The Collatz tree as a Hilbert hotel:a proof of the 3x + 1 conjecture

The Collatz conjecture maintains that each natural number $n$ is a node in the Collatz tree with a root path to $1$ generated by iterations of the Collatz function, which connects even $n$ to $n/2$ and odd $n$ to $3n+1$. The conjecture holds if a binary subtree $T_{\ge 0}$ exists with as nodes $\textit{all}$ branching numbers constituting the congruence classes ${\left[4,16\right]}_{18}$ with density $2/18$. For each branching number, the $\textit{upward}$ and $\textit{rightward}$ function generate arrows to two branching numbers in the directions $2n$ and $(n-1)/3$. In the automorphism graph of tree $T_{\ge 0}$. The iterates and conjugates of these two functions build outer paths from tree $T_{\ge 0}$ at its root node to two infinite sequences of nodes containing nested subtrees and disjoint cotrees. The nodes and conjugate inner arrows within each subtree and within each cotree correspond one-to-one to the nodes and arrows within tree $T_{\ge 0}$. The cumulative density of the congruence classes of numbers in disjoint cotrees of $T_{\ge 0}$ amounts to the density $2/18$ of all branching numbers, thus proving the conjecture.Comment: Collatz paths between branching classes $[4,16]_18$ and partitioning of the binary tree $T_{ge 0}$ into disjoint cotrees now validate in v5 the density proofs in previous versions (v1 31 Aug 2000

The Detrimental Effects of Oxytocin-Induced Conformity on Dishonesty in Competition

Justifications may promote unethical behavior because they constitute a convenient loophole through which people can gain from immoral behavior and preserve a positive self-image at the same time. A justification that is widely used is rooted in conformity: Unethical choices become more permissible because one's peers are expected to make the same unethical choices. In the current study, we tested whether an exogenous alteration of conformity led to a lower inclination to adhere to a widely accepted norm (i.e., honesty) under the pressure of competition. We took advantage of the well-known effects of intranasally applied oxytocin on affiliation, in-group conformity, and in-group favoritism in humans. We found that conformity was enhanced by oxytocin, and this enhancement had a detrimental effect on honesty in a competitive environment but not in a noncompetitive environment. Our findings contribute to recent evidence showing that competition may lead to unethical behavior and erode moral values

Psoriasis and 5HT-R2C Gene Polymorphism: Association between Clinical, Demographic and Therapeutic Parameters in the Turkish Population

We aimed to investigate the relationship between single nucleotide polymorphism (SNP) in the promoter region of the 5-HT-R2C gene and stress-related disease psoriasis in the Turkish population. The putative association between the 5-HTR2C variant (rs6318 Cys23Ser allele) and patients with psoriasis was investigated. 100 patients with psoriasis and 100 age-sex matched, unrelated healthy subjects representing the control group were included in the study. The PCR-RFLP method was used for genotyping the 5-HTR2C variation. There was no statistically difference in terms of genotype distributions and allele frequencies between the control subjects and patients with psoriasis (P=0.360 and P=0.439, respectively). The comparison between the presence and absence of the 5-HTR2C gene rs6318 G allele within the determined clinical subsets resulted in a significant difference with regard to treatment methodology only when conventional therapy and one or more medical therapy was compared (P=0.021). This study is the first clinical study to investigate the association between 5-HTR2C polymorphism and psoriasis. The role of the 5-HTR2C gene should be examined with more parameters in a larger case series

A comparison among PCNL, Miniperc and Ultraminiperc for lower calyceal stones between 1 and 2 cm: A prospective, comparative, multicenter and randomised study

Background: Conventional Percutaneous Lithotripsy (PCNL) has been an effective, successful and easy approach for especially > 1 cm sized calyceal stones however risks of complications and nephron loss are inevitable. Our aim is to compare the efficacy and safety of PCNL, MiniPerc (MP) and UltraMiniPerc (UMP) for lower calyceal stones between 1 and 2 cm with a multicenter prospective randomized study. Methods: Between January 2015 and June 2018, 132 consecutive patients with single lower calyceal stone were enrolled. Patients were randomized in three groups; A: PCNL; B: MP; C: UMP. 44 patients for the Group A, 47 for Group B and 41 for Group C. Exclusion criterias were the presence of coagulation impairments, age of < 18 or > 75, presence of infection or serious comorbidities. Patients were controlled with computerized tomography scan after 3 months. A negative CT or an asymptomatic patient with stone fragments < 3 mm size were the criteria to assess the stone-free status. Patient characteristics, stone free rates (SFR) s, complications and re-treatment rates were analyzed. Results: The mean stone size were 16.38, 16.82 and 15.23 mm respectively in Group A, B and C(p = 0.34). The overall SFR was significantly higher in Group A (86.3%) and B (82.9%) as compared to Group C (78%)(p < 0.05). The re-treatment rate was significantly higher in Group C (12.1%) and complication rates was higher in Group A (13.6%) as compared to others(p < 0.05). The hospitalization was significantly shorter in Group C compared to Group A (p = 0.04). Conclusions: PCNL and MP showed higher efficacy than UMP to obtain a better SFR. Auxiliary and re-treatment rates were higher in UMP. On the other hand for such this kind of stones PCNL had more complications. Overall evaluation favors MP as a better indication in stones 1-2 cm size

A prospective multicenter randomized comparison between Holmium Laser Enucleation of the Prostate (HoLEP) and Thulium Laser Enucleation of the Prostate (ThuLEP)

Purpose: To compare intra and perioperative parameters between HoLEP and ThuLEP in the treatment of benign prostatic hyperplasia and to evaluate clinical and functional outcomes of the two procedures with a 12-month follow-up. Methods: A prospective randomized study was performed on 236 consecutive patients who underwent ThuLEP (n = 115), or HoLEP (n = 121) in three different centers. Intra and perioperative parameters were analyzed: operative time, enucleated tissue weight, irrigation volume, blood loss, catheterization time, hospital stay and complications. Patients were evaluated preoperatively and 3 and 12 months postoperatively with the international prostate symptom score (IPSS), the quality of life (QoL) score, post-void residual volume (PVR), PSA and maximum flow rate (Qmax). Results: Preoperative variables in each study arm did not show any significant difference. Compared to HoLEP, ThuLEP showed similar operative time (63.69 vs 71.66 min, p = 0.245), enucleated tissue weight (48.84 vs 51.13 g, p = 0.321), catheterization time (1.9 vs 2.0 days, p = 0.450) and hospital stay (2.2 vs 2.8 days, p = 0.216), but resulted in less haemoglobin decrease (0.45 vs 2.77 g/dL, p = 0.005). HoLEP presented a significantly higher number of patients with postoperative acute urinary retention and stress incontinence. No significant differences were found in PSA, Qmax, PVR, IPSS and QoL score during follow-up. Conclusion: ThuLEP and HoLEP both relieved lower urinary tract symptoms equally, with high efficacy and safety. ThuLEP detemined reduced blood loss and early postoperative complications. Catheterization time, enucleated tissue, hospital stay, operative time and follow-up parameters did not show any significant difference

Global warming shifts the composition of the abundant bacterial phyllosphere microbiota as indicated by a cultivation dependent and independent study of the grassland phyllosphere of a long-term warming field-experiment

Plant science