EXCEL VBA-BASED USER DEFINED FUNCTIONS FOR HIGHLY PRECISE COLEBROOK’S PIPE FLOW FRICTION APPROXIMATIONS: A COMPARATIVE OVERVIEW

This review paper gives Excel functions for highly precise Colebrook’s pipe flow friction approximations developed by users. All shown codes are implemented as User Defined Functions – UDFs written in Visual Basic for Applications – VBA, a common programming language for MS Excel spreadsheet solver. Accuracy of the friction factor computed using nine to date the most accurate explicit approximations is compared with the sufficiently accurate solution obtained through an iterative scheme which gives satisfying results after sufficient number of iterations. The codes are given for the presented approximations, for the used iterative scheme and for the Colebrook equation expressed through the Lambert W-function (including its cognate Wright ω-function). The developed code for the principal branch of the Lambert W-function has additional and more general application for solving different problems from variety branches of engineering and physics. The approach from this review paper automates computational processes and speeds up manual tasks.

Growth of Alfavaca-Cravo in Response to Different Levels of Shade and Tiririca Density

The species Ocimum gratissimum L. is widely utilized in food, cosmetics, and folk medicine, and is also an important source of essential oils. Understanding its behavior in response to environmental conditions is of paramount importance to improving crop management methods. In this context, the following study aimed to evaluate the effects of shade, and of competition with weeds (Cyperus rotundus L.), on the growth of Ocimum gratissimum L. The experimental design adopted was randomized blocks, in a 5 × 5 factorial scheme, with 5 levels of shading (48%, 75%, 77%, 83% and 90%) and 5 densities of Cyperus rotundus L. (0, 5, 10, 15 and 20 per pot), with 4 repetitions. The variables analyzed were main stem height (MSH), diameter of stem base (DSB), number of leaves on the principal branch (NL), number of ramifications (NR), chlorophyll index of leaves (CIL), foliar area (FA), dry mass of the aerial part of the medicinal species (DMAPm), dry mass of the aerial part of the weed species (DMAPw) and essential oil content (EOC). The results demonstrate that the Ocimum gratissimum L. plants presented compatible tolerance responses to up to 70% shading, and that competition with Cyperus rotundus L. was detrimental in a density above 13 plants per pot in interaction with shading. The highest dry mass production and, consequently, the highest oil yield, were obtained from the 48% shading treatment.

Excel VBA-Based User Defined Functions for Highly Precise Colebrook’s Pipe Flow Friction Approximations: A Comparative Overview

This review paper gives Excel functions for highly precise Colebrook’s pipe flow friction approximations developed by users. All shown codes are implemented as User Defined Functions – UDFs written in Visual Basic for Applications – VBA, a common programming language for MS Excel spreadsheet solver. Accuracy of the friction factor computed using nine to date the most accurate explicit approximations is compared with the sufficiently accurate solution obtained through an iterative scheme which gives satisfying results after sufficient number of iterations. The codes are given for the presented approximations, for the used iterative scheme and for the Colebrook equation expressed through the Lambert W-function (including its cognate Wright ω-function). The developed code for the principal branch of the Lambert W-function has additional and more general application for solving different problems from variety branches of engineering and physics. The approach from this review paper automates computational processes and speeds up manual tasks.

Entire functions, PT-symmetry and Voros’s quantization scheme

In this paper, A. Avila's theoremon convergence of the exact quantization scheme of A.~Vo\-rosis related to the reality proofs of eigenvalues of certain $PT$-symmetricboundary value problems.As a result, a special caseof a conjecture of C. Bender, S. Boettcherand P. Meisinger on reality of eigenvalues is proved.In particular the following Theorem~2 is proved:{\sl Consider the eigenvalue problem$$-w''+(-1)^\ell(iz)^mw=\lambda w,$$where $m\geq 2$ is real, and $(iz)^m$ is the principal branch,$(iz)^m>0$ when $z$ is on the negative imaginary ray,with boundary conditions $w(te^{i\beta})\to 0,\ t\to\infty,$where$ \beta=\pi/2\pm\frac{\ell+1}{m+2}\pi.$If $\ell=2$, and $m\geq 4$, then all eigenvalues are positive.}\

Affinity maturation in a human humoral response to influenza hemagglutinin

Affinity maturation of the B cell antigen receptor (BCR) is a conserved and crucial component of the adaptive immune response. BCR lineages, inferred from paired heavy- and light-chain sequences of rearranged Ig genes from multiple descendants of the same naive B cell precursor (the lineages’ unmutated common ancestor, “UCA”), make it possible to reconstruct the underlying somatic evolutionary history. We present here an extensive structural and biophysical analysis of a lineage of BCRs directed against the receptor binding site (RBS) of subtype H1 influenza virus hemagglutinin (HA). The lineage includes 8 antibodies detected directly by sequencing, 3 in 1 principal branch and 5 in the other. When bound to HA, the heavy-chain third complementarity determining region (HCDR3) fits with an invariant pose into the RBS, but in each of the 2 branches, the rest of the Fab reorients specifically, from its position in the HA-bound UCA, about a hinge at the base of HCDR3. New contacts generated by the reorientation compensate for contacts lost as the H1 HA mutated during the time between the donor’s initial exposure and the vaccination that preceded sampling. Our data indicate that a “pluripotent” naive response differentiated, in each branch, into 1 of its possible alternatives. This property of naive BCRs and persistence of multiple branches of their progeny lineages can offer broader protection from evolving pathogens than can a single, linear pathway of somatic mutation.

ON THE ACCURACY OF ASYMPTOTIC APPROXIMATIONS TO THE LOG-GAMMA AND RIEMANN–SIEGEL THETA FUNCTIONS

We give bounds on the error in the asymptotic approximation of the log-Gamma function $\ln \unicode[STIX]{x1D6E4}(z)$ for complex $z$ in the right half-plane. These improve on earlier bounds by Behnke and Sommer [Theorie der analytischen Funktionen einer komplexen Veränderlichen, 2nd edn (Springer, Berlin, 1962)], Spira [‘Calculation of the Gamma function by Stirling’s formula’, Math. Comp.25 (1971), 317–322], and Hare [‘Computing the principal branch of log-Gamma’, J. Algorithms25 (1997), 221–236]. We show that $|R_{k+1}(z)/T_{k}(z)|<\sqrt{\unicode[STIX]{x1D70B}k}$ for nonzero $z$ in the right half-plane, where $T_{k}(z)$ is the $k$th term in the asymptotic series, and $R_{k+1}(z)$ is the error incurred in truncating the series after $k$ terms. We deduce similar bounds for asymptotic approximation of the Riemann–Siegel theta function $\unicode[STIX]{x1D717}(t)$. We show that the accuracy of a well-known approximation to $\unicode[STIX]{x1D717}(t)$ can be improved by including an exponentially small term in the approximation. This improves the attainable accuracy for real $t>0$ from $O(\exp (-\unicode[STIX]{x1D70B}t))$ to $O(\exp (-2\unicode[STIX]{x1D70B}t))$. We discuss a similar example due to Olver [‘Error bounds for asymptotic expansions, with an application to cylinder functions of large argument’, in: Asymptotic Solutions of Differential Equations and Their Applications (ed. C. H. Wilcox) (Wiley, New York, 1964), 16–18], and a connection with the Stokes phenomenon.

On Constructing Two-Point Optimal Fourth-Order Multiple-Root Finders with a Generic Error Corrector and Illustrating Their Dynamics

With an error corrector via principal branch of themth root of a function-to-function ratio, we propose optimal quartic-order multiple-root finders for nonlinear equations. The relevant optimal order satisfies Kung-Traub conjecture made in 1974. Numerical experiments performed for various test equations demonstrate convergence behavior agreeing with theory and the basins of attractions for several examples are presented.

The Medici Family

Members of the Medici family were arguably the most-conspicuous social climbers of the Renaissance period. In the 15th century the principal branch of the family acquired great wealth from banking and commerce and used it to exercise political influence in the Florentine republic, but in the 16th century the republic gave way to a principate, with the Medici as dukes of Florence and grand dukes of Tuscany, a transformation made possible by the election of Medici popes. Whether as citizens or as princes, posterity has placed so much emphasis on their cultural patronage that they have often been cast as central figures of the Renaissance as a cultural phenomenon. This article opens with General Overviews, Reference Works, Collections of Papers, and Digital Resources, all of which span various generations of the family’s history, but then follows the example of so many works in those opening sections by taking a chronological approach to the subject. The section on the Earlier Medici takes the story up to the death of Giovanni di Bicci de’ Medici in 1429. Thereafter, the article traces the family’s rising economic and political fortunes in the Generation of Cosimo il Vecchio, the initial reaction against their anti-republican instincts in the Generation of Piero il Gottoso, and the more determined but ultimately futile opposition in the Generation of Lorenzo il Magnifico. From that point the story is much more complicated, in part because there was a genuine difference of opinion about whether republican Florence was better off with or without the Medici, and in part because that portion of the dynasty known as the line of Cafaggiolo dwindled to a clerically-led rump. Those clerics were nevertheless the key to what happened next. The first Medici pope, Leo X, obtained titles of nobility for his kinsmen, and the second, Clement VII, ensured that his niece Caterina married into the ruling French house of Valois and that Alessandro de’ Medici, regardless of his illegitimate birth, became the first duke of Florence. Leo is featured among the Children of Lorenzo il Magnifico; Caterina/Catherine and Alessandro, among the Other Descendants of Lorenzo il Magnifico to 1537, the year that began with Alessandro’s assassination by his kinsman Lorenzino de’ Medici. If anything, when Florence rejected its republican past and embraced a dynastic present and future, it created a model that other states followed: many a feature of what came to be regarded as the ancien régime could be seen first in the Tuscany of Cosimo I, Francesco I, and Ferdinando I. Their title may have been inflated from duke of Florence to grand duke of Tuscany, but by the Generations of Cosimo II and Ferdinando II, their realm was becoming a somewhat Ruritanian shadow of its former self, while the economic and political initiative was assumed by the Atlantic powers.

Parameters Affecting Dendrite Growth of Fe-C Alloy in a Forced Flow

The phase-field model coupling with the concentration field and flow field is used to simulate the dendrite growth during isothermal solidification of Fe-C alloy in a forced flow. The effects of noise amplitude and interface thickness on the dendrite growth are studied. The results indicate that with noise amplitude increasing, the secondary dendrite arm average space(SDAAS) on the the upstream of the lateral principal branch decreases, but the dendrite tip velocity remained about the same. With an increase in the interface thickness, the principal and secondary branch of dendrite degenerated, the equilibrium morphology of the crystal changes from developed dendrite to compact dendrite, the dendrite tip solute concentration decreases first, then increases slowly.