We consider a parabolic-type PDE with diffusion given by fractional Laplacian operator and quadratic nonlinearity of the 'gradient' solution, convoluted singular term b. Our first result is well-posedness for this problem: show existence uniqueness (local in time) mild solution. The main about blow-up said particular we find sufficient conditions on initial datum b to ensure solution finite time.

OBJECTIVE Hypertension and diabetes are both risk factors for cognitive decline, and individuals with both might have an especially high risk. We therefore examined linear and nonlinear (quadratic) associations of 24-h blood pressure (BP) with cognitive performance in participants with and without type 2 diabetes. We also tested the association of nocturnal dipping status with cognitive perform...

Let D = 1 be a positive non-square integer and let δ = √ D or 1+ √ D 2 be a real quadratic irrational with trace t = δ + δ and norm n = δδ. Let γ = P+δ Q be a quadratic irrational for positive integers P and Q. Given a quadratic irrational γ, there exist a quadratic ideal Iγ = [Q, δ + P ] and an indefinite quadratic form Fγ(x, y) = Q(x−γy)(x−γy) of discriminant Δ = t − 4n. In the first section,...

We establish character sum bounds of the form ∣∣∣∣ ∑ a≤x≤a+H b≤y≤b+H χ(x + ky) ∣∣∣∣ < p−τH2, where χ is a nontrivial character (mod p), p 1 4 +ε < H < p, and |a|, |b| < p H. As an application, we obtain that given k ∈ Z\{0}, x + k is a quadratic non-residue (mod p) for some 1 ≤ x < p 1 2e. Introduction. Let k be a nonzero integer. Let p be a large prime and let H ≤ p. We are interested in the c...

The classic paradigm for MRI requires a homogeneous B(0) field in combination with linear encoding gradients. Distortions are produced when the B(0) is not homogeneous, and several postprocessing techniques have been developed to correct them. Field homogeneity is difficult to achieve, particularly for short-bore magnets and higher B(0) fields. Nonlinear magnetic components can also arise from ...

For an elliptic curve over the rationals, Goldfeld’s conjecture [4] asserts that the analytic rank ords=1 L(Ed/Q, s) of quadratic twists Ed of E is positive for squarefree d’s with density 1/2. In other words, the analytic rank of E goes up in quadratic extensions Q( √ d)/Q half of the time. In particular, for every E/Q there are (a) infinitely many quadratic extensions where the rank goes up, ...

In this paper, I discuss various examples arising from different areas in psychometrics. I will show that they have a common background and can be solved using a constrained quadratic optimisation algorithm developed by Albers et al. (2008a,b).

For any irreducible quadratic polynomial f(x) in Z[x] we obtain the estimate log l.c.m. (f(1), . . . , f(n)) = n log n + Bn + o(n) where B is a constant depending on f .

This paper provides four alternative suucient condition-sets, ensuring that a patch of a parametric tensor-product B-spline surface is locally convex. These conditions are at most tri-quadratic with respect to the control points of the surface.