![]() ![]() By the time he published his Institutionum calculi integralis (three volumes, 1768-1770), he had a full com-mand of the solutions of first-order linear differential equations with constant coefficients. Euler's views on trigonometry matured and in 1748 in his Introductio in analysin infinitorum, he introduced the trigonometic functions on the unit circle just the way we introduced them today. Previously there were only trigonometric "lines" in a circle. ![]() Supplementary Material: ġ Abstract In 1735, Euler found the differential equation k 4 d 4 y d 4 x = y to be "rather slippery." In 1739, he "rather unexpectedly" found the full solution, a solution involving trigonometric functions. A Logical Formalization of the Notion of Interval Dependency: Towards Reliable Intervalizations of Quantifiable Uncertainties. Cite as: Hend Dawood and Yasser Dawood (2019). That being so, our concern is to shed new light on some fundamental problems of interval mathematics and to take one small step towards paving the way for developing alternate dependency-aware interval theories and computational methods. Moreover, on the strength of the generality of the logical apparatus we adopt, the results of this article are not only about classical intervals, but they are meant to apply also to any possible theory of interval arithmetic. ![]() A novelty of this formalization is the expression of interval dependency as a logical predicate (or relation) and thereby gaining the advantage of deducing its fundamental properties in a merely logical manner. This article, therefore, is devoted to presenting a complete systematic formalization of the notion of interval dependency, by means of the notions of Skolemization and quantification dependence. Here, we attempt to answer this long-standing question. Although the notion of interval dependency is widely used in the interval and fuzzy literature, it is only illustrated by example, without explicit formalization, and no attempt has been made to put on a systematic basis its meaning, that is, to indicate formally the criteria by which it is to be characterized. This, naturally, confronts us with the question: Formally, what is interval dependency? Is it a meta-concept or an object-ingredient of interval and fuzzy computations? In other words, what is the fundamental defining properties that characterize the notion of interval dependency as a formal mathematical object? Since the early works on interval mathematics by John Charles Burkill and Rosalind Cecily Young in the dawning of the twentieth century, this question has never been touched upon and remained a question still today unanswered. A main drawback of interval mathematics, though, is the persisting problem known as the "interval dependency problem". One approach proved to be most fundamental and reliable in coping with quantifiable uncertainties is interval mathematics. Through our encounters with the physical world, it reveals itself to us as systems of uncertain quantifiable properties. Progress in scientific knowledge discloses an increasingly paramount use of quantifiable properties in the description of states and processes of the real-world physical systems. That it increases with the area of contact. It also yields a formula for this force which shows For the bottom case, this approach yieldsĪ downward force that increases with depth, which contrasts to AP but is inĪgreement to experiments. The subtle possibility, missed in literature, of these forces to be distinct when measured in different places is pointed out.Ī mathematical derivation of the force exerted by an \emph. By comparing this force to the weight of the displaced fluid we show that the above question admits a negative answer as long as these forces are measured in the same place. Would this approximation be essential for that law to be valid? In this note, starting from a surface integral of the pressure forces exerted by the fluid, we obtain a volume integral for the buoyant force valid for nonuniform gravitational fields. Whenever this topic is treated in physics and engineering textbooks, a uniform gravitational field is assumed, which is a good approximation near the surface of the Earth. It is an easy task to apply the divergence theorem to show that this force agrees to that predicted by the well-known Archimedes' principle, namely an upward force whose magnitude equals the weight of the displaced liquid. When an arbitrarily-shaped body is fully immersed in a liquid in equilibrium, it gets from the liquid a non-null hydrostatic force known as buoyant force. ![]()
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