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We develop the notions of connections and curvature for general Lie-Rinehart algebras without using smoothness assumptions on the base space. We present situations when a connection exists. E.g., this is the case when the underlying module is finitely generated. We show how the group of module automorphism acts as gauge transformations on the space of connections. When the underlying module is projective we define a version of the Chern character reproducing results of Hideki Ozeki. We discuss various examples of flat connections and the associated Maurer-Cartan equations. We provide examples of Levi-Civita connections on singular varieties and singular differential spaces with non-zero Riemannian curvature. The main observation is that for quotient singularities, even though the metric degenerates along strata, the poles of the Christoffel symbols are removable.

Grain boundaries are dominant imperfections in nanocrystalline materials that form a complex 3-dimensional (3D) network. Solute segregation to grain boundaries is strongly coupled to the grain boundary character, which governs the stability and macroscopic properties of nanostructured materials. Here, we develop a 3-dimensional transmission electron microscopy and atom probe tomography correlation framework to retrieve the grain boundary character and composition at the highest spatial resolution and chemical sensitivity by correlating four-dimensional scanning precession electron diffraction tomography (4D-SPED) and atom probe tomography (APT) on the same sample. We obtain the 3D grain boundary habit plane network and explore the preferential segregation of Cu and Si in a nanocrystalline Ni-W alloy. The correlation of structural and compositional information reveals that Cu segregates predominantly along high angle grain boundaries and incoherent twin boundaries, whereas Si segregation to low angle and incommensurate grain boundaries is observed. The novel full 3D correlative approach employed in this work opens up new possibilities to explore the 3D crystallographic and compositional nature of nanomaterials. This lays the foundation for both probing the true 3D structure-chemistry at the sub-nanometer scale and, consequentially, tailoring the macroscopic properties of advanced nanomaterials.

This paper addresses, for the first time, a key aspect of the phenomenology of Compact Symmetric Objects (CSOs) -- the characteristics of their radio spectra. We present a radio-spectrum description of a complete sample of high luminosity CSOs (CSO-2s), which shows that they exhibit the \textit{complete} range of spectral types, including flat-spectrum sources ($\alpha \ge -0.5$), steep-spectrum sources ($\alpha < -0.5$), and peaked-spectrum sources. We show that there is no clear correlation between spectral type and size, but there is a correlation between the high-frequency spectral index and both object type and size. We also show that, to avoid biasing the data and to understand the various classes of jetted-AGN involved, the complete range of spectral types should be included in studying the general phenomenology of compact jetted-AGN, and that complete samples must be used, selected over a wide range of frequencies. We discuss examples that demonstrate these points. We find that the high-frequency spectral indices of CSO-2s span $-1.3 <\alpha_{\rm hi} < -0.3$, and hence that radio spectral signatures cannot be used to discriminate definitively between CSO-2s, binary galactic nuclei, and millilensed objects, unless they have $\alpha_{\rm hi} >-0.3$.

Given an affine Poisson algebra, that is singular one may ask whether there is an associated symplectic form. In the smooth case the answer is obvious: for the symplectic form to exist the Poisson tensor has to be invertible. In the singular case, however, derivations do not form a projective module and the nondegeneracy condition is more subtle. For a symplectic singularity one may naively ask if there is indeed an analogue of a symplectic form. We examine an example of a symplectic singularity, namely the double cone, and show that here such a symplectic form exists. We use the naive de Rham complex of a Lie-Rinehart algebra. Our analysis of the double cone uses Gr\"obner bases calculations. We also give an alternative construction of the symplectic form that generalizes to categorical quotients of cotangent lifted representations of finite groups. We use the same formulas to construct a symplectic form on the simple cone, seen as a Poisson differential space and generalize the construction to linear symplectic orbifolds. We present useful auxiliary results that enable to explicitly determine generators for the module of derivations an affine variety. The latter may be understood as a differential space., Comment: The paper has been greatly revised and expanded and has now 14 pages. Coauthor Christopher Seaton has been added. A mistake in Section 4 has been corrected

We give an explicit solution formula for the polynomial regression problem in terms of Schur polynomials and Vandermonde determinants. We thereby generalize the work of Chang, Deng, and Floater to the case of model functions of the form $\sum _{i=1}^{n} a_{i} x^{d_{i}}$ for some integer exponents $d_{1} >d_{2} >\dotsc >d_{n} \geq 0$ and phrase the results using Schur polynomials. Even though the solution circumvents the well-known problems with the forward stability of the normal equation, it is only of practical value if $n$ is small because the number of terms in the formula grows rapidly with the number $m$ of data points. The formula can be evaluated essentially without rounding., Comment: 12 pages, 2 figures

This note is about the observation that the various transition formulas between bases of trigonometric polynomials can be expressed in terms binomial coefficients. More specifically, we write the entries of the Chebyshev matrices $ T$ and $ U$ in terms of binomial coefficients. We remark that the inverses of certain submatrices of the Chebyshev matrix $ T$ correspond to classical power reduction formulas for trigonometric functions. Therefore the entries of $ T^{-1}$ are expressible in terms of binomial coefficients and powers of two. The inverses of certain submatrices of the Chebyshev matrix $ U$ appear in power reduction formulas involving the Catalan triangles. We could not spot them in the literature. As a corollary the Catalan triangles are interpreted as inverses of rather natural matrices of binomial coefficients. We explain the matrix inversions in terms of Riordan arrays. We solve the integral $ \int _{0}^{2\pi }\cos^{2m}( x)\sin^{2n}( x) dx$ and deduce a hypergeometric identity from the Fourier expansion of $\cos^{2m}(x)$ and $ \sin^{2n}( x)$. As a corollary we prove that super Catalan numbers are integers. We emphasize that from the point of view of these base changes between the trigonometric polynomials it is more natural to work with $2\cos(x)$ and $2\sin(x)$ instead of $\cos(x)$ and $\sin(x)$, since the transition matrices become invertible over the integers. We do a similar analysis for the spread polynomials. This enables us to make a conjecture of Goh and Wildberger more precise. In this exposition we give a mostly self-contained account of the matter, the prerequisites just being four semesters of calculus, linear algebra, elementary complex variables and the residue theorem.

We call a reductive complex group $G$ quasi-toral if $G^0$ is a torus. Let $G$ be quasi-toral and let $V$ be a faithful $1$-modular $G$-module. Let $N$ (the shell) be the zero fiber of the canonical moment mapping $\mu\colon V\oplus V^*\to\mathfrak{g}^*$. Then $N$ is a complete intersection variety with rational singularities. Let $M$ denote the categorical quotient $N/\!\!/ G$. We show that $M$ determines $V\oplus V^*$ and $G$, up to isomorphism, if $\operatorname{codim}_N N_\mathrm{sing}\geq 4$. If $\operatorname{codim}_NN_\mathrm{sing}=3$, the lowest possible, then there is a process to produce an algebraic (hence quasi-toral) subgroup $G'\subset G$ and a faithful $1$-modular $G'$-submodule $V'\subset V$ with shell $N'$ such that $\operatorname{codim}_{N'}(N')_\mathrm{sing}\geq 4$. Moreover, there is a $G'$-equivariant morphism $N'\to N$ inducing an isomorphism $N'/\!\!/ G'\xrightarrow{\sim} N/\!\!/ G$. Thus, up to isomorphism, $M$ determines $V'\oplus (V')^*$ and $G'$, hence also $N'$. We establish similar results for real shells and real symplectic quotients associated to unitary modules for compact Lie groups., Comment: 24 pages

I present a universal method, called pivotal condensation, for calculating stoichiometric factors of chemical reactions. It can be done by hand, even for rather complicated reactions. The main trick, which I call kernel pivotal condensation (ker pc), to calculate the kernel of a matrix might be of independent interest. The discussion is elaborated for matrices with entries in a principal ideal domain $R$. The ker pc calculates a basis with coefficients in $R$ for the kernel of a matrix, seen as e $Q$-vector space, where $Q$ is the quotient field of $R$. If $W$ is a free saturated $R$-submodule of $R^n$ I address the question how to modify an $R$-basis of the $Q$-vector subspace $Q\otimes _R W$ over the quotient field $Q$ to obtain a basis of the $R$-module $W$. I also indicate how one can solve inhomogeneous linear systems, invert matrices and determine the four subspaces using pivotal condensation. I formulate the balancing by inspection method that is widely used to reduce the size of a linear system arising in chemical balancing in mathematical language., Comment: This is version 2. Minor errors have been corrected

Let $\boldsymbol{k}$ be a field of characteristic zero and $A=\boldsymbol{k}[x_{1},...,x_{n}]/I$ with $I=(f_{1},...,f_{k})$ be an affine algebra. We study Nambu-Poisson brackets on $A$ of arity $m\geq 2$, focusing on the case when $m$ is even. We construct an $L_{\infty}$-algebroid on the cotangent complex $\mathbb{L}_{A|\boldsymbol{k}}$, generalizing previous work on the case when $A$ is a Poisson algebra. This structure is referred to as the higher form brackets. The main tool is a $P_{\infty}$-structure on a resolvent $R$ of $A$. These $P_{\infty}$- and $L_{\infty}$-structures are merely $\mathbb Z_2$-graded for $m\neq 2$. We discuss several examples and propose a method to obtain new ones that we call the outer tensor product. We compare our higher form brackets with the form bracket of Vaisman. We introduce the notion of a Lie-Rinehart $m$-algebra, the form bracket of a Nambu-Poisson bracket of even arity being an example. We find a flat Nambu connection on the conormal module., Comment: 29 pages

Machine learning has made important headway in helping to improve the treatment of quantum many-body systems. A domain of particular relevance are correlated inhomogeneous systems. What has been missing so far is a general, scalable deep-learning approach that would enable the rapid prediction of spatial densities for strongly correlated systems in arbitrary potentials. In this work, we present a straightforward scheme, where we learn to predict densities using convolutional neural networks trained on random potentials. While we demonstrate this approach in 1D and 2D lattice models using data from numerical techniques like Quantum Monte Carlo, it is directly applicable as well to training data obtained from experimental quantum simulators. We train networks that can predict the densities of multiple observables simultaneously and that can predict for a whole class of many-body lattice models, for arbitrary system sizes. We show that our approach can handle well the interplay of interference and interactions and the behaviour of models with phase transitions in inhomogeneous situations, and we also illustrate the ability to solve inverse problems, finding a potential for a desired density., Comment: 7 pages, 3 figures

The field equations are proposed for the third rank tensor field with the hook Young diagram. The equations describe the irreducible spin two massless representation in any $d\geq 3$. The starting point of the construction is the linearised system of Einstein equations which includes the Nordstr\"om equation. This equation, being considered irrespectively to the rest of the Einstein system, corresponds to the topological field theory. The general solution is a pure gauge, modulo topological modes which we neglect in this article. We find the sequence of the reducible gauge transformations for the linearised Nordstr\"om equation, with the hook tensor being the initial gauge symmetry parameter. By substituting the general solution of the Nordstr\"om equation into the rest of the Einstein's system, we arrive at the field equations for the hook tensor. The degree of freedom number count confirms, it is the spin two theory., Comment: 19 pages; minor corrections, reference added, to appear in European Physical Journal C

Monolayers of two-dimensional van der Waals materials exhibit novel electronic phases distinct from their bulk due to the symmetry breaking and reduced screening in the absence of the interlayer coupling. In this work, we combine angle-resolved photoemission spectroscopy and scanning tunneling microscopy/spectroscopy to demonstrate the emergence of a unique insulating 2 x 1 dimer ground state in monolayer 1T-IrTe2 that has a large band gap in contrast to the metallic bilayer-to-bulk forms of this material. First-principles calculations reveal that phonon and charge instabilities as well as local bond formation collectively enhance and stabilize a charge-ordered ground state. Our findings provide important insights into the subtle balance of interactions having similar energy scales that occurs in the absence of strong interlayer coupling, which offers new opportunities to engineer the properties of 2D monolayers.

We compute univariate and multigraded Hilbert series of invariants and covariants of representations of the circle and orthogonal group $\operatorname{O}_2$. The multigradings considered include the maximal grading associated to the decomposition of the representation into irreducibles as well as the bigrading associated to a cotangent-lifted representation, or equivalently, the bigrading associated to the holomorphic and antiholomorphic parts of the real invariants and covariants. This bigrading induces a bigrading on the algebra of on-shell invariants of the symplectic quotient, and the corresponding Hilbert series are computed as well. We also compute the first few Laurent coefficients of the univariate Hilbert series, give sample calculations of the multigraded Laurent coefficients, and give an example to illustrate the extension of these techniques to the semidirect product of the circle by other finite groups. We describe an algorithm to compute each of the associated Hilbert series., Comment: v2: 26 pages, corrected typos and error in Theorem 3.14, improved exposition

Let $G$ be a complex reductive group and $V$ a $G$-module. There is a natural moment mapping $\mu\colon V\oplus V^*\to\mathfrak{g}^*$ and we denote $\mu^{-1}(0)$ (the shell) by $N_V$. We use invariant theory and results of Musta\c{t}\u{a} [Mus01] to find criteria for $N_V$ to have rational singularities and for the categorical quotient $N_V /\!\!/ G$ to have symplectic singularities, the latter results improving upon [HSS20]. It turns out that for ``most'' $G$-modules $V$, the shell $N_V$ has rational singularities. For the case of direct sums of classical representations of the classical groups, $N_V$ has rational singularities and $N_V /\!\!/ G$ has symplectic singularities if $N_V$ is a reduced and irreducible complete intersection. Another important special case is $V=p\,\mathfrak{g}$ (the direct sum of $p$ copies of the Lie algebra of $G$) where $p\geq 2$. We show that $N_V$ has rational singularities and that $N_V /\!\!/ G$ has symplectic singularities, improving upon results of [Bud19], [AA16], [Kap19] and [GH20]. Let $\pi=\pi_1(\Sigma)$ where $\Sigma$ is a closed Riemann surface of genus $p\geq 2$. Let $G$ be semisimple and let $\operatorname{Hom}(\pi,G)$ and $\mathscr X\!(\pi,G)$ be the corresponding representation variety and character variety. We show that $\operatorname{Hom}(\pi,G)$ is a complete intersection with rational singularities and that $\mathscr X\!(\pi,G)$ has symplectic singularities. If $p>2$ or $G$ contains no simple factor of rank $1$, then the singularities of $\operatorname{Hom}(\pi,G)$ and $\mathscr X\!(\pi,G)$ are in codimension at least four and $\operatorname{Hom}(\pi,G)$ is locally factorial. If, in addition, $G$ is simply connected, then $\mathscr X\!(\pi,G)$ is locally factorial., Comment: 21 pages. v2: 22 pages, clarified and improved arguments in Section 3, improved exposition and minor corrections. v3: 30 pages, reformatted, minor corrections. v4, 31 pages, improved exposition and minor corrections following referee's remarks

Let $n\ge 1$ and $A$ be a commutative algebra of the form $\boldsymbol k[x_1,x_2,\dots, x_n]/I$ where $\boldsymbol k$ is a field of characteristic $0$ and $I\subseteq \boldsymbol k[x_1,x_2,\dots, x_n]$ is an ideal. Assume that there is a Poisson bracket $\{\:,\:\}$ on $S$ such that $\{I,S\}\subseteq I$ and let us denote the induced bracket on $A$ by $\{\:,\:\}$ as well. It is well-known that $[\mathrm d x_i,\mathrm d x_j]:=\mathrm d\{x_i,x_j\}$ defines a Lie bracket on the $A$-module $\Omega_{A|\boldsymbol k}$ of K\"ahler differentials making $(A,\Omega_{A|\boldsymbol k})$ a Lie-Rinehart pair. Recall that $A$ is regular if and only if $\Omega_{A|\boldsymbol k}$ is projective as an $A$-module. If $A$ is not regular, the cotangent complex $\mathbb L_{A|\boldsymbol k}$ may serve as a replacement for the $A$-module $\Omega_{A|\boldsymbol k}$. We prove that there is a structure of an $L_\infty$-algebroid on $\mathbb L_{A|\boldsymbol k}$, compatible with the Lie-Rinehart pair $(A,\Omega_{A|\boldsymbol k})$. The $L_\infty$-algebroid on $\mathbb L_{A|\boldsymbol k}$ actually comes from a $P_\infty$-algebra structure on the resolvent of the morphism $k[x_1,x_2,\dots, x_n]\to A$. We identify examples when this $L_\infty$-algebroid simplifies to a dg Lie algebroid. For aesthetic reasons we concentrate on cases when $ \boldsymbol k[x_1,x_2,\dots, x_n]$ carries a (possibly nonstandard) $\mathbb Z_{\ge 0}$-grading and both $I$ and $\{\:,\:\}$ are homogeneous., Comment: 32 pages, 2 tables. V2: We corrected an error in the bracket table of the invariants of the Kleinian singularity E_7 (which does not appear in the published version in IMRN, https://doi.org/10.1093/imrn/rnac170) that was identified by William Osnayder Clavijo Esquivel; we express our appreciation for identifying this error

We compute the Hilbert series of the graded algebra of real regular functions on a linear symplectic quotient by the $2$-torus as well as the first four coefficients of the Laurent expansion of this Hilbert series at $t = 1$. We describe an algorithm to compute the Hilbert series as well as the Laurent coefficients in explicit examples., Comment: 22 pages

The discovery of interaction-driven insulating and superconducting phases in moir\'e van der Waals heterostructures has sparked considerable interest in understanding the novel correlated physics of these systems. While a significant number of studies have focused on twisted bilayer graphene, correlated insulating states and a superconductivity-like transition up to 12 K have been reported in recent transport measurements of twisted double bilayer graphene. Here we present a scanning tunneling microscopy and spectroscopy study of gate-tunable twisted double bilayer graphene devices. We observe splitting of the van Hove singularity peak by ~20 meV at half-filling of the conduction flat band, with a corresponding reduction of the local density of states at the Fermi level. By mapping the tunneling differential conductance we show that this correlated system exhibits energetically split states that are spatially delocalized throughout the different regions in the moir\'e unit cell, inconsistent with order originating solely from onsite Coulomb repulsion within strongly-localized orbitals. We have performed self-consistent Hartree-Fock calculations that suggest exchange-driven spontaneous symmetry breaking in the degenerate conduction flat band is the origin of the observed correlated state. Our results provide new insight into the nature of electron-electron interactions in twisted double bilayer graphene and related moir\'e systems., Comment: 24 pages, 5 figures

Observing solute hydrogen (H) in matter is a formidable challenge, yet, enabling quantitative imaging of H at the atomic-scale is critical to understand its deleterious influence on the mechanical strength of many metallic alloys that has resulted in many catastrophic failures of engineering parts and structures. Here, we report on the APT analysis of hydrogen (H) and deuterium (D) within the nanostructure of an ultra-high strength steel with high resistance to hydrogen embrittlement. Cold drawn, severely deformed pearlitic steel wires (Fe-0.98C-0.31Mn-0.20Si-0.20Cr-0.01Cu-0.006P-0.007S wt.%, {\epsilon}=3.1) contains cementite decomposed during the pre-deformation of the alloy and ferrite. We find H and D within the decomposed cementite, and at some interfaces with the surrounding ferrite. To ascertain the origin of the H/D signal obtained in APT, we explored a series of experimental workflows including cryogenic specimen preparation and cryogenic-vacuum transfer from the preparation into a state-of-the-art atom probe. Our study points to the critical role of the preparation, i.e. the possible saturation of H-trapping sites during electrochemical polishing, how these can be alleviated by the use of an outgassing treatment, cryogenic preparation and transfer prior to charging. Accommodation of large amounts of H in the under-stoichiometric carbide likely explains the resistance of pearlite against hydrogen embrittlement.

Deep Neural Networks for image classification have been found to be vulnerable to adversarial samples, which consist of sub-perceptual noise added to a benign image that can easily fool trained neural networks, posing a significant risk to their commercial deployment. In this work, we analyze adversarial samples through the lens of their contributions to the principal components of each image, which is different than prior works in which authors performed PCA on the entire dataset. We investigate a number of state-of-the-art deep neural networks trained on ImageNet as well as several attacks for each of the networks. Our results demonstrate empirically that adversarial samples across several attacks have similar properties in their contributions to the principal components of neural network inputs. We propose a new metric for neural networks to measure their robustness to adversarial samples, termed the (k,p) point. We utilize this metric to achieve 93.36% accuracy in detecting adversarial samples independent of architecture and attack type for models trained on ImageNet.