Commutativity and collinearity: a historical case study of the interconnection of mathematical ideas. Part II

This paper investigates the discovery of an intriguing and fundamental connection between the famous but apparently unrelated mathematical work of two late third-century mathematicians. This link went unnoticed for well over 1500 years until the publication of two groundbreaking but again ostensibly unrelated works by two German mathematicians at the close of the nineteenth century. In this, the second and final part of the paper, we continue our examination of the chain of mathematical events and the related development of mathematical disciplines, without which the connection might never have been noticed in the first place.     

Common-value all-pay auctions with asymmetric information

We study two-player common-value all-pay auctions in which the players have ex-ante asymmetric information represented by finite connected partitions of the set of states of nature. Our focus is on a family of such auctions in which no player has an information advantage over his opponent. We find sufficient conditions for the existence of equilibrium with monotone strategies, and show that such an equilibrium is unique. We further show that the ex-ante distribution of equilibrium effort is the same for every player (and hence the players’ expected efforts are equal), although their expected payoffs are different and they do not have the same ex-ante probability of winning.     

Poly(ester anhydride)s prepared by the insertion of ricinoleic acid into poly(sebacic acid)

New degradable poly(ester anhydride)s were prepared by the melt polycondensation of diacid oligomers of poly(sebacic acid) (PSA) transesterified with ricinoleic acid. The transesterification of PSA with ricinoleic acid to form oligomers was conducted via a melt bulk reaction between a high molecular weight PSA and ricinoleic acid. A systematic study on the synthesis, characterization, degradation in vitro, drug release, and stability of these polymers was performed. Polymers with weight‐average molecular weights of 2000–60,000 and melting temperatures of 24–77 °C were obtained for PSA containing 20–90% (w/w) ricinoleic acid. NMR and IR analyses indicated the formation of ester bonds along the polyanhydride backbone. These new degradable copolymers have potential use as drug carriers. © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 1059–1069, 2003     

Electrodynamic radiation reaction and general relativity

We argue that the well-known problem of the instabilities that are associated with the self-forces (radiation reaction forces) in classical electrodynamics are possibly stabilized by the introduction of gravitational forces via general relativity.     

Null infinity, H-space and equations of motion

We discuss the existence, arising by analogy to that in algebraically special space-times, of a unique asymptotically shear-free congruence in any asymptotically flat space-time. Associated with it is a unique complex analytic curve in H-space. The surprising potential physical significance of this curve is discussed. Fifth Award in the 2006 Essay Competition of the Gravity Research Foundation.     

Equivalence between bargaining sets and the core in simple games

We investigate the equivalence between several notions of bargaining sets which occur in the literature and the core of simple games.     

A note on asymptotically flat spaces. II

After brief reviews of the Geroch and spin-coefficient formalism approaches to null infinity, we present a dictionary which translates between the two formalisms.     

Frequency Response Envelopes of a Family of Uncertain Discrete-Time Systems

An efficient technique is presented for the calculation of the exact amplitude and phase frequency response envelopes (Bode envelopes) of an infinite impulse response filter (i.e., a real rational function in a complex variable) with interval uncertainty in the coefficients. In the first stage, the order of complexity is reduced from 2ⁿ to n², using the structure of the value set. In the second stage, the problem is completely solved by calculating the frequency response of a finite number of pertinent transfer functions with fixed coefficients. The computations involved in this stage consist of finding a number of zeros of certain polynomials plus some combinatorics. The algorithm extends easily to complex coefficient filters with box uncertainties. Also, the problem trivializes for complex filters with disk uncertainties. However, this latter case leads to extremely simple and completely analytic bounds for the frequency response envelopes of both real and complex coefficient filters.     

Maxwell's equations,linear gravity,and twistors

A detailed outline is presented of several convergent points of view connecting the self-dual and anti-self-dual fields with their free data. This is done for the Maxwell and for linearized gravity as exemplifying the approaches. The Sparling equation provides one tool of great power and characterizes one approach. The twistor theory of Penrose yields another equally powerful point of view. The links between these two basic approaches given in this paper provide a unification that allows workers and others with interest in this area to proceed more readily toward the goal of understanding the full nonlinear Einstein equations.This work has been partially supported by a grant from the NSF.     

Engineering encodable lanthanide-binding tags into loop regions of proteins

Lanthanide-binding tags (LBTs) are valuable tools for investigation of protein structure, function, and dynamics by NMR spectroscopy, X-ray crystallography, and luminescence studies. We have inserted LBTs into three different loop positions (denoted L, R, and S) of the model protein interleukin-1β (IL1β) and varied the length of the spacer between the LBT and the protein (denoted 1?3). Luminescence studies demonstrate that all nine constructs bind Tb3+ tightly in the low nanomolar range. No significant change in the fusion protein occurs from insertion of the LBT, as shown by two X-ray crystallographic structures of the IL1β-S1 and IL1β-L3 constructs and for the remaining constructs by comparing the 1H?15N heteronuclear single-quantum coherence NMR spectra with that of the wild-type IL1β. Additionally, binding of LBT-loop IL1β proteins to their native binding partner in vitro remains unaltered. X-ray crystallographic phasing was successful using only the signal from the bound lanthanide. Large residual dipolar couplings (RDCs) could be determined by NMR spectroscopy for all LBT-loop constructs and revealed that the LBT-2 series were rigidly incorporated into the interleukin-1β structure. The paramagnetic NMR spectra of loop-LBT mutant IL1β-R2 were assigned and the Δχ tensor components were calculated on the basis of RDCs and pseudocontact shifts. A structural model of the IL1β-R2 construct was calculated using the paramagnetic restraints. The current data provide support that encodable LBTs serve as versatile biophysical tags when inserted into loop regions of proteins of known structure or predicted via homology modeling.