Estimate of the maximal cycle length in the graph of polynomial transformation of Galois–Eisenstein ring

The paper is concerned with polynomial transformations of a finite commutative local principal ideal of a ring (a finite commutative uniserial ring, a Galois–Eisenstein ring). It is shown that in the class of Galois–Eisenstein rings with equal cardinalities and nilpotency indexes over Galois rings there exist polynomial generators for which the period of the output sequence exceeds those of the output sequences of polynomial generators over other rings.

AN ALGORITHM FOR GENERATING NATURAL COLOR IMAGES FROM FALSE COLOR USING SPECTRAL TRANSFORMATION TECHNIQUE WITH HIGHER POLYNOMIAL ORDER

<p><strong>Abstract.</strong> Satellite imageries in True color composite or Natural Color composite (NCC) serves the best combination for visual interpretation. Red, Green and Infrared channels form false color composite which might not be as useful as NCC to a non-remote sensing professional. As blue band is affected by large atmospheric scattering, satellites like IRS-LISS IV, SPOT do not have blue band. To generate NCC from such satellite data blue band must be simulated. Existing algorithms of spectral transformation do not provide robust coefficients leading to wrong NCC colors especially in water bodies. To achieve more robust coefficients, we have proposed new algorithm to generate NCC for IRS-LISS IV data using second order polynomial regression technique. Second order polynomial transformation functions consider even minor variability present in the image as compared to 1st order so that the derived coefficients are adjustable to accommodate spatial and temporal variability while generating NCC. In this study, Sentinel-2 image was used for deriving coefficients with blue band as dependent and green, red and infrared as independent variables. Simulated Sentinel band showed high accuracy with correlation of 0.93 and 0.97 for two test sites. Using the same coefficients, blue band was simulated for LISS-IV which also showed good correlation of 0.90 with sentinel original blue band. On comparing LISS-IV simulated NCC with simulated NCC from other algorithms, it was observed that higher order polynomial transformation was able to achieve higher accuracy especially for water bodies where expected color is green. Thus, proposed algorithms can be used for transforming false color image to natural color images.</p>

On the structure of digraphs of polynomial transformations over finite commutative rings with unity

The paper describes structural characteristics of the digraph of an arbitrary polynomial transformation of a finite commutative ring with unity. A classification of vertices of the digraph is proposed: cyclic elements, initial elements, and branch points are described. Quantitative results on such objects and heights of vertices are given. Besides, polynomial transformations are shown to have cycles whose lengths coincide with the lengths of cycles of the induced polynomial transformation over the field R/ℜ, where ℜ is the radical of the finite commutative local ring R.

First passage probability assessment of stationary non-Gaussian process using the third-order polynomial transformation

In this article, an analytical moment-based procedure is developed for estimating the first passage probability of stationary non-Gaussian structural responses for practical applications. In the procedure, an improved explicit third-order polynomial transformation (fourth-moment Gaussian transformation) is proposed, and the coefficients of the third-order polynomial transformation are first determined by the first four moments (i.e. mean, standard deviation, skewness, and kurtosis) of the structural response. The inverse transformation (the equivalent Gaussian fractile) of the third-order polynomial transformation is then used to map the marginal distributions of a non-Gaussian response into the standard Gaussian distributions. Finally, the first passage probabilities can be calculated with the consideration of the effects of clumping crossings and initial conditions. The accuracy and efficiency of the proposed transformation are demonstrated through several numerical examples for both the “softening” responses (with wider tails than Gaussian distribution; for example, kurtosis > 3) and “hardening” responses (with narrower tails; for example, kurtosis < 3). It is found that the proposed method has better accuracy for estimating the first passage probabilities than the existing methods, which provides an efficient and rational tool for the first passage probability assessment of stationary non-Gaussian process.

On the concept of a filtered bundle

We present the notion of a filtered bundle as a generalization of a graded bundle. In particular, we weaken the necessity of the transformation laws for local coordinates to exactly respect the weight of the coordinates by allowing more general polynomial transformation laws. The key examples of such bundles include affine bundles and various jet bundles, both of which play fundamental roles in geometric mechanics and classical field theory. We also present the notion of double filtered bundles which provide natural generalizations of double vector bundles and double affine bundles. Furthermore, we show that the linearization of a filtered bundle — which can be seen as a partial polarization of the admissible changes of local coordinates — is well defined.