Topological and statistical properties of nonlinear force-free fields

We use our semi-analytic solution of the nonlinear force-free field equation to construct three-dimensional magnetic fields that are applicable to the solar corona and study their statistical properties for estimating the degree of braiding exhibited by these fields. We present a new formula for calculating the winding number and compare it with the formula for the crossing number. The comparison is shown for a toy model of two helices and for realistic cases of nonlinear force-free fields; conceptually the formulae are nearly the same but the resulting distributions calculated for a given topology can be different. We also calculate linkages, which are useful topological quantities that are independent measures of the contribution of magnetic braiding to the total free energy and relative helicity of the field. Finally, we derive new analytical bounds for the free energy and relative helicity for the field configurations in terms of the linking number. These bounds will be of utility in estimating the braided energy available for nano-flares or for eruptions.Comment: 22 pages, 8 figures, 1 table, to appear in a special issue of Advances in Space Research on the "Dynamic Sun

Addendum to `Fake Projective Planes'

The addendum updates the results presented in the paper `Fake Projective Plane, Invent Math 168, 321-370 (2007)' and makes some additions and corrections. The fake projective planes are classified into twenty six classes. Together with a recent work of Donald Cartwright and Tim Steger, there is now a complete list of fake projective planes. There are precisely one hundred fake projective planes as complex surfaces classified up to biholomorphism.Comment: A more refined classification is given in the new versio

A Technique for Duplicating a Fixed Complete Denture

The fixed implant prosthesis often requires repair due to fracture and/or wear of acrylic resin teeth. Repair of such a prosthesis, although simple, requires retaining the prosthesis to be revised in the laboratory. In order to reveneer the implant prosthesis, a provisional restoration should be fabricated for the patient that provides similar function and esthetics. This article presents a technique for duplicating the existing fixed implant-supported prosthesis prior to repair/reveneer, thus providing a smooth transition from the provisional to the definitive prosthesis

Biomaterial Properties of Titanium in Dentistry

Background Among various dental materials and their successful restorative uses, titanium provides an excellent example of integrating science and technology involving multiple disciplines of dentistry including biomaterials, prosthodontics and surgical sciences. Titanium and its alloys have emerged as a material of choice for dental implants fulfilling all requirements biologically, chemically and mechanically. Several excellent reviews have discussed the properties of titanium and its surface characteristics that render it biocompatible. However, in most patients, titanium implants are used alongside several other metals. Presence of different metals in the same oral environment can alter the properties of titanium. Other influencing factors include intra-oral pH, salivary content, and effect of fluorides. Highlight This review discusses the effect of the above-mentioned conditions on the properties of titanium and its alloys. An extensive literature search encompassing the properties of titanium in an altered oral environment and its interaction with other restorative materials is presented. Specific conditions that could cause titanium to corrode, specifically due to interaction with other dental materials used in oral rehabilitation, as well as methods that can be employed for passivation of titanium are discussed. Conclusion This review presents an overview of the properties of titanium that are vital for its use in implant dentistry. From a restorative perspective, interaction between implant restoration metals, intra-oral fluorides and pH may cause titanium to corrode. Therefore, in order to avoid the resulting deleterious effects, an understanding of these interactions is important for long-term prognosis of implant restorations

On the Gaussian Many-to-One X Channel

In this paper, the Gaussian many-to-one X channel, which is a special case of general multiuser X channel, is studied. In the Gaussian many-to-one X channel, communication links exist between all transmitters and one of the receivers, along with a communication link between each transmitter and its corresponding receiver. As per the X channel assumption, transmission of messages is allowed on all the links of the channel. This communication model is different from the corresponding many-to-one interference channel (IC). Transmission strategies which involve using Gaussian codebooks and treating interference from a subset of transmitters as noise are formulated for the above channel. Sum-rate is used as the criterion of optimality for evaluating the strategies. Initially, a $3 \times 3$ many-to-one X channel is considered and three transmission strategies are analyzed. The first two strategies are shown to achieve sum-rate capacity under certain channel conditions. For the third strategy, a sum-rate outer bound is derived and the gap between the outer bound and the achieved rate is characterized. These results are later extended to the $K \times K$ case. Next, a region in which the many-to-one X channel can be operated as a many-to-one IC without loss of sum-rate is identified. Further, in the above region, it is shown that using Gaussian codebooks and treating interference as noise achieves a rate point that is within $K/2 -1$ bits from the sum-rate capacity. Subsequently, some implications of the above results to the Gaussian many-to-one IC are discussed. Transmission strategies for the many-to-one IC are formulated and channel conditions under which the strategies achieve sum-rate capacity are obtained. A region where the sum-rate capacity can be characterized to within $K/2-1$ bits is also identified.Comment: Submitted to IEEE Transactions on Information Theory; Revised and updated version of the original draf