Psychological Research of the Ability to Foreign Languages Acquisition

Along with psychological approaches and concepts of teaching foreign languages, concepts of mastering a foreign language, built on the ideas of appropriation and interpretation of a foreign language linguistic experience, have spread in psychological science. While not being opposed to theories of teaching foreign languages, these concepts and approaches, however, have some differences from them, which also require analysis and interpretation. To avoid misunderstandings in the use of the most general concepts from the field of psychology of mastering foreign languages, in their study the authors clarify them, guided by the literary materials that make up its scientific paradigm. The authors analysed the use of non-standard techniques to facilitate the acquisition of foreign languages.In their research, the authors tested the hypothesis that in the conditions of personality-oriented dialogical teaching of a foreign language, personal characteristics that complicate foreign language communication are significantly optimised. As a result of the experimental work, the hypothesis was confirmed; still, in an even more strengthened form: the significant connection between the motivation of learning and the level of development of abilities for languages is fundamentally destroyed. In other words, in the conditions of student-centred learning, insufficient development of potential language abilities ceases to be a factor that negatively affects the motivation of learning.

Strengthened convexity of positive operator monotone decreasing functions

We prove a strengthened form of convexity for operator monotone decreasing positive functions defined on the positive real numbers. This extends Ando and Hiai's work to allow arbitrary positive maps instead of states (or the identity map), and functional calculus by operator monotone functions defined on the positive real numbers instead of the logarithmic function.

Newton and Bouligand derivatives of the scalar play and stop operator

We prove that the play and the stop operator possess Newton and Bouligand derivatives, and exhibit formulas for those derivatives. The remainder estimate is given in a strengthened form, and a corresponding chain rule is developed. The construction of the Newton derivative ensures that the mappings involved are measurable.

THE ELIMINATION OF ATOMIC CUTS AND THE SEMISHORTENING PROPERTY FOR GENTZEN’S SEQUENT CALCULUS WITH EQUALITY

We study various extensions of Gentzen’s sequent calculus obtained by adding rules for equality. One of them is singled out as particularly natural and shown to satisfy full cut elimination, namely, also atomic cuts can be eliminated. Furthermore we tell apart the extensions that satisfy full cut elimination from those that do not and establish a strengthened form of the nonlenghtening property of Lifschitz and Orevkov.

Squared Chromatic Number Without Claws or Large Cliques

Let $G$ be a claw-free graph on $n$ vertices with clique number $\unicode[STIX]{x1D714}$, and consider the chromatic number $\unicode[STIX]{x1D712}(G^{2})$ of the square $G^{2}$ of $G$. Writing $\unicode[STIX]{x1D712}_{s}^{\prime }(d)$ for the supremum of $\unicode[STIX]{x1D712}(L^{2})$ over the line graphs $L$ of simple graphs of maximum degree at most $d$, we prove that $\unicode[STIX]{x1D712}(G^{2})\leqslant \unicode[STIX]{x1D712}_{s}^{\prime }(\unicode[STIX]{x1D714})$ for $\unicode[STIX]{x1D714}\in \{3,4\}$. For $\unicode[STIX]{x1D714}=3$, this implies the sharp bound $\unicode[STIX]{x1D712}(G^{2})\leqslant 10$. For $\unicode[STIX]{x1D714}=4$, this implies $\unicode[STIX]{x1D712}(G^{2})\leqslant 22$, which is within 2 of the conjectured best bound. This work is motivated by a strengthened form of a conjecture of Erdős and Nešetřil.

Uniformity, universality, and computability theory

We prove a number of results motivated by global questions of uniformity in computabi- lity theory, and universality of countable Borel equivalence relations. Our main technical tool is a game for constructing functions on free products of countable groups. We begin by investigating the notion of uniform universality, first proposed by Montalbán, Reimann and Slaman. This notion is a strengthened form of a countable Borel equivalence relation being universal, which we conjecture is equivalent to the usual notion. With this additional uniformity hypothesis, we can answer many questions concerning how countable groups, probability measures, the subset relation, and increasing unions interact with universality. For many natural classes of countable Borel equivalence relations, we can also classify exactly which are uniformly universal. We also show the existence of refinements of Martin’s ultrafilter on Turing invariant Borel sets to the invariant Borel sets of equivalence relations that are much finer than Turing equivalence. For example, we construct such an ultrafilter for the orbit equivalence relation of the shift action of the free group on countably many generators. These ultrafilters imply a number of structural properties for these equivalence relations.

On Independent Sets in Graphs with Given Minimum Degree

We consider numbers and sizes of independent sets in graphs with minimum degree at leastd. In particular, we investigate which of these graphs yield the maximum numbers of independent sets of different sizes, and which yield the largest random independent sets. We establish a strengthened form of a conjecture of Galvin concerning the first of these topics.