Total Least Squares (TLS) is a method for treating an overdetermined system of linear equations Ax ≈ b, where both the matrix A and the vector b are contaminated by noise. In practical situations, the linear system is often ill-conditioned. For example, this happens when the system is obtained via discretization of ill-posed problems such as integral equations of the first kind (see e.g., [7] a...

The adaptive linear quadratic Gaussian control problem, where the linear transformation of the state A and the linear transformation of the control B are unknown, is solved assuming only that (A; B) is controllable and (A; Q 1 ) is observable, where Q 1 determines the quadratic form for the state in the integrand of the cost functional. A weighted least squares algorithm is modified by using a ...

We present a Krylov subspace–type projection method for a quadratic matrix polynomial λ2I − λA − B that works directly with A and B without going through any linearization. We discuss a special case when one matrix is a low rank perturbation of the other matrix. We also apply the method to solve quadratically constrained linear least squares problem through a reformulation of Gander, Golub, and...

is a (nonconstant) arithmetic progression of positive integers. We consider a general binary quadratic form ax2 + bxy + cy' ( a , b , c E Z ) and ask the question "Can the form ax' + hxy + ry' represen1 every inleger in 1he arithmetic progression kNo + 1 for any natural numbers k and l?" In a sampling of books containing a discussion of binary quadratic forms [2]-[9], we did not find this qustl...

Some symmetries of time-dependent Schrödinger equations for inverse quadratic, linear, and quadratic potentials have been systematically examined by using a method suitable to the problem. Especially, the symmetry group for the case of the linear potential turns out to be a semi-direct product SL(2, R) j s T2(R) of the SL(2, R) with a two-dimensional real translation group T2(R). Here, the time...

A sampling-based optimization method for quadratic functions is proposed. Our method approximately solves the following n-dimensional quadratic minimization problem in constant time, which is independent of n: z∗ = minv∈Rn〈v, Av〉 + n〈v,diag(d)v〉 + n〈b,v〉, where A ∈ Rn×n is a matrix and d, b ∈ R are vectors. Our theoretical analysis specifies the number of samples k(δ, ) such that the approximat...

Let d be the discriminant of an imaginary quadratic field. Let a, b, c be integers such that b − 4ac = d, a > 0, gcd(a, b, c) = 1. The value of |η ( (b + √ d)/2a ) | is determined explicitly, where η(z) is Dedekind’s eta function η(z) = e ∞ ∏ m=1 (1− e) ( im(z) > 0 ) .

In this paper, we prove that for any polynomial function f of fixed degree without multiple roots, the probability that all the (f(x+ 1), f(x+ 2), ..., f(x+ κ)) are quadratic non-residue is ≈ 1 2 . In particular for f(x) = x + ax+ b corresponding to the elliptic curve y = x + ax+ b, it implies that the quadratic residues (f(x + 1), f(x + 2), . . . in a finite field are sufficiently randomly dis...