∀i, yi = [0, ..., 0 } {{ } j−1 , 1, 0, ..., 0 } {{ } c−j ] ⇒ xi W0 = x̄j W0, (1) where y i is the i-th row of the true cluster assignment matrix Y and x̄j is the mean of the data that belongs to class j. Denote X̄c = [x̄1, ..., x̄c]. Note that X̄c = XY Σ, where Σ ∈ Rc×c is a diagonal matrix with the i-th diagonal element as 1/ni, ni is the number of the data that belongs to class i. Then rank(X̄ c W0)...

The distributions of products and ratios of random variables are of interest in many areas of the sciences. In this paper, the exact distributions of the product | XY | and the ratio | X/Y | are derived when X and Y are Student’s t and Bessel function random variables distributed independently of each other.

Here G is considered as an algebraic group defined over F . The Lie algebra g may be identified with Mn(F ), with Lie bracket [X,Y ] = XY − Y X . The group T is isomorphic to the algebraic torus Gm, T (F ) is isomorphic to (F), and the Lie algebra t is isomorphic to F. The character group X(T ) is the lattice of algebraic homomorphisms T → Gm, the cocharacter group that of algebraic homomorphis...

Let U = Tri(A,M,B) be a triangular algebra, where A, B are unital algebras over a field F and M is a faithful (A,B)-bimodule. Assume that ξ ∈ F and L : U → U is a map. It is shown that, under some mild conditions, L is linear and satisfies L([X, Y ]) = [L(X), Y ] + [X,L(Y )] for any X,Y ∈ U with [X, Y ] = XY − Y X = 0 if and only if L(X) = φ(X) + ZX + f(X) for all A, where φ is a linear derivat...

The row model is used to study the commensurate-incommensurate (C-IC) and isotropic (FFTXY) transitions of the frustrated 2D XY model on the triangular lattice. New relevant variables clarify the physics of these transitions: phase and chiral variables are coupled so that spin waves generate long range polar interactions. The resulting dielectric constant diverges at the transition. A single tr...

The XY Z spin chain is considered in the framework of the generalized algebraic Bethe ansatz developed by Takhtajan and Faddeev. The sum of norms of the Bethe vectors is computed and expressed in the form of a Jacobian. This result corresponds to the Gaudin hypothesis for the XY Z spin chain.

Let X and Y be two nonnegative and dependent random variables following a generalized Farlie–Gumbel–Morgenstern distribution. In this short note, we study the impact of a dependence structure of X and Y on the tail behavior of XY . We quantify the impact as the limit, as x → ∞, of the quotient of Pr(XY > x) and Pr(XY ∗ > x), where X and Y ∗ are independent random variables identically distribut...

We point out that the claim of strong universality of [1] is incorrect, as it contradicts known rigorous results. AMS 2000 subject classification: 82B20, 82B26, 60K35.

We found the eigenvalues of the transfer matrices for the 1-D Hubbard model and for the coupled XY model with twisted boundary condition by using the analytic Bethe Ansatz method. Under a particular condition the two models have the same Bethe Ansatz equations. We have also proved that the periodic 1-D Hubbard model is exactly equal to the coupled XY model with nontrivial twisted boundary condi...