Locally finite profinite rings

We investigate the structure of locally finite profinite rings. We classify (Jacobson-) semisimple locally finite profinite rings as products of complete matrix rings of bounded cardinality over finite fields, and we prove that the Jacobson radical of any locally finite profinite ring is nil of finite nilexponent. Our results apply to the context of small compact $G$-rings, where we also obtain a description of possible actions of $G$ on the underlying ring.Comment: 17 page

Around Podewski's conjecture

A long-standing conjecture of Podewski states that every minimal field is algebraically closed. It was proved by Wagner for fields of positive characteristic, but it remains wide open in the zero-characteristic case. We reduce Podewski's conjecture to the case of fields having a definable (in the pure field structure), well partial order with an infinite chain, and we conjecture that such fields do not exist. Then we support this conjecture by showing that there is no minimal field interpreting a linear order in a specific way; in our terminology, there is no almost linear, minimal field. On the other hand, we give an example of an almost linear, minimal group $(M,<,+,0)$ of exponent 2, and we show that each almost linear, minimal group is elementary abelian of prime exponent. On the other hand, we give an example of an almost linear, minimal group $(M,<,+,0)$ of exponent 2, and we show that each almost linear, minimal group is torsion.Comment: 16 page

Generalized locally compact models for approximate groups

We give a proof of the existence of generalized definable locally compact models for arbitrary approximate subgroups via an application of topological dynamics in model theory. Our construction is simpler and shorter than the original one obtained by Hrushovski in ``Beyond the Lascar group'', and it uses only basic model theory (mostly spaces of types and realizations of types). The main tools are Ellis groups from topological dynamics considered for suitable spaces of types. However, we need to redevelop some basic theory of topological dynamics for suitable ``locally compact flows'' in place of (compact) flows. We also prove that the generalized definable locally compact model which we constructed is universal in an appropriate category. We note that the main result yields structural information on definable generic subsets of definable groups, with a more precise structural result for generics in the universal cover of $\textrm{SL}_2(\mathbb{R})$

Maximal stable quotients of invariant types in NIP theories

For a NIP theory $T$, a sufficiently saturated model $\mathfrak{C}$ of $T$, and an invariant (over some small subset of $\mathfrak{C}$) global type $p$, we prove that there exists a finest relatively type-definable over a small set of parameters from $\mathfrak{C}$ equivalence relation on the set of realizations of $p$ which has stable quotient. This is a counterpart for equivalence relations of the main result of the paper "On maximal stable quotients of definable groups in NIP theories" by M. Haskel and A. Pillay which shows the existence of maximal stable quotients of type-definable groups in NIP theories. Our proof adapts the ideas of the proof of this result, working with relatively type-definable subsets of the group of automorphisms of the monster model as defined in the paper "On first order amenability" by E. Hrushovski, K. Krupinski, and A. Pillay

Amenability, connected components, and definable actions

We study amenability of definable groups and topological groups, and prove various results, briefly described below. Among our main technical tools, of interest in its own right, is an elaboration on and strengthening of the Massicot-Wagner version of the stabilizer theorem, and also some results about measures and measure-like functions (which we call means and pre-means). As an application we show that if $G$ is an amenable topological group, then the Bohr compactification of $G$ coincides with a certain ``weak Bohr compactification'' introduced in [24]. In other words, the conclusion says that certain connected components of $G$ coincide: $G^{00}_{topo} = G^{000}_{topo}$. We also prove wide generalizations of this result, implying in particular its extension to a ``definable-topological'' context, confirming the main conjectures from [24]. We also introduce $\bigvee$-definable group topologies on a given $\emptyset$-definable group $G$ (including group topologies induced by type-definable subgroups as well as uniformly definable group topologies), and prove that the existence of a mean on the lattice of closed, type-definable subsets of $G$ implies (under some assumption) that $cl(G^{00}_M) = cl(G^{000}_M)$ for any model $M$. Thirdly, we give an example of a $\emptyset$-definable approximate subgroup $X$ in a saturated extension of the group $\mathbb{F}_2 \times \mathbb{Z}$ in a suitable language (where $\mathbb{F}_2$ is the free group in 2-generators) for which the $\bigvee$-definable group $H:=\langle X \rangle$ contains no type-definable subgroup of bounded index. This refutes a conjecture by Wagner and shows that the Massicot-Wagner approach to prove that a locally compact (and in consequence also Lie) ``model'' exists for each approximate subgroup does not work in general (they proved in [29] that it works for definably amenable approximate subgroups).Comment: Version 3 contains the material in Sections 2, 3, and 5 of version 1. Following the advice of editors and referees we have divided version 1 into two papers, version 3 being the first. The second paper is entitled "On first order amenability

Peripheral blood concentrations of vascular endothelial growth factor and its soluble receptors (R1 and R2) in patients with adrenal cortex tumours treated by surgery

Wstęp: Neoangiogeneza należy do kluczowych mechanizmów patologicznych w przebiegu choroby nowotworowej gruczołów dokrewnych, w tym kory nadnerczy. Naczyniowo-śródbłonkowy czynnik wzrostu (VEGF, vascular endothelial growth factor), po aktywacji specyficznych receptorów w komórkach endothelium, wykazuje działanie angiogenne, mitogenne i zwiększa przepuszczalność ścian naczyń krwionośnych. Celem pracy było zbadanie stężeń VEGF, sVEGFR1 i sVEGFR2 we krwi obwodowej u chorych z guzami kory nadnerczy o charakterze łagodnym i złośliwym poddanych adrenalektomii. Materiał i metody: Przed leczeniem operacyjnym zbadano krew u 41 pacjentów z guzami kory nadnerczy oraz u 10 osób zdrowych bez zmian hormonalnych i obrazowych (USG/CT) nadnerczy (grupa kontrolna). Po adrenalektomii zbadano ponownie krew u 16 chorych. Wyniki: Stężenia VEGF, VEGFR1 i VEGFR2 zbadano w osoczu krwi przed i po 30 dniach od operacji metodą ELISA. Przed operacją stężenia VEGF we krwi nie różniły się pomiędzy całą grupą pacjentów z guzami kory nadnerczy a grupą kontrolną. Po leczeniu chirurgicznym średnie stężenia VEGF zmniejszyły się w całej grupie operowanych chorych i w podgrupie z gruczolakami kory. Stężenia VEGF R1 przed operacją były wyższe tylko w grupie chorych z zespołem Conna, a po adrenalektomii obniżyły się tyko w podgrupie osób z gruczolakami kory. Stężenia VEGFR2 nie różniły się pomiędzy wszystkimi badanymi grupami oraz przed i po operacji. Wnioski: W praktyce klinicznej oznaczanie stężeń VEGF, VEGFR1 i VEGFR2 we krwi obwodowej u chorych z nowotworami nadnerczy nie pozwala na rozpoznanie guzów kory nadnerczy o charakterze złośliwym.Introduction: Neoangiogenesis appears to be an important event in tumour invasion and in the formation of metastases in many endocrine-related human cancers. Vascular endothelial growth factor (VEGF) is a glycoprotein with potent angiogenic, mitogenic and vascular permeability-enhancing activities specific for endothelial cells and acts through VEGF receptors. The aim of the study was to evaluate the plasma blood concentrations of VEGF, sVEGFR1, and sVEGFR2 in patients with benign and malignant adrenal tumours treated by surgery. Material and methods: We studied the blood before surgery of 41 patients with adrenal cortex tumours and 10 normal subjects without hormonal or CT/USG pathology of the adrenal glands (controls). We studied the blood after adrenalectomy of 16 patients with tumours of the adrenal cortex. Results: Concentrations of VEGF, sVEGFR1 and sVEGFR2 in blood plasma before as well as 30 days after surgery were evaluated by ELISA. VEGF blood concentrations before surgery did not differ in the patients with the cortical tumours as compared to the controls. After surgery VEGF concentrations decreased among the patients, taken in total, with adrenal cortex tumours and cortical adenomas. Before surgery sVEGFR1 blood concentrations were increased in the patients with Conn&#8217;s syndrome only in comparison with the controls. After surgery, sVEGFR1 concentrations decreased significantly in the group with cortical adenomas only. Before and after surgery sVEGFR2 blood concentrations did not differ between the groups of patients studied and the controls. Conclusions: Peripheral blood concentrations of VEGF and its receptors cannot be clinically valuable markers that discriminate between benign and malignant adrenocortical tumours before and after adrenalectomy