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).     

Stable white light emission from an externally modified organic light-emitting device

A blue organic light-emitting device, based on an iridium phosphorescent dopant in a polyvinylcarbazole host, has been modified by the addition of an external CaS:Eu inorganic phosphor layer. By incorporating a surfactant in the phosphor mixture, a uniform coating could be achieved by drop-casting. The resulting hybrid device exhibited white light emission, with Commission Internationale de l’Eclairage, CIE (x, y) coordinates of x = 0.32, y = 0.35. No significant change in these coordinates was observed for current densities in the range 25–510 A m^?2. The maximum power efficiencies of the white device was 2.3 lm W^?1 at a brightness of 254 cd m^?2.     

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).     

An Algorithm for the Quillen–Suslin Theorem for Quotients of Polynomial Rings by Monomial Ideals

This paper presents an algorithm for the Quillen–Suslin Theorem for quotients of polynomial rings by monomial ideals, that is, quotients of the form A = k [ x₀, . . . ,x_n ] / I, with I a monomial ideal and k a field. Vorst proved that finitely generated projective modules over such algebras are free. Given a finitely generated module P, described by generators and relations, the algorithm tests whether P is projective, in which case it computes a free basis forP .     

Growth and electrical properties of highly (0 0 1)-oriented Pb(Zr0.52Ti0.48)O3 thin films on amorphous TiN buffered Si(1 0 0)

Pb(Zr_0.52Ti_0.48)O₃ (PZT) ferroelectric thin films with LaNiO₃ (LNO) as bottom electrodes have been grown on amorphous TiN buffered Si(1 0 0) substrates by pulsed laser deposition. It was found that highly (0 0 1)-oriented LNO films could be obtained even if TiN underlayers were amorphous. XRD analyses showed that the subsequently deposited PZT films were also preferentially (0 0 1)-oriented due to the template effect of the perovskite structured LNO films. Dielectric constant of the PZT thin films remained almost constant with frequency in the range from 10³ to 10⁶ Hz, and tangent loss was as small as 0.02 at high frequencies. The remnant polarization and coercive field of an Au/PZT/LNO capacitor were typically 20 μC/cm² and 30 kV/cm, respectively. C–V and I–V characteristics revealed the capacitance and leakage current variations with applied voltage were asymmetric when the bottom electrode was negatively as well as positively biased, indicating that ferroelectric/electrode interfaces and space charges play an important role in the electrical properties of ferroelectric capacitors.     

Small molecules with pyridine backbone modified with carbazole,fluorine and bromine for white light-emitting diode applications

Seven compounds with pyridine as the backbone modified by carbazole moiety, bromine atom and fluorine atom were synthesized. Compounds 1, 2, 3 with bromo substitution at the 2-position and carbazole modification at the 5-position of pyridine emit not only a sharp blue singlet fluorescence but also a wide banded excimer-based orange emission. The two colors coming from a single molecule can be used to fabricate a simplified white light emitting device. The electroluminescence based on 1 and 2 exhibits white-light emission with CIE coordinates of x = 0.25 and y = 0.30 for 1 and x = 0.33 and y = 0.37 for 2 at high current densities, very close to pure white emission. In addition, the role of bromo-substitution at pyridine is concluded to be essential to generate molecular interaction thus an excimer emission.     

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.     

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.     

Solving the Word Problem for Two Classes of Non-associative Rings by Rewriting

Using the notion of modified completion given in Widiger, A. (1998, Deciding degree-four-identities for alternative rings by rewriting. In Bronstein, M., Grabmeier, J., Weispfenning, V. eds, Symbolic Rewriting Techniques, PCS 15, pp. 277–288. Birkhäuser-Verlag), it is shown that the word problem for the varieties of non-associative rings defined by (xy)z = y(zx) and (xy)z = y(xz) respectively can be decided by rewriting.     

Ergodic distribution for a fuzzy inventory model of type (s, S) with gamma distributed demands

In this study, a stochastic process (X(t)), which describes a fuzzy inventory model of type (s, S) is considered. Under some weak assumptions, the ergodic distribution of the process X(t) is expressed by a fuzzy renewal function U(x). Then, membership function of the fuzzy renewal function U(x) is obtained when the amount of demand has a Gamma distribution with fuzzy parameters. Finally, membership function and alpha cuts of fuzzy ergodic distribution of this process is derived by using extension principle of L. Zadeh.     

Integrating Factors for Second-order ODEs

A systematic algorithm for building integrating factors of the form μ(x, y), μ(x, y^′) or μ(y, y^′) for second-order ODEs is presented. The algorithm can determine the existence and explicit form of the integrating factors themselves without solving any differential equations, except for a linear ODE in one subcase of the μ (x, y) problem. Examples of ODEs not having point symmetries are shown to be solvable using this algorithm. The scheme was implemented in Maple, in the framework of the ODEtools package and its ODE-solver. A comparison between this implementation and other computer algebra ODE-solvers in tackling non-linear examples from Kamke's book is shown.     

Gröbner Bases Applied to Finitely Generated Field Extensions

Using a constructive field-ideal correspondence it is shown how to compute the transcendence degree and a (separating) transcendence basis of finitely generated field extensionsk (x) / k(g), resp. how to determine the (separable) degree if k(x) / k(g) is algebraic. Moreover, this correspondence is used to derive a method for computing minimal polynomials and deciding field membership. Finally, a connection between certain intermediate fields of k(x) / k(g) and a minimal primary decomposition of a suitable ideal is described. For Galois extensions the field-ideal correspondence can also be used to determine the elements of the Galois group.     

Phase equilibria in the Ge-Mn-Ti ternary system at 973 K, 1073 K and 1173 K

Phase relationships in the Ge-Mn-Ti ternary system have been studied through alloy samples approach. Assisted with Electron Probe Microanalysis (EPMA) and X-ray diffraction (XRD) techniques, isothermal sections at 973 K, 1073 K and 1173 K of this system were constructed and existence of 2 ternary phases, i.e. GeMnTi and Ge₂MnTi, were confirmed. In addition, remarkable ternary solubilities in some binary compounds were detected, e.g. Ge in Mn₂Ti and Mn in Ge₅Ti₆ can be up to 15 at% and 50 at% at 1173 K, respectively. Furthermore, the substitution of Ti by Mn atoms in Ge₅Ti₆ was confirmed with Rietveld refinement results of solid solutions, Ge₅(Mn_0.30Ti_0.70)₆ and Ge₅(Mn_0.67Ti_0.33)₆.     

Weyl Closure of a Linear Differential Operator

We study the Weyl closure Cl(L)  = K(x)〈∂〉L ∩ D for an operator L of the first Weyl algebra D = K〈x, ∂〉. We give an algorithm to compute Cl(L) and we describe its initial ideal under the order filtration. Our main application is an algorithm for constructing a Jordan–Hölder series for a holonomic D -module and a formula for its length. Using the closure, we also reproduce a result ofStrömbeck (1978), who described the initial ideals of left ideals of D under the order filtration, and a result ofCannings and Holland (1994), who described the isomorphism classes of right ideals of D.     

Modeling of the energies and splitting of the Qx and Qy bands in positional isomers of zinc pyridinoporphyrazines by TDDFT approach: Can TDDFT help distinguishing the structural isomers?

Electronic structures, energies and splitting of the Q_x and Q_y bands for positional isomers of zinc mono-, di-, tri-, and tetra pyridinoporphyrazines as well as parent zinc phthalocyanine were investigated using density functional theory (DFT) and time-dependent (TD) DFT approaches. The influence of the Hartree–Fock exchange on excited state energies and Q_x and Q_y bands splitting were studied using GGA BP86 and hybrid B3LYP and PBE1PBE exchange-correlation functionals. Solvent effects were estimated using the polarized continuum model (PCM) approach and cyclohexane, toluene, or DMSO as solvents. It was found that general trends in the Q_x and Q_y band energies and splitting correlate very well with the available experimental data on pyridinoporphyrazines and follow the trends in HOMO–LUMO and HOMO–LUMO + 1 energy gaps as well as LUMO–LUMO + 1 splitting. TDDFT trends allow estimation of the Q_x and Q_y band energies and splitting in unknown tripyridinoporphyrazines and in individual positional isomers of tetrapyridinoporphyrazines.     

A new red colour emitting phosphor – ZnS:Mn co-doped with Ba for electroluminescent (EL) displays

A new Mn activated Ba co-doped ZnS phosphor shows an interesting shift in the spectral emission from amber-yellow (570 nm) to red (620 nm) colour. To realize the phosphor material physics, attempts were made to fine tune the band gap of ZnS by co-doping Ba within the concentration range 2.50–10 mol%. Ba co-doping in ZnS host was achieved by sintering BaSO₄ with ZnS at 900 °C by carbothermal reduction method. Measurement of the CIE (Commission International deEchairge) chromaticity colour coordinates (x, y) of Zn_0.99S:Mn_0.01 and Ba_0.10Zn_0.89S:Mn_0.01 phosphors shown to be (0.53, 0.47) and (0.65, 0.32), respectively.     

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.     

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.     

Monitoring the tool wear,surface roughness and chip formation occurrences using multiple sensors in turning

Tool wear, chip formation and surface roughness of workpiece under different cutting conditions have been investigated in machining using acoustic emission (AE) and vibration signature in turning. The investigation has shown that the AE and vibration components can effectively respond to the different occurrences in turning including tool wear and surface roughness. The AE has shown a very significant response to the tool wear progression whereas the resultant vibration (V) represented the surface roughness in turning. The vibration components V_x, V_y and V_z described the chip formation type and are found to have the most significant response to the change of feed, depth of cut and cutting speed respectively. The amplitude of vibration components, V_x, V_y and V_z increased with the increase of feed rate, depth of cut and cutting speed respectively. Even though the frequency of different signal components fluctuated at the different stages of tool wear and at different cutting conditions, the frequency of vibration components was always within a band of 98–40 kHz, and the AE has varied between 51 kHz and 620 kHz.