Rendezvous of Two Robots with Constant Memory

We study the impact that persistent memory has on the classical rendezvous problem of two mobile computational entities, called robots, in the plane. It is well known that, without additional assumptions, rendezvous is impossible if the entities are oblivious (i.e., have no persistent memory) even if the system is semi-synchronous (SSynch). It has been recently shown that rendezvous is possible even if the system is asynchronous (ASynch) if each robot is endowed with O(1) bits of persistent memory, can transmit O(1) bits in each cycle, and can remember (i.e., can persistently store) the last received transmission. This setting is overly powerful. In this paper we weaken that setting in two different ways: (1) by maintaining the O(1) bits of persistent memory but removing the communication capabilities; and (2) by maintaining the O(1) transmission capability and the ability to remember the last received transmission, but removing the ability of an agent to remember its previous activities. We call the former setting finite-state (FState) and the latter finite-communication (FComm). Note that, even though its use is very different, in both settings, the amount of persistent memory of a robot is constant. We investigate the rendezvous problem in these two weaker settings. We model both settings as a system of robots endowed with visible lights: in FState, a robot can only see its own light, while in FComm a robot can only see the other robot's light. We prove, among other things, that finite-state robots can rendezvous in SSynch, and that finite-communication robots are able to rendezvous even in ASynch. All proofs are constructive: in each setting, we present a protocol that allows the two robots to rendezvous in finite time.Comment: 18 pages, 3 figure

Moebius Structure of the Spectral Space of Schroedinger Operators with Point Interaction

The Schroedinger operator with point interaction in one dimension has a U(2) family of self-adjoint extensions. We study the spectrum of the operator and show that (i) the spectrum is uniquely determined by the eigenvalues of the matrix U belonging to U(2) that characterizes the extension, and that (ii) the space of distinct spectra is given by the orbifold T^2/Z_2 which is a Moebius strip with boundary. We employ a parametrization of U(2) that admits a direct physical interpretation and furnishes a coherent framework to realize the spectral duality and anholonomy recently found. This allows us to find that (iii) physically distinct point interactions form a three-parameter quotient space of the U(2) family.Comment: 16 pages, 2 figure

Mathematical analysis of a model of river channel formation.

The study of overland flow of water over an erodible sediment leads to a coupled model describing the evolution of the topographic elevation and the depth of the overland water film. The spatially uniform solution of this model is unstable, and this instability corresponds to the formation of rills, which in reality then grow and coalesce to form large-scale river channels. In this paper we consider the deduction and mathematical analysis of a deterministic model describing river channel formation and the evolution of its depth. The model involves a degenerate nonlinear parabolic equation (satisfied on the interior of the support of the solution) with a super-linear source term and a prescribed constant mass. We propose here a global formulation of the problem (formulated in the whole space, beyond the support of the solution) which allows us to show the existence of a solution and leads to a suitable numerical scheme for its approximation. A particular novelty of the model is that the evolving channel self-determines its own width, without the need to pose any extra conditions at the channel margin

Spherically symmetric analysis on open FLRW solution in non-linear massive gravity

We study non-linear massive gravity in the spherically symmetric context. Our main motivation is to investigate the effect of helicity-0 mode which remains elusive after analysis of cosmological perturbation around an open Friedmann-Lemaitre-Robertson-Walker (FLRW) universe. The non-linear form of the effective energy-momentum tensor stemming from the mass term is derived for the spherically symmetric case. Only in the special case where the area of the two sphere is not deviated away from the FLRW universe, the effective energy momentum tensor becomes completely the same as that of cosmological constant. This opens a window for discriminating the non-linear massive gravity from general relativity (GR). Indeed, by further solving these spherically symmetric gravitational equations of motion in vacuum to the linear order, we obtain a solution which has an arbitrary time-dependent parameter. In GR, this parameter is a constant and corresponds to the mass of a star. Our result means that Birkhoff's theorem no longer holds in the non-linear massive gravity and suggests that energy can probably be emitted superluminously (with infinite speed) on the self-accelerating background by the helicity-0 mode, which could be a potential plague of this theory.Comment: 16 pages, references added; minor modification to match version published in JCA

Exploration of finite dimensional Kac algebras and lattices of intermediate subfactors of irreducible inclusions

We study the four infinite families KA(n), KB(n), KD(n), KQ(n) of finite dimensional Hopf (in fact Kac) algebras constructed respectively by A. Masuoka and L. Vainerman: isomorphisms, automorphism groups, self-duality, lattices of coideal subalgebras. We reduce the study to KD(n) by proving that the others are isomorphic to KD(n), its dual, or an index 2 subalgebra of KD(2n). We derive many examples of lattices of intermediate subfactors of the inclusions of depth 2 associated to those Kac algebras, as well as the corresponding principal graphs, which is the original motivation. Along the way, we extend some general results on the Galois correspondence for depth 2 inclusions, and develop some tools and algorithms for the study of twisted group algebras and their lattices of coideal subalgebras. This research was driven by heavy computer exploration, whose tools and methodology we further describe.Comment: v1: 84 pages, 13 figures, submitted. v2: 94 pages, 15 figures, added connections with Masuoka's families KA and KB, description of K3 in KD(n), lattices for KD(8) and KD(15). v3: 93 pages, 15 figures, proven lattice for KD(6), misc improvements, accepted for publication in Journal of Algebra and Its Application

Helium Nova on a Very Massive White Dwarf -- A Light Curve Model of V445 Puppis (2000) Revised

V445 Pup (2000) is a unique object identified as a helium nova. Color indexes during the outburst are consistent with those of free-free emission. We present a free-free emission dominated light curve model of V445 Pup on the basis of the optically thick wind theory. Our light curve fitting shows that (1) the white dwarf (WD) mass is very massive (M_WD \gtrsim 1.35 M_\sun), and (2) a half of the accreted matter remains on the WD, both of which suggest that the increasing WD mass. Therefore, V445 Pup is a strong candidate of Type Ia supernova progenitor. The estimated distance to V445 Pup is now consistent with the recent observational suggestions, 3.5 < d < 6.5 kpc. A helium star companion is consistent with the brightness of m_v=14.5 mag just before the outburst, if it is a little bit evolved hot (\log T (K) \gtrsim 4.5) star with the mass of M_He \gtrsim 0.8 M_\sun. We then emphasize importance of observations in the near future quiescent phase after the thick circumstellar dust dissipates away, especially its color and magnitude to specify the nature of the companion star. We have also calculated helium ignition masses for helium shell flashes against various helium accretion rates and discussed the recurrence period of helium novae.Comment: 8 pages including 12 figures, to appear in Ap

A new clue to the transition mechanism between optical high and low states of the supersoft X-ray source RX J0513.9-6951, implied from the recurrent nova CI Aquilae 2000 outburst model

We have found a new clue to the transition mechanism between optical high/X-ray off and optical low/X-ray on states of the LMC supersoft X-ray source RX J0513.9-6951. A sharp ~1 mag drop is common to the CI Aql 2000 outburst. These drops are naturally attributed to cessation of optically thick winds on white dwarfs. A detailed light-curve analysis of CI Aql indicates that the size of a disk drastically shrinks when the wind stops. This causes ~1-2 mag drop in the optical light curve. In RX J0513.9-6951, the same mechanism reproduces sharp ~1 mag drop from optical high to low states. We predict this mechanism also works on the transition from low to high states. Interaction between the wind and the companion star attenuates the mass transfer and drives full cycles of low and high states.Comment: 9 pages including 5 figures, to appear in the Astrophysical Journa

Entanglement Measures for Intermediate Separability of Quantum States

We present a family of entanglement measures R_m which act as indicators for separability of n-qubit quantum states into m subsystems for arbitrary 2 \leq m \leq n. The measure R_m vanishes if the state is separable into m subsystems, and for m = n it gives the Meyer-Wallach measure while for m = 2 it reduces, in effect, to the one introduced recently by Love et al. The measures R_m are evaluated explicitly for the GHZ state and the W state (and its modifications, the W_k states) to show that these globally entangled states exhibit rather distinct behaviors under the measures, indicating the utility of the measures R_m for characterizing globally entangled states as well.Comment: 8 pages, 8 figure

On the disappearance of a cold molecular torus around the low-luminosity active galactic nucleus of NGC 1097

We used the Atacama Large Millimeter/submillimeter Array (ALMA) to map the CO(3-2) and the underlying continuum emissions around the type 1 low-luminosity active galactic nucleus (LLAGN; bolometric luminosity $\lesssim 10^{42}$ erg~s$^{-1}$) of NGC 1097 at $\sim 10$ pc resolution. These observations revealed a detailed cold gas distribution within a $\sim 100$ pc of this LLAGN. In contrast to the luminous Seyfert galaxy NGC 1068, where a $\sim 7$ pc cold molecular torus was recently revealed, a distinctively dense and compact torus is missing in our CO(3-2) integrated intensity map of NGC 1097. Based on the CO(3-2) flux, the gas mass of the torus of NGC 1097 would be a factor of $\gtrsim 2-3$ less than that found for NGC 1068 by using the same CO-to-H$_2$ conversion factor, which implies less active nuclear star formation and/or inflows in NGC 1097. Our dynamical modeling of the CO(3-2) velocity field implies that the cold molecular gas is concentrated in a thin layer as compared to the hot gas traced by the 2.12 $\mu$m H$_2$ emission in and around the torus. Furthermore, we suggest that NGC 1097 hosts a geometrically thinner torus than NGC 1068. Although the physical origin of the torus thickness remains unclear, our observations support a theoretical prediction that geometrically thick tori with high opacity will become deficient as AGNs evolve from luminous Seyferts to LLAGNs.Comment: 9 pages, 5 figures. Accepted for publication in ApJ

Structural and superconducting properties of MgB2−x_{2-x}Bex_x

We prepared MgB$_{2-x}$Be$_{x}$ ($x=0$, 0.2, 0.3, 0.4, and 0.6) samples where B is substituted with Be. MgB$_{2}$ structure is maintained up to $x=0.6$. In-plane and inter-plane lattice constants were found to decrease and increase, respectively. Superconducting transition temperature $T_{c}$ decreases with $x$. We found that the $T_{c}$ decrease is correlated with in-plane contraction but is insensitive to carrier doping, which is consistent with other substitution studies such as Mg$_{1-x}$Al$_{x}$B$_{2}$ and MgB$_{2-x}$C$_{x}$. Implication of this work is discussed in terms of the 2D nature of $\sigma$-band.Comment: 3 pages,4 figures, to be published in Phys. Rev.