The local polynomial hull near a degenerate CR singularity: Bishop discs revisited

Let be a smooth real surface in and let be a point at which the tangent plane is a complex line. How does one determine whether or not is locally polynomially convex at such a p-i.e. at a CR singularity? Even when the order of contact of with at p equals 2, no clean characterisation exists; difficulties are posed by parabolic points. Hence, we study non-parabolic CR singularities. We show that the presence or absence of Bishop discs around certain non-parabolic CR singularities is completely determined by a Maslov-type index. This result subsumes all known facts about Bishop discs around order-two, non-parabolic CR singularities. Sufficient conditions for Bishop discs have earlier been investigated at CR singularities having high order of contact with . These results relied upon a subharmonicity condition, which fails in many simple cases. Hence, we look beyond potential theory and refine certain ideas going back to Bishop.

A Nash Bargaining Approach to Retention Enhancing Bid Optimization in Sponsored Search Auctions with Discrete Bids

Bid optimization is now becoming quite popular in sponsored search auctions on the Web. Given a keyword and the maximum willingness to pay of each advertiser interested in the keyword, the bid optimizer generates a profile of bids for the advertisers with the objective of maximizing customer retention without compromising the revenue of the search engine. In this paper, we present a bid optimization algorithm that is based on a Nash bargaining model where the first player is the search engine and the second player is a virtual agent representing all the bidders. We make the realistic assumption that each bidder specifies a maximum willingness to pay values and a discrete, finite set of bid values. We show that the Nash bargaining solution for this problem always lies on a certain edge of the convex hull such that one end point of the edge is the vector of maximum willingness to pay of all the bidders. We show that the other endpoint of this edge can be computed as a solution of a linear programming problem. We also show how the solution can be transformed to a bid profile of the advertisers.

A Fast Linear Separability Test by Projection of Positive Points on Subspaces

A geometric and non parametric procedure for testing if two finite set of points are linearly separable is proposed. The Linear Separability Test is equivalent to a test that determines if a strictly positive point h > 0 exists in the range of a matrix A (related to the points in the two finite sets). The algorithm proposed in the paper iteratively checks if a strictly positive point exists in a subspace by projecting a strictly positive vector with equal co-ordinates (p), on the subspace. At the end of each iteration, the subspace is reduced to a lower dimensional subspace. The test is completed within r ≤ min(n, d + 1) steps, for both linearly separable and non separable problems (r is the rank of A, n is the number of points and d is the dimension of the space containing the points). The worst case time complexity of the algorithm is O(nr3) and space complexity of the algorithm is O(nd). A small review of some of the prominent algorithms and their time complexities is included. The worst case computational complexity of our algorithm is lower than the worst case computational complexity of Simplex, Perceptron, Support Vector Machine and Convex Hull Algorithms, if d

Polynomial Approximation, Local Polynomial convexity, and Degenerate CR Singularities- II

We provide some conditions for the graph of a Holder-continuous function on (D) over bar, where (D) over bar is a closed disk in C, to be polynomially convex. Almost all sufficient conditions known to date - provided the function (say F) is smooth - arise from versions of the Weierstrass Approximation Theorem on (D) over bar. These conditions often fail to yield any conclusion if rank(R)DF is not maximal on a sufficiently large subset of (D) over bar. We bypass this difficulty by introducing a technique that relies on the interplay of certain plurisubharmonic functions. This technique also allows us to make some observations on the polynomial hull of a graph in C(2) at an isolated complex tangency.

Rice LHS1/OsMADS1 Controls Floret Meristem Specification by Coordinated Regulation of Transcription Factors and Hormone Signaling Pathways

SEPALLATA (SEP) MADS box transcription factors mediate floral development in association with other regulators. Mutants in five rice (Oryza sativa) SEP genes suggest both redundant and unique functions in panicle branching and floret development. LEAFY HULL STERILE1/OsMADS1, from a grass-specific subgroup of LOFSEP genes, is required for specifying a single floret on the spikelet meristem and for floret organ development, but its downstream mechanisms are unknown. Here, key pathways and directly modulated targets of OsMADS1 were deduced from expression analysis after its knockdown and induction in developing florets and by studying its chromatin occupancy at downstream genes. The negative regulation of OsMADS34, another LOFSEP gene, and activation of OsMADS55, a SHORT VEGETATIVE PHASE-like floret meristem identity gene, show its role in facilitating the spikelet-to-floret meristem transition. Direct regulation of other transcription factor genes like OsHB4 (a class III homeodomain Leu zipper member), OsBLH1 (a BEL1-like homeodomain member), OsKANADI2, OsKANADI4, and OsETTIN2 show its role in meristem maintenance, determinacy, and lateral organ development. We found that the OsMADS1 targets OsETTIN1 and OsETTIN2 redundantly ensure carpel differentiation. The multiple effects of OsMADS1 in promoting auxin transport, signaling, and auxin-dependent expression and its direct repression of three cytokinin A-type response regulators show its role in balancing meristem growth, lateral organ differentiation, and determinacy. Overall, we show that OsMADS1 integrates transcriptional and signaling pathways to promote rice floret specification and development.