We give a description of non-growing subsets in linear groups, which extends the Product theorem for simple groups of Lie type. We also give an account of various related aspects of growth in linear groups.

There are several results in the literature concerning p-groups G with a maximal elementary abelian normal subgroup of rank k due to Thompson, Mann and others. Following an idea of Sambale we obtain bounds for the number of generators etc. of a 2-group G in terms of k, which were previously known only for p>2. We also prove a theorem that is new even for odd primes. Namely, we show that if G has a maximal elementary abelian normal subgroup of rank k, then for any abelian subgroup A the Frattini subgroup Φ(A) can be generated by 2k elements (3k when p=2). The proof of this rests upon the following result of independent interest: If V is an n-dimensional vector space, then any commutative subalgebra of End(V) contains a zero algebra of codimension at most n.

Liebeck, Nikolov, and Shalev conjectured that for every subset A of a finite simple group S with |A|>1, there exist O( log|S| / log|A| ) conjugates of A whose product is S. This paper is a companion to [Lifshitz: Completing the proof of the Liebeck-Nikolov-Shalev conjecture] and together they prove the conjecture. In this paper we prove the conjecture in the regime where |A|>|S|^c for an absolute constant c>0. We also prove that the following Skew Product Theorem holds for all finite simple groups. Namely we show that either the product of two conjugates of A has size at least |A|^1.49, or S is the product of boundedly many conjugates of A.

We list four types of planar curves such that arrangements of their translates are (locally) combinatorially equivalent to an arrangement of lines. We find them by characterising diffeomorphisms ϕ:R2→R2 and continuous curves C⊂R2 such that ϕ(t+C) is a line for all t∈R2. There are exactly five maps satisfying (at least locally) this condition. Two of them define the same curve, so we have four different curves. These can be used to define norms giving constructions with Ω(n4/3) unit distances among n points in the plane.

Planar point sets with many triple lines (which contain at least three distinct points of the set) have been studied for 180 years, started with Jackson and followed by Sylvester. Green and Tao (Discret Comput Geom 50(2):409–468, 2013) have shown that the maximum possible number of triple lines for an n -element set is ⌊ n(n - 3)/6⌋ + 1 . Here we address the related problem of describing the structure of the asymptotically near-optimal configurations, i.e., of those for which the number of straight lines which go through three or more points has a quadratic (i.e., best possible) order of magnitude. We pose the problem whether such point sets must always be related to cubic curves. To support this conjecture we settle various special cases; some of them are also related to the four-in-a-line problem of Erdős.

Artificial intelligence (AI) has the potential to improve breast cancer screening; however, prospective evidence of the safe implementation of AI into real clinical practice is limited. A commercially available AI system was implemented as an additional reader to standard double reading to flag cases for further arbitration review among screened women. Performance was assessed prospectively in three phases: a single-center pilot rollout, a wider multicenter pilot rollout and a full live rollout. The results showed that, compared to double reading, implementing the AI-assisted additional-reader process could achieve 0.7–1.6 additional cancer detection per 1,000 cases, with 0.16–0.30% additional recalls, 0–0.23% unnecessary recalls and a 0.1–1.9% increase in positive predictive value (PPV) after 7–11% additional human reads of AI-flagged cases (equating to 4–6% additional overall reading workload). The majority of cancerous cases detected by the AI-assisted additional-reader process were invasive (83.3%) and small-sized (≤10 mm, 47.0%). This evaluation suggests that using AI as an additional reader can improve the early detection of breast cancer with relevant prognostic features, with minimal to no unnecessary recalls. Although the AI-assisted additional-reader workflow requires additional reads, the higher PPV suggests that it can increase screening effectiveness.

Around twenty years ago Ghys conjectured that finite subgroups of the diffeomorphism group of a compact smooth manifold M have an abelian normal subgroup of index at most a(M), where a(M) depends only on M. First we construct a family of counterexamples to this conjecture including, for example, the product space T^2× S^2. Following the first appearance of our counterexample on the arXiv Ghys put forward a revised conjecture, which predicts only the existence of a nilpotent normal subgroup of index at most n(M). Our main result is the proof of the revised Ghys conjecture. More generally, we show that the same result holds for homeomorphism groups of not necessarily compact topological manifolds with finitely generated homology groups. Our proofs are based on finite group theoretic results which provide a general strategy for proving similar Jordan-type theorems.

We show that there are five types of planar curves such that arrangements of its translates are combinatorially equivalent to an arrangement of lines. These curves can be used to define norms giving constructions with many unit distances among points in the plane.

Background: After the outbreak of the corona virus disease-19 (COVID-19) pandemic, teledermatology was implemented in the Hungarian public healthcare system for the first time. Our objective was to assess aggregated diagnostic agreements and to determine the effectiveness of an asynchronous teledermatology system for skin cancer screening. Methods: This retrospective single-center study included cases submitted for teledermatology consultation during the first wave of the COVID-19 pandemic. Follow-up of the patients was performed to collect the results of any subsequent personal examination. Results: 749 patients with 779 lesions were involved. 15 malignant melanomas (9.9%), 78 basal cell carcinomas (51.3%), 21 squamous cell carcinomas (13.8%), 7 other malignancies (4.6%) and 31 actinic keratoses (20.4%) were confirmed. 87 malignancies were diagnosed in the high-urgency group (42.2%), 49 malignancies in the moderate-urgency group (21.6%) and 16 malignancies in the low-urgency group (4.6%) (p < 0.0001). Agreement of malignancies was substantial for primary (86.3%; κ = 0.647) and aggregated diagnoses (85.3%; κ = 0.644). Agreement of total lesions was also substantial for primary (81.2%; κ = 0.769) and aggregated diagnoses (87.9%; κ = 0.754). Conclusions: Our findings showed that asynchronous teledermatology using a mobile phone application served as an accurate skin cancer screening system during the first wave of the COVID-19 pandemic.

If a finite p-group G acts continuously on a compact topological manifold M then, with some bound C depending on M alone, G has a subgroup H of index at most C such that the H-action on M has at most C stabilizer subgroups. This result plays a crucial role in the proof of a deep conjecture of Ghys.

Összefoglaló. Bevezetés: A térdízületnek ultrafriss osteochondralis allograft segítségével történő részleges ortopédiai rekonstrukciója képalkotó vizsgálatokon alapuló pontos tervezést igényel, mely folyamatban a morfológia felismerésére képes mesterséges intelligencia nagy segítséget jelenthet. Célkitűzés: Jelen kutatásunk célja a porc morfológiájának MR-felvételen történő felismerésére alkalmas mesterséges intelligencia kifejlesztése volt. Módszer: A feladatra legalkalmasabb MR-szekvencia meghatározása és 180 térd-MR-felvétel elkészítése után a mesterséges intelligencia tanításához manuálisan és félautomata szegmentálási módszerrel bejelölt porckontúrokkal tréninghalmazt hoztunk létre. A mély convolutiós neuralis hálózaton alapuló mesterséges intelligenciát ezekkel az adatokkal tanítottuk be. Eredmények: Munkánk eredménye, hogy a mesterséges intelligencia képes a meghatározott szekvenciájú MR-felvételen a porcnak a műtéti tervezéshez szükséges pontosságú bejelölésére, mely az első lépés a gép által végzett műtéti tervezés felé. Következtetés: A választott technológia – a mesterséges intelligencia – alkalmasnak tűnik a porc geometriájával kapcsolatos feladatok megoldására, ami széles körű alkalmazási lehetőséget teremt az ízületi terápiában. Orv Hetil. 2021; 162(9): 352–360. Summary. Introduction: The partial orthopedic reconstruction of the knee joint with an osteochondral allograft requires precise planning based on medical imaging reliant; an artificial intelligence capable of determining the morphology of the cartilage tissue can be of great help in such a planning. Objective: We aimed to develop and train an artificial intelligence capable of determining the cartilage morphology in a knee joint based on an MR image. Method: After having determined the most appropriate MR sequence to use for this project and having acquired 180 knee MR images, we created the training set for the artificial intelligence by manually and semi-automatically segmenting the contours of the cartilage in the images. We then trained the neural network with this dataset. Results: As a result of our work, the artificial intelligence is capable to determine the morphology of the cartilage tissue in the MR image to a level of accuracy that is sufficient for surgery planning, therefore we have made the first step towards machine-planned surgeries. Conclusion: The selected technology – artificial intelligence – seems capable of solving tasks related to cartilage geometry, creating a wide range of application opportunities in joint therapy. Orv Hetil. 2021; 162(9): 352–360.

Let $$\mathbb R(C)$$ be the function field of a smooth, irreducible projective curve over $$\mathbb R$$. Let X be a smooth, projective, geometrically irreducible variety equipped with a dominant morphism f onto a smooth projective rational variety with a smooth generic fibre over $$\mathbb R(C)$$. Assume that the cohomological obstruction introduced by Colliot-Thélène is the only one to the local-global principle for rational points for the smooth fibres of f over $$\mathbb R(C)$$-valued points. Then we show that the same holds for X, too, by adopting the fibration method similarly to Harpaz–Wittenberg.

We prove an incidence theorem for points and curves in the complex plane. Given a set of m points in ℝ 2 and a set of n curves with k degrees of freedom, Pach and Sharir proved that the number of point-curve incidences is O( m^k/2k - 1n^2k - 2/2k - 1 + m + n) . We establish the slightly weaker bound O_ε( m^k/2k - 1 + εn^2k - 2/2k - 1 + m + n) on the number of incidences between m points and n (complex) algebraic curves in ℂ 2 with k degrees of freedom. We combine tools from algebraic geometry and differential geometry to prove a key technical lemma that controls the number of complex curves that can be contained inside a real hypersurface. This lemma may be of independent interest to other researchers proving incidence theorems over ℂ.

A 2009/2010-es időmérleg adatai alapján megvizsgáljuk a háztartásban végzett nem fizetett munka, illetve az e munka révén előállított javak és szolgáltatások fogyasztásának koreloszlásait nemek szerint. Kis leegyszerűsítéssel elvégezzük a háztartásgazdaságot leíró háztartási szatellitszámla elsődleges demográfiai elemzését. Eredményeink a férfiak és nők közötti munkamegosztás jól dokumentált különbségeit pontosítják. A női korprofilra sokkal erőteljesebben jellemző a kétcsúcspontú eloszlás. A nemek közötti különbség a gyermeknevelési életkorokban a legnagyobb, de minden 10 évesnél idősebb korcsoportban jelen van. Mindezt figyelembe véve, a háztartáson belüli időtranszferek igazi nyertesei nem a nemek, hanem a korcsoportok közötti újraelosztásban keresendők. A 30 és 53 év közötti férfiak nettó nyújtói és nem kedvezményezettjei az időtranszfereknek. Egy átlagos 54 év feletti férfi 1,3 havi keresetnek megfelelő értékű időtranszfert kap – egy átlagos gyermek azonban 0 és 17 év között 5,6 havi keresetnyit. A háztartásgazdaság nem csak nemek között, hanem életkorok között mozgat transzfereket.

A basic problem with the ever-changing Hungarian retirement rules has been that they created excessive shares of gainers and of losers. Certain workers with long (and continuous) employment could retire well below the normal retirement age (NRA) with full benefit. Other workers, with fragmented and therefore short employment had to work until reaching the ever rising NRA. A peculiar consequence of these rules is the strong negative correlation between the retirement age and the length of contribution. Moving in the direction of a fair system, like the Nonfinancial Defined Contribution system, would improve sustainability and fairness.

A tanulmany a nyugdijrendszer azon ket szabalyaval foglalkozik, amely meghatarozza, hogyan novekszik a kezdőnyugdij: 1. a befizetett jarulektol es 2. a nyugdijba vonulasi eletkortol fuggően. Vizmertekul az eszmei szamla szolgalhat, amely egy eletbiztositasi- eletjaradekos keny szer meg ta kari tasi rendszert utanoz. Altalanosabban rugalmas korhatarrol beszelunk, ha mind a szolgalati idő, mind a nyugdijba vonulasi kor emelese jelentősen noveli a nyugdijjaradekot, s ezzel osztonoz a nyugdijba vonulas idejenek rugalmas megvalasztasara. Hibas elkepzelesek miatt ez a megkozelites Magyarorszagon csak nagyon torz formaban ervenyesul. Korabban egy engedekeny szabaly szerint az atlag folotti szolgalati idő eseten az előrehozott nyugdijba vonulas nem csokkentette a nyugdijat. Jelenleg ket kiegeszitő szabalyt kell kiemelnunk a) a kivetelező: 2011 ota a Nők40 kereteben minden nő, akinek a jogviszonya eleri a 40 evet, teljes előrehozott nyugdijat kap, b) a merevitő: 2012 ota az a) kategorian kivul senki sem mehet a korhatar alatt nyugdijba - meg csokkentett előrehozott nyugdijjal sem. Bevonva a keresetek es a nyugdij kozotti kapcsolatot megteremtő szabalyt is, a rugalmas esethez kepest ertekeljuk a korabbi es a jelenlegi szabalyok melletti nyerteseket es veszteseket.* Journal of Economic Literature (JEL) kod: H55, I14, J26.

We prove that if L is a finite simple group of Lie type and A a symmetric set of generators of L, then A grows i.e |AAA| > |A|^(1+epsilon) where epsilon depends only on the Lie rank of L, or AAA=L. This implies that for a family of simple groups L of Lie type the diameter of any Cayley graph is polylogarithmic in |L|. Combining our result on growth with known results of Bourgain,Gamburd and Varj\'u it follows that if LAMBDA is a Zariski-dense subgroup of SL(d,Z) generated by a finite symmetric set S, then for square-free moduli m which are relatively prime to some number m_0 the Cayley graphs Gamma(SL(d,m),pi_m(S)) form an expander family.

A tanulmany a nyugdijrendszer azon ket szabalyaval foglalkozik, amely meghatarozza, hogyan novekszik a kezdőnyugdij: 1. a befizetett jarulektol es 2. a nyugdijba vonulasi eletkortol fuggően. Vizmertekul az eszmei szamla szolgalhat, amely egy eletbiztositasi- eletjaradekos keny szer meg ta kari tasi rendszert utanoz. Altalanosabban rugalmas korhatarrol beszelunk, ha mind a szolgalati idő, mind a nyugdijba vonulasi kor emelese jelentősen noveli a nyugdijjaradekot, s ezzel osztonoz a nyugdijba vonulas idejenek rugalmas megvalasztasara. Hibas elkepzelesek miatt ez a megkozelites Magyarorszagon csak nagyon torz formaban ervenyesul. Korabban egy engedekeny szabaly szerint az atlag folotti szolgalati idő eseten az előrehozott nyugdijba vonulas nem csokkentette a nyugdijat. Jelenleg ket kiegeszitő szabalyt kell kiemelnunk a) a kivetelező: 2011 ota a Nők40 kereteben minden nő, akinek a jogviszonya eleri a 40 evet, teljes előrehozott nyugdijat kap, b) a merevitő: 2012 ota az a) kategorian kivul senki sem mehet a korhatar alatt nyugdijba - meg csokkentett előrehozott nyugdijjal sem. Bevonva a keresetek es a nyugdij kozotti kapcsolatot megteremtő szabalyt is, a rugalmas esethez kepest ertekeljuk a korabbi es a jelenlegi szabalyok melletti nyerteseket es veszteseket.* Journal of Economic Literature (JEL) kod: H55, I14, J26.

Though the Hungarian pension system has been suffering from many erroneous rules, in the present paper we confine our attention to the rules of retirement in Hungary since 1990. In every pension system, there exist two rules which determine how the lifetime contribution (which is approximately proportional to the years of contributions) and the retirement age influence the benefit amount, respectively. As a benchmark, we use the system of nonfinancial defined contributions, simulating a mandatory life insurance and life annuity system. More generally, we speak of flexible retirement if adding a year to the contributions or the retirement age strongly increases the retirement benefit, opening the way to the flexible choice of the retirement age. Due to erroneous concepts, flexibility has only functioned in a very imperfect form in Hungary. Before 2011/2012, an exemption rule completed the two foregoing rules: if somebody had above the critical value (35–40) of years of contributions, he/she could use early retirement without significant benefit reduction. Since 2011/2012 two other rules have completed these rules: (a) as an exception, since 2011, rule Females 40 has rewarded any woman who had at least 40 years of rights with a full benefit; (b) as a rigid rule, since 2012, nobody could have retired before reaching the statutory retirement age except for category (a). Taking into account the dependence of monthly benefits on the lifetime average valorized wages, we assess the gainers and losers of the past and the present systems.