Bounds for the Roots of Lacunary Polynomials

A polynomial P(X)  = X^d + a_d − 1X^d − 1 + ⋯ is called lacunary when a_d − 1 =  0. We give bounds for the roots of such polynomials with complex coefficients. These bounds are much smaller than for general polynomials.     

Approximate Solutions of Polynomial Equations

In this paper, we introduce “approximate solutions" to solve the following problem: given a polynomial F(x, y) over Q, where x represents an n -tuple of variables, can we find all the polynomials G(x) such that F(x, G(x)) is identically equal to a constant c in Q ? We have the following: let F(x, y) be a polynomial over Q and the degree of y in F(x, y) be n. Either there is a unique polynomial g(x)  ∈ Q [ x ], with its constant term equal to 0, such that F(x, y)  = ∑_j = 0ⁿc_j(y − g(x))^jfor some rational numbers c_j, hence, F(x, g(x)  + a)  ∈ Q for all a ∈ Q, or there are at most t distinct polynomials g₁(x),⋯ , g_t(x), t ≤ n, such that F(x, g_i(x))  ∈ Q for 1  ≤ i ≤ t. Suppose that F(x, y) is a polynomial of two variables. The polynomial g(x) for the first case, or g₁(x),⋯ , g_t(x) for the second case, are approximate solutions of F(x, y), respectively. There is also a polynomial time algorithm to find all of these approximate solutions. We then use Kronecker’s substitution to solve the case of F(x, y).     

A simple semi-analytical model for remote estimation of chlorophyll-a in turbid waters: Validation

Accurate assessment of phytoplankton chlorophyll-a (chla) concentrations in turbid waters by means of remote sensing is challenging due to the optical complexity of case 2 waters. We have applied a recently developed model of the form [R_rs^? 1(λ₁) ? R_rs^? 1(λ₂)] × R_rs(λ₃) where R_rs(λ_i) is the remote-sensing reflectance at the wavelength λ_i, for the estimation of chla concentrations in turbid waters. The objectives of this paper are (a) to validate the three-band model as well as its special case, the two-band model R_rs^? 1(λ₁) × R_rs(λ₃), using datasets collected over a considerable range of optical properties, trophic status, and geographical locations in turbid lakes, reservoirs, estuaries, and coastal waters, and (b) to evaluate the extent to which the three-band model could be applied to the Medium Resolution Imaging Spectrometer (MERIS) and two-band model could be applied to the Moderate Resolution Imaging Spectroradiometer (MODIS) to estimate chla in turbid waters.The three-band model was calibrated and validated using three MERIS spectral bands (660–670 nm, 703.75–713.75 nm, and 750?757.5 nm), and the 2-band model was tested using two MODIS spectral bands (λ₁ = 662–672, λ₃ = 743–753 nm). We assessed the accuracy of chla prediction in four independent datasets without re-parameterization (adjustment of the coefficients) after initial calibration elsewhere. Although the validation data set contained widely variable chla (1.2 to 236 mg m^? 3), Secchi disk depth (0.18 to 4.1 m), and turbidity (1.3 to 78 NTU), chla predicted by the three-band algorithm was strongly correlated with observed chla (r² > 0.96), with a precision of 32% and average bias across data sets of ? 4.9% to 11%. Chla predicted by the two-band algorithm was also closely correlated with observed chla (r² > 0.92); however, the precision declined to 57%, and average bias across the data sets was 18% to 50.3%. These findings imply that, provided that an atmospheric correction scheme for the red and NIR bands is available, the extensive database of MERIS and MODIS imagery could be used for quantitative monitoring of chla in turbid waters.     

On the Practical Solution of Genus Zero Diophantine Equations

Let f(X, Y) be an absolutely irreducible polynomial with integer coefficients such that the curve defined by the equation f(X, Y)  =  0 is of genus 0 having at least three infinite valuations. This paper describes a practical general method for the explicit determination of all integer solutions of the diophantine equation f(X, Y)  =  0. Some elaborated examples are given.     

Solving Genus Zero Diophantine Equations with at Most Two Infinite Valuations

Let f(X, Y) be an absolutely irreducible polynomial with integer coefficients such that the curve defined by the equation f(X, Y)  =  0 is of genus 0 having at most two infinite valuations. This paper describes a practical general method for the explicit determination of all integer solutions of the diophantine equation f(X, Y)  =  0. Several elaborated examples are given. Furthermore, a necessary and sufficient condition for a curve of genus 0 to have infinitely many integer points is obtained.     

Mason: morphological simplification

The traditional rounding and filleting morphological filters are biased. Hence, as r grows, the rounding R_r (S) of S shrinks and the filleting F_r (S) grows. A shape S is r-regular when R_r (S) = F_r (S) = S. The combinations F_r (R_r (S)) and R_r (F_r (S)) produce nearly r-regular shapes, but retain a bias: F_r (R_r (S)) is usually smaller than S and R_r (F_r (S)) is larger. To overcome this bias, we propose a new filter, called Mason. The r-mortar M_r (S) of S is F_r (S)–R_r (S), and the stability of a point P with respect to S is the smallest value of r for which P belongs to M_r (S). Stability provides important information about the shape’s imbedding that cannot be obtained through traditional topological or differential analysis tools. F_r (R_r (S)) and R_r (F_r (S)) only affect space in M_r (S). For each maximally connected component of M_r (S), Mason performs either F_r (R_r (S)) or R_r (F_r (S)), choosing the combination that alters the smallest portion of that component. Hence, Mason acts symmetrically on the shape and on its complement. Its output is guaranteed to have a smaller symmetric difference with the original shape than that of either combination F_r (R_r (S)) or R_r (F_r (S)). Many previously proposed shape simplification algorithms were focused on reducing the combinatorial storage or processing costs of a shape at the expense of the smoothness and regularity or altered the shape in regular portions that did not exhibit any high frequency complexity. Mason is the first shape simplification operator that is independent of the particular representation and offers the advantage of preserving portions of the boundary of S that are regular at the desired scale.     

On identifying codes that are robust against edge changes

Assume that G = (V, E) is an undirected graph, and C ⊆ V. For every v ∈ V, denote I_r(G; v) = {u ∈ C: d(u,v) ≤ r}, where d(u,v) denotes the number of edges on any shortest path from u to v in G. If all the sets I_r(G; v) for v ∈ V are pairwise different, and none of them is the empty set, the code C is called r-identifying. The motivation for identifying codes comes, for instance, from finding faulty processors in multiprocessor systems or from location detection in emergency sensor networks. The underlying architecture is modelled by a graph. We study various types of identifying codes that are robust against six natural changes in the graph; known or unknown edge deletions, additions or both. Our focus is on the radius r = 1. We show that in the infinite square grid the optimal density of a 1-identifying code that is robust against one unknown edge deletion is 1/2 and the optimal density of a 1-identifying code that is robust against one unknown edge addition equals 3/4 in the infinite hexagonal mesh. Moreover, although it is shown that all six problems are in general different, we prove that in the binary hypercube there are cases where five of the six problems coincide.     

Tetrel bonds between PySiX3 and some nitrogenated bases: Hybridization,substitution, and cooperativity

Ab initio calculations have been performed to study the influence of hybridization, substitution, and cooperativity on the tetrel bond in the complexes of PySiX₃ (Py = pyridine and X = halogen). The tetrel bond becomes stronger in the order of p-PySiF₃⋯NCH(sp) < p-PySiF₃⋯NHCH₂(sp²) < p-PySiF₃⋯NH₃(sp³). The electron-donating group in the electron donor strengthens the tetrel bond, and this effect is mainly achieved through electrostatic and polarization interactions. Tetrel bonding displays cooperative effects with triel bonding and chalcogen bonding, characterized by shorter binding distances and greater electron densities. The cooperative effects between triel/chalcogen bond and tetrel bond have been analyzed by molecular electrostatic potentials and charge transfer. Energy decomposition indicates that many-body effects are mainly caused by polarization energy. The geometries of Si⋯N interaction and its applications in crystal materials have been characterized and evidenced by a CSD research.     

Composing Power Series Over a Finite Ring in Essentially Linear Time

Fix a finite commutative ringR. Letuandvbe power series overR, withv(0) = 0. This paper presents an algorithm that computes the firstnterms of the compositionu(v), given the firstnterms ofuandv, inn^1 + o(1)ring operations. The algorithm is very fast in practice whenRhas small characteristic.     

Fast Constructive Recognition of a Black Box Group Isomorphic toSnorAnusing Goldbach’s Conjecture

The well-known Goldbach Conjecture (GC) states that any sufficiently large even number can be represented as a sum of two odd primes. Although not yet demonstrated, it has been checked for integers up to 10¹⁴. Using two stronger versions of the conjecture, we offer a simple and fast method for recognition of a gray box group G known to be isomorphic to S_n(or A_n) with knownn ≥  20, i.e. for construction1of an isomorphism from G toS_n (or A_n). Correctness and rigorous worst case complexity estimates rely heavily on the conjectures, and yield times of O([ρ + ν + μ ] n log²n) or O([ ρ + ν + μ ] n logn / loglog n) depending on which of the stronger versions of the GC is assumed to hold. Here,ρ is the complexity of generating a uniform random element of G, ν is the complexity of finding the order of a group element in G, and μ is the time necessary for group multiplication in G. Rigorous lower bound and probabilistic approach to the time complexity of the algorithm are discussed in the Appendix.     

A study of nonlinear dispersive equations with solitary-wave solutions having compact support

With the use of Adomian decomposition method, the prototypical, genuinely nonlinear K(m,n) equation, u_t+(u^m)_x+(uⁿ)_xxx=0, which exhibits compactons — solitons with finite wavelength — is solved exactly. Two numerical illustrations, K(2,2) and K(3,3), are investigated to illustrate the pertinent features of the proposed scheme. The technique is presented in a general way so that it can be used in nonlinear dispersive equations.     

Tighter Lower Bounds for Nearest Neighbor Search and Related Problems in the Cell Probe Model

We prove new lower bounds for nearest neighbor search in the Hamming cube. Our lower bounds are for randomized, two-sided error, algorithms in Yao's cell probe model. Our bounds are in the form of a tradeoff among the number of cells, the size of a cell, and the search time. For example, suppose we are searching among n points in the d dimensional cube, we use poly(n,d) cells, each containing poly(d, log n) bits. We get a lower bound of Ω(d/log n) on the search time, a significant improvement over the recent bound of Ω(log d) of Borodin et al. This should be contrasted with the upper bound of O(log log d) for approximate search (and O(1) for a decision version of the problem; our lower bounds hold in that case). By previous results, the bounds for the cube imply similar bounds for nearest neighbor search in high dimensional Euclidean space, and for other geometric problems.     

Algorithmic Computation of de Rham Cohomology of Complements of Complex Affine Varieties

Let X = Cⁿ. In this paper we present an algorithm that computes the de Rham cohomology groups H_dRⁱ(U,C ) where U is the complement of an arbitrary Zariski-closed set Y in X. Our algorithm is a merger of the algorithm given inOaku and Takayama (1999), who considered the case where Y is a hypersurface, and our methods from Walther (1999) for the computation of local cohomology. We further extend the algorithm to compute de Rham cohomology groups with supports H_{dR, Z}ⁱ(U,C ) where again U is an arbitrary Zariski-open subset of X and Z is an arbitrary Zariski-closed subset of U. Our main tool is a generalization of the restriction process from Oaku and Takayama (in press) to complexes of modules over the Weyl algebra. The restriction rests on an existence theorem onV_d -strict resolutions of complexes that we prove by means of an explicit construction via Cartan–Eilenberg resolutions. All presented algorithms are based on Gröbner basis computations in the Weyl algebra and the examples are carried out using the computer system Kan by Takayama (1999).     

Calibration and validation of hyperspectral indices for the estimation of broadleaved forest leaf chlorophyll content,leaf mass per area,leaf area index and leaf canopy biomass

This article aims at finding efficient hyperspectral indices for the estimation of forest sun leaf chlorophyll content (CHL, µg cm_leaf^? 2), sun leaf mass per area (LMA, g_{dry matter} m_leaf^? 2), canopy leaf area index (LAI, m²_leaf m_soil^? 2) and leaf canopy biomass (B_leaf, g_{dry matter} m_soil^? 2). These parameters are useful inputs for forest ecosystem simulations at landscape scale. The method is based on the determination of the best vegetation indices (index form and wavelengths) using the radiative transfer model PROSAIL (formed by the newly-calibrated leaf reflectance model PROSPECT coupled with the multi-layer version of the canopy radiative transfer model SAIL). The results are tested on experimental measurements at both leaf and canopy scales. At the leaf scale, it is possible to estimate CHL with high precision using a two wavelength vegetation index after a simulation based calibration. At the leaf scale, the LMA is more difficult to estimate with indices. At the canopy scale, efficient indices were determined on a generic simulated database to estimate CHL, LMA, LAI and Bleaf in a general way. These indices were then applied to two Hyperion images (50 plots) on the Fontainebleau and Fougères forests and portable spectroradiometer measurements. They showed good results with an RMSE of 8.2 µg cm^? 2 for CHL, 9.1 g m^? 2 for LMA, 1.7 m² m^? 2 for LAI and 50.6 g m^? 2 for Bleaf. However, at the canopy scale, even if the wavelengths of the calibrated indices were accurately determined with the simulated database, the regressions between the indices and the biophysical characteristics still had to be calibrated on measurements. At the canopy scale, the best indices were: for leaf chlorophyll content: ND_chl = (ρ₉₂₅ ? ρ₇₁₀)/(ρ₉₂₅ + ρ₇₁₀), for leaf mass per area: ND_LMA = (ρ₂₂₆₀ ? ρ₁₄₉₀)/(ρ₂₂₆₀ + ρ₁₄₉₀), for leaf area index: D_LAI = ρ₁₇₂₅ ? ρ₉₇₀, and for canopy leaf biomass: ND_Bleaf = (ρ₂₁₆₀ ? ρ₁₅₄₀)/(ρ₂₁₆₀ + ρ₁₅₄₀).     

An Efficient Algorithm for the k-Pairwise Disjoint Paths Problem in Hypercubes

A graph G(V, E) (|V|⩾2k) satisfies property A_k if, given k pairs of distinct nodes (s₁, t₁), …, (s_k, t_k) of V(G), there are k mutually node-disjoint paths, one connecting s_i and t_i for each i, 1⩽i⩽k. A necessary condition for any graph to satisfy A_k is that it is (2k−1)-connected. Hypercubes are important interconnection topologies for parallel computation and communication networks. It has been known that hypercubes of dimension n (which are n-connected) satisfy A_⌈n/2⌉. In this paper we give an algorithm which, given k=⌈n/2⌉ pairs of distinct nodes (s₁, t₁), …, (s_k, t_k) in the n-dimensional hypercube, finds the k disjoint paths of length at most n+⌈log n⌉+1 in O(n² log* n) time.     

Dielectric and piezoelectric properties of [0 0 1] and [0 1 1]-poled relaxor ferroelectric PZN–PT and PMN–PT single crystals

The properties of PZN–PT and PMN–PT single crystals of varying compositions and orientations have been investigated. Among the various compositions studied, [0 0 1]-optimally poled PZN-(6–7)%PT and PMN-30%PT exhibit superior dielectric and piezoelectric properties, with K^T ≈ 6800–8000, d₃₃ ≈ 2800 pC/N, d₃₁ ≈ −(1200–1800) pC/N for PZN-(6–7)%PT; and K^T = 7500–9000, d₃₃ = 2200–2500 pC/N and d₃₁ = −(1100–1400) pC/N for PMN-30%PT. These two compositions are also fairly resistant to over-poling. The [0 0 1]-poled electromechanical coupling factors (k₃₃, k₃₁ and k_t) are relatively insensitive to crystal composition. [0 1 1]-optimally poled PZN-7%PT single crystal also exhibits extremely high d₃₁ values of up to −4000 pC/N with k₃₁ ≈ 0.90–0.96. While [0 1 1]-poled PZN-7%PT single crystal becomes over-poled with much degraded properties when poled at and above 0.6 kV/mm, PZN-6%PT crystal shows no signs of over-poling even when poled to 2.0 kV/mm. The presence of a certain amount (i.e., 10–15%) of orthorhombic phase in a rhombohedral matrix has been found to be responsible for the superior transverse piezoelectric properties of [0 1 1]-optimally poled PZN-(6–7)%PT. The present work shows that flux-grown PZN–PT crystals exhibit superior and consistent properties and improved over-poling resistance to flux-grown PMN–PT crystals and that, for or a given crystal composition, flux-grown PMN–PT crystals exhibit superior over-poling resistance to their melt-grown counterparts.     

Linear Differential Operators for Polynomial Equations

Given a squarefree polynomial P ∈ k₀[ x,y ], k₀a number field, we construct a linear differential operator that allows one to calculate the genus of the complex curve defined by P =  0 (when P is absolutely irreducible), the absolute factorization of P over the algebraic closure of k₀, and calculate information concerning the Galois group of P over ___ k₀(x) as well as overk₀ (x).     

Artificial neural network-based model for predicting VO2max from a submaximal exercise test

The goal of this study is to develop an accurate artificial neural network (ANN)-based model to predict maximal oxygen uptake (VO₂max) of fit adults from a single stage submaximal treadmill jogging test. Participants (81 males and 45 females), aged from 17 to 40 years, successfully completed a maximal graded exercise test (GXT) to determine VO₂max. The variables; gender, age, body mass, steady-state heart rate and jogging speed are used to build the ANN prediction model. Using 10-fold cross validation on the dataset, the average values of standard error of estimate (SEE), Pearson’s correlation coefficient (r) and multiple correlation coefficient (R) of the model are calculated as 1.80 ml kg^?1 min^?1, 0.95 and 0.93, respectively. Compared with the results of the other prediction models in literature that were developed using Multiple Linear Regression Analysis, the reported values of SEE, r and R in this study are considerably more accurate.