R-matrices in Rime

We replace the ice Ansatz on matrix solutions of the Yang-Baxter equation by a weaker condition which we call "rime". Rime solutions include the standard Drinfeld-Jimbo R-matrix. Solutions of the Yang--Baxter equation within the rime Ansatz which are maximally different from the standard one we call "strict rime". A strict rime non-unitary solution is parameterized by a projective vector. We show that this solution transforms to the Cremmer-Gervais R-matrix by a change of basis with a matrix containing symmetric functions in the components of the parameterizing vector. A strict unitary solution (the rime Ansatz is well adapted for taking a unitary limit) is shown to be equivalent to a quantization of a classical "boundary" r-matrix of Gerstenhaber and Giaquinto. We analyze the structure of the elementary rime blocks and find, as a by-product, that all non-standard R-matrices of GL(1|1)-type can be uniformly described in a rime form. We discuss then connections of the classical rime solutions with the Bezout operators. The Bezout operators satisfy the (non-)homogeneous associative classical Yang--Baxter equation which is related to the Rota-Baxter operators. We classify the rime Poisson brackets: they form a 3-dimensional pencil. A normal form of each individual member of the pencil depends on the discriminant of a certain quadratic polynomial. We also classify orderable quadratic rime associative algebras. For the standard Drinfeld-Jimbo solution, there is a choice of the multiparameters, for which it can be non-trivially rimed. However, not every Belavin-Drinfeld triple admits a choice of the multiparameters for which it can be rimed. We give a minimal example.Comment: 50 pages, typos correcte

Characteristic Angles in the Wetting of an Angular Region: Surface Shape

The shape of a liquid surface bounded by an acute or obtuse planar angular sector is considered by using classical analysis methods. For acute angular sectors the two principal curvatures are of the order of the (fixed) mean curvature. But for obtuse sectors, the principal curvatures both diverge as the vertex is approached. The power-law divergence becomes stronger with increasing opening angle. Possible implications of this contrasting behavior are suggested.Comment: 19 pages, 9 figures, LaTeX; submitted to The European Physics Journal E; v2: Introduction was revised (a number of references added), minor changes to the main part (mostly typos), former Implications subsection was almost entirely rewritten and is now called Experimental Realizations (experimental results and two figures added); v3: Introduction was slightly modified, four references added; v4: Title was modified, section Calculation was significantly modified (subsections Bounary Problem and Horizontal Solution almost entirely rewritten, minor changes to the other subsections), subsection Curvature in section Discussion was revised, one reference adde

Surgical treatment of the perihilar cholangiocarcinoma with portal vein invasion

Background. Perichilar cholangiocarcinoma is a rare type of malignant neoplasm and is 3-7 cases per 100,000 population. Surgical method is the only radical method of treatment, allowing to improve long-term survival results. One of the important and characteristic features of perihilar cholangiocarcinoma is tumor invasion to the area of the portal vein bifurcation, which occurs in 30–45% of cases. Portal vein invasion is the one of the main causes of perihilar cholangiocarcinoma irresectability. However, innovative surgical technologies allow resection of the liver with resection and reconstruction of the portal vein with acceptable mortality. The aim. The aim of our study was to asses results of surgical treatment of perihilar cholangiocarcinoma with (Group 1) and without (Group 2) portal vein invasion. Materials and methods. From 2003 to January 2023 in the Department of Surgery and Liver Transplantation of the Ukrainian National Institute of Surgery and Transplantation, 208 patients with perihilar cholangiocarcinoma underwent major extended liver resections. We compared 93 (46%) patients who received extended liver resection with portal vein resection (Group 1) with 115 (54%) patients who underwent liver resections without vascular reconstructions (Group 2). The average Ca 19–9 in the group 1 was 288 (8 – 1000) U/ml, in the group 2 –262 (10 – 612) U/ml. The level of total bilirubin in patients of the group 1 was 312 (43 – 621) mcmol/l, in the group 2 – 267 (10 – 612) mcmol/l. In view of this, in the preoperative period, 190 (91,3%) patients underwent decompression of the bile ducts, using percutaneous transhepatic cholangiostomy (PTBD) or retrograde endobiliary stenting. For patients with small remnant liver volume less than 40 %, in 80(38,5%) cases we did preoperative PVE of a resected part of the liver. In 9 cases we made simultaneous PVE and PTBD. When choosing the volume of surgical intervention, we proceeded from the tumor type of Bismuth-Corlette classification, invasion into the portal vessels and the depth of the liver lesion. The portal vein reconstruction was in all cases performed in an “end-to-end”. In all cases we made extended lymphadenectomy. Results. All complications were classified according to the Dindo-Clavien classification. Postoperative mortality in the main group was 11.5%. The overall 1, 3, 5-year survival in the group 1 was 96%, 68,3%, 57,4%, respectively. 1, 3, 5-year survival rate in the comparison group 2 was 98,4%, 76,7%, 47,3%, respectively. Conclusions. Aggressive tactics of surgical treatment of perihilar cholangiocarcinoma provides maximum radicality, allows to increase resectability in case of tumor invasion of the portal vein with acceptable mortality and long-term survival

Bidirectional imperfect quantum teleportation with a single Bell state

We present a bidirectional modification of the standard one-qubit teleportation protocol, where both Alice and Bob transfer noisy versions of their qubit states to each other by using single Bell state and auxiliary (trigger) qubits. Three schemes are considered: the first where the actions of parties are governed by two independent quantum random triggers, the second with single random trigger, and the third as a mixture of the first two. We calculate the fidelities of teleportation for all schemes and find a condition on correlation between trigger qubits in the mixed scheme which allows us to overcome the classical fidelity boundary of 2/3. We apply the Choi-Jamiolkowski isomorphism to the quantum channels obtained in order to investigate an interplay between their ability to transfer the information, entanglement-breaking property, and auxiliary classical communication needed to form correlations between trigger qubits. The suggested scheme for bidirectional teleportation can be realized by using current experimental tools.Comment: 8 pages, 4 figures; published versio

Empirical evaluation of accuracy of mathematical software used for availability assessment of fault-tolerant computer systems

Dependability assessment is typically based on complex probabilistic models. Markov and semi-Markov models are widely used to model dependability of complex hardware/software architectures. Solving such models, especially when they are stiff, is not trivial and is usually done using sophisticated mathematical software packages. We report a practical experience of comparing the accuracy of solutions stiff Markov models obtained using well known commercial and research software packages. The study is conducted on a contrived but realistic cases study of computer system with hardware redundancy and diverse software under the assumptions that the rate of failure of software may vary over time, a realistic assumption. We observe that the disagreement between the solutions obtained with the different packages may be very significant. We discuss these findings and directions for future research

Phase transition for the frog model

We study a system of simple random walks on graphs, known as frog model. This model can be described as follows: There are active and sleeping particles living on some graph G. Each active particle performs a simple random walk with discrete time and at each moment it may disappear with probability 1-p. When an active particle hits a sleeping particle, the latter becomes active. Phase transition results and asymptotic values for critical parameters are presented for Z^d and regular trees