The paper deals with asymptotics for a class of arithmetic functions which describe the value distribution of the greatest-common-divisor function. Typically, they are generated by a Dirichlet series whose analytic behavior is determined by the factor ζ2(s)ζ(2s − 1). Furthermore, multivariate generalizations are considered.
This paper deals with the average order of arithmetic functions a(n) with a generating function of the shape (1.1) wheref, and gj possess certain characteristic properties of the zera-function. Under fairly general conditions, a lower bound for the error term in the asymptotic formula for the corresponding Dirichlet summatory function is established. Finally, several specific applications to functions from elementary number theory and to arithmetic problems in number fields are discussed.
In this supposition, C ∞ can be replaced by C K with some K ∈ N sufficiently large. The whole somewhat technical condition is only stated to ensure the validity of the estimate (1.3), which in turn depends on the asymptotic expansion for the Fourier coefficients of the indicator function of √ x D (see Hlawka [6], [7]).
Von Bertalanffy proposed the differential equation m′(t) = p × m(t)a − q × m(t) for the description of the mass growth of animals as a function m(t) of time t. He suggested that the solution using the metabolic scaling exponent a = 2/3 (Von Bertalanffy growth function VBGF) would be universal for vertebrates. Several authors questioned universality, as for certain species other models would provide a better fit. This paper reconsiders this question. Based on 60 data sets from literature (37 about fish and 23 about non-fish species) it optimizes the model parameters, in particular the exponent 0 ≤ a < 1, so that the model curve achieves the best fit to the data. The main observation of the paper is the large variability in the exponent, which can vary over a very large range without affecting the fit to the data significantly, when the other parameters are also optimized. The paper explains this by differences in the data quality: variability is low for data from highly controlled experiments and high for natural data. Other deficiencies were biologically meaningless optimal parameter values or optimal parameter values attained on the boundary of the parameter region (indicating the possible need for a different model). Only 11 of the 60 data sets were free of such deficiencies and for them no universal exponent could be discerned.
Background Longitudinal studies of tumor volume have used certain named mathematical growth models. The Bertalanffy-Pütter differential equation unifies them: It uses five parameters, amongst them two exponents related to tumor metabolism and morphology. Each exponent-pair defines a unique three-parameter model of the Bertalanffy-Pütter type, and the above-mentioned named models correspond to specific exponent-pairs. Amongst these models we seek the best fitting one. Method The best fitting model curve within the Bertalanffy-Pütter class minimizes the sum of squared errors (SSE). We investigate also near-optimal model curves; their SSE is at most a certain percentage (e.g. 1%) larger than the minimal SSE. Models with near-optimal curves are visualized by the region of their near-optimal exponent pairs. While there is barely a visible difference concerning the goodness of fit between the best fitting and the near-optimal model curves, there are differences in the prognosis, whence the near-optimal models are used to assess the uncertainty of extrapolation. Results For data about the growth of an untreated tumor we found the best fitting growth model which reduced SSE by about 30% compared to the hitherto best fit. In order to analyze the uncertainty of prognosis, we repeated the search for the optimal and near-optimal exponent-pairs for the initial segments of the data (meaning the subset of the data for the first n days) and compared the prognosis based on these models with the actual data (i.e. the data for the remaining days). The optimal exponent-pairs and the regions of near-optimal exponent-pairs depended on how many data-points were used. Further, the regions of near-optimal exponent-pairs were larger for the first initial segments, where fewer data were used. Conclusion While for each near optimal exponent-pair its best fitting model curve remained close to the fitted data points, the prognosis using these model curves differed widely for the remaining data, whence e.g. the best fitting model for the first 65 days of growth was not capable to inform about tumor size for the remaining 49 days. For the present data, prognosis appeared to be feasible for a time span of ten days, at most.
Introduction: A large body of literature aims at identifying growth models that fit best to given mass-at-age data. The von Bertalanffy-Pütter differential equation is a unifying framework for the study of growth models. Problem: The most common growth models used in poultry science literature fit into this framework, as these models correspond to different exponent-pairs (e.g., Brody, Gompertz, logistic, Richards, and von Bertalanffy models). Here, we search for the optimal exponent-pairs ( a and b ) amongst all possible exponent-pairs and expect a significantly better fit of the growth curve to concrete mass-at-age data. Method: Data fitting becomes more difficult, as there is a large region of nearly optimal exponent-pairs. We therefore develop a fully automated optimization method, with computation time of about 1 to 2 wk per data-set. For the proof of principle, we applied it to literature data about 217 male meat-type chickens, Athens Canadian Random Bred, that were reared under controlled conditions and weighed 28 times during a time span of 170 D. Results: We compared 2 methods of data fitting, least squares using the sum of squared errors ( SSE ), which is common in literature, and a variant using the sum of squared log-errors SSElog . For these data, the optimal exponent-pairs were (0.43, 4.06) for SSE = 2,208.6 (31% improvement over literature values for the residual standard deviation) and (0.89, 0.93) for SSElog = 0.04599. Both optimal exponents were clearly distinct from the exponent-pairs of the common models in literature. This finding was reinforced by considering the region of nearly optimal exponents. Discussion: We explain, why we recommend using SSElog for data fitting and we discuss prognosis, where data from the first 8 wk of growth would not be enough.
In this article we investigate the number A (t) of lattice points in x/t ~ where is a convex body in R s (s/> 3) which has a smooth boundary with nonzero Gaussian curvature throughout, and t is a large real parameter. We establish an asymptotic formula A (t) = Vt s/2 + 0 (t x(s~) (V the volume of ~) which improves upon a classic result of E. HLAWXA [5].
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