In the e-commerce era, recommendation systems were introduced to share customer experience and comments. At the same time, there is a need for Ecommerce entities to join their recommender system databases to enhance the reliability toward prospective customers and also to maximize the precision of target marketing. However, there will be a privacy disclosure hazard while joining recommender sys...

The interphase excess free energy a due to an interphase boundary (IPB) is calculated in the Ising model using the Scalar Product (SP) method. Different from the "sum" method calculation of a based on the boundary profile, the SP approach skips the profile and directly evaluates a from the equilibrium properties of the homogeneous phases meeting at the boundary. Using a series of Cluster Variat...

Jack polynomials are a remarkable family of polynomials in n variables x = (x1, · · · , xn) with coefficients in the field F := Q(α) where α is an indeterminate. They arise naturally in several statistical, physical, combinatorial, and representation theoretic considerations. The symmetric polynomials ([M1], [St], [LV], [KS]) Jλ = J (α) λ are indexed by partitions λ = (λ1, · · · , λn) where λ1 ...

In this paper, we have investigated operator product expansion (OPE) for the thermal correlator of the two scalar currents. In extending OPE to finite temperature, some complications arise. One of these complications is the breakdown of Lorentz invariance by the choice of the reference frame. Due to the residual O(3) symmetry, more operators of the same dimension appear in the OPE than those at...

Let A be a unital C -algebra with involution represented in a Hilbert space H, G the group of invertible elements of A, U the unitary group of A, G the set of invertible selfadjoint elements of A, Q = f" 2 G : " = 1g the space of re ections and P = Q \ U . For any positive a 2 G consider the a-unitary group Ua = fg 2 G : a g a = g g, i.e., the elements which are unitary with respect to the scal...

Finite dimensional linear spaces (both complex and real) with indefinite scalar product [·, ·] are considered. Upper and lower bounds are given for the size of an indecomposable matrix that is normal with respect to this scalar product in terms of specific functions of v = min{v − , v+}, where v− (v+) is the number of negative (positive) squares of the form [x, x]. All the bounds except for one...

We investigate the effect of structure-preserving perturbations on the solution to a linear system, matrix inversion, and distance to singularity. Particular attention is paid to linear and nonlinear structures that form Lie algebras, Jordan algebras and automorphism groups of a scalar product. These include complex symmetric, pseudo-symmetric, persymmetric, skewsymmetric, Hamiltonian, unitary,...

The recent investigation of privacy-preserving data mining has been motivated by the growing concern about the privacy of individuals when their data is stored, aggregated, and mined for information. In an effort towards practical algorithms for privacy-preserving data mining solutions, we analyze and implement solutions to an important primitive: the privacy-preserving scalar product of two ve...

The canonical correlation or Kubo-Mori scalar product on the state space of a finite quantum system is a natural generalization of the classical Fisher metric. This metric is induced by the von Neumann entropy or the relative entropy of the quantum mechanical states. An important conjecture of Petz that the scalar curvature of the state space with Kubo-Mori scalar product as Riemannian metric i...

The ability to perform scalar products of two vectors, each known to a different party, is a central problem in privacy preserving data mining and other multi party computation problems. Ongoing search for both efficient and secure scalar product protocols has revealed that this task is not easy. In this paper we show that, indeed, scalar products can never be made secure in the information the...