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We undertake a detailed study of the L 2 discrepancy of 2-dimensional Korobov lattices and their irrational analogues, either with or without symmetrization. We give a full characterization of such lattices with optimal L 2 discrepancy in terms of the continued fraction partial quotients, and compute the precise asymptotics whenever the continued fraction expansion is explicitly known, such as for quadratic irrationals or Euler's number e. In the metric theory, we find the asymptotics of the L 2 discrepancy for almost every irrational, and the limit distribution for randomly chosen rational and irrational lattices.

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We present a new method for approximating two-body interatomic potentials from existing ab initio data based on representing the unknown function as an analytic continued fraction. In this study, our method was first inspired by a representation of the unknown potential as a Dirichlet polynomial, i.e., the partial sum of some terms of a Dirichlet series. Our method allows for a close and computationally efficient approximation of the ab initio data for the noble gases Xenon (Xe), Krypton (Kr), Argon (Ar), and Neon (Ne), which are proportional to r - 6 and to a very simple d e p t h = 1 truncated continued fraction with integer coefficients and depending on n - r only, where n is a natural number (with n = 13 for Xe, n = 16 for Kr, n = 17 for Ar, and n = 27 for Neon). For Helium (He), the data is well approximated with a function having only one variable n - r with n = 31 and a truncated continued fraction with d e p t h = 2 (i.e., the third convergent of the expansion). Also, for He, we have found an interesting d e p t h = 0 result, a Dirichlet polynomial of the form k 1 6 - r + k 2 48 - r + k 3 72 - r (with k 1 , k 2 , k 3 all integers), which provides a surprisingly good fit, not only in the attractive but also in the repulsive region. We also discuss lessons learned while facing the surprisingly challenging non-linear optimisation tasks in fitting these approximations and opportunities for parallelisation.

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Enzyme behaviour is characterised in the laboratory using diluted solutions of enzyme. However, in vivo processes usually occur at [S T ] ≈ [E T ] ≈ K m . Furthermore, the study of enzyme action involves characterisation of inhibitors and their mechanisms. However, to date, there have been no reports proposing mathematical expressions that can be used to describe enzyme activity at high enzyme concentration apart from the simplest single substrate, irreversible case. Using a continued fraction approach, equations can be easily derived for the most common cases in monosubstrate reactions, such as irreversible or reversible reactions and effector (inhibitor or activator) kinetic interactions. These expressions are an extension of the classical Michaelis-Menten equations. A first analysis using these expressions permits to deduce some differences at high versus low enzyme concentration, such as the greater effectiveness of allosteric inhibitors compared to catalytic ones. Also, they can be used to understand catalyst saturation in a reaction. Although they can be linearised, these equations also show differences that need to be taken into account. For example, the different meaning of line intersection points in Dixon plots. All in all, these expressions may be useful tools for modelling in vivo and biotechnological processes.

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In this paper, we study the relation between the function J 4 1 , 0 , which arises from a quantum invariant of the figure-eight knot, and Sudler's trigonometric product. We find J 4 1 , 0 up to a constant factor along continued fraction convergents to a quadratic irrational, and we show that its asymptotics deviates from the universal limiting behavior that has been found by Bettin and Drappeau in the case of large partial quotients. We relate the value of J 4 1 , 0 to that of Sudler's trigonometric product, and establish asymptotic upper and lower bounds for such Sudler products in response to a question of Lubinsky.

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Logging while drilling (LWD) plays a crucial role in geo-steering, which can determine the formation boundary and resistivity in real time. In this study, an efficient inversion, which can accurately invert formation information in real time on the basis of fast-forward modeling, is presented. In forward modeling, the Gauss-Legendre quadrature combined with the continued fraction method is used to calculate the response of the LWD instrument in a layered formation. In inversion modeling, the Levenberg-Marquardt (LM) algorithm, combined with the line search method of the Armijo criterion, are used to minimize the cost function, and a constraint algorithm is added to ensure the stability of the inversion. A positive and negative sign is added to the distance parameter to determine whether the LWD instrument is located above or below the formation boundary. We have carried out a series of experiments to verify the accuracy of the inversion. The experimental results suggest that the forward algorithm can make the infinite integral of the Bessel function rapidly converge, and accurately obtain the response of the LWD instrument in a layered formation. The inversion can accurately determine the formation resistivity and boundary in real time. This is significant for geological exploration.

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Geometrical studies of phyllotactic patterns deal with the centric or cylindrical models produced by ideal lattices. van Iterson (Mathematische und mikroskopisch - anatomische Studien über Blattstellungen nebst Betrachtungen über den Schalenbau der Miliolinen, Verlag von Gustav Fischer, Jena, 1907) suggested a centric model representing ideal phyllotactic patterns as disk packings of Bernoulli spiral lattices and presented a phase diagram now called Van Iterson's diagram explaining the bifurcation processes of their combinatorial structures. Geometrical properties on disk packings were shown by Rothen & Koch (J. Phys France, 50(13), 1603-1621, 1989). In contrast, as another centric model, we organized a mathematical framework of Voronoi tilings of Bernoulli spiral lattices and showed mathematically that the phase diagram of a Voronoi tiling is graph-theoretically dual to Van Iterson's diagram. This paper gives a review of two centric models for disk packings and Voronoi tilings of Bernoulli spiral lattices.

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Objective To explore the anti-HBV activity of anodonta polysaccharides (AP) and dose-effect relationship in vitro.Methods HepG2.2.15 cells were cultured in vitro and incubated at 37℃ for nine days with AP at a dilution ratio of 1∶10.The expression of HBsAg and HBeAg were detected using ELISA and HBV-DNA copies were detected by real-time fluorescent quantitative PCR.Based on Thiele-type continued-fraction interpolation method,the anti-HBV activity of AP was studied,and the IC50 and the maximum inhibition rate were calculated.Results AP had significant inhibitory effect on the expression of HBsAg and HBeAg in HepG2.2.15 cells in vitro,as well as HBV DNA replication.By Thiele-type continued-fraction interpolation the equations of dose-effect relationship were obtained to determine the maximum inhibition rates of AP on HBeAg and HBsAg secretion being 47.7％ and 56.4％,and the IC50 inhibiting the expression of HBeAg being 143.7mg/L.AP was also able to inhibit HBV-DNA replication and the maximum inhibition rate was 17.8％ with the same method above.Conclusion Anodonta polysaccharides have anti-HBV activity.The thiele-type continued-fraction interpolation method is simple and practical and could be used as a new method for the analysis of drug activity.

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This paper uses the Continued Fraction Expansion (CFE) method for analog realization of fractional order differ-integrator and few special classes of fractional order (FO) controllers viz. Fractional Order Proportional-Integral-Derivative (FOPID) controller, FO[PD] controller and FO lead-lag compensator. Contemporary researchers have given several formulations for rational approximation of fractional order elements. However, approximation of the controllers studied in this paper, due to having fractional power of a rational transfer function, is not available in analog domain; although its digital realization already exists. This motivates us for applying CFE based analog realization technique for complicated FO controller structures to get equivalent rational transfer functions in terms of the controller tuning parameters. The symbolic expressions for rationalized transfer function in terms of the controller tuning parameters are especially important as ready references, without the need of running CFE algorithm every time and also helps in the synthesis of analog circuits for such FO controllers.

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Accurate, efficient, automatic methods for computing the complex error function to any precision are detailed and implemented in an American Standard FORTRAN subroutine. A six significant figure table of erfc z, e x 2 erfc z, and e z 2 erfc(-z) is included for z in polar coordinate form with the modulus of z ranging from 0 to 9. The argand diagram is given for erf z.