Matrix characterization of linear codes with arbitrary Hamming weight hierarchy

The support of an [n, k] linear code C over a finite field F_q is the set of all coordinate positions such that at least one codeword has a nonzero entry in each of these coordinate position. The rth generalized Hamming weight d_r(C), 1 ⩽ r ⩽ k, of C is defined as the minimum of the cardinalities of the supports of all [n, r] subcodes of C. The sequence (d₁(C), d₂(C), … , d_k(C)) is called the Hamming weight hierarchy (HWH) of C. The HWH, d_r(C) = n − k + r;  r = 1, 2 , …, k, characterizes maximum distance separable (MDS) codes. Therefore the matrix characterization of MDS codes is also the characterization of codes with the HWH d_r(C) = n − k + r; r = 1, 2, … , k. A linear code C with systematic check matrix [I∣P], where I is the (n − k) × (n − k) identity matrix and P is a (n − k) × k matrix, is MDS iff every square submatrix of P is nonsingular. In this paper we extend this characterization to linear codes with arbitrary HWH. Using this result, we characterize Near-MDS codes, Near-Near-MDS (N²-MDS) codes and A^μ-MDS codes. The MDS-rank of C is the smallest integer η such that d_η+1 = n − k + η + 1 and the defect vector of C with MDS-rank η is defined as the ordered set {μ₁(C), μ₂(C), μ₃(C), … , μ_η(C), μ_η+1(C)}, where μ_i(C) = n − k + i − d_i(C). We call C a dually defective code if the defect vector of the code and its dual are the same. We also discuss matrix characterization of dually defective codes. Further, the codes meeting the generalized Greismer bound are characterized in terms of their generator matrix. The HWH of dually defective codes meeting the generalized Greismer bound are also reported.     

Magnetohydrodynamic stationary and oscillatory convective stability in a mushy layer during binary alloy solidification

For the case of solidification of a bottom cooled binary alloy, the magnetohydrodynamic stationary and oscillatory convective stability in the mushy layer is investigated analytically using normal mode linear stability analysis. In the limit of large Stefan number (S_t), a near–eutectic approximation with large far field temperature is considered in the present research. To ascertain the instability in the mushy layer, the strength of the superimposed magnetic field is so chosen that it corresponds to a given mush Hartmann number (Ha_m) of the problem. The results are presented for various values of mush Hartmann numbers in the range, 0 ≤ Ha_m ≤ 50. The critical Rayleigh number for stationary convection shows a linear relationship with increasing Ha_m. The magnetohydrodynamic effect imparts a stabilizing influence during stationary convection. In comparison to that of the stationary convective mode, the oscillatory mode appears to be critically susceptible at higher values of β (β = S_t/℘² ϒ², ℘ is the compositional ratio, ϒ = 1 + S_t/℘), and vice versa for lower β values. Analogous to the behavior for stationary convection, the magnetic field also offers a stabilizing effect in oscillatory convection and thus influences global stability of the mushy layer. Increasing magnetic strength shows reduction in the wavenumber and in the number of rolls formed in the mushy layer.     

Delayed transitions in non-linear replicator networks: About ghosts and hypercycles

In this paper we analyze delayed transition phenomena associated to extinction thresholds in a mean field model for hypercycles composed of three and four units, respectively. Hence, we extend a previous analysis carried out with the two-membered hypercycle [see Sardanyés J, Solé RV. Ghosts in the origins of life? Int J Bifurcation Chaos 2006;16(9), in press]. The models we analyze show that, after the tangent bifurcation, these hypercycles also leave a ghost in phase space. These ghosts, which actually conserve the dynamical properties of the coalesced coexistence fixed point, delay the flows before hypercycle extinction. In contrast with the two-component hypercycle, both ghosts show a plateau in the delay as ϕ → 0, thus displacing the power-law dependence to higher values of ϕ, in which the scaling law is now given by τ ∼ ϕ^β, with β = −1/3 (where τ is the delay and ϕ = ϵ − ϵ_c, the parametric distance above the extinction bifurcation point). These results suggest that the presence of the ghost is a general property of hypercycles. Such ghosts actually cause a memory effect which might increase hypercycle survival chances in fluctuating environments.     

Net-convergence and weak separation axioms in (L, M)-fuzzy topological molecular lattices

In this paper, we used the concept of (L, M)-fuzzy remote neighborhood system to study and establish the convergence theory of molecular nets. Next, we introduce the T_i-axioms (i = ?1, 0, 1, 2) in (L, M)-fuzzy topological molecular lattices, and discuss some of their characterizations. Finally, we show that the T_i-axioms (i = ?1, 0, 1, 2) are preserved under homeomorphisms.     

Electrostatics in fractal geometry: Fractional calculus approach

An electric field in a composite dielectric with a fractal charge distribution is obtained in the spherical symmetry case. The method is based on the splitting of a composite volume into a fractal volume V_d ～ r^d with the fractal dimension d and a complementary host volume V_h = V₃ ? V_d. Integrations over these fractal volumes correspond to the convolution integrals that eventually lead to the employment of the fractional integro-differentiation.     

Unsteady magnetohydrodynamic Hartmann–Couette flow and heat transfer in a Darcian channel with Hall current,ionslip, viscous and Joule heating effects: Network numerical solutions

A theoretical study of unsteady magnetohydrodynamic viscous Hartmann–Couette laminar flow and heat transfer in a Darcian porous medium intercalated between parallel plates, under a constant pressure gradient is presented. Viscous dissipation, Joule heating, Hall current and ionslip current effects are included as is lateral mass flux at both plates. The dimensionless conservation equations for the primary (x¹-direction), secondary (z¹-direction) momentum and also energy conservation equation are derived and solved using a computational technique known as Network Simulation Methodology (NSM). Velocity distributions (u¹, w¹) and temperature distribution (T¹) at the channel centre (y¹ = 0) over time (t¹) are studied graphically for the effects of Darcy number (Da), Hartmann number (Ha), transpiration (N_t), Hall current parameter (Be), ionslip parameter (Bi), pressure gradient parameter (dP/dx¹) with Prandtl number prescribed at 7.0 (electrically conducting water), Eckert number held constant at 0.25 (heat convection from the plates to the fluid) and Reynolds number (Re) fixed at 5.0 (for Re < 10, Darcian model is generally valid). Increasing Darcy number causes an increase in temperature, T¹; values are however significantly reduced for the higher Hartmann number case (Ha = 10). For the case of low transpiration (i.e. N_t = 1 which corresponds to weak suction at the upper plate and weak injection at the lower plate), both primary velocity (u¹) and secondary velocity (w¹) are increased with a rise in Darcy number (owing to a simultaneous decrease in Darcian porous drag); temperature T¹ is also increased considerably with increasing Da. However, for stronger transpiration (N_t = 10), magnitudes of u¹, w¹ and T¹ are significantly reduced and also significant overshoots are detected prior to the establishment of steady state flow. With increasing Hall current parameter, Be, (for the purely fluid regime i.e. Da → ∞), primary velocity is considerably increased, whereas secondary velocity is reduced; temperatures are decreased in the early stages of flow but effectively increased in the steady state with increasing Be. With strong Darcian drag present (Da = 0.01 i.e. very low permeability), magnitudes of u¹, w¹ and T¹ are considerably reduced and temperatures are found to be reduced for all t¹, with increasing Hall current effect (Be). Increasing ionslip current parameter (Bi) increases primary velocity (u¹), decreases secondary velocity (w¹) and also temperature (T¹) for all time (t¹), in the infinite permeability case (Da → ∞). For weakly Darcian flow, ionslip parameter (Bi) has a much reduced effect on the velocity distributions. Temperature, T¹ is strongly increased with a rise in pressure gradient parameter, dP¹/dx¹, as is primary velocity (u¹); however, secondary velocity (w¹) is reduced. The present study has applications in hybrid magnetohydrodynamic (MHD) energy generators, materials processing, geophysical hydromagnetics, etc.     

Exponential stability of delayed fuzzy cellular neural networks with diffusion

The exponential stability of delayed fuzzy cellular neural networks (FCNN) with diffusion is investigated. Exponential stability, significant for applications of neural networks, is obtained under conditions that are easily verified by a new approach. Earlier results on the exponential stability of FCNN with time-dependent delay, a special case of the model studied in this paper, are improved without using the time-varying term condition: dτ(t)/dt < μ.     

Existence and asymptotics of traveling waves for nonlocal diffusion systems

In this paper, we deal with the existence and asymptotic behavior of traveling waves for nonlocal diffusion systems with delayed monostable reaction terms. We obtain the existence of traveling wave front by using upper-lower solutions method and Schauder’s fixed point theorem for c > c₁(τ) and using a limiting argument for c = c₁(τ). Moreover, we find a priori asymptotic behavior of traveling waves with the help of Ikehara’s Theorem by constructing a Laplace transform representation of a solution. Especially, the delay can slow the minimal wave speed for ?₂f(0, 0) > 0 and the delay is independent of the minimal wave speed for ?₂f(0, 0) = 0.     

Multifractal rainfall extremes: Theoretical analysis and practical estimation

We study the extremes generated by a multifractal model of temporal rainfall and propose a practical method to estimate the Intensity–Duration–Frequency (IDF) curves. The model assumes that rainfall is a sequence of independent and identically distributed multiplicative cascades of the beta-lognormal type, with common duration D. When properly fitted to data, this simple model was found to produce accurate IDF results [Langousis A, Veneziano D. Intensity–duration–frequency curves from scaling representations of rainfall. Water Resour Res 2007;43. doi:10.1029/2006WR005245]. Previous studies also showed that the IDF values from multifractal representations of rainfall scale with duration d and return period T under either d → 0 or T → ∞, with different scaling exponents in the two cases. We determine the regions of the (d, T)-plane in which each asymptotic scaling behavior applies in good approximation, find expressions for the IDF values in the scaling and non-scaling regimes, and quantify the bias when estimating the asymptotic power-law tail of rainfall intensity from finite-duration records, as was often done in the past. Numerically calculated exact IDF curves are compared to several analytic approximations. The approximations are found to be accurate and are used to propose a practical IDF estimation procedure.     

Conditional contractivity of Runge–Kutta methods for nonlinear differential equations with many variable delays

This paper deals with conditional contractivity properties of Runge–Kutta (RK) methods with variable step-size applied to nonlinear differential equations with many variable delays (MDDEs). The concepts of CRN_m(ω, H)- and BN_f(μ, ?)-stability are introduced. It is shown that the numerical solution produced by a BN_f(μ, ?)-stable Runge–Kutta method with an appropriate interpolation is contractive. In particular, these results are also novel for nonlinear differential equations with many constant delays or single variable delay. To obtain BN_f(μ, ?)-stable methods, (k, l)-algebraically stable Runge–Kutta methods are also investigated.     

Simulación numérica del flujo turbulento de aire con gotas dispersas de agua a través de separadores de torres de refrigeración

This work presents a numerical study on the turbulent flow of air with dispersed water droplets in separators of mechanical cooling towers. The averaged Navier-Stokes equations are discretised through a finite volume method, using the Fluent and Phoenics codes, and alternatively employing the turbulence models k ? ?, k ? ω and the Reynolds stress model, with low-Re version and wall enhanced treatment refinements. The results obtained are compared with numerical and experimental results taken from the literature. The degree of accuracy obtained with each of the considered models of turbulence is stated. The influence of considering whether or not the simulation of the turbulent dispersion of droplets is analyzed, as well as the effects of other relevant parameters on the collection efficiency and the coefficient of pressure drop. Focusing on four specific eliminators (‘Belgian wave’, ‘H1-V’, ‘L-shaped’ and ‘Zig-zag’), the following ranges of parameters are outlined: 1 ≤ U_e ≤ 5 m/s for the entrance velocity, 2 ≤ D_p ≤ 50 μm for the droplet diameter, 650 ≤ Re ≤ 8.500 for Reynolds number, and 0.05 ≤ P_i ≤ 5 for the inertial parameter. Results reached alternately with Fluent and Phoenics codes are compared. The best results correspond to the simulations performed with Fluent, using the SST k ? ω turbulence model, with values of the dimensionless scaled distance to wall y⁺ in the range 0.2 to 0.5. Finally, correlations are presented to predict the conditions for maximum collection efficiency (100 %), depending on the geometric parameter of removal efficiency of each of the separators, which is introduced in this work.     

A semi-Markov process with an inverse Gaussian distribution as sojourn time

Let X_n denote the state of a device after n repairs. We assume that the time between two repairs is the time τ taken by a Wiener process {W(t), t ? 0}, starting from w₀ and with drift μ < 0, to reach c ∈ [0, w₀). After the nth repair, the process takes on either the value X_n?1 + 1 or X_n?1 + 2. The probability that X_n = X_n?1 + j, for j = 1, 2, depends on whether τ ? t₀ (a fixed constant) or τ > t₀. The device is considered to be worn out when X_n ? k, where k ∈ {1, 2, …}. This model is based on the ones proposed by Rishel (1991) [1] and Tseng and Peng (2007) [2]. We obtain an explicit expression for the mean lifetime of the device. Numerical methods are used to illustrate the analytical findings.     

The Exponent of Discrepancy Is at Least 1.0669

LetP⊂[0, 1]^dbe ann-point set and letw: P→[0, ∞) be a weight function withw(P)=∑_z∈P w(z)=1. TheL₂-discrepancy of the weighted set (P, w) is defined as theL₂-average ofD(x)=vol(B_x)−w(P∩B_x) overx∈[0, 1]^d, where vol(B_x) is the volume of thed-dimensional intervalB_x=∏^d_k=1 [0, x_k). The exponent of discrepancyp* is defined as the infimum of numberspsuch that for all dimensionsd⩾1 and allε>0 there exists a weighted set of at mostKε^−ppoints in [0, 1]^dwithL₂-discrepancy at mostε, whereK=K(p) is a suitable number independent ofεandd. Wasilkowski and Woźniakowski proved thatp*⩽1.4779, by combining known bounds for the error of numerical integration and using their relation toL₂-discrepancy. In this note we observe that a careful treatment of a classical lower- bound proof of Roth yieldsp*⩾1.04882, and by a slight modification of the proof we getp*⩾1.0669. Determiningp* exactly seems to be quite a difficult problem.     

Interpolation by Integer-Valued Polynomials

LetRbe a Krull ring with quotient fieldKanda₁,…,a_ninR. If and only if thea_iare pairwise incongruent mod every height 1 prime ideal of infinite index inRdoes there exist for all valuesb₁,…,b_ninRan interpolating integer-valued polynomial, i.e., anf ∈ K[x] withf(a_i) = b_iandf(R) ⊆ R.IfSis an infinite subring of a discrete valuation ringR_vwith quotient fieldKanda₁,…,a_ninSare pairwise incongruent mod allM^k_v ∩ Sof infinite index inS, we also determine the minimald(depending on the distribution of thea_iamong residue classes of the idealsM^k_v ∩ S) such that for allb₁,…,b_n ∈ R_vthere exists a polynomialf ∈ K[x] of degree at mostdwithf(a_i) = b_iandf(S) ⊆ R_v.     

Space–time symmetry violation,configuration mixing model and E-infinity theory

We have studied the time reversal symmetry violation on the bases of the configuration mixing model and E-infinity theory. With the use of the Cabibbo angle approximation, we have presented the transformation matrix in terms of the golden ratio (?), and shown that the time reversal symmetry violation is described by the configuration mixing of the unstable and stable manifolds (W^u, W^s). The magnitude of the mixing for the weak interaction field is given by the expression sin² θ_T(theor) ≈ sin⁴ θ_C(theor) ≈ (?)¹² = 3.105 × 10^?3, which is compared to the Kaon decay experiment ～2.3 × 10^?3. We have also discussed the space–time symmetry violation by using the CPT theorem.     

Exact solutions for the rotational flow of a generalized Maxwell fluid between two circular cylinders

Unsteady flow of an incompressible generalized Maxwell fluid between two coaxial circular cylinders is studied by means of the Laplace and finite Hankel transforms. The motion of the fluid is produced by the rotation of cylinders around their common axis. The solutions that have been obtained, written in integral and series form in terms of the generalized G_a,b,c(·, t)-functions, are presented as a sum of the Newtonian solutions and the corresponding non-Newtonian contributions. They satisfy all imposed initial and boundary conditions and for λ → 0 reduce to the solutions corresponding to the Newtonian fluids performing the same solution. Furthermore, the corresponding solutions for ordinary Maxwell fluids are also obtained for β = 1. Finally, in order to reveal some relevant physical aspects of the obtained results, the diagrams of the velocity field ω(r, t) have been depicted against r and t for different values of the material and fractional parameters.     

Simple regression in view of elliptical models

For the simple linear model Y = θ1 + βx + ? where the error vector follows the elliptically contoured distribution, we consider the unrestricted, restricted, preliminary test and shrinkage estimators for the intercept parameter, θ when it is suspected that the slope parameter β may be β_o. The exact bias and MSE expressions are derived and the mean-square relative efficiency is taken to determine the relative dominance properties of the proposed estimators in comparison. In the continuation, the optimal level of significance of the preliminary test estimator is tabulated and some graphical result are also displayed.     

Quantum Complexity of Integration

It is known that quantum computers yield a speed-up for certain discrete problems. Here we want to know whether quantum computers are useful for continuous problems. We study the computation of the integral of functions from the classical Hölder classes F^k, α_d on [0, 1]^d and define γ by γ=(k+α)/d. The known optimal orders for the complexity of deterministic and (general) randomized methods are comp(F^k, α_d, ε)≍ε^−1/γ and comp^random(F^k, α_d, ε)≍ε^−2/(1+2γ). For a quantum computer we prove comp^quant_query(F^k, α_d, ε)≍ε^−1/(1+γ) and comp^quant(F^k, α_d, ε)⩽Cε^−1/(1+γ)(log ε⁻¹)^1/(1+γ). For restricted Monte Carlo (only coin tossing instead of general random numbers) we prove comp^coin(F^k, α_d, ε)⩽Cε^−2/(1+2γ)(log ε⁻¹)^1/(1+2γ). To summarize the results one can say that   • there is an exponential speed-up of quantum algorithms over deterministic (classical) algorithms, if γ is small;   • there is a (roughly) quadratic speed-up of quantum algorithms over randomized classical methods, if γ is small.     

Effects of CO2 concentration and moisture content of sugar-free media on the tissue-cultured plantlets in a large growth chamber

The dynamic fluctuations of CO₂ concentration in the tissue culture growth chamber after transplantation of petunia, chrysanthemum and tomato plantlets were recorded with a real-time control system to determine the critical CO₂ concentration levels of 35 μl l^?1 at which CO₂ enrichment is needed. The experimental data showed that the tissue-cultured plantlets of petunia, chrysanthemum and tomato had the same CO₂ concentration dynamics. The results indicated that CO₂ enrichment was proper on the second day after transplantation. Petunia plantlets were used to conduct experiments under PPFD of 80 μmol m^?2 s^?1, and CO₂ concentrations of 350 ± 50 μl l^?1, 650 ± 50 μl l^?1 and 950 ± 50 μl l^?1 as well as medium moisture contents of 60%, 70% and 80%, with the result that plantlets grew better under CO₂ concentration of 650 ± 50 μl l^?1 than under the other two concentrations with all the different media water contents. Three media water contents under the same CO₂ concentration produced plantlets with the same quality. The impacts of CO₂ concentrations on plantlets are more important than those of the media water contents. Sugar-free tissue culture, as compared with the conventional culture, showed that CO₂ enrichment to 350 ± 50 μl l^?1 can promote the growth of the cultured plantlets. Sugar-free tissue culture produced healthy plantlets with thick roots, almost equivalent to the common plantlets.     

On chaotic set-valued discrete dynamical systems

Let (X, d) be a metric space and let f: (X, d) → (X, d) be a continuous map. In this note we investigate the relationships between the chaoticity of some set-valued discrete dynamical systems associated to f (collective chaos) and the chaoticity of f (individual chaos).