David Benko

A New Approach to Hilbert's Third Problem

1. INTRODUCTION. If two polygons have the same area, it is always possible to decompose one of them into a finite number of polygons that can be rearranged to form the second polygon. This is the well-known Bolyai-Gerwien theorem [3, pp. 49–56]. One might ask whether this is true in space for polyhedra. In fact, F. Bolyai and Gauss had already asked this around 1844. Hilbert raised this question again: it was the third of his celebrated list of twenty-three problems in 1900 [11]. The negative answer was given by Max Dehn in 1902 [6]. Let F, F 1 ,. .. , F k be polyhedra. By writing F = F 1 + · · · + F k we mean that the interiors of the polyhedra F 1 ,. .. , F k are pairwise disjoint and F = F 1 ∪ · · · ∪ F k. Polyhedra F and G are equidecomposable if the polyhedron F can be suitably decomposed into a finite number of pieces that can be reassembled to give the polyhedron G.

A Weierstrass-type theorem for homogeneous polynomials

By the celebrated Weierstrass Theorem the set of algebraic polynomials is dense in the space of continuous functions on a compact set in R d . In this paper we study the following question: does the density hold if we approximate only by homogeneous polynomials? Since the set of homogeneous polynomials is nonlinear this leads to a nontrivial problem. It is easy to see that: 1) density may hold only on star-like 0-symmetric surfaces; 2) at least 2 homogeneous polynomials are needed for approximation. The most interesting special case of a star-like surface is a convex surface. It has been conjectured by the second author that functions continuous on 0-symmetric convex surfaces in R d can be approximated by a pair of homogeneous polynomials. This conjecture is not resolved yet but we make substantial progress towards its positive settlement. In particular, it is shown in the present paper that the above conjecture holds for 1) d = 2, 2) convex surfaces in R d with C 1+ǫ boundary.

Numerical approximation for singular second order differential equations

We consider numerical approximation of solutions of singular second order differential equations. In particular, we study the backward (or implicit) Euler method. We prove results concerning consistency, global error and stability. We show that the global error is linear with respect to the step size. Numerical results are also given, which demonstrate the linear convergence and we compare the numerical results with known approximations.

The Basel Problem as a Telescoping Series.

Summary The celebrated Basel Problem, that of finding the infinite sum 1 + 1/4 + 1/9 + 1/16 + …, was open for 91 years. In 1735 Euler showed that the sum is π2/6. Dozens of other solutions have been found. We give one that is short and elementary.

The full Markov-Newman inequality for Müntz polynomials on positive intervals

For a function f defined on an interval [a, b] let ∥f∥ [a,b] := sup{|f(x)|: x ∈ [a,b]}. The principal result of this paper is the following Markov-type inequality for Muntz polynomials. Theorem. Let n > 1 be an integer. Let λ 0 , λ 1 ,...,λ n be n + 1 distinct real numbers. Let 0 < a < b. Then formula math. formula math. where the supremum is taken for all Q ∈ span{x λ 0,x λ 1,.., x λ n} (the span is the linear span over R).

Asymptotic Bounds on the Integrity of Graphs and Separator Theorems for Graphs

In this paper we study the integrity of certain graph families. These include planar graphs, graphs with a given genus, graphs on the

Sets with Interior Extremal Points for the Markoff Inequality

d

The Basel Problem as a Rearrangement of Series

-dimensional integer lattice

A New Angle on the Fermat–Torricelli Point

\mathbb{Z}^d

Approximation by weighted polynomials

, and graphs that have no