The main goal of this note is to introduce another second-order differenceequation where every nontrivial solution is of minimal period 5, namelythe difference equation:xn+1 =1 + xn−1xnxn−1 − 1, n = 1, 2, 3, . . .with initial conditions x0 and x1 any real numbers such that x0x1 6= 1.

We deal with Perazzo 3-folds in P4, i.e. hypersurfaces X=V(f)⊂P4 of degree d defined by a homogeneous polynomial f(x0,x1,x2,u,v)=p0(u,v)x0+p1(u,v)x1+p2(u,v)x2+g(u,v), where p0,p1,p2 are algebraically dependent but linearly independent forms d−1 u,v, and g is form u,v d. have vanishing hessian and, hence, the associated graded Artinian Gorenstein algebra Af fails strong Lefschetz Property. In th...

Let p be prime and q|p − 1. Suppose xq ≡ a(mod p) has a solution. We estimate the size of the smallest solution x0 with 0 < x0 < p. We prove that |x0| p3/2q−1 log p. By applying the Burgess character sum estimates, and estimates of certain exponential sums due to Bourgain, Glibichuk and Konyagin, we derive refinements of our result.

We study Nevai’s condition that for orthogonal polynomials on the real line, Kn(x, x0)Kn(x0, x0)−1 dρ(x) → δx0 , where Kn is the Christoffel–Darboux kernel. We prove that it holds for the Nevai class of a finite gap set uniformly on the spectrum, and we provide an example of a regular measure on [−2,2] where it fails on an interval.

1 The Sullivan model 1.1 Rational homotopy theory of spaces We will restrict our attention to simply-connected spaces. Much of this goes through with nilpotent spaces, but this will keep things easier and less technical. Definition 1. A 1-connected space X is said to be rational if either of the following equivalent conditions holds: 1. π∗X forms a graded Q-vector space. 2. H̃∗X forms a graded Q...

Let T be a tree and let x0 be a vertex of T . T is called a superstar with central vertex x0 if T − x0 is a union of paths. The General Inverse Eigenvalue Problem for certain trees is partially answered. Using this description, some superstars are presented for which the problem of ordered multiplicity lists and the Inverse Eigenvalue Problem are not equivalent.

We relate a part of the abelian étale fundamental group of curves over local fields to the component group of the Néron model of the jacobian. We apply the result to the modular curve X0(p)/Qp to show that the unramified abelian covering X1(p) → X0(p) (Shimura covering) uses up all the possible ramification over the special fiber of X0(p).