arXiv subjects
Publications and source records attributed to E. Paal.
By using the discrete Poisson equations the star-triangle (external) equivalence in conducting networks is considered and the Kennelly famous transformation formulae [Kennelly A E 1899 Electrical World and Engineer 34, 413] are explicitly restated.
Operadic Lax representations for the harmonic oscillator are used to construct the quantum counterparts of some 3d real Lie algebras in Bianchi classification. The Jacobians of these quantum algebras are studied. It is conjectured that the tangent algebras of these quantum algebras are the Heisenberg algebra. From this it follows that the volume element in $\mathbb{R}^{3}$ is quantized by $|(x,y,z)|=4\sqrt{2}(2n+1)$, ($n=0,1,2,\dots$). Thus, the elementary (minimal) length in this model is $l_{min}=2^{5/6}$.
Operadic Lax representations for the harmonic oscillator are used to construct the quantum counterparts of some real three dimensional Lie algebras. The Jacobi operators of these quantum algebras are studied in semiclassical approximation.
Operadic Lax representations for the harmonic oscillator are used to construct the quantum counterparts of three-dimensional real Lie algebras. The Jacobi operators of these quantum algebras are explicitly calculated.
Operadic Lax representations for the harmonic oscillator are used to construct the quantum counterparts of 3d real Lie algebras. The Jacobians of the corresponding quantum algebras are calculated.
It is clarified how cohomologies and Gerstenhaber algebras can be associated with linear pre-operads (comp algebras). Their relation to mechanics and operadic physics is concisely discussed.
The deformation equation and its integrability condition (Bianchi identity) of a non-associative deformation in operad algebra are found. Their relation to the theory of gravity is discussed.
The essential parts of the operad algebra are concisely presented, which should be useful when confronting with the operadic physics. It is also clarified how the Gerstenhaber algebras can be associated with the linear pre-operads (comp algebras). Their relation to mechanics is concisely discussed. A hypothesis that the Feynman diagrams are observables is proposed.
We consider basic algebraic constructions associated with an abstract pre-operad, such as a $\smile$-algebra, total composition $\bul$, pre-coboundary operator $\de$ and tribraces $\{\cdot,\cdot,\cdot\}$. A derivation deviation of the pre-coboundary operator over the tribraces is calculated in terms of the $\smile$-multiplication and total composition.
We consider basic algebraic constructions associated with an abstract pre-operad, such as a $\smile$-algebra, total composition $\bul$, pre-coboundary operator $\de$, tribraces $\{\cdot,\cdot,\cdot\}$ and tetrabraces $\{\cdot,\cdot,\cdot,\cdot\}$. A derivation deviation of the pre-coboundary operator over the tetrabraces is calculated in terms of the $\smile$-multiplication and tribraces.
In 1963 [Ann. of Math. {\bf 78}, 267-288], Gerstenhaber invented a \emph{comp(osition)} calculus in the Hochschild complex of an associative algebra. In this paper, the first steps of the Gerstenhaber theory are exposed in an abstract (comp system) setting. In particular, as in the Hochschild complex, a graded Lie algebra and a pre-coboundary operator can be associated to every comp system. A derivation deviation of the pre-coboundary operator over the total composition is calculated in two ways, (the long) one of which is essentially new and can be seen as an example and elaboration of the auxiliary variables method proposed by Gerstenhaber in the early days of the comp calculus.