Oscar M. Mansfield

An obituary for businessman Oscar M. Mansfield

Embedded Cobordism Categories and Spaces of Manifolds

Galatius, Madsen, Tillmann and Weiss have identified the homotopy type of the classifying space of the cobordism category with objects (d-1)-dimensional manifolds embedded in R^\infty. In this paper we apply the techniques of spaces of manifolds, as developed by the author and Galatius, to identify the homotopy type of the cobordism category with objects (d-1)-dimensional submanifolds of a fixed background manifold M. There is a description in terms of a space of sections of a bundle over M associated to its tangent bundle. This can be interpreted as a form of Poincare duality, relating a space of submanifolds of M to a space of functions on M

Rotational Surfaces in S^3 with constant mean curvature

Very recently Ben Andrews and Haizhong Li showed that every embedded cmc torus in the three dimensional sphere is axially symmetric. There is a two-parametric family of axially symmetric cmc surfaces; more precisely, for every real number H and every C > 2 (H+\sqrt{1+H^2}) there is an axially symmetry surface \Sigma_{H,C} with mean curvature H. In 2010, Perdomo showed that for every H between cot(\pi/m) and (m^2-2)/(2(m^2-1)^1/2), there exists an embedded axially symmetric example with non constant principal curvatures that is invariant under the ciclic group Z_m. Andrews and Li, showed that these examples are the only non-isoparametric embedded examples in the family when H>0. In this paper we study those examples in the family with H<0. We prove that there are no embedded examples in the family when H<0 and we also prove that for every integer m>2 there is a properly immersed example in this family that contains a great circle and is invariant under the ciclic group Z_m. We will say that these examples contain the axis of symmetry. Finally we show that every non-isoparametric surface \Sigma_{H,C} is either properly immersed invariant under the ciclic group Z_m for some integer m>1 or it is dense in the region bounded by two isoparametric tori if the surface \Sigma_{H,C} does not contain the axis of symmetry or it is dense in the region bounded by a totally umbilical surface if the surface \Sigma_{H,C} contains the axis of symmetry.Comment: 13 pages, 8 figure

Piety and Politics: Catholic Revival and the Generation of 1905-1914 in France

Reviewed Book: Cohan, Paul M. Piety and Politics: Catholic Revival and the Generation of 1905-1914 in France. New York: Garland Pub, 1987

Certificate of Death: Hutson, Oscar M.

State of Florida death certificate for Oscar M. Hutson, age 42. Handwritten notes on back

Message Authentication Code over a Wiretap Channel

Message Authentication Code (MAC) is a keyed function $f_K$ such that when Alice, who shares the secret $K$ with Bob, sends $f_K(M)$ to the latter, Bob will be assured of the integrity and authenticity of $M$. Traditionally, it is assumed that the channel is noiseless. However, Maurer showed that in this case an attacker can succeed with probability $2^{-\frac{H(K)}{\ell+1}}$ after authenticating $\ell$ messages. In this paper, we consider the setting where the channel is noisy. Specifically, Alice and Bob are connected by a discrete memoryless channel (DMC) $W_1$ and a noiseless but insecure channel. In addition, an attacker Oscar is connected with Alice through DMC $W_2$ and with Bob through a noiseless channel. In this setting, we study the framework that sends $M$ over the noiseless channel and the traditional MAC $f_K(M)$ over channel $(W_1, W_2)$. We regard the noisy channel as an expensive resource and define the authentication rate $\rho_{auth}$ as the ratio of message length to the number $n$ of channel $W_1$ uses. The security of this framework depends on the channel coding scheme for $f_K(M)$. A natural coding scheme is to use the secrecy capacity achieving code of Csisz\'{a}r and K\"{o}rner. Intuitively, this is also the optimal strategy. However, we propose a coding scheme that achieves a higher $\rho_{auth}.$ Our crucial point for this is that in the secrecy capacity setting, Bob needs to recover $f_K(M)$ while in our coding scheme this is not necessary. How to detect the attack without recovering $f_K(M)$ is the main contribution of this work. We achieve this through random coding techniques.Comment: Formulation of model is change