Cologne (kəˈloʊn ; Köln kœln; Kölle ˈkœlə) is the largest city of the German western state of North Rhine-Westphalia (NRW) and the fourth-most populous city of Germany with nearly 1.1 million inhabitants in the city proper and over 3.1 million people in the urban region. Centered on the left (west) bank of the Rhine, Cologne is about southeast of NRW's state capital Düsseldorf and northwest of Bonn, the former capital of West Germany.
The city's medieval Catholic Cologne Cathedral (Kölner Dom) is the third-tallest church and tallest cathedral in the world. It was constructed to house the Shrine of the Three Kings and is a globally recognized landmark and one of the most visited sights and pilgrimage destinations in Europe. The cityscape is further shaped by the Twelve Romanesque churches of Cologne, and Cologne is famous for Eau de Cologne, that has been produced in the city since 1709, and "cologne" has since come to be a generic term.
Cologne was founded and established in Germanic Ubii territory in the 1st century CE as the Roman Colonia Agrippina, hence its name. Agrippina was later dropped (except in Latin), and Colonia became the name of the city in its own right, which developed into modern German as Köln. Cologne, the French version of the city's name, has become standard in English as well. Cologne functioned as the capital of the Roman province of Germania Inferior and as the headquarters of the Roman military in the region until occupied by the Franks in 462. During the Middle Ages the city flourished as being located on one of the most important major trade routes between east and western Europe (including the Brabant Road, Via Regia and Publica). Cologne was a free imperial city of the Holy Roman Empire and one of the major members of the trade union Hanseatic League. It was one of the largest European cities in medieval and renaissance times.
Prior to World War II, the city had undergone occupations by the French (1794–1815) and the British (1918–1926), and was part of Prussia beginning in 1815.
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Aachen ('ɑːxən , ˈaːxn̩; Oche ˈɔːxə; French and traditional English: Aix-la-Chapelle) is the 13th-largest city in North Rhine-Westphalia and the 27th-largest city of Germany, with around 249,000 inhabitants. It is the westernmost city in Germany, and borders Belgium and the Netherlands to the west, the triborder area. It is located between Maastricht (NL) and Liège (BE) in the west, and Bonn and Cologne in the east. The Wurm River flows through the city, and together with Mönchengladbach, Aachen is the only larger German city in the drainage basin of the Meuse.
Cologne (kəˈloʊn ; Köln kœln; Kölle ˈkœlə) is the largest city of the German western state of North Rhine-Westphalia (NRW) and the fourth-most populous city of Germany with nearly 1.1 million inhabitants in the city proper and over 3.1 million people in the urban region. Centered on the left (west) bank of the Rhine, Cologne is about southeast of NRW's state capital Düsseldorf and northwest of Bonn, the former capital of West Germany. The city's medieval Catholic Cologne Cathedral (Kölner Dom) is the third-tallest church and tallest cathedral in the world.
Germany, officially the Federal Republic of Germany, is a country in Central Europe. It is the second-most populous country in Europe after Russia, and the most populous member state of the European Union. Germany is situated between the Baltic and North seas to the north, and the Alps to the south. Its 16 constituent states are bordered by Denmark to the north, Poland and the Czech Republic to the east, Austria and Switzerland to the south, and France, Luxembourg, Belgium, and the Netherlands to the west.
Explores the stability of Ordinary Differential Equations, focusing on solution dependence, critical data, linearization, and control of nonlinear systems.