Blended infinite elements for paraboblic boundary value problems

A common method for numerically approximating two-point parabolic boundary value problems of the form u_t = L[u]+f(u) defined of the semi-infinite strip S = [0, 1]×[0, ∞] is to first discretize the spatial operator in the differential equation and then solve for the time evolution. Such an approach typically involves solving a system of algebriaic equations at a sequence of time steps. In this paper we take a different approach and subdivide S into a collection of semi-infinite substrips S_i = [x_i, x_i+1]×[0, ∞], and use blending function techniques to derive finite parameter functions e_i(x, t) defined on S_i. Spectral matching methods are used in deriving e_i to ensure that (u ? e_i) can be made small on S_i. Galerkin's method, with associated integration sover the entire space-time domain S, is then used to generate approximations to u(x, t) based upon the so defined infinite element (e_i, S_i). Approximations are hence found for all (x, t) in S by solving one well structed system of algebraic equations. We apply the method to several linear and non-linear problms.     

Toric Varieties Hirzebruch Surfaces and Error-Correcting Codes

 For any integral convex polytope in ℝ there is an explicit construction of an error-correcting code of length (q-1)² over the finite field 𝔽 _q , obtained by evaluation of rational functions on a toric surface associated to the polytope. The dimension of the code is equal to the number of integral points in the given polytope and the minimum distance is determined using the cohomology and intersection theory of the underlying surfaces. In detail we treat Hirzebruch surfaces. Received: August 21, 2000; revised version: September 3, 2002     

Fast Parallel Algorithms for Matrix Reduction to Normal Forms

 We investigate fast parallel algorithms to compute normal forms of matrices and the corresponding transformations. Given a matrix B in ℳ _n,n (K), where K is an arbitrary commutative field, we establish that computing a similarity transformation P such that F=P ^-1 BP is in Frobenius normal form can be done in ?C ² _K . Using a reduction to this first problem, a similar fact is then proved for the Smith normal form S(x) of a polynomial matrix A(x) in ℳ _n,m (K[x]); to compute unimodular matrices U(x) and V(x) such that S(x)=U(x)A(x)V(x) can be done in ?C ² _K . We get that over concrete fields such as the rationals, these problems are in ?C ². Using our previous results we have thus established that the problems of computing transformations over a field extension for the Jordan normal form, and transformations over the input field for the Frobenius and the Smith normal form are all in ?C ² _K . As a corollary we establish a polynomial-time sequential algorithm to compute transformations for the Smith form over K[x]. Received: February 29, 1996; revised version: August 29, 1997     

High-pressure behavior of the crystal structure of the fullerene molecular complex with ferrocene C60·{Fe(C5H5)2}2

Abstract

The crystal structure of the molecular complex C₆₀·{Fe(C₅H₅)₂}₂ was studied by single crystal X-ray diffraction (XRD) analysis at pressures up to 5?GPa using the diamond anvil cell (DAC) technique. The XRD data and subsequent structural analysis clearly show that there is no pressure-induced polymerization in fullerene layers in pressure range studied. The reciprocal unit cell volume V₀/V is a smooth and monotonous function of pressure and fits well to the Murnaghan equation of state (V₀/V) ^B'=?{1?+?P·(B'/B₀)}, where V₀ is the volume at ambient pressure, B₀=8.7?GPa and B'=10.5 are the bulk modulus and its derivative, respectively. Pressure dependence of the shortest distance between the C₅H₅ ring of the ferrocene molecule and the center of the nearest fullerene molecule is linear, while the slope changes at 2.2?GPa indicating on the intermolecular interactions crossover. Other relevant parameter, the Fe-C bond lengths of ferrocene, sensitive to iron charge state, gradually increases. This peculiarity can a sign of pressure-induced partial charge transfer between the donor ferrocene and acceptor fullerene molecules.     

Structural features and energy of small water clusters

Collisions between water clusters (H₂O)_n, n=27, with relative velocities V=1, 3, or 10 km/s and various initial temperatures, were studied by the molecular dynamics method within the framework of a polarization model. It was found that the clusters may either stick together (at V=1 km/s), break into fragments (V=3 km/s), or form compressed (excited) molecules (V=10 km/s) dissociating into H⁺ and OH⁻. Inside the cluster, the charge is transferred by the H⁺ hopping mechanism. Upon fragmentation, the clusters separate into H⁺(H₂O)_i and OH⁻(H₂O)_j ions.     

A new relaxation method for obtaining the lowest eigenvalue and eigenvector of a matrix equation

The matrix eigenvalue problem Hu_i = λ_i u_i is considered. It is shown that when a new approximate vector v⁽ⁿ⁺¹⁾ to u₁ (the eigenvector of the lowest eigenvalue) is computed from the present one v⁽ⁿ⁾ by the relation v(n+1) = (1? αH + βH²) v⁽ⁿ⁾ or v⁽ⁿ⁺¹⁾ = (1? αH + βH² – γH³) v⁽ⁿ⁾, the convergence rate is at least double that of the gradient method which corresponds to set β = γ = 0. Moreover, by choosing parameters α, β, or γ properly, one can get about three to five times faster convergence rate than that of the latter method, for H having very small γ₂–γ₁ and very large λ_N (the largest eigenvalue), further modifications are suggested. The relation with the Richardson method is also discussed.     

A predictor–corrector-type technique for the approximate parameterization of intersection curves

We describe a method to approximate a segment of the intersection curve of two implicitly defined surfaces by a rational parametric curve. Starting from an initial solution, the method applies predictor and corrector steps in order to obtain the result. Based on a preconditioning of the two given surfaces, the corrector step is formulated as an optimization problem, where the objective function approximates the integral of the squared Euclidean distance of the curve to the intersection curve. An SQP-type method is used to solve the optimization problem numerically. Two different predictor steps, which are based on simple extrapolation and on a differential equation, are formulated. Error bounds are needed in order to certify the accuracy of the result. In the case of the intersection of two algebraic surfaces, we show how to bound the Hausdorff distance between the intersection curve (an algebraic space curve) and its rational approximation.     

Two-Bands Superconductivity with Intra- and Interband Pairing for Synthetic Superlattices

We consider a model for superconductivity in a two-band superconductor, having an anisotropic electronic structure made of two partially overlapping bands with a first hole-like and a second electron-like Fermi surface. In this pairing scenario, driven by the interplay between interband V _i,j and intraband V _i,i pairing terms, we have solved the two gap equations at the critical temperature T=T _c and calculate T _c and the chemical potential μ as a function of the number of carriers n for various values of pairing interactions, V _1,1, V _2,2, and V _1,2. The results show the complexity of the physics of condensates with multiple order parameters with the chemical potential near band edges.     

Algorithms for Computing Sparse Shifts for Multivariate Polynomials

In this paper, we investigate the problem of finding t-sparse shifts for multivariate polynomials. Given a polynomial f∈ℱ[x ₁, x ₂, …, x _n ] of degree d, and a positive integer t, we consider the problem of representing f(x) as a ?-linear combination of the power products of u _i where u _i = x _i −b _i for some b _i ∈?, an extension of ℱ, for i = 1, …, n, i.e., f = ∑ _j F _j u ^αj , in which at most t of the F _j are non-zero. We provide sufficient conditions for uniqueness of sparse shifts for multivariate polynomials, prove tight bounds on the degree of the polynomial being interpolated in terms of the sparsity bound t and a bound on the size of the coefficients of the polynomial in the standard representation, and describe two new efficient algorithms for computing sparse shifts for a multivariate polynomial. Received: January 30, 1996; revised version: January 15, 2000     

Inorganic Nano Light‐Emitting Transistor: p‐Type Porous Silicon Nanowire/n‐Type ZnO Nanofilm

An inorganic nano light‐emitting transistor (INLET) consisting of p‐type porous Si nanowires (PoSiNWs) and an n‐type ZnO nanofilm was integrated on a heavily doped p‐type Si substrate with a thermally grown SiO₂ layer. To verify that modulation of the Fermi level of the PoSiNWs is key for switchable light emitting, I–V and electroluminescent characteristics of the INLET are investigated as a function of gate bias (V _g). As the V _g is changed from 0 V to ?20 V, the current level and light‐emission intensity in the orange–red range increase by three and two times, respectively, with a forward bias of 20 V in the p–n junction, compared to those at a V _g of 0 V. On the other hand, as the V _g approaches 10 V, the current level decreases and the emission intensity is reduced and then finally switched off. This result arises from the modulation of the Fermi level of the PoSiNWs and the built‐in potential at the p–n junction by the applied gate electric field.     

Control of threshold voltage of organic field-effect transistors by space charge polarization

We demonstrate organic field-effect transistors (OFETs) with an ion-dispersed polymer for the gate dielectrics. By applying external electric field (V_ex), the dispersed ions can migrate by electrophoresis and separated ion pairs form space charge polarization in the gate dielectrics. After V_ex was applied, the drain current is increased over 7 times and threshold voltage is decreased from − 12.9 V to − 2.9 V. The shift direction of V_th is controllable by the polarity of the V_ex. Results of ultraviolet/visible differential absorption study reveal that the active layer of OFETs is charged not only electrostatically but electrochemically with increasing the time after V_ex was applied.     

Renewal-type equations for a general repair process

We consider a component which undergoes instantaneous general repairs on failure. Suppose that V_i = ϕ(V_i−1 + Y_i), where V_i is the virtual age immediately after the ith failure, Y_i is the time between the (i − 1)th and ith failures, ϕ(·) is a specified repair functional, and V₀ = s. We derive integral equations for the repair density, and also for the joint density of repairs with respect to chronological age and virtual age on failure. In the asymptotic case, approximations are obtained for the mean and variance of virtual age on failure, virtual age after repair, and time between failures. Two policies are then considered. The first, an upgraded repair strategy employs minimal repairs until the component reaches a specified age, thereafter repairs restore the virtual age to that prescribed level. The second policy considered is the Kijima Type 2 model which is not amenable to the usual g-renewal analysis that is used in the Type 1 model. The repair density is obtained by numerical solution of the relevant integral equation. Finally, approximations for the asymptotic moments are found to be in close agreement with simulated results. © 1997 John Wiley & Sons, Ltd.     

Observation of Fibre Laser Instabilities Through Modulation of the Cavity Optical Length

Abstract

A Nd-doped, single-mode, double-clad fibre laser (FL) is coiled around a resonant piezoelectric ceramic ring; this induces modulation of the cavity optical length and two distinct classes of laser intensity instabilities are observed as the ring drive voltage V _P-P is increased. The first class involves a slowly varying intensity, with characteristic time scale in the microsecond range. For fixed drive-voltage frequency f _m ≈ 20 kHz, we consistently observe subharmonics f _m/2 and f _m/4 in the FL output intensity spectrum; noise in the system precludes the observation of higher subharmonic bifurcations. Small-amplitude random relaxation oscillations of the unmodulated FL are strongly amplified as V _P-P is increased and transition to a state of large-amplitude chaotic relaxation oscillations occurs for V _P-P ≈ 4 V; a value of 2·6 has been measured for the correlation dimension of the chaotic attractor. The second class of laser instability occurs simultaneously for V _P-P ? 10 V, and the FL output consists of a train of pulses about 2 ns wide at a repetition rate close to the cavity frequency f _cav ≈ 15 MHz and modulated by the slowly varying chaotic envelope described above.     

Adaptive Distribution-free Regression Methods and their Applications

We consider the model Y = βx + Z, where the random variable Z has a continuoustype distribution that can be badly skewed, contaminated, or censored. To test the hypothesis H ₀ : β = β₀, we use the distribution-free statistic K(β₀) = Σc(Q _i )a(R _i ), where c(·) and a(·) are increasing score functions and Q _i and R _i are the respective ranks of x _i and y _i – β₀ x _i . The score functions c(·) and a(·) can be adapted or chosen after observing the data without destroying the distribution-free nature of the test. A Monte Carlo study is presented which illustrates the excellent performance of an adaptive test when a wide range of distributions is considered for the residuals. Interval and point estimates of sβ can be found by employing the “inverse” of the testing procedure. These results are used to find estimates of the percentile lines. Two examples are given which involve lifetimes of electric motor insulation and grade point averages of beginning university students, respectively.     

Localization of an algebraic hypersurface by the exclusion algorithm

We describe a new algorithm for the localization of an algebraic hypersurfaceV inR ⁿ or Cⁿ. This algorithm computes a decreasing sequence of closed sets whose intersection isV. In the particular case of an hypersurface without any point at infinity, the notion of the asymptotic cone is used to determine a compact set containing this hypersurface. We give also a numerical version of this algorithm.     

ZnSe<Cu> Luminescence at High Doping Levels

ZnSe was doped with Cu via thermal diffusion from Cu foil in the course of chemical vapor deposition. The Cu distribution over the deposit was studied, and the Cu solubility in ZnSe was determined as a function of temperature. Copper was shown to prevent oxygen incorporation from the gas phase. The microstructure and homogeneity of ZnSe were studied by scanning electron microscopy. The cathodoluminescence (CL) spectra were measured on fracture surfaces in order to elucidate the origin of the Cu-related emission centers. Heavy doping with Cu was shown to gives rise to a long-wavelength CL band, which shifts from 525 to 550 nm as the temperature is raised from 80 to 300 K. The associated changes in oxygen-related self-activated luminescence were correlated with the Cu and O concentrations. To gain more detailed insight into the origin of the green Cu-related emission in p-type ZnSe, we examined its photoexcitation spectrum, temperature-dependent peak position, quenching, and variation with excitation intensity. The conclusion is drawn that the longer wavelength CL bands are due to electron transitions from the Zn_i ^·· donor level to the impurity band. The CL spectrum of heavily doped ZnSe shows no emissions between 800 and 1400 nm, related to the V _Se and V _Zn vacancies.     

Phase-pure Na3V2(PO4)2F3 embedded in carbon matrix through a facile polyol synthesis as a potential cathode for high performance sodium-ion batteries

In this study, a pseudo-layered Na super-ionic conductor of Na₃V₂(PO₄)₂F₃ (NVPF)/C cathode for sodium-ion batteries is prepared successfully using a facile polyol refluxing process without any impurity phases. The X-ray diffraction and Rietveld refinement results confirm that NVPF possesses tetragonal NASICON-type lattice with a space group of P₄₂/mnm. In this preparative method, polyol is utilized as a solvent as well as a carbon source. The presence of nanosized NVPF particles in the carbon network is confirmed by field-emission scanning electron microscopy (FE-SEM) and high-resolution transmission electron microscopy (HR-TEM). The existence of carbon is analyzed by Raman scattering and elemental analysis. When applied as a Na-storage material in a potential window of 2.0–4.3 V, the electrode exhibits two flat voltage plateaus at 3.7 and 4.2 V with an electrochemically active V³⁺/V⁴⁺ redox couple. In addition, Na₃V₂(PO₄)₂F₃/C composite achieved a retention capacity of ~ 88% even after 1,500 cycles at 15 C. Moreover, at high current densities of 30 and 50 C, Na₃V₂(PO₄)₂F₃/C cathode retains the specific discharge capacities of 108.4 and 105.9 mAh·g^–1, respectively, revealing the structural stability of the material prepared through a facile polyol refluxing method.

     

1-Perfect Uniform and Distance Invariant Partitions

Let F ⁿ be the n-dimensional vector space over ℤ₂. A (binary) 1-perfect partition of F ⁿ is a partition of F ⁿ into (binary) perfect single error-correcting codes or 1-perfect codes. We define two metric properties for 1-perfect partitions: uniformity and distance invariance. Then we prove the equivalence between these properties and algebraic properties of the code (the class containing the zero vector). In this way, we characterize 1-perfect partitions obtained using 1-perfect translation invariant and not translation invariant propelinear codes. The search for examples of 1-perfect uniform but not distance invariant partitions enabled us to deduce a non-Abelian propelinear group structure for any Hamming code of length greater than 7. Received: March 6, 2000; revised version: November 30, 2000     

Regularizing algorithm for inverting the Abel equation

The article presents a regularizing algorithm for solving the Abel equation using information on the statistics of the error of measurement of the right-hand side of the equation.Notation (r), f(x) solution and right-hand side of the Abel equation, respectively - f_i value of the right-hand side measured at point x_i - _i uncertainty of the i-th measurement - n number of measurements of the right-hand side - V correlation matrix of the uncertainty of measurement - smoothing parameter - S_n(x) interpolating spline - S_n,(x) smoothing spline - a_i, b_i, c_i, d_i coefficients of the smoothing spline - (r) regularized solution of the Abel equation - e() discrepancy vector - Sp[V] trace of the matrix V Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 39, No. 2, pp. 270–274, August, 1980.     

Image force effects on trapezoidal barrier parameters in metal-insulator-metal tunnel junctions

A one-dimensional trapezoidal barrier potential U₀(x) with and without the image potential U_i(x) was used to model three tunnel junctions: Al-Al₂O₃-Au (-Ag, -Cu) and calculate the junction I-V curves. As the effective trapezoidal barrier parameters are determined by fitting the calculated I-V curves to the experimental ones, it is found that if the I-V curves are calculated based on the barrier potential U₀(x)+U_i(x), the parameters of the actual barrier U₀(x)−U_i(x) are obtained. Thus, the parameters of the actual trapezoidal barrier U₀(x)+U_i(x) can be obtained by extrapolating. The results show that the area of the actual U₀(x)+U_i(x) is smaller than that of the U₀(x) and for the Au-, Ag- and Cu-junction the area reduction is − 6.9, −5.4 and −4.2%, respectively. The variation of the barrier area reduction with the electrodes can be ascribed to the diffusion and oxidation reaction in the region of the junction interfaces, which results in smeared and indistinct interfaces and an increase in the actual distance between the metal mirrors of the two electrodes.