A new algebraic framework is introduced based on stochastic matrices. Together with several new operators, the set of stochastic matrices is shown to constitute a vector space, an inner-product space, and an associative algebra. The new zero vector is the uniform probability distribution and linear addition is akin to statistical independence. This new stochastic algebra allows Markov chains an...

We introduce and study a 2–parameter family of unitarily invariant probability measures on the space of infinite Hermitian matrices. We show that the decomposition of a measure from this family on ergodic components is described by a determinantal point process on the real line. The correlation kernel for this process is explicitly computed. At certain values of parameters the kernel turns into...

A real square matrix satisfies the weak Hawkins-Simon condition if its leading principal minors are positive (the condition was first studied by the French mathematician Maurice Potron). Three characterizations are given. Simple sufficient conditions ensure that the condition holds after a suitable reordering of columns. A full characterization of this set of matrices should take into account t...

The large N spectrum of the quantum mechanical hamiltonian of two hermitean matrices in a harmonic potential is studied in a framework where one of the matrices is treated exactly and the other is treated as a creation operator impurity in the background of the first matrix. For the free case, the complete set of invariant eigenstates and corresponding energies are obtained. When g YM interacti...

We provide a graphical calculus for computing averages of tensor network diagrams with respect to the distribution random vectors containing independent uniform complex phases. Our method exploits order structure partially ordered set block permutations. A similar is developed consisting signs, based on combinatorics even partitions. employ our extend some results by Johnston and MacLean family...

We show that the commutator of lepton mass matrices is invariant under terrestial matter effects in the four-neutrino mixing scheme. A set of modelindependent sum rules for neutrino masses, which may be generalized to hold for an arbitrary number of neutrino families, are for the first time uncovered. Useful sum rules for the rephasing-invariant measures of leptonic CP violation have also been ...

In a graph or complex network, communities and anti-communities are node sets whose modularity attains extremely large values, positive and negative, respectively. We consider the simultaneous detection of communities and anti-communities, by looking at spectral methods based on various matrixbased definitions of the modularity of a vertex set. Invariant subspaces associated to extreme eigenval...

A distribution-based classification approach and a set of recently developed texture measures are applied to rotation-invariant texture classification. The performance is compared to that obtained with the well-known circular-symmetric autoregressive random field (CSAR) model approach. A difficult classification problem of 15 different Brodatz textures and seven rotation angles is used in exper...

Abstract: I follow a lesson learnt from Heisenberg’s matrix Quantum Mechanics to study the property of lepton flavor mixing and CP violation. I show that the commutator of lepton mass matrices defined in vacuum keeps invariant under terrestrial matter effects within the four-neutrino mixing scheme. A set of model-independent sum rules for neutrino masses are obtained, and they can be generalize...

A bistochastic matrix B of size N is called unistochastic if there exists a uni-tary U such that B ij = |U ij | 2 for i, j = 1,. .. , N. The set U 3 of all unistochastic matrices of order N = 3 forms a proper subset of the Birkhoff polytope, which contains all bistochastic (doubly stochastic) matrices. We compute the volume of the set U 3 with respect to the flat (Lebesgue) measure and analytic...