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We consider a sequence of n, n≥3, zero (0) - one (1) Markov-dependent trials. We focus on k-tuples of 1s; i.e. runs of 1s of length at least equal to a fixed integer number k, 1≤ k≤ n. The statistics denoting the number of k-tuples of 1s, the number of 1s in them and the distance between the first and the last k-tuple of 1s in the sequence, are defined. The work provides, in a closed form, the exact conditional joint distribution of these statistics given that the number of k-tuples of 1s in the sequence is at least two. The case of independent and identical 0−1 trials is also covered in the study. A numerical example illustrates further the theoretical results. [ABSTRACT FROM AUTHOR]

We study asymptotical probabilities of first order and monadic second order properties of Bernoulli random graph G ( n , n − a ) . The random graph obeys FO (MSO) zero-one k -law ( k is a positive integer) if, for any first order (monadic second order) formulae with quantifier depth at most k , it is true for G ( n , n − a ) with probability tending to 0 or to 1. Zero-one k -laws are well studied only for the first order language and a < 1 . We obtain new zero-one k -laws (both for first order and monadic second order languages) when a > 1 . Proofs of these results are based on the earlier studies of first order equivalence classes and our study of monadic second order equivalence classes. The respective results are of interest by themselves. [ABSTRACT FROM AUTHOR]

This paper introduces a DC/AC converter, which can be connected in parallel with a diode rectifier for regenerative applications. The DC/AC converter is supposed to transmit regenerative energy to the power grid when a motor is braking. Isolation transformers are not needed in the topology, which can reduce the size and cost. An analysis of the zero-order current existing in the system is carried out. In addition, algorithms to minimize the zero-order current, control the power factor and keep the DC bus voltage stable are discussed. A 55kW industrial prototype is built to verify the proposed analysis and control strategies. [ABSTRACT FROM AUTHOR]

We provide sufficient conditions for the validity of a dichotomy, i.e. zero-one law, between recurrence and transience of general frog models. In particular, the results cover frog models with i.i.d. numbers of frogs per site where the frog dynamics are given by quasi-transitive Markov chains or by random walks in a common random environment including super-critical percolation clusters on $${\mathbb {Z}}^d$$ . We also give a sufficient and almost sharp condition for recurrence of uniformly elliptic frog processes on $${\mathbb {Z}}^d$$ . Its proof uses the general zero-one law. [ABSTRACT FROM AUTHOR]

We develop the inhomogeneous counterpart to some key aspects of the story of the Duffin-Schaeffer Conjecture (1941). Specifically, we construct counterexamples to a number of candidates for a sans-monotonicity version of Szüsz's inhomogeneous (1958) version of Khintchine's Theorem (1924). For example, given any real sequence , we build a divergent series of non-negative reals such that for any , almost no real number is inhomogeneously -approximable with inhomogeneous parameter . Furthermore, given any second sequence not intersecting the rational span of , and assuming a dynamical version of Erdős' Covering Systems Conjecture (1950), we can ensure that almost every real number is inhomogeneously -approximable with any inhomogeneous parameter . Next, we prove a positive result that is near optimal in view of the limitations that our counterexamples impose. This leads to a discussion of natural analogues of the Duffin-Schaeffer Conjecture and Duffin-Schaeffer Theorem (1941) in the inhomogeneous setting. As a step toward these, we prove versions of Gallagher's Zero-One Law (1961) for inhomogeneous approximation by reduced fractions. [ABSTRACT FROM AUTHOR]

We investigate $k$ -connectivity in secure wireless sensor networks under the random pairwise key predistribution scheme with unreliable links. When wireless communication links are modeled as independent on-off channels, this amounts to analyzing a random graph model formed by intersecting a random $K$ -out graph and an Erdős-Rényi graph. We present conditions on how to scale the parameters of this intersection model so that the resulting graph is $k$ -connected with probability approaching to one (resp. zero) as the number of nodes gets large. The resulting zero-one law is shown to improve and sharpen the previous result on the 1-connectivity of the same model. We also provide numerical results to support our analysis. [ABSTRACT FROM PUBLISHER]

The presented paper is devoted to the asymptotical behavior of first-order properties of the Erdős–Rényi random graph. In previous works the zero–one k -law was proved. This law describes asymptotical behavior of first-order properties which are expressed by formulae with a quantifier depth bounded by k . The random graph G ( N , N − α ) obeys the law if α ∈ ( 0 , 1 / ( k − 2 ) ) . In this work we find new values of α , which are close to 1, such that G ( N , N − α ) obeys the zero–one k -law and, therefore, extend the previous result. [ABSTRACT FROM AUTHOR]

We introduce a new random key predistribution scheme for securing heterogeneous wireless sensor networks. Each of the n sensors in the network is classified into r classes according to some probability distribution \boldsymbol \mu =\\mu 1,\ldots ,\mu r\ . Before deployment, a class- i sensor is assigned Ki cryptographic keys that are selected uniformly at random from a common pool of P keys. Once deployed, a pair of sensors can communicate securely if and only if they have a key in common. We model the communication topology of this network by a newly defined inhomogeneous random key graph. We establish scaling conditions on the parameters P and \K1,\ldots , Kr\ so that this graph: 1) has no isolated nodes and 2) is connected, both with high probability. The results are given in the form of zero-one laws with the number of sensors n growing unboundedly large; critical scalings are identified and shown to coincide for both graph properties. Our results are shown to complement and improve those given by Godehardt et al. and Zhao et al. for the same model, therein referred to as the general random intersection graph. [ABSTRACT FROM PUBLISHER]

This paper is concerned with situations in which the uncertain measure of an event can only be zero or one, and the uncertain zero-one laws are derived within the framework of uncertainty theory that can be seen as the counterpart of Kolmogorov zero-one law and Borel-Cantelli lemma, which can be used as a tool for solving some problems concerning almost sure convergence of uncertain sequence. [ABSTRACT FROM AUTHOR]

The Le Bars conjecture (2001) states that the binomial random graph G(n, 12) obeys the zero-one law for existential monadic sentences with two first-order variables. This conjecture is disproved. Moreover, it is proved that there exists an existential monadic sentence with a single monadic variable and two first-order variables whose truth probability does not converge. [ABSTRACT FROM AUTHOR]

We study connectivity property in the superposition of random key graph on random geometric graph. For this class of random graphs, we establish a new version of a conjectured zero-one law for graph connectivity as the number of nodes becomes unboundedly large. The results reported here strengthen recent work by the Krishnan et al. [ABSTRACT FROM AUTHOR]

We study the secure and reliable connectivity of wireless sensor networks. Security is assumed to be ensured by the random pairwise key predistribution scheme of Chan, Perrig, and Song, and unreliable wireless links are represented by independent ON/OFF channels. Modeling the network by an intersection of a random $K$ -out graph and an Erdős–Rényi graph, we present scaling conditions (on the number of nodes $n$ , the scheme parameter $K$ , and the probability $p$ of a wireless channel being on), such that the resulting graph contains no nodes with a degree less than $k$ with high probability. Results are given in the form of zero–one laws with $n$ getting large, and are shown to improve the previous results by Yağan and Makowski on the absence of isolated nodes (i.e., absence of nodes with degree zero) in the same model. Through simulations, the established zero–one laws are also shown to hold for the property of $k$ -connectivity, i.e., the property that graph remains connected despite the deletion of any $k-1$ nodes or edges. [ABSTRACT FROM PUBLISHER]

We prove analogues of some classical results from Diophantine approximation and metric number theory (namely Dirichlet's theorem and the Duffin–Schaeffer theorem) in the setting of diagonal Diophantine approximation, i.e. approximating elements of R × Q p 1 × ⋯ × Q p r by elements of the diagonal embedding of Q into this space. [ABSTRACT FROM AUTHOR]

The limit probabilities of the first-order properties of a random graph in the Erdős-Rényi model G( n, n), α ∈ (0, 1], are studied. Earlier, the author obtained zero-one k-laws for any positive integer k ≥ 3, which describe the behavior of the probabilities of the first-order properties expressed by formulas of quantifier depth bounded by k for α in the interval (0, 1/( k − 2)] and k ≥ 4 in the interval (1 − 1/2, 1). This result is improved for k = 4. Moreover, it is proved that, for any k ≥ 4, the zero-one k-law does not hold at the lower boundary of the interval (1 − 1/2, 1). [ABSTRACT FROM AUTHOR]

The article presents a study that determines zero-one laws for random graphs. It describes the method of the study which considers the Erdös-Renyi random graph model. It outlines the result of the study which indicates the proofs of zero-one laws for sequences of random graphs following assertion about the existence of a winning strategy for duplicator in the Ehrenfeucht game.

We study zero-one laws for random distance graph with vertices in {−1, 0, 1} depending on a set of parameters. We give some conditions under which sequences of random distance graphs obey or do not obey the zero-one law. [ABSTRACT FROM AUTHOR]

We explore the scaling properties for graph connectivity in random threshold graphs. In the many node limit, we provide a complete characterization for the existence and type of the underlying zero-one laws, and identify the corresponding critical scalings. These results are consequences of well-known facts in Extreme Value Theory concerning the asymptotic behavior of running maxima on i.i.d. random variables. In the important special case of exponentially distributed fitness, we show that the (essentially unique) critical scaling which ensures a power-law degree distribution, does not result in graph connectivity in the asymptotically almost sure (a.a.s.) sense. [ABSTRACT FROM AUTHOR]

We consider a collection of n independent points which are distributed on the unit interval [0,1] according to some probability distribution function F. Two nodes communicate with each other if their distance is less than some given threshold value. When F admits a density f which is strictly positive on [0,1], we give conditions on f under which the property of graph connectivity for the induced geometric random graph obeys a very strong zero-one law when the transmission range is scaled appropriately with n large. The very strong critical threshold is identified. This is done by applying a version of the method of first and second moments. [ABSTRACT FROM AUTHOR]

Let X be a second countable locally compact Abelian group, and let μ be a Gaussian measure in the sense of Bernstein on X. Under the assumption that the connected component of zero of X contains a finite number of elements of order 2, we prove that μ is a convolution of a Gaussian measure, the Haar distribution of a compact subgroup of X, and a signed measure supported in the subgroup of X generated by elements of order 2. We describe the support of μ on an arbitrary group X. We prove that if the connected component of zero of X has finite dimension, then the zero-one law holds for μ under the assumption that μ has no idempotent factors. [ABSTRACT FROM AUTHOR]

Obtaining high-quality feasible solution is the core and the major difficulty in solving security-constrained unit commitment (SCUC) problems. This paper presents a systematic method for constructing feasible solutions to SCUC problem based on a group of analytical feasibility conditions. The feasibility check is performed based on the analytical necessary conditions such that most of infeasible UC states can be identified without solving LP problem. If a UC state is infeasible, it is adjusted with the possibly minimal operating cost increase based on the cost information. This UC adjusting issue is formulated as a zero-one programming problem and a branch and bound (B&B) method is established based on these feasibility conditions. Numerical testing is performed for a 31-bus system, an IEEE 24-bus system, and an IEEE 118-bus system. The testing results suggest that over 95% of infeasible UC states are identified by the analytical necessary conditions. The near-optimal feasible schedules for SCUC problem can be obtained efficiently by the proposed method. The feasible schedules obtained are compared with those obtained from mixed integer programming-based method in the IEEE 118-bus system. It is shown that the new method can produce competitive results in terms of solution quality and computational efficiency. [ABSTRACT FROM PUBLISHER]

Discussed here are criteria for the existence of continuous components in the spectra of operators with random potential. First, the essential condition for the Simon-Wolff criterion is shown to be measurable at infinity. By implication, for the i.i.d. case and more generally potentials with the K-property, the criterion is boosted by a zero-one law. The boosted criterion, combined with tunneling estimates, is then applied for sufficiency conditions for the presence of continuous spectrum for random Schrödinger operators. The general proof strategy that this yields is modeled on the resonant delocalization arguments by which continuous spectrum in the presence of disorder was previously established for random operators on tree graphs. In another application of the Simon-Wolff rank-one analysis we prove the almost sure simplicity of the pure point spectrum for operators with random potentials of conditionally continuous distribution.© 2015 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]

In this paper we prove that the unboundedness of nonconstant homogeneous stochastic processes with independent increments is a natural corollary of the Kolmogorov zero-one law. The known proof of this statement is based on a deep preliminary investigation of stochastic processes with independent increments. [ABSTRACT FROM AUTHOR]

A random graph G( n, p) is said to obey the (monadic) zero-one k-law if, for any monadic formula of quantifier depth k, the probability that it is true for the random graph tends to either zero or one. In this paper, following J. Spencer and S. Shelah, we consider the case p = n . It is proved that the least k for which there are infinitely many α such that a random graph does not obey the zero-one k-law is equal to 4. [ABSTRACT FROM AUTHOR]