On Intrinsic Bounds in the Nullstellensatz

 Let k be a field and  f ₁, . . . ,  f _s be non constant polynomials in k[X ₁, . . . , X _n ] which generate the trivial ideal. In this paper we define an invariant associated to the sequence  f ₁, . . . ,  f _s : the geometric degree of the system. With this notion we can show the following effective Nullstellensatz: if δ denotes the geometric degree of the trivial system  f ₁, . . . ,  f _s and d :=max _j  deg( f _j ), then there exist polynomials p ₁, . . . , p _s ∈k[X ₁, . . . , X _n ] such that 1=∑ _j  p _j  f _j and deg p _j   f _j ≦3n ²δd. Since the number δ is always bounded by (d+1) ^n-1 , one deduces a classical single exponential upper bound in terms of d and n, but in some cases our new bound improves the known ones. Received November 24, 1995, revised version January 19, 1996     

Truncated Gröbner Bases for Integer Programming

 The toric ideal I _A of a matrix A=(a ₁, . . . ,  a _n )∈ℤd ^× n is the kernel of the monoid algebra map π^ _A  : k[x ₁, . . . ,  x _n ]→k[t ^±1 ₁, . . . ,  t ^±1 _d ], defined as x _j ?t ^aj . It was shown in [4] that the reduced Gr?bner basis of I _A , with respect to the weight vector c, can be used to solve all integer programs minimize {cx : Ax=b, x∈ℕ ⁿ }, denoted IP _A,b,c,= , as b varies. In this paper we describe the construction of a truncated Gr?bner basis of I _A with respect to c, that solves IP _A,b,c,= for a fixed b. This is achieved by establishing the homogeneity of I _A with respect to a multivariate grading induced by A. Depending on b, the truncated Gr?bner basis may be considerably smaller than the entire Gr?bner basis of I _A with respect to c. For programs of the form maximize{cx : Ax≦b, x≦u, x∈ℕ  ⁿ } in which all data are non-negative, this algebraic method gives rise to a combinatorial algorithm presented in [17]. Received: April 10, 1995; revised version: February 6, 1996     

Bounds on Minimum Distance for Linear Codes over GF(5)

Let [n, k, d; q]-codes be linear codes of length n, dimension k and minimum Hamming distance d over GF(q). Let d ₅(n, k) be the maximum possible minimum Hamming distance of a linear [n, k, d; 5]-code for given values of n and k. In this paper, forty four new linear codes over GF(5) are constructed and a table of d ₅(n, k) k≤ 8, n≤ 100 is presented.     

On the Lattice Structure of Pseudorandom Numbers Generated over Arbitrary Finite Fields

Marsaglia's lattice test for congruential pseudorandom number generators modulo a prime is extended to a test for generators over arbitrary finite fields. A congruential generator η₀,η₁,…, generated by η _n =g(n), n = 0, 1,…, passes Marsaglia's s-dimensional lattice test if and only if s≤ deg(g). It is investigated how far this conditin holds true for polynomials over arbitrary finite fields F _q , particularly for polynomials of the form g _d (x)=α(x+β) ^d +γ, α, β, γ∈F _q , α≠ 0, 1 ≤d≤q− 1. Received: December 13, 1999; revised version: November 17, 2000     

Cryptanalysis of ‘Less Short’ RSA Secret Exponents

 In some applications of RSA, it is desirable to have a short secret exponent d. Wiener [6], describes a technique to use continued fractions (CF) in a cryptanalytic attack on an RSA cryptosystem having a ‘short’ secret exponent. Let n=p ⋅ q be the modulus of the system. In the typical case that G=gcd(p−1, q−1) is small. Wiener’s method will give the secret exponent d when d does not exceed (approximately) n ^1/4. Here, we describe a general method to compute the CF-convergents of the continued fraction expansion of the same number as in Wiener (which has denominator d ⋅ G) up to the point where the denominator of the CF-convergent exceeds approximately n ^1/4. When d<n ^1/4 this technique determines d, p, and q as does Wiener’s method. For larger values of d there is still information available on the secret key. An estimate is made of the remaining workload to determine d, p and q. Roughly speaking this workload corresponds to an exhaustive search for about 2r+8 bit, where r=ln₂ d/n ^1/4. Received: September 30, 1996; revised version: March 7, 1997     

Electron-beam enhanced fusion of binary metals and alloying in melts

Binary mixing and alloying during electron-beam fusion in bi-metal joints have been investigated and presented in this paper. In case of beam-power densities greater than about 10¹⁰ Wm⁻², rapid fusion of metals is accompanied by violent agitation of the melts. The resulting mass transport in the melt is described here from an enhanced mass diffusion coefficientD =D _s(W′/a ₀ η)^1/2, whereD _s is mass diffusivity in solid,W′ is beam-power density,a ₀ is acceleration in melt growth andη is the coefficient of viscosity. Moreover, a uniformly mixed central-widthw′ is predicted in the fusion zone (total widthw), where,w′ is significantly dependent onη andw throughW′. For these conditions, the continuity of mass transport in stationary and travel modes of the source has been solved using line and plane source models and conservation principles. The predictions from these calculations have been validated through quantitative analyses of experiments.     

Some Power Mappings with Low Differential Uniformity

 Differentially uniform power mappings of the form f (x)=x ^d over GF(p ⁿ ) are considered. We construct an infinite family of 2-uniform mappings in the binary case. In the nonbinary case we give two large families of k-uniform mappings with low values of k. We also show how to construct families of sequences from differentially 1-uniform power mappings, which have parameters as good as the best presently known comparable families of sequences. Received: November 4, 1996; revised version: February 14, 1997     

Thermal-lens study of thermo-optical properties of tellurite glasses

Mode-mismatched Thermal Lens (TL) measurements were performed in 70TeO₂–19WO₃–7Na₂O–4Nb₂O₅ (% mol) tellurite glasses doped with either Er³⁺ or Tm³⁺ and co-doped with Er³⁺/Tm³⁺ ions. Thermo-optical parameters (D, K, ds/dQ and ds/dT) were obtained in function of thulium concentrations (0.39–1.6) × 10 ²⁰ ions/cm³. For Er³⁺/Tm³⁺ co-doped tellurite glasses, D and K values are practically independent of the Tm³⁺ concentrations used in this study. The average values of D and ds/dT obtained for tellurite glasses are: (3.1 ± 0.2) × 10⁻³ cm²/s and (16 ± 3) × 10⁻⁶ K⁻¹, respectively.     

Reduction of Np(V) with Fe(II) in weakly acidic solutions containing H<Subscript>2</Subscript>C<Subscript>2</Subscript>O<Subscript>4</Subscript>

The kinetics of the reaction of Np(V) with Fe(II) in dilute perchloric and nitric acid solutions containing H₂C₂O₄ was studied by spectrophotometry. In the range pH 1–2, the reaction rate is described by the equation d[Np(V)]/dt = k[Np(V)][Fe(II)][H₂C₂O₄]²[H⁺]^−1.6, k = 182 mol^−1.4 l^1.4 s⁻¹. The activation energy in the range 25–45°C is 26 kJ mol⁻¹. The reaction mechanism involves formation of Fe(II) and Np(V) oxalate complexes, followed by their reaction with the participation of the H⁺ ion.     

Anomalously strong small-q electron-phonon coupling in cuprates due to dielectricity

We consider the role of the enormous ionic dielectric constant ε₀≈ 30–50 in the cuprates. The ionically-screened Thomas-Fermi screening parameter ~q_TF = [4θe²N(E_F)/εin0 ^1/2 is extremely small, namely 0.2-0.3 A^-1. The electron-phonon coupling constant I(q,w) is found to be anomalously large forq ≈ ~q_TF and smallco, accounting for the CDW observed in overdoped BSCCO with wavevector 0.24 A^-1. Using the Eliashberg theory we derive from this I(q,w) a maximumT _c of about 200 K. The small electron-phonon scattering angle △θ = ~q _{F ≈} 0.3 rad acounts for the observedd-wave pairing, as originating from a phonon-mediated mechananism.     

Satellite peaks in the scattering of p-polarized light from a randomly rough film on a perfectly conducting substrate

Abstract

The diffuse scattering of p-polarized light from a one-dimensional, randomly rough dielectric film deposited on a planar, perfectly conducting surface is studied by means of small-amplitude perturbation theory. The dielectric constant ε_d and the thickness d of the dielectric film are chosen in such a way that in the absence of roughness the scattering system supports N (≥ 2) guided modes whose wavenumbers are q ₁ (ω), …,q _N(ω) at the frequency ω of the light incident at an angle θ_o. We investigate the occurrence of satellite peaks, in addition to the enhanced back-scattering peak, in the angular distribution of the intensity of the diffusely scattered light, at angles θ_s(n, m) given by sin θ_s(n, m)= ? sin θ₀± (c/ω) [Q_n(ω) ? q_m(ω)] for n, m=1, …, N(n ≠ m). These satellite peaks are multiple-scattering effects due to degenerate timereversal symmetry.     

Primitive Normal Bases with Prescribed Trace

Let E be a finite degree extension over a finite field F = GF(q), G the Galois group of E over F and let a∈F be nonzero. We prove the existence of an element w in E satisfying the following conditions: - w is primitive in E, i.e., w generates the multiplicative group of E (as a module over the ring of integers). - the set {w ^g ∣g∈G} of conjugates of w under G forms a normal basis of E over F. - the (E, F)-trace of w is equal to a. This result is a strengthening of the primitive normal basis theorem of Lenstra and Schoof [10] and the theorem of Cohen on primitive elements with prescribed trace [3]. It establishes a recent conjecture of Morgan and Mullen [14], who, by means of a computer search, have verified the existence of such elements for the cases in which q≤ 97 and n≤ 6, n being the degree of E over F. Apart from two pairs (F, E) (or (q, n)) we are able to settle the conjecture purely theoretically. Received: June 15, 1998; revised version: December 2, 1998     

Travelling waves in autocatalytic chemical systems with decay: bounds on existence

The existence of solutions to the travelling-wave equations governed by an autocatalytic reaction of order p (p ≥ 1) and an autocatalytic decay step of order q (q ≥ 1) are examined in the limit of large p. Two cases are treated, q of O(1) and . In the first case, an upper bound k _crit is found for k for the existence of a solution, where k is a dimensionless measure of the strength of the decay step. In the second case, an upper bound on k is also found when q < p. For q ≥ p, there is no upper bound on k and solutions exist for all (positive) values of k.     

Piezoelectric,ferroelectric and mechanical properties of lead zirconate titanate/zinc oxide nanowhisker ceramics

New lead zirconate titanate/zinc oxide nanowhisker (PZT/ZnO_w) ceramics were fabricated by a conventional solid state processing and their structures, piezoelectric, ferroelectric and mechanical properties were studied. Both the PZT perovskite and ZnO phases can be observed from the X-ray diffraction patterns. The grain size of ceramics is reduced due to the ZnO_w addition. The incorporation of ZnO_w into the PZT ceramics improves the strength and toughness, while deteriorates the piezoelectric and ferroelectric properties. For the PZT/ZnO_w ceramics with 1–2 wt% ZnO_w, the mechanical properties become optimum, meanwhile maintain good piezoelectric and ferroelectric properties: σ _c = 376–484 MPa, σ _f = 115–121 MPa, K _IC = 1.41–1.54 MPa m^1/2, d ₃₃ = 442–490 pC/N, k _p = 0.54–0.55, ε _r = 3,322–3,980, Q _m = 99–101, tanδ = 1.6%–1.7%, P _r = 21.5–26.9 μC/cm² and E _c = 8.1–8.6 kV/cm.     

Goodness-of-fit tests in parametric regression based on the estimation of the error distribution

Consider a heteroscedastic regression model Y=m(X)+σ(X)ε, where m(X)=E(Y|X) and σ ²(X)=Var (Y|X) are unknown, and the error ε is independent of the covariate X. We propose a new type of test statistic for testing whether the regression curve m(⋅) belongs to some parametric family of regression functions. The proposed test statistic measures the distance between the empirical distribution function of the parametric and of the nonparametric residuals. The asymptotic theory of the proposed test is developed, and the proposed testing procedure is illustrated by means of a small simulation study and the analysis of a data set.     

Computation of the relaxation and creep functions of elastomers from harmonic shear modulus

The purpose of this paper is to compute the relaxation and creep functions from the data of shear complex modulus, G ^∗(iν). The experimental data are available in the frequency window ν∈[ν_min,ν_max] in terms of the storage G′(ν) and loss G″(ν) moduli. The loss factor h( n) = \fracG"( n)G￠(n)\eta( \nu) = \frac{G'( \nu )}{G'(\nu )} is asymmetrical function. Therefore, a five-parameter fractional derivative model is used to predict the complex shear modulus, G ^∗(iν). The corresponding relaxation spectrum is evaluated numerically because the analytical solution does not exist. Thereby, the fractional model is approximated by a generalized Maxwell model and its rheological parameters (G _k ,τ _k ,N) are determined leading to the discrete relaxation spectrum G(t) valid in time interval corresponding to the frequency window of the input experimental data. Based on the deterministic approach, the creep compliance J(t) is computed on inversing the relaxation function G(t).     

Bounds for traces in complete intersections and degrees in the Nullstellensatz

In this paper we obtain an effective Nullstellensatz using quantitative considerations of the classical duality theory in complete intersections. Letk be an infinite perfect field and let f₁,...,f n–rk[X₁,...,X_n] be a regular sequence with d:=max_j deg f_j. Denote byA the polynomial ringk [X₁,..., X_r] and byB the factor ring k[X₁,...,X_n]/(f₁,...,f_n r); assume that the canonical morphism AB is injective and integral and that the Jacobian determinant with respect to the variables X_r+1,...,X_n is not a zero divisor inB. Let finally B^*:=Hom_A(B, A) be the generator of B^* associated to the regular sequence.We show that for each polynomialf the inequality deg (¯f) dⁿ r(+1) holds (¯fdenotes the class off inB and is an upper bound for (n–r)d and degf). For the usual trace associated to the (free) extensionA B we obtain a somewhat more precise bound: deg Tr(¯f) dⁿ r degf. From these bounds and Bertini's theorem we deduce an elementary proof of the following effective Nullstellensatz: let f₁,..., f_s be polynomials in k[X₁,...,X_n] with degrees bounded by a constant d2; then 1 (f₁,..., f_s) if and only if there exist polynomials p₁,..., p_sk[X₁,..., X_n] with degrees bounded by 4n(d+ 1)ⁿ such that 1=_ip_if_i. in the particular cases when the characteristic of the base fieldk is zero ord=2 the sharper bound 4ndⁿ is obtained.Partially supported by UBACYT and CONICET (Argentina)     

A response-modeling alternative to surrogate models for support in computational analyses

Often, the objectives in a computational analysis involve characterization of system performance based on some function of the computed response. In general, this characterization includes (at least) an estimate or prediction for some performance measure and an estimate of the associated uncertainty. Surrogate models can be used to approximate the response in regions where simulations were not performed. For most surrogate modeling approaches, however, (1) estimates are based on smoothing of available data and (2) uncertainty in the response is specified in a point-wise (in the input space) fashion. These aspects of the surrogate model construction might limit their capabilities.One alternative is to construct a probability measure, G(r), for the computer response, r, based on available data. This “response-modeling” approach will permit probability estimation for an arbitrary event, E(r), based on the computer response. In this general setting, event probabilities can be computed: prob(E)=∫_rI(E(r))dG(r) where I is the indicator function. Furthermore, one can use G(r) to calculate an induced distribution on a performance measure, pm. For prediction problems where the performance measure is a scalar, its distribution F_pm is determined by: F_pm(z)=∫_rI(pm(r)z)dG(r). We introduce response models for scalar computer output and then generalize the approach to more complicated responses that utilize multiple response models.     

Ferroelectric and piezoelectric properties of [(Bi0.98La0.02Na1?xLix)0.5]0.94Ba0.06TiO3 lead-free ceramics

Lead-free ceramics [(Bi_0.98La_0.02Na_1−x Li _x )_0.5]_0.94Ba_0.06TiO₃ have been prepared by an ordinary sintering technique and their ferroelectric and piezoelectric properties have been studied. The results of X-ray diffraction reveal that Li⁺, Ba²⁺, and La³⁺ diffuse into the Bi_0.5Na_0.5TiO₃ lattices to form a new solid solution with a pure perovskite structure. The partial substitution of Li⁺ lowers the coercive field E _c and improves the remanent polarization P _r. Because of the larger P _r and lower E _c, the ceramic with x = 0.10 exhibits optimum piezoelectric properties: d ₃₃ = 212 pC/N and k _P = 36.1%. The partial substitution of Li⁺ for Na⁺ shifts the depolarization temperature T _d toward low temperature. The ceramics exhibit relaxor characteristic, which is probably resulted from the cation disordering in the 12-fold coordination sites. The temperature dependences of the ferroelectric and dielectric properties suggest that the ceramics contain both the polar and non-polar regions near/above T _d, which cause the polarization hysteresis loop become deformed and the ceramics become depolarized.     

Retardation of grain growth in electrodeposited Cu by an electric field

The application of an external dc electric field E = 5 kV/cm during the annealing of electrodeposited Cu foil at 150–195 °C retarded grain growth. The time dependence of the grain size both with and without the field was