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Portland stone

Portland Stone or Portland Stone Formation is a limestone formation from the Tithonian stage of the Jurassic period quarried on the Isle of Portland, England. The quarries are cut in beds of white-grey limestone separated by chert beds, it has been used extensively as a building stone throughout the British Isles, notably in major public buildings in London such as St Paul's Cathedral and Buckingham Palace. Portland Stone is exported to many countries—being used for example in the United Nations headquarters building in New York City. Portland Stone formed in a marine environment, on the floor of a shallow, sub-tropical sea near land; when seawater is warmed by the sun, its capacity to hold dissolved gas is reduced. Calcium and bicarbonate ions within the water are able to combine, to form calcium carbonate as a precipitate; the process of limescale build up in a kettle in hard-water areas is similar. Calcium carbonate is the principal constituent of most limestones. Billions of minute crystals of precipitated calcium carbonate accumulated forming lime mud which covered the sea floor.

Small particles of sand or organic detritus, such as shell fragments, formed a nucleus, which became coated with layers of calcite as they were rolled around in the muddy micrite. Portland stone measures 3.5 on the Mohs scale of mineral hardness. The calcite accumulated around the fragments of shell in concentric layers, forming small balls; this process is similar to the way in which a snowball grows in size as it is rolled around in the snow. Over time, countless billions of these balls, known as "ooids" or "ooliths", became cemented together by more calcite, to form the oolitic limestone, called Portland stone; the degree of cementation in Portland stone is such that the stone is sufficiently well cemented to allow it to resist weathering, but not so well cemented that it can't be worked by masons. This is one of the reasons why Portland stone is so favoured as a monumental and architectural stone. Dr Geoff Townson conducted three years doctoral research on the Portlandian, being the first to describe the patch-reef facies and Dorset-wide sedimentation details.

Dr Ian West of the School of Ocean and Earth Sciences at Southampton University completed a detailed geological survey of Withies Croft Quarry before the Portland Beds were quarried by Albion Stone plc. Stone has been quarried on Portland since Roman times and was being shipped to London in the 14th century. Extraction as an industry began in the early 17th century, with shipments to London for Inigo Jones' Banqueting House. Wren's choice of Portland for the new St Paul's Cathedral was a great boost for the quarries and established Portland as London's choice of building stone; the island was connected by railway to the rest of the country from 1865. Albion Stone PLC has been quarrying and mining Portland stone since 1984. Portland Stone Firms Ltd have been quarrying Portland stone since 1994. Jordans is part of the Inmosthay Quarry in the centre of the Island, which includes Fancy Beach; the quarry has been worked since the late 19th century. Albion Stone leases the southern section from The Crown Estate and purchased the northern part of the site in 2006.

The majority of the southern reserves lie under the grounds of the local cricket club. To avoid disturbing the site at surface level, the company has applied and received permission to extract the stone using mining rather than quarrying techniques; the reserves to the north will be quarried using the diamond bladed cutting machines, hydro bags and wire saws to shape the blocks. This process avoids the use of dusty and noisy blasting as the primary extraction method, thereby protecting the surrounding environment, designated as a Site of Special Scientific Interest. Albion Stone PLC now extract all their stone through mining which reduces the impact on the environment and local residence. Jordan's Mine is the biggest mine on Portland. Bowers Quarry has been operational since the late 18th century, it has been leased from The Crown Estate since 1979, in 2002 it became the site of the first Portland stone mine by Albion Stone PLC. Extraction from this site is now underground, with the original Bowers Mine in the extreme southern end of the quarry and the High Wall Extraction on the eastern and south east boundaries.

High Wall Extraction is a series of small mines that extract otherwise wasted stone that sits between the final faces of the quarry and the actual boundary of the site. Stonehills Mine is the first new mine on Portland. Albion Stone Plc began the process to open this mine began in 2015 and reserves are estimated to last for 50 years. Coombefield Quarry, located near Southwell has been open cast quarried over the last 80 years and is one of three owned quarries by Portland Stone Firms Ltd, the largest landholder on the Island; the quarry is nearing the end of its life and will be regenerated as a holiday caravan park to boost local tourism on the Island. Perryfield Quarry is found towards the middle of the Island and being open cast quarried. There are over 20 years of reserves left, owned by Portland Stone Firms. Open cast quarrying provides quicker extraction of raw block dimension stone whilst maintaining its integrity; the majority of buildings in London today use Portland, quarried using the same methods over the last

Kohoj

Kohojकोहोज किल्ला is a medieval military fortification located near Palghar in Palghar district in Maharashtra, India. Kohoj is located about 104 km from Mumbai off the Mumbai - Ahmedabad Highway. Upon reaching Manor you have to take a right turn under a newly built flyover and take the Manor - Wada road, it is around 10 km from the right turn. A few minutes drive on this road and one can see the Kohoj fort on your right. At the bottom of the fort is a lake known as'Pazhar'; the fort is said to be about 800 years old and finds a mention in the Purandar treaty along with 22 other forts which were surrendered by Shivaji Maharaj to the Mughals on 11 June 1665. It was an important fort for keeping a look on the nearby coast and thus to keep the territory safe and secure. Kohoj Fort has remained abandoned for centuries. Kohoj has a moderately difficult gradient and it takes about two and a half hours to reach the main plateau from Vaghote village, situated at the base of the fort. Access to Kohoj Fort is via the main trek trail at which breaks from the Kohoj Fort road near the Saltwater lake on the Wada-Manor road.

On reaching the plateau of the fort, a temple of Lord Shankar can be seen, which has now been rehabilitated. There were ten rock cut cisterns on the fort Two cisterns. One of the routes descends from the left side of the temple, where seven adjoined cisterns are located. One of the cisterns bears clean water; the other two cisterns have been the remaining four being clogged. Few remnants in dilapidated state can be seen to the right of the temple. Fortification is seen at some of the places on the fort; the other way stretches from the right side of the temple up the hill. Three large cisterns can be seen on this route, one of, clogged and the other two contain water; this water can be used for drinking. An idol of Lord Hanumana can be seen near these cisterns. A man-shaped pinnacle created due to wind-erosion is one of the best natural sites on the fort. Different shapes of this pinnacle are evident. A temple of Lord Krishna is situated ahead along this route. To reach the highest point it takes another 10–15 minutes.

The height of the fort is about 3200 feet. From the top of the fort are seen Tandulwadi fort, Takmak fort, Asheri fort, Mahalaxmi pinnacle and the Arabian sea. There is no accommodation on the fort; the temple of Lord Shiva can hardly accommodate two persons at a time

Pontryagin duality

In mathematics in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact abelian groups, such as R, the circle, or finite cyclic groups. The Pontryagin duality theorem itself states that locally compact abelian groups identify with their bidual; the subject is named after Lev Semenovich Pontryagin who laid down the foundations for the theory of locally compact abelian groups and their duality during his early mathematical works in 1934. Pontryagin's treatment relied on the group being second-countable and either discrete; this was improved to cover the general locally compact abelian groups by Egbert van Kampen in 1935 and André Weil in 1940. Pontryagin duality places in a unified context a number of observations about functions on the real line or on finite abelian groups: Suitably regular complex-valued periodic functions on the real line have Fourier series and these functions can be recovered from their Fourier series.

Moreover, any function on a finite group can be recovered from its discrete Fourier transform. The theory, introduced by Lev Pontryagin and combined with the Haar measure introduced by John von Neumann, André Weil and others depends on the theory of the dual group of a locally compact abelian group, it is analogous to the dual vector space of a vector space: a finite-dimensional vector space V and its dual vector space V* are not isomorphic, but the endomorphism algebra of one is isomorphic to the opposite of the endomorphism algebra of the other: End ≅ End op, via the transpose. A group G and its dual group G ^ are not in general isomorphic, but their endomorphism rings are opposite to each other: End ≅ End op. More categorically, this is not just an isomorphism of endomorphism algebras, but a contravariant equivalence of categories – see categorical considerations. A topological group is called locally compact if the underlying topological space is locally compact and Hausdorff. Examples of locally compact abelian groups include finite abelian groups, the integers, the real numbers, the circle group T, the p-adic numbers.

For a locally compact abelian group G, the Pontryagin dual is defined to be the group G ^ of continuous group homomorphisms from G to the circle group T. That is, G ^:= Hom ⁡ G ^ can be endowed with the topology given by uniform convergence on compact sets. For example, Z ^ = T, R ^ = R, T ^ = Z. Theorem. There is a canonical isomorphism G ≅ G ^ ^ between any locally compact abelian group G and its double dual. Canonical means that there is a defined map ev G: G → G ^ ^; the canonical isomorphism is defined on x ∈ G as follows: ev G ⁡ = i.e. ev G ⁡:= χ ∈ T. In other words, each group element x is identified to the evaluation character on the dual; this is analogous to the canonical isomorphism between a finite-dimensional vecto