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Lemniscatic Elliptic Function
In mathematics, a lemniscatic elliptic function is an elliptic function related to the arc length of a lemniscate of Bernoulli studied by Giulio Carlo de' Toschi di Fagnano in 1718. It has a square period lattice and is closely related to the Weierstrass elliptic function when the Weierstrass invariants satisfy g2 = 1 and g3 = 0. In the lemniscatic case, the minimal half period ω1 is real and equal to where Γ is the gamma function. The second smallest half period is pure imaginary and equal to iω1. In more algebraic terms, the period lattice is a real multiple of the Gaussian integers. The constants e1, e2, and e3 are given by The case g2 = a, g3 = 0 may be handled by a scaling transformation. However, this may involve complex numbers. If it is desired to remain within real numbers, there are two cases to consider: a > 0 and a < 0. The period parallelogram is either a square or a rhombus.
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  • 24 Oct 2022
Topic Review
Deformation (Engineering)
In engineering, deformation refers to the change in size or shape of an object. Displacements are the absolute change in position of a point on the object. Deflection is the relative change in external displacements on an object. Strain is the relative internal change in shape of an infinitesimally small cube of material and can be expressed as a non-dimensional change in length or angle of distortion of the cube. Strains are related to the forces acting on the cube, which are known as stress, by a stress-strain curve. The relationship between stress and strain is generally linear and reversible up until the yield point and the deformation is elastic. The linear relationship for a material is known as Young's modulus. Above the yield point, some degree of permanent distortion remains after unloading and is termed plastic deformation. The determination of the stress and strain throughout a solid object is given by the field of strength of materials and for a structure by structural analysis. Engineering stress and engineering strain are approximations to the internal state that may be determined from the external forces and deformations of an object, provided that there is no significant change in size. When there is a significant change in size, the true stress and true strain can be derived from the instantaneous size of the object. In the figure it can be seen that the compressive loading (indicated by the arrow) has caused deformation in the cylinder so that the original shape (dashed lines) has changed (deformed) into one with bulging sides. The sides bulge because the material, although strong enough to not crack or otherwise fail, is not strong enough to support the load without change. As a result, the material is forced out laterally. Internal forces (in this case at right angles to the deformation) resist the applied load. The concept of a rigid body can be applied if the deformation is negligible.
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Topic Review
Yamhad Dynasty
Template:Royal house The Yamhad dynasty was an ancient Amorite royal family founded in c. 1810 BC by Sumu-Epuh of Yamhad who had his capital in the city of Aleppo. Started as a local dynasty, the family expanded its influence through the actions of its energetic ruler Yarim-Lim I who turned it into the most influential family in the Levant through both diplomatic and military tools. At its height the dynasty controlled most of northern Syria and the modern Turkish province of Hatay with a cadet branch ruling in the city of Alalakh (Land of Mukish). The dynasty was ousted during a short Hittite occupation of Aleppo in the beginning of the 16th century BC but was restored and expanded the kingdom again before being driven out of Aleppo by the Mitannians in c. 1524. Idrimi a member of the dynasty was able to conquer Alalakh leaving his descendents to rule until the last of them was dethroned by the Hittite king Suppiluliuma I in c. 1344 BC.
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Lake Rotoma
Lake Rotoma is the fourth largest lake of the 11 lakes in the Rotorua Lakes district, or the Hot Lakes district as it was known in the early decades of the 20th century. The Rotorua lakes are located in New Zealand's North Island in the Bay of Plenty Region. Lake Rotoma is the easternmost in the chain of three lakes to the northeast of Lake Rotorua The other two are Lake Rotoiti and Lake Rotoehu. Rotoma is located halfway between the city of Rotorua and town of Whakatane. Lake Rotoma has a high water quality with visibility up to around 13 metres deep. The lake has a maximum depth of 83 metres in the northern part and 73.5 metres in the southern part.
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Acute and Obtuse Triangles
An acute triangle (or acute-angled triangle) is a triangle with three acute angles (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one obtuse angle (greater than 90°) and two acute angles. Since a triangle's angles must sum to 180° in Euclidean geometry, no Euclidean triangle can have more than one obtuse angle. Acute and obtuse triangles are the two different types of oblique triangles — triangles that are not right triangles because they have no 90° angle.
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Power Engineering
Power engineering, also called power systems engineering, is a subfield of electrical engineering that deals with the generation, transmission, distribution and utilization of electric power, and the electrical apparatus connected to such systems. Although much of the field is concerned with the problems of three-phase AC power – the standard for large-scale power transmission and distribution across the modern world – a significant fraction of the field is concerned with the conversion between AC and DC power and the development of specialized power systems such as those used in aircraft or for electric railway networks. Power engineering draws the majority of its theoretical base from electrical engineering.
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  • 24 Oct 2022
Topic Review
Hamzanama
The Hamzanama (Persian/Urdu: حمزه نامه Hamzenâme, Epic of Hamza) or Dastan-e-Amir Hamza (Persian/Urdu: داستان امیر حمزه Dâstâne Amir Hamze, "Adventures of Amir Hamza") narrates the legendary exploits of Amir Hamza, or Hamza ibn Abdul-Muttalib, an uncle of Muhammad. Most of the stories are extremely fanciful, "a continuous series of romantic interludes, threatening events, narrow escapes, and violent acts". The Hamzanama chronicles the fantastic adventures of Hamza as he and his band of heroes fight against the enemies of Islam. The stories, from a long-established oral tradition, were written down in Persian, the language of the courts of Persianate societies, in multiple volumes presumably in the era of Mahmud of Ghazni. In the West, the work is best known for the enormous illustrated manuscript commissioned by the Mughal emperor Akbar about 1562. The text augmented the story, as traditionally told in dastan performances. The dastan (storytelling tradition) about Amir Hamza persists far and wide up to Bengal and Arakan, as the Mughal Empire controlled those territories. The longest version of the Hamzanama exists in Urdu and contains 46 volumes in approximately over 45,000 pages.
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Cygnus CRS Orb-3
Cygnus CRS Orb-3, also known as Orbital Sciences CRS Flight 3 or Orbital 3, was an attempted flight of Cygnus, an automated cargo spacecraft developed by United States-based company Orbital Sciences, on 28 October 2014. The mission was intended to launch at 6:22 PM that evening. This flight, which would have been its fourth to the International Space Station and the fifth of an Antares launch vehicle, resulted in the Antares rocket exploding seconds after liftoff.
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Dukkha
Dukkha (/ˈduːkə/; Pāli; Sanskrit: duḥkha; Tibetan: སྡུག་བསྔལ་ sdug bsngal, pr. "duk-ngel") is an important Buddhist concept, commonly translated as "suffering", "pain", "unsatisfactoriness" or "stress". It refers to the fundamental unsatisfactoriness and painfulness of mundane life. It is the first of the Four Noble Truths. The term is also found in scriptures of Hinduism, such as the Upanishads, in discussions of moksha (spiritual liberation).
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Kepler-86
Kepler-86, PH2 or KIC 12735740 (2MASS 19190326+5157453), is a G-type star 1,130 ly (350 pc) distant within the constellation Cygnus. Roughly the size and temperature of the Sun, PH2 gained prominence when it was known to be the host of one of 42 planet candidates detected by the Planet Hunters citizen science project in its second data release. The candidate orbiting around PH2, known as PH2 b, had been determined to have a spurious detection probability of only 0.08%, thus effectively confirming its existence as a planet. Located in its parent star's habitable zone, PH2 b (or Kepler-86b) is a "Jupiter-size" gas giant which may have a natural satellite suitable for hosting life. The report of the confirmed detection of PH2 b was submitted on January 3, 2013. It was discovered by amateur Pole Rafał Herszkowicz using his laptop and access to the Internet project with data from the Kepler space observatory.
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