Zeros of Jones Polynomials for Families of Knots and Links

We calculate Jones polynomials $V_L(t)$ for several families of alternating knots and links by computing the Tutte polynomials $T(G,x,y)$ for the associated graphs $G$ and then obtaining $V_L(t)$ as a special case of the Tutte polynomial. For each of these families we determine the zeros of the Jones polynomial, including the accumulation set in the limit of infinitely many crossings. A discussion is also given of the calculation of Jones polynomials for non-alternating links.Comment: 30 pages, latex, 9 postscript figures; minor rewording on a reference, no changes in result

Designing structured tight frames via an alternating projection method

Tight frames, also known as general Welch-bound- equality sequences, generalize orthonormal systems. Numerous applications - including communications, coding, and sparse approximation- require finite-dimensional tight frames that possess additional structural properties. This paper proposes an alternating projection method that is versatile enough to solve a huge class of inverse eigenvalue problems (IEPs), which includes the frame design problem. To apply this method, one needs only to solve a matrix nearness problem that arises naturally from the design specifications. Therefore, it is the fast and easy to develop versions of the algorithm that target new design problems. Alternating projection will often succeed even if algebraic constructions are unavailable. To demonstrate that alternating projection is an effective tool for frame design, the paper studies some important structural properties in detail. First, it addresses the most basic design problem: constructing tight frames with prescribed vector norms. Then, it discusses equiangular tight frames, which are natural dictionaries for sparse approximation. Finally, it examines tight frames whose individual vectors have low peak-to-average-power ratio (PAR), which is a valuable property for code-division multiple-access (CDMA) applications. Numerical experiments show that the proposed algorithm succeeds in each of these three cases. The appendices investigate the convergence properties of the algorithm

Construction of equiangular signatures for synchronous CDMA systems

Welch bound equality (WBE) signature sequences maximize the uplink sum capacity in direct-spread synchronous code division multiple access (CDMA) systems. WBE sequences have a nice interference invariance property that typically holds only when the system is fully loaded, and, to maintain this property, the signature set must be redesigned and reassigned as the number of active users changes. An additional equiangular constraint on the signature set, however, maintains interference invariance. Finding such signatures requires equiangular side constraints to be imposed on an inverse eigenvalue problem. The paper presents an alternating projection algorithm that can design WBE sequences that satisfy equiangular side constraints. The proposed algorithm can be used to find Grassmannian frames as well as equiangular tight frames. Though one projection is onto a closed, but non-convex, set, it is shown that this algorithm converges to a fixed point, and these fixed points are partially characterized

The best of both worlds? Online ties and the alternating use of social network sites in the context of migration

While an ever-growing body of research is concerned with user behavior on individual social network sites (SNSs)—mostly Facebook—studies addressing an alternating use of two or more SNS are rare. Here, we investigate the relationship between alternating SNS use and social capital in the context of migration. Alternating SNS use avoids some of the problems associated with large networks located on one site; in particular the management of different social or cultural spheres. Not only does this strategy hold potential for increased social capital, it also provides a particular incentive for migrants faced with the challenge of staying in touch with back home and managing a new social environment. Two survey studies are presented that focus on the relationship between alternating SNS use and online ties in a migrant context involving Indian nationals. Study 1 looked at migration within India, whereas Study 2 compared international with domestic SNS users. In both studies, alternating SNS use added to the prediction of online network size and accounted for differences in network size found for migrant and non-migrant users. Differences were due to the number of peripheral ties, rather than core ties. Findings suggest that alternating SNS use may constitute a compensatory strategy that helps to overcome lower levels of socializing represented through a single SNS

CDMA signature sequences with low peak-to-average-power ratio via alternating projection

Several algorithms have been proposed to construct optimal signature sequences that maximize the sum capacity of the uplink in a direct-spread synchronous code division multiple access (CDMA) system. These algorithms produce signatures with real-valued or complex-valued entries that generally have a large peak-to-average power ratio (PAR). This paper presents an alternating projection algorithm that can design optimal signature sequences that satisfy PAR side constraints. This algorithm converges to a fixed point, and these fixed points are partially characterized