Abstract. For any bounded regular domain Ω of a real analytic Riemannian manifold M , we denote by λk(Ω) the k-th eigenvalue of the Dirichlet Laplacian of Ω. In this paper, we consider λk and as a functional upon the set of domains of fixed volume in M . We introduce and investigate a natural notion of critical domain for this functional. In particular, we obtain necessary and sufficient condit...

Quantum fields propagating on a curved spacetime are investigated in terms of microlocal analysis. We discuss a condition on the wave front set for the corresponding n-point distributions, called “microlocal spectrum condition” (μSC). On Minkowski space, this condition is satisfied as a consequence of the usual spectrum condition. Based on Radzikowski’s determination of the wave front set of th...

In this paper we obtain several results on the rank properties of Hadamard matrices (including Sylvester Hadamard matrices) as well as (generalized) Hadamard matrices. These results are used to show that the classes of (generalized) Sylvester Hadamard matrices and of generalized pseudo-noise matrices are equivalent, i.e., they can be obtained from each other by means of row/column permutations....

The Z2s-additive and Z2Z4-additive codes are subgroups of Z n 2 and Z α 2 × Z β 4 , respectively. Both families can be seen as generalizations of linear codes over Z2 and Z4. A Z2s-linear (resp. Z2Z4-linear) Hadamard code is a binary Hadamard code which is the Gray map image of a Z2s-additive (resp. Z2Z4-additive) code. It is known that there are exactly ⌊ t−1 2 ⌋ and ⌊ t 2⌋ nonequivalent Z2Z4-...

Applications of well-known matrix theory reveal some interesting and possibly useful invariant properties of the real Hadamard matrix and transform (including the Walsh matrix and transform). Subject to certain conditions that can be fulfilled for many orders of the matrix, the space it defines can be decomposed into two invariant subspaces defined by two real, singular, mutually orthogonal (al...

A balanced incomplete block design (BIBD) [1] with parameters 2-(v, b, r, k, λ) (short 2-(v, k, λ)) is a pair (V,B) where V is a v-set (elements are called points) and B is a collection of b k-subsets (elements are called blocks) of V such that each point is contained in exactly r blocks and any pair of points is contained in exactly λ blocks. A Hadamard matrix of order n is an n × n (1,−1)-mat...

Hadamard (or complete $CAT(0)$) spaces are complete, non-positive curvature, metric spaces. Here, we prove a nonlinear ergodic theorem for continuous non-expansive semigroup in these spaces as well as a strong convergence theorem for the commutative case. Our results extend the standard non-linear ergodic theorems for non-expansive maps on real Hilbert spaces, to non-expansive maps on Ha...

In this paper, the concepts of well-posednesses and Hadamard well-posedness for a family of mixed variational inequalities are studied. Also, some metric characterizations of them are presented and some relations between well-posedness and Hadamard well-posedness of a family of mixed variational inequalities is studied. Finally, a relation between well-posedness for the family of mixed variatio...

Non-affine groups acting doubly transitively on a Hadamard matrix have been classified by Ito. Implicit in this work is a list of Hadamard matrices with non-affine doubly transitive automorphism group. We give this list explicitly, in the process settling an old research problem of Ito and Leon. We then use our classification to show that the only cocyclic Hadamard matrices developed form a dif...

‎In this paper‎, ‎we shall establish some extended Simpson-type inequalities‎ ‎for differentiable convex functions and differentiable concave functions‎ ‎which are connected with Hermite-Hadamard inequality‎. ‎Some error estimates‎ ‎for the midpoint‎, ‎trapezoidal and Simpson formula are also given‎.