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In this study, we have developed and implemented a new technology capable of probabilistically selecting phase arrival times and determining the initial polarity of seismic waveforms without the requirement of prior information. In this new method, the arrival time is determined through an eigen-equation associated with the probability distribution of the noise level, which is then used to calculate the probability of the polarity. We have tested this method using synthetic waveforms as well as records from well-established databases. The results demonstrate a high degree of concurrence with manually picked arrival times and polarities (98% accuracy) in the local seismic catalog. This suggests that the proposed method can provide consistent and unified judgments in phase picking tasks. In comparison, this method has shown comparable reliability to existing neural-network-based AI methods while maintaining greater portability due to its lack of dependence on training data. Graphical abstract

A randomly generated fuzzy matrix refers to a fuzzy matrix in which the values of elements belong to the sample space of a [0,1]-random variable that follows a certain probability distribution. This paper studies the max–min transitive closure of two-type randomly generated fuzzy matrices: Bernoulli and classical probabilistic models. By introducing the concept of superposed fuzzy matrices, we investigate the probability distribution of the transitive closure of randomly generated fuzzy matrices for two probabilistic models. First, we presented the arithmetic operation rules for the superposed fuzzy relations. The expected value of the randomly generated Bernoulli fuzzy matrix transition closure was studied. A direct calculation method for the randomly generated fuzzy matrix transitive closure of the classical probability model was provided. Finally, the errors between the direct calculation method and the traditional transitive closure calculation method were compared.

A theorem of Hoffman gives an upper bound on the independence ratio of regular graphs in terms of the minimum λmin of the spectrum of the adjacency matrix. To complement this result we use random eigenvectors to gain lower bounds in the vertex-transitive case. For example, we prove that the independence ratio of a 3-regular transitive graph is at least $q=\\frac {1}{2}-\\frac{3}{4\\pi}arccos\\left ( \\frac{1-\\lambda_{min}{4} \\right )$. The same bound holds for infinite transitive graphs: we construct factor of i.i.d. independent sets for which the probability that any given vertex is in the set is at least q − o(1). We also show that the set of the distributions of factor of i.i.d. processes is not closed w.r.t. the weak topology provided that the spectrum of the graph is uncountable.

In many design settings where model violations are present, a \"stochastic\" minimaxity for many standard randomization procedures is demonstrated. This result requires no special analytic properties of the loss function and estimators. Next, under the squared loss and with the restriction to the use of linear estimators, a recipe for finding a randomized strategy is given. As a special case, randomizing an A-optimal design in the standard manner and using the least squares estimates yields a minimax strategy in most cases. These results generalize some aspects of Wu (1981).

Efficient two-factor balanced designs are constructed by using balanced arrays of strength two with indices λ(x, y) = λ0 if x = y or λ1 if x ≠ y. Balanced arrays of this type are constructed from doubly transitive permutation groups and partly resolvable orthogonal arrays. Some two-factor balanced design with all the main effects unconfounded with block effects are constructed.