Groups containing locally maximal product-free sets of size 4

Every locally maximal product-free set S in a finite group G satisfies G=S∪SS∪S−1S∪SS−1∪S−−√, where SS={xy∣x,y∈S}, S−1S={x−1y∣x,y∈S}, SS−1={xy−1∣x,y∈S} and S−−√={x∈G∣x2∈S}. To better understand locally maximal product-free sets, Bertram asked whether every locally maximal product-free set S in a finite abelian group satisfy |S−−√|≤2|S|. This question was recently answered in the negation by the current author. Here, we improve some results on the structures and sizes of finite groups in terms of their locally maximal product-free sets. A consequence of our results is the classification of abelian groups that contain locally maximal product-free sets of size 4, continuing the work of Street, Whitehead, Giudici and Hart on the classification of groups containing locally maximal product-free sets of small sizes. We also obtain partial results on arbitrary groups containing locally maximal product-free sets of size 4, and conclude with a conjecture on the size 4 problem as well as an open problem on the general case.

Generalised Iwasawa invariants and the growth of class numbers

We study the generalised Iwasawa invariants of {\mathbb{Z}_{p}^{d}}-extensions of a fixed number field K. Based on an inequality between ranks of finitely generated torsion {\mathbb{Z}_{p}[\kern-2.133957pt[T_{1},\dots,T_{d}]\kern-2.133957pt]}-modules and their corresponding elementary modules, we prove that these invariants are locally maximal with respect to a suitable topology on the set of {\mathbb{Z}_{p}^{d}}-extensions of K, i.e., that the generalised Iwasawa invariants of a {\mathbb{Z}_{p}^{d}}-extension {\mathbb{K}} of K bound the invariants of all {\mathbb{Z}_{p}^{d}}-extensions of K in an open neighbourhood of {\mathbb{K}}. Moreover, we prove an asymptotic growth formula for the class numbers of the intermediate fields in certain {\mathbb{Z}_{p}^{2}}-extensions, which improves former results of Cuoco and Monsky. We also briefly discuss the impact of generalised Iwasawa invariants on the global boundedness of Iwasawa λ-invariants.

Orbital shadowing property on chain transitive sets for generic diffeomorphisms

Let f : M → M be a diffeomorphism on a closed smooth n(≥ 2) dimensional manifold M. We show that C1 generically, if a diffeomorphism f has the orbital shadowing property on locally maximal chain transitive sets which admits a dominated splitting then it is hyperbolic.

Comparative Thermodynamic Analysis of Kalina and Kalina Flash Cycles for Utilizing Low-Grade Heat Sources

The Kalina flash cycle (KFC) is a novel, recently proposed modification of the Kalina cycle (KC) equipped with a flash vessel. This study performs a comparative analysis of the thermodynamic performance of KC and KFC utilizing low-grade heat sources. How separator pressure, flash pressure, and ammonia mass fraction affect the system performance is systematically and parametrically investigated. Dependences of net power and cycle efficiencies on these parameters as well as the mass flow rate, heat transfer rate and power production at the cycle components are analyzed. For a given set of separator pressure and ammonia mass fraction, there exists an optimum flash pressure making exergy efficiency locally maximal. For these pressures, which are higher for higher separator pressure and lower ammonia mass fraction, KFC shows better performance than KC both in net power and cycle efficiencies. At higher ammonia mass fraction, however, the difference is smaller. While the maximum power production increases with separator pressure, the dependence is quite weak for the maximum values of both efficiencies.

Local behavior of Iwasawa’s invariants

Let [Formula: see text] be a number field, let [Formula: see text] denote a fixed rational prime. We study the local behavior of Iwasawa’s invariants as functions on the set [Formula: see text] of all [Formula: see text]-extensions of [Formula: see text]. With respect to a certain topology on [Formula: see text] that takes care of ramification, we prove that for each [Formula: see text] the [Formula: see text]-invariant of [Formula: see text] is locally maximal among the [Formula: see text]-invariants, and we give sufficient conditions for the [Formula: see text]-invariant to be locally maximal (e.g., a vanishing [Formula: see text]-invariant). This concerns a question raised by R. Greenberg in 1973. Our main result also provides information about [Formula: see text]- (and even [Formula: see text]-) invariants in the case of a nonvanishing [Formula: see text]-invariant. The main tool used in the proof is a new result based on the stabilization of certain ranks, which considerably generalizes a theorem of T. Fukuda.