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Vol.23 No.3
Abstract
We formulate the additive entropy of a natural number in terms of the additive partition function, and show that its multiplicative entropy is directly related to the multiplicative partition function. We give a practical formula for the multiplicative entropy of natural numbers with two prime factors. We use this formula to analyze the comparative density of additive and multiplicative entropy, prove that this density converges to zero as the number tends to infinity, and empirically observe this asymptotic behavior.
Abstract
We show how to evaluate the cubic continued fraction <TEX>$G(e^{-{\pi}\sqrt{n}})$</TEX> and <TEX>$G(-e^{-{\pi}\sqrt{n}})$</TEX> for n = 4<sup>m</sup>, 4<sup>−m</sup>, 2 · 4<sup>m</sup>, and 2<sup>−1</sup> · 4<sup>−m</sup> for some nonnegative integer m by using modular equations of degree 9. We then find some explicit values of them.
Abstract
Ebadian et al. proved the Hyers-Ulam stability of bimultipliers and Jordan bimultipliers in C*-ternary algebras by using the fixed point method. Under the conditions in the main theorems for bimultipliers, we can show that the related mappings must be zero. Moreover, there are some mathematical errors in the statements and the proofs of the results. In this paper, we correct the statements and the proofs of the results, and prove the corrected theorems by using the direct method.
Abstract
Let <TEX>$M_{1}f(x,y):=\frac{3}{4}f(x+y)-\frac{1}{4}f(-x-y)+\frac{1}{4}f(x-y)+\frac{1}{4}f(y-x)-f(x)-f(y)$</TEX>, <TEX>$M_{2}f(x,y):=2f(\frac{x+y}{2})+f(\frac{x-y}{2})+f(\frac{y-x}{2})-f(x)-f(y)$</TEX>. Using the direct method, we prove the Hyers-Ulam stability of the additive-quadratic ρ-functional inequalities (0.1) <TEX>$N(M_{1}f(x,y),t){\geq}N({\rho}M_{2}f(x,y),t)$</TEX> where ρ is a fixed real number with |ρ| < 1, and (0.2) <TEX>$N(M_{2}f(x,y),t){\geq}N({\rho}M_{1}f(x,y),t)$</TEX> where ρ is a fixed real number with |ρ| < <TEX>$\frac{1}{2}$</TEX>.
Abstract
In this paper, we introduce functional equations in G-normed spaces and we prove the Hyers-Ulam stability of an additive-quadratic-cubic-quartic functional equation in complete G-normed spaces by using the fixed point method.
Abstract
Using the fixed point method, we prove the Hyers-Ulam stability of an additive-quadratic-cubic-quartic functional equation in matrix fuzzy normed spaces.
Abstract
As a generalization of an inflection point, we consider a point P on a smooth plane curve C of degree m at which another curve C' of degree n meets C with high intersection multiplicity. Especially, we deal with the existence of two curves of degree m and n meeting at the unique point.
Abstract
In this paper, we solve the additive ρ-functional inequalities (0.1) ||f(2x-y)+f(y-x)-f(x)|| <TEX>$\leq$</TEX> ||<TEX>${\rho}(f(x+y)-f(x)-f(y))$</TEX>||, where ρ is a fixed complex number with |ρ| < 1, and (0.2) ||f(x+y)-f(x)-f(y)|| <TEX>$\leq$</TEX> ||<TEX>${\rho}(f(2x-y)-f(y-x)-f(x))$</TEX>||, where ρ is a fixed complex number with |ρ| < <TEX>$\frac{1}{2}$</TEX>. Using the direct method, we prove the Hyers-Ulam stability of the additive ρ-functional inequalities (0.1) and (0.2) in β-homogeneous F-spaces.