Price caps

Price-cap regulation of AT&T, which became effective on July 1, 1989, is an example of an idea that made its way from economic theory to institutional practice; in this case the process took about seven years. We describe both price-cap regulation and its predecessor — rate-of-return regulation — with particular regard to their incentive properties. We then give the assumptions and conclusions of a theoretical study (in fact a principal-agent model) that bears on the likely effectiveness of price-cap regulation. In the concluding section we describe some aspects of the progress of this idea from theory to practice, and draw tentative conclusions about the conditions that made it possible for this progress to be successfully completed.The views expressed here are the authors', and not necessarily those of AT&T Bell Laboratories.     

Construction of caps by means of caps in complementary subspaces

A construction of caps is given which yields in particular caps with a free pair of points. Applying this construction, we meet the bound of Farr and Lisoněk [J. Farr, P. Lisoněk, Caps with free pairs of points, J. Geom. 85 (2006) 35-41] for caps with a free pair of points in , q even.     

Conics and caps

In this article, we begin with arcs in PG(2, q ⁿ ) and show that they correspond to caps in PG(2n, q) via the André/Bruck?CBose representation of PG(2, q ⁿ ) in PG(2n, q). In particular, we show that a conic of PG(2, q ⁿ ) that meets ?_?? in x points corresponds to a (q ⁿ ?+?1 ? x)-cap in PG(2n, q). If x?=?0, this cap is the intersection of n quadrics. If x?=?1 or 2, this cap is contained in the intersection of n quadrics and we discuss ways of extending these caps. We also investigate the structure of the n quadrics.     

Bounds on affine caps

We present a short and elementary proof for an upper bound on caps in affine spaces. The bound generalizes and strengthens a result by Meshulam. The proof is inspired by Meshulam's Fourier transform method without making explicit use of the Fourier transform. © 2002 Wiley Periodicals, Inc. J Combin Designs 10: 111–115, 2002; DOI 10.1002/jcd.10000     

Binary codes and caps

The connection between maximal caps (sometimes called complete caps) and certain binary codes called quasi-perfect codes is described. We provide a geometric approach to the foundational work of Davydov and Tombak who have obtained the exact possible sizes of large maximal caps. A new self-contained proof of the existence and the structure of the largest maximal nonaffine cap in ℙG(n, 2) is given. Combinatorial and geometric consequences are briefly sketched. Some of these, such as the connection with families of symmetric-difference free subsets of a finite set will be developed elsewhere. © 1998 John Wiley & Sons, Inc. J Combin Designs 6: 275–284, 1998     

Quantum codes from caps

Caps in a finite projective geometry over GF(4) are used for the construction of some quantum error-correcting codes, including an optimal ?27,13,5? code.     

Singular Lagrangian manifolds and their Lagrangian maps

The paper examines the singularity theory of Lagrangian manifolds and its connection with variational calculus, classification of Coxeter groups, and symplectic topology. We consider the application of the theory to the problem of going past an obstacle, to partial differential equations, and to the analysis of singularities of ray systems.Translated from Itogi Nauki i Tekhniki, Seriya Sovermennye Problemy Matematiki, Noveishie Dostizheniya, Vol. 33, pp. 55–112, 1988.     

A new construction of caps

In this paper we present a general method to construct caps in higher-dimensional projective spaces. As an application, for q≥8 even we obtain caps in PG(5,q) larger than the caps known so far, and a new class of caps of size (q+1)(q²+3) for q≥7 odd.     

Intrinsic invariants of cross caps

It is classically known that generic smooth maps of \(\varvec{R}^2\) into \(\varvec{R}^3\) admit only isolated cross cap singularities. This suggests that the class of cross caps might be an important object in differential geometry. We show that the standard cross cap \(f_{\mathrm{std }}(u,v)=(u,uv,v^2)\) has non-trivial isometric deformations with infinite-dimensional freedom. Since there are several geometric invariants for cross caps, the existence of isometric deformations suggests that one can ask which invariants of cross caps are intrinsic. In this paper, we show that there are three fundamental intrinsic invariants for cross caps. The existence of extrinsic invariants is also shown.     

Lagrangian non-intersections

No Abstract. . This research was supported by the Israel Science Foundation (grant No. 205/02 *). Received: December 2004 Revision: March 2005 Accepted: March 2005     

Computer search for small complete caps

A computer search method for finding small complete caps inPG(d,q) is considered. The problem is formulated as a combinatorial optimization problem to which record-to-record travel, a stochastic local search method, is applied. Several new complete caps are found and upper bounds on the number of points in the smallest complete caps inPG(d,q) are tabulated for small parameters.This research was supported by the Academy of Finland.     

A characteristic property of spherical caps

It is proved that given H ≥ 0 and an embedded compact orientable constant mean curvature H surface M included in the half space z ≥ 0, not everywhere tangent to z = 0 along its boundary , the inequality