Algebraic geometry

Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques from commutative algebra, for solving geometrical problems about these sets of zeros; the fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, parabolas, hyperbolas, cubic curves like elliptic curves, quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the curve and relations between the curves given by different equations.

Algebraic geometry occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex analysis and number theory. A study of systems of polynomial equations in several variables, the subject of algebraic geometry starts where equation solving leaves off, it becomes more important to understand the intrinsic properties of the totality of solutions of a system of equations, than to find a specific solution. In the 20th century, algebraic geometry split into several subareas; the mainstream of algebraic geometry is devoted to the study of the complex points of the algebraic varieties and more to the points with coordinates in an algebraically closed field. Real algebraic geometry is the study of the real points of an algebraic variety. Diophantine geometry and, more arithmetic geometry is the study of the points of an algebraic variety with coordinates in fields that are not algebraically closed and occur in algebraic number theory, such as the field of rational numbers, number fields, finite fields, function fields, p-adic fields.

A large part of singularity theory is devoted to the singularities of algebraic varieties. Computational algebraic geometry is an area that has emerged at the intersection of algebraic geometry and computer algebra, with the rise of computers, it consists of algorithm design and software development for the study of properties of explicitly given algebraic varieties. Much of the development of the mainstream of algebraic geometry in the 20th century occurred within an abstract algebraic framework, with increasing emphasis being placed on "intrinsic" properties of algebraic varieties not dependent on any particular way of embedding the variety in an ambient coordinate space. One key achievement of this abstract algebraic geometry is Grothendieck's scheme theory which allows one to use sheaf theory to study algebraic varieties in a way, similar to its use in the study of differential and analytic manifolds; this is obtained by extending the notion of point: In classical algebraic geometry, a point of an affine variety may be identified, through Hilbert's Nullstellensatz, with a maximal ideal of the coordinate ring, while the points of the corresponding affine scheme are all prime ideals of this ring.

This means that a point of such a scheme may be either a subvariety. This approach enables a unification of the language and the tools of classical algebraic geometry concerned with complex points, of algebraic number theory. Wiles' proof of the longstanding conjecture called Fermat's last theorem is an example of the power of this approach. In classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, meaning the set of all points that satisfy one or more polynomial equations. For instance, the two-dimensional sphere of radius 1 in three-dimensional Euclidean space R3 could be defined as the set of all points with x 2 + y 2 + z 2 − 1 = 0. A "slanted" circle in R3 can be defined as the set of all points which satisfy the two polynomial equations x 2 + y 2 + z 2 − 1 = 0, x + y + z = 0. First we start with a field k. In classical algebraic geometry, this field was always the complex numbers C, but many of the same results are true if we assume only that k is algebraically closed.

We consider the affine space of dimension n over denoted An. When one fixes a coordinate system, one may identify An with kn; the purpose of not working with kn is to emphasize that one "forgets" the vector space structure that kn carries. A function f: An → A1 is said to be polynomial if it can be written as a polynomial, that is, if there is a polynomial p in k such that f = p for every point M with coordinates in An; the property of a function to be polynomial does not depend on the choice of a coordinate system in An. When a coordinate system is chosen, the regular functions on the affine n-space may be identified with the ring of polynomial functions in n variables over k. Therefore, the set of the

Naval Aviation Museum (India)

The Naval Aviation Museum is a military museum located in Bogmalo, 6 km from Vasco da Gama, India. This museum contains exhibits; the museum is divided into an outdoor exhibit and a two-storey indoor gallery. The museum was inaugurated in October 1998 and is one of the two military aviation museums in India, the other being the Indian Air Force Museum, Palam in Delhi; the Naval Aviation Museum is the only of its kind in Asia. The outdoor exhibit is a huge park that visitors can walk through and see decommissioned aircraft that saw service with the Navy; some of them date back to before the 1940s. A small shed displays various aircraft engines. A total of 13 aircraft are on display. Short Sealand Mk 2 - IN 106 is the only surviving example in India and one of three known surviving Sealands in the world; the Sealand was the first aircraft type to be inducted after the establishment of the Directorate of Naval Aviation in 1953. They were phased out in 1965. Fairey Firefly TT Mk1 - IN 112 is the sole surviving example in India of the British WW2-era carrier-borne fighter and anti-submarine aircraft, acquired in May 55 for target towing purposes.

HAL HT-2 - BX 748. The Navy used the HT-2 primary trainers from 1956 to 1964; the current example on display has IAF markings. de Havilland Vampire T-55 - IN 149. The T55 two seater variant of the Vampire was procured in September 1957 to train Naval airmen on Jet aircraft before the Navy inducted its Sea Hawks. Hawker Sea Hawk FGA Mk 100 - IN 234; the Sea Hawks entered service along with INS Vikrant, India's first Aircraft carrier and served for two decades before being replaced by the Sea Harriers. Breguet Alizé - IN 202; the Alize was the Navy's first carrier based Anti Submarine and Maritime Surveillance aircraft and entered service in 1961. de Havilland Dove - IN 124. The Dove was procured from the Indian Air Force in 1965 to replace the Short Sealand that were being phased out. HAL Chetak - IN 475 entered service with INS Vikrant in 1961 as a Search and Rescue helicopter. Hughes Hu-300 - IN 083; the Hughes Hu300 two seater helicopters were inducted in 1971 for ab-initio training of helicopter pilots and were phased out in the mid 1980s.

Westland Sea King Mk 42. - IN 505 was procured in 1970 to fulfill an Anti Submarine Warfare role in the Navy. Lockheed L1049G Super Constellation - IN 315; the main attraction among the outdoor exhibits, this Lockheed L-1049G was delivered to Air India in 1955 and named "Rani of Ellora". It was transferred to the Indian Air Force in 1961 on to the Naval Air Arm in 1976 and retired in 1983. Kamov Ka-25 - IN 573; the Ka-25s, commissioned in 1980 were for Anti-Submarine Warfare with secondary surveillance and Search and Rescue duties. Sea Harrier FRS.51 - IN 621. The single seater Sea Harrier on display was delivered in 1991 and was based both on the INS Vikrant as well as the INS Viraat. Inside, visitors can read and learn about key battles the Indian Air and Naval forces have participated in; the indoor gallery is divided into special rooms designated for a certain topic. Some of them are'armament', which show military weapons that are attached to ships. On display are many rare and vintage photographs and documents that show several important periods in Naval Aviation History from 1959 onwards.

One gallery holds massive replicas of the INS Vikrant and INS Viraat. Another gallery holds a variety of bombs, torpedoes and cannons used by the Indian Navy; the Museum lies on Naval property adjoining the South side of Dabolim Airport, on Bogmalo Road off National Highway 17. Admission to the Naval Aviation Museum costs 20 Rupees for adults. Cameras would be charged separately; the museum is open on all days from 10:00 am -- 5:00 pm, except Monday. Naval Aviation Museum Pictures

Norby Chabert

Norbert Nolty Chabert, known as Norby Chabert, is a former political consultant from Houma, a former Republican member of the Louisiana State Senate for District 20 in Terrebonne and Lafourche parishes. From 2009 until 2020, he held the same seat occupied from 1980 to 1991 by his late father, Leonard J. Chabert, from 1992 to 1996 by his older brother, Marty James Chabert, both Democrats. Chabert's parents, Leonard Chabert and the former Viona Lapeyrouse, were born in Chauvin in Terrebonne Parish, he was born and reared on the banks of Bayou Petite Caillou in the community of Lil Caillou in Terrebonne Parish. He graduated in 1994 from South Terrebonne High School in Bourg. In 2001, Norby Chabert received his Bachelor of Arts degree in government from Nicholls State University in Thibodaux, Louisiana. Prior to his entering the Senate, he was a political consultant, a businessman, as associate director of marketing at Nicholls State, he is affiliated with the Chamber of Commerce. An active sportsman, he resides in "Old Houma" near Maple Avenue Park.

He describes his religious views as "Christian." Norbert Chabert himself was a Democrat until March 2011, when that same month he joined Jody Amedee of Gonzales in switching to GOP allegiance. The changes produced a numerical Republican majority in the upper legislative chamber; the preceding year, nearly 47 percent of Chabert's constituents who voted in the special election cast ballots for the defeated Republican candidate. Chabert won the Senate seat in special election in the summer of 2009 to succeed Democrat Reggie Paul Dupre, Jr. Chabert ran second in the contest with 4,359 votes. Republican Brent Callais led with 5,055 votes. A second Democrat, Damon J. Baldone polled the critical 3,957 votes. In the second round of balloting, Chabert defeated Callais, 9,576 to 8,050. In 2010, Chabert voted to require sonograms for women contemplating abortion, he vote against allowing insurance companies to cover elective abortions. Chabert voted to impose penalties to those participating in cockfighting.

He supported making the state tobacco tax permanent. In 2010, Chabert voted 55 percent of the time with the Louisiana Association of Business and Industry; the Louisiana Family Forum gave him a 78 percent rating. In 2011, Chabert voted against the congressional redistricting bill because Houma and Thibodaux have been split between the First and Third districts. Chabert ran unopposed for a second term in the Senate in the nonpartisan blanket primary held on October 22, 2011, he won his third Senate term in 2015 with 11,921 votes in a race against his fellow Republican Michael "Mike" Fesi, who polled 9,944 votes. The remaining 1,456 ballots went to Mark Atzenhoffer. On February 2, 2013, the Chaberts, Leonard J. Marty J. and Norbert N. were inducted into the Louisiana Political Museum and Hall of Fame in Winnfield, along with several other individuals, including former Sheriff Leonard R. "Pop" Hataway of Grant Parish, the late State Senator Charles C. Barham, George Dement, the former mayor of Bossier City