Abstract:
In this study, a novel procedure is presented for control and analysis of a group of autonomous agents with point mass dynamics achieving flocking motion by using a fuzzy-logic-based attractive/repulsive function. Two cooperative control laws are proposed for a group of autonomous agents to achieve flocking formations related to two different centers (mass center and geometric center) of the flock. The first one is designed for flocking motion guided at mass center and the other for geometric center. A virtual agent is introduced to represent a group objective for tracking purposes. Smooth graph Laplacian is introduced to overcome the difficulties in theoretical analysis. A new fuzzy-logic-based attractive/repulsive function is proposed for separation and cohesion control among agents. The theoretical results are presented to indicate the stability (separation, collision avoidance and velocity matching) of the control systems. Finally, simulation example is demonstrated to validate the theoretical results.

Let C be a plane convex body. For arbitrary
points , a,b ∈E^{ n}denote by │ab│ the Euclidean length of the line-segment ab. Let a_{1}b_{1} be a longest chord of C parallel to the
line-segment ab. The relative distance d_{c}(a,b) between the points a and b is the ratio of the Euclidean distance between a and b to the half of the Euclidean distance between a_{1} and b_{1}. In this note we prove the
triangle inequality in E^{2} with the relative metric d_{c( }^{.}_{,}^{.}_{)}, and apply this inequality to
show that 6≤l(P)≤8, where l(P) is the perimeter of the convex polygon P measured in the metric d_{p}

Abstract:
A new scheme for overcoming losses with incoherent optical gain in a quantum-coherent left-handed atomic vapor is suggested. In order to obtain low-loss, lossless or active left-handed media (LHM), a pump field, which aims at realizing population inversion of atomic levels, is introduced into a four-level atomic system. Both analytical and numerical results are given to illustrate that such an atomic vapor can exhibit intriguing electric and magnetic responses required for achieving simultaneously negative permittivity and permeability (and hence a gain-assisted quantum-coherent negative refractive index would emerge). The quantum-coherent left-handed atomic vapor presented here could have four fascinating characteristics: i) three-dimensionally isotropic negative refractive index, ii) doublenegative atomic medium at visible and infrared wavelengths, iii) high-gain optical amplification, and iv) tunable negative refractive index based on quantum coherent control. Such a three-dimensionally isotropic gain medium with negative refractive index at visible and infrared frequencies would have a potential application in design of new quantum optical and photonic devices, including superlenses for perfect imaging and subwavelength focusing.

Abstract:
A physically interesting {\it effective rest mass} of photons in electromagnetic media, which is independent of wave frequency $\omega$, is defined in the present paper. It is verified that this frequency-independent effective rest mass of photons can be easily read off from the optical refractive index squared $n^{2}(\omega) $ of commonly-seen electromagnetic media. As an illustrative example, we extract the frequency-independent effective rest mass of photons from $n^{2}(\omega) $ in the {\it two time derivative Lorentz material} (2TDLM) model. The connection between effective rest mass and electromagnetic parameters of electric permittivity ($\epsilon $) and magnetic permeability ($\mu $) in left-handed media is also briefly discussed.

Abstract:
The connection between the quantum-vacuum geometric phases (which originates from the vacuum zero-point electromagnetic fluctuation) and the non-normal product procedure is considered in the present Letter. In order to investigate this physically interesting geometric phases at quantum-vacuum level, we suggest an experimentally feasible scheme to test it by means of a noncoplanarly curved fiber made of gyrotropic media. A remarkable feature of the present experimental realization is that one can easily extract the nonvanishing and nontrivial quantum-vacuum geometric phases of left- and/or right- handed circularly polarized light from the vanishing and trivial total quantum-vacuum geometric phases.

Abstract:
In this Letter the expression for the refractive index of de Broglie wave in the presence of a potential field is obtained and based on this, the physical meanings of negative index of refraction is revealed. We demonstrate that the electromagnetic wave propagation in a left-handed medium with negative refractive index behaves just like that of antiphotons, which is required of the complex vector field theory. It is believed that the complex vector field theory is helpful in considering the wave propagation and photonic band gap structure in the left-handed medium photonic crystals with a periodicity in negative and positive index of refraction.

Abstract:
Three related topics on the quantum-vacuum geometric phases in a noncoplanarly curved optical fiber is presented: (i) a brief review: the investigation of vacuum effect and its experimental realization; (ii) the sequence of ideas of geometric phases in the fiber; (iii) three derivations of effective Hamiltonian that describes the wave propagation of photon field in a curved fiber.

Abstract:
This paper demonstrates that there is much similarity in the mathematical formalisms between the optical constants of artificial electromagnetic media (such as chiral media, left-handed media, photonic crystals and EIT media) and some physical phenomena in field theory, including general relativity, quantum mechanics, energy band theory, etc. The significance of such comparisons lies in that: (i) the unification in mathematical descriptions shows that many physical phenomena and effects, which seem to have no connections between them, actually share almost the same mathematical structures; (ii) it can provide clue to us on suggesting more new effects which is similar in mathematical descriptions to the familiar phenomena in other areas.

Abstract:
The propagation of monomode photons inside a coiled optical fibre was regarded as a time-dependent quantum evolution process, which gives rise to a geometric phase. It is well known that the investigation of non-adiabatic geometric phases ought to be performed only in the Schr\"{o}dinger picture. So, the projections of photon spin operators onto the fixed frame of reference is discussed in this paper. In addition, we also treat the non-normal-order spin operators and consider the potential effects (e.g., quantum-vacuum geometric phases) of quantum fluctuation fields arising in a curved optical fibre. The quantum-vacuum geometric phase, which is of physical interest, can be deducted by using the operator normal product, and the doubt of validity and universality for the normal-normal procedure applied to time-dependent quantum systems is thus proposed. In the Appendix, the discussion of possible experimental realizations of quantum-vacuum geometric phases is briefly presented.

Abstract:
The connections between the anti-shielding effect, negative absolute temperature and superluminal light propagation in both the instantaneously reversed electric field and the left-handed media are considered in the present paper. The instantaneous inversion of the exterior electric field may cause the electric dipoles into the state of negative absolute temperature and therefore give rise to a negative effective mass term of electromagnetic field (i. e., the electromagnetic field propagating inside the negative-temperature medium will acquire an imaginary rest mass), which is said to result in the potential superluminality effect of light propagation in this anti-shielding dielectric. In left-handed media, such phenomena may also arise.