In mathematical logic, the Peano axioms known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly unchanged in a number of metamathematical investigations, including research into fundamental questions of whether number theory is consistent and complete; the need to formalize arithmetic was not well appreciated until the work of Hermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about the successor operation and induction. In 1881, Charles Sanders Peirce provided an axiomatization of natural-number arithmetic. In 1888, Richard Dedekind proposed another axiomatization of natural-number arithmetic, in 1889, Peano published a simplified version of them as a collection of axioms in his book, The principles of arithmetic presented by a new method; the Peano axioms contain three types of statements.

The first axiom asserts the existence of at least one member of the set of natural numbers. The next four are general statements about equality; the next three axioms are first-order statements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final axiom is a second order statement of the principle of mathematical induction over the natural numbers. A weaker first-order system called Peano arithmetic is obtained by explicitly adding the addition and multiplication operation symbols and replacing the second-order induction axiom with a first-order axiom schema; when Peano formulated his axioms, the language of mathematical logic was in its infancy. The system of logical notation he created to present the axioms did not prove to be popular, although it was the genesis of the modern notation for set membership and implication Peano maintained a clear distinction between mathematical and logical symbols, not yet common in mathematics. Peano was unaware of Frege's work and independently recreated his logical apparatus based on the work of Boole and Schröder.

The Peano axioms define the arithmetical properties of natural numbers represented as a set N or N. The non-logical symbols for the axioms consist of a constant symbol 0 and a unary function symbol S; the first axiom states that the constant 0 is a natural number: The next four axioms describe the equality relation. Since they are logically valid in first-order logic with equality, they are not considered to be part of "the Peano axioms" in modern treatments; the remaining axioms define the arithmetical properties of the natural numbers. The naturals are assumed to be closed under a single-valued "successor" function S. Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number; this choice is arbitrary. However, because 0 is the additive identity in arithmetic, most modern formulations of the Peano axioms start from 0. Axioms 1, 6, 7, 8 define a unary representation of the intuitive notion of natural numbers: the number 1 can be defined as S, 2 as S, etc. However, considering the notion of natural numbers as being defined by these axioms, axioms 1, 6, 7, 8 do not imply that the successor function generates all the natural numbers different from 0.

Put differently, they do not guarantee that every natural number other than zero must succeed some other natural number. The intuitive notion that each natural number can be obtained by applying successor sufficiently to zero requires an additional axiom, sometimes called the axiom of induction; the induction axiom is sometimes stated in the following form: In Peano's original formulation, the induction axiom is a second-order axiom. It is now common to replace this second-order principle with a weaker first-order induction scheme. There are important differences between the second-order and first-order formulations, as discussed in the section § Models below; the Peano axioms can be augmented with the operations of addition and multiplication and the usual total ordering on N. The respective functions and relations are constructed in set theory or second-order logic, can be shown to be unique using the Peano axioms. Addition is a function, it is defined recursively as: a + 0 = a, a + S = S. For example: a + 1 = a + S by definition = S using = S, using a

The Cellular Jail known as Kālā Pānī, was a colonial prison in the Andaman and Nicobar Islands, India. The prison was used by the British for the express purpose of exiling political prisoners to the remote archipelago. Many notable independence activists, including Batukeshwar Dutt, Yogendra Shukla and Vinayak Damodar Savarkar, were imprisoned here during the struggle for India's independence. Today, the complex serves as a national memorial monument. Although the prison complex itself was constructed between 1896 and 1906, the British had been using the Andaman Islands as a prison since the days in the immediate aftermath of the revolt of 1857. Shortly after the rebellion was suppressed, the British executed many rebels; those who survived were exiled for life to the Andamans to prevent their re-offending. Two hundred rebels were transported to the islands under the custody of the jailer David Barry and Major James Pattison Walker, a military doctor, warden of the prison at Agra. Another 733 from Karachi arrived in April, 1868.

In 1863, the Rev. Henry Fisher Corbyn, of the Bengal Ecclesiastical Establishment, was sent out there and he set up the'Andamanese Home' there, a repressive institution albeit disguised as a charitable one. Rev. Corbyn was posted in 1866 as Vicar to St. Luke's Church and died there and is buried at the Old Christian Cemetery, Abbottabad. More prisoners arrived from Burma as the settlement grew. Anyone who belonged to the Mughal royal family, or who had sent a petition to Bahadur Shah Zafar during the Rebellion was liable to be deported to the islands; the remote islands were considered to be a suitable place to punish the independence activists. Not only were they isolated from the mainland, the overseas journey to the islands threatened them with loss of caste, resulting in social exclusion; the convicts could be used in chain gangs to construct prisons and harbour facilities. Many died in this enterprise, they served to colonise the island for the British. By the late 19th century the independence movement had picked up momentum.

As a result, the number of prisoners being sent to the Andamans grew and the need for a high-security prison was felt. From August 1889 Charles James Lyall served as home secretary in the Raj government, was tasked with an investigation of the penal settlement at Port Blair, he and A. S. Lethbridge, a surgeon in the British administration, concluded that the punishment of transportation to the Andaman Islands was failing to achieve the purpose intended and that indeed criminals preferred to go there rather than be incarcerated in Indian jails. Lyall and Lethbridge recommended that a "penal stage" should exist in the transportation sentence, whereby transported prisoners were subjected to a period of harsh treatment upon arrival; the outcome was the construction of the Cellular Jail, described as "a place of exclusion and isolation within a more broadly constituted remote penal space." The construction of the prison started in 1896 and was completed in 1906. The original building was a puce-coloured brick building.

The bricks used to build the building were brought from Burma. The building had seven wings, at the centre of which a tower served as the intersection and was used by guards to keep watch on the inmates; the wings radiated from the tower in straight lines, much like the spokes of a bicycle wheel. Each of the seven wings had three stories upon completion. There were a total of 696 cells; each cell was 4.5 by 2.7 metres in size with a ventilator located at a height of 3 metres. The name, "cellular jail", derived from the solitary cells which prevented any prisoner from communicating with any other; the spokes were so designed such that the face of a cell in a spoke saw the back of cells in another spoke. This way, communication between prisoners was impossible, they were all in solitary confinement. "The British Raj sent Indian dissidents and mutineers to a remote island penal colony in an'experiment' that involved torture, medical tests, forced labour and, for many, death." It is estimated that of the total 80,000 political prisoners, the British Raj held at the Kalapani, a few survived.

Solitary confinement was implemented as the British government desired to ensure that political prisoners and revolutionaries be isolated from one another. The Andaman island served as the ideal setting for the government to achieve this. Most prisoners of the Cellular Jail were independence activists; some inmates were Fazl-e-Haq Khairabadi, Yogendra Shukla, Batukeshwar Dutt, Babarao Savarkar, Vinayak Damodar Savarkar, Sachindra Nath Sanyal, Bhai Parmanand, Sohan Singh, Subodh Roy and Trailokyanath Chakravarty Several revolutionaries were tried in the Alipore Case, such as Barindra Kumar Ghose, the surviving companion of Bagha Jatin, was transferred to Berhampore Jail in Bengal, before his mysterious death in 1924. The Savarkar brothers and Vinayak, did not know that they were in different cells in the same jail for two years. In March 1868, 238 prisoners tried to escape. By April they were all caught. One committed suicide and of the remainder Superintendent Walker ordered 87 to be hanged. Among the records of the Government of India's Home Department, we found the Empire's response in its Orders to Provincial Governors and Chief Commissioners.

"Very Secret: Regarding security prisoners who hunger strike, every effort should be made to prevent the incidents from being reported, no concessions to be given to the prisoners who must be kept alive. Manual methods of restraint are best mechanical when the patient resists." Hunger strikes by the inmates

Senior trooper is a rank used by several state police agencies within the United States and in some world militaries. It is the third class in the progressive series of state trooper ranks, it is a step below master trooper, yet above trooper first class. For some agencies, the insignia for this position consists of a gold colored'ST' collar pin worn on the wearer's right lapel while others bear a'senior trooper' plate, located below their nametag; the title of address is senior trooper. Senior troopers take on the responsibility to use preventive measures in dealing with accidents and crime, enforce the law, conduct highway patrol and perform related administrative duties in order to protect citizens, public highways, property of the State or Commonwealth. Senior troopers provide mentoring and training to newly assigned troopers, they have jurisdiction in all parts of the State or Commonwealth they carry a weapon. Senior troopers are required to be knowledgeable on criminal and traffic laws, skilled in the use of firearms and operation of an issued vehicle.

Troopers who complete ten years of satisfactory or exceptional service are promoted to the rank of senior trooper. However, it is possible for a trooper to become a senior trooper without ten years of experience. Consideration for movement to senior trooper, master trooper, senior special agent is based on years of experience, performance evaluation, weight control, educational achievement. In some agencies, becoming a senior trooper is not a rise in rank but does include a pay raise. In the Texas Highway Patrol, senior troopers are instead positions only attained with 20 years of service, differing from other organizations where the position is given with 10 or less years of service. Usage in other agencies or countries may vary. In the United States, state agencies are referred to highway patrol; the rank of Senior Trooper is used by the following state agencies within the United States: Louisiana State Police Maryland State Police North Carolina State Highway Patrol South Carolina Highway Patrol Texas Highway Patrol Virginia State Police West Virginia State Police Master trooper Trooper first class Police ranks of the United States National Association of Police Organizations website

Ahzee or at times Kevin Ahzee or DJ Ahzee is an American DJ and producer from New York, NY. Signed to BIP Records and House Garden Records and in the United States to Ultra Music, Ahzee has gained fame in Europe France and Belgium, through "Born Again" that charted in both countries, he has released "Drums" jointly with Belgian DJ diMaro. In October 2014, he was nominated for "Best Club Hit of the Year" for his single "Born Again" during the NRJ DJ Awards held at MICS - Monaco International Clubbing Show, he remains popular in France and in Belgium with his 2015 hit "Make a Wish" charting in Belgian Flanders chart. Addict Official website Facebook

Niagara District Secondary School was a public secondary school located in Niagara-on-the-Lake, Canada. Opened in 1957 by the Niagara Town and Township High School Board, NDSS featured a specialized arts program focusing on the dramatic arts, visual arts, music, it offered the possibility of earning a specialized DNA arts certificate upon graduation. The school is arranged in a P shaped configuration and has a large track located northeast of the school. Parking is available along Niagara Stone East-West Line; the school is situated in rural site with residential areas located to the southwest in Virgil and northeast in Niagara-on-the-Lake. When NDSS was open enrolment was made from graduates from feeder schools in the area: Parliament Oak Public School Virgil Public School Colonel John Butler Public School St. Michael's Catholic School St. David's Public School Brockview Public School Around 2008 the Ministry of Education began the process of determining the future of the school. Based on declining enrollment, a decision was made to proceed with closing the school at the end of 2010 academic year after enrollment failed to reach the threshold of 350 as outline in 2009 ARC process along with complicated battle between the community and the District School Board of Niagara.

The school was hampered by the aging population and drop in younger residents in the area when compared with other schools in the area. Students once served by the school were re-directed to Laura Secord Secondary School in St. Catharines and A. N. Myer Secondary School in Niagara Falls, Ontario. Located at 1875 Niagara Stone Road, it is now used as Niagara-on-the-Lake Resource Centre by the school board. On Tuesday, January 27, 2015 the town of Niagara-on-the-Lake and the District School Board of Niagara reached an agreement for the town to purchase the 26-ache Niagara District Secondary School site for $1.67 million. The former NDSS property will now remain owned by the town and the sale for the piece of land will close on February 27, 2015, it is now the site of the Royal Elite International Academy. Malin Åkerman, a Swedish-born Canadian film actress and model who attended the school Bill Danychuk, Canadian Football League player and all-star offensive lineman

Robert Hausmann was a notable 19th-century German cellist who premiered important works by Johannes Brahms and Max Bruch. He taught at the Berlin Königliche Hochschule für Müsik. Robert Hausmann was born in Harz, in present-day Saxony-Anhalt, Germany, his paternal grandfather, Friedrich Ludwig Hausmann was a Professor of Mineralogy at the University of Göttingen, his father Friedrich Ludolf Hausmann was involved in mining in the mineral-rich Harz mountains. The Hausmann family had played a prominent role in civic and cultural life of the city of Hanover since the eighteenth century. Robert's great-uncle Bernhard was an important art collector and amateur violinist whose memoirs, Erinnerungen aus dem 80 jährigen Leben eines hannoverschen Burgers Hannover provide a detailed account of his many activities during an eventful period in Hanover's history. Robert entered the Gymnasium in Brunswick at age eight in 1861, where his cello studies proceeded under Theodor Müller, the cellist of the Müller Quartet, one of the earliest professional string quartets in Germany.

In 1869 he was one of the first pupils of the Berlin Hochschule für Musik,where he studied under Müller's nephew Wilhelm Müller, under the general guidance of the violinist Joseph Joachim. Joachim introduced him to the great Italian cellist and teacher Carlo Alfredo Piatti, who taught him in London in 1871 and at his estate at Cadenabbia on Lake Como, Italy, he joined the string quartet of Count Hochberg in Silesia from 1871 to 1876, when he was appointed instructor of cello at the Berlin Hochschule. He became principal instructor on the retirement of Wilhelm Müller in 1879, was named Professor in 1884. From 1879 until Joseph Joachim's death in 1907 he was the cellist of the Joachim Quartet, alongside Joachim himself, Henrich De Ahna, replaced by Karel Halíř and Emanuel Wirth, he was always a active chamber music player, renowned in Germany, in Europe more and in London. Hausmann performed in Britain every year, beginning in 1876, until a month before his death. In 1879-80, Charles Villiers Stanford wrote a Cello Concerto in D minor for Robert Hausmann.

This followed Hausmann's premiering in England Stanford's Cello Sonata, Op. 9, in 1879. It was the first significant British cello sonata. In 1881 Hausmann premiered Max Bruch's Kol Nidrei, Op. 47, dedicated to him. Bruch wrote this in response to a longstanding request from Hausmann to write a piece for cello and orchestra to match those he had written for violin and orchestra. Bruch consulted Hausmann about bowings and other technical aspects. Bruch's Canzone in B-flat, Op. 55, Vier Stücke, Op. 70, are dedicated to Hausmann. Hausmann's most significant association was with Johannes Brahms. After they first played together in 1883, he was a frequent guest among Brahms's circle of friends who had private performances in their homes; the Cello Sonata No. 2 in F major, Op. 99, was both dedicated to and premiered by him in Vienna, with the composer at the piano. He popularized the Cello Sonata No. 1 in E minor, Op. 38. In November 1891 he participated in the first private performance, in Meiningen, of the Clarinet Trio in A minor, Op. 114, with Richard Mühlfeld on clarinet and Brahms on piano.

The following month they had a triumph with the public premiere in Berlin. In March 1892 he introduced the work with Mühlfeld and Fanny Davies. Further, Hausmann was part of the Berlin premieres of Brahms's Piano Trio in C minor Op. 101, the Quintet in G Major, Op. 111, the Clarinet Quintet, Op. 115. Most Hausmann and Joseph Joachim were the two soloists for whom Brahms wrote the Double Concerto in A minor, op. 102, his last orchestral work. Brahms worked with both of them on the piece before its premiere in Wiesbaden on 18 October 1887; the critic Eduard Hanslick wrote that Hausmann was "in no way inferior to Joachim." As part of the Joachim Quartet, Hausmann championed all of Brahms's chamber music for strings, which were featured on the Quartet's Berlin concert series for over thirty years. Besides the Quartet, Hausmann was a founding member of a piano trio group, made up of his colleagues at the Hochschule, Heinrich de Ahna and the pianist Heinrich Barth, they started a subscription concert series in Berlin that lasted from 1878 until 1907.

Their concert season was similar to the Quartet's, ending in March. Their first years of playing were in the Singakademie. However, in 1889 they started playing "popular chamber music evenings" in the much larger Philharmonie, where they filled the 2000-plus seats. In 1894 Hausmann married Helene von Maybach, daughter of the Prussian Minister of Commerce Albert von Maybach, they had two children: Luise and Friedrich Georg, who became a musician. Hausmann published editions of the Bach Cello Suites and the Mendelssohn Cello Sonatas and Variations Concertantes in D major, Op. 17. His students included Friedrich Koch, Wallingford Riegger, Philipp Roth, Percy Such, Hugo Dechert, Otto Lüdemann, Lucy Campbell, Arthur Williams, others. See: List of music students by teacher: G to J#Robert Hausmann, he played a fine Stradivarius cello from 1724, still known as the "Hausmann" Strad. He acquired it from his great uncle's son, the cello virtu