On the combinatorial rank of the quantum groups of type G2

In this paper, we calculate the combinatorial rank of the positive part [Formula: see text] of the multiparameter version of the small Lusztig quantum group, where [Formula: see text] is a simple Lie algebra of type [Formula: see text]. Supposing that the main parameter of quantization [Formula: see text] has multiplicative order [Formula: see text], where [Formula: see text] is finite, [Formula: see text], we prove that the combinatorial rank equals 3.

Skyrme forces and isotopic dependence of evaporation residue cross-section in the decay of 252,254−256No∗ formed in 204,206,207,208Pb+48Ca reactions

The fusion evaporation residue (ER) cross-sections [Formula: see text] for the decay of compound nucleus (CN) [Formula: see text]No[Formula: see text] via 1[Formula: see text]–4[Formula: see text] decay channels, synthesized in [Formula: see text]Pb[Formula: see text][Formula: see text][Formula: see text][Formula: see text]Ca reaction, are studied, including deformations [Formula: see text] for cold-optimum orientations [Formula: see text] at various [Formula: see text]Ca excitation energies of [Formula: see text] to 45[Formula: see text]MeV. For the nuclear interaction potentials, we use the Skyrme energy density functional (SEDF)-based on semi-classical extended Thomas Fermi (ETF) approach under frozen density approximation on our earlier study of fusion ER cross-section for the decay of [Formula: see text]No[Formula: see text], via 1[Formula: see text]–4[Formula: see text] decay channels synthesized in [Formula: see text]Pb[Formula: see text][Formula: see text][Formula: see text][Formula: see text]Ca reaction based on the Dynamical cluster-decay model (DCM) using the pocket formula for nuclear proximity potential in which the above reaction was investigated by using hot-optimum orientations. In this work the above reaction has been investigated in two parts, in the first part, [Formula: see text][Formula: see text]MeV is used for cold elongated configurations and in the second part, [Formula: see text][Formula: see text]MeV is used for hot compact configurations. The Skyrme forces used here are the old SIII, and new GSkI and KDE0(v1) given for both normal and isospin-rich nuclei, with densities added in frozen density approximation. Interestingly, independent of the Skyrme force used, the DCM gives an excellent fit within one-parameter fitting of [Formula: see text] to the measured data on fusion ER for cold fusion. Of all the three Pb-isotopes and three [Formula: see text] considered, at each [Formula: see text], the [Formula: see text] is largest for compound system with mass numbers 256 and 254, and smallest for 252, which means that the neutrons emission occur earliest for 256, then 255 followed by 254 and finally for 252, in complete agreement with experimental data. The possible fusion–fission (ff) and quasi-fission (qf) mass-regions of fragments on DCM are also predicted. The DCM with Skyrme forces is further used to look for all the possible target-projectile ([Formula: see text]-[Formula: see text]) combinations forming the “cold” CN [Formula: see text]No[Formula: see text] at the CN excitation energy of [Formula: see text] for “optimum cold” configurations. The fusion ER cross-sections, for the proposed new reactions in synthesizing the CN [Formula: see text]No[Formula: see text], are also estimated for the future experiments.

Fourth Amendment Textualism

The Fourth Amendment’s prohibition of “unreasonable searches” is one of the most storied constitutional commands. Yet after decades of Supreme Court jurisprudence, a coherent definition of the term “search” remains surprisingly elusive. Even the justices know they have a problem. Recent opinions only halfheartedly apply the controlling “reasonable expectation of privacy” test and its wildly unpopular cousin, “third-party doctrine,” with a few justices in open revolt. These fissures hint at the Court’s openness to a new approach. Unfortunately, no viable alternatives appear on the horizon. The justices themselves offer little in the way of a replacement. And scholars’ proposals exhibit the same complexity, subjectivity, and illegitimacy that pervade the status quo. This Article proposes a shift toward simplicity. Buried underneath the doctrinal complexity of the past fifty years is a straightforward constitutional directive. A three-part formula, derived from the constitutional text, deftly solves the Fourth Amendment “search” conundrums that continue to beguile the Court. This textualist approach offers clarity and legitimacy, both long missing from “search” jurisprudence. And by generating predictable and sensible answers, the proposed framework establishes clear boundaries for police investigation while incentivizing legislators to add additional privacy protections where needed.

Flower-like MoS2 Nanospheres: A Promising Material with Good Microwave Absorption Property in the Frequency Range of 8.2–12.4 GHz

MoS2 nanospheres have been synthesized by a hydrothermal method at various reaction times using sodium molybdate and thioacetamide as the sulfur source and reducing agent, respectively. The prepared MoS2 nanospheres have been characterized by XRD, SEM and vector network analyzer, respectively. Results show that the prepared MoS2 samples were the flower-like nanospheres and the powder size increased with increasing reaction time. The real part ([Formula: see text]) and imaginary part ([Formula: see text]) of permittivity of the prepared MoS2 nanospheres increased as the reaction time increased initially and decreased afterwards. When the reaction time was 20[Formula: see text]h, the [Formula: see text] and [Formula: see text] reached the maximum values, 8.25–8.05 and 4.1–3.85, respectively. The minimum reflection loss value of [Formula: see text][Formula: see text]dB was achieved at 8.3[Formula: see text]GHz for the synthesized MoS2 nanospheres with reaction time of 16[Formula: see text]h. All the samples with matching thicknesses of 2–4[Formula: see text]mm presented wider effective absorption bandwidths ([Formula: see text][Formula: see text]dB) in the frequency range of 8.2–12.4[Formula: see text]GHz.

First-principles studies on electronic structures and optical properties of two-dimensional Sn1−xTi(Zr)xS2 alloys

Through first-principles calculations we study the electronic structures and optical properties of two-dimensional (2D) Sn[Formula: see text]Ti(Zr)[Formula: see text]S2 alloys. The results indicate that the band gap value of Sn[Formula: see text]Ti(Zr)[Formula: see text]S2 alloys is decreased continuously when Ti(Zr) concentration is increased, which is very beneficial to optoelectronic devices applications. Moreover, the static dielectric constant is increased when the Ti(Zr) concentration is increased in the 2D Sn[Formula: see text]Ti(Zr)[Formula: see text]S2 alloys. In addition, we also calculate the imaginary part [Formula: see text] dispersion of Sn[Formula: see text]Ti(Zr)[Formula: see text]S2 alloys along the plane with different Ti(Zr) concentrations. The threshold energy values decrease with increasing Ti(Zr) concentrations in the Sn[Formula: see text]Ti(Zr)[Formula: see text]S2 ternary alloys. Moreover, the calculations of formation energy also indicate that these 2D alloys can be fabricated under some experimental conditions. These results suggest that Ti(Zr) substituting Sn atom is an efficient way to tune the band gap and optical properties of 2D SnS2 nanosheets.

The Weyl curvature tensor, Cotton–York tensor and gravitational waves: A covariant consideration

[Formula: see text] covariant approach to cosmological perturbation theory often employs the electric part ([Formula: see text]), the magnetic part ([Formula: see text]) of the Weyl tensor or the shear tensor ([Formula: see text]) in a phenomenological description of gravitational waves. The Cotton–York tensor is rarely mentioned in connection with gravitational waves in this approach. This tensor acts as a source for the magnetic part of the Weyl tensor which should not be neglected in studies of gravitational waves in the [Formula: see text] formalism. The tensor is only mentioned in connection with studies of “silent model” but even there the connection with gravitational waves is not exhaustively explored. In this study, we demonstrate that the Cotton–York tensor encodes contributions from both electric and magnetic parts of the Weyl tensor and in directly from the shear tensor. In our opinion, this makes the Cotton–York tensor arguably the natural choice for linear gravitational waves in the [Formula: see text] covariant formalism. The tensor is cumbersome to work with but that should negate its usefulness. It is conceivable that the tensor would equally be useful in the metric approach, although we have not demonstrated this in this study. We contend that the use of only one of the Weyl tensor or the shear tensor, although phenomenologically correct, leads to loss of information. Such information is vital particularly when examining the contribution of gravitational waves to the anisotropy of an almost-Friedmann–Lamitre–Robertson–Walker (FLRW) universe. The recourse to this loss is the use Cotton–York tensor.

Structure of states for which each localized dynamics reduces to a localized subdynamics

We consider a bipartite quantum system [Formula: see text] (including parties [Formula: see text] and [Formula: see text]), interacting with an environment [Formula: see text] through a localized quantum dynamics [Formula: see text]. We call a quantum dynamics [Formula: see text] localized if, e.g. the party [Formula: see text] is isolated from the environment and only [Formula: see text] interacts with the environment: [Formula: see text], where [Formula: see text] is the identity map on the part [Formula: see text] and [Formula: see text] is a completely positive (CP) map on the both [Formula: see text] and [Formula: see text]. We will show that the reduced dynamics of the system is also localized as [Formula: see text], where [Formula: see text] is a CP map on [Formula: see text], if and only if the initial state of the system-environment is a Markov state. We then generalize this result to the two following cases: when both [Formula: see text] and [Formula: see text] interact with a same environment, and when each party interacts with its local environment.

Monte Carlo simulation on the diffusion of polymer in narrow periodical channels

Diffusion of polymer in narrow periodical channels, patterned alternately into part [Formula: see text] and part [Formula: see text] with the same length [Formula: see text], was studied by using Monte Carlo simulation. The interaction between polymer and channel [Formula: see text] is purely repulsive, while that between polymer and channel [Formula: see text] is attractive. Results show that the diffusion of polymer is remarkably affected by the periodicity of channel, and the diffusion constant [Formula: see text] changes periodically with the polymer length [Formula: see text]. At the peaks of [Formula: see text], the projected length of polymer along the channel is an even multiple of [Formula: see text], and the diffusion of polymer in periodical channel is nearly the same as that of polymer in homogeneous channel. While at the valleys of [Formula: see text], the projected length of polymer is an odd multiple of [Formula: see text], and polymer is in a trapped state for a long time and it rapidly jumps to other trapped regions during the diffusion process. The physical mechanisms are discussed from the view of polymer–channel interaction energy landscape.

Study of improved confinement by a stepwise increase of the input heating power for tokamak plasmas

This paper is an extension of the brief study by Sarah Douglas et al. [Phys. Plasmas 20 (2013) 114504] where in the study a sinusoidal perturbation of the heating power has been studied. In this paper a stepwise increase of the heating power and its influence on the [Formula: see text]–[Formula: see text] transition are studied. Using a function, [Formula: see text] for the transition of input heating power for tokamak plasmas, i.e. the addition of the perturbation, [Formula: see text], to constant power [Formula: see text] is shown to promote the confinement, leading to the [Formula: see text]–[Formula: see text] transition at a lower value of [Formula: see text], as compared to the case of constant [Formula: see text] without the [Formula: see text] perturbation. It is seen that the input heating power [Formula: see text] that consists of constant part [Formula: see text] in addition to a function [Formula: see text] provides the [Formula: see text]–[Formula: see text] transition for relatively small [Formula: see text] and much wider range values of [Formula: see text] as compared to Sarah Douglas et al. [Phys. Plasmas 20 (2013) 114504].

APPLICABILITY OF A REPRESENTATION FOR THE MARTIN'S REAL-PART FORMULA IN MODEL-INDEPENDENT ANALYSES

Using a novel representation for the Martin's real-part formula without the full scaling property, an almost model-independent description of the proton–proton differential cross-section data at high energies (19.4 GeV–62.5 GeV) is obtained. In the impact parameter and eikonal frameworks, the extracted inelastic overlap function presents a peripheral effect (tail) above 2 fm and the extracted opacity function is characterized by a zero (change of sign) in the momentum transfer space, confirming results from previous model-independent analyses. Analytical parametrization for these empirical results are introduced and discussed. The importance of investigations on the inverse problems in high-energy elastic hadron scattering is stressed and the relevance of the proposed representation is commented. A short critical review on the use of Martin's formula is also presented.