This article presents methods and algorithms for the computation of isogenies of degree ℓn. Some of these methods are obtained using recurrence equations and generating functions. A standard multiplication based algorithm for computation of isogeny of degree ℓn has time complexity equal to O(n2 M (n log n)), where M(N) denotes the cost of integers of size N multiplication. The memory complexity of this algorithm is equal to O (n log (n log (n))). In this article are presented algorithms for:
where optimality in this context means that, for the given parameters, no other strategy exists that requires fewer operations for computation of isogeny.
Also this article presents a method using generating functions for obtaining the solutions of sequences (um) and (cm) where cm denotes the cost of computations of isogeny of degree ℓum for given costs p; q of ℓ-isogeny computation and ℓ-isogeny evaluation. These solutions are also used in the construction of the algorithms presented in this article.
Polish exploration and exploitation of marine resources of Antarctic waters date back to the reconnaissance cruise of the Sea Fisheries Institutes (SFI) r/v Profesor Siedlecki in 1974. Since 1975, a co-operation between the Institute of Ecology, Polish Academy of Sciences (PAS) at Dziekanów Leśny and SFI in Gdynia with participation of the University of Agriculture in Szczecin, Faculty Marine Fisheries and Food Technology (UA) was established. Fishing fleets of the Polish Deep-Sea Fisheries Companies Odra, Dalmor and Gryf, since 1976 were operating in the Atlantic sector of Antarctic waters, south of the convergence.
In the summer of 1977/78, materials for determining the age and growth rate of Pseudochaenichthys georgianus Norman, 1937 were obtained in the South Georgia region. The 11—51 cm long fishes were of the age 1+ to 13+. The curvilinear relationship between the body length and the otolith radius took an S shape. In the first four years annual increments are almost identical and start to decrease in the following years. Spawning may occur in the 5th or 6th year of life, and hatching of larvae from eggs laid in autumn (IV—VI) takes place at the end of winter (IX of X). Larvae and juvenile fish lead an exclusively pelagic life, while older fish feed in near-bottom layers: when resources are scare there they seek food in the pelagial.
The work presents the results of examinations concerning the influence of various amounts of home scrap additions on the porosity of
castings made of MgAl9Zn1 alloy. The fraction of home scrap in the metal charge ranged from 0 to 100%. Castings were pressure cast by
means of the hot-chamber pressure die casting machine under the industrial conditions in one of the domestic foundries. Additionally, for
the purpose of comparison, the porosity of specimens cut out directly of the MgAl9Zn1 ingot alloy was also determined. The examinations
consisted in the qualitative assessment of porosity by means of the optical microscopy and its quantitative determination by the method of
weighting specimens in air and in water. It was found during the examination that the porosity of castings decreases with an increase in the
home scrap fraction in the metal charge. The qualitative examinations confirmed the beneficial influence of the increased home scrap
fraction on the porosity of castings. It was concluded that the reusing of home scrap in a foundry can be a good way of reduction of costs
related to the production of pressure castings.
The work presents the results of examinations concerning the influence of various amounts of home scrap additions on the properties of
castings made of MgAl9Zn1 alloy. The fraction of home scrap in the metal charge ranged from 0 to 100%. Castings were pressure cast by
means of the hot-chamber pressure die casting machine under the industrial conditions in one of the domestic foundries. The examinations
consisted in the determination of the following properties: tensile strength Rm, yield strength Rp0.2, and the unit elongation A5, all being
measured during the static tensile test. Also, the hardness measurements were taken by the Brinell method. It was found that the
mechanical properties (mainly the strength properties) are being improved up to the home scrap fraction of 50%. Their values were
increased by about 30% over this range. Further rise in the home scrap content, however, brought a definite decrease in these properties.
The unit elongation A5 exhibited continual decrease with an increase in the home scrap fraction in the metal charge. A large growth of
hardness was noticed for the home scrap fraction increasing up to the value of 50%. Further increasing the home scrap percentage,
however, did not result in a significant rise of the hardness value any more.
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