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If $A=(A,\oplus,\odot)$ is a $\lambda$-homomorphic brace with $(A,\oplus)=\mathbb{Z}^2$, then the operations in this brace are given by formulas \begin{align*}\begin{pmatrix}a_1\\a_2\end{pmatrix}\oplus\begin{pmatrix}b_1\\b_2\end{pmatrix}=\begin{pmatrix}a_1+b_1\\a_2+b_2\end{pmatrix},&&\begin{pmatrix}a_1\\a_2\end{pmatrix}\odot\begin{pmatrix}b_1\\b_2\end{pmatrix}=\begin{pmatrix}a_1\\a_2\end{pmatrix}+\varphi^{a_1}\psi^{a_2}\begin{pmatrix}b_1\\b_2\end{pmatrix}, \end{align*} where $\varphi,\psi\in{\rm GL}_2(\mathbb{Z})$ are cpecific matrices which depend on $A$. Not every pair $(\varphi,\psi)$ lead to a brace. In the present paper we find all possible pairs $(\varphi,\psi)$ of matrices from ${\rm GL}_2(\mathbb{Z})$ which lead to $\lambda$-homomorphic braces with $(A,\oplus)=\mathbb{Z}^2$. The obtained result gives the full classification of $\lambda$-homomorphic braces on $\mathbb{Z}^2$ which was started by Bardakov, Neshchadim and Yadav in [J. Pure App. Algebra, V. 226, N. 6, 2022, 106961].

Recently, the n$^3$He collaboration reported a measurement of the parity-violating (PV) proton directional asymmetry $A_{\mathrm {PV}} = (1.55\pm 0.97~\mathrm {(st\ at)} \pm 0.24~\mathrm {(sys)})\times 10^{-8}$ in the capture reaction of ${}^3$He$(\vec {n},{\mathrm p}){}^3$H at meV incident neutron energies. The result increased the limited inventory of precisely measured and calculable PV observables in few-body systems required to further understand the structure of hadronic weak interaction. In this letter, we report the experimental and theoretical investigation of a parity conserving (PC) asymmetry $A_{\mathrm {PC}}$ in the same reaction (the first ever measured PC observable at meV neutron energies). As a result of S- and P-wave mixing in the reaction, the $A_{\mathrm {PC}}$ is inversely proportional to the neutron wavelength $\lambda$. The experimental value is $(\lambda\times A_{\mathrm {PC}})\equiv\beta= (-1.97 \pm 0.28~\mathrm{(stat)}\pm 0.12~\mathrm{(sys)}) \times 10^{-6}$ Amstrongs. We present results for a theoretical analysis of this reaction by solving the four-body scattering problem within the hyperspherical harmonic method. We find that in the ${}^3$He$(\vec {n},{\mathrm p}){}^3$H reaction, $A_{\mathrm {PC}}$ depends critically on the energy and width of the close $0^-$ resonant state of ${}^4$He, resulting in a large sensitivity to the spin-orbit components of the nucleon-nucleon force and even to the three-nucleon force. The analysis of the accurately measured $A_{\mathrm {PC}}$ and $A_{\mathrm {PV}}$ using the same few-body theoretical models gives essential information needed to interpret the PV asymmetry in the ${}^3$He$(\vec {n}, {\mathrm p}){}^3$H reaction., Comment: 8 pages, 3 figures

A researcher is interested in what sample size is needed to get the required significance of the same test, assuming exactly the same situation that was in the study with the non-significant result. We propose a simple solution to the problem., Comment: 14 pages including Appendix

We report a measurement of two energy-weighted gamma cascade angular distributions from polarized slow neutron capture on the ${}^{35}$Cl nucleus, one parity-odd correlation proportional to $\vec{s_{n}} \cdot \vec{k_{\gamma}}$ and one parity-even correlation proportional to $\vec{s_{n}} \cdot \vec{k_{n}} \times \vec{k_{\gamma}}$. A parity violating asymmetry can appear in this reaction due to the weak nucleon-nucleon (NN) interaction which mixes opposite parity S and P-wave levels in the excited compound $^{36}$Cl nucleus formed upon slow neutron capture. If parity-violating (PV) and parity-conserving (PC) terms both exist, the measured differential cross section can be related to them via $\frac{d\sigma}{d\Omega}\propto1+A_{\gamma, PV}\cos\theta+A_{\gamma,PC}\sin\theta$. The PV and PC asymmetries for energy-weighted gamma cascade angular distributions for polarized slow neutron capture on $^{35}$Cl averaged over the neutron energies from 2.27~meV to 9.53~meV were measured to be $A_{\gamma,PV}=(-23.9\pm0.7)\times 10^{-6}$ and $A_{\gamma,PC}=(0.1\pm0.7)\times 10^{-6}$. These results are consistent with previous experimental results. Systematic errors were quantified and shown to be small compared to the statistical error. These asymmetries in the angular distributions of the gamma rays emitted from the capture of polarized neutrons in $^{35}$Cl were used to verify the operation and data analysis procedures for the NPDGamma experiment which measured the parity-odd asymmetry in the angular distribution of gammas from polarized slow neutron capture on protons.

We provide an overview of the xFitter open-source software package, review the general capabilities of the program, and highlight applications relevant to the Snowmass study. An updated version of the program (2.2.0) is available on CERN GitLab, a and this has been updated to a C++ codebase with enhanced and extended features. We also discuss some of the ongoing and future code developments that may be useful for precision studies. We survey recent analyses performed by the xFitter developers' team including: W and Z production, photon PDFs, Drell-Yan forward-backward asymmetry studies, resummation of small-x contributions, heavy quark production, constraints on the strange PDF, determination of the pion PDF, and determination of the pion Fragmentation Functions. Finally, we briefly summarize selected applications of xFitter in the literature. The xFitter program is a versatile, flexible, modular, and comprehensive tool that can provide impact studies for possible future facilities. We encourage the use of xFitter, and welcome new contributions from the community., Comment: 20 pages, 12 figures. Contribution to the DPF Snowmass Community Planning Exercise

We present an experimental study of the turn-on delay in pulsed mid-infrared quantum cascade lasers. We report the unexpectedly long delay time depending on the pumping current, which does not agree with conventional theoretical predictions for step-like excitation. Similar behavior has been observed in InP- and InAs-based QCLs emitting near 8${\mu}$m. Numerical simulations performed using a model based on rate equations for excitation by current pulses with non-zero rise time provide fair agreement with our observations.

We consider the process of catastrophic expansion of a spacelike wormhole after a violation of its equilibrium state. The dynamics of deformation of the comoving reference frame is investigated. We show that the deformation has a very specific anisotropic feature. The statement made earlier by other authors, that in the process of expanding the wormhole connecting two universes these universes ultimately unite into one universe, is not correct. We show that the transverse size of the wormhole (its throat) increases and the length of the corridor decreases which does not correspond to the de Sitter model.

We report the first precision measurement of the parity-violating asymmetry in the direction of proton emission with respect to the neutron spin, in the reaction $^{3}\mathrm{He}(\mathrm{n},\mathrm{p})^{3}\mathrm{H}$, using the capture of polarized cold neutrons in an unpolarized active $^3\rm{He}$ target. The asymmetry is a result of the weak interaction between nucleons, which remains one of the most poorly understood aspects of electro-weak theory. The measurement provides an important benchmark for modern effective field theory (EFT) calculations. Measurements like this are necessary to determine the spin-isospin structure of the hadronic weak interaction. Our asymmetry result is $A_{PV} = \left( 1.58 \pm 0.97 ~\mathrm{(stat)} \pm 0.24~\mathrm{(sys)}\right)\times10^{-8}$, which has the smallest uncertainty of any parity-violating asymmetry measurement so far.

Significant progress has been made to experimentally determine a complete set of the parity-violating (PV) weak-interaction amplitudes between nucleons. In this paper we describe the design, construction and operation of the n$^3$He experiment that was used to measure the PV asymmetry $A_{\mathrm{PV}}$ in the direction of proton emission in the reaction $\vec{\mathrm{n}} + {^3}\mathrm{He} \rightarrow {^3}\mathrm{H} + \mathrm{p}$, using the capture of polarized cold neutrons in an unpolarized gaseous $^3\mathrm{He}$ target. This asymmetry has was recently calculated \cite{Viviani,Viviani2}, both in the traditional style meson exchange picture, and in effective field theory (EFT), including two-pion exchange. The high precision result (published separately) obtained with the experiment described herein forms an important benchmark for hadronic PV (HPV) theory in few-body systems, where precise calculations are possible. To this day, HPV is still one of the most poorly understood aspects of the electro-weak theory. The calculations estimate the size of the asymmetry to be in the range of $(-9.4 \rightarrow 3.5)\times 10^{-8}$, depending on the framework or model. The small size of the asymmetry and the small overall goal uncertainty of the experiment of $\delta A_{\mathrm{PV}} \simeq 1\times10^{-8}$ places strict requirements on the experiment, especially on the design of the target-detector chamber. In this paper we describe the experimental setup and the measurement methodology as well as the detailed design of the chamber, including results of Garfield++ and Geant4 simulations that form the basis of the chamber design and analysis. We also show data from commissioning and production and define the systematic errors that the chamber contributes to the measured $A_{\mathrm{PV}}$. We give the final uncertainty on the measurement.

The possibility of relatively fast neutron oscillations into a mirror neutron state is not excluded experimentally when a mirror magnetic field is considered. Direct searches for the disappearance of neutrons into mirror neutrons in a controlled magnetic field have previously been performed using ultracold neutrons, with some anomalous results reported. We describe a technique using cold neutrons to perform a disappearance and regeneration search, which would allow us to unambiguously identify a possible oscillation signal. An experiment using the existing General Purpose-Small Angle Neutron Scattering instrument at the High Flux Isotope Reactor at Oak Ridge National Laboratory will have the sensitivity to fully explore the parameter space of prior ultracold neutron searches and confirm or refute previous claims of observation. This instrument can also conclusively test the validity of recently suggested oscillation-based explanations for the neutron lifetime anomaly., Comment: 5 pages, 4 figures, Proceedings of the International Workshop on Particle Physics at Neutron Sources (PPNS 2018)