This paper is concerned with the approximation of the effective conductivity σ(A) associated to an elliptic operator ∇xA(x, η)∇x where for x ∈ R d, d ≥ 1, A(x, η) is a bounded elliptic random symmetric d × d matrix and η takes value in an ergodic probability space. Writing AN (x, η) the periodization of A(x, η) on the torus T d N of dimension d and side N we prove that η-a.s. lim N→+∞ σ(A (x, η...

On a complex manifold $(M,J)$, we interpret symplectic and pseudo-K\"ahler structures as forms with respect to which $J$ is, respectively, symmetric skew-symmetric. We classify on 4-dimensional Lie algebras. develop method for constructing hypersymplectic from the above data. This allows us obtain an example of structure 4-step nilmanifold.

We establish several discrepancy and isoperimetric inequalities for directed graphs by considering the associated random walk. We show that various isoperimetric parameters, as measured by the stationary distribution of the random walks, including the Cheeger constant and discrepancy, are related to the singular values of the normalized probability matrix and the normalized Laplacian. Further, ...

An oriented graph Gσ is a simple undirected graph G with an orientation, which assigns to each edge a direction so that Gσ becomes a directed graph. G is called the underlying graph of Gσ and we denote by S(Gσ) the skew-adjacency matrix of Gσ and its spectrum Sp(Gσ) is called the skew-spectrum of Gσ. In this paper, the coefficients of the characteristic polynomial of the skew-adjacency matrix S...

Introduction A notion of quasideterminants for matrices over a noncommutative skew-field was introduced in [GR], [GR1], [GR2]. It proved its effectiveness for many areas including noncommutative symmetric functions [GKLLRT], noncommutative The main property of quasideterminants is a " heredity principle " : let A be a square matrix over a skew-field and (A ij) be its block decomposition into su...

The rank 4 locus of a general skew-symmetric 7 × 7 matrix gives the pfaffian variety in P which is not defined as a complete intersection. Intersecting this with a general P6 gives a Calabi–Yau manifold. An orbifold construction seems to give the 1-parameter mirror-family of this. However, corresponding to two points in the 1-parameter family of complex structures, both with maximally unipotent...

Let S(Gσ) be the skew-adjacency matrix of an oriented graph Gσ . The skew energy of Gσ is the sum of all singular values of its skew-adjacency matrix S(Gσ). This paper first establishes an integral formula for the skew energy of an oriented graph. Then, it determines all oriented graphs with minimal skew energy among all connected oriented graphs on n vertices with m (n ≤ m < 2(n− 2)) arcs, whi...

The large N limit of some matrix integrals over the orthogonal group O(N) and its relation with those pertaining to the unitary group U(N) are reexamined. It is proved that limN→∞ N −2 ∫ DO expN Tr JO is half the corresponding function in U(N), with a similar relation for limN→∞ ∫ DO expN Tr (AOBO), for A and B both symmetric or both skew symmetric. Large N limit of integrals over the orthogona...

The inverse eigenvalue problem of constructing symmetric positive semidefinite matrixD written as D ≥ 0 and real-valued skew-symmetric matrix G i.e., G −G of order n for the quadratic pencilQ λ : λMa λ D G Ka, whereMa > 0,Ka ≥ 0 are given analytical mass and stiffness matrices, so that Q λ has a prescribed subset of eigenvalues and eigenvectors, is considered. Necessary and sufficient condition...

in this paper, some elementary operations on triangular fuzzynumbers (tfns) are defined. we also define some operations on triangularfuzzy matrices (tfms) such as trace and triangular fuzzy determinant(tfd). using elementary operations, some important properties of tfms arepresented. the concept of adjoints on tfm is discussed and some of theirproperties are. some special types of tfms (e.g. pu...