We consider solutions of the KP hierarchy which are elliptic functions $x=t_1$. It is known that their poles as $t_2$ move particles Calogero-Moser model. extend this correspondence to level hierarchies and find Hamiltonian $H_k$ model governs dynamics with respect $k$-th hierarchical time. The Hamiltonians obtained coefficients expansion spectral curve near marked point in Baker-Akhiezer funct...

Abstract Starting from the 3-wave interaction equations in 2+1 dimensions (i.e., two space and one time dimension), we complexify independent variables, thus doubling number of real hence work 4+2 dimensions: $$x_1$$ x 1 , $$x_2$$ 2 $$y_1$$ y</m...

We establish an asymptotic formula for the number of lattice points in sets $$\begin{aligned} {\textbf{S}}_{h_1, h_2, h_3}(\lambda ): =\left\{ x\in {\mathbb {Z}}_+^3:\lfloor h_1(x_1)\rfloor +\lfloor h_2(x_2)\rfloor h_3(x_3)\rfloor =\lambda \right\} \quad \text {with}\quad \lambda \in {Z}}_+; \end{aligned}$$ where functions $$h_1, h_3$$ are constant multiples regularly varying form $$h(x):=x^c\e...

Recently Galashin, Grinberg, and Liu introduced the refined dual stable Grothendieck polynomials, which are symmetric functions in $x=(x_1,x_2,\dots)$ with additional parameters $t=(t_1,t_2,\dots)$. The polynomials defined as a generating function for reverse plane partitions of given shape. They interpolate between Schur by Lam Pylyavskyy 2007. Flagged more version where lower upper bounds ent...

In this paper, we study the separtion axioms $T_0,T_1,T_2$ and $T_{5/2}$ on topological and semitopological residuated lattices and we show that they are equivalent on topological residuated lattices. Then we prove that for every infinite cardinal number $alpha$, there exists at least one nontrivial Hausdorff topological residuated lattice of cardinality $alpha$. In the follows, we obtain some ...

Petroleum fluid has been extensively studied at low magnetic fields by Nuclear Magnetic Resonance (NMR) Spectroscopy, but high field NMR studies are rarely found in this area. The aim of study is to determine the proton spin-lattice relaxation rate (1/T1), T1 relaxivity (R1), spin-spin (1/T2) and T2 (R2) paraffinic CH2 gamma CH3 peaks. For purpose, crude oil samples were taken from 3 separate w...

There has been considerable recent progress in identifying candidate materials for the transverse-field Ising chain (TFIC), a paradigmatic model quantum criticality. Here, we study local spin dynamical structure factor of different components disordered region TFIC. We show that low-frequency dynamics spins Ising- and directions have strikingly distinctive temperature dependencies. This leads t...

We prove that given a measure preserving system \begin{document}$ (X,\mathcal{B},\mu,T_1,\dots, T_d) $\end{document} with commuting, ergodic transformations id="M2">\begin{document}$ T_i such id="M3">\begin{document}$ T_iT_j^{-1} are for all id="M4">\begin{document}$ i \neq j $\end{document}</i...

The $k$-token graph $T_k(G)$ is the whose vertices are $k$-subsets of a $G$, with two adjacent if their symmetric difference an edge $G$. We explore when well-covered graph, that is, all its maximal independent sets have same cardinality. For bipartite graphs we classify well-covered. arbitrary show $T_2(G)$ well-covered, then girth $G$ at most four. include upper and lower bounds on independen...