Integral Solutions of the Ternary Cubic Equation 6(x^2+y^2 )-11xy=288z^3

The Ternary cubic Diophantine Equation represented by૟(࢞ ࢟ + ૛ ࢠૡૡ = ૛࢟࢞૚૚ − (૛ ૜ is analyzed for its infinite number of non-zero integral solutions. A few interesting among the solutions are also discussed. Keywords: Diophantine equation, Integral solutions, cubic equation with three unknowns, Ternary equation.

RINGS: Almost a ring, semiring, zero, integral domain

RING, commutative ring, almost a ring, semiring, zero ring, zero property, zero divisors, domain, integral domain, and their underlying definitions are presented in this white paper (knowledge base).

Compositional scalar-on-function regression between geochemical composition and particle size distribution

<p>To describe the relationship between the distribution of particle sizes in soil (particle size distribution, PSD) and the geochemical composition of sediment samples, specific attributes of the variables need to be considered.  In this case, the explanatory variable can be described in form of the probability density function while the response is a real variable represented by log-ratios of the original chemical concentrations. Due to the relative character of density functions, an adequate methodology must be used to satisfy their specific properties. Here, the Bayes space methodology was employed, specifically the centred logratio (clr) transformation played the role to represent the PSDs (densities) in the standard $L^2$ space which is suitable for multivariate statistical methods, including regression. The idea of smoothing splines was used to represent the discretized input densities while fulfilling the zero-integral constraint imposed by the clr transformation. The resulting regression parameters (densities) can be interpreted in both the original and clr space, however, in the latter the interpretation is more straightforward. The newly developed regression model, called compositional scalar-on-function regression was then used for real-world geological data consisting of samples from four loess-paleosol sequences (LPS) in the Czech Republic (Brodek u Přerova, Dobšice, Ivaň, Rozvadovice). The regression modeling allows to distinguish local effects on the PSD and elemental composition of loess, which were not apparent by the standard approach where the PSD and compositions are usually plotted separately. The high mixing capacity of the aeolian transport caused a similar mineral and chemical composition, despite the different source areas of the studied LPSs. Local variability in the PSDs and distribution of selected elements in different grain fractions reflect some microclimatic features, especially the annual precipitation totals, which affected the particle size distribution of dust material blown by wind as well as the intensity of subsequent post-deposition and pedogenic processes.</p>

Segmentation of retinal blood vessels using normalized Gabor filters and automatic thresholding

Although computerized retinal image blood vessel segmentation has been extensively researched, there is still room for improvement in the quality of the segmented images. Since retinal image analysis is still widely used in the diagnosis of diabetic retinopathy, efficient and accurate image characterization techniques are required. Previous work has mainly focused on improving segmentation accuracy rates with little regard to the false positives that are produced by illumination variation. This research work presents a hybrid approach towards the segmentation of retinal blood vessels. New approaches towards the reduction of background illumination variation are proposed using normalized Gabor filtering. These are the base-offset encoding and a modified version of an existing zero-integral kernel technique. The valley emphasis automatic thresholding scheme is used to segment the Gabor response images. Experiments are conducted on the DRIVE and STARE retinal image data sets. Accuracy rates of up to 94% are achieved through the zero-integral and base offset methods. This is comparable with results from literature, where the same data sets are segmented using other classification techniques. The median-offset method is found to most effectively reduce background illumination variation.

Stability of traveling waves of the Keller–Segel system with logarithmic sensitivity

Proceeding with a series of works (Refs. 12, 23–25) by the authors, this paper establishes the nonlinear asymptotic stability of traveling wave solutions of the Keller–Segel system with nonzero chemical diffusion and linear consumption rate, where the right asymptotic state of cell density is vacuum (zero) and the initial value is a perturbation with zero integral from the spatially shifted traveling wave. The main challenge of the problem is various singularities caused by the logarithmic sensitivity and the vacuum asymptotic state, which are overcome by a Hopf–Cole type transformation and the weighted energy estimates with an unbounded weight function introduced in the paper.

Observations on the Ternary Quadratic Equation X2=24α2+Y2

The Ternary Quadratic Equation X2=24α2+Y2 is Considered.Employing its Non-Zero Integral Solutions, Relations among a few Special Polygonal Numbers are Determined.

A two-dimensional vortex condensate at high Reynolds number

We investigate solutions of the two-dimensional Navier–Stokes equation in a $\lrm{\pi} \ensuremath{\times} \lrm{\pi} $ square box with stress-free boundary conditions. The flow is steadily forced by the addition of a source $\sin nx\sin ny$ to the vorticity equation; attention is restricted to even $n$ so that the forcing has zero integral. Numerical solutions with $n= 2$ and $6$ show that at high Reynolds numbers the solution is a domain-scale vortex condensate with a strong projection on the gravest mode, $\sin x\sin y$. The sign of the vortex condensate is selected by a symmetry-breaking instability. We show that the amplitude of the vortex condensate has a finite limit as $\nu \ensuremath{\rightarrow} 0$. Using a quasilinear approximation we make an analytic prediction of the amplitude of the condensate and show that the amplitude is determined by viscous selection of a particular solution from a family of solutions to the forced two-dimensional Euler equation. This theory indicates that the condensate amplitude will depend sensitively on the form of the dissipation, even in the undamped limit. This prediction is verified by considering the addition of a drag term to the Navier–Stokes equation and comparing the quasilinear theory with numerical solution.

Rigidity properties of Anosov optical hypersurfaces

We consider an optical hypersurface Σ in the cotangent bundle τ:T*M→M of a closed manifold M endowed with a twisted symplectic structure. We show that if the characteristic foliation of Σ is Anosov, then a smooth 1-form θ on M is exact if and only if τ*θ has zero integral over every closed characteristic of Σ. This result is derived from a related theorem about magnetic flows which generalizes our previous work [N. S. Dairbekov and G. P. Paternain. Longitudinal KAM cocycles and action spectra of magnetic flows. Math. Res. Lett.12 (2005), 719–729]. Other rigidity issues are also discussed.