In this text the construction process of the real numbers is described. Starting from a set satisfying the Peano Axioms we construct succesive the integers Z, the rational numbers Q and then nally the real numbers R. All results are given without proofs as the intention of this text is only to outline the construction process.

Recently, some combinatorial properties for the the Catalan-Larcombe-French numbers have been proved by Sun and Wu, and Zhao. Recently, Z. W. Sun conjectured that the root of the Catalan-Larcombe-French numbers is log-concave. In this paper, we confirm Sun's conjecture by establishing the lower and upper bound for the ratios of the Catalan-Larcombe-French numbers.

We define a Carmichael number of order m to be a composite integer n such that nth-power raising defines an endomorphism of every Z/nZ-algebra that can be generated as a Z/nZ-module by m elements. We give a simple criterion to determine whether a number is a Carmichael number of order m, and we give a heuristic argument (based on an argument of Erdős for the usual Carmichael numbers) that indic...

The main purpose of this paper is to present some properties of the meromorphic solutions of complex difference equation of the form ∑ λ∈I αλ z ∏n ν 1f z cν lλ,ν / ∑ μ∈J βμ z ∏n ν 1f z cν mμ,ν R z, f z , where I {λ lλ,1, lλ,2, . . . , lλ,n | lλ,ν ∈ N ∪ {0}, ν 1, 2, . . . , n} and J {μ mμ,1,mμ,2, . . . , mμ,n | mμ, ν ∈ N ∪ {0}, ν 1, 2, . . . , n} are two finite index sets, cν ν 1, 2, . . . , n a...

We define a Carmichael number of order m to be a composite integer n such that nth-power raising defines an endomorphism of every Z/nZalgebra that can be generated as a Z/nZ-module by m elements. We give a simple criterion to determine whether a number is a Carmichael number of order m, and we give a heuristic argument (based on an argument of Erdős for the usual Carmichael numbers) that indica...

We define a Carmichael number of order m to be a composite integer n such that nth-power raising defines an endomorphism of every Z/nZ-algebra that can be generated as a Z/nZ-module by m elements. We give a simple criterion to determine whether a number is a Carmichael number of order m, and we give a heuristic argument (based on an argument of Erdős for the usual Carmichael numbers) that indic...

BACKGROUND National Strategic Plans (NSPs) for HIV have become foundational documents that frame responses to HIV. Both Global Fund and PEPFAR require coordination with NSPs as a component of their operations. Despite the role of NSPs in country planning, no rigorous assessment of NSP targets and performance outcomes exists. We performed a quantitative analysis of the quality of NSP indicators ...

We report measurements of the temperature variance σ(2)(z,r) and frequency power spectrum P(f,z,r) (z is the distance from the sample bottom and r the radial coordinate) in turbulent Rayleigh-Bénard convection (RBC) for Rayleigh numbers Ra = 1.6 × 10(13) and 1.1 × 10(15) and for a Prandtl number Pr ≃ 0.8 for a sample with a height L = 224 cm and aspect ratio D/L=0.50 (D is the diameter). For z/...

We study α-adic expansions of numbers, that is to say, left infinite representations of numbers in the positional numeration system with the base α, where α is an algebraic conjugate of a Pisot number β. Based on a result of Bertrand and Schmidt, we prove that a number belongs to Q(α) if and only if it has an eventually periodic α-expansion. Then we consider α-adic expansions of elements of the...

The information expression and modeling of decision-making are critical problems in the fuzzy decision theory method. However, existing trapezoidal neutrosophic numbers (TrNNs) Z-numbers (NZNs) their multicriteria (MDM) methods reveal insufficiencies, such as without considering reliability measures TrNN continuous NZN. To overcome it is necessary that one needs to propose (TrNZNs), aggregation...