End Of Primer Research Articles

On the discrete boundary extension of mappings in terms of prime ends

On the discrete boundary extension of mappings in terms of prime ends

Parameterized family of annular homeomorphisms with pseudo-circle attractors

In this paper we construct a paramaterized family of annular homeomorphisms with Birkhoff-like rotational attractors that vary continuously with the parameter, are all homeomorphic to a unique topological object, called the R.H. Bing's pseudo-circle, yet display an interesting boundary dynamics. Namely, in the constructed family of homeomorphisms the outer prime ends rotation number vary continuously with the parameter through the interval [0,1/2]. This, in particular, answers a question from Boroński et al. (2020) [15]. Furthermore, these attractors preserve the induced Lebesgue measure from the circle and have strong measure-theoretic and statistical properties. To show main results of the paper we first prove a result of an independent interest, that Lebesgue-measure preserving circle maps generically satisfy the crookedness condition which implies that generically the inverse limits of Lebesgue measure-preserving circle maps are the pseudo-solenoids. For degree one circle maps, this implies that the generic inverse limit in this context is the pseudo-circle.

On prime end distortion estimates of mappings with the Poletsky condition in domains with the Poincar´e inequality

This article is devoted to the study of mappings with bounded andfinite distortion defined in some domain of the Euclidean space. Weconsider mappings that satisfy some upper estimates for thedistortion of the modulus of families of paths, where the order ofthe modulus equals to $p,$ $n-1<p\leqslant n.$ The main problemstudied in the manuscript is the investigation of the boundarybehavior of such mappings, more precisely, the distortion of thedistance under mappings near boundary points. The publication isprimarily devoted to definition domains with ``bad boundaries'', inwhich the mappings not even have a continuous extension to theboundary in the Euclidean sense. However, we introduce the conceptof a quasiconformal regular domain in which the specified continuousextension is valid and the corresponding distance distortionestimates are satisfied; however, both must be understood in thesense of the so-called prime ends. More precisely, such estimateshold in the case when the mapping acts from a quasiconformal regulardomain to an Ahlfors regular domain with the Poincar\'e inequality.The consideration of domains that are Ahlfors regular and satisfythe Poincar\'e inequality is due to the fact that, lower estimates forthe modulus of families of paths through the diameter of thecorresponding sets hold in these domains. (There are the so-calledLoewner-type estimates). We consider homeomorphisms and mappingswith branching separately. The main analytical condition under whichthe results of the paper were obtained is the finiteness of theintegral averages of some majorant involved in the defining modulusinequality under infinitesimal balls. This condition includes thesituation of quasiconformal and quasiregular mappings, because forthem the specified majorant is itself bounded in a definitiondomain. Also, the results of the article are valid for more generalclasses for which Poletsky-type upper moduli inequalities aresatisfied, for example, for mappings with finite length distortion.

On the Equicontinuity of Generalized Quasiconformal Mappings by Prime Ends

We study mappings satisfying the inverse Poletsky type inequality with respect to a non-conformal modulus. It was prove that such mappings, with some additional conditions, have a continuous extension to the boundary in terms of prime ends and form equicontinuous families of mappings.

On the application of one modulus inequality to the mapping theory

The authors study mappings that satisfy some estimate of the distortion of the modulus of families of paths. Under certain conditions on the domains between which the mappings act, we established that, these mappings are Hölder logarithmic continuous at the boundary points. It is known that, the Hölder continuity is established for many classes of mappings, say quasiconformal and quasiregular mappings. In this regard, it is possible to point to the classical distortion estimates by Martio-Rickman-Väisälä type, as well as the estimates related to the modern classes of mappings with finite distortion. In particular, V.I. Ryazanov together with R.R. Salimov and E.O. Sevost'yanov established local distortion estimates for plane and spatial mappings under FMO condition, or under the Lehto-type integral condition. Recently, the second co-author have obtained Hölder logarithmic continuity for the studied class at points of the unit sphere. This article considers the situation of similar mappings of different domains, not only the unit sphere. Namely, we consider mappings between quasiextremal distance domains (QED-domains) and convex domains. Note that, quasiextremal distance domains introduced by Gehring and Martio are structures in which the modulus of families of paths is metrically related to the diameter of sets. Also, convex domains are involved in the formulation of the main result; we consider mappings that surjectively act onto them. In addition, the article contains the formulations and proofs for some other results on this topic. We consider several more cases in detail, in particular when: 1) the definition domain is a domain with a locally quasiconformal boundary, and the image domain is a bounded convex domain; 2) the definition domain is a regular domain in the sense of prime ends, and the image domain is a bounded convex domain; 3) the mapping acts between the QED-domain and the bounded convex domain and has a fixed point. In all three cases, the mapping is Hölder logarithmic continuous; moreover, in case 2), which refers to prime ends, logarithmic continuity should also be understood in terms of prime ends.

On the boundary extension of mappings on Riemannian surfaces in terms of prime ends

On the boundary extension of mappings on Riemannian surfaces in terms of prime ends

On the boundary extension of mappings on Riemannian surfaces in terms of prime ends

On the boundary extension of mappings on Riemannian surfaces in terms of prime ends

Prime ends dynamics on invariant Peano continua

The dynamics of a planar homeomorphism h is simple on any non-separating Peano continuum K that is invariant under h. This means that all limit sets on K are either fixed points or periodic orbits. The map h induces a homeomorphism h⁎ on the space of prime ends associated to K. The goal of this paper is to show that in some cases the dynamics on prime ends can have a certain complexity. We construct a dissipative homeomorphism with attractor K and h⁎ a Denjoy map.

On the Inverse Poletsky Inequality in Metric Spaces and Prime Ends

On the Inverse Poletsky Inequality in Metric Spaces and Prime Ends

On compact classes of solutions of Dirichlet problem in simply connected domains

The article is devoted tocompactness of solutions of the Dirichlet problem for the Beltramiequation in some simply connected domain. In terms of prime ends, wehave proved corresponding results for the case when the maximaldilatations of these solutions satisfy certain integral constraints.The first section is devoted to a presentation of well-knowndefinitions that are necessary for the formulation of the mainresults. In particular, here we have given a definition of a primeend corresponding to N\"{a}kki's concept. The research tool that wasused to establish the main results is the method of moduli forfamilies of paths. In this regard, in the second section we studymappings that satisfy upper bounds for the distortion of themodulus, and in the third section, similar lower bounds. The mainresults of these two sections include the equicontinuity of thefamilies of mappings indicated above, which is obtained underintegral restrictions on those characteristics. The proof of themain theorem is done in the fourth section and is based on thewell-known Stoilow factorization theorem. According to this, an opendiscrete solution of the Dirichlet problem for the Beltrami equationis a composition of some homeomorphism and an analytic function. Inturn, the family of these homeomorphisms is equicontinuous(Section~2). At the same time, the equicontinuity of the family ofcorresponding analytic functions in composition with some(auxiliary) homeomorphisms reduces to using the Schwartz formula, aswell as the equicontinuity of the family of corresponding inversehomeomorphisms (Section~3).

On Equicontinuity of the Families of Mappings with One Normalization Condition in Terms of Prime Ends

On Equicontinuity of the Families of Mappings with One Normalization Condition in Terms of Prime Ends

Unraveling the miRnome of Nicotiana rustica (Aztec tobacco) - A Genomewide computational assessment

MicroRNAs (miRNAs) are endogenous small non-coding RNAs, known as chief regulators for cellular growth and development in plants. Nicotiana rustica (N. rustica), also known as Aztec tobacco is the second most widely used lucrative crop for tobacco production across the globe. Since the miRNAs of Nicotiana rustica have never been reported, we employed a genome-wide computational approach which resulted in 101 potential candidates of miRNAs, belonging to 73 families, followed by stringent filtration criteria. Statistical analysis confirmed the occurrence of uracil as the dominant initial biased nucleotide base at 5 prime ends of mature miRNAs that may lead to a significant role in miRNA biogenesis and or miRNA mediated gene regulation. Due to unavailability of target sequences, the RNAseq data of N. rustica leaf, flower, stem, and root were retrieved, assembled de novo and concatenated into a clustered customized transcript dataset. The study validates miR160, miR393, miR397, miR403 & miR529 miRNAs and confirmed their targets through experimentally proven literature evidence. These miRNAs can modulate the biological role in transcription regulation, defense response, transporter activity, hormonal signaling and different protein kinase activities. Collectively, this study reports putative miRNAs and their role in Aztec tobacco development and stress response.

On equicontinuity of families of mappings between Riemannian surfaces with respect to prime ends

Given a domain of some Riemannian surface,we consider questions related to the possibility of a continuousextension to the boundary of one class of Sobolev mappings. It isproved that such maps have a continuous boundary extension in termsof prime ends, and under some additional restrictions their familiesare equicontinuous at inner and boundary points of the domain. Wehave separately considered the cases of homeomorphisms and mappingswith branching.

P347: COOPERATION OF TLX3 AND FLT3-ITD IN T-CELL ACUTE LYMPHOBLASTIC LEUKEMIA IS ENHANCED BY TLE4 INACTIVATION

Background: One fifth of T-cell acute lymphoblastic leukemia (T-ALL) cases is defined by the aberrant expression of TLX3. Being similar to the closely related TLX1, its expression is mainly associated with mutations in the JAK/STAT pathway (Liu et al., Nat Genetics 2017), which have been shown to cooperate with TLX1 in driving T-ALL (Vanden Bempt et al., Cancer Cell 2018). However, it is not known whether TLX3 has the same oncogenic role as TLX1. Additionally, TLX3 expression is also regularly found with FLT3-ITD mutations, which are virtually absent in TLX1-positive cases (Liu et al., Nat Genetics 2017), suggesting there are some important functional differences between TLX3 and TLX1. Aims: To set up a model of cooperation between TLX3 and FLT3-ITD and to elucidate the oncogenic role of TLX3 in T-ALL. Methods: We performed in vivo bone marrow transplant (BMT) assays by transducing lineage-negative cells derived from C57BL/6 mice with the respective oncogenes using retroviruses. Pro-T cell growth curves were performed in the absence of interleukin-7 (Il7) as described previously (Van Thillo et al., Nat Communications 2021). Fluorescence activated cell sorting (FACS) staining was performed on the MACSQuant VYB (Miltenyi Biotec) or Fortessa (BD). RNA was prepared at the KU Leuven Genomics Core and 3’ prime end sequencing (QuantSeq) was performed. Differential gene expression analyses were conducted using the DeSeq2 R package (v1.22.0). Results: The combination of TLX3 and FLT3-ITD in a constitutive BMT assay resulted in a rapid leukemia with an immature myeloid immunophenotype (CD11b+, Gr1-, CD4-, CD8-). The leukemia cells were positive for both FLT3-ITD (GFP) and TLX3 (mCHERRY), indicating cooperation between the two oncogenes in vivo. Similarly, in our in vitro pro-T-cell model, the TLX3 and FLT3-ITD double-positive cells outcompeted the single-positive cells after withdrawal of Il7. Next, we performed RNA-sequencing in CreER (estrogen receptor) pro-T cells 24 hours after activation of TLX3 expression by adding tamoxifen. This showed both up- and downregulation of many genes, including Tle4 among the downregulated genes. Interestingly, TLE4 expression was also lowest in the TLX3-subgroup of patients (Liu. et al. Nat Genetics 2017). TLE proteins are transcriptional co-repressors that can interact with the engrailed-homology 1 (Eh1) domain of transcription factors (Riz et al. Biochem Biophys Res Commun 2009). Therefore, we inserted a point mutation in the Eh1 domain of TLX3, resulting in a phenylalanine to glutamic acid substitution at position 18, to impair this interaction. Strikingly, TLX3(F18E) in combination with FLT3-ITD led to a proliferative advantage in pro-T cells after the omission of Il7 compared to wild-type TLX3. On top of that, cells with TLX3(F18E) and FLT3-ITD were able to grow in the absence of stem cell factor (Scf). Summary/Conclusion: We demonstrate that TLX3 cooperates with FLT3-ITD in vivo and in vitro in a pro-T-cell context. TLX3 downregulates the co-repressor Tle4 and inhibiting the interaction between TLX3 and Tle4 reinforces the proliferative capacity of pro-T cells in the absence of growth factors.

Parametrized family of pseudo-arc attractors: Physical measures and prime end rotations.

The main goal of this paper is to study topological and measure‐theoretic properties of an intriguing family of strange planar attractors. Building toward these results, we first show that any generic Lebesgue measure‐preserving map f generates the pseudo‐arc as inverse limit with f as a single bonding map. These maps can be realized as attractors of disc homeomorphisms in such a way that the attractors vary continuously (in Hausdorff distance on the disc) with the change of bonding map as a parameter. Furthermore, for generic Lebesgue measure‐preserving maps f the background Oxtoby–Ulam measures induced by Lebesgue measure for f on the interval are physical on the disc and in addition there is a dense set of maps f defining a unique physical measure. Moreover, the family of physical measures on the attractors varies continuously in the weak* topology; that is, the parametrized family is statistically stable. We also find an arc in the generic Lebesgue measure‐preserving set of maps and construct a family of disk homeomorphisms parametrized by this arc which induces a continuously varying family of pseudo‐arc attractors with prime ends rotation numbers varying continuously in [0,1/2]. It follows that there are uncountably many dynamically non‐equivalent embeddings of the pseudo‐arc in this family of attractors.

Evaluation of the Simple Sequence Repeats to Work as a Phase Variation with the Neisseria meningitidis Genome

Background: The most crucial mechanism of genetic variation in N. meningitidis is the slipped strand mispairing, this mechanism generates Phase variation using simple sequence repeat (SSR) and is commonly used by the N. meningitidis to escape the immune system despite its function in eradicating the pathogenic and commensal bacteria. Some of simple sequence repeats (SSRs) that located within the genome works as phase variation while other SSRs have no role in generating phase variation mechanisms. Therefore,
 Aim: the main goal of the current in silico study was to detect the probability of SSR to enroll with phase variation for the entire N. meningitidis genome.
 Methods: Different criteria were used to judge SSR as it works in phase variation and these criteria were taken from the current literature. These criteria involve the Z score value of the synonymous shuffling model and Markov model of SSR, the position of SSR in the gene or the promoter, instability and polymorphism of SSR in genomes of different strains, and the length of SSR.
 Results: The positive Z score value of SSR, SSR being variable in length among genomes of different strains and SSR location in 3 prime end of a gene or in the promoter indicates that the SSR generates phase variation in a particular gene.
 Conclusion: 67 out of 327 putative phase variable genes located on the N. meningitidis genome were determined to fit these criteria. We assume, therefore, that SSR in these genes may be connected with phase variation mechanism. We recommend that experimental evidence should be generated to confirm these findings.

Boundary Extensions of Mappings Satisfying the Poletsky Inverse Inequality in Terms of Prime Ends

Boundary Extensions of Mappings Satisfying the Poletsky Inverse Inequality in Terms of Prime Ends

On the global behavior of inverse mappings in terms of prime ends

The paper is devoted to the study of mappings with finite distortion, actively studied recently. For mappings whose inverse satisfy the Poletsky inequality, the results on boundary behavior in terms of prime ends are obtained. In particular, it was proved that the families of the indicated mappings are equicontinuous at the points of the boundary if a certain function determining the distortion of the modulus of families of paths under the mappings is integrable.

Caratheodory theorem about prime ends on Riemann surfaces

The present paper is a continuation of our research that was devoted to the theory of the boundary behavior of mappings on Riemann surfaces. Here we develop the theory of the boundary behavior of the mappings in the class FLD (mappings with finite length distortion) first introduced for the Euclidean spaces in the article of Martio--Ryazanov--Srebro--Yakubov at 2004 and then included in the known monograph of these authors in the modern mapping theory at 2009. As it was shown in the recent papers of Kovtonyuk-Petkov-Ryazanov at 2017, such mappings, generally speaking, are not mappings in the Sobolev classes because their first partial derivatives can be not locally integrable. At the same time, this class is a natural generalization of the well-known significant classes of isometries and quasi--isometries. We obtain here a series of criteria in terms of dilatations for the homeomorphic extension of the mappings with finite length distortion between domains on Riemann surfaces to the completions of the domains by prime ends of Caratheodory. Here we start from the general criterion in Lemma 1 in terms of singular functional parameters and then derive on this basis many other criteria. In particular, Lemma 1 implies Theorem 1 with a criterion of the Lehto type and Corollary 1 shows that the conclusion holds, if the dilatation grows not quickly than logarithm of the hyperbolic distance at every boundary point. The next consequence in Theorem 2 gives an integral criterion of the Orlicz type and Corollary 2 says on simple integral conditions of the exponential type. Further, Theorem 3 and Remark 2 contain criteria in terms of singular integrals of the Calderon--Zygmund type. The other criterion in Theorem 4 is the existence of a dominant for the dilatation in the class FMO (functions with finite mean oscillation), i.e., having a finite mean deviation from its mean value over infinitesimal discs centered at boundary points. In other words, the latter means that such a dominant has a finite dispersion over the given infinitesimal discs. In particular, the latter leads to Corollary 3 on a dominant in the well--known class BMO (bounded mean oscillation) by John--Nirenberg and to a simple criterion in Corollary 4 on finiteness of the average of the dilatation over infinitesimal disks centered at boundary points.

On Dirichlet problem for Beltrami equations in finitely connected domains

Boundary value problems for the Beltrami equations are due to the famous Riemann dissertation (1851) in the simplest case of analytic functions and to the known works of Hilbert (1904, 1924) and Poincare (1910) for the corresponding Cauchy--Riemann system. Of course, the Dirichlet problem was well studied for uniformly elliptic systems, see, e.g., \cite{Boj} and \cite{Vekua}. Moreover, the corresponding results on the Dirichlet problem for degenerate Beltrami equations in the unit disk can be found in the monograph \cite{GRSY}. In our article \cite{KPR1}, see also \cite{KPR3} and \cite{KPR5}, it was shown that each generalized homeomorphic solution of a Beltrami equation is the so-called lower $Q-$homeomorphism with its dilatation quotient as $Q$ and developed on this basis the theory of the boundary behavior of such solutions. In the next papers \cite{KPR2} and \cite{KPR4}, the latter made possible us to solve the Dirichlet problem with continuous boundary data for a wide circle of degenerate Beltrami equations in finitely connected Jordan domains, see also [\citen{KPR5}--\citen{KPR7}]. Similar problems were also investigated in the case of bounded finitely connected domains in terms of prime ends by Caratheodory in the papers [\citen{KPR9}--\citen{KPR10}] and [\citen{P1}--\citen{P2}]. Finally, in the present paper, we prove a series of effective criteria for the existence of pseudo\-re\-gu\-lar and multi-valued solutions of the Dirichlet problem for the degenerate Beltrami equations in arbitrary bounded finitely connected domains in terms of prime ends by Caratheodory.