Given a continuous (or just measurable) real-valued function on [0, 1] and a closed subset E aS, [0, 1], denote by f| (E) the restriction of f to E. f| (E) can be "better behaved" than f and we discuss the e...
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Given a continuous (or just measurable) real-valued function on [0, 1] and a closed subset E aS, [0, 1], denote by f| (E) the restriction of f to E. f| (E) can be "better behaved" than f and we discuss the existence, for every f, or every f in some class, of sets E that are substantial in terms of fractal dimension, such that f| (E) has bounded total variation, or is monotone, or satisfies a given modulus of continuity.
For a function f : [0, 1] -> R, we consider the set E( f) of points at which f cuts the real axis. Given f : [0, 1] -> R and a Cantor set D subset of [0, 1] with {0, 1} subset of D, we obtain conditions equivale...
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For a function f : [0, 1] -> R, we consider the set E( f) of points at which f cuts the real axis. Given f : [0, 1] -> R and a Cantor set D subset of [0, 1] with {0, 1} subset of D, we obtain conditions equivalent to the conjunction f is an element of C[0, 1] (or f is an element of C-infinity[0, 1]) and D subset of E(f). This generalizes some ideas of Zabeti. We observe that, if f is continuous, then E(f) is a closed nowhere dense subset of f(-1)[continuous]. Additionally, if Int f(-1)[continuous] = empty set, each x is an element of {0, 1} boolean AND E(f) is an accumulation point of E(f). Our main result states that, for a closed nowhere dense set F subset of [0, 1] with each x is an element of {0, 1} boolean AND F being an accumulation point of F, there exists f is an element of C-infinity[0, 1] such that F = E(f) = f(-1)[continuous]. (C) 2021 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.
It is consistent that for every function f:R x R --> R there is an uncountable set A subset of or equal to R and two continuous functions f(0), f(1) :D(A) --> R such that f(alpha, beta) is an element of (f(0)(al...
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It is consistent that for every function f:R x R --> R there is an uncountable set A subset of or equal to R and two continuous functions f(0), f(1) :D(A) --> R such that f(alpha, beta) is an element of (f(0)(alpha, beta), f(1)(alpha, beta)} for every (alpha, beta) is an element of A(2), alpha not equal beta. (C) 2000 Elsevier Science B.V. All rights reserved.
In this paper, we relax the control condition of convergence of SP-iteration presented by Phuengrattana and Suantai (J. Comput. Appl. Math. 235:3006-3014, 2011). We compare the rate of convergence of Mann, Ishikawa an...
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In this paper, we relax the control condition of convergence of SP-iteration presented by Phuengrattana and Suantai (J. Comput. Appl. Math. 235:3006-3014, 2011). We compare the rate of convergence of Mann, Ishikawa and Noor iterations from another point of view and come to a different conclusion. Finally, we provide a numerical example for Ishikawa and Noor iterations, which supports our theoretical results. MSC: 47H05, 47H07, 47H10.
Various problems of continuum mechanics [1–3] for partial differentialequations [4, p. 158; 5, p. 18] and a number of other problems [6–8] can be reduced to specialcases of the integral equation x(t, s) =fl(t, s, τ...
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Various problems of continuum mechanics [1–3] for partial differentialequations [4, p. 158; 5, p. 18] and a number of other problems [6–8] can be reduced to specialcases of the integral equation x(t, s) =fl(t, s, τ )x(τ, s)dτ +fm(t, s, σ)x(t, σ)dσ + ff n(t,s, τ, σ)x(τ, σ)dτ dσ + f(t, s) ≡ (Kx)(t, s) + f(t, s), where T and S are sets of finiteLebesgue measure in Rn and Rm, respectively; D = T ×S; t, τ ∈ T; s, σ ∈ S; l : D×T → R, m :D×S → R, and n : D×D → R are measurable functions; f : D → R is a continuous function; and theintegrals are treated in the Lebesgue sense.
Let K subset of Omega subset of or equal to R(n), where K is polar and compact and Omega is a domain with Green function G(Omega)(.,.) We characterize those subsets E of Omega/K which have the following property: Ever...
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Let K subset of Omega subset of or equal to R(n), where K is polar and compact and Omega is a domain with Green function G(Omega)(.,.) We characterize those subsets E of Omega/K which have the following property: Every positive continuous function on K can be written as Sigma(k) lambda(k)G(Omega)(x(k),.), where x(k) is an element of E and lambda(k) > 0 for each k.
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