We present a tight bound on the exact maximum complexity of Minkowski sums of polytopes in R. In particular, we prove that the maximum number of facets of the Minkowski sum of k polytopes with m1,m2, . . . ,mk facets respectively is bounded from above by

The subject of this thesis is the theory of nonholomorphic modular forms of non-integral weight, and its applications to arithmetical functions involving Dedekind sums and Kloosterman sums. As was discovered by Andre Weil, automorphic forms of non-integral weight correspond to invariant funtions on Metaplectic groups. We thus give an explicit description of Meptaplectic groups corresponding to ...

In a statistical database, the query-answering system should leave unanswered sum-queries that could lead to the disclosure of confidential data. To this end, each sum-query and previously answered sum-queries should be audited. We give a general framework for controlling the amount of information released when sum-queries are answered, both from the viewpoint of the user and from the viewpoint...

While the idea of a matching pennies game may seem contrived, it is merely the simplest example of a general class of zero-sum games, where the total payoff of the players is constant regardless of the outcome. Consequently gains for one player can only come from losses of the other. For this reason, zero-sum games will rarely have a pure strategy Nash equilibrium. Examples would be chess, or m...

Let A be a finite, nonempty subset of an abelian group. We show that if every element is sum two other elements, then has zero-sum subset. That is, (finite, nonempty) sum-full group not zero-sum-free.

It is shown that complexity of implementation of prefix sums of m variables (i.e. functions x1 ◦ . . . ◦ xi, 1 ≤ i ≤ m) by circuits of depth dlog2 me in the case m = 2n is exactly 3.5 · 2 − (8.5 + 3.5(n mod 2))2bn/2c + n + 5. As a consequence, for an arbitrary m an upper bound (3.5 − o(1))m holds. In addition, an upper bound ( 3 3 11 − o(1) ) m for complexity of the minimal depth prefix circuit...

Let z be a real quadratic irrational. We compare the asymptotic behavior of Dedekind sums S(pk, qk) belonging to convergents pk/qk of the regular continued fraction expansion of z with that of Dedekind sums S(sj/tj) belonging to convergents sj/tj of the negative regular continued fraction expansion of z. Whereas the three main cases of this behavior are closely related, a more detailed study of...

We present a simple yet rigorous theory of integration that is based on two axioms rather than on a construction involving Riemann sums. With several examples we demonstrate how to set up integrals in applications of calculus without using Riemann sums. In our axiomatic approach even the proof of the existence of the definite integral (which does use Riemann sums) becomes slightly more elegant ...