Splittability and 1-amalgamability of permutation classes

A permutation class $C$ is splittable if it is contained in a merge of two of its proper subclasses, and it is 1-amalgamable if given two permutations $\sigma$ and $\tau$ in $C$, each with a marked element, we can find a permutation $\pi$ in $C$ containing both $\sigma$ and $\tau$ such that the two marked elements coincide. It was previously shown that unsplittability implies 1-amalgamability. We prove that unsplittability and 1-amalgamability are not equivalent properties of permutation classes by showing that the class $Av(1423, 1342)$ is both splittable and 1-amalgamable. Our construction is based on the concept of LR-inflations, which we introduce here and which may be of independent interest.Comment: 17 pages, 7 figure

Evaluation of the slag regime and desulphurization of steel with synthetic slag containing Cr2O3

The paper focuses on an evaluation of the slag regime and steel desulphurisation in a ladle with the help of synthetic slags based on Al2O3 containing various mass amounts of Cr2O3 varying from 0.3 % to 3.0 %. The aim of the plant experiments was to assess the achieved results when using two types of synthetic slag focusing on the course of desulphurisation, an analysis of the achieved chemical composition of the slag and an evaluation of the influence of oxide Cr2O3 on the increase in the chromium amounts in steel during secondary metallurgy. During the evaluation of the slag regime in the ladle, samples of steel from various technological nodes for a determination of the desulphurisation degree and the achieved chromium amounts in steel were taken. Samples of slags for the evaluation of the achieved chemical composition and amounts of Cr2O3 in the slag were also taken. The results mentioned in this paper represent the basic information about the possibilities of applying synthetic slags with increased amounts of Cr2O3 within secondary metallurgy.Web of Science4896597

Evaluation of influence of briquetted synthetic slags on slag regime and process of steel desulphurization

This paper presents the industrial results of evaluation of efficiency of synthetic slags during the treatment of steel at the equipment of the secondary metallurgy under conditions of the steel plant VÍTKOVICE HEAVY MACHINERY a.s. The aim of the heats was to assess the influence of the briquetted and sintered synthetic slags based on Al2O3 aiming the course of the steel desulphurization and slag regime during the production and treatment of steel grade 42CrMo4 with the technology EAF→LF→ VD. Within the plant experiments, basic parameters influencing the steel desulphurization and slag regime were monitored: desulphurization degree, basicity, content of easily reducible oxides, proportion of CaO/Al2O3 and Mannesmann’s index. Obtained results allowed to compare the steel desulphurization and to evaluate the slag regime. It was proved that the synthetic slag presenting the briquetted mixture of secondary corundum raw materials can adequately replace the synthetic slag created from the sintered mixture of natural raw materials.W pracy tej przedstawiono przemysłowe wyniki oceny efektywności zastosowania żużli syntetycznych podczas obróbki pozapiecowej stali w zakładzie VÍTKOVlCE HEAVY MACHINERY a.s. Celem badań był a ocena wpływu brykietówanych i spiekanych żużli syntetycznych opartych na AI2O3 - na przebieg odsiarczania stali i żużla podczas produkcji i obróbki stali 42CrMo4. zgodnie z technologią EAF → LF → VD. Podczas eksperymentów w zakładzie, monitorowano podstawowe parametry wpływające na odsiarczania stali i żużla: stopień odsiarczania, zasadowość, zawartość tlenków łatwo redukowałnych, stosunek CaO/Al2O3 oraz indeks Mannesniann’a. Uzyskane wyniki pozwoliły na porównanie stopnia odsiarczania stali w kontekście zastosowanych żużli. Wykazano, że syntetyczny żużel składający się z brykietowanej mieszaniny wtórnego korundu, może zastąpić syntetyczny żużel otrzymywany ze spieku mieszaniny surowców naturalnych.Web of Science59281380

Griddings of Permutations and Hardness of Pattern Matching

We study the complexity of the decision problem known as Permutation Pattern Matching, or PPM. The input of PPM consists of a pair of permutations ? (the "text") and ? (the "pattern"), and the goal is to decide whether ? contains ? as a subpermutation. On general inputs, PPM is known to be NP-complete by a result of Bose, Buss and Lubiw. In this paper, we focus on restricted instances of PPM where the text is assumed to avoid a fixed (small) pattern ?; this restriction is known as Av(?)-PPM. It has been previously shown that Av(?)-PPM is polynomial for any ? of size at most 3, while it is NP-hard for any ? containing a monotone subsequence of length four. In this paper, we present a new hardness reduction which allows us to show, in a uniform way, that Av(?)-PPM is hard for every ? of size at least 6, for every ? of size 5 except the symmetry class of 41352, as well as for every ? symmetric to one of the three permutations 4321, 4312 and 4231. Moreover, assuming the exponential time hypothesis, none of these hard cases of Av(?)-PPM can be solved in time 2^o(n/log n). Previously, such conditional lower bound was not known even for the unconstrained PPM problem. On the tractability side, we combine the CSP approach of Guillemot and Marx with the structural results of Huczynska and Vatter to show that for any monotone-griddable permutation class ?, PPM is polynomial when the text is restricted to a permutation from ?