The identity component of the leaf preserving diffeomorphism group is perfect

It is shown that for any smooth foliated manifold the identity component of the group of all leaf preserving diffeomorphisms is perfect. This result generalizes in a sense a well-known theorem of Thurston.     

The only convex body with extremal distance from the ball is the simplex

We can extend the Banach-Mazur distance to be a distance between non-symmetric sets by allowing affine transformations instead of linear transformations. It was proved in [J] that for every convex bodyK we haved(K, D)≤n. It is proved that ifK is a convex body in ℝ ⁿ such thatd(K, D)=n, thenK is a simplex. This article is an M.Sc. thesis written under the supervision of E. Gluskin and V.D. Milman at Tel Aviv University. Partially supported by a G.I.F. grant.     

An explicit extension formula of bounded holomorphic functions from analytic varieties to strictly convex domains

We consider a strictly convex domain D ⁿ and m holomorphic functions, φ₁,…, φ_m, in a domain . We set V = {z ε Ω: φ₁(z) = ··· = φ_m(z) = 0}, M = V ∩ D and ∂M = V ∩ ∂D. Under the assumptions that the variety V has no singular point on ∂M and that V meets ∂D transversally we construct an explicit kernel K(ζ, z) defined for ζ ε ∂M and z ε D so that the integral operator Ef(z) = ∝ _{ζ ε ∂M} f(ζ) K(ζ, z) (z ε D), defined for f ε H^∞(M) (using the boundary values f(ζ) for a.e. ζ ε ∂M), is an extension operator, i.e., Ef(z) = f(z) for z ε M and furthermore E is a bounded operator from H^∞ to H^∞(D).     

An asymptotic formula for the minimal capacity among sets of equal area

We give an asymptotic formula for

An example concerning the translative kissing number of a convex body

This article presents a class of convex bodies inE ^d (d≥3) where their maximum kissing numbers in translative packings are larger than their maximum kissing numbers in lattice packings. This work was supported by the Austrian Academic Exchange Service.     

An algorithm for determining if the inverse of a strictly diagonally dominant Stieltjes matrix is strictly ultrametric

Summary It was recently shown that the inverse of a strictly ultrametric matrix is a strictly diagonally dominant Stieltjes matrix. On the other hand, as it is well-known that the inverse of a strictly diagonally dominant Stieltjes matrix is a real symmetric matrix with nonnegative entries, it is natural to ask, conversely, if every strictly diagonally dominant Stieltjes matrix has a strictly ultrametric inverse. Examples show, however, that the converse is not true in general, i.e., there are strictly diagonally dominant Stieltjes matrices in ^n×n (for everyn3) whose inverses are not strictly ultrametric matrices. Then, the question naturally arises if one can determine which strictly diagonally dominant Stieltjes matrices, in ^n×n (n3), have inverses which are strictly ultrametric. Here, we develop an algorithm, based on graph theory, which determines if a given strictly diagonally dominant Stieltjes matrixA has a strictly ultrametric inverse, where the algorithm is applied toA and requires no computation of inverse. Moreover, if this given strictly diagonally dominant Stieltjes matrix has a strictly ultrametric inverse, our algorithm uniquely determines this inverse as a special sum of rank-one matrices.Research supported by the National Science FoundationResearch supported by the Deutsche Forschungsgemeinschaft     

New bounds on the average distance from the Fermat–Weber center of a planar convex body

The Fermat–Weber center of a planar body Q is a point in the plane from which the average distance to the points in Q is minimal. We first show that for any convex body Q in the plane, the average distance from the Fermat–Weber center of Q to the points in Q is larger than , where Δ(Q) is the diameter of Q. This proves a conjecture of Carmi, Har-Peled and Katz. From the other direction, we prove that the same average distance is at most . The new bound substantially improves the previous bound of due to Abu-Affash and Katz, and brings us closer to the conjectured value of . We also confirm the upper bound conjecture for centrally symmetric planar convex bodies.     

An extension of Glimm's method to inhomogeneous strictly hyperbolic systems of conservation laws by “weaker than weak” solutions of the Riemann problem

We construct a generalized solution of the Riemann problem for strictly hyperbolic systems of conservation laws with source terms, and we use this to show that Glimm's method can be used directly to establish the existence of solutions of the Cauchy problem. The source terms are taken to be of the form a^′G, and this enables us to extend the method introduced by Lax to construct general solutions of the Riemann problem. Our generalized solution of the Riemann problem is “weaker than weak” in the sense that it is weaker than a distributional solution. Thus, we prove that a weak solution of the Cauchy problem is the limit of a sequence of Glimm scheme approximate solutions that are based on “weaker than weak” solutions of the Riemann problem. By establishing the convergence of Glimm's method, it follows that all of the results on time asymptotics and uniqueness for Glimm's method (in the presence of a linearly degenerate field) now apply, unchanged, to inhomogeneous systems.     

An example from power residues of the critical problem of Crapo and Rota

A natural density arising from the author's recent work on a generalization of Artin's conjecture for primitive roots is shown to be essentially the characteristic polynomial of a geometric lattice, as defined by Crapo and Rota. Necessary and sufficient conditions are obtained for the vanishing of this density.     

An example of a space L on which the Hardy-Littlewood maximal operator is not bounded

We give an example of a function p such that the Hardy-Littlewood maximal operator is not bounded on the generalized Lebesgue space L^p(x).     

Reduction of bi-Hamiltonian systems and separation of variables: An example from the Boussinesq hierarchy

We discuss the Boussinesq system with the stationary time t₅ within a general framework of stationary flows of n-Gel'fand-Dickey hierarchies. A careful use of the bi-Hamiltonian structure can provide a set of separation coordinates for the corresponding Hamilton-Jacobi equations. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 122, No. 2, pp. 219–238, February, 2000.