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On further properties of minimal size in hypergroups of type U on the right
No full textIn this paper, we prove that the minimum size of a hypergroup of type U on the right which contains a proper subsemihypergroup of type U on the right of size 6 and a subgroup of size ≥ 2 is 18. We study some properties of hypergroups of type U on the right of size 2p that have a cyclic subgroup of size p. Moreover, we give a more general answer to a problem left open in [11]
An "open loop" control methodology for sound reduction inside a car
No full textIn this paper we consider the problem of reducing the sound generated by the engine inside a car enclosure by using piezoelectric actuators located on the car's roof and microphones placed in correspondence of the passengers. A preliminary analysis shows that the frequency responses from the actuators to the microphones are characterized by a very large band and exhibit a high modal density with many resonance peaks. Based on these facts, the design of a feedback control law could originate spillover phenomena, which will manifest through the generation of high frequency sounds even more unpleasent than the original noise. Therefore we propose the design of a controller based on an open loop control philosophy; that is the control action is designed in such a way that the spectrum of the control signals is contained into the disturbance band. Since the noise we deal with is not of a classical type (pure sinusoidal or multi-tonal) but is a wide band disturbance, classical open loop control techniques, based on model inversion, cannot be implemented; to this end the concept of Generalized Frequency Response, which allows us to deal with wide band signals, is introduced in the paper. Numerical simulations show the benefits of the proposed control technique
Preface to the Special Issue on “Hypergroup Theory and Algebrization of Incidence Structures”
Full text linkThis work contains the accepted papers of a Special Issue of the MDPI journal Mathematics entitled “Hypergroup Theory and Algebrization of Incidence Structure” [...
Some applications of algebraic entropy to the proof of Milnor-Wolf theorem
Full text linkWe discuss the growth function of a finitely generated cascade and its
connection to the growth function of its related semi-direct product
(Conjecture 1.9). The results is applied for simpler proof of well-known
results in the realm of geometric group theory. We show that the finitely
generated cascades on nilpotent groups obey the dichotomy rule (only
polynomial and exponential growth are possible)
G-Hypergroups: Hypergroups with a Group-Isomorphic Heart
Full text linkHypergroups can be subdivided into two large classes: those whose heart coincide with the entire hypergroup and those in which the heart is a proper sub-hypergroup. The latter class includes the family of 1-hypergroups, whose heart reduces to a singleton, and therefore is the trivial group. However, very little is known about hypergroups that are neither 1-hypergroups nor belong to the first class. The goal of this work is to take a first step in classifying G-hypergroups, that is, hypergroups whose heart is a nontrivial group. We introduce their main properties, with an emphasis on G-hypergroups whose the heart is a torsion group. We analyze the main properties of the stabilizers of group actions of the heart, which play an important role in the construction of multiplicative tables of G-hypergroups. Based on these results, we characterize the G-hypergroups that are of type U on the right or cogroups on the right. Finally, we present the hyperproduct tables of all G-hypergroups of size not larger than 5, apart of isomorphisms
Commutativity and Completeness Degrees of Weakly Complete Hypergroups
No full textWe introduce a family of hypergroups, called weakly complete, generalizing the construction of complete hypergroups. Starting from a given group G, our construction prescribes the β-classes of the hypergroups and allows some hyperproducts not to be complete parts, based on a suitably defined relation over G. The commutativity degree of weakly complete hypergroups can be related to that of the underlying group. Furthermore, in analogy to the degree of commutativity, we introduce the degree of completeness of finite hypergroups and analyze this degree for weakly complete hypergroups in terms of their β-classes
1-hypergroups of small sizes
Full text linkIn this paper, we show a new construction of hypergroups that, under appropriate conditions, are complete hypergroups or non-complete 1-hypergroups. Furthermore, we classify the 1-hypergroups of size 5 and 6 based on the partition induced by the fundamental relation β. Many of these hypergroups can be obtained using the aforesaid hypergroup construction
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