On Repdigits as Sums of Fibonacci and Tribonacci Numbers

In this paper, we use Baker’s theory for nonzero linear forms in logarithms of algebraic numbers and a Baker-Davenport reduction procedure to find all repdigits (i.e., numbers with only one distinct digit in its decimal expansion, thus they can be seen as the easiest case of palindromic numbers, which are a ”symmetrical” type of numbers) that can be written in the form Fn+Tn, for some n≥1, where (Fn)n≥0 and (Tn)n≥0 are the sequences of Fibonacci and Tribonacci numbers, respectively.

A direct correlation between commissural orientation and annular shape in bicuspid aortic valves: a new anatomical and computed tomography classification

OBJECTIVES The shape of the aortic annulus is still under debate. Recent findings suggest a possible gradual spectrum of circularity from tricuspid aortic valves (TAVs), to type 1 bicuspid aortic valves (BAVs) to type 0 BAVs. BAVs have been recently classified in a symmetrical (type A), asymmetrical (type B) or very asymmetrical (type C) phenotype according to the commissural orientation (CO) (160°–180°, 140°–159° and 120°–139°, respectively). The aim of this study is to verify in BAVs the correlation between the aortic annular shape and the CO of valve cusps and to suggest a new anatomical and geometric classification of BAVs based on CO and annular shape. METHODS We retrospectively selected 191 consecutive patients who underwent both electrocardiography-gated computed tomography scan of the aortic root and transthoracic echocardiography between January 2016 and June 2019. The population was divided into 2 groups: 54 BAVs and 137 TAVs. We analysed the subgroup of BAV patients and divided them into group A, group B and group C depending on the CO. The shape of the aortic annulus was considered ‘circular’ or ‘elliptic’ according to the ellipticity index (EI).We studied the possible correlation between CO and annular shape in BAVs. We also analysed the subgroup of TAV patients studying their annular shape and EI. RESULTS After univariate linear regression, BAV patients showed a significant correlation between the CO and the EI (R = −0, 445, R2 = 0, 198, P = 0.001). As the CO decreases, the EI increases and approaches an elliptical shape. After grouping BAVs according to the CO, a mean EI of 1.10 ± 0.07 was found in group A, 1.13 ± 0.08 in group B, 1.18 ± 0.07 in group C, P = 0.0097 indicating a gradual spectrum of ellipticity with the decrease of CO. TAVs subgroup showed a mean EI of 1.27 ± 0.09, suggesting that the more the CO is reduced in BAVs, the more the annulus probably tends towards the very elliptical shape of TAVs. CONCLUSIONS This study shows a linear correlation between CO and annular shape in BAVs. In particular, the aortic annulus follows a continuous spectrum of ellipticity depending on the CO. These findings lay the groundwork for a new anatomical classification of BAVs based on CO and annular shape.

Suprascapular Notch Asymmetry: A Study on 311 Patients

The most important risk factor of suprascapular nerve entrapment is probably the shape of the suprascapular notch (SSN). The aim of the study was to perform a radiological study of the symmetry of SSN. Included in the study were 311 patients (137 women and 174 men) who underwent standard computed tomography investigation of the chest. A total of 622 computed tomography scans of scapulae were retrospectively analyzed to classify suprascapular notches into five types. Suprascapular notch was recognized as a symmetrical feature in 53.45% of the patients. Symmetry was more frequently seen in females (54.0% versus 52.9%), but not to any significant degree (P=0.8413). Type III was the most commonly noted symmetrical feature (66.9%) and type II was less common (0.6%). Type III was the most symmetrical type of suprascapular notch, occurring significantly more often as a symmetrical feature in comparison with type I (P<0.0001), type II (P=0.00137), or type IV (P=0.001). Our investigation did not show that the suprascapular notch is a symmetrical feature. However, symmetry was recognized more frequently in the case of type III SSN. No significant differences in symmetry were found with regard to sex.

Regiospecific synthesis of lanthanum(III) and neodymium(III) triple-decker (tetrakis-meso-(3-bromophenyl)-porphyrinato)(crownphthalocyaninates)

The triple-decker complexes of symmetrical type [m Br 4 TPP ] Ln [(15 C 5)4 Pc ] Ln [m Br 4 TPP ] ( Ln -m TD ; Ln = La , Nd ; [(15 C 5)4 Pc ] = tetra-(15-crown-5)-phthalocyaninato-ligand; [m Br 4 TPP ] = tetrakis-meso-(3-bromophenyl)-porphyrinato-ligand) are synthesized for the first time with 43% and 36% yield. The applicability of the previously developed selective one-step synthetic approach for the preparation of the mentioned complexes is demonstrated and the limitations, determined by meta-substitution of meso-aryl-fragments, are revealed. The spectral features of the obtained complexes is determined by means of UV-vis and NMR spectroscopy. The hindered rotation of porphyrin meso-substituents in the obtained complexes and formation of statistical set of atropisomers are demonstrated.