Mutual Exclusion Constraints with floor operators

In this paper a special type of nonlinear marking specifications called stair generalized mutual exclusion constraints (stair-GMECs) is defined. A stair-GMEC can be represented by an inequality whose left-hand is a linear combination of floor functions. Stair-GMECs have higher modeling power than classical GMECs and can model legal marking sets that cannot be defined by OR–AND GMECs. We propose two algorithms to enforce a stair-GMEC as a closed-loop net, in which the control structure is composed by a residue counter, remainder counters, and duplicate transitions. We also show that the proposed control structure is maximally permissive since it prevents all and only the illegal trajectories of a plant net. This approach can be applied to both bounded and unbounded nets. Several examples are proposed to illustrate the approach

Tip-Enhanced Fluorescence Microscopy at 10 Nanometer Resolution

We demonstrate unambiguously that the field enhancement near the apex of a laser-illuminated silicon tip decays according to a power law that is moderated by a single parameter characterizing the tip sharpness. Oscillating the probe in intermittent contact with a semiconductor nanocrystal strongly modulates the fluorescence excitation rate, providing robust optical contrast and enabling excellent background rejection. Laterally encoded demodulation yields images with <10 nm spatial resolution, consistent with independent measurements of tip sharpness

Big Data and the Internet of Things

Advances in sensing and computing capabilities are making it possible to embed increasing computing power in small devices. This has enabled the sensing devices not just to passively capture data at very high resolution but also to take sophisticated actions in response. Combined with advances in communication, this is resulting in an ecosystem of highly interconnected devices referred to as the Internet of Things - IoT. In conjunction, the advances in machine learning have allowed building models on this ever increasing amounts of data. Consequently, devices all the way from heavy assets such as aircraft engines to wearables such as health monitors can all now not only generate massive amounts of data but can draw back on aggregate analytics to "improve" their performance over time. Big data analytics has been identified as a key enabler for the IoT. In this chapter, we discuss various avenues of the IoT where big data analytics either is already making a significant impact or is on the cusp of doing so. We also discuss social implications and areas of concern.Comment: 33 pages. draft of upcoming book chapter in Japkowicz and Stefanowski (eds.) Big Data Analysis: New algorithms for a new society, Springer Series on Studies in Big Data, to appea

Levinson's theorem for the Schr\"{o}dinger equation in two dimensions

Levinson's theorem for the Schr\"{o}dinger equation with a cylindrically symmetric potential in two dimensions is re-established by the Sturm-Liouville theorem. The critical case, where the Schr\"{o}dinger equation has a finite zero-energy solution, is analyzed in detail. It is shown that, in comparison with Levinson's theorem in non-critical case, the half bound state for $P$ wave, in which the wave function for the zero-energy solution does not decay fast enough at infinity to be square integrable, will cause the phase shift of $P$ wave at zero energy to increase an additional $\pi$.Comment: Latex 11 pages, no figure and accepted by P.R.A (in August); Email: [email protected], [email protected]

Investigation of electrical properties for cantilever-based piezoelectric energy harvester

In the present era, the renewable sources of energy, e.g., piezoelectric materials are in great demand. They play a vital role in the field of micro-electromechanical systems, e.g., sensors and actuators. The cantilever-based piezoelectric energy harvesters are very popular because of their high performance and utilization. In this research-work, an energy harvester model based on a cantilever beam with bimorph PZT-5A, having a substrate layer of structural steel, was presented. The proposed energy scavenging system, designed in COMSOL Multiphysics, was applied to analyze the electrical output as a function of excitation frequencies, load resistances and accelerations. Analytical modeling was employed to measure the output voltage and power under pre-defined conditions of acceleration and load resistance. Experimentation was also performed to determine the relationship between independent and output parameters. Energy harvester is capable of producing the maximum power of 1.16 mW at a resonant frequency of 71 Hz under 1g acceleration, having load resistance of 12 k Omega. It was observed that acceleration and output power are directly proportional to each other. Moreover, the investigation conveys that the experimental results are in good agreement with the numerical results. The maximum error obtained between the experimental and numerical investigation was found to equal 4.3%