This paper examines two major results concerning the symmetric group, Sn. The first result, Landau’s theorem, gives an asymptotic formula for the maximum order of an element in Sn, and the second, Dixon’s Theorem, settled an open conjecture concerning the probability that two randomly selected elements of Sn will generate An or Sn.

rock sawability prediction and determination of effective parameters are important in the cost estimation and the planning of the stone factories. an accurate estimation of rock sawability and performance prediction of sawing machine helps to make the planning of the rock sawing projects more efficient. it is necessary to predict the rock sawability and determine the effective parameters to ach...

1 Deviation bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1 Markov and generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Class of sub-Gaussian random variables . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.1 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2....

The paper is concerned with stochastic approximation procedures having three main characteristics: truncations with random moving bounds, a matrix valued random step-size sequence, and a dynamically changing random regression function. We study convergence and rate of convergence. Main results are supplemented with corollaries to establish various sets of sufficient conditions, with the main em...