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The Jordan-Wigner fermionization for the one-dimensional Bariev model of three coupled XY chains is formulated. The Lax operator in terms of fermion operators and the quantum R-matrix are presented explicitly. Furthermore, the graded reflection equations and their solutions are discussed.

Let X = (xij) and Y = (yij) be generic n by n matrices and Z = XY − Y X. Let S = k[x11, . . . , xnn, y11, . . . , ynn], where k is a field, let I be the ideal generated by the entries of Z and let R = S/I . We give a conjecture on the first syzygies of I , show how these can be used to give a conjecture on the canonical module of R. Using this and the Hilbert series of I we give a conjecture on...

In this paper we prove that the coordinate ring of the pinched Veronese (i.e k[X, XY, XY , Y , XZ, Y Z, XZ, Y Z, Z] ⊂ k[X, Y, Z]) is Koszul. The strategy of the proof is the following: we can consider a presentation S/I where S = k[X1, . . . , X9]. Using a distinguished weight ω, it’s enough to show that S/inωI is Koszul. We write inωI as J + H where J is generated by a Gröbner basis of quadric...

Following the work of Cline and Samorodnitsky (Stoch. Process. Their Appl. 49(1):75–98, 1994), we reexamine the subexponentiality of the product of two random variables, X and Y, which are independent and have distributions F and G, respectively. The main result is the following: If F belongs to the class A [that is to say, F is subexponential and lim supx→∞ F(vx)/F(x) < 1 holds for some v > 1]...

We find a full system of invariants with respect to gauge transformations L → g−1Lg for third-order hyperbolic linear partial differential operators on the plane. The operators are considered in a normalized form, in which they have the symbol SymL = (pX + qY )XY for some non-zero bivariate functions p and q. For this normalized form, explicit formulae are given. The paper generalizes a previou...

Let X and Y be two independent nonnegative random variables, of which X has a distribution belonging to the class L(γ ) or S(γ ) for some γ ≥ 0 and Y is unbounded. We study how their product XY inherits the tail behavior of X. Under some mild technical assumptions we prove that the distribution of XY belongs to the class L(0) or S(0) accordingly. Hence, the multiplier Y builds a bridge between ...

We study the equations for the quasi-momenta which characterize the wave-functions in the Bethe ansatz for the XXX-Heisenberg model. We show in a simple analytical fashion, that the usual " string hypothesis " incorrectly predicts the number of real solutions and the 1