Ill-posed problems arise in many areas of science and engineering. Their solutions, if they exist, are very sensitive to perturbations the data. To reduce this sensitivity, original problem may be replaced by a minimization with fidelity term regularization term. We consider kind, which is square ?2-norm discrepancy qth power ?q-norm size computed solution measured some manner. interested situa...

This paper describes parallel matrix transpose algorithms on distributed memory concurrent processors. We assume that the matrix is distributed over a P Q processor template with a block scattered data distribution. P , Q, and the block size can be arbitrary, so the algorithms have wide applicability. The communication schemes of the algorithms are determined by the greatest common divisor (GCD...

We consider the techniques behind the current best algorithms for matrix multiplication. Our results are threefold. (1) We provide a unifying framework, showing that all known matrix multiplication running times since 1986 can be achieved from a single very natural tensor the structural tensor Tq of addition modulo an integer q. (2) We show that if one applies a generalization of the known tech...

On the half line 0; 1) we study rst order diierential operators of the form B 1 i d dx + Q(x); where B := B 1 0 0 ?B 2 ; B 1 ; B 2 2 M(n; C) are self{adjoint positive deenite matrices and Q : R + ! M(2n; C); R + := 0; 1); is a continuous self{adjoint oo{diagonal matrix function. We determine the self{adjoint boundary conditions for these operators. We prove that for each such boundary value pro...

where detξ denotes the determinant of ξ, and (C4) W (PξQ) = W (ξ) for all ξ ∈ M and all P,Q ∈ SO(3) for N = 3, with SO(3) := {Q ∈ M : QQ = QQ = I3 and detQ = 1}, where I3 denotes the identity matrix in M 3×3 and Q is the transposed matrix of Q. (In fact, (C4) is an additional condition which is not related to (2). However, it means that W is frame-indifferent, i.e., W (Pξ) = W (ξ) for all ξ ∈ M...

An (n, k) linear code C of length n over GF(q) = Fq, a Galios field of order q, where q is a prime, is a fc-dimensional linear subspace of F"q, where F"q denotes the space of all n-tuples over GF(q). A generator matrix G of this code is a k x n matrix whose rows form a basis of C. The parity-check matrix H of this code is an(n — fc) x n matrix such that Hv = 0 for all vectors veC. The row space...

Let Γ denote a distance-regular graph with classical parameters (D, b, α, β) and b 6= 1, α = b − 1. The condition on α implies that Γ is formally self-dual. For b = q we use the adjacency matrix and dual adjacency matrix to obtain an action of the q-tetrahedron algebra ⊠q on the standard module of Γ. We describe four algebra homomorphisms into ⊠q from the quantum affine algebra Uq(ŝl2); using t...

where detξ denotes the determinant of ξ, and (C4) W (PξQ) = W (ξ) for all ξ ∈ M and all P,Q ∈ SO(3) for N = 3, with SO(3) := {Q ∈ M : QQ = QQ = I3 and detQ = 1}, where I3 denotes the identity matrix in M 3×3 and Q is the transposed matrix of Q. (In fact, (C4) is an additional condition which is not related to (2). However, it means that W is frame-indifferent, i.e., W (Pξ) = W (ξ) for all ξ ∈ M...

Least squares approximation of a symmetric matrix C by a symmetric positive definite matrix Ĉ of rank at most p is a classical problem. It is typically solved by computing an eigen-decomposition of C and by truncating the eigen-decomposition by only using the eigenvectors and associated with the min(p, q) largest positive eigenvalues of C. Here q is the number of positive eigenvalues. Thus if q...

Semidefinite Programming Formulation By representer theorem: f(x) = ∑n l=1 αlK (Xi, x) under the condition that the function has an SoS representation, i.e., f(x) = φ(x)>Qφ(x) for some Q 0. Define a d × n matrix Φ = [φ(X1) · · ·φ(Xn)] and an n × n diagonal matrix A = diag(α) = diag(α1, . . . , αn). We have Q = ΦAΦ>. Q is d × d, but has rank n, which can be much smaller than d. The constraint on...