Optimal Fixing of the Bandwidth-Parameter for the Empirical Regression1,2

The estimator ?₀(x) of the regression r(x) = E (Y | × = x) from measured points (x_i, y_i), i = 1(1) n, of a continuous two-dimensional random variable (X, Y) with unknown continuous density function f(x, y) and with moments up to the second order can be made with the help of a density estimation f?₀(x, y) (see e.g. SCHMERLING and PEIL, 1980). Here f?₀(x, y) still contains free parameters (so-called band-width-parameters), the values of which have to be optimally fixed in the concrete case. This fixing can be done by using a modification of the maximum-likelihood principle including jackknife techniques. The parameter values can be also found from the estimators for r(x). Here the cross-validation principle can be applied. Some numerical aspects of these possibilities for optimally fixing the bandwidth-parameter are discussed by means of examples. If ?₀(x) is used as a smoothing operator for time series the optimal choice of the parameter values is dependent on the purpose of application of the smoothed time series. The fixing will then be done by considering the so-called filter-characteristic of ?C₀(x).     

The decay of genetic variability in geographically structured populations. II

The ultimate rate of approach to equilibrium in the infinite stepping-stone model is calculated. The analysis is restricted to a single locus in the absence of selection, and every mutant is assumed to be new to the population. Let f(t, x) be the probability that two homologous genes separated by the vector x in generation t are the same allele. It is supposed that $\text{f(}\text{0, x}\text{) = O(x}{}^{\mathrm{?2?\eta }}\text{), η > 0,}\text{as}\text{x ≡ ¦}\text{x}\text{¦ → ∞}$. In the absence of mutation, f(t, x) tends to unity at the rate $\text{t}{}^{\mathrm{<mtext>?1</mtext><mtext>2</mtext>}}$ in one dimension and (ln t)^?1 in two dimensions. Thus, the loss of genetic variability in two dimensions is so slow that evolutionary forces not considered in this model would supervene long before a two-dimensional natural population became completely homogeneous. If the mutation rate, u, is not zero f(t, x) asymptotically approaches equilibrium at the rate $\text{(1 ? u)}{}^{\mathrm{2t}}\text{t}{}^{\mathrm{<mtext>?3</mtext><mtext>2</mtext>}}$ in one dimension and (1 ? u)^2tt^?1(lnt)^?2 in two dimensions. Integral formulas are presented for the spatial dependence of the deviation of f(t, x) from its stationary value as t → ∞, and for large separations this dependence is shown to be (const + x) in one dimension and (const + ln x) in two dimensions. All the results are the same for the Malécot model of a continuously distributed population provided the number of individuals per colony is replaced by the population density. The relatively slow algebraic and logarithmic rates of convergence for the infinite habitat contrast sharply with the exponential one for a finite habitat.     

Energetics of Shortening Muscles in Twitches and Tetanic Contractions : II. Force-Determined Shortening Heat

The extra heat liberation accompanying muscular shortening, the force-determined shortening heat, is defined as the difference between the heat produced when shortening occurs and that produced in an isometric contraction developing the same amount of force and performing the same amount of internal work. Based on this definition, the initial energy production in twitches and tetanic contractions (E) is given by E = A + f (P, t) + α_Fx + W, where A is the activation heat, f(P, t), the tension-related heat (a heat production associated with the development and maintenance of tension), α_Fx, the force-determined shortening heat, and W, the external work. It is demonstrated that this equation accurately accounts for the time-course of heat evolution and the total initial energy production in both twitches and tetani at 0°C. The force-determined shortening heat is liberated, during shortening, in direct proportion to (a) the distance shortened, and (b) the force against which shortening occurs. The normalized value of the force-determined shortening heat coefficient, α_F/P_o, is the same in both the twitch and the tetanus. Finally, this formulation of the muscle's energy production also accounts for the total energy production in afterload isotonic twitches at 20°C, where a Fenn effect is not demonstrable.     

A New Functional Equation Analogous to CAUCHT-PEXIDER Functional Equation and its Application

The CAUCHY-PEXIDER functional equation H (x±y)=F(x) G(y) is generalized to the form H ((x^c±y^c)^1/c) = F(x) G(y), c≠0, assuming the function H(x) possesses a measurable majorant on a set of positive measure. The result is used to obtain a characterization of WEIBULL distribution. This functional equation is generalized to functions of vector variables.     

Chemotactic collapse for the Keller-Segel model

 This work is concerned with the system (S) {u _t =Δu − χ∇ (u∇v) for x∈Ω, t>0Γ v _t =Δv+(u−1) for x∈Ω, t>0 where Γ, χ are positive constants and Ω is a bounded and smooth open set in ℝ². On the boundary ∂Ω, we impose no-flux conditions: (N) ∂u∂n =∂v∂n =0 for x∈∂ Ω, t>0 Problem (S), (N) is a classical model to describe chemotaxis corresponding to a species of concentration u(x, t) which tends to aggregate towards high concentrations of a chemical that the species releases. When completed with suitable initial values at t=0 for u(x, t), v(x, t), the problem under consideration is known to be well posed, locally in time. By means of matched asymptotic expansions techniques, we show here that there exist radial solutions exhibiting chemotactic collapse. By this we mean that u(r, t) →Aδ(y) as t→T for some T<∞, where A is the total concentration of the species. Received 9 March 1995; received in revised form 25 December 1995     

Energetics of Contraction in Phasic and Tonic Skeletal Muscles of the Chicken

Comparative energetics of chicken latissimus dorsi muscles, tonic anterior (ALD) and phasic posterior (PLD), were investigated by measuring initial heat production. Heat components were analyzed in terms of the equation: E = A + W + α_F(L) + f(P, t) As the muscles were stretched by increments, heat produced in isometric twitches and tetani decreased in a linear fashion. Two processes are involved: one tension independent, the activation heat, or A; and the other tension dependent, W_i + α_F(L) + f(P, t). In twitches, A, per unit tension, is equivalent in the PLD and ALD. Tension-dependent heat, per unit tension, is greater in the PLD due to W_i; but tension-time-related heat, f(P, t), per unit tension, is similar in both muscles. In tetanic contractions, differences in A and f(P, t), per unit tension, are attributed to the greater V_max in the PLD. The differences in the energetics of isometric contractions in the PLD and ALD, therefore, can be explained by inherent differences in tension development, compliance, and myosin and reticular ATPase activities. Data from isotonic twitches were quantified by means of the equivalent tension technique. Both muscles exhibited an extra heat associated with shortening, α_F(L). In the PLD, the ratio α_F/P_ot is greater; it is load independent and ½ the value of a/P_o in both muscles. Enthalpy efficiency, W_e + W_i/E, is comparable in both muscles. A Fenn effect is observed only when isotonic energy liberation is compared to a decreasing isometric energy expenditure base line.     

A model of the complex response of Staphylococcus aureus to methicillin

It is widely accepted that β-lactam antimicrobials cause cell death through a mechanism that interferes with cell wall synthesis. Later studies have also revealed that β-lactams modify the autolysis function (the natural process of self-exfoliation of the cell wall) of cells. The dynamic equilibrium between growth and autolysis is perturbed by the presence of the antimicrobial. Studies with Staphylococcus aureus to determine the minimum inhibitory concentration (MIC) have revealed complex responses to methicillin exposure. The organism exhibits four qualitatively different responses: homogeneous sensitivity, homogeneous resistance, heterogeneous resistance and the so-called ‘Eagle-effect’. A mathematical model is presented that links antimicrobial action on the molecular level with the overall response of the cell population to antimicrobial exposure. The cell population is modeled as a probability density function F(x,t) that depends on cell wall thickness x and time t. The function F(x,t) is the solution to a Fokker-Planck equation. The fixed point solutions are perturbed by the antimicrobial load and the advection of F(x,t) depends on the rates of cell wall synthesis, autolysis and the antimicrobial concentration. Solutions of the Fokker-Planck model are presented for all four qualitative responses of S. aureus to methicillin exposure.     

Lanthanide iodides as promoters of acetonitrile amination. Molecular structure of MeC(NH)NHPr, MeC(NH)NHBu and {Dy[MeC(NH)NEt2]6}I3

Amination of acetonitrile by the amines MeNH₂, PrⁿNH₂, PrⁱNH₂, Bu^tNH₂, and Et₂NH is efficiently promoted by the lanthanide iodides LnI₂ (Ln = Nd, Dy, Tm), LnI₃ (Ln = Pr, Nd, Dy) and LnI₃(THF)₃ (Ln = Pr, Nd, Dy). The formed mono- and N,N′-disubstituted amidines MeC(NH)NHR (R = Prⁱ, Bu^t), MeC(NH)NEt₂, MeC(NR)NHR (R = Me, Prⁿ) were isolated mainly as the complexes with starting iodide of general composition LnI₂(amidine)_x (1) or LnI₃(amidine)_x (2) (x = 3-8). In the products 1, which evidently are the mixtures of LnI²⁺, and LnI₃ derivatives, the metal exists in trivalent state but one of the ligands actually is amidinate anion. A part of the generated amidines remains in the reaction solutions in free form. Heating of the 1 and 2 in vacuum at 150-200 °C affords corresponding amidine and the complexes with reduced amount of the amidine ligands LnI₂(amidine)_y (3) or LnI₃(amidine)_y (4) (y = 2-3). The products 3 and 4 displayed the same catalytic activity in the acetonitrile-amine cross-coupling as the initial iodides. SmI₂ and especially YbI₂ revealed lower activity. The structure of isopropylacetamidine (5), tert-butylacetamidine (6) and {Dy[MeC(NH)NEt₂]₆}I₃(MeCN) (7) were determined by X-ray diffraction analysis.     

How do spinal segments move?

PurposeTo study and clarify the kinematics of spinal segments following cyclic torques causing axial rotation (T_z (t)), lateral-flexion (T_x (t)), flexion/extension (T_y (t)).MethodsA 6D--Measurement of location, alignment, and migration of the instantaneous helical axis (IHA) as a function of rotational angle in cervical, thoracic, and lumbar segments subjected to axially directed preloads.ResultsIHA retained an almost constant alignment, but migrated along distinct centrodes.Thoracic segmentsIHA was almost parallel to T_z (t), T_x (t), or T_y (t), stationary for T_x (t) or T_y (t), and migrating for T_z (t) along dorsally opened bows. IHA locations hardly depended on the position or size of axial preload.Lumbar segmentsIHA was also almost parallel to T_z (t), T_x (t), or T_y (t). In axial rotation IHA-migration along wide, ventrally or dorsally bent bows depending on segmental flexional/extensional status. Distances covered: 20–60 mm. In lateral-flexion: IHA-migration to the left/right joint and vice versa. In flexion/extension IHA-migration from the facets to the centre of the disc.Cervical segmentsIn flexion/flexion IHA was almost stationary for and parallel to T_y (t). In axial rotation or lateral-flexion IHA intersected T_z (t)/T_x (t) under approximately ?30°/+30°.ConclusionsGenerally joints alternate in guidance. Lumbar segments: in axial rotation and lateral-flexion parametrical control of IHA-position and IHA-migration by axial preload position. Cervical segments: kinematical coupling between axial rotation and lateral-flexion.The IHA-migration guided by the joints should be taken into account in the design of non-fusion implants. FE-calculations of spinal mechanics and kinematics should be based on detailed data of curvature morphology of the articulating surfaces of the joint facets.     

No-escape regions and oscillations in second order predator-prey recurrences

We are concerned with the second order recurrence x _n+1 = x _n f(x _n, y _n), y _n+1 = y _n g(x _n, y _n), where n N ₀, x ₀ > 0, y ₀ > 0, and the reproduction rates f and g simulate predator-prey interaction. Under conditions on the sign of f – 1 and g–1 we show the existence of a nontrivial no-escape region D, i.e. a compact set D {(x, y): x > 0, y > 0} which is invariant under the recurrence and has the property that every sequence enters D after finitely many steps. Under further conditions on the shape of the isoclines {f = 1} and {g = 1} and on the stationary points {f = 1} {g = 1} we are able to show the existence of sustained oscillations.This work has been supported by the Deutsche Forschungsgemeinschaft     

Barium acylpyrazolonate derivatives stabilized by O- and N-donor ligands: synthesis, spectral and structural characterization

New Ba acylpyrazolonate derivatives of formula [Ba(Q)₂(tetraglyme)] (HQ = HQ^tbu = 1-Ph-3-Me-4-C(O)CH₂Bu^t-5-pyrazolone; tetraglyme = 2,5,8,11,14-pentaoxapentadecane), [Ba(Q)₂(tmeda)₂] (HQ = HQ^tbu or HQ^piv, where HQ^piv = 1-Ph-3-Me-4-C(O)Bu^t-5-pyrazolone; tmeda = N,N,N′,N′-tetramethylethylenediamine), [Ba(Q)₂(pmdien)(H₂O)] (HQ = HQ^tbu or HQ^piv; pmdien = N,N,N′,N″,N″-pentamethyldiethylenetriamine) and [Ba(Q)₂(pmdien)(Meim)] (HQ = HQ^tbu or HQ^piv; Meim = 1-methylimidazole) have been synthesized and analytically and spectroscopically characterized. They are mononuclear air and solution stable compounds containing an eight-coordinated barium atom. The X-ray crystal structures of the hydrates [Ba(Q^tbu)₂(pmdien)(H₂O)] and [Ba(Q^piv)₂(pmdien)(H₂O)] show the water molecule directly bonded to Ba and involved in intermolecular H-bonding network. In the X-ray crystal structure of [Ba(Q^piv)₂(pmdien)(Meim)], the Meim ligand substitutes the water of previous derivatives and decreases the strength of intermolecular interactions, lowering the melting point. The derivative [Ba(Q^tbu)₂(pmdien)(NEt₃)] has been prepared from interaction of [Ba(Q^tbu)₂(pmdien)(H₂O)] with excess triethylamine in acetonitrile.     

Electrostatic charging of an electrically insulated metal body streamlined by a plasma flow

Electrostatic charging of an electrically insulated metal body streamlined by a plasma flow is investigated theoretically. The physical processes occurring in the vicinity of the body are considered. The electric currents resulting in the charging or discharging of the body are determined. The electric charge Q(t), electric field E(t), and potential φ(t) of a metal sphere are estimated taking into account the adopted conditions and parameters of the problem. The physical phenomena that take place in the vicinity of a charged body after Q(t) reaches a certain threshold value are discussed. The obtained values of Q(t), E(t), and φ(t) are compared with the corresponding values for a body streamlined by an atmospheric air flow.     

Novel triorganotin(IV) complexes of β-diketonates bearing two heterocycles in their structures

Novel triorganotin(IV) derivatives of β-diketonate Q ligands (HQ in general, in detail HQ^fur = 1-phenyl-3-methyl-4-(2-furancarbonyl)-pyrazol-5-one, HQ^thi = 1-phenyl-3-methyl-4-(2-thienylcarbonyl)-pyrazol-5-one) of general formula (Q)SnR₃·xH₂O (R = Ph, x = 0; R = Buⁿ or Me, x = 1) have been synthesized and spectroscopically and thermally characterized. Triphenyltin(IV) complexes have been isolated as anhydrous compounds while trialkyltin(IV) are always monohydrated. The structures of (Q^fur)SnPh₃ and (Q^thi)SnMe₃(OH₂) are recorded. The tin atoms are five-coordinate in both. In the first, the pyrazolonate ligand behaves as an O,O′-bidentate; there are two similar but independent molecules in the structure. In the quasi-trigonal-bipyramidal environments, Sn-O(acyl) are 2.478(3), 2.364(3), Sn-O(pyrazolonate) 2.050(2), 2.079(2), Sn-C 2.123(4)-2.162(3) Å with the longer O(acyl) and a phenyl group quasi-trans (O-Sn-C 162.5(1), 160.8(1)°). In (Q^thi)SnMe₃(OH₂), the three methyl groups are equatorial (Sn-C 2.1259(9)-2.1380(8) Å); Sn-O(Q^thi,OH₂) are 2.2143(5), 2.3350(6) Å, O-Sn-O 175.36(2)°. Trimethyltin(IV) derivatives decompose on heating with release of H₂O and SnMe₄ and formation of (Q)₂SnMe₂. Decomposition occurs also within two days after dissolution of (Q)SnMe₃(OH₂) in chloroform.     

Three azido-, alk/phenoxido- and acetato-bridged tetranuclear nickel complexes featuring defective double-cubane

Three new tetranuclear nickel(II) complexes of general formula [Ni₄(L)₂(N₃)₂(CH₃COO)₂(CH₃O)₂]₂·xCH₃OH·yH₂O (HL = HL¹, HL² and HL³; x = 0, y = 1 for 1; x = 2, y = 0 for 2 and x = 2, y = 4 for 3) were synthesized and characterized by single crystal X-ray diffraction and magnetic measurements. Single crystal X-ray studies reveal that all three complexes exhibit similar tetranuclear face-shared defective double-cubane structure, having azido-, alk/phenoxido- and acetato-bridges. Magnetic susceptibility measurements on the complexes in the range of 300-2 K indicate ferromagnetic coupling between the metal ions. The slightly different magnetic behaviors observed are probably caused by subtle structural differences between the respective [Ni₄O₄N₂] cores induced by ligand variation.     

Bounds on the total population for species governed by reaction-diffusion equations in arbitrary two-dimensional regions

Analysis based on the integration of differential inequalities is employed to derive upper and lower bounds on the total populationN(t) = ∫ _R θ(x ₁,x ₂,t) dx ₁ dx ₂ of a biological species with an area-density distribution function θ=θ(x ₁,x ₂,t) (≥0) governed by a reaction-diffusion equation of the form ∂θ/∂t =D∇²θ +fθ −gθ ⁿ⁺¹ whereD (>0),n (>0),f andg are constant parameters, θ=0 at all points on the boundary ∂R of an (arbitrary) two-dimensional regionR, and the initial distribution (θ(_{x 1},x ₂, 0) is such thatN(0) is finite. Forg≥0 withR the entire two-dimensional Euclidean space, a lower bound onN(t) is obtained, showing in particular thatN(∞) is bounded below by a finite positive quantity forf≥0 andn>1. An upper bound onN(t) is obtained for arbitrary bounded or unbounded)R withn=1,f andg negative, and ∫ _R θ(x ₁,x ₂, 0)² dx ₁ dx ₂ sufficiently small in magnitude, implying that the population goes to extinction with increasing values of the time,N(∞)=0. Forg≥0 andR of finite area, the analysis yields upper bounds onN(t), predicting eventual extinction of the population if eitherf≤0 or if the area ofR is less than a certain grouping of the parameters in cases for whichf is positive. These results are directly applicable to biological species with distributions satisfying the Fisher equation in two spatial dimensions and to species governed by certain specialized population models.     

Photoinhibition and xanthophyll cycle activity in bayberry (Myrica rubra) leaves induced by high irradiance

The effect of high irradiance (HI, photosynthetically active photon flux density of 1 300 μmol m⁻² s⁻¹) on net photosynthetic rate (P _N), chlorophyll fluorescence parameters, and xanthophyll cycle components were studied in fruit tree bayberry leaves. HI induced the photoinhibition and inactivation of photosystem 2 (PS2) reaction centres (RCs), which was characterized by decreased P _N, maximum yield of fluorescence after dark adaptation (F_m), photochemical efficiency of PS2 (F_v/F_m) and quantum yield of PS2 (Φ_PS2), and increased reduction state of Q_A (1-q_P) and non-photochemical quenching (NPQ). Initial fluorescence (F₀) showed a decrease after the first 2 h, and subsequently increased from the third hour exposure to HI. Furthermore, a greater increase in the ratio (F_i-F₀)/(F_p-F₀) which is an expression of the proportion of the Q_B non-reducing PS2 centres, whereas a remarked decrease in the slope of F_i to F_p which represents the rate of Q_A reduction was observed in leaves after HI exposure. Additionally, HI caused an increase in the pool size of the xanthophyll cycle pigments and sustained elevated contents of zeaxanthin (Z), antheraxanthin (A), and de-epoxidation state (DES) at the end of the irradiation period. During HI, decreased F_m, F_v/F_m, Φ_PS2, NPQ, slope of F_i to F_p, V+A+Z, and DES, and increased F₀, 1-q_P, ratio (F_i-F₀)/(F_p-F₀), and V were observed in dithiothreitol (DTT)-fed leaves compared to control ones under the same conditions. Hence photoinhibition caused by HI in bayberry was probably attributed to inactivation of PS2 RCs, and photoprotection from photodamage were mainly related to the xanthophyll cycle-dependent heat dissipation in excess photons.     

On washout in nonlinear compartmental systems

Let x(t) be a solution of a compartmental system. If, for some compartment j, x_j(t)→0 as t→∞, then we say that the compartment j washes out. We show that a compartment washes out if it always reaches (along a fixed path) either the environment or another compartment for which there is no return path. Additional criteria, particularly regarding exponential convergence, are also presented. Examples are drawn from tracer kinetics, enzyme reactions, and epidemic models.