Strategy and Complexity of The Game of Squares

We introduce a game called Squares where the single player ispresented with a pattern of black and white squares and hasto reduce the pattern to white by making as few moves as possible.We present a method for solving the game, and show that thefollowing problem is NP-complete. PROBLEM 1 (Squares-Solvability). Given a pattern X and kN, canX be solved in k or less moves? We demonstrate a reduction to this problem from Not-All-Equal-3SAT.We also present another NP-complete problem that Squares-Solvabilitycan be reduced to. 1991 Mathematics Subject Classification 68Q25.     

A new line-search step based on the Weierstrass \wp-function for minimizing a class of logarithmic barrier functions

Summary. We define the notion of self-concordance of order two for the restriction of a logarithmic barrier function to a given line. Based on this notion we prove an inner approximation of the domain of , as well as a lower bound of the distance from a point $t$ to the minimum of . These results provide the theoretical tools to develop a simple and efficient search step for finding the minimum of the barrier function along a given line. The new bound on the size of the line-search step is better than the optimal bound known for the case of a self-concordant function (of order one). We conclude with some numerical examples that illustrate the promise of the new line-search step. Received May 24, 1993 / Revised version received February 1994     

The single-comparator method in thermal neutron activation analysis extended for some (n,p) reactions

From experimental studies of the rate of the reactions⁴⁷Ti(n, p)⁴⁷Sc,⁴⁸Ti(n, p)⁴⁸Sc and⁵⁸Ni(n, p)⁵⁸Co in four nuclear reactors, it is concluded that for the irradiation positions of the light water moderated reactors BR2 (Mol, Belgium) and HFR (Petten, Netherlands) a simple empirical relation exists between the fast neutron flux on the one hand and the thermal and epithermal neutron flux on the other. The graphite moderated reactor BR1 (Mol, Belgium) and the heavy water reactor FRJ2 (Jülich, Federal Republic of Germany), however, have relatively much lower fast fluxes and their irradiation facilities do not obey the empirical relation determined.     

Ag2PbO2, Optimierung der Kristallzüchtung und Verfeinerung der Kristallstruktur

Ag₂PbO₂, Optimization of Crystal Growth and Crystal Structure Refinement Large single crystals of Ag₂PbO₂ (1×1×7 mm) suitable for measurements of anisotropic physical properties have been grown. The refinement of the crystal structure (I2/c; crystal size 0.08 × 0.08 × 0.31 mm; a = 6.0775(9), b = 8.704(1), c = 6.553(1) Å, β = 93.545(8)°; Z = 4; 1235 diffractometer data; R = 0.038) confirms the positions of the heavy atoms, and allows for an unambiguous determination of the oxygen coordinates.     

Comment on “The electron‐pair origin of anti‐aromaticity: Spectroscopic manifestations”

In the article by Zilberg and Haas, “The Electron‐Pair Origin of Anti‐aromaticity: Spectroscopic Manifestations,” the relative sign of the two Kekulé valence bond functions, R and L, in conjugated cyclic hydrocarbons was discussed. It was proposed that in the ground‐state wave function of aromatic compounds, the two functions contribute with like sign, while in the ground state of anti‐aromatic compounds, the two functions contribute with opposite sign. In this Comment, it is shown that the two functions enter with like sign also into the ground‐state wave function of anti‐aromatic compounds. Furthermore, it was argued that resonance tends to (de)stabilize a symmetric ground‐state geometry in case of the (anti‐)aromatic compounds. The expression derived by Zilberg and Haas for the stabilization energy shows an unusual dependence on the ring size and distortion coordinate. An alternative formula is derived for the stabilization energy, in which the energy depends quadratically on the distortion coordinate. Without further numerical calculations, it is not possible to predict whether this term will (de)stabilize a symmetric geometry of the ground state of (anti‐)aromatic molecules. Rather, we are led to believe that the influence of term in question on the geometric stability may be small, thus not providing the main reason for the geometric distortion of anti‐aromatic compounds. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007