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January 1, 2006
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Let μ ( t ) be the number of particles at time t of a continuous-time critical branching process. It is known that the probability of non-extinction of the process at time t Q ( t ) = P { μ ( t ) > 0 | μ (0) = 1} → 0 as t → ∞. Hence it follows that Q m 0 = P { μ ( t ) > 0 | μ (0) = m } ∼ mQ ( t ) → 0 for any m = 2,3, . . . Let for any integer m > r ≥ 1 In this paper, we prove that Q mr ( t ) ∼ ( m − r ) Q ( t ) as t → ∞ for any critical continuous-time Markov branching process. Earlier, this result was obtained for branching processes with finite variation of the number of particles.
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Every Boolean function of n variables is identified with a function F : Q → P , where Q = GF (2 n ), P = GF (2). A. Youssef and G. Gong showed that for n = 2 λ there exist functions F which have equally bad approximations not only by linear functions (that is, by functions tr ( μx ), where μ ∈ Q * and tr: Q → P is the trace function), but also by proper monomial functions (functions tr( μx δ ), where ( δ , 2 n − 1) = 1). Such functions F were called hyper-bent functions ( HB functions, HBF ), and for any n = 2 λ a non-empty class of HBF having the property F (0) = 0 was constructed. This class consists of the functions F ( x ) = such that the equation F ( x ) = 1 has exactly (2 λ − 1)2 λ −1 solutions in Q . In the present paper, we give some essential restrictions on the parameters of an arbitrary HBF showing that the class of HBF is far less than that of bent functions. In particular, we show that any HBF is a bent function having the degree of nonlinearity λ , and for some n (for instance, if λ > 2 and 2 λ − 1 is prime, or λ ∈ {4,9,25,27}) the class of HBF is exhausted by the functions F ( x ) = described by A. Youssef and G. Gong. For n = 4, in addition to 10 HBF listed above there exist 18 more HBF with property F (0) = 0. The question of whether there exist other hyper-bent functions for n > 4 remains open.
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We describe the classes of all modules and all rings such that any system of linear equations over them is solvable if and only if it is concordant with linear relations.
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We analyse the distribution of the number of solutions of a system of random linear equations over GF ( q ) in the set of vectors which have a given number of nonzero coordinates and in some subsets of this set. We deduce sufficient conditions for convergence of the distribution to the Poisson law, as well as to some other limit distributions related to this law, and to the standard normal law. Here we extend the results which the author has proved earlier for the case q = 2.
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In this paper, we consider algorithms to pack rectangles into a strip. As the main result we present an algorithm that packs rectangles online and for which the ratio of expected wasted area to expected occupied area tends to zero as the number of rectangles increases.
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We consider a problem to pack rectangles into several semi-infinite strips of certain widths. We suggest two simply realised online algorithms, that is, algorithms which pack rectangles just at the moments of their arrivals. It is shown that the accuracy of the former algorithm cannot be approximated by any absolute constant. The latter algorithm guarantees a constant multiplicative accuracy, and the obtained estimate of the multiplicative accuracy is unimprovable.
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On the set of all infinite binary sequences, we consider the simplest form of algorithmic reducibility, namely, the Boolean reducibility. Each set Q of Boolean functions which contains a selector function and is closed with respect to the superposition operation of special kind generates the Q -reducibility and Q -degrees, the sets of Q -equivalent sequences. The Q -degree of a sequence α characterises the relative 'informational complexity' of the sequence α , in a sense, Q is a set of operators of information retrieval from infinite sequences. In this paper, we study the partially ordered sets ℒ Q of all Q -degrees for the most important classes Q of Boolean functions. We investigate the positions of periodic and narrow Q -degrees in ℒ Q , find the number of minimal elements and atoms and also the initial segments isomorphic to given finite lattices.