Codimension two CR singular submanifolds and extensions of CR functions

Let $M \subset {\mathbb{C}}^{n+1}$, $n \geq 2$, be a real codimension two CR singular real-analytic submanifold that is nondegenerate and holomorphically flat. We prove that every real-analytic function on $M$ that is CR outside the CR singularities extends to a holomorphic function in a neighborhood of $M$. Our motivation is to prove the following analogue of the Hartogs-Bochner theorem. Let $\Omega \subset {\mathbb{C}}^n \times {\mathbb{R}}$, $n \geq 2$, be a bounded domain with a connected real-analytic boundary such that $\partial \Omega$ has only nondegenerate CR singularities. We prove that if $f \colon \partial \Omega \to {\mathbb{C}}$ is a real-analytic function that is CR at CR points of $\partial \Omega$, then $f$ extends to a holomorphic function on a neighborhood of $\overline{\Omega}$ in ${\mathbb{C}}^n \times {\mathbb{C}}$.Comment: 16 pages, 1 figure, fixed typos, updated references. To appear in Journal of Geometric Analysi

On Lewy extension for smooth hypersurfaces in Cn×R{\mathbb C}^n \times {\mathbb R}

We prove an analogue of the Lewy extension theorem for a real dimension $2n$ smooth submanifold $M \subset {\mathbb C}^{n}\times {\mathbb R}$, $n \geq 2$. A theorem of Hill and Taiani implies that if $M$ is CR and the Levi-form has a positive eigenvalue restricted to the leaves of ${\mathbb C}^n \times {\mathbb R}$, then every smooth CR function $f$ extends smoothly as a CR function to one side of $M$. If the Levi-form has eigenvalues of both signs, then $f$ extends to a neighborhood of $M$. Our main result concerns CR singular manifolds with a nondegenerate quadratic part $Q$. A smooth CR $f$ extends to one side if the Hermitian part of $Q$ has at least two positive eigenvalues, and $f$ extends to the other side if the form has at least two negative eigenvalues. We provide examples to show that at least two nonzero eigenvalues in the direction of the extension are needed.Comment: 22 pages; new examples and minor typo fixe

Extension of CR functions from boundaries in Cn×R{\mathbb C}^n \times {\mathbb R}

Let $\Omega \subset {\mathbb C}^n \times {\mathbb R}$ be a bounded domain with smooth boundary such that $\partial \Omega$ has only nondegenerate elliptic CR singularities, and let $f \colon \partial \Omega \to {\mathbb C}$ be a smooth function that is CR at CR points of $\partial \Omega$ (when $n=1$ we require separate holomorphic extensions for each real parameter). Then $f$ extends to a smooth CR function on $\bar{\Omega}$, that is, an analogue of Hartogs-Bochner holds. In addition, if $f$ and $\partial \Omega$ are real-analytic, then $f$ is the restriction of a function that is holomorphic on a neighborhood of $\bar{\Omega}$ in ${\mathbb C}^{n+1}$. An immediate application is a (possibly singular) solution of the Levi-flat Plateau problem for codimension 2 submanifolds that are CR images of $\partial \Omega$ as above. The extension also holds locally near nondegenerate, holomorphically flat, elliptic CR singularities.Comment: 20 pages, to appear in Indiana Univ. Math. J; fixed several typo

Diversification of Fish Culture and Exports among Major Shrimp-producing Countries of Asia: A Spatial and Temporal Analysis

Asian countries contribute nearly 90 per cent to global production of culture fisheries. Diversification of species is not happening in fish culture because of the high value shrimp commands in the export market. Shrimp farming is on the increase in all the major shrimp-producing Asian countries. Higher the growth in shrimp production, higher is the instability in production due to production risks associated with ecosystem complexities and disease incidences. Most of the shrimp-producing countries have been moving towards specialization of shrimp production for export because of the high price that the shrimp asserts in the international market. A comparison between the Asian countries for the period 1976-2002 has revealed that Malaysia has maintained low growth and low instability and higher diversification, and Bangladesh and Vietnam have diversified in fish exports. In a country-level comparison, India, Indonesia, Thailand and Philippines have shown moderate growth in production and export of shrimp. The instability associated with their shrimp production has also been moderate. Diversification in fish production as well as exports has been suggested for assured income to farmers and the country. It has been pointed that replacing, supplementing or rotating the shrimp farming with culture of other high-value fish species like Asian sea bass (Lates calcarifer), grouper (Epinephelus spp), mullets (Mugil spp) and milk fish (Chanos spp) culture may ease the risks in mono-cropping of shrimp.Agricultural and Food Policy,

Domestic fish marketing in India - changing structure, conduct, performance and policies

This study has been conducted in all the major coastal states and some selected inland states to understand the domestic marketing of fish in India. The total marketing costs of auctioneer, wholesaler, retailer, vendor, marine fishermen cooperative society and contractor/freshwater fishermen cooperative society have been found to be Re 0.98, Rs 8.89, Rs 6.61, Rs 4.50, Rs 6.00 and Rs 3.51, respectively. The marketing efficiencies for Indian major carps (IMC), sardine and seer fish have been found to vary from 34 per cent to 74 per cent, depending on the length of market channel. The marketing efficiency has been found more in the case of marine species than freshwater species, since the latter travel longer distances from the point of production to consumption centre, passing many intermediaries as compared to the former. The fisherman’s share in consumer’s rupee has shown variations across species, marketing channels and markets. The infrastructure facilities at most of the surveyed landing centres, fishing harbours and wholesale and retail markets have been found grossly inadequate and poorly maintained. The study has highlighted the need for formulating a uniform market policy for fishes for easy operation and regulation so that the country’s fish production is efficiently managed and delivered to the consuming population, ensuring at the same time remunerative prices to the fishers.Marketing,

Wireless strain sensing system for assessing condition of civil infrastructure facilities

Bežični senzori i senzorske mreže postaju zamjena za tradicionalne sustave za praćenje ponašanja konstrukcija. Njihova je prednost u nižoj cijeni ugradnje jer nema potrebe za polaganjem dugih kabela između senzora i sustava za prikupljanje podataka. U radu se opisuju istraživanja provedena u svrhu ocjenjivanja učinkovitosti bežičnih senzora za mjerenje relativnih deformacija. Dan je primjer primjene bežičnog senzorskog sustava pri mjerenju ponašanja konstrukcije željezničkog mosta te je napravljena usporedba s primjenom klasičnih sustava.Wireless sensors and sensor networks are emerging as substitutes for traditional structural monitoring systems. Their benefit lies in a lower cost of installation because extensive wiring is no longer required between sensors and the data acquisition system. Studies carried out to evaluate performance of wireless strain measurement units are described in this paper. An example is given of a wireless system used for measuring behaviour of a railway bridge, and comparison with traditional systems is made