Theta Bodies for Polynomial Ideals

Inspired by a question of Lov\'asz, we introduce a hierarchy of nested semidefinite relaxations of the convex hull of real solutions to an arbitrary polynomial ideal, called theta bodies of the ideal. For the stable set problem in a graph, the first theta body in this hierarchy is exactly Lov\'asz's theta body of the graph. We prove that theta bodies are, up to closure, a version of Lasserre's relaxations for real solutions to ideals, and that they can be computed explicitly using combinatorial moment matrices. Theta bodies provide a new canonical set of semidefinite relaxations for the max cut problem. For vanishing ideals of finite point sets, we give several equivalent characterizations of when the first theta body equals the convex hull of the points. We also determine the structure of the first theta body for all ideals.Comment: 26 pages, 3 figure

Approximate cone factorizations and lifts of polytopes

In this paper we show how to construct inner and outer convex approximations of a polytope from an approximate cone factorization of its slack matrix. This provides a robust generalization of the famous result of Yannakakis that polyhedral lifts of a polytope are controlled by (exact) nonnegative factorizations of its slack matrix. Our approximations behave well under polarity and have efficient representations using second order cones. We establish a direct relationship between the quality of the factorization and the quality of the approximations, and our results extend to generalized slack matrices that arise from a polytope contained in a polyhedron

On the local stability of semidefinite relaxations

We consider a parametric family of quadratically constrained quadratic programs (QCQP) and their associated semidefinite programming (SDP) relaxations. Given a nominal value of the parameter at which the SDP relaxation is exact, we study conditions (and quantitative bounds) under which the relaxation will continue to be exact as the parameter moves in a neighborhood around the nominal value. Our framework captures a wide array of statistical estimation problems including tensor principal component analysis, rotation synchronization, orthogonal Procrustes, camera triangulation and resectioning, essential matrix estimation, system identification, and approximate GCD. Our results can also be used to analyze the stability of SOS relaxations of general polynomial optimization problems.Comment: 23 pages, 3 figure

From horticulture to psychonautics: an analysis of online communities discussing and trading plants with psychotropic properties.

This study is a spinoff of the cross-disciplinary project “FloraGuard: Tackling the Illegal Trade in Endangered Plants”, and focuses on the analysis of online forums dedicated to the discussion and the trades of plant species, often highly endangered in nature, that are sought after for their psychotropic properties. The study sheds light on the interesting but overlooked area of the intersection of environmental crimes, illegal online trades, and drug use. Some species of conservation concern have known psychoactive/analgesic properties; as these properties are now openly and broadly discussed in specialised online communities, attention is required both as regards the potential for health-related harms suffered by reckless users, and for environmental-related harms for the species in question

Lifts of convex sets and cone factorizations

In this paper we address the basic geometric question of when a given convex set is the image under a linear map of an affine slice of a given closed convex cone. Such a representation or 'lift' of the convex set is especially useful if the cone admits an efficient algorithm for linear optimization over its affine slices. We show that the existence of a lift of a convex set to a cone is equivalent to the existence of a factorization of an operator associated to the set and its polar via elements in the cone and its dual. This generalizes a theorem of Yannakakis that established a connection between polyhedral lifts of a polytope and nonnegative factorizations of its slack matrix. Symmetric lifts of convex sets can also be characterized similarly. When the cones live in a family, our results lead to the definition of the rank of a convex set with respect to this family. We present results about this rank in the context of cones of positive semidefinite matrices. Our methods provide new tools for understanding cone lifts of convex sets.Comment: 20 pages, 2 figure

Chromosome Studies and Karyotype Analysis of some Triploid Banana (Musa Species) Cultivars of AAA Genomic Group

Bananas are the highly evolved, oldest fruits known to mankind. The Cavendish group cultivars are popular commercial varieties. AAA genomic group cultivars are said to have evolved from the wild AA Musa acuminata species by natural hybridization and polyploidization and these vigorous triploids were selected by man for cultivation. Basic cytological studies on banana are comparatively few due to the plant's complex nature. In this report, karyo-morphological studies on five AAA Cavendish group cultivars i.e. Robusta, Dwarf Cavendish, Grand Naine, Gros Michel and Red banana are reported. All the five cultivars had similar karyotype, except cv. Robusta. Total chromosome length was highest in Red banana and lowest in cv. Gros Michel

Towards conceptualizing child wellbeing in India: The need for a paradigm shift

Globally, there is a vast array of social indicators, many of these specifically oriented to the lives, experience and needs of children. This approach is much more advanced in developed economies and rich countries, where the focus has widened and shifted progressively towards a full recognition of the nonmonetary dimensions of child wellbeing. At present, there would appear to be a propitious academic, activist and policy conjuncture for the widening of the discourse on child deprivation in India. This environment is created partly by the emerging reporting requirements and exhortations of the international development regime. But it is also fuelled by dissatisfaction over the inability of the existing methodologies, dominated by the reductionist monetary poverty line approach, to provide a meaningful intellectual or operational frame for contending with issues of child wellbeing in a holistic manner. The basic argument of this paper is that a double paradigm shift is urgently necessary: from mainstream approaches which tend to focus overwhelmingly on the material poverty and deprivation experienced by some children, deemed by definition to be those in households-in-poverty, to one that widens the field of vision to include both material and non-material dimensions of wellbeing of all children. Clearly, fresh epistemological and methodological challenges will have to be met with innovative and creative responses. It is time for India to catch up with best practices in rich countries, and given the impressive dimensions of India's academic and professional infrastructure, this should not be an unrealistic goal

From Poverty to Wellbeing: Alternative Approaches to the Recognition of Child Deprivation in India

Ways of seeing influence ways of doing; so there is much to be gained potentially by a thorough stock-taking and interrogation of the habitual methods and techniques employed in the field of child poverty measurement in India. The basic argument of this paper is that a paradigm shift is urgently necessary: from the mainstream approach which tends to focus overwhelmingly on the material poverty and deprivation experienced by some children, deemed by definition to be those in households-in-poverty, to one that widens the field of vision to include both material and non-material dimensions of wellbeing of all children. Such a shift carries significant implications for modes of conceptualization and recognition; for the focus and substantive content of analysis, for the choice of methods and tools, for the framing and design of policies and interventions, and more generally for the scope of debates and discourse pertaining to the development rights of children

Ploidy analysis among Citrus mutants using leaf meristematic tissue

A promising method for preparing metaphase spread for counting the number of chromosomes from the emerging shoot tissue is described in this report. In the present study, we adopted enzymatic digestion of shoot tips to analyse the chromosome number. The chromosomes in metaphase stage of cell division are highly condensed and easy to count in routine cytological technique. Even the morphological features like position of centromere can be seen in metaphase. In prophase it may not be clear as the chromosomes are getting ready for cell division. In enzymatic digestion even the prophase chromosomes are visible, which can be counted. Hence enzymatic digestion technique is more efficient in citrus as compared to acid digestion method as the citrus crop is a perennial crop with small-sized chromosomes. Furthermore, the sample collection in the field was easy and actively growing vegetative flush was available throughout the year. This technique was attempted in the tissue culture lab of ICAR- CCRI in various in vito and in vivo ploidy induction experiments in Citrus sinensis Osbeck (Sweet orange cv. mosambi), C. reticulata Blanco (Nagpur mandarin) and C. jambhiri Lush (Rough lemon), for confirmation of diploidy (2n=2x=18), triploidy (2n=3x=27), tetraploid (2n=4x=36), hexaploid (2n=6x=54)