Invariants of Triangular Lie Algebras

Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular matrices of any finite size over the real/complex number field. In the paper invariants ('generalized Casimir operators') are found for three classes of Lie algebras, namely those which are either strictly or non-strictly triangular, and for so-called special upper triangular Lie algebras. Algebraic algorithm of [J. Phys. A: Math. Gen., 2006, V.39, 5749; math-ph/0602046], developed further in [J. Phys. A: Math. Theor., 2007, V.40, 113; math-ph/0606045], is used to determine the invariants. A conjecture of [J. Phys. A: Math. Gen., 2001, V.34, 9085], concerning the number of independent invariants and their form, is corroborated.Comment: LaTeX2e, 16 pages; misprints are corrected, some proofs are extende

The atomic structure of large-angle grain boundaries Σ5\Sigma 5 and Σ13\Sigma 13 in YBa2Cu3O7−δ{\rm YBa_2Cu_3O_{7-\delta}} and their transport properties

We present the results of a computer simulation of the atomic structures of large-angle symmetrical tilt grain boundaries (GBs) $\Sigma 5$ (misorientation angles \q{36.87}{^{\circ}} and \q{53.13}{^{\circ}}), $\Sigma 13$ (misorientation angles \q{22.62}{^{\circ}} and \q{67.38}{^{\circ}}). The critical strain level $\epsilon_{crit}$ criterion (phenomenological criterion) of Chisholm and Pennycook is applied to the computer simulation data to estimate the thickness of the nonsuperconducting layer ${\rm h_n}$ enveloping the grain boundaries. The ${\rm h_n}$ is estimated also by a bond-valence-sum analysis. We propose that the phenomenological criterion is caused by the change of the bond lengths and valence of atoms in the GB structure on the atomic level. The macro- and micro- approaches become consistent if the $\epsilon_{crit}$ is greater than in earlier papers. It is predicted that the symmetrical tilt GB $\Sigma5$ \theta = \q{53.13}{^{\circ}} should demonstrate a largest critical current across the boundary.Comment: 10 pages, 2 figure

Key exchange with the help of a public ledger

Blockchains and other public ledger structures promise a new way to create globally consistent event logs and other records. We make use of this consistency property to detect and prevent man-in-the-middle attacks in a key exchange such as Diffie-Hellman or ECDH. Essentially, the MitM attack creates an inconsistency in the world views of the two honest parties, and they can detect it with the help of the ledger. Thus, there is no need for prior knowledge or trusted third parties apart from the distributed ledger. To prevent impersonation attacks, we require user interaction. It appears that, in some applications, the required user interaction is reduced in comparison to other user-assisted key-exchange protocols

Analytical method for determining the stationary thermal fields in layered structures

Запропоновано метод розрахунку двовимірних стаціонарних, періодичних по просторовій координаті, теплових полів у багатошарових плитах. На верхній і нижній межах плити температура описується парними періодичними функціями з однаковими періодами. На спільних межах шарів виконується умова неперервності температурного поля і рівність теплових потоків. Шукані температури в кожному із шарів записано у вигляді тригонометричних рядів по косинусах. Для забезпечення виконання умов на спільних межах шарів пропонується модифікація методу матриць податливості. Сформульовано алгоритм розв’язання задачі та показано, що спосіб дає точний розв’язок задачі для будь-якої скінченої кількості шарів. Наведено приклади результатів числових досліджень для різних граничних умов. Проведено порівняльний аналіз і зроблено висновки.The method of two-dimensional thermal stationary fields’ calculation in multilayer plates is proposed. Thermal fields are considered periodical along spatial value. The temperature of the upper and lower limits is described by pair periodic functions with similar periods. Continuity condition of thermal field and thermal flow equality is realized within layer limits. Found temperatures of the layers are expressed in trigonometric series cosines. There are two free constants of differential equations solution about amplitude to every layer and harmonic. The method of compliance matrices is proposed for realizing conditions within layer limits. Two auxiliary sequences are introduced for every layer. These sequences are connected with temperature and thermal flow on the upper layer limit. They realize thermal field distribution within layer. The author proved that the elements of one of these sequences are expressed by the elements of another sequence in this layer, and appropriate coefficient of Fourier series of the lower plate limit. Recurrence relations are built for the coefficients of these dependences. These dependences allow calculating the coefficients in accordance with geometrical and physical properties of the plate’s layers, beginning with the lower one. Algorithm of task solution is stated. The author stresses that if the functions describing the upper and lower plate’s limits spread out into the complete Fourier series, then the proposed method provides accurate task solution for any complete quantity of layers. The main advantage of this method is that its labor coefficient rises slowly with layer growth. The results of numerical experiments show the influence of geometrical and physical parameters on the heat distribution in a two-layer plate. Just shows graphs constructed for different conditions at the external borders of the plate. Influence of heat conductivity factor changing in the middle layer of three-dimensional plate on heat distribution within plate is analyzed. Three-dimensional temperature graphs are built. The conclusion has been drawn

Analytical method for determining the stationary thermal fields in layered structures

Запропоновано метод розрахунку двовимірних стаціонарних, періодичних по просторовій координаті, теплових полів у багатошарових плитах. На верхній і нижній межах плити температура описується парними періодичними функціями з однаковими періодами. На спільних межах шарів виконується умова неперервності температурного поля і рівність теплових потоків. Шукані температури в кожному із шарів записано у вигляді тригонометричних рядів по косинусах. Для забезпечення виконання умов на спільних межах шарів пропонується модифікація методу матриць податливості. Сформульовано алгоритм розв’язання задачі та показано, що спосіб дає точний розв’язок задачі для будь-якої скінченої кількості шарів. Наведено приклади результатів числових досліджень для різних граничних умов. Проведено порівняльний аналіз і зроблено висновки.The method of two-dimensional thermal stationary fields’ calculation in multilayer plates is proposed. Thermal fields are considered periodical along spatial value. The temperature of the upper and lower limits is described by pair periodic functions with similar periods. Continuity condition of thermal field and thermal flow equality is realized within layer limits. Found temperatures of the layers are expressed in trigonometric series cosines. There are two free constants of differential equations solution about amplitude to every layer and harmonic. The method of compliance matrices is proposed for realizing conditions within layer limits. Two auxiliary sequences are introduced for every layer. These sequences are connected with temperature and thermal flow on the upper layer limit. They realize thermal field distribution within layer. The author proved that the elements of one of these sequences are expressed by the elements of another sequence in this layer, and appropriate coefficient of Fourier series of the lower plate limit. Recurrence relations are built for the coefficients of these dependences. These dependences allow calculating the coefficients in accordance with geometrical and physical properties of the plate’s layers, beginning with the lower one. Algorithm of task solution is stated. The author stresses that if the functions describing the upper and lower plate’s limits spread out into the complete Fourier series, then the proposed method provides accurate task solution for any complete quantity of layers. The main advantage of this method is that its labor coefficient rises slowly with layer growth. The results of numerical experiments show the influence of geometrical and physical parameters on the heat distribution in a two-layer plate. Just shows graphs constructed for different conditions at the external borders of the plate. Influence of heat conductivity factor changing in the middle layer of three-dimensional plate on heat distribution within plate is analyzed. Three-dimensional temperature graphs are built. The conclusion has been drawn

All solvable extensions of a class of nilpotent Lie algebras of dimension n and degree of nilpotency n-1

We construct all solvable Lie algebras with a specific n-dimensional nilradical n_(n,2) (of degree of nilpotency (n-1) and with an (n-2)-dimensional maximal Abelian ideal). We find that for given n such a solvable algebra is unique up to isomorphisms. Using the method of moving frames we construct a basis for the Casimir invariants of the nilradical n_(n,2). We also construct a basis for the generalized Casimir invariants of its solvable extension s_(n+1) consisting entirely of rational functions of the chosen invariants of the nilradical.Comment: 19 pages; added references, changes mainly in introduction and conclusions, typos corrected; submitted to J. Phys. A, version to be publishe