Rationalizing Denominators Using Gröbner Bases

The problem of rationalizing denominators for two types of fractions is discussed in the paper. By using the theory and algorithms of Gröbner bases, we first introduce a method to rationalize the denominators of fractions with square root and cube root, and then, for the denominators with higher radical of the general form, the problem of rationalizing denominators is converted into the related problem of finding the minimal polynomials. Some interesting results and an executable algorithm for rationalizing the denominator of these type fractions are presented. Furthermore, an example is also established to illustrate the effectiveness of the algorithm.

An Effective Solution of the Cube-Root Truly Nonlinear Oscillator: Extended Iteration Procedure

The cube-root truly nonlinear oscillator and the inverse cube-root truly nonlinear oscillator are the most meaningful and classical nonlinear ordinary differential equations on behalf of its various applications in science and engineering. Especially, the oscillators are used widely in the study of elastic force, structural dynamics, and elliptic curve cryptography. In this paper, we have applied modified Mickens extended iteration method to solve the cube-root truly nonlinear oscillator, the inverse cube-root truly nonlinear oscillator, and the equation of pendulum. Comparison is made among iteration method, harmonic balance method, He’s amplitude-frequency formulation, He’s homotopy perturbation method, improved harmonic balance method, and homotopy perturbation method. After comparison, we analyze that modified Mickens extended iteration method is more accurate, effective, easy, and straightforward. Also, the comparison of the obtained analytical solutions with the numerical results represented an extraordinary accuracy. The percentage error for the fourth approximate frequency of cube-root truly nonlinear oscillator is 0.006 and the percentage error for the fourth approximate frequency of inverse cube-root truly nonlinear oscillator is 0.12.

k-super cube root cube mean labeling of graphs

Consider a graph G with |V (G)| = p and |E(G)| = q and let f : V (G) → {k, k + 1, k + 2, . . . p + q + k − 1}} be an injective function. The induced edge labeling f ∗ for a vertex labeling f is defined by f ∗ (e) = for all e = uv ∈ E(G) is bijective. If f(V (G)) ∪ {f ∗ (e) : e ∈ E(G)} = {k, k + 1, k + 2, . . . , p + q + k − 1}, then f is called a k-super cube root cube mean labeling. If such labeling exists, then G is a k-super cube root cube mean graph. In this paper, I introduce k-super cube root cube mean labeling and prove the existence of this labeling to the graphs viz., triangular snake graph Tn, double triangular snake graph D(Tn), Quadrilateral snake graph Qn, double quadrilateral snake graph D(Qn), alternate triangular snake graph A(Tn), alternate double triangular snake graph AD(Tn), alternate quadrilateral snake graph A(Qn), & alternate double quadrilateral snake graph AD(Qn).

An Evaluation of Woody Biomass and Pulpwood Market Competition within a Range of Procurement Distances

In the Southern United States, a rising number of biomass facilities have created new market opportunities for forest landowners, consulting foresters, and loggers, which could increase the competition between the biomass market and pulpwood market for forest biomass. Thus, comparing the profits from conventional roundwood harvesting and biomass harvesting within a range of procurement distances could be crucial to make a harvest decision. In this study, we considered two harvesting systems: conventional and biomass. We developed a decision support tool to predict and compare the final stumpage value from both harvesting systems based on the stand and site conditions, market conditions, and distance to the nearest market. We grew (simulated) loblolly pine (Pinus taeda) plantations to six different thinning ages (12, 14, 16, 18, 20, and 22 yr) at five different site indices (17, 20, 23, 26, and 29 m at a base age of 25 yr) using the PTAEDA4.0 software. Different models were fitted and evaluated for certain training and validating criteria. In both harvesting systems, we select the cube root-transformed model as the best model. Using the models, we predict that the utilization of logging residues and pulpwood as wood chips may yield a higher return to the landowner when the delivered price of the wood chips is comparable to the delivered price of the pulpwood and within the same procurement distance. The selected models thus serve as a decision support tool to inform stakeholders to further maximize their economic return from timber harvesting operations by selecting the most profitable option.

Oil Spills or Look-Alikes? Classification Rank of Surface Ocean Slick Signatures in Satellite Data

Linear discriminant analysis (LDA) is a mathematically robust multivariate data analysis approach that is sometimes used for surface oil slick signature classification. Our goal is to rank the effectiveness of LDAs to differentiate oil spills from look-alike slicks. We explored multiple combinations of (i) variables (size information, Meteorological-Oceanographic (metoc), geo-location parameters) and (ii) data transformations (non-transformed, cube root, log10). Active and passive satellite-based measurements of RADARSAT, QuikSCAT, AVHRR, SeaWiFS, and MODIS were used. Results from two experiments are reported and discussed: (i) an investigation of 60 combinations of several attributes subjected to the same data transformation and (ii) a survey of 54 other data combinations of three selected variables subjected to different data transformations. In Experiment 1, the best discrimination was reached using ten cube-transformed attributes: ~85% overall accuracy using six pieces of size information, three metoc variables, and one geo-location parameter. In Experiment 2, two combinations of three variables tied as the most effective: ~81% of overall accuracy using area (log transformed), length-to-width ratio (log- or cube-transformed), and number of feature parts (non-transformed). After verifying the classification accuracy of 114 algorithms by comparing with expert interpretations, we concluded that applying different data transformations and accounting for metoc and geo-location attributes optimizes the accuracies of binary classifiers (oil spill vs. look-alike slicks) using the simple LDA technique.

Highly magmatic breakup LIP centers – a revisit to the East Greenland VRM

Decompressional release of magma at continental triple rift breakup LIP centers, above mantle plume stems, result in highly magmatic settings. As a particularly well exposed example, it is proposed that the East Greenland coastal dyke swarm preserves a structural record of how dyke dilations versus tectonic extension increased upon approaching its Kangerlussuaq triple rift center. Such more magmatic breakup is reflected by how abruptly its volcanic rifted margin transitions into 100% dykes, and in this case up to 100 km farther inland than its geophysically determined continent-ocean boundary. Correspondingly high magma flux through an igneous Kap Edward Holm center sustained the lateral injection of up to 150 km-long dykes, evidenced by increased cut-off dyke thicknesses - below which there is an anomalously low abundance of thinner dykes - that conform to the cube root of their thermal arrest distance. Only the thickest and thereby longest dyke injections linked up with a more southerly located igneous Imilik center of an en echelon offset dyke swarm, the complex transition into which is also addressed. This highly magmatic central plumbing system is further compared to similar volcanic zones across Iceland and other post-Paleozoic breakup LIPs, in order to contextualize its importance.

Genetic Improvement of Data for Maths Functions

We use continuous optimisation and manual code changes to evolve up to 1024 Newton-Raphson numerical values embedded in an open source GNU C library glibc square root sqrt to implement a double precision cube root routine cbrt, binary logarithm log2 and reciprocal square root function for C in seconds. The GI inverted square root x -1/2 is far more accurate than Quake’s InvSqrt, Quare root. GI shows potential for automatically creating mobile or low resource mote smart dust bespoke custom mathematical libraries with new functionality.

Using electrical resistance tomography (ERT) and computational fluid dynamics (CFD) to study the mixing of pseudoplastic fluids with a SCABA 6SRGT impeller

The objective of this work is to use electrical resistance tomography (ERT) and computational fluid dynamic (CFD) modeling to investigate the flow field generated by a Scaba 6SRGT impeller in the agitation of the xanthan solution, as a pseudoplastic fluid with yield stress. ERT provides a non-destructive technique to measure, in three dimensions, the concentration fields inside the mixing tanks. Using ERT, the impeller flow pattern, the dimensions of the cavern formed and the mixing time in the agitation of xanthan solutions were evaluated. The sizes of cavern measured using ERT were in good agreement with that calculated using Elson's model (cylindrical model). ERT provides both overall mixing time using 1264 probes (316 probes for each plan) and local mixing time using 4 selected probes or pixels. The dimensionless mixing times obtained from ERT were correlated well with the Moo-Young correlation, confirming that increased impeller speeds decreases the mixing times. The 3D flow field generated by a Scaba 6SRGT impeller and tracer homogenization in the agitation of xanthan gum were also simulated using the commercial CFD package (FLUENT). The experimental torque measurements were used to validate the numerical simulations. The validated CFD model provided useful information regarding the impeller pumping capacity and flow pattern, the velocity profiles, the formation of cavern around the impeller, and the mixing time. CFD results show good qualitative as well as quantitative agreement with the experimental results and theory. The sizes of cavern measure using CFD were in good agreement with that calculated using Elson's model. The local mixing times predicted from CFD simulations agreed well with literature in a way that mixing times varied inversely with the cube root of the power consumed per unit volume of the solution. CFD under predicted the local mixing times measured using ERT by 11-47%.

Using electrical resistance tomography (ERT) and computational fluid dynamics (CFD) to study the mixing of pseudoplastic fluids with a SCABA 6SRGT impeller

The objective of this work is to use electrical resistance tomography (ERT) and computational fluid dynamic (CFD) modeling to investigate the flow field generated by a Scaba 6SRGT impeller in the agitation of the xanthan solution, as a pseudoplastic fluid with yield stress. ERT provides a non-destructive technique to measure, in three dimensions, the concentration fields inside the mixing tanks. Using ERT, the impeller flow pattern, the dimensions of the cavern formed and the mixing time in the agitation of xanthan solutions were evaluated. The sizes of cavern measured using ERT were in good agreement with that calculated using Elson's model (cylindrical model). ERT provides both overall mixing time using 1264 probes (316 probes for each plan) and local mixing time using 4 selected probes or pixels. The dimensionless mixing times obtained from ERT were correlated well with the Moo-Young correlation, confirming that increased impeller speeds decreases the mixing times. The 3D flow field generated by a Scaba 6SRGT impeller and tracer homogenization in the agitation of xanthan gum were also simulated using the commercial CFD package (FLUENT). The experimental torque measurements were used to validate the numerical simulations. The validated CFD model provided useful information regarding the impeller pumping capacity and flow pattern, the velocity profiles, the formation of cavern around the impeller, and the mixing time. CFD results show good qualitative as well as quantitative agreement with the experimental results and theory. The sizes of cavern measure using CFD were in good agreement with that calculated using Elson's model. The local mixing times predicted from CFD simulations agreed well with literature in a way that mixing times varied inversely with the cube root of the power consumed per unit volume of the solution. CFD under predicted the local mixing times measured using ERT by 11-47%.