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

Deduction of the quantum numbers of low-lying states of 6-nucleon systems based on symmetry

The inherent nodal structures of the wavefunctions of 6-nucleon systems have been investigated. The existence of a group of six low-lying states dominated by L=0 has been deduced. The spatial symmetries of these six states are found to be mainly {4,2} and {2,2,2}.Comment: 8 pages, no figure

Laboratory studies of photodissociation processes relevant to the formation of cometary radicals

The strength of the C2(d 3 Pi g yields a 3 Pi u) Swan band emission in the spectra of cometary comae identifies this species as a prominent constituent of the coma gas. It was previously suggested that the formation of cometary C2 proceeds via the secondary photolysis of the C2H radical. The detection of C2H in the interstellar medium and the recent analysis of the radial variation in C2(delta V=O) surface brightness of Comet Halley support the postulate that C2 is a third-generation molecule. Measurement of the C2 and C2H translational energy distributions produced from the multiphoton dissociation (MPD) of acetylene at 193 nm are identified . Time-resolved FTIR emission studies of the nascent C2H radical formed in the C2H2 yields C2H + H reaction verify that this species is produced both vibrationally and electronically excited. A survey of the internal energy distributions of the C2 fragments produced from the MPD of acetylene using a high intensity ArF laser is currently in progress in the laboratory. Recent experiments have focused on the measurement of rotational energy distribution for the C2(A 1 Pi u, a 3 Pi u) fragments. The C2(a 3 Pi u) detection capability is currently being improved by performing this experiment in a molecular beam, thus allowing for discrimination between initial emission and laser-induced fluorescence (LIF). Although the experiments performed to date provide considerable evidence in support of C2H yields C2 + H reaction, there is an important distinction to be made when comparing the laboratory conditions to those typically found in comets. The C2H radicals generated in the laboratory experiments are formed vibrationally and/or electronically excited. Any rotationally/vibrationally excited C2H present in cometary comae will quickly undergo radiative relaxation in the infrared to their lowest rotational and vibrational state. Experiments are currently under way to confirm the cometary formation of C2 via the VUV dissociation of cold C2H