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Let $G$ be a group coacting on an Artin-Schelter regular algebra $A$ homogeneously and inner-faithfully. When the identity component $A_e$ is also Artin-Schelter regular, providing a generalization of the Shephard-Todd-Chevalley Theorem, we say that $G$ is a dual reflection group for $A$. We give two examples of dual reflection groups of order 16, and study algebraic and geometric properties of three associated Artin-Schelter regular algebras of dimension four., Comment: Comments welcome

All checkerboard surfaces for a given knot in $S^3$ are related by isotopy and "kinking" and "unkinking" moves, which change the surfaces' Goeritz matrices like this: $G\leftrightarrow G\oplus [\pm1]=\left[\begin{smallmatrix} G&\mathbf{0}\\ \mathbf{0}^T&\pm1 \end{smallmatrix}\right]$. We call two symmetric integer matrices "kink-equivalent" if they are related by "kinking'' and "unkinking'' moves $G\leftrightarrow G\oplus [\pm1]$ and unimodular congruence. We prove constructively that every nonsingular symmetric integer matrix is kink-equivalent to a positive-definite matrix and to a negative-definite matrix, and we give bounds on the number of moves required. This has several implications, e.g. every knot in $S^3$ is "alternating up to fake unkinking moves" and every simply connected, closed, topological 4-manifold with nonsingular intersection pairing has a positive blow-up that is homeomorphic to a negative blow-up of a positive-definite, simply connected, closed, topological 4-manifold., Comment: 19 pages, 9 figures, comments welcome

For a finite group $G$ and $\Bbbk$ an algebraically closed field of characteristic zero we consider Artin-Schelter regular algebras $A$ on which the Drinfeld double $D(G)$ acts inner faithfully, and its associated algebras of invariants $A^{D(G)}$. Explicit computations for the cases when $G$ is the (generalized) quaternion group of order 8 and 16 are given.

Let $(R,\mathfrak{m},\Bbbk)$ be a regular local ring of dimension 3. Let $I$ be a Gorenstein ideal of $R$ of grade 3. It follows from a result of Buchsbaum and Eisenbud that there is a skew-symmetric matrix of odd size such that $I$ is generated by the sub-maximal pfaffians of this matrix. Let $J$ be the ideal obtained by multiplying some of the pfaffian generators of $I$ by $\mathfrak{m}$; we say that $J$ is a trimming of $I$. In a previous work, the first author and A. Hardesty constructed an explicit free resolution of $R/J$ and computed a DG algebra structure on this resolution. They utilized these products to analyze the Tor algebra of such trimmed ideals. Missing from their result was the case where $I$ is five generated. In this paper we address this case.

Let $\Bbbk$ be a field and let $I$ be a monomial ideal in the polynomial ring $Q=\Bbbk[x_1,\ldots,x_n]$. In her thesis, Taylor introduced a complex which provides a finite free resolution for $Q/I$ as a $Q$-module. Later, Gemeda constructed a differential graded structure on the Taylor resolution. More recently, Avramov showed that this differential graded algebra admits divided powers. We generalize each of these results to monomial ideals in a skew polynomial ring $R$. Under the hypothesis that the skew commuting parameters defining $R$ are roots of unity, we prove as an application that as $I$ varies among all ideals generated by a fixed number of monomials of degree at least two in $R$, there is only a finite number of possibilities for the Poincar\'{e} series of $\Bbbk$ over $R/I$ and for the isomorphism classes of the homotopy Lie algebra of $R/I$ in cohomological degree larger or equal to two.

In this article we study a theory of support varieties over a skew complete intersection $R$, i.e. a skew polynomial ring modulo an ideal generated by a sequence of regular normal elements. We compute the derived braided Hochschild cohomology of $R$ relative to the skew polynomial ring and show its action on $\mathrm{Ext}_R(M,N)$ is noetherian for finitely generated $R$-modules $M$ and $N$ respecting the braiding of $R$. When the parameters defining the skew polynomial ring are roots of unity we use this action to define a support theory. In this setting applications include a proof of the Generalized Auslander-Reiten Conjecture and that $R$ possesses symmetric complexity.

The La-based '214' cuprates host several symmetry breaking phases including superconductivity, charge and spin order in the form of stripes, and a structural othorhombic-to-tetragonal phase transition. Therefore, these materials are an ideal system to study the effects of uniaxial stress onto the various correlations that pervade the cuprate phase diagram. We report resonant x-ray scattering experiments on $\textrm{La}_{1.475}\textrm{Nd}_{0.4}\textrm{Sr}_{0.125}\textrm{Cu}\textrm{O}_{4}$ (LNSCO-125) that reveal a significant response of charge stripes to uniaxial tensile-stress of $\sim$ 0.1 GPa. These effects include a reduction of the onset temperature of stripes by $\sim$ 50 K, a 29 K reduction of the low-temperature orthorhombic-to-tetragonal transition, competition between charge order and superconductivity, and a preference for stripes to form along the direction of applied stress. Altogether, we observe a dramatic response of the electronic properties of LNSCO-125 to a modest amount of uniaxial stress., Comment: 6 pages, 3 figures, Supplemental material available upon request

Comets are small bodies thought to contain the most pristine material in the solar system. However, since their formation 4.5 Gy ago, they have been altered by different processes. While not exposed to much electromagnetic radiation, they experience intense particle radiation. Galactic cosmic rays and solar energetic particles have a broad spectrum of energies and interact with the cometary surface and subsurface; they are the main source of space weathering for a comet in the Kuiper Belt or in the Oort cloud; and also affect the ice prior to the comet agglomeration. While low energy particles interact only with the cometary surface, the most energetic ones deposit a significant amount of energy down to tens of meters. This interaction can modify the isotopic ratios in cometary ices and create secondary compounds through radiolysis, such as O2 and H2O2 (paper II: Maggiolo et al., 2020). In this paper, we model the energy deposition of energetic particles as a function of depth using a Geant4 application modified to account for the isotope creation process. We quantify the energy deposited in cometary nucleus by galactic cosmic rays and solar energetic particles. The consequences of the energy deposition on the isotopic and chemical composition of cometary ices and their implication on the interpretation of cometary observations, notably of 67P/Churyumov Gerasimenko by the ESA/Rosetta spacecraft, will be discussed in Paper II.

The $\beta^-$ decay of $^{47}$K to $^{47}$Ca is an appropriate mechanism for benchmarking interactions spanning the $sd$ and $pf$ shells, but current knowledge of the $\beta^-$-decay scheme is limited. We have performed a high-resolution, high-efficiency study of the $\beta^-$-decay of $^{47}$K with the GRIFFIN spectrometer at TRIUMF-ISAC. The study revealed 48 new transitions, a more precise value for the $^{47}$K half-life (17.38(3)~s), and new spin and parity assignments for eight excited states. Levels placed for the first time here raise the highest state observed in $\beta^-$ decay to within 568(3) keV of the $Q$-value and confirm the previously measured large $\beta^-$-decay branching ratios to the low-lying states. Previously unobserved $\beta^-$-feeding to 3/2$^+$ states between 4.5 and 6.1~MeV excitation energy was identified with a total $\beta^-$-feeding intensity of 1.29(2)\%. The sum of the $B(GT)$ values for these states indicates that the $1s_{1/2}$ proton hole strength near this excitation energy is comparable to the previously known $1s_{1/2}$ proton and neutron hole strengths near 2.6 MeV.

This study presents two different machine learning approaches for the modeling of hydrodynamic force on particles in a particle-laden multiphase flow. Results from particle-resolved direct numerical simulations (PR-DNS) of flow over a random array of stationary particles for eight combinations of particle Reynolds number ($Re$) and volume fraction ($\phi$) are used in the development of the models. The first approach follows a two step process. In the first flow prediction step, the perturbation flow due to a particle is obtained as an axisymmetric superposable wake using linear regression. In the second force prediction step, the force on a particle is evaluated in terms of the perturbation flow induced by all its neighbors using the generalized Fax\'en form of the force expression. In the second approach, the force data on all the particles from the PR-DNS simulations is used to develop an artificial neural network (ANN) model for direct prediction of force on a particle. Due to the unavoidable limitation on the number of fully resolved particles in the PR-DNS simulations, direct force prediction with the ANN model tends to over-fit the data and performs poorly in the prediction of test data. In contrast, due to the millions of grid points used in the PR-DNS simulations, accurate flow prediction is possible, which then allows accurate prediction of particle force. This hybridization of multiphase physics and machine learning is particularly important, since it blends the strength of each, and the resulting pairwise interaction extended point-particle (PIEP) model cannot be developed by either physics or machine learning alone.

Fluid flow around a random distribution of stationary spherical particles is a problem of substantial importance in the study of dispersed multiphase flows. In this paper we present a machine learning methodology using Generative Adversarial Network framework and Convolutional Neural Network architecture to recreate particle-resolved fluid flow around a random distribution of monodispersed particles. The model was applied to various Reynolds number and particle volume fraction combinations spanning over a range of [2.69, 172.96] and [0.11, 0.45] respectively. Test performance of the model for the studied cases is very promising.

We report on high-statistics data from the $\beta^-$ decay of the $^{46}$K $J^{\pi}$ = 2$^-$ ground state taken with the GRIFFIN spectrometer located at the TRIUMF-ISAC facility. In total, 199 $\gamma$ rays and 42 excited states were placed in the level scheme, and from the observed $\beta$ feeding and angular correlations of pairs of cascading $\gamma$ rays, it was possible to assign spins and parities to excited states and determine mixing ratios for selected $\gamma$ rays. The level structure of $^{46}$Ca is compared to theoretical predictions from a microscopic valence-space Hamiltonian derived from two- (NN) and three-nucleon (3N) forces. These calculations are in reasonable agreement with the experimental data and indicate that the protons in this region are not as inert as would be expected for semi-magic nuclei.

For each nontrivial semisimple Hopf algebra $H$ of dimension sixteen over $\mathbb{C}$, the smallest dimension inner-faithful representation of $H$ acting on a quadratic AS regular algebra $A$ of dimension 2 or 3, homogeneously and preserving the grading, is determined. Each invariant subring $A^H$ is determined. When $A^H$ is also AS regular, thus providing a generalization of the Chevalley-Shephard-Todd Theorem, we say that $H$ is a reflection Hopf algebra for $A$., Comment: Revised version. To appear in Algebras and Representation Theory

We present two noncommutative algebras over a field of characteristic zero that each posses a family of actions by cyclic groups of order $2n$, represented in $n \times n$ matrices, requiring generators of degree $3n$.

Differential graded algebra techniques have played a crucial role in the development of homological algebra, especially in the study of homological properties of commutative rings carried out by Serre, Tate, Gulliksen, Avramov, and others. In this article, we extend the construction of the Koszul complex and acyclic closure to a more general setting. As an application of our constructions, we shine some light on the structure of the Ext algebra of quotients of skew polynomial rings by ideals generated by normal elements. As a consequence, we give a presentation of the Ext algebra when the elements generating the ideal form a regular sequence, generalizing a theorem of Bergh and Oppermann. It follows that in this case the Ext algebra is noetherian, providing a partial answer to a question of Kirkman, Kuzmanovich and Zhang.

We apply the bi-moment determinant method to compute a representation of the matrix product algebra -- a quadratic algebra satisfied by the operators $\mathbf{d}$ and $\mathbf{e}$ -- for the five parameter ($\alpha$, $\beta$, $\gamma$, $\delta$ and $q$) Asymmetric Simple Exclusion Process. This method requires an $LDU$ decomposition of the ``bi-moment matrix''. The decomposition defines a new pair of basis vectors sets, the `boundary basis'. This basis is defined by the action of polynomials $\{P_n\}$ and $\{Q_n\}$ on the quantum oscillator basis (and its dual). Theses polynomials are orthogonal to themselves (ie.\ each satisfy a three term recurrence relation) and are orthogonal to each other (with respect to the same linear functional defining the stationary state). Hence termed `bi-orthogonal'. With respect to the boundary basis the bi-moment matrix is diagonal and the representation of the operator $\mathbf{d}+\mathbf{e}$ is tri-diagonal. This tri-diagonal matrix defines another set of orthogonal polynomials very closely related to the the Askey-Wilson polynomials (they have the same moments).

Let $(Q,\mathfrak n,k)$ be a commutative local Noetherian ring, $f_1,\dots, f_c$ a $Q$-regular sequence in $\mathfrak n$, and $R=Q/(f_1,\dots,f_c)$. Given a complex of finitely generated free $R$-modules, we give a construction of a complex of finitely generated free $Q$-modules having the same homology. A key application is when the original complex is an $R$-free resolution of a finitely generated $R$-module. In this case our construction is a sort of converse to a construction of Eisenbud-Shamash yielding a free resolution of an $R$-module $M$ over $R$ given one over $Q$.

Let $H$ be a semisimple Hopf algebra acting on an Artin-Schelter regular algebra $A$, homogeneously, inner-faithfully, preserving the grading on $A$, and so that $A$ is an $H$-module algebra. When the fixed subring $A^H$ is also AS regular, thus providing a generalization of the Chevalley-Shephard-Todd Theorem, we say that $H$ is a reflection Hopf algebra for $A$. We show that each of the semisimple Hopf algebras $H_{2n^2}$ of Pansera, and $\mathcal{A}_{4m}$ and $\mathcal{B}_{4m}$ of Masuoka is a reflection Hopf algebra for an AS regular algebra of dimension 2 or 3., Comment: Some minor corrections

We establish a version of Kn\"{o}rrer's Periodicity Theorem in the context of noncommutative invariant theory. Namely, let $A$ be a left noetherian AS-regular algebra, let $f$ be a normal and regular element of $A$ of positive degree, and take $B=A/(f)$. Then there exists a bijection between the set of isomorphism classes of indecomposable non-free maximal Cohen-Macaulay modules over $B$ and those over (a noncommutative analog of) its second double branched cover $(B^\#)^\#$. Our results use and extend the study of twisted matrix factorizations, which was introduced by the first three authors with Cassidy. These results are applied to the noncommutative Kleinian singularities studied by the second and fourth authors with Chan and Zhang., Comment: Numerous typos fixed, removed unnecessary finite order hypothesis

Gamma-Ray Infrastructure For Fundamental Investigations of Nuclei, GRIFFIN, is a new high-efficiency $\gamma$-ray spectrometer designed for use in decay spectroscopy experiments with low-energy radioactive ion beams provided by TRIUMF's Isotope Separator and Accelerator (ISAC-I) facility. GRIFFIN is composed of sixteen Compton-suppressed large-volume clover-type high-purity germanium (HPGe) $\gamma$-ray detectors combined with a suite of ancillary detection systems and coupled to a custom digital data acquisition system. The infrastructure and detectors of the spectrometer as well as the performance characteristics and the analysis techniques applied to the experimental data are described.

It is currently unknown how common life is on exoplanets, or how long planets can remain viable for life. To date, we have a superficial notion of habitability, a necessary first step, but so far lacking an understanding of the detailed interaction between stars and planets over geological timescales, dynamical evolution of planetary systems, and atmospheric evolution on planets in other systems. A planet mass, net insolation, and atmospheric composition alone are insufficient to determine the probability that life on a planet could arise or be detected. The latter set of planetary considerations, among others, underpin the concept of the habitable zone (HZ), defined as the circumstellar region where standing bodies of liquid water could be supported on the surface of a rocky planet. However, stars within the same spectral class are often treated in the same way in HZ studies, without any regard for variations in activity among individual stars. Such formulations ignore differences in how nonthermal emission and magnetic energy of transient events in different stars affect the ability of an exoplanet to retain its atmosphere.In the last few years there has been a growing appreciation that the atmospheric chemistry, and even retention of an atmosphere in many cases, depends critically on the high-energy radiation and particle environments around these stars. Indeed, recent studies have shown stellar activity and the extreme space weather, such as that created by the frequent flares and coronal mass ejections (CMEs) from the active stars and young Sun, may have profoundly affected the chemistry and climate and thus habitability of the early Earth and terrestrial type exoplanets. The goal of this white paper is to identify and describe promising key research goals to aid the field of the exoplanetary habitability for the next 20 years., Comment: 6 pages, the white paper submitted to the US National Academy of Sciences call on Exoplanet Science Strategy

When $A = \mathbb{k}[x_1, \ldots, x_n]$ and $G$ is a small subgroup of $\operatorname{GL}_n(\mathbb{k})$, Auslander's Theorem says that the skew group algebra $A \# G$ is isomorphic to $\operatorname{End}_{A^G}(A)$ as graded algebras. We prove a generalization of Auslander's Theorem for permutation actions on $(-1)$-skew polynomial rings, $(-1)$-quantum Weyl algebras, three-dimensional Sklyanin algebras, and a certain graded down-up algebra. We also show that certain fixed rings $A^G$ are graded isolated singularities in the sense of Ueyama.

Let $R$ be a commutative local ring. We study the subcategory of the homotopy category of $R$-complexes consisting of the totally acyclic $R$-complexes. In particular, in the context where $Q\to R$ is a surjective local ring homomorphism such that $R$ has finite projective dimension over $Q$, we define an adjoint pair of functors between the homotopy category of totally acyclic $R$-complexes and that of $Q$-complexes, which are analogous to the classical adjoint pair between the module categories of $R$ and $Q$. We give detailed proofs of the adjunction in terms of the unit and counit. As a consequence, one obtains a precise notion of approximations of totally acyclic $R$-complexes by totally acyclic $Q$-complexes.

We provide formulas for computing the discriminant of noncommutative algebras over central subalgebras in the case of Ore extensions and skew group extensions. The formulas follow from a more general result regarding the discriminants of certain twisted tensor products. We employ our formulas to compute automorphism groups for examples in each case.

The notion of a matrix factorization was introduced by Eisenbud in the commutative case in his study of bounded (periodic) free resolutions over complete intersections. In this work, we extend the notion of (homogeneous) matrix factorizations to regular normal elements of connected graded algebras over a field. Next, we relate the category of twisted matrix factorizations to an element over a ring and certain Zhang twists. We also show that in the AS-regular setting, every sufficiently high syzygy module is the cokernel of some twisted matrix factorization. Furthermore, we show that in this setting there is an equivalence of categories between the homotopy category of twisted matrix factorizations and the singularity category of the hypersurface, following work of Orlov.

We give a short overview of the DarkLight detector concept which is designed to search for a heavy photon A' with a mass in the range 10 MeV/c^2 < m(A') < 90 MeV/c^2 and which decays to lepton pairs. We describe the intended operating environment, the Jefferson Laboratory free electon laser, and a way to extend DarkLight's reach using A' --> invisible decays., Comment: 8 pages, 4 figures, contributed to the Community Summer Study 2013 "Snowmass on the Mississippi" in the New, Light, Weakly Coupled Particles (NLWCP) subgroup of the Intensity Frontier

Let A be a connected graded noncommutative monomial algebra. We associate to A a finite graph \Gamma(A) called the CPS graph of A. Finiteness properties of the Yoneda algebra Ext_A(k,k) including Noetherianity, finite GK dimension, and finite generation are characterized in terms of \Gamma(A). We show these properties, notably finite generation, can be checked by means of a terminating algorithm.

Let Z_K be the moment angle complex associated to a simplicial complex K, with the canonical torus T-action. In this paper, we prove that, for any possibly disconnected subgroup G of T, G-equivariant cohomology of Z_K over the integer Z is isomophic to the Tor-module Tor_{H(BR;Z)}(Z[K],Z) as graded modules, where Z_[K] is the Stanley-Reisner ring of K. Based on this, we prove that the surjectivity of the natural map H_T(Z_K;Z) to H_G(Z_K;Z) is equivalent to the vanishing of Tor^{H(BR;Z)}_1(Z[K],Z). Since the integral cohomology of various toric orbifolds can be identified with H_G(Z_K;Z), we studied the conditions for the cohomology of a toric orbifold to be a quotient of its equivariant cohomology by linear terms., Comment: 21 papges. Comments are welcome

In this paper, we discuss the connected sum K_1#^Z K_2 of simplicial complexes K_1 and K_2, as well as define the notion of a strong connected sum. Geometrically, the connected sum is motivated by Lerman's symplectic cut applied to a toric orbifold, and algebraically, it is motivated by the connected sum of rings introduced by Ananthnarayan-Avramov-Moore. We show that the Stanley-Reisner ring of a connected sum K_1#^Z K_2 is the connected sum of the Stanley-Reisner rings of K_1 and K_2 along the Stanley-Reisner ring of the intersection of K_1 and K_2. The strong connected sum K_1 #^Z K_2 is defined in such a way that when K_1 and K_2 are Gorenstein, and Z is a suitable subset of the intersection of K_1 and K_2, then the Stanley-Reisner ring of the connected sum is Gorenstein, by the work of Ananthnarayan-Avramov-Moore. These algebraic computations can be interpreted in terms of the equivariant cohomology of moment angle complexes and we also describe the symplectic cut of a toric orbifold in terms of moment angle complexes., Comment: 14 pages

We study H. Dao's invariant $\eta_c^R$ of pairs of modules defined over a complete intersection ring $R$ of codimension $c$ having an isolated singularity. Our main result is that $\eta_c^R$ vanishes for all pairs of modules when $R$ is a {\em graded} complete intersection ring of codimension $c > 1$ having an isolated singularity. A consequence of this result is that all pairs of modules over such a ring are $c$-$\Tor$-rigid., Comment: 14 pages. Appendix added

A new construction of rings is introduced, studied, and applied. Given surjective homomorphisms $R\to T\gets S$ of local rings, and ideals in $R$ and $S$ that are isomorphic to some $T$-module $V$, the \emph{connected sum} $R#_TS$ is defined to be the local ring obtained by factoring out the diagonal image of $V$ in the fiber product $R\times_TS$. When $T$ is Cohen-Macaulay of dimension $d$ and $V$ is a canonical module of $T$, it is proved that if $R$ and $S$ are Gorenstein of dimension $d$, then so is $R#_TS$. This result is used to study how closely an artinian ring can be approximated by Gorenstein rings mapping onto it. It is proved that when $T$ is a field the cohomology algebra $\Ext^*_{R#_kS}(k,k)$ is an amalgam of the algebras $\Ext^*_{R}(k,k)$ and $\Ext^*_{S}(k,k)$ over isomorphic polynomial subalgebras generated by one element of degree 2. This is used to show that when $T$ is regular, the ring $R#_TS$ almost never is complete intersection., Comment: This version includes a new theorem (Theorem 1.8), in which results due to D'Anna and Shapiro are completed and strengthened. Other changes to the text are minor. To appear in Crelle's J

Let R be an isolated hypersurface singularity, and let M and N be finitely generated R-modules. As R is a hypersurface, the torsion modules of M against N are eventually periodic of period two (i.e., Tor_i^R(M,N) is isomorphic to Tor_{i+2}^R(M,N) for i sufficiently large). Since R has only an isolated singularity, these torsion modules are of finite length for i sufficiently large. The theta invariant of the pair (M,N) is defined by Hochster to be length(Tor_{2i}^R(M,N)) - length(Tor_{2i+1}^R(M,N)) for i sufficiently large. H. Dao has conjectured that the theta invariant is zero for all pairs (M,N) when R has even dimension and contains a field. This paper proves this conjecture under the additional assumption that R is graded with its irrelevant maximal ideal giving the isolated singularity. We also give a careful analysis of the theta pairing when the dimension of R is odd, and relate it to a classical pairing on the smooth variety Proj(R)., Comment: 20 pages

Let $S$ and $T$ be local rings with common residue field $k$, let $R$ be the fiber product $S \times_k T$, and let $M$ be an $S$-module. The Poincar\'e series $P^R_M$ of $M$ has been expressed in terms of $P^S_M$, $P^S_k$ and $P^T_k$ by Kostrikin and Shafarevich, and by Dress and Kr\"amer. Here, an explicit minimal resolution, as well as theorems on the structure of $\Ext_R(k,k)$ and $\Ext_R(M,k)$ are given that illuminate these equalities. Structure theorems for the cohomology modules of fiber products of modules are also given. As an application of these results, we compute the depth of cohomology modules over a fiber product., Comment: 13 pages