A threshold for the Maker‐Breaker clique game

We study the Maker‐Breaker k‐clique game played on the edge set of the random graph G(n, p). In this game, two players, Maker and Breaker, alternately claim unclaimed edges of G(n, p), until all the edges are claimed. Maker wins if he claims all the edges of a k‐clique; Breaker wins otherwise. We determine that the threshold for the graph property that Maker can win this game is at , for all k > 3, thus proving a conjecture from Ref. [Stojakovi? and Szabó, Random Struct Algor 26 (2005), 204–223]. More precisely, we conclude that there exist constants such that when the game is Maker's win a.a.s., and when it is Breaker's win a.a.s. For the triangle game, when k = 3, we give a more precise result, describing the hitting time of Maker's win in the random graph process. We show that, with high probability, Maker can win the triangle game exactly at the time when a copy of K₅ with one edge removed appears in the random graph process. As a consequence, we are able to give an expression for the limiting probability of Maker's win in the triangle game played on the edge set of G(n, p). © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 318–341, 2014     

Nonperiodic Long-Range Order for Fast-Decaying Interactions at Positive Temperatures

We present the first example of an exponentially decaying interaction which gives rise to nonperiodic long-range order at positive temperatures.     

Study of Mn laser ablation in methane atmosphere

Laser ablation of Mn target in vacuum and in the presence of CH₄ was studied under 308 nm laser irradiation. Time-resolved emission using gated detection and scanning monochromator and absorption using the cavity ring-down spectroscopy were used to study vaporized plume. In the CH₄ atmosphere we observed transitions identified as C₂ and MnH bands, while these spectral features were not detected in emission spectra. This is a clear evidence of importance in combining both spectroscopic techniques in laser vaporized plume study.     

Effects of acute gamma-irradiation on extracellular adenine nucleotide hydrolysis in developing rat brain

Cell membrane is highly sensitive to irradiation which, acting directly or indirectly, may disturb functions of constitutive proteins including membrane enzymes. Plasma membrane surface-located enzyme chain of ecto-nucleotide triphospho diphosphohydrolases (NTPDases) and 5′-nucleotidase are involved in termination of cell purinergic signalization by hydrolyzing extracellular, excitatory adenosine triphosphate (ATP), as well as nucleotide di-, and mono-phosphate (ADP and AMP) to neuroprotective adenosine. Extracellular ATP, ADP, and AMP hydrolyzes were examined in purified synaptic plasma membranes after whole-body acute irradiation. All measurements were done 24 h after irradiation of developing (15-, 30-day-old) and adult (90-day-old) rats with low (50 cGy) and high (2 Gy) dose of gamma-rays. Both, high and low doses inhibited nucleotide hydrolyses in 15-day-old rats; in 30-day-old rats low dose of radiation inhibited ADP and AMP hydrolyses while high dose inhibited only ATP hydrolyse. In adult rats high dose induced no effects, while low dose stimulated nucleotides hydrolyses. According to obtained results it was concluded that ecto-nucleotidases of young rats are more sensitive to irradiation, since even low dose induces inhibition of ecto-nucleotidases activities. Ionizing radiation, by decreasing brain nucleotide hydrolyses in developing rats, induces accumulation of ATP and decreases production of adenosine in synaptic cleft which could be neurocytotoxic. On the contrary, in adult rats low dose of radiation stimulates NTPDase and 5′-nucleotidase activity and protective adenosine production which indicates protective and adaptive mechanisms developed in adult brain neuronal cells. The article is published in the original.     

A higher order family for the simultaneous inclusion of multiple zeros of polynomials

Starting from a suitable fixed point relation, a new family of iterative methods for the simultaneous inclusion of multiple complex zeros in circular complex arithmetic is constructed. The order of convergence of the basic family is four. Using Newtons and Halleys corrections, we obtain families with improved convergence. Faster convergence of accelerated methods is attained with only few additional numerical operations, which provides a high computational efficiency of these methods. Convergence analysis of the presented methods and numerical results are given. AMS subject classification 65H05, 65G20, 30C15     

Ground-state entropy of the q-state Potts antiferromagnet in a magnetic field

We present a simple method to obtain reliable ground-state entropies of the q-state Potts antiferromagnet in an external magnetic field. As an example, the ground-state entropy for the triangular lattice is established for all q. In the particular case q = 2, our method gives results which coincide with the first-order approximation obtained by the corner transfer matrix method.     

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The parabolic equation as a limiting case of a certain elliptic equation

Summary We consider the Dirichlet problem for the equation L_ε(u) ≡ u_xx+ɛu_yy++A(x, y)u_x−B(y)u_y+C(x, y)u=F(x, y) where B(y)>0 and ɛ is a small positive parameter. An asymptotic formula is proved, from which it follows that in a suitable part of the domain of definition u(x, y, ɛ)→U(x, y) as ɛ→0+, where U(x, y) is the solution of the corresponding boundary - value problem for the reduced equation L₀(U)≡U_xx+A(x, y)U_x−B(y)U+C(x, y)U=F(x, y). To Enrico Bompiani on his scientific Jubilee.