Values Of Λ1 Research Articles

Effect of growth rate and Mg content on dendrite tip characteristics of Al–Cu–Mg ternary alloys

An experimental analysis is presented to correlate the secondary dendrite arm spacing λ2 and dendrite tip radius R with growth rate V and Mg content C0-Mg of Al–Cu–Mg ternary alloys. Under constant temperature gradient G (4.84±0.13 K mm−1), a series of directional solidification experiments were performed at five different growth rates V (16.7–83.3 μm/s) and five different Mg contents C0-Mg in Al–5 wt.% Cu–(0.5–5) wt.% Mg alloys. Solid–liquid interface was investigated from the longitudinal sections of the quenched samples, and λ2 and R were measured on the dendrite tips. The dependencies of λ2 and R on V and C0-Mg were determined. The experimental results showed that the values of λ2 and R decrease as V and C0-Mg increase at a constant G. The present exponent values related to V are found to be slightly lower than the values of the theoretical models and previous experimental works; however, C0-Mg exponent values are found to be much lower than the theoretical models and previous experimental works. The ratio of the secondary dendrite arm spacing to the dendrite tip radius is 2.09±0.15, in good agreement with the scaling law. At a constant C0-Mg, the values of VR2 were found to slightly increase with the ascending V. However, as C0-Mg increases, the values of VR2 decrease.

Stability criterion for determination of primary arm spacing

A new expression of the primary arm spacing is analytically evaluated assuming that the observed cell/primary arm spacing defines a stable cell/dendrite configuration. Two mechanisms determining the existence of a stable configuration have been previously identified. In the lower cell/dendrite spacing range, there is the mechanism of overgrown which limits inferiorly the stable cell dendrite spacing. In the upper cell/dendrite spacing range, the mechanism of dendrite division is identified. One expects at the intersection of these two unstable regimes to find a stable dendrite array configuration defined by the primary arm spacing λ1. A quasi-steady state columnar dendrite array, whose one dendrite is slightly perturbed, is considered. Using this particular configuration, it is possible to derive a characteristic time for each mechanism described above. The primary arm spacing is derived assuming that stability of the dendritic array is insured only if these both characteristic times are of the same order of magnitude. The value of λ1 is confronted with experimental results. The comparison is excellent and improves previous models predictions.

An experimentalist's view of the physics of rubber elasticity

An experimentalist's view of the physics of rubber elasticity

Strange quark, tachyonic gluon masses and [formula omitted] from hadronic tau decays

We apply, to the new OPAL data on V+A strange spectral functions, a suitable combination of QCD spectral sum rules directly sensitive to the combination of strange quark (ms) and tachyonic gluon (λ2) masses. Using the mean value of λ2 from a sum rule analysis of different channels, we deduce the invariant strange quark mass mˆs=(106−37+33) MeV to order αs3, which leads to the running mass m¯s(2 GeV)=(93−32+29) MeV. We also obtain an interesting though less accurate estimate of the CKM angle: |Vus|=0.217±0.019.

New Invariant of DNA Sequences.

For a DNA sequence with n bases, one can always associate it with an n × n nonnegative real symmetric matrix whose diagonal entries are zero. Once the matrix is given, its leading eigenvalue is usually calculated and used as an invariant to characterize the DNA sequence. Let M be such a matrix, and λ1 its leading eigenvalue. Then (1/n)||M||m1 and ||M||F are the lower and upper bounds of λ1, respectively. Since their arithmetic average is an approximate value of λ1 and simpler for calculation, we can use it as an alternative invariant to characterize the DNA sequence. The utility of the new parameter is illustrated on the DNA sequences of five species: human, chimpanzee, mouse, rat, and gallus.

Second‐order boundary value problems with nonhomogeneous boundary conditions (I)

AbstractWe study the nonlinear boundary value problem with nonhomogeneous multi‐point boundary condition Sufficient conditions are found for the existence of solutions of the problem based on the existence of lower and upper solutions with certain relation. Using this existence result, under some assumptions, we obtain explicit ranges of values of λ1 and λ2 with which the problem has a solution, has a positive solution, and has no solution, respectively. Furthermore, we prove that the whole plane for λ1 and λ2 can be divided into two disjoint connected regions ΛE and ΛN such that the problem has a solution for (λ1, λ2) ∈ ΛE and has no solution for (λ1, λ2) ∈ ΛN. We also show that under different assumptions, the problem has a solution for all (λ1, λ2) ∈ ℝ2. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Bounding the largest eigenvalue of trees in terms of the largest vertex degree

Let λ1(G) denote the largest eigenvalue of the adjacency matrix and let μ1(G) denote the largest eigenvalue of the Laplacian matrix of a graph G. It is well known that if a graph G has the largest vertex degree Δ≠0 thenΔ⩽λ1(G)⩽ΔandΔ+1⩽μ1(G)⩽2Δ.Thus the gap between the maximum and minimum value of λ1(G) and μ1(G) in the class of graphs with fixed Δ is Θ(Δ). In this note we show that in the class of trees with fixed Δ this gap is just Θ(Δ). Namely, we show that if a tree T has the largest vertex degree Δ thenλ1(T)<2Δ−1andμ1(T)<Δ+2Δ−1.New bounds are an improvement for Δ⩾3.

On the complete solution of the Sturm–Liouville problem (d2X/dx2)+λ2X=0 over a closed interval

We discuss the sets of orthogonal functions that form solutions to the Sturm–Liouville problem for the equation d2X/dx2+λ2X=0 and for the general unmixed boundary conditions over a closed interval of the variable x. The conditions for the presence of the solutions with λ2⩽0 are specifically considered and their necessity for completeness of a set of eigenfunctions. Their implications are discussed for three examples from mathematical physics, showing that although for some problems the solutions, corresponding to the negative values of λ2, may reflect physically unusual boundary conditions, their presence is necessary in the general solution for the drift diffusion equation where they may represent stationary or growing in time solutions.

Exterior stokes flows with stick-slip boundary conditions

Steady two-dimensional creeping flows induced by line singularities in the presence of an infinitely long circular cylinder with stick-slip boundary conditions are examined. The singularities considered here include a rotlet, a potential source and a stokeslet located outside a cylinder and lying in a plane containing the cylinder axis. The general exterior boundary value problem is formulated and solved in terms of a stream function by making use of the Fourier expansion method. The solutions for various singularity driven flows in the presence of a cylinder are derived from the general results. The stream function representation of the solutions involves a definite integral whose evaluation depends on a non-dimensional slip parameter λ1. For extremal values, λ1 = 0 and λ1 = 1, of the slip parameter our results reduce to solutions of boundary value problems with stick (no-slip) and perfect slip conditions, respectively.The slip parameter influences the flow patterns significantly. The plots of streamlines in each case show interesting flow patterns. In particular, in the case of a single rotlet/stokeslet (with axis along y-direction) flows, eddies are observed for various values of λ1. The flow fields for a pair of singularities located on either side of the cylinder are also presented. In these flows, eddies of different sizes and shapes exist for various values of λ1 and the singularity locations. Plots of the fluid velocity on the surface show locations of the stagnation points on the surface of the cylinder and their dependencies on λ1 and singularity locations.

A sharp bound for the ratio of the first two Dirichlet eigenvalues of a domain in a hemisphere of 𝕊ⁿ

For a domain Ω \Omega contained in a hemisphere of the n n –dimensional sphere S n \mathbb {S}^n we prove the optimal result λ 2 / λ 1 ( Ω ) ≤ λ 2 / λ 1 ( Ω ⋆ ) \lambda _2/\lambda _1(\Omega ) \le \lambda _2/\lambda _1(\Omega ^{\star }) for the ratio of its first two Dirichlet eigenvalues where Ω ⋆ \Omega ^{\star } , the symmetric rearrangement of Ω \Omega in S n \mathbb {S}^n , is a geodesic ball in S n \mathbb {S}^n having the same n n –volume as Ω \Omega . We also show that λ 2 / λ 1 \lambda _2/\lambda _1 for geodesic balls of geodesic radius θ 1 \theta _1 less than or equal to π / 2 \pi /2 is an increasing function of θ 1 \theta _1 which runs between the value ( j n / 2 , 1 / j n / 2 − 1 , 1 ) 2 (j_{n/2,1}/j_{n/2-1,1})^2 for θ 1 = 0 \theta _1=0 (this is the Euclidean value) and 2 ( n + 1 ) / n 2(n+1)/n for θ 1 = π / 2 \theta _1=\pi /2 . Here j ν , k j_{\nu ,k} denotes the k k th positive zero of the Bessel function J ν ( t ) J_{\nu }(t) . This result generalizes the Payne–Pólya–Weinberger conjecture, which applies to bounded domains in Euclidean space and which we had proved earlier. Our method makes use of symmetric rearrangement of functions and various technical properties of special functions. We also prove that among all domains contained in a hemisphere of S n \mathbb {S}^n and having a fixed value of λ 1 \lambda _1 the one with the maximal value of λ 2 \lambda _2 is the geodesic ball of the appropriate radius. This is a stronger, but slightly less accessible, isoperimetric result than that for λ 2 / λ 1 \lambda _2/\lambda _1 . Various other results for λ 1 \lambda _1 and λ 2 \lambda _2 of geodesic balls in S n \mathbb {S}^n are proved in the course of our work.

Cahn–Hilliard theory with triple-parabolic free energy. I. Nucleation and growth of a stable crystalline phase

Nucleation and growth of a stable crystalline phase are described in the framework of a single-order-parameter Cahn–Hilliard theory. A piecewise parabolic free energy-order parameter relationship composed of three parabolas is adopted with a negative curvature coefficient (λ1) for the central part. An analytical solution of the problem is presented. The work of formation of critical fluctuations, the temperature coefficient of their interfacial free energy, and the Tolman length are found to be sensitive to the value of λ1, whereas the steady-state growth rate is rather insensitive. It is demonstrated that for systems of known free-energy order parameter relationship, the triple-parabola approximation is useful in obtaining qualitative and semiquantitative results for nucleation and growth rates.

Efficient approximate solutions to micromagnetic problems

The sixfold integral of magnetostatics ∫∫∫∫∫∫ div M{r}div M{r′}/|r′−r|d3r′ d3r is replaced by a local pseudo interaction λ2∫∫∫ div2 M{r}d3r in the expression for the micromagnetic energy. A magnetic configuration M{λ,r} is found by minimization of this micromagnetic energy for each choice of λ2. The sixfold integral is then evaluated for a few values of λ2. These are used to interpolate between those values to find a constrained minimum of the micromagnetic energy with the sixfold integrals. The method is applied to the description of the intersection of a Bloch wall with the surface of a thick crystal for magnetically soft materials characterized by large saturation magnetization Ms and small uniaxial anisotropy Ku; that is, Ms2&amp;gt;Ku.

Charge-Transfer Complexes of Tetracyanoethylene with Alkyl and Aryl Derivatives of the Halogens

Spectral characteristics (λCT, Δν1/2, and e) and association constants (in dichloromethane) of charge-transfer (CT) complexes of tetracyanoethylene (T) with donors containing C−X linkages (X = F, Cl, Br, and I) are presented. The donor orbitals from which CT transitions originate are identified by correlation of λCT and Δν1/2 values of CT bands of the complexes with the ionization bands of the photoelectron spectra of the donor molecules. CT energies of the complexes are influenced by intramolecular conjugative and inductive effects between halogen atoms and alkyl and aryl groups. Haloalkanes form weak complexes with T through interaction of the px and py orbitals of the halogen with the π* LUMO of T. Monoiodoalkane−T complexes exhibit two spectral bands arising from the transfer of nonbonded electrons from the 5p orbitals of iodine which are split into 2E1/2 and 2E3/2 states through spin−orbit coupling. λ1 and λ2 values of monoiodoalkanes increase progressively with increasing bulk of the alkyl group. Ha...

A kinetic analysis of the primary charge separation in bacterial photosynthesis. Energy gaps and static heterogeneity

Abstract We consider the energetics, the mechanism and the implications of static heterogeneity for the primary electron transfer (ET) from the electronically excited singlet state of the bacteriochlorophyll dimer (1O∗) in the bacterial photosynthetic reaction center (RC) and some of its mutants. The energetics of the primary ET was inferred from an analysis of the experimental free energy relation (at T = 300 K) between the short-time decay rates of 1P∗ and the oxidation potentials of the dimer (P) for a series of single site “good” mutants, for which geometrical changes are minimized and perturbations of the prosthetic groups of the accessory bacteriochlorophyll (B) and of the bacteriopheophytin (H) by the mutants are minor. This analysis resulted in the reasonable value of λ1 = 800 ± 250 cm−1 for the (mutant invariant) medium reorganization energy and ΔG10(N) = −480 ± 180 cm−1 for the energy gap for the native (N) RC. The low value of ΔG10(N) implies that the dominant room temperature ET mechanism for the native RC involves sequential ET. Next, we have explored the effects of heterogeneity on the primary ET by model calculations for the parallel sequential-superexchange mechanism, which is subjected to Gaussian energy distributions of the energies of the P+B−H and P+BH− ion pair states (with a width parameter of σ = 400 cm−1). The modelling of the heterogeneous kinetics by varying the (mean) energy gap ΔG1 between P+B−H and 1P∗ was performed to elucidate the temporal decay of 1P∗ and the ET quantum yield in “good” mutants, to explore the gross feature of primary ET in a triple hydrogen bonded mutant and to characterize some of the temperature dependence of the primary ET. The most pronounced manifestations of heterogeneity within the native RC and its single site mutants (ΔG1 = −900 to 300 cm−1) are the nonexponential temporal decay probabilities for 1P∗, which exhibit long-time tails, with heterogeneity effects being marked (in the classical limit) when σ(ΔG1 + λ1) > λ1kBT. When ΔG1 ⪢ σ (i.e., ΔG1 ⩾ 1000 cm−1), the relaxation rate of 1P∗ is slow, being dominated by the dimer internal conversion rate, with the effects of heterogeneity being less marked, as is the case for the triple hydrogen bond mutant. Regarding mechanistic issues, our kinetic modelling implies that at room temperature, primary ET in the native RC and its single site mutants is dominated by the sequential route and only the triple mutant exhibits a marked contribution of the superexchange route. At low temperature (T = 20 K), ET in the native RC is still dominated by the sequential route (with a small (i.e., ∼ 10%) superexchange contribution being manifested in its long-time decay), for single site mutants there is an interplay between sequential and superexchange routes, while superexchange dominates ET in the triple mutant. The heterogeneous parallel sequential-superexchange mechanism is of intrinsic significance to insure the stability of primary photosynthetic ET for different native and mutagenetically modified RCs over a broad temperature domain.

Differential angular correlation of the 1408–122 keV cascade and the g factor of the 122 keV 2+ level of 152Sm

Time-differential angular-correlation measurements on the 1408–122 keV cascade of 152Sm were performed in liquid EuCl3 and Eu(NO3)3 sources at a time resolution of 450 ps. The results obtained are λ2 = (8.3 ± 2.0) × 107 s−1, A2(0) = 0.218 ± 0.008 in SmCl3 and λ2 = (2.5 ± 2.1) × 107 s−1, A2(0) = 0.206 ± 0.018 in Sm(NO3)3. Integral angular-correlation measurements on the 244–122, 964–122, and 1112–122 keV cascades were also performed in the same chloride and nitrate sources using a NaI(Tl)–HPGe system. Using the values of λ2, the unperturbed angular-correlation coefficients for these cascades are determined and therefrom the mixing ratio of the λ transitions are found to be δ(1408) = 0.05 ± 0.01, [Formula: see text] and [Formula: see text]. The magnetic moment of the 122 keV 2+ level was remeasured and a value of μ = 0.80 ± 0.06 nm was obtained.

Spectral correlation of the components of a two-frequency tunable laser

An investigation was made of the statistical properties of the radiation spectrum of a pulsed tunable laser emitting at two wavelengths simultaneously, λ1 and λ2. The correlation between the instantaneous and average values of λ1 and λ2 was studied, and it was found that under conditions of asymmetric limitations of the field in the resonator and a dynamic lens induced by the pump radiation in the active medium, the change in the average wavelengths of the components is characterized by a high correlation coefficient K > 0.8. These results offer the prospect of improved accuracy in measurements involving two-frequency lasers.

Second-derivative spectrophotometric determination of mixtures of phenylpropanolamine hydrochloride and dextromethorphan hydrobromide in some pharmaceutical preparations

A second-derivative spectrophotometric method was developed for the simultaneous determination of phenylpropanolamine hydrochloride and dextromethorphan hydrobromide in the presence of one another without prior separation. Second-derivative spectra of sample solutions in methanol were scanned between 220 and 320 nm. Because of differences in molar absorptivities of the two compounds, different d2A/dλ2 ordinate scales were used for each compound. For phenylpropanolamine, the largest amplitude in the derivative spectrum was measured between the maximum at 260 nm and its minimum at 257 nm (2D260,257). For dextromethorphan, the peak amplitude in the derivative spectrum was measured between the maximum at 291.5 nm and its minimum at 287.5 nm (2D291.5,287.5). The 2Dλ1,λ2 values for each compound were not affected by the presence of the other compound, and were linear in the ranges 20–350 μg ml−1 phenylpropanolamine hydrochloride (r = 0.9996) and 16–200 μg ml−1 dextromethorphan hydrobromide (r = 0.9992). Average recoveries (±SD; n = 5) were: from simulated capsules, 99.6 ± 1.3% phenylpropanolamine hydrochloride and 98.7 ± 2.4% dextromethorphan hydrobromide; and from simulated lozenges, 98.0 ± 2.4% phenylpropanolamine hydrochloride 99.9 ± 2.0% dextromethorphan hydrobromide. The method was designed to allow single-unit assay for the normal dosages used. Application of the method to solid commercial formulations was made.

Rate of precipitation scavenging of nitrates on central Long Island

By comparing deposition of nitrate in successive hours of precipitation samples taken over a six year period at Brookhaven National Laboratory (BNL), Long Island, New York, we have estimated scavenging coefficients in air, λ, for nitrate under a variety of meteorological and seasonal conditions. The square root transformation of the scavenging coefficients λ½ yields an approximately normal distribution; the mean value is 1.1 × 10−4 s−1, in general agreement with previous experimental and theoretical estimates. An interesting property of the distribution of the scavenging coefficients is that the ratio of the mean value of λ½ to its standard deviation varies little from 2.4 for various subsets of data; this property may limit the choice of stochastic models which describe the scavenging process. A trend analysis of the relationship between scavenging coefficients and rainfall rate indicates that the scavenging coefficient of nitrate does not increase for rainfall rates ≤3.2 mm h−1. The indicated increase at greater rainfall rates is qualitatively similar to model predictions but is based upon a small number of data points. The mean scavenging coefficient for winter rain events is greater than for snow.

Coherent states of the real symplectic group in a complex analytic parametrization. I. Unitary-operator coherent states

In the present series of papers, the coherent states of Sp(2d,R), corresponding to the positive discrete series irreducible representations 〈λd+n/2,...,λ1+n/2〉 encountered in physical applications, are analyzed in detail with special emphasis on those of Sp(4,R) and Sp(6,R). The present paper discusses the unitary-operator coherent states, as defined by Klauder, Perelomov, and Gilmore. These states are parametrized by the points of the coset space Sp(2d,R)/H, where H is the stability group of the Sp(2d,R) irreducible representation lowest weight state, chosen as the reference state, and depends upon the relative values of λ1,...,λd, subject to the conditions λ1≥λ2≥ ⋅ ⋅ ⋅ ≥λd≥0. A parametrization of Sp(2d,R)/H corresponding to a factorization of the latter into a product of coset spaces Sp(2d,R)/U(d) and U(d)/H is chosen. The overlap of two coherent states is calculated, the action of the Sp(2d,R) generators on the coherent states is determined, and the explicit form of the unity resolution relation satisfied by the coherent states in the representation space of the irreducible representation is obtained. The Hilbert space of analytic functions arising from the coherent state representation is studied in detail. Finally, some applications of the formalism developed in the present paper are outlined. In particular, its relevance to the study of boson realizations of the Sp(2d,R) algebra is stressed.

An extension of the Leslie matrix model to include a variable immature period

AbstractAn extension to the Leslie matrix is presented in which the age of transformation from immature to adult has a log—normal distribution. The major effect of this is shown to be on the second largest eigenvalue. The ratio of the largest to the second largest eigenvalue |λ1/λ2|, which is an index of the rate of approach to the stable age distribution, is greater in the new model, even though the value of λ1 is effectively the same. The differences in the models are most pronounced where the population is subjected to a harvesting regime.