Directional nature of Goodman–Kruskal gamma and some consequences: identity of Goodman–Kruskal gamma and Somers delta,and their connection to Jonckheere–Terpstra test statistic

Although usually taken as a symmetric measure, G is shown to be a directional coefficient of association. The direction in G is not related to rows or columns of the cross-table nor the identity of the variables to be a predictor or a criterion variable but, instead, to the number of categories in the scales. Under the conditions where there are no tied pairs in the dataset, G equals Somers’ D so directed that the variable with a wider scale (X) explains the response pattern in the variable with a narrower scale (g), that is, D(g│X). Hence, G = G(g│X) = D(g│X) but G ≠ D(X│g) and G ≠ D(symmetric). If there are tied pairs, the estimates by G = G(g│X) are more liberal in comparison with those by D(g│X). Algebraic relation of G and D with Jonckheere–Terpstra test statistic (JT) is derived. Because of the connection to JT, G = G(g│X) and D = D(g│X) indicate the proportion of logically ordered test-takers in the item after they are ordered by the score. It is strongly recommendable that gamma should not be used as a symmetric measure, and it should be used directionally only when willing to explain the behaviour of a variable with a narrower scale by the variable with a wider scale. This fits well with the measurement modelling settings.

     

Pinning Energy of High-Tc (Bi, Pb)2Sr2Ca2Cu3Oy

The flux pinning potential energy of silver (Ag) sheathed high-T _c Bi-2223 tapes determined from measured current densities, J(T,H), is reported in this paper. Measurements of the transport current density, J, of the Ag sheathed Bi-2223 tapes were performed with an applied magnetic field directions parallel (H‖ab) and perpendicular (H‖c) to the tape’s board surface from 0 to 3 T, at temperatures from 23 to 77 K. To obtain the pinning potential energy of the Bi-2223 superconductor, an empirical scaling of J(T,H) is introduced in this study. Under the flux dynamics, the effective pinning energy, U _eff(T,H,J), can be formulated using the measured current density, J(T,H). Furthermore, the pinning potential energies, U(T,H), for both H‖ab and H‖c were quantitatively determined by the scaling of J(T,H). The two energies, U(T,H), exhibit as a function that is inversely proportional to both temperature and logarithmic magnetic field. The magnitude of U(T,H) for H‖ab is three or more times larger than that for H‖c under the considered conditions. The U(T,H) is sensitive to the temperature, but insensitive to the magnetic field. This characteristic of insensitiveness towards magnetic field is an advantage for applications on the conditions of high current density, high magnetic field, and low temperature.     

C 1-stably ℒ p shadowable homoclinic classes

Let p be a hyperbolic saddle of diffeomorphism f on closed manifold M and H(p, f) be the homoclinic class associated with it. In this article, we introduce the notion of C ¹-stably ? _p shadowing and prove that if f is C ¹-stably ? _p shadowable on a homoclinic class H(p, f) then, H(p, f) has a dominated splitting. Moreover, we prove that if f is C ¹-stably Lipschitz ? _p shadowable on H(p, f) and H(p, f)-germ of f is expansive then the homoclinic class is hyperbolic.     

Pressure- and Field-induced Non-Fermi-liquid Behaviors in a Heavy-Fermion Antiferromagnet Ce7Ni3

We review recent investigations of the heavy-fermion antiferromagnet Ce₇Ni₃ with a geometrically frustrated structure under hydrostatic pressures and magnetic fields. Below T _N1=1.9 K, an incommensurate spin-density-wave (SDW) develops, and another antiferromagnetic transition occurs at T _N2 = 0.7 K. At a rather low critical pressure (P _c ), P _c = 0.39 GPa, both T _N1 and T _N2 vanish, and the specific heat divided by temperature (C/T) exhibits –lnT dependence (i.e., non-Fermi liquid behavior). At P = 0.43 GPa > P _c , a T-independent behavior of C/T, i.e., Fermi-liquid behavior, recovers below 0.2 K. However, the magnetic susceptibility continues to increase down to 0.09 K at 0.43 GPa, which is not consistent with a conventional Fermi-liquid theory. On the other hand, upon applying magnetic fields B along the hexagonal c axis, T _N1 decreases and vanishes at 0.3 T. Magnetoresistance, specific-heat, and magnetization measurements reveal that a field-induced magnetic (FIM) phase appears in a B–T region for B _{∥ c} > 0.7 T and T < 0.5 K. Neutron diffraction experiments indicate that the magnetic unit cell in the c plane for the FIM phase is treble that of the chemical unit cell. Moreover, the intensity of the magnetic reflection remains even in the region between the FIM phase and SDW phase. This observation indicates the presence of large spin fluctuations in the c-plane associated with the magnetic frustration, which should be responsible for the non-Fermi-liquid behavior of Ce₇Ni₃.     

The Distribution of the Maximum Sum of Rank

In this paper a c-sample slippage analogue of the Wilcoxon [11] test is considered. Given a sample of size n for each of c populations, the test rejects the hypothesis that the c populations are identical when max_1≤i≤c σ _k r _ik > λ, where r _i1, …, r _in are the ranks of the observations from the i-th population in the combined sample of size cn. The small and large sample distributions of the test statistic are derived. Tables of the exact distribution are given for c = 2(1)5, n = 2(1)5. Tables of critical values are given for c = 2(1)6, n = 2(1)8 for values of α = 0.001, 0.005, 0.01, 0.025, 0.05, 0.10, and 0.20.     

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     

Numerical solution of linear two-point boundary problems via the fundamental-matrix method

A numerical method is presented for the-solution of linear systems of differential equations with initial-value or two-point boundary conditions. For y ′(x) = A (x) y (x) + f (x) the domain of interest [a,b] is divided into an appropriate number L of subintervals. The coefficient matrix A (x) is replaced by its value A_k at a point x_k within the Kth subinterval, thus replacing the original system by the L discretized systems y k(x) = A _k y _k(x) + f _k(x), k = 1,2,…, L. The fundamental matrix solution Φ_k(x, x_k) over each subinterval is found by computing the eigenvalues and eigenvectors of each A _k. By matching the solutions y _k(x) at the L – 1 equispaced grid points defining the limits of the subintervals and the boundary conditions, the two-point problem is reduced to solving a system of linear algebraic equations for the matching constants characterizing the different y _k(x). The values of y ₁(a) and y _L(b) are used to calculate the missing boundary conditions. For initial-value problems this method is equivalent to a one-step method for generating approximate solutions. By means of a coordinate transformation, as in the multiple shooting method,¹ the method becomes particularly suitable for stiff systems of linear ordinary differential equations. Five examples are discussed to illustrate the viability of the method.     

Propagation and far-field beam quality of M × N Hermite–Gaussian beams propagating through atmospheric turbulence

The analytical propagation equation of M×N Hermite–Gaussian (H–G) beams, which are combined incoherently and propagate through atmospheric turbulence, is derived, which enables us to study their propagation properties and far-field beam quality. The propagation of M×N Gaussian beams through atmospheric turbulence and M×N H–G beams in free space are treated as special cases of our general result. The power in the bucket (PIB), β-parameter and Strehl ratio are chosen as the parameters characterizing the beam quality in the far field. The dependence of PIB, β-parameter and Strehl ratio of M×N H–G beams through atmospheric turbulence on the refraction index structure constant C _n ², beam numbers M, N, mode indices m, n and separate distances x_d , y_d is illustrated numerically and interpreted physically. It is found that M×N H–G beams are less sensitive to the effects of turbulence than M×N Gaussian beams.     

Bayes estimation of the general hazard rate model

In reliability theory and life testing models, the life time distributions are often specified by choosing a relevant hazard rate function. Here a general hazard rate function h(t)=a+bt^c−1, where c, a, b are constants greater than zero, is considered. The parameter c is assumed to be known. The Bayes estimators of (a,b) based on the data of type II/item-censored testing without replacement are obtained. A large simulation study using Monte Carlo Method is done to compare the performance of Bayes with regression estimators of (a,b). The criterion for comparison is made based on the Bayes risk associated with the respective estimator. Also, the influence of the number of failed items on the accuracy of the estimators (Bayes and regression) is investigated. Estimations for the parameters (a,b) of the linearly increasing hazard rate model h(t)=a+bt, where a, b are greater than zero, can be obtained as the special case, letting c=2.     

A Note on Estimation from a Type I Extreme-Value Distribution

If the random variable T has the ta-o-parameter Weibull distribution with cumulative distribution function F(t; θ, K) = 1 – exp[–(t/θ) ^k ], where θ is the scale parameter and K is the shape parameter, then the random vatiable X = In T has the Type I extreme-value distribution of smallest values with cumulative distribution function F(x; u, b) = 1 – exp {–exp [(x – u)/b}, where u = In θ is the location parameter (mode) and b = 1/K is the scale parameter. It is therefore possible to obtain the maximum-likelihood estimator û _mn | b of u, based on the first m order statistics of a sample of size n, when b is known, by a simple transformation of the corresponding estimator of θ when K is known. Use is made of the fact that û _mn | b = In _mn | K, where 2m( _mn | K) ^k /θ ^k has the chi-square distribution with 2m degrees of freedom, to set confidence bounds on u. The probability density function of û _mn | b which for given m is the same for any n ≥ m, is obtained by a simple transformation of that of _mn | K. Integration yields expressions, involving digamma and trigamma functions, for the bias E = E[(û _mn |b) – u] and the variance V = V(û _mn | b). By subtracting the bias E](û _mn |b) – u] from û _mn |b, one obtains an unbiased estimator û|b which has the same variance as the maximum-likelihood estimator. Values of E/b(6DP) and of V/b ²(6DP) are tabulated for m = 1(1)100. The use of the table is discussed and illustrated by a numerical example.     

Matrix-Product Codes over ? q

Codes C ₁ ,…,C _M of length n over ? _q and an M × N matrix A over ? _q define a matrix-product code C = [C ₁ …C _M ] ·A consisting of all matrix products [c ₁ … c _M ] ·A. This generalizes the (u|u+v)-, (u+v+w|2u+v|u)-, (a+x|b+x|a+b+x)-, (u+v|u-v)- etc. constructions. We study matrix-product codes using Linear Algebra. This provides a basis for a unified analysis of |C|, d(C), the minimum Hamming distance of C, and C ^⊥. It also reveals an interesting connection with MDS codes. We determine |C| when A is non-singular. To underbound d(C), we need A to be `non-singular by columns (NSC)'. We investigate NSC matrices. We show that Generalized Reed-Muller codes are iterative NSC matrix-product codes, generalizing the construction of Reed-Muller codes, as are the ternary `Main Sequence codes'. We obtain a simpler proof of the minimum Hamming distance of such families of codes. If A is square and NSC, C ^⊥ can be described using C ₁ ^⊥, …,C _M ^⊥ and a transformation of A. This yields d(C ^⊥). Finally we show that an NSC matrix-product code is a generalized concatenated code. Received: July 20, 1999; revised version: August 27, 2001     

Three-dimensional analysis for explosions of a propane-air mixture in cylindrical vessels

A mathematically simple spatial difference method has been applied for analysing three-dimensionally, and for illustrating graphically, the process of the development of a flame after the propane-air stoichiometric mixture is ignited. The calculated results show that the mathematical simulation can well express the process of mixture explosion in cylindrical vessels, and can evaluate the effects of laminar swirl flow on the flame development. It is concluded that the swirl motion deforms the flame front and accelerates the flame enlargement. Described also is a comparison of swirl flame calculation, with and without the centripetal effect caused by the difference in densities between burnt and unburnt gases.List of symbols A, A _f , A _K , A _W area, flame surface area, flame element area, and wall area - b _e heat penetration coefficient - C _d coefficient - C _p molar specific heat at constant pressure - G mass of gas - H cylinder height - h enthalpy and molar enthalpy - L effective length - M _j molecular weight - n, N normal unit vector and normal vector to flame surface - p pressure - q heat input per unit mass - r _c flame center location - r, R radius and vessel radius - R universal gas constant - Re Reynolds number - s _b , s _u flame speed and burning velocity - t time measured from spark-ignition - U _c inward moving velocity of flame center - u _u compression velocity of the unburnt gas just ahead of the flame front - U(u, v, w) velocity vector - v _i , V _i specific volume and volume of region-i - x _j mole fraction of specie-_j - x mass fraction of burnt gas - X(x, Y, z), X(x, r, ) orthogonal and cylindrical coordinate systems - Y _j mass fraction of specie-_j - Y mass fraction of burnt gas - small amount of change - change from the initial state Suffixes ij flame element number - i region or state - j chemical species     

A control scheme for monitoring the frequency and magnitude of an event

While many control charts have been developed for monitoring the time interval (t) between the occurrences of an event, many other charts are employed to examine the magnitude (x) of the event. These two types of control charts have usually been investigated and applied separately with limited syntheses in conventional statistical process control (SPC) methods. This article presents an SPC method for simultaneously monitoring the time interval t and magnitude x. It, essentially, combines a t chart and an x chart, and is therefore referred to as a t&x chart. The new chart is more effective than an individual t chart or individual x chart for detecting the out-of-control status of the event, in particular for detecting downward shifts (sparse occurrence and/or small magnitude). More profound is that, compared with an individual t or x chart, the detection effectiveness of the t&x chart is more invariable against different types of shifts, i.e. t shift, x shift and joint shift in t and x. The t&x chart has demonstrated its potential not only for manufacturing systems, but also for non-manufacturing sectors such as supply chain management, office administration and health care industry.     

An alternative optimal solution technique for a single-vendor single-buyer integrated production inventory model

For the single-vendor single-buyer integrated production inventory problem, the lot is transferred by some researchers in shipments of equal and/or unequal sizes. For a given total number (n) of shipments, the minimal number (m) of unequal sized shipments is selected satisfying a constraint. Due to the complex dependence of m on n, there remained a conjecture about the convexity of the total cost (C) in n. Consequently, for a known n, lower and upper bounds on the minimal total cost are imposed and this cost is obtained through a search over n based on the convexity of C in the lot size (Q). Here it is demonstrated that m thus selected might not be the minimal and a method for obtaining the minimal m is developed. A formula for the minimal n based on the convexity of C in n (given m and Q) is also derived. Integrating the ways of finding the minimal m and n, along with the convexity of C in Q, an alternative minimal cost solution technique is presented. Finally, a comparative theoretical analysis of the methods along with numerical illustration is carried out to show the efficiency of our method.     

Theoretical aspects of charge correlations in θ-(BEDT-TTF)2X

Abstract

A review is given on the theoretical studies of charge correlations in θ-(BEDT-TTF)₂X. Various studies show that within a purely electronic model on the θ-type lattice with on-site U and nearest neighbor V_p and V_c interactions, the diagonal stripe, c-axis three-fold, and the vertical stripe charge correlations are favored in the regime V_p<V_c, V_p～V_c, and V_p>V_c, respectively. In the realistic parameter regime of V_p～V_c, there is competition between the c-axis three fold state and diagonal stripe state. Since these are different from the experimentally observed a-axis three fold and the horizontal stripe charge correlations, additional effects have to be included in order to understand the experiments. The electron–lattice coupling, which tends to distort the lattice into the θ_d-type, is found to favor the horizontal stripe state, suggesting that the occurrence of this stripe ordering in the actual materials may not be of purely electronic origin. On the other hand, distant electron–electron interactions have to be considered in order to understand the a-axis three fold correlation, whose wave vector is close to the nesting vector of the Fermi surface. These studies seem to suggest that the minimal model to understand the charge correlation in θ-(BEDT-TTF)₂X may be more complicated than expected. Future problems regarding the competition between different types of charge correlations are discussed.     

Skew product cycles with rich dynamics: From totally non-hyperbolic dynamics to fully prevalent hyperbolicity

We introduce a two-parameter family of ‘partially hyperbolic’ skew products (G_{a, t})_{a > 0, t ∈ [ ? ε, ε]} maps with one dimensional centre direction. In this family, the parameter a models the central dynamics and the parameter t the unfolding of cycles (that occurs for t = 0). The parameter a also measures the ‘central distortion’ of the systems: for small a, the distortion of the systems is small and it increases and goes to infinity as a → ∞. The family (G_{a, t}) displays some of the main characteristic properties of the unfolding of heterodimensional cycles as intermingled homoclinic classes of different indices and secondary bifurcations via collision of hyperbolic homoclinic classes. For a ∈ (0, log?2), the dynamics of (G_{a, t}) is always non-hyperbolic after the unfolding of the cycle. However, for a > log?4 intervals of t-parameters corresponding to hyperbolic dynamics appear and turn into totally prevalent as a → ∞ (the density of ‘hyperbolic parameters’ goes to 1 as a → ∞). The dynamics of the maps G_{a, t} is described using a family of iterated function systems modelling the dynamics in the one-dimensional central direction.     

Criteria on existence of horseshoes near homoclinic tangencies of arbitrary orders

Consider (m + 1)-dimensional, m ≥ 1, diffeomorphisms that have a saddle fixed point O with m-dimensional stable manifold W^s(O), one-dimensional unstable manifold W^u(O), and with the saddle value σ different from 1. We assume that W^s(O) and W^u(O) are tangent at the points of some homoclinic orbit and we let the order of tangency be arbitrary. In the case when σ < 1, we prove necessary and sufficient conditions of existence of topological horseshoes near homoclinic tangencies. In the case when σ > 1, we also obtain the criterion of existence of horseshoes under the additional assumption that the homoclinic tangency is simple.     

Mixed-State Specific Heat of the Type-II Superconductor Nb0.77Zr0.23 in Magnetic Fields up toB c2

In order to document the behavior of the mean-field mixed-state specific heat of an isotropic. strongly type-II superconductor (i.e., with a large value of the Ginzburg parameterk), and to provide a basis for comparison with high-temperature superconductors, we measured the specific heatC of the alloy Nb_0.77Zr_0.23 withT _c = 10.8K, B _c2 (0) = 7.9T, in magnetic fieldsB = 0, 0.2, 1.0, 12, 2.0, 2.4. 3.0, 3.3. 4.0. 4.4, 4.8, 5.2, 6.0. 6.6, 7.2 and 10 T. The values of the upper critical fieldB _c2 ( T), thermodynamic critical fieldB _c (T), Ginzburg parameterk(T), and coefficient γ(B) = lim_{T0(C(T. B)/T)} are derived from the specific heat data and found to be in agreement with the GLAG theory in the dirty limit. The behavior of the mixed-state specific heat is analyzed in terms ofC _el /T,∂(C _el /T)/∂B, and∂(C _el /T)/∂T vs. Tcurves, whereC _el is the electronic contribution to the specific heat.     

Accelerated Life Testing for a Class of Linear Hazard Rate Type Distributions

Accelerated life testing for distributions with hazard rate functions of the form r(t) = Ag(t) + Bh(t) are considered. Let V ₁, …, V _k be stress levels larger than V ₀—the stress level under normal conditions [V ₀ > 0]—and let a(v) be a nondecreasing function on (0, ∞). We discuss a generalization of the common accelerated models (the power rule model and the Arrhenius model) by assuming that the hazard rate under the stress level V, is of the form (a(V _t )) ^P (Ag(t) + Bh(t)). The maximum likelihood estimators of A, B and P for complete and censored samples are studied. The estimation procedure reduces to a solution of one equation with one unknown parameter. The estimation procedure under the assumption of aging is also described. The asymptotic variance-covariance matrix is given.     

Tables of the Distribution of the Mann-Whitney-Wilcoxon U-Statistic Under Lehmann Alternatives

A simple recursive formula is used to generate the distribution of the MWW U-statistic under alternatives of the form K ₁: G – F ^k ,1 ≠ k < 0, and K ₂: G – 1? (1 ? F) ^k , 1 ≠ k > 0, and it is pointed out that many of these alternatives are of interest in life testing, reliability testing and related fields. Tables are given for the case G = 1 – (1 ? F) ^k for 4 ≤ n, m ≤ 8 and k = 2,3,4,6,9,12 and it is shown that these tables can also be used when G = F ^k and when 0 < k < 1.