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Документ Implicit linear nonhomogeneous difference equation over Z with a random right-hand side(Харків : Фізико-технічний інститут низьких температур ім. Б. І. Вєркіна НАН України., 2022) Gefter, S. L.; Piven’, A. L.Let {fn}∞ n=0 be a sequence of independent identically distributed integer valued random variables which are defined on a probability space (Ω, F, P). We assume that these variables have a non-degenerate distribution. Let a and b be integers, b 6= 0, ±1, and let a be not divisible by b. For every ω ∈ Ω, we consider the implicit first-order linear nonhomogeneous difference equa tion bxn+1 + axn = fn(ω), n = 0, 1, 2, . . .. It is proved that the probability that there exists an integer solution of this implicit difference equation is equal to zero. Hence, under the random choice of integers f0, f1, f2, . . ., the implicit linear difference equation bxn+1 + axn = fn, n = 0, 1, 2, . . ., has no solutions in integers. We also prove that if a and b are co-prime integers, then the solvability set for this difference equation is an uncountable dense meagre set in the space of all sequences of integers.Документ Some class of nonlinear partial differential equations in the ring of copolynomials over a commutative ring(Lausanne-London -Madrid-Beijing : Frontiers Media S.A., 2024-11-22) Gefter, Sergiy L; Piven, Aleksey L.We study the copolynomials, i.e., K-linear mappings from the ring of polynomials K[x] into the commutative ring K. With the help of the Cauchy–Stieltjes transform of a copolynomial, we introduce and examine a multiplication of copolynomials. We investigate the Cauchy problem related to the nonlinear partial differencial equation ∂u/∂t = aum0 • (∂u/∂x)m1 • (∂2u/∂x2)m2 • (∂3u/∂x3)m3 , m0, m1, m2, m3 ∈ N0, ∑3 (j=0) mj > 0, a ∈ K in the ring of copolynomials. To find a solution, we use the series of powers of the δ-function. As examples, we consider the Cauchy problem with the Euler-Hopf equation ∂u/∂t + u•(∂u/∂x) = 0, for a Hamilton–Jacobi type equation ∂u/∂t = (∂u/∂x)2 , and for the Harry Dym equation ∂u/∂t = u3 • (∂3u/∂x3).Документ Теоретико-множественная семантика одной системы "обыденной логики"(Москва : Издательство "Наука", ГРВЛ, 1987) Калюжный, Владимир Николаевич; Калюжний, Володимир МиколайовичПобудовано формальну семантику для обчислення повсякденної логіки, побудованої Ю.І. Левіним.Документ Приближенное решение некорректно поставленных задач(2017-11-14) Иевлев, И.И.Рассматривается приближенный метод решения некорректно поставленных задач, опирающийся на понятие (a,b)-устойчивости Гавурина М.К.Документ Spatial Structures in Magnetizable Fluids(Journal of Mathematical Sciences, 2012-01-02) Patsegon, N.F.; Popova, L.N.; Patsegon, N.; Popova, L.The conditions of existence and stability of spatial structures of ferroparticles of the type of isolated and periodic strata that are formed in fixed layers of magnetizable fluid in a constant homogeneous magnetic field are studied.