A numerical algorithm for nth root

If λ \lambda is a positive real number strictly less than log ⁡ 3 \operatorname {log}3 , there is a positive number V λ V_\lambda such that every orientable hyperbolic 3 3 -manifold of volume greater than V λ V_\lambda admits λ \lambda as a Margulis number. If λ > ( log ⁡ 3 ) / 2 \lambda >(\operatorname {log}3)/2 , such a V λ V_\lambda can be specified explicitly and is bounded above by \[ λ ( 6 + 880 log ⁡ 3 − 2 λ log ⁡ 1 log ⁡ 3 − 2 λ ) , \lambda \bigg (6+\frac {880}{\operatorname {log}3-2\lambda } \operatorname {log}{1\over \operatorname {log}3-2\lambda }\bigg ), \] where log \operatorname {log} denotes the natural logarithm. These results imply that for λ > log ⁡ 3 \lambda >\operatorname {log}3 , an orientable hyperbolic 3 3 -manifold that does not have λ \lambda as a Margulis number has a rank- 2 2 subgroup of bounded index in its fundamental group and in particular has a fundamental group of bounded rank. Again, the bounds in these corollaries can be made explicit if λ > ( log ⁡ 3 ) / 2 \lambda >(\operatorname {log}3)/2 .

Consistent formulations of direct and iterative methods of pile group analysis are presented. The iterative methods are derived from the same equilibrium and compatibility equations as the direct methods but are cast in different forms. These new iterative methods highlight the shortcomings of some existing iterative methods of analysis, although conceptually they are similar. Unlike the direct methods of analysis in which the stiffness matrix of the pile group system is generally fully coupled, the individual pile stiffness matrix can be uncoupled in the iterative methods and requires only a minimal amount of computer core storage. The accuracy of the iterative methods is compared with that of the direct method, and generally good agreement can be obtained. Field tests on the axial and flexural response of pile groups are studied and show that, in general, the computed response compares favourably with the field measurements. L'article présente des formulations logiques de méthodes directes et itératives pour l'analyse des groupes de pieux. Les méthodes itératives découlent des mêmes équations d'équilibres et de compatibilité que les méthodes directes, mais elles prennent des formes différentes. Ces nouvelles méthodes itératives accentuent les insuffisances de quelques méthodes itératives d'analyse à présent disponibles, malgré une certaine ressemblance de conception. En contraste avec la méthode directe d'analyse, dans laquelle les matrices de rigidité du système représenté par le groupe de pieux sont en général pleinement couplées, la matrice de rigidité du pieu individuel peut être découplée dans les méthodes itératives et n'a besoin que d'une quantité minimale de mémorisation par tores magnétiques dans l'ordinateur. Une comparaison de la précision des méthodes itératives avec celle de la méthode directe indique qu'en général l'accord est bon. Une étude des essais in situ sur la réponse axiale et de flexion des groupes de pieux démontre que généralement la réponse calculée s'accorde bien avec les mesures en place

In this paper we investigate the behaviour of the solutions of the following close-to-cyclic system of two difference equations with maximum: x n + 1 = max { A , y n p x n − 1 q } , y n + 1 = max { B , x n p y n − 1 q } where n = 0 , 1 , … , the coefficients A , B are positive real numbers, thevexponents p , q are positive real numbers such that p > q + 1 , and the initial values x − 1 , x 0 , y − 1 , y 0 , are positive real numbers.

In the eighteenth century, Landen, Lagrange and Gauss studied a function of two positive real numbers that has become known as the arithmetic-geometric mean (AGM). In the nineteenth century, Borchardt generalized the AGM to a function of any 2n(n = 1,2,3,…) positive real numbers. In this paper, we generalize the AGM to a function of any even number of positive real numbers. If M(a, b) is the original AGM then M(a, b) is concave in the pair (a, b) of positive numbers and log M(eα, eβ) is convex in the pair (α,β) of real numbers; all our generalizations of the AGM behave similarly. We generalize this analysis extensively.

CHAPTER V - SOLUTION OF EQUATIONS OF LOW ORDERS AND EXTRACTION OF ROOTS

Load balancing is performed to achieve the optimal use of the existing computational resources as much as possible whereby none of the resources remains idle while some other resources are being utilized. Balanced load distribution can be achieved by the immigration of the load from the source nodes which have surplus workload to the comparatively lightly loaded destination nodes. Applying load balancing during run time is called dynamic load balancing (DLB). DLB can be realized both in a direct or iterative manner according to the execution node selection. In iterative methods, the final destination node is determined through several iteration steps, while in direct methods it is selected in one step. This paper presents the randomized hydrodynamic load balancing (RHLB) method which is a hybrid method that takes advantage of both direct and iterative methods. Using random load migration as a direct method, RHLB approach intends to solve the problems derived from the exceptional instantaneous load rises, and diffuse the surplus workload to relatively free resources. Besides, using hydrodynamic approach as an iterative method, RHLB aims to consume minimum possible system resources to balance the common workload distributions. The results of the experiments designate that, RHLB outruns other iterative based methods in terms of both balance quality and the total time of the load balancing process.

In this paper, we treat first-order algebraic differential equations whose coefficients belong to a certain type of function field.In the particular case where the coefficients are rational functions, our main result states that for any given sector S in the plane, there exists a positive real number N, depending only on the equation and the angle opening of S, such that any solution y(z), which is meromorphic in S and satisfies the condition z~"y -» co as z -> co in S, must be of the form exp J" czm(\ +o(l)) in subsectors, where c and m are constants.(From this, we easily obtain a similar representation for analytic solutions in S, which are not identically zero, and for which zKy -> 0 as z -» oo in S, where the positive real number K again depends only on the equation and the angle opening of S.)

If α is a positive real number then, for each set E in R3, we definewhere U(ρ, E) is any countable collection of convex sets, each of diameter at most ρ, whose union covers E, and Aα is a positive real number. Then the convex α-dimensional measure Λα(E), of E, is . In this paper we shall only be considering the cases α = 1,2 where, as is usual, we take A1 = 1 and A2 = ¼π. The symbols ,Λs(E) will denote the spherical (circular) measure of E, i.e. when the coverings are restricted to being collections of spheres (circles), and E(z) is the intersection of a set E (in R3) with the plane (x, y) at z.

A period-specific realization of linear continuous-time systems

The study was directed at extraction, bioactivity analysis and ascertaining phytochemical composition of the leaf and root extracts of Acanthus montanus. Acanthus montanus (Acanthaceae family) is a shrub that is popularly grown in tropical countries of the world. The plant materials were harvested from Choba community in Rivers State, Nigeria. Extracts of Acanthus montanus (28.3, 19.45 and 25.5 g of n-hexane, ethylacetate and ethanol leaf extracts; and 3.45, 11.32 and 12.44 g of n-hexane, ethylacetate and ethanol root extracts respectively) were obtained through sequential maceration and were separately subjected to in vitro bioassay against a wide range of pathogenic microbes using the agar well diffusion method. Fifteen microorganisms were employed in the investigation, eight bacteria which included Salmonella typhi, Salmonella paratyphi, Pseudomonas aeruginosa, Shigella sonniea, Shigella dysentariae, Escherichia coli, Staphylococcus aureus, Staphylococcus pyogenes and seven fungi which included Candida albicans, Sacharomyces cerevisiae, Rhizopus eligastus, Aspergillus flavus, Aspergillus fumigates, Fusa equisetti and Aspergillus niger. The crude extracts exhibited high microbial growth inhibitory activities with zones of inhibition which ranged between 14 and 34 mm, though there was moderate (19 mm zone of inhibition), or no activity observed for the n-hexane leaf and root extracts. All test organisms were sensitive to the extracts. Standard analytical methods and Gas Chromatography - Flame Ionization Detector (GC-FID) were used for analysis and quantification of the phytochemicals present in the extracts. Major phytochemical contents of the leaf and root extracts included flavonoids (86% - 52.66%), alkaloids (10.66% - 3.25%) and other phenolic compounds (20.52% - 6.38%). In conclusion, the leaf and root extracts of Acanthus montanus are broad-spectrum antimicrobial agents also rich in flavonoids.

n= Zx,+ i=1 The principal tool in the proof is the Hardy-Littlewood circle method, incorporating results and techniques of a powerful new iterative method developed by Vaughan and Wooley ([Va3], [Va4], [VW1], [Wol], [VW2]). In section 2, an algorithm developed in [Fo] for optimizing the parameters in mixed power mean value theorems is generalized and analyzed. Section 3 details a more sophisticated method of generating mixed power mean value theorems, by a limited adaptation of the iterative method itself. The form of these estimates offers many advantages over those of section 2, and provides the key to the elimination of the 16th power from (1.1). These mean value theorems are then applied to the proof of Theorem 1 in section 4. The tools developed here are applicable to a wide range of mixed power representation problems, and we briefly illustrate in section 5 the application to the problem of determining the number of terms required to represent all large n, when the lowest power used is a kth power instead of a square. Throughout, n is a large natural number whose representation as a sum of mixed powers is at issue, and E is an arbitrarily small positive real number. Constants implied by the Landau Oand Vinogradov <<-symbols may depend on E or k. For a real number x, write e(x) for e27iX, [x] for the greatest integer not exceeding x and llxll for the distance from x to the nearest integer. Unless otherwise specified, lowercase Latin letters denote natural numbers and Greek letters denote real numbers. Let A(P, R) denote the set of natural numbers not exceeding P with no

Purpose – The linear regression technique is widely used to determine empirical parameters of fatigue life profile while the results may not continuously depend on experimental data. Thus Tikhonov-Morozov method is utilized here to regularize the linear regression results and consequently reduces the influence of measurement noise without notably distorting the fatigue life distribution. The paper aims to discuss these issues. Design/methodology/approach – Tikhonov-Morozov regularization method would be shown to effectively reduce the influences of measurement noise without distorting the fatigue life distribution. Moreover since iterative regularization methods are known to be an attractive alternative to Tikhonov regularization, four gradient iterative methods called as simple iteration, minimum error, steepest descent and conjugate gradient methods are examined with an appropriate initial guess of regularized coefficients. Findings – It has been shown that in case of sparse fatigue life measurements, linear regression results may not have continuous dependence on experimental data and measurement error could lead to misinterpretations of the solution. Therefore from engineering safety point of view, utilizing regularization method could successfully reduce the influence of measurement noise without significantly distorting the fatigue life distribution. Originality/value – An excellent initial guess for mixed iterative-direct algorithm is introduced and it has been shown that the combination of Newton iterative approach and Morozov discrepancy principle is one of the interesting strategies for determination of regularization parameter having an excellent rate of convergence. Moreover since iterative methods are known to be an attractive alternative to Tikhonov regularization, four gradient descend methods are examined here for regularization of the linear regression problem. It has been found that all of gradient decent methods with an appropriate initial guess of regularized coefficients have an excellent convergence to Tikhonov-Morozov regularization results.

Objective: This study seeks to identify a more sensitive smear microscopy method for the detection of Mycobacterium tuberculosis in low-resource centers. Materials and methods: One hundred and forty sputum specimens were collected and direct smears were prepared as the specimen was submitted. The specimens were afterward digested/decontaminated using the modified Petroff’s method. Both smear methods were stained by the Ziehl-Neelsen technique and examined. All specimens (direct and concentrated) were cultured on a Lowenstein-Jensen medium and results were considered the gold standard to calculate sensitivity. Results: Out of 140 specimens, 27 (19%) and 34 (24%) were positive by direct and concentrated methods respectively. The number of culture positives by the direct and concentrated method was 26 (19%) and 31(22%) respectively. The sensitivity of direct and concentrated methods was 92.3% and 100% respectively. The negative culture and contamination rate was minimum with the concentrated method. The diagnostic accuracy of direct and concentrated methods was 88.6% and 97.1%. Conclusion: In summary, results showed concentrated techniques were able to detect more true positive Mycobacterium tuberculosis than the direct smear method. Therefore, the study region with AFB smear microscopy can utilize the concentrated smear method for AFB smear microscopy to improve the case-finding strategy.

Most of researches for large-scale parallel structural analysis using finite element method have focused on iterative solution methods since direct solution methods generally have many difficulties and disadvantages for large-scale problems. However, due to the numerical robustness of direct methods that guarantees the solution to be obtained within estimated time, direct methods are much more desirable for general application of large-scale structural analysis, as well as due to ease of use which makes most of structural engineers prefer direct methods. In this research, we propose a parallel direct solver for large-scale finite element analysis that can overcome most of difficulties and disadvantages of direct methods by using domain-wise approach for direct methods. By using the proposed solver with our own structural analysis code, we can show good scalability as a direct method and can solve the largest problem ever solved by direct solvers.

Algorithmic choice is essential in any problem domain to realizing optimal computational performance. Multigrid is a prime example: not only is it possible to make choices at the highest grid resolution, but a program can switch techniques as the problem is recursively attacked on coarser grid levels to take advantage of algorithms with different scaling behaviors. Additionally, users with different convergence criteria must experiment with parameters to yield a tuned algorithm that meets their accuracy requirements. Even after a tuned algorithm has been found, users often have to start all over when migrating from one machine to another. We present an algorithm and autotuning methodology that address these issues in a near-optimal and efficient manner. The freedom of independently tuning both the algorithm and the number of iterations at each recursion level results in an exponential search space of tuned algorithms that have different accuracies and performances. To search this space efficiently, our autotuner utilizes a novel dynamic programming method to build efficient tuned algorithms from the bottom up. The results are customized multigrid algorithms that invest targeted computational power to yield the accuracy required by the user. The techniques we describe allow the user to automatically generate tuned multigrid cycles of different shapes targeted to the user's specific combination of problem, hardware, and accuracy requirements. These cycle shapes dictate the order in which grid coarsening and grid refinement are interleaved with both iterative methods, such as Jacobi or Successive Over-Relaxation, as well as direct methods, which tend to have superior performance for small problem sizes. The need to make choices between all of these methods brings the issue of variable accuracy to the forefront. Not only must the autotuning framework compare different possible multigrid cycle shapes against each other, but it also needs the ability to compare tuned cycles against both direct and (non-multigrid) iterative methods. We address this problem by using an accuracy metric for measuring the effectiveness of tuned cycle shapes and making comparisons over all algorithmic types based on this common yardstick. In our results, we find that the flexibility to trade performance versus accuracy at all levels of recursive computation enables us to achieve excellent performance on a variety of platforms compared to algorithmically static implementations of multigrid. Our implementation uses PetaBricks, an implicitly parallel programming language where algorithmic choices are exposed in the language. The PetaBricks compiler uses these choices to analyze, autotune, and verify the PetaBricks program. These language features, most notably the autotuner, were key in enabling our implementation to be clear, correct, and fast. Copyright 2009 ACM.