G-Hypergroups: Hypergroups with a Group-Isomorphic Heart

Hypergroups 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.

Constructing cubic curves with involutions

In 1888, Heinrich Schroeter provided a ruler construction for points on cubic curves based on line involutions. Using Chasles’ Theorem and the terminology of elliptic curves, we give a simple proof of Schroeter’s construction. In addition, we show how to construct tangents and additional points on the curve using another ruler construction which is also based on line involutions. As an application of Schroeter’s construction we provide a new parametrisation of elliptic curves with torsion group $$\mathbb {Z}/2\mathbb {Z}\times \mathbb {Z}/8\mathbb {Z}$$ Z / 2 Z × Z / 8 Z and give some configurations with all their points on a cubic curve.

On theory of finite subsets monoid for one torsion abelian group

Ранее был доказан следующий результат: если абелева группа $\gG$ не является группой кручения, то теория моноида ее конечных подмножеств позволяет интерпретировать элементарную арифметику. В настоящей работе мы приводим пример, который показывает, что аналогичный результат можно получить и, по крайней мере, для некоторых групп кручения. Earlier it was proved the following claim. Let $\gG$ be a non-torsion abelian group and $\gG$ be the semigroup of finite subsets of $\gG$. Then elementary arithmetic can be interpreted in $\gG^*$, so the theory of $\gG^*$ is undecidable. Here we prove the same result for one torsion group, the multiplicative group of all roots of unity.

A NOTE ON GROUP RINGS WITH TRIVIAL UNITS

Let R be a ring with identity of characteristic two and G a nontrivial torsion group. We show that if the units in the group ring $RG$ are all trivial, then G must be cyclic of order two or three. We also consider the case where R is a commutative ring with identity of odd prime characteristic and G is a nontrivial locally finite group. We show that in this case, if the units in $RG$ are all trivial, then G must be cyclic of order two. These results improve on a result of Herman et al. [‘Trivial units for group rings with G-adapted coefficient rings’, Canad. Math. Bull.48(1) (2005), 80–89].

Clinical-ultrasound and clinical-morphological characteristics of adnexal torsion in girls and adolescents

Clinical-ultrasound and clinical-morphological characteristics of adnexal torsion to the improvement of early diagnosis and development of optimal ways of organ-preserving surgical tactics of treatment have been determined. An analysis was carried out of 71 patients with adnexal torsion (group 1– 30 girls of 2–12 years, group 2 – 41 adolescents of 13–17 years). The main clinical signs of adnexal torsion were nonspecific and similar to the well-known clinic of acute abdomen. Ultrasound examination with color doppler mapping (CDM) allowed to suspect adnexal torsion in 44 (62 %) patients. The main echographic features of adnexal torsion were: ovarian enlargement, non-typical location and a change in the structure of ovary, the presence of a free liquid in the cavity of a small pelvis or in the abdominal cavity, «a symptom of springs» (a twisted vascular leg), a decrease or absence of blood flow in CDM. The torsion of intact uterine appendages is installed in 29 patients (twice as often in group 1). In 42 cases were detected the torsion of the uterine appendages, compromised by the presence of tumors and tumor-like formations (twice as often in group 2). In both groups, 36 (50.7 %) organ-preserving and 35 (49.3 %) radical operations were carried out. According to the results of morphological examination of the removed tissues, the following data were obtained: follicular cysts – 15, paratubal cysts – 8, corpus luteum cysts – 8, teratomas – 5, serous cystadenomas – 2, mucinous cystadenoma – 1, paraovarian cysts – 3. Differential diagnosis of adnexal torsion in girls and adolescents should include a thorough collection of anamnestic data, taking into account the features of the clinical course of the disease, the results of a comprehensive examination with an assessment of ultrasound data from CDM, computer and magnetic resonance imaging, which will contribute to the advanced adequate selection of tactics of urgent gynecological intervention. The method of choice for the treatment of adnexal torsion is a minimally invasive surgical intervention – a laparoscopy with organ-preserving operations. Keywords: adnexal torsion, girls, ultrasound and morphological characteristics.

Protective effects of curcumin on chromatin quality, sperm parameters, and apoptosis following testicular torsion-detorsion in mice

Objective: The chief outcome of testicular torsion in clinical and experimental contexts is testicular ischemia. Curcumin, a compound with anti-inflammatory and antioxidant properties, has fascinated researchers and clinicians for its promise in the treatment of fertility diseases. Methods: Thirty-five fully grown male mice were randomly classified into five groups: control, sham, testicular torsion, treatment group 1 (testicular torsion+short-term curcumin), and treatment group 2 (testicular torsion+long-term curcumin). Thirty-five days later, spermatozoa from the right cauda epididymis were analyzed with regard to count and motility. Toluidine blue (TB), aniline blue (AB), and chromomycin A3 (CMA3) staining assays were used to evaluate the sperm chromatin integrity. In addition, the terminal deoxynucleotidyl transferase-mediated deoxyuridine triphosphate nick-end labeling (TUNEL) test was used to assess apoptosis.Result: Treatment group 1 exhibited a remarkably elevated sperm count compared to the testicular torsion group. Additionally, notably lower sperm motility was found in the testicular torsion group compared to the control, treatment 1, and treatment 2 groups. Staining (CMA3, AB, and TB) and the TUNEL test indicated significantly greater testicular torsion in the torsion group compared to the control group (p<0.05). The data also revealed notably lower results of all sperm chromatin assays and lower apoptosis in both treatment groups relative to the testicular torsion group (p<0.05). Significantly elevated (p<0.05) AB and TB results were noted in treatment group 1 compared to treatment group 2.Conclusion: Curcumin can compensate for the harmful effects of testicular ischemia and improve sperm chromatin quality in mice.

Application value of real-time shear wave elastography in differential diagnosis of testicular torsion

Aim: To evaluate the clinical value of real-time shear wave elastography (SWE) in differential diagnosis of testicular torsion and acute orchiditis.Material and methods: During a 3-year period, 14 cases of testicular torsion and 16 cases of acute orchiditis met the inclusion criteria. Young’s modulus maximum hardness (Emax) of testicular capsule region, middle testicular parenchyma, warped spermatic segment or inferior spermatic segment was measured in each group. SWE “stiff ring sign” of testis refers to the appearance of a red ring in the testicular capsule area and “stiff knot sign” of spermatic cord refers to the appearance of a red knot in the lower segment of the spermatic cord.Results: Emax value of the testicular capsule in the torsion group was higher than in the acute inflammation group (138.76±58.27 vs 16.40±4.71 kPa, p=0.0001). Emax value in the middle parenchyma of the testis showed no statistically significant difference between groups (p=0.053). Emax value in the twisted spermatic segment was higher than that in the lower spermatic segment with acute inflammation (166.61±60.07 vs 14.14±4.93, p=0.0001). In the torsion group, 12 testicular capsule areas showed “stiff ring sign” and all twisted segments of spermatic cord showed “stiff knot sign” but no signs were found in the inflammatory group.Conclusions: “Stiff ring sign” of testis, “stiff knot sign” of spermatic cord, high stiffness of the testicular capsule and in the twisted spermatic segment are the typical SWE findings of testicular torsion, with important clinical value in the differential diagnosis of testicular torsion and acute orchiditis.

Torsion of rational elliptic curves over different types of cubic fields

Let [Formula: see text] be an elliptic curve defined over [Formula: see text], and let [Formula: see text] be the torsion group [Formula: see text] for some cubic field [Formula: see text] which does not occur over [Formula: see text]. In this paper, we determine over which types of cubic number fields (cyclic cubic, non-Galois totally real cubic, complex cubic or pure cubic) [Formula: see text] can occur, and if so, whether it can occur infinitely often or not. Moreover, if it occurs, we provide elliptic curves [Formula: see text] together with cubic fields [Formula: see text] so that [Formula: see text].

On growth of generalized Grigorchuk's overgroups

Grigorchuk's Overgroup G˜, is a branch group of intermediate growth. It contains the first Grigorchuk's torsion group G of intermediate growth constructed in 1980, but also has elements of infinite order. Its growth is substantially greater than the growth of G. The group G, corresponding to the sequence (012)∞=012012…, is a member of the family {Gω|ω∈Ω={0,1,2}N} consisting of groups of intermediate growth when sequence ω is not eventually constant. Following this construction we define the family {G˜ω,ω∈Ω} of generalized overgroups. Then G˜=G˜(012)∞ and Gω is a subgroup of G˜ω for each ω∈Ω. We prove, if ω is eventually constant, then G˜ω is of polynomial growth and if ω is not eventually constant, then G˜ω is of intermediate growth.