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Let $R$ be a ring with identity and $\delta(R)$ denote the Zhou radical of $R$. A ring $R$ is called {\it $\delta$-reversible} if for any $a$, $b \in R$, $ab = 0$ implies $ba \in \delta(R)$. In this paper, we give some properties of $\delta$-reversible rings. We examine some extensions of $\delta$-reversible rings.

The Mitsch order is already known as a natural partial order for semigroups and rings. The purpose of this paper is to further study of the Mitsch order on modules by investigating basic properties via endomorphism rings. And so this study also contribute to the results related to the orders on rings. As a module theoretic analog of the Mitsch order, we show that this order is a partial order on arbitrary modules. Among others, lattice properties of the Mitsch order and the relations between the Mitsch order and the other well-known orders, such as, the minus order, the Jones order, the direct sum order and the space pre-order on modules are studied. In particular, we prove that the minus order is the Mitsch order and we supply an example to show that the converse does not hold in general.

We present new additive results for the group inverse in a Banach algebra under certain perturbations. The upper bound of $\|(a+b)^{\#}-a^d\|$ is thereby given. These extend the main results in [X. Liu, Y. Qin and H. Wei, Perturbation bound of the group inverse and the generalized Schur complement in Banach algebra, Abstr. Appl. Anal., 2012, 22 pages. DOI:10.1155/2012/629178]., Comment: 17 pages

In this paper, we introduce a class of rings which is a generalization of reflexive rings and $J$-reversible rings. Let $R$ be a ring with identity and $J(R)$ denote the Jacobson radical of $R$. A ring $R$ is called {\it $J$-reflexive} if for any $a$, $b \in R$, $aRb = 0$ implies $bRa \subseteq J(R)$. We give some characterizations of a $J$-reflexive ring. We prove that some results of reflexive rings can be extended to $J$-reflexive rings for this general setting. We conclude some relations between $J$-reflexive rings and some related rings. We investigate some extensions of a ring which satisfies the $J$-reflexive property and we show that the $J$-reflexive property is Morita invariant., Comment: appears in Mathematica Bohemica

In this paper, we study on the primeness and semiprimeness of a Morita context related to the rings and modules. Necessary and sufficient conditions are investigated for an ideal of a Morita context to be a prime ideal and a semiprime ideal. In particular, we determine the conditions under which a Morita context is prime and semiprime.

We employ a variety of ab initio methods including Faddeev-Yakubovsky equations, No-Core Configuration Interaction Approach, Coupled-Cluster Theory and In-Medium Similarity Renormalization Group to perform a comprehensive analysis of the nucleon-deuteron elastic and breakup reactions and selected properties of light and medium-mass nuclei up to 48Ca using the recently constructed semilocal coordinate-space regularized chiral nucleon-nucleon potentials. We compare the results with those based on selected phenomenological and chiral EFT two-nucleon potentials, discuss the convergence pattern of the chiral expansion and estimate the achievable theoretical accuracy at various chiral orders using the novel approach to quantify truncation errors of the chiral expansion without relying on cutoff variation. We also address the robustness of this method and explore alternative ways to estimate the theoretical uncertainty from the truncation of the chiral expansion., Comment: 34 pages, 15 figures, 11 tables

The exotic $^9$He nucleus, which presents one of the most extreme neutron-to-proton ratios, belongs to the $N=7$ isotonic chain famous for the phenomenon of ground-state parity inversion with decreasing number of protons. Consequently, it would be expected to have an unnatural (positive) parity ground state similar to $^{11}$Be and $^{10}$Li. Despite many experimental and theoretical investigations, its structure remains uncertain. Apart from the fact that it is unbound, other properties including the spin and parity of its ground state and the very existence of additional low-lying resonances are still a matter of debate. In this work we study the properties of $^9$He by analyzing the $n+^8$He continuum in the context of the ab initio no-core shell model with continuum (NCSMC) formalism with chiral interactions as the only input. The NCSMC is a state-of-the-art approach for the ab initio description of light nuclei. With its capability to predict properties of bound states, resonances, and scattering states in a unified framework, the method is particularly well suited for the study of unbound nuclei such as $^9$He. Our analysis produces an unbound $^9$He nucleus. Two resonant states are found at the energies of ${\sim}1$ and ${\sim}3.5$ MeV, respectively, above the $n+^8$He breakup threshold. The first state has a spin-parity assignment of $J^{\pi} = {1/2}^-$ and can be associated with the ground state of $^9$He, while the second, broader state has a spin-parity of ${3/2}^-$. No resonance is found in the ${1/2}^+$ channel, only a very weak attraction. We find that the $^9$He ground-state resonance has a negative parity and thus breaks the parity-inversion mechanism found in the $^{11}$Be and $^{10}$Li nuclei of the same $N=7$ isotonic chain.

We derive a microscopic optical potential for intermediate energies using ab initio translationally invariant nonlocal one-body nuclear densities computed within the no-core shell model (NCSM) approach utilizing two- and three-nucleon chiral interactions as the only input. The optical potential is derived at first-order within the spectator expansion of the non-relativistic multiple scattering theory by adopting the impulse approximation and using the same chiral nucleon-nucleon interaction as that used to compute densities. The ground state local and nonlocal densities of 4,6,8He, 12C, and 16O are calculated and applied to optical potential construction. The differential cross sections and the analyzing powers for the elastic proton scattering off of these nuclei are then calculated for different values of the incident proton energy. The impact of nonlocality and the COM removal is discussed. The use of nonlocal densities has a substantial impact on the differential cross sections and improves agreement with experiment in comparison to results generated with the local densities especially for light nuclei.

How does nature hold together protons and neutrons to form the wide variety of complex nuclei in the universe? Describing many-nucleon systems from the fundamental theory of quantum chromodynamics has been the greatest challenge in answering this question. The chiral effective field theory description of the nuclear force now makes this possible but requires certain parameters that are not uniquely determined. Defining the nuclear force needs identification of observables sensitive to the different parametrizations. From a measurement of proton elastic scattering on $^{10}$C at TRIUMF and ab initio nuclear reaction calculations we show that the shape and magnitude of the measured differential cross section is strongly sensitive to the nuclear force prescription., Comment: 7 pages, 6 figures

The weakly bound exotic ${}^{11}$Be nucleus, famous for its ground-state parity inversion and distinct n+ ${}^{10}$Be halo structure, is investigated from first principles using chiral two- and three-nucleon forces. An explicit treatment of continuum effects is found to be indispensable. We study the sensitivity of the ${}^{11}$Be spectrum to the details of the three-nucleon force and demonstrate that only certain chiral interactions are capable of reproducing the parity inversion. With such interactions, the extremely large E1 transition between the bound states is reproduced. We compare our photodisintegration calculations to conflicting experimental data and predict a distinct dip around the $3/2^-_1$ resonance energy. Finally, we predict low-lying $3/2^+$ and $9/2^+$ resonances that are not or not sufficiently measured in experiments., Comment: 6 pages, 5 figures, plus Supplemental Material

We present a nucleus-dependent valence-space approach for calculating ground and excited states of nuclei, which generalizes the shell-model in-medium similarity renormalization group to an ensemble reference with fractionally filled orbitals. Because the ensemble is used only as a reference, and not to represent physical states, no symmetry restoration is required. This allows us to capture 3N forces among valence nucleons with a valence-space Hamiltonian specifically targeted to each nucleus of interest. Predicted ground-state energies from carbon through nickel agree with results of other large-space ab initio methods, generally to the 1\% level. In addition, we show that this new approach is required in order to obtain convergence for nuclei in the upper $p$ and $sd$ shells. Finally, we address the $1^+$/$3^+$ ground-state inversion problem in $^{22}\text{Na}$ and $^{46}\text{V}$. This approach extends the reach of ab initio nuclear structure calculations to essentially all light- and medium-mass nuclei., Comment: 6 pages, 5 figures

Starting from a set of different two- and three-nucleon interactions from chiral effective field theory, we use the importance-truncated no-core shell model for ab initio calculations of excitation energies as well as electric quadrupole (E2) and magnetic dipole (M1) moments and transition strengths for selected p-shell nuclei. We explore the sensitivity of the excitation energies to the chiral interactions as a first step towards and systematic uncertainty propagation from chiral inputs to nuclear structure observables. The uncertainty band spanned by the different chiral interactions is typically in agreement with experimental excitation energies, but we also identify observables with notable discrepancies beyond the theoretical uncertainty that reveal insufficiencies in the chiral interactions. For electromagnetic observables we identify correlations among pairs of E2 or M1 observables based on the ab initio calculations for the different interactions. We find extremely robust correlations for E2 observables and illustrate how these correlations can be used to predict one observable based on an experimental datum for the second observable. In this way we circumvent convergence issues and arrive at far more accurate results than any direct ab initio calculation. A prime example for this approach is the quadrupole moment of the first 2^+ state in C-12, which is predicted with an drastically improved accuracy., Comment: 11 pages, 8 figures

The description of nuclei starting from the constituent nucleons and the realistic interactions among them has been a long-standing goal in nuclear physics. In addition to the complex nature of the nuclear forces, with two-, three- and possibly higher many-nucleon components, one faces the quantum-mechanical many-nucleon problem governed by an interplay between bound and continuum states. In recent years, significant progress has been made in ab initio nuclear structure and reaction calculations based on input from QCD-employing Hamiltonians constructed within chiral effective field theory. After a brief overview of the field, we focus on ab initio many-body approaches - built upon the No-Core Shell Model - that are capable of simultaneously describing both bound and scattering nuclear states, and present results for resonances in light nuclei, reactions important for astrophysics and fusion research. In particular, we review recent calculations of resonances in the $^6$He halo nucleus, of five- and six-nucleon scattering, and an investigation of the role of chiral three-nucleon interactions in the structure of $^9$Be. Further, we discuss applications to the $^7$Be$(p,\gamma)^8$B radiative capture. Finally, we highlight our efforts to describe transfer reactions including the $^3$H$(d,n)^4$He fusion., Comment: Contribution to the Special Physica Scripta Edition - 40 year anniversary - Nobel Prize '75, 71 pages, 29 figures

We discuss the approximate inclusion of three-nucleon interactions into ab initio nuclear structure calculations using a multi-reference formulation of normal ordering and Wick's theorem. Following the successful application of single-reference normal ordering for the study of ground states of closed-shell nuclei, e.g., in coupled-cluster theory, multi-reference normal ordering opens a path to open-shell nuclei and excited states. Based on different multi-determinantal reference states we benchmark the truncation of the normal-ordered Hamiltonian at the two-body level in no-core shell-model calculations for p-shell nuclei, including 6-Li, 12-C, and 10-B. We find that this multi-reference normal-ordered two-body approximation is able to capture the effects of the 3N interaction with sufficient accuracy, both, for ground-state and excitation energies, at the computational cost of a two-body Hamiltonian. It is robust with respect to the choice of reference states and has a multitude of applications in ab initio nuclear structure calculations of open-shell nuclei and their excitations as well as in nuclear reaction studies., Comment: 6 pages, 4 figures, v2: update to published version

An {\em ab initio} (i.e., from first principles) theoretical framework capable of providing a unified description of the structure and low-energy reaction properties of light nuclei is desirable to further our understanding of the fundamental interactions among nucleons, and provide accurate predictions of crucial reaction rates for nuclear astrophysics, fusion-energy research, and other applications. In this contribution we review {\em ab initio} calculations for nucleon and deuterium scattering on light nuclei starting from chiral two- and three-body Hamiltonians, obtained within the framework of the {\em ab initio} no-core shell model with continuum. This is a unified approach to nuclear bound and scattering states, in which square-integrable energy eigenstates of the $A$-nucleon system are coupled to $(A-a)+a$ target-plus-projectile wave functions in the spirit of the resonating group method to obtain an efficient description of the many-body nuclear dynamics both at short and medium distances and at long ranges., Comment: 9 pages, 5 figures, proceedings of the 21st International Conference on Few-Body Problems in Physics

There has been significant recent progress in solving the long-standing problems of how nuclear shell structure and collective motion emerge from underlying microscopic inter-nucleon interactions. We review a selection of recent significant results within the ab initio No Core Shell Model (NCSM) closely tied to three major factors enabling this progress: (1) improved nuclear interactions that accurately describe the experimental two-nucleon and three-nucleon interaction data; (2) advances in algorithms to simulate the quantum many-body problem with strong interactions; and (3) continued rapid development of high-performance computers now capable of performing $20 \times 10^{15}$ floating point operations per second. We also comment on prospects for further developments., Comment: Invited paper presented at NTSE-2014 and published online in the proceedings (see footnote on p.1)

In this paper, we introduce a class of quasipolar rings which is a generalization of $J$-quasipolar rings. Let $R$ be a ring with identity. An element $a \in R$ is called {\it $\delta$-quasipolar} if there exists $p^2 = p\in comm^2(a)$ such that $a + p$ is contained in $\delta(R)$, and the ring $R$ is called {\it $\delta$-quasipolar} if every element of $R$ is $\delta$-quasipolar. We use $\delta$-quasipolar rings to extend some results of $J$-quasipolar rings. Then some of the main results of $J$-quasipolar rings are special cases of our results for this general setting. We give many characterizations and investigate general properties of $\delta$-quasipolar rings., Comment: Submitted for publication

In this paper, we introduce a class of $J$-quasipolar rings. Let $R$ be a ring with identity. An element $a$ of a ring $R$ is called {\it weakly $J$-quasipolar} if there exists $p^2 = p\in comm^2(a)$ such that $a + p$ or $a-p$ are contained in $J(R)$ and the ring $R$ is called {\it weakly $J$-quasipolar} if every element of $R$ is weakly $J$-quasipolar. We give many characterizations and investigate general properties of weakly $J$-quasipolar rings. If $R$ is a weakly $J$-quasipolar ring, then we show that (1) $R/J(R)$ is weakly $J$-quasipolar, (2) $R/J(R)$ is commutative, (3) $R/J(R)$ is reduced. We use weakly $J$-quasipolar rings to obtain more results for $J$-quasipolar rings. We prove that the class of weakly $J$-quasipolar rings lies between the class of $J$-quasipolar rings and the class of quasipolar rings. Among others it is shown that a ring $R$ is abelian weakly $J$-quasipolar if and only if $R$ is uniquely clean., Comment: Submitted for publication

We apply improved nucleon-nucleon potentials up to fifth order in chiral effective field theory, along with a new analysis of the theoretical truncation errors, to study nucleon-deuteron (Nd) scattering and selected low-energy observables in 3H, 4He, and 6Li. Calculations beyond second order differ from experiment well outside the range of quantified uncertainties, providing truly unambiguous evidence for missing three-nucleon forces within the employed framework. The sizes of the required three-nucleon force contributions agree well with expectations based on Weinberg's power counting. We identify the energy range in elastic Nd scattering best suited to study three-nucleon force effects and estimate the achievable accuracy of theoretical predictions for various observables., Comment: 5 pages, 5 figures

This paper describes some applications of GPU acceleration in ab initio nuclear structure calculations. Specifically, we discuss GPU acceleration of the software package MFDn, a parallel nuclear structure eigensolver. We modify the matrix construction stage to run partly on the GPU. On the Titan supercomputer at the Oak Ridge Leadership Computing Facility, this produces a speedup of approximately 2.2x - 2.7x for the matrix construction stage and 1.2x - 1.4x for the entire run., Comment: 9 pages, 4 figures

We extend the recently proposed ab initio no-core shell model with continuum to include three-nucleon (3N) interactions beyond the few-body domain. The extended approach allows for the assessment of effects of continuum degrees of freedom as well as of the 3N force in ab initio calculations of structure and reaction observables of p- and lower-sd-shell nuclei. As first application we concentrate on energy levels of the 9Be system for which all excited states lie above the n-8Be threshold. For all energy levels, the inclusion of the continuum significantly improves the agreement with experiment, which was an issue in standard no-core shell model calculations. Furthermore, we find the proper treatment of the continuum indispensable for reliable statements about the quality of the adopted 3N interaction from chiral effective field theory. In particular, we find the 1/2+ resonance energy, which is of astrophysical interest, in good agreement with experiment., Comment: 6 pages, 5 figures

We solve three-nucleon Faddeev equations with nucleon-nucleon and three-nucleon forces derived consistently in the framework of chiral perturbation theory at next-to-next-to-next-to-leading order in the chiral expansion. In this first investigation we include only matrix elements of the three-nucleon force for partial waves with the total two-nucleon (three-nucleon) angular momenta up to 3 (5/2). Low-energy neutron-deuteron elastic scattering and deuteron breakup reaction are studied. Emphasis is put on Ay puzzle in elastic scattering and cross sections in symmetric-space-star and neutron-neutron quasi-free-scattering breakup configurations, for which large discrepancies between data and theory have been reported., Comment: 22 pages, 7 figures

We use the newly developed Multi-Reference In-Medium Similarity Renormalization Group to study all even isotopes of the calcium and nickel isotopic chains, based on two- plus three-nucleon interactions derived from chiral effective field theory. We present results for ground-state and two-neutron separation energies and quantify their theoretical uncertainties. At shell closures, we find excellent agreement with Coupled Cluster results obtained with the same Hamiltonians. Our results highlight the importance of the chiral 3N interaction to obtain a correct reproduction of experimental energy trends, and their subtle impact on the location of the neutron drip lines in the Ca and Ni chains. At the same time, we uncover and discuss deficiencies of the input Hamiltonians which need to be addressed by the next generation of chiral interactions., Comment: 7 pages, 5 figures

We report ab initio calculations for neutron drops in a 10 MeV external harmonic-oscillator trap using chiral nucleon-nucleon plus three-nucleon interactions. We present total binding energies, internal energies, radii and odd-even energy differences for neutron numbers N = 2 - 18 using the no-core shell model with and without importance truncation. Furthermore, we present total binding energies for N = 8, 16, 20, 28, 40, 50 obtained in a coupled-cluster approach. Comparisons with Green's Function Monte Carlo results, where available, using Argonne v8' with three-nucleon interactions reveal important dependences on the chosen Hamiltonian., Comment: 7 pages, 5 figures

We investigate selected static and transition properties of $^{12}$C using ab initio No-Core Shell Model (NCSM) methods with chiral two- and three-nucleon interactions. We adopt the Similarity Renormalization Group (SRG) to assist convergence including up to three-nucleon (3N) contributions. We examine the dependences of the $^{12}$C observables on the SRG evolution scale and on the model-space parameters. We obtain nearly converged low-lying excitation spectra. We compare results of the full NCSM with the Importance Truncated NCSM in large model spaces for benchmarking purposes. We highlight the effects of the chiral 3N interaction on several spectroscopic observables. The agreement of some observables with experiment is improved significantly by the inclusion of 3N interactions, e.g., the B(M1) from the first $J^\pi T = 1^+ 1$ state to the ground state. However, in some cases the agreement deteriorates, e.g., for the excitation energy of the first $1^+ 0$ state, leaving room for improved next-generation chiral Hamiltonians., Comment: 11 pages, 9 figures

We present the first ab initio calculations for p-shell single-Lambda hypernuclei. For the solution of the many-baryon problem, we develop two variants of the no-core shell model with explicit $\Lambda$ and $\Sigma^+$, $\Sigma^0$, $\Sigma^-$ hyperons including $\Lambda$-$\Sigma$ conversion, optionally supplemented by a similarity renormalization group transformation to accelerate model-space convergence. In addition to state-of-the-art chiral two- and three-nucleon interactions, we use leading-order chiral hyperon-nucleon interactions and a recent meson-exchange hyperon-nucleon interaction. We validate the approach for s-shell hypernuclei and apply it to p-shell hypernuclei, in particular to $^7_\Lambda$Li, $^9_\Lambda$Be and $^{13}_\Lambda$C. We show that the chiral hyperon-nucleon interactions provide ground-state and excitation energies that agree with experiment within the cutoff dependence. At the same time we demonstrate that hypernuclear spectroscopy provides tight constraints on the hyperon-nucleon interactions and we discuss the impact of induced hyperon-nucleon-nucleon interactions., Comment: 6 pages, 4 figures

We present the first ab initio construction of valence-space Hamiltonians for medium-mass nuclei based on chiral two- and three-nucleon interactions using the in-medium similarity renormalization group. When applied to the oxygen isotopes, we find experimental ground-state energies are well reproduced, including the flat trend beyond the drip line at 24O. Similarly, natural-parity spectra in 21,22,23,24O are in agreement with experiment, and we present predictions for excited states in 25,26O. The results exhibit a weak dependence on the harmonic-oscillator basis parameter and reproduce spectroscopy within the standard sd valence space., Comment: 6 pages, 5 figures, published version

The description of structural and dynamical properties of nuclei starting from the fundamental interaction between nucleons has been a long-standing goal in nuclear physics. The ab initio No-Core Shell Model combined with the Resonating-Group Method (NCSM/RGM) is capable of addressing both structural and reaction properties of light-nuclei. While promising results have already been achieved starting from a two-body Hamiltonian, a truly realistic prediction of nuclear observables requires the treatment of the three-nucleon interaction. Using similarity-renormalization-group evolved two- and three-nucleon interactions, we will present recent applications to n-4He scattering process when accounting for the chiral two- plus three-nucleon interaction versus the chiral two-nucleon interaction. We compare our results to phase shifts obtained from R-matrix analysis of data up to 16 MeV neutron energy, below the d-3H threshold., Comment: 4 pages, 2 figures, Proceedings of the 22nd European Conference on Few-Body Problems in Physics, Krakow, Poland, 9-13 September 2013

We present the first ab initio calculations of nuclear ground states up into the domain of heavy nuclei, spanning the range from 16-O to 132-Sn based on two- plus three-nucleon interactions derived within chiral effective field theory. We employ the similarity renormalization group for preparing the Hamiltonian and use coupled-cluster theory to solve the many-body problem for nuclei with closed sub-shells. Through an analysis of theoretical uncertainties resulting from various truncations in this framework, we identify and eliminate the technical hurdles that previously inhibited the step beyond medium-mass nuclei, allowing for reliable validations of nuclear Hamiltonians in the heavy regime. Following this path we show that chiral Hamiltonians qualitatively reproduce the systematics of nuclear ground-state energies up to the neutron-rich Sn isotopes., Comment: 5 pages, 5 figures

We discuss the building blocks for a consistent inclusion of chiral three-nucleon (3N) interactions into ab initio nuclear structure calculations beyond the lower p-shell. We highlight important technical developments, such as the similarity renormalization group (SRG) evolution in the 3N sector, a JT-coupled storage scheme for 3N matrix elements with efficient on-the-fly decoupling, and the importance truncated no-core shell model with 3N interactions. Together, these developments make converged ab initio calculations with explicit 3N interactions possible also beyond the lower p-shell. We analyze in detail the impact of various truncations of the SRG-evolved Hamiltonian, in particular the truncation of the harmonic-oscillator model space used for solving the SRG flow equations and the omission of the induced beyond-3N contributions of the evolved Hamiltonian. Both truncations lead to sizable effects in the upper p-shell and beyond and we present options to remedy these truncation effects. The analysis of the different truncations is a first step towards a systematic uncertainty quantification of all stages of the calculation., Comment: 21 pages, 16 figures, 2 tables

We generalize the coupled-cluster (CC) approach with singles, doubles, and the non-iterative treatment of triples termed $\Lambda$CCSD(T) to Hamiltonians containing three-body interactions. The resulting method and the underlying CC approach with singles and doubles only (CCSD) are applied to the medium-mass closed-shell nuclei O16, O24, and Ca40. By comparing the results of CCSD and $\Lambda$CCSD(T) calculations with explicit treatment of three-nucleon interactions to those obtained using an approximate treatment in which they are included effectively via the zero-, one-, and two-body components of the Hamiltonian in normal-ordered form, we quantify the contributions of the residual three-body interactions neglected in the approximate treatment. We find these residual normal-ordered three-body contributions negligible for the $\Lambda$CCSD(T) method, although they can become significant in the lower-level CCSD approach, particularly when the nucleon-nucleon interactions are soft., Comment: 21 pages, 3 figures

We extend the ab initio no-core shell model/resonating-group method to include three-nucleon (3N) interactions for the description of nucleon-nucleus collisions. We outline the formalism, give algebraic expressions for the 3N-force integration kernels, and discuss computational aspects of two alternative implementations. The extended theoretical framework is then applied to nucleon-4He scattering using similarity-renormalization-group (SRG) evolved nucleon-nucleon plus three-nucleon potentials derived from chiral effective field theory. We analyze the convergence properties of the calculated phase shifts and explore their dependence upon the SRG evolution parameter. We include up to six excited states of the 4He target and find significant effects from the inclusion of the chiral 3N force, e.g., it enhances the spin-orbit splitting between the 3/2- and 1/2- resonances and leads to an improved agreement with the phase shifts obtained from an accurate R-matrix analysis of the five-nucleon experimental data. We find remarkably good agreement with measured differential cross sections at various energies, while analyzing powers manifest larger deviations from experiment for certain energies and angles., Comment: 18 pages, 12 figures