Let G be a finite group with derived subgroup of rank r. We prove that |G : Z2(G)| ≤ |G |. Motivated by the results of I. M. Isaacs in [2] we show that if G is capable then |G : Z(G)| ≤ |G| . This answers a question of L. Pyber. We prove that if G is a capable p-group then the rank of G/Z(G) is bounded above in terms of the rank of G′.

We prove sharp L estimates for oscillatory integral and Fourier integral operators for which the associated canonical relation C ⊂ T ∗ΩL × T ∗ΩR projects to T ∗ΩL and to T ∗ΩR with corank one singularities of type ≤ 2. This includes two-sided cusp singularities. Applications are given to operators with one-sided swallowtail singularities such as restricted X-ray transforms for well-curved line ...

We denote by A the subclass of H a, 1 with the usual normalization f 0 f ′ 0 − 1 0, and denote by A the subclass ofA satisfying the condition h z h′ z / 0 for z ∈ U \ {0}. Let f and F be members of H. The function f is said to be subordinate to F, or F is said to be superordinate to f, if there exists a functionw analytic in Uwithw 0 0 and |w z | < 1 for z ∈ U and such that f z g w z z ∈ U . In...

This paper presents a new multi-criteria decision-making (MCDM) method, namely the ratio product model (RPM). We first overview two popular aggregating models: weighted sum (WSM) and (WPM). Then, we argue that models suffer from some fundamental issues mainly due to ignoring nature of alternatives' scores with respect criteria importance weights criteria. Building on notion compositional data a...

Markov chain Monte Carlo is a key computational tool in Bayesian statistics, but it can be challenging to monitor the convergence of an iterative stochastic algorithm. In this paper we show that diagnostic $\widehat{R}$ Gelman and Rubin (1992) has serious flaws. Traditional will fail correctly diagnose failures when heavy tail or variance varies across chains. propose alternative rank-based fix...

We prove sharp L estimates for oscillatory integral and Fourier integral operators for which the associated canonical relation C ⊂ T ∗ΩL × T ∗ΩR projects to T ∗ΩL and to T ∗ΩR with corank one singularities of type ≤ 2. This includes two-sided cusp singularities. Applications are given to operators with one-sided swallowtail singularities such as restricted X-ray transforms for well-curved line ...

The demand for brown sugar products often falls below the level of production, resulting in unsold goods when market surpasses production capacity. This paper addresses challenge faced by many businesses estimating yields. Another issue, apart from uncertainty, is presence a dataset with significant nominal range. study focuses on specific producing company Indonesia. To address estimation prob...

We prove sharp L estimates for oscillatory integral and Fourier integral operators for which the associated canonical relation C ⊂ T ∗ΩL × T ∗ΩR projects to T ∗ΩL and to T ∗ΩR with corank one singularities of type ≤ 2. This includes two-sided cusp singularities. Applications are given to operators with one-sided swallowtail singularities such as restricted X-ray transforms for well-curved line ...