When are translations of P-positions of Wythoff's game P-positions?

We study the problem whether there exist variants of {\sc Wythoff}'s game whose $\P$-positions, except for a finite number, are obtained from those of {\sc Wythoff}'s game by adding a constant $k$ to each $\P$-position. We solve this question by introducing a class \{\W_k\}_{k \geq 0} of variants of {\sc Wythoff}'s game in which, for any fixed $k \geq 0$, the $\P$-positions of \W_k form the set $\{(i,i) | 0 \leq i < k\}\cup \{(\lfloor \phi n \rfloor + k, \lfloor \phi^2 n \rfloor + k) | n\ge 0\}$, where $\phi$ is the golden ratio. We then analyze a class \{\T_k\}_{k \geq 0} of variants of {\sc Wythoff}'s game whose members share the same $\P$-positions set $\{(0,0)\}\cup \{(\lfloor \phi n \rfloor + 1, \lfloor \phi^2 n \rfloor + 1) | n \geq 0 \}$. We establish several results for the Sprague-Grundy function of these two families. On the way we exhibit a family of games with different rule sets that share the same set of $\P$-positions

Inhomogeneity-Induced Casimir Transport of Nanoparticles

This letter proposes a scheme for transporting nanoparticles immersed in a fluid, relying on quantum vacuum fluctuations. The mechanism lies in the inhomogeneity-induced lateral Casimir force between a nanoparticle and a gradient metasurface, and the relaxation of the conventional Dzyaloshinski\v{i}-Lifshitz-Pitaevski\v{i} constraint, which allows quantum levitation for a broader class of material configurations. The velocity for a nanosphere levitated above a grating is calculated and can be up to a few microns per minute. The Born approximation gives general expressions for the Casimir energy which reveal size-selective transport. For any given metasurface, a certain particle-metasurface separation exists where the transport velocity peaks, forming a "Casimir passage". The sign and strength of the Casimir interactions can be tuned by the shapes of liquid-air menisci, potentially allowing real-time control of an otherwise passive force, and enabling interesting on-off or directional switching of the transport process.Comment: 7 figure

Symmetry Constraints and the Electronic Structures of a Quantum Dot with Thirteen Electrons

The symmetry constraints imposing on the quantum states of a dot with 13 electrons has been investigated. Based on this study, the favorable structures (FSs) of each state has been identified. Numerical calculations have been performed to inspect the role played by the FSs. It was found that, if a first-state has a remarkably competitive FS, this FS would be pursued and the state would be crystal-like and have a specific core-ring structure associated with the FS. The magic numbers are found to be closely related to the FSs.Comment: 13 pages, 5 figure

Distribution of localized states from fine analysis of electron spin resonance spectra of organic semiconductors: Physical meaning and methodology

We develop an analytical method for the processing of electron spin resonance (ESR) spectra. The goal is to obtain the distributions of trapped carriers over both their degree of localization and their binding energy in semiconductor crystals or films composed of regularly aligned organic molecules [Phys. Rev. Lett. v. 104, 056602 (2010)]. Our method has two steps. We first carry out a fine analysis of the shape of the ESR spectra due to the trapped carriers; this reveals the distribution of the trap density of the states over the degree of localization. This analysis is based on the reasonable assumption that the linewidth of the trapped carriers is predetermined by their degree of localization because of the hyperfine mechanism. We then transform the distribution over the degree of localization into a distribution over the binding energies. The transformation uses the relationships between the binding energies and the localization parameters of the trapped carriers. The particular relation for the system under study is obtained by the Holstein model for trapped polarons using a diagrammatic Monte Carlo analysis. We illustrate the application of the method to pentacene organic thin-film transistors.Comment: 14 pages, 11 figure

Antiferromagnetism in semiconducting KFe0.85Ag1.15Te2 single crystals

We have synthesized single crystals of K1.00(3)Fe0.85(2)Ag1.15(2)Te2.0(1). The materials crystallizes in the ThCr2Si2 structure with I4/mmm symmetry and without K and Fe/Ag deficiencies, unlike in KxFe2-ySe2 and KxFe2-yS2. In contrast to theoretical prediction for higher Tc in KFe2Te2, KFe0.85Ag1.15Te2 is a semiconductor with long-range antiferromagnetic transition at TN = 35 K.Comment: 4 pages, 4 figure