Ropelength of tight polygonal knots

A physical interpretation of the rope simulated by the SONO algorithm is presented. Properties of the tight polygonal knots delivered by the algorithm are analyzed. An algorithm for bounding the ropelength of a smooth inscribed knot is shown. Two ways of calculating the ropelength of tight polygonal knots are compared. An analytical calculation performed for a model knot shows that an appropriately weighted average should provide a good estimation of the minimum ropelength for relatively small numbers of edges.Comment: 27 pages, to appear in "Physical and Numerical Models in Knot Theory and their Application to the Life Sciences

Continuous unitary transformation approach to the Kondo-Majorana interplay

We analyze a setup composed of a correlated quantum dot (QD) coupled to one metallic lead and one end of topological chain hosting a Majorana zero mode (MZM). In such a hybrid structure, a leakage of the MZM into the region of the QD competes with the Kondo resonance appearing as a consequence of the spin-exchange interactions between the dot and the lead. In the work, we use the nontrivial technique called the continuous unitary transformation (CUT) to analyze this competition. Using the CUT technique, we inspect the influence of the coupling between the QD and the chain on effective exchange interactions and calculate the resultant Kondo temperature.Comment: 6 Pages, 5 figure

Length of the tightest trefoil knot

The physical sense of tight knots provided by the SONO algorithm is discussed. A method allowing one to predict their length is presented. An upper bound for the minimum length of a smooth trefoil knot is determined

Mangiferin and Hesperidin Transdermal Distribution and Permeability through the Skin from Solutions and Honeybush Extracts (Cyclopia sp.)—A Comparison Ex Vivo Study

Polyphenolic compounds—mangiferin and hesperidin—are, among others, the most important secondary metabolites of African shrub Cyclopia sp. (honeybush). The aim of this study was to compare the percutaneous absorption of mangiferin and hesperidin from solutions (water, ethanol 50%, (v/v)) and extracts obtained from green and fermented honeybush (water, ethanol 50%, (v/v)). Research was performed with the Bronaugh cells, on human dorsal skin. The mangiferin and hesperidin distributions in skin layers (stratum corneum, epidermis, and dermis) and in acceptor fluid (in every 2, 4, 6, and 24 h) were evaluated by HPLC–Photodiode Array Coulometric and Coulometric Electrochemical Array Detection. The transdermal distribution of hesperidin was also demonstrated by fluorescence microscopy. Results indicated that mangiferin and hesperidin were able to cross the stratum corneum and penetrate into the epidermis and dermis. An advantage of hesperidin penetration into the skin from the water over ethanol solution was observed (451.02 ± 14.50 vs. 357.39 ± 4.51 ng/cm2), as well as in the mangiferin study (127.56 ± 9.49 vs. 97.23 ± 2.92 ng/cm2). Furthermore, mangiferin penetration was more evident from nonfermented honeybush ethanol extract (189.85 ± 4.11 ng/cm2) than from solutions. The permeation of mangiferin and hesperidin through the skin to the acceptor fluid was observed regardless of whether the solution or the honeybush extract was applied. The highest ability to permeate the skin was demonstrated for the water solution of hesperidin (250.92 ± 16.01 ng/cm2), while the hesperidin occurring in the extracts permeated in a very low capacity. Mangiferin from nonfermented honeybush ethanol extract had the highest ability to permeate to the acceptor fluid within 24 h (152.36 ± 8.57 ng/cm2)