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The symmetry approach to the classification of evolution integrable partial differential equations (see, for example~\cite{MikShaSok91}) produces an infinite series of functions, defined in terms of the right hand side, that are conserved densities of any equation having infinitely many infinitesimal symmetries. For instance, the function $\frac{\partial f}{\partial u_{x}}$ has to be a conserved density of any integrable equation of the~KdV type~$u_t=u_{xxx}+f(u,u_x)$. This fact imposes very strong conditions on the form of the function~$f$. In this paper we construct similar canonical densities for equations of the Boussinesq type. In order to do that, we write the equations as evolution systems and generalise the formal diagonalisation procedure proposed in \cite{MSY} to these systems.

In this note we are dealing with a particular class of quadratic algebras -- the so-called quantum matrix algebras. The well-known examples are the algebras of quantized functions on classical Lie groups (the RTT algebras). We consider the problem of constructing some projectors on homogenous components of such algebras, which are analogs of the usual symmetrizers. The main objective of this note is to present a method, which hopefully enables one to construct symmetrizers on all homogenous components of the RTT algebras. We illustrate the construction by two low-dimensional examples. A way of extending this method onto other quantum matrix algebras is also discussed.

We find all non-abelian generalizations of $\text{P}_1 - \text{P}_6$ Painlev\'e systems such that the corresponding autonomous system obtained by freezing the independent variable is integrable. All these systems have isomonodromic Lax representations.

As data rates for multi-gigabit serial interfaces within multi-node compute systems approach and exceed 10 Gigabits per second (Gbps), board-to-board and chip-to-chip optical signaling solutions become more attractive, particularly for longer (e.g. 50-100 cm) links. The transition to optical signaling will potentially allow new high performance compute (HPC) system architectures that benefit from characteristics unique to optical links. To examine these characteristics, we built and tested several optical demonstration vehicles; one based on dense wavelength division multiplexing (DWDM), and others based on multiple point-to-point links carried across multimode fibers. All test vehicles were constructed to evaluate applicability to a multi-node compute system. Test results, combined with data from recent research efforts are summarized and compared to equivalent electrical links and the advantages and design characteristics unique to optical signaling are identified., Comment: 25 pages, 18 figures, DesignCon 2011

All Hamiltonian non-abelian Painlev\'e systems of ${\rm{P}}_{1}-{\rm{P}}_{6}$ type with constant coefficients are found. For ${\rm{P}}_{1}-{\rm{P}}_{5}$ systems, we replace an appropriate inessential constant parameter with a non-abelian constant. To prove the integrability of new ${\rm{P}}_{3}^{\prime}$ and ${\rm{P}}_{5}$ systems thus obtained, we find isomonodromic Lax pairs for them., Comment: Some typos were fixed

We study non-abelian systems of Painlev\'e type. To derive them, we introduce an auxiliary autonomous system with the frozen independent variable and postulate its integrability in the sense of the existence of a non-abelian first integral that generalizes the Okamoto Hamiltonian. All non-abelian P6-P2-systems with such integrals are found. A coalescence limiting scheme is constructed for these non-abelian Painlev\'e systems. This allows us to construct an isomonodromic Lax pair for each of them.

The highest critical mass of neutron stars (NSs) was reviewed in the context of equation of state and observationsl results/ It was predicted that the maximum NS mass (MNS) exists in the range MNS ~ 2.2-2.9 M_Sun. However, recent observations of gravitationsl waves and other studies had suggested the higher mass limit of NSs, MNS ~ 3/2 M_Sun. The NS mass up to the value of MNS ~ 2 M_Sun is well understood, and with such a mass value it was meaning ful to discuss the "mass gap" ("m-gap") between the NS and black hole (BH) collapsars.The "m-gap" exists in between the highest mass of NS and the lowest mass of BH collapsars (Mm-gap ~ 2-5 M_Sun). in the mass distribution, the maximim population of NSs and BHs is located at MNS = 1.4 M_Sun and MBH = 6.7 M_Sun, respectively. However, recent ofservational results predicted filling the "m-gap" by the compact objects. In this paper, the concept of gravidynamics was reported to resolve the problem of peak likelihood value of gravitational mass at Mpeak = 6.7 M_Sun and the "m-gap" (Mm-gap ~ 2-5 M_Sun). This concept was based on a non-metric scalar-tensor model of gravitational interaction with localizable field energy. The gravidynamics model shows the total mass (MQ) of a compact relativistic object filled with matter of quark-gluon plasma of the radius r* = GMQ/c2 ~ 10 km, consistent with the "m-gap". It was conceptualized that the total measurable gravitational mass of such an extremely dense object consists of both matter and field, which is described by scalar-tensor components. This model is also useful for predicting the collapsars within the "m-gap"., Comment: 25 pages, 4 figures, accepted for publication in Astrophysical Bulletin, vol.77. arXiv admin note: text overlap with arXiv:1309.6635, arXiv:1006.2834, arXiv:1909.05804 by other authors

For all non-equivalent matrix systems of Painlev\'e-4 type found by authors in arXiv:2107.11680, isomonodromic Lax pairs are presented. Limiting transitions from these systems to matrix Painlev\'e-2 equations are found.

Relations between triple Jordan systems and integrable multi-component models of the modified Korteveg--de Vries type are established. The most general model is related to a pair consisting of a triple Jordan system and a skew-symmetric bilinear operation. If this operation is a Lie bracket, then we arrive at the Lie-Jordan algebras, Comment: 22 pages. arXiv admin note: substantial text overlap with arXiv:1804.06396

A complete classification of isotropic vector equations of the geometric type that possess higher symmetries is proposed. New examples of integrable multi-component systems of the geometric type and their auto-Backlund transformations are found., Comment: 18 pages

We provide a concise introduction to the symmetry approach to integrability. Some results on integrable evolution and systems of evolution equations are reviewed. Quasi-local recursion and Hamiltonian operators are discussed. We further describe non-abelian integrable equations, especially matrix (ODE and PDE) systems. Some non-evolutionary integrable equations are studied using a formulation of formal recursion operators that allows to study non-diagonalisable systems of evolution equations., Comment: To be published in: Nonlinear Systems and Their Remarkable Mathematical Structures, Vol. 2, N. Euler and M.C. Nucci (editors), CRC Press, Boca Raton, 2019. arXiv admin note: text overlap with arXiv:1711.10624

The first part of the book is devoted to the symmetry approach to classification of scalar integrable evolution PDEs with two independent variables. In the second part systems of evolution equations with polynomial homogeneous right-hand side are considered. Such systems are associated with associative algebras and, more generally, with non-associative algebraic structures, whose structural constants are defined by the coefficients of the system under consideration. We explore which structures correspond to polynomial integrable models. Integrable systems polynomial in derivatives are described in geometric terms. The third part we consider integrable vector izotropic and anizotropic systems., Comment: 149 pages

The survey is devoted to algebraic structures related to integrable ODEs and evolution PDEs. A description of Lax representations is given in terms of vector space decomposition of loop algebras into a direct sum of Taylor series and a complementary subalgebra. Examples of complementary subalgebras and corresponding integrable models are presented. In the framework of the bi-Hamiltonian approach compatible associative algebras related affine Dynkin diagrams are considered. A bi-Hamiltonian origin of the classical elliptic Calogero-Moser models is revealed., Comment: 107 pages

We consider second order differential operators $P$ with polynomial coefficients that preserve the vector space $V_k$ of polynomials of degrees not greater then $k$. We assume that the metric associated with the symbol of $P$ is flat and that the operator $P$ is potential. In the case of two independent variables we obtain some classification results and find polynomial forms for the elliptic $A_2$ and $G_2$ Calogero-Moser Hamiltonians and for the elliptic Inosemtsev model., Comment: 14 pages

The potential of the $A_2$ quantum elliptic model (3-body Calogero-Moser elliptic model) is defined by the pairwise three-body interaction through Weierstrass $\wp$-function and has a single coupling constant. A change of variables has been found, which are $A_2$ elliptic invariants, such that the potential becomes a rational function, while the flat space metric as well as its associated vector are polynomials in two variables. It is shown that the model possesses the hidden $sl(3)$ algebra - the Hamiltonian is an element of the universal enveloping algebra $U_{sl(3)}$ for arbitrary coupling constant - thus, it is equivalent to $sl(3)$-quantum Euler-Arnold top. The integral, in a form of the third order differential operator with polynomial, is constructed explicitly, being also an element of $U_{sl(3)}$. It is shown that there exists a discrete sequence of the coupling constants for which a finite number of polynomial eigenfunctions, up to a (non-singular) gauge factor occur. The potential of the $G_2$ quantum elliptic model (3-body Wolfes elliptic model) is defined by the pairwise and three-body interactions through Weierstrass $\wp$-function and has two coupling constants. A change of variables has been found, which are $G_2$ elliptic invariants, such that the potential becomes a rational function, while the flat space metric as well as its associated vector are polynomials in two variables. It is shown the model possesses the hidden $g^{(2)}$ algebra. It is shown that there exists a discrete family of the coupling constants for which a finite number of polynomial eigenfunctions up to a (non-singular) gauge factor occur., Comment: 18 pages, the solution of G_2 elliptic model added, title changed (slightly) and also abstract, some references added

We discuss associative analogues of classical Yang-Baxter equation meromorphically dependent on parameters. We discover that such equations enter in a description of a general class of parameter-dependent Poisson structures and double Lie and Poisson structures in sense of M. Van den Bergh. We propose a classification of all solutions for one-dimensional associative Yang-Baxter equations., Comment: 18 pages, LATEX2, ws-ijgmmp style. Few typos corrected, aknowledgements added

The complete lists of vector hyperbolic equations on the sphere that have integrable third order vector isotropic and anisotropic symmetries are presented. Several new integrable hyperbolic vector models are found. By their integrability we mean the existence of vector Backlund transformations depending on a parameter. For all new equations such transformations are constructed., Comment: Submitted in Lett. Math. Phys

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We present optical photometric and low-resolution spectroscopic observations of the Type II plateau supernova (SN) 2008in, which occurred in the outskirts of the nearly face-on spiral galaxy M 61. Photometric data in the X-rays, ultraviolet and near-infrared bands have been used to characterize this event. The SN field was imaged with the ROTSE-IIIb optical telescope about seven days before the explosion. This allowed us to constrain the epoch of the shock breakout to JD = 2454825.6. The duration of the plateau phase, as derived from the photometric monitoring, was ~ 98 days. The spectra of SN 2008in show a striking resemblance to those of the archetypal low-luminosity IIP SNe 1997D and 1999br. A comparison of ejecta kinematics of SN 2008in with the hydrodynamical simulations of Type IIP SNe by Dessart et al. (2010) indicates that it is a less energetic event (~ 5$\times10^{50}$ erg). However, the light curve indicates that the production of radioactive $^{56}$Ni is significantly higher than that in the low-luminosity SNe. Adopting an interstellar absorption along the SN direction of $A_V$ ~ 0.3 mag and a distance of 13.2 Mpc, we estimated a synthesized Ni mass of ~ 0.015 $M_{\odot}$. Employing semi-analytical formulae (Litvinova & Nadezhin 1985), we derived a pre-SN radius of ~ 126$R_{\odot}$, an explosion energy of ~ 5.4$\times10^{50}$ erg and a total ejected mass of ~ 16.7$M_{\odot}$. The latter indicates that the zero age main-sequence mass of the progenitor did not exceed 20$M_{\odot}$. Considering the above properties of SN 2008in, and its occurrence in a region of sub-solar metallicity ([O/H] ~ 8.44 dex), it is unlikely that fall-back of the ejecta onto a newly formed black hole occurred in SN 2008in. We therefore favor a low-energy explosion scenario of a relatively compact, moderate-mass progenitor star that generates a neutron star., Comment: Accepted for publication in the Astrophysical Journal

We consider a special class of linear and quadratic Poisson brackets related to ODE systems with matrix variables. We investigate general properties of such brackets, present an example of a compatible pair of quadratic and linear brackets and found the corresponding hierarchy of integrable models, which generalizes the two-component Manakov's matrix system in the case of arbitrary number of matrices., Comment: 9 pages, latex

We present BV RI photometric and low-resolution spectroscopic investigation of a type II core-collapse supernova (SN) 2008gz, which occurred in a star forming arm and within a half-light radius (solar metallicity region) of a nearby spiral galaxy NGC 3672. The SN event was detected late and a detailed investigation of its light curves and spectra spanning 200 days suggest that it is an event of type IIP similar to archetypal SNe 2004et and 1999em. However, in contrast to other events of its class, the SN 2008gz exhibits rarely observed V magnitude drop of 1.5 over the period of a month during plateau to nebular phase. Using 0.21 mag of Av as a lower limit and a distance of 25.5 Mpc, we estimate synthesized $^{56}$Ni mass of 0.05 \pm 0.01 M* and a mid-plateau Mv of -16.6 \pm 0.2 mag. The photospheric velocity is observed to be higher than that was observed for SN 2004et at similar epochs, indicating explosion energy was comparable to or higher than SN 2004et. Similar trend was also seen for the expansion velocity of H-envelopes. By comparing its properties with other well studied events as well as by using a recent simulation of pre-SN models of Dessart, Livne & Waldman (2010), we infer an explosion energy range of 2 - 3 x 10$^{51}$ erg and this coupled with the observed width of the forbidden [O I] 6300-6364 {\AA} line at 275 days after the explosion gives an upper limit for the main-sequence (non-rotating, solar metallicity) progenitor mass of 17 M*. Our narrow-band H{\alpha} observation, taken nearly 560 days after the explosion and the presence of an emission kink at zero velocity in the Doppler corrected spectra of SN indicate that the event took place in a low luminosity star forming H II region., Comment: Accepted for publication in MNRAS

We describe the results that have so far been obtained in the classification problem for integrable (2+1)-dimensional systems of hydrodynamic type. The systems of Gibbons--Tsarev type are the most fundamental here. A whole class of integrable (2+1)-dimensional models is related to each such system. We present the known GT systems related to algebraic curves of genus g=0 and g=1 and also a new GT system corresponding to algebraic curves of genus g=2. We construct a wide class of integrable models generated by the simplest GT system, which was not considered previously because it is in a sense trivial., Comment: 47 pages, no figures

We find a general solution to the unique 7th order ODE admitting ten dimensional group of contact symmetries. The integral curves of this ODE are rational contact curves in $\PP^3$ which give rise to rational plane curves of degree six. The moduli space of these curves is a real form of the homogeneous space $Sp(4)/SL(2)$., Comment: 6 pages, 2 figures. Final version - to appear in JGP

We construct integrable pseudopotentials with an arbitrary number of fields in terms of elliptic generalization of hypergeometric functions in several variables. These pseudopotentials yield some integrable (2+1)-dimensional hydrodynamic type systems. An interesting class of integrable (1+1)-dimensional hydrodynamic type systems is also generated by our pseudopotentials., Comment: 19 pages, latex

We construct integrable pseudopotentials with an arbitrary number of fields in terms of generalized hypergeometric functions. These pseudopotentials yield some integrable (2+1)-dimensional hydrodynamic type systems. An interesting class of integrable (1+1)-dimensional hydrodynamic type systems is also generated by our pseudopotentials., Comment: Latex, 28 pages, new material on integrable 3-dimensional hydrodynamic type systems is added

A certain class of integrable hydrodynamic type systems with three independent and N dependent variables is considered. We choose the existence of a pseudopotential as a criterion of integrability. It turns out that the class of integrable systems having pseudopotentials with movable singularities is described by a functional equation, which can be solved explicitly. This allows us to construct interesting examples of integrable hydrodynamic systems for arbitrary N., Comment: 12 pages, latex