2. Group theory and graphs with large girth. 3. Expanders and superconcentrators. 4. Character sums and pseudo-random graphs.

Abstract. An easy explicit construction is given for a full and

Abstract. In this paper we present for every d ≥ 2 and every local field F of positive characteristic, explicit constructions of Ramanujan complexes which are quotients of the Bruhat-Tits building Bd(F) associated with PGLd(F). 1.

In this paper we give an explicit construction of a system of parametric equations describing a discrete valuation over k((X 1 ; : : : ; Xn )). This amounts to finding a parameter and a eld of coefficients. We devote section 1 to finding an element of value 1, that is, the parameter. The field

Given a solution of the Kadomtsev-Petviashvili equation we explicitly construct a solution of the modified Kadomtsev-Petviashvili equation related to one another by a generalized Miura transformation. The construction is modeled after a previous treatment of the modified Kortewegde Vries case

The existence of extremal combinatorial objects, such as Ramsey graphs and expanders, is often shown using the probabilistic method. It is folklore that pseudo-random generators can be used to obtain explicit constructions of these objects, if the test that the object is extremal can be implemented

In this paper we give an explicit construction of a system of parametric equations describing a discrete valuation over k((X1,..., Xn)). This amounts to finding a parameter and a field of coefficients. We devote section 1 to finding an element of value 1, that is, the parameter. The field

-point function turns out to be significantly simpler than the original direct calculation. This calculation also provides an explicit construction of an N = 2 component of an N = 4 five-point nilpotent covariant that violates U(1) Y symmetry. 1 UMR 5108 associ'ee `a l'Universit'e de Savoie 1

We study tight wavelet frames associated with given refinable functions which are obtained with the unitary extension principles. All possible solutions of the corresponding matrix equations are found. It is proved that the problem of the extension may be always solved with two framelets. In particular, if symbols of the refinable functions are polynomials (rational functions), then the corresponding framelets with polynomial (rational) symbols can be found.

Quantum expanders are a natural generalization of classical expanders. These objects were introduced and studied by [1, 3, 4]. In this note we show how to construct explicit, constant-degree quantum expanders. The construction is essentially the classical Zig-Zag expander construction of [5