We investigate Lifshits-tail behaviour of the integrated density of states for a wide class of Schrödinger operators with positive random potentials. The setting includes alloy-type and Poissonian random potentials. The considered (single-site) impurity potentials f : Rd → [0,∞[ decay at infinity in an anisotropic way, for example, f(x1, x2) ∼ (|x1|α1 +|x2|α2 )−1 as |(x1, x2)| → ∞. As is expect...

We give a reformulation of a six-parameter family of coupled Painlevé VI systems with affine Weyl group symmetry of type D (1) 6 . We also study some Hamiltonian structures of this system. 0. Introduction In [4], we proposed a 6-parameter family of four-dimensional coupled Painlevé VI systems with affine Weyl group symmetry of type D (1) 6 . This system can be considered as a genelarization of ...

Abstract Background Thalassemia is common in the Iranian population, and it must be considered in the differential diagnosis of the microcytic hypochromic anemia. The molecular analysis of β-thalassemia is necessary for prenatal molecular diagnosis. Α-thalassemia caused by loss of function of either one of the two duplicated α-globin genes or in less frequent non deletion mutations mostly loc...

Proof. (Projection Theorem) Showing that the existence of minimizer implies that V is closed is left as an exercise. So we assume that V is closed. For f ∈ H, let α := infv∈V ‖v − f‖. It is a little easier to work with this in an equivalent form, α2 = infv∈V ‖v − f‖2. Thus, for every ε > 0 there is a vε ∈ V such that α2 ≤ ‖vε − f‖2 < α2 + ε. By choosing ε = 1/n, where n is a positive integer, w...

Let φ : R → R be a continuously differentiable function on an interval J ⊂ R and let α = (α1, α2) be a point with algebraically conjugate coordinates such that the minimal polynomial P of α1, α2 is of degree ≤ n and height ≤ Q. Denote by Mn φ (Q, γ, J) the set of such points α such that |φ(α1)− α2| ≤ c1Q −γ . We show that for a real 0 < γ < 1 and any sufficiently large Q there exist positive va...

with a1, a2 > 0. (Indeed any rank 2 matrix can be written as RA with A as above, and R orthogonal. Hence any affine transformation can be written as h ◦ f with f as claimed and g a rigid motion.) Hence, by Caratheodory uniqueness theorem it is sufficient to show that λ2(ASu(α1, α2)) = |det(A)|λ2(Su(α1, α2)) for all u = (u1, u2) ∈ R, αi ∈ R+ (3) Su(α, β) ≡ { x = (x1, x2) ∈ R : x1 ∈ (u1, u1 + α1]...

The rostral ventrolateral medulla (RVLM) controls sympathetic tone via excitatory bulbospinal neurons. It is also the main target of α2-adrenoceptor (α2-AR) agonists used for treatment of hypertension. In this study, we examined the synaptic mechanisms by which α2-AR agonists may inhibit the activity of RVLM bulbospinal neurons. We recorded selectively from RVLM bulbospinal neurons in brain ste...

We find the greatest value α1 and α2 , and the least values β1 and β2 , such that the double inequalities α1C(a,b)+(1−α1)A(a,b) < T (a,b) < β1C(a,b)+(1−β1)A(a,b) and α2/A(a,b)+(1−α2)/C(a,b) < 1/T (a,b) < β2/A(a,b)+(1−β2)/C(a,b) hold for all a,b > 0 with a = b . As applications, we get new bounds for the complete elliptic integral of the second kind. Here, C(a,b) = (a2 +b2)/(a+b) , A(a,b) = (a+b...

Let B1 and B2 be disjoint separable algebraic extensions of a field F, and let B = B1B2 be their composite. Let α1 be an element of B1 and α2 be an element of B2. Suppose α1 and α2 are primitive (resp. normal, resp. subnormal). We investigate the question of when α1 +α2 and α1α2 are necessarily primitive (resp. normal, resp. subnormal) elements of B. (A normal element of a Galois extension is d...

Proof. (Projection Theorem) Showing that the existence of minimizer implies that V is closed is left as an exercise. So we assume that V is closed. For f ∈ H, let α := infv∈V ‖v − f‖. It is a little easier to work with this in an equivalent form, α2 = infv∈V ‖v − f‖2. Thus, for every ε > 0 there is a vε ∈ V such that α2 ≤ ‖vε − f‖2 < α2 + ε. By choosing ε = 1/n, where n is a positive integer, w...