Topics on n-ary algebras

We describe the basic properties of two n-ary algebras, the Generalized Lie Algebras (GLAs) and, particularly, the Filippov (or n-Lie) algebras (FAs), and comment on their n-ary Poisson counterparts, the Generalized Poisson (GP) and Nambu-Poisson (N-P) structures. We describe the Filippov algebra cohomology relevant for the central extensions and infinitesimal deformations of FAs. It is seen that semisimple FAs do not admit central extensions and, moreover, that they are rigid. This extends the familiar Whitehead's lemma to all $n\geq 2$ FAs, n=2 being the standard Lie algebra case. When the n-bracket of the FAs is no longer required to be fully skewsymmetric one is led to the n-Leibniz (or Loday's) algebra structure. Using that FAs are a particular case of n-Leibniz algebras, those with an anticommutative n-bracket, we study the class of n-Leibniz deformations of simple FAs that retain the skewsymmetry for the first n-1 entires of the n-Leibniz bracket.Comment: 11 page

Cohomology of Filippov algebras and an analogue of Whitehead's lemma

We show that two cohomological properties of semisimple Lie algebras also hold for Filippov (n-Lie) algebras, namely, that semisimple n-Lie algebras do not admit non-trivial central extensions and that they are rigid i.e., cannot be deformed in Gerstenhaber sense. This result is the analogue of Whitehead's Lemma for Filippov algebras. A few comments about the n-Leibniz algebras case are made at the end.Comment: plain latex, no figures, 29 page

Cartan-Weyl 3-algebras and the BLG Theory I: Classification of Cartan-Weyl 3-algebras

As Lie algebras of compact connected Lie groups, semisimple Lie algebras have wide applications in the description of continuous symmetries of physical systems. Mathematically, semisimple Lie algebra admits a Cartan-Weyl basis of generators which consists of a Cartan subalgebra of mutually commuting generators H_I and a number of step generators E^\alpha that are characterized by a root space of non-degenerate one-forms \alpha. This simple decomposition in terms of the root space allows for a complete classification of semisimple Lie algebras. In this paper, we introduce the analogous concept of a Cartan-Weyl Lie 3-algebra. We analyze their structure and obtain a complete classification of them. Many known examples of metric Lie 3-algebras (e.g. the Lorentzian 3-algebras) are special cases of the Cartan-Weyl 3-algebras. Due to their elegant and simple structure, we speculate that Cartan-Weyl 3-algebras may be useful for describing some kinds of generalized symmetries. As an application, we consider their use in the Bagger-Lambert-Gustavsson (BLG) theory.Comment: LaTeX. 34 pages.v2. deleted some distracting paragraphs in the introduction to bring more out the main results of the paper. typos corrected and references adde

Experimental investigation of the deepening of the combustion front into peat layers different in botanical composition

The deepening of the center of combustion into peat layers of different botanical compositions (pine-cotton grass and grass-sphagnum peats), typical for the Tomsk region, was investigated experimentally. Peats were ignited from a model ground forest fire initiated by firing of a needle-litter layer. As a result of laboratory investigations, the change in the temperature in the bulk of peat samples with time was determined and analyzed, and the rates of their combustion in the horizontal and vertical directions were estimated. It was established that a fire penetrates deep into a layer of grass-sphagnum peat, containing more than 70% of combustion conductors in its composition, more rapidly as compared to that of pine-cotton grass peat. The rates of combustion of grass-sphagnum peat in the vertical and horizontal directions are larger by 20 and 22%, respectively, than those of pine-cotton grass peat, which is evidently due to the botanical composition of grass-sphagnum peat and the random arrangement of components in its layers

Experimental investigation of the deepening of the combustion front into peat layers different in botanical composition

The deepening of the center of combustion into peat layers of different botanical compositions (pine-cotton grass and grass-sphagnum peats), typical for the Tomsk region, was investigated experimentally. Peats were ignited from a model ground forest fire initiated by firing of a needle-litter layer. As a result of laboratory investigations, the change in the temperature in the bulk of peat samples with time was determined and analyzed, and the rates of their combustion in the horizontal and vertical directions were estimated. It was established that a fire penetrates deep into a layer of grass-sphagnum peat, containing more than 70% of combustion conductors in its composition, more rapidly as compared to that of pine-cotton grass peat. The rates of combustion of grass-sphagnum peat in the vertical and horizontal directions are larger by 20 and 22%, respectively, than those of pine-cotton grass peat, which is evidently due to the botanical composition of grass-sphagnum peat and the random arrangement of components in its layers

Simulation of fuel bed ignition by wildland firebrands

A 3-D mathematical model of fuel bed (FB) ignition initiated by glowing firebrands originating during wildland fires is proposed. In order to test and verify the model, a series of experiments was conducted to determine the FB ignition time by a single pine bark and twig firebrand (Pinus sylvestris). Irrespective of the pine bark sample sizes and experimental conditions, the ignition of the FB was not observed. Conversely, pine twigs, under certain parameters, ignited the FB in the range of densities (60–105 kg m−3) and with the airflow velocity of ≥2 m s−1. The results of the mathematical modelling have shown that a single pine bark firebrand ≤5 cm long with a temperature of T ≤ 1073 K does not ignite in the flaming mode the FB, and only the thermal energy of larger particles is sufficient for flaming ignition of the adjacent layers of the FB. The analysis of the results has shown that the firebrand length is a major factor in the initiation of ignition. Comparison of the calculated and observed FB ignition times by a single firebrand have shown that our modelling accords well with the experimental results

Pelvic floor reconstruction after total exenteration for radiation injuries of the pelvic organs (clinical case)

The objective is to present the clinical case of pelvic floor reconstruction after total exenteration.Clinical case. The formation of the combined fistula after surgical treatment and combined radiation therapy for uterine body cancer is presented in female patient (57 years old). The rectum and bladder were removed, the perineal tissue was excised and a vulvectomy was performed. The rectus abdominis muscle was mobilized with epigastric artery and moved to the perineum. The wound was healed by primary tension. The patient lives a full life, adapted, and does not require analgesic therapy.Conclusion. The treatment and diagnosis tactics for regional combined fistulas are not standardized, and surgical treatment is the method of choice, especially in cases where the patient has already had radiation therapy for malignancy and relapse