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Quantum machine learning has gained considerable attention as quantum technology advances, presenting a promising approach for efficiently learning complex data patterns. Despite this promise, most contemporary quantum methods require significant resources for variational parameter optimization and face issues with vanishing gradients, leading to experiments that are either limited in scale or lack potential for quantum advantage. To address this, we develop a general-purpose, gradient-free, and scalable quantum reservoir learning algorithm that harnesses the quantum dynamics of neutral-atom analog quantum computers to process data. We experimentally implement the algorithm, achieving competitive performance across various categories of machine learning tasks, including binary and multi-class classification, as well as timeseries prediction. Effective and improving learning is observed with increasing system sizes of up to 108 qubits, demonstrating the largest quantum machine learning experiment to date. We further observe comparative quantum kernel advantage in learning tasks by constructing synthetic datasets based on the geometric differences between generated quantum and classical data kernels. Our findings demonstrate the potential of utilizing classically intractable quantum correlations for effective machine learning. We expect these results to stimulate further extensions to different quantum hardware and machine learning paradigms, including early fault-tolerant hardware and generative machine learning tasks., Comment: 10 + 14 pages, 4 + 7 figures

We discuss how the diffraction theory of a single translation bounded measure or a family of such measures can be understood within the framework of unitary group representations. This allows us to prove an orthogonality feature of measures whose diffractions are mutually singular. We apply this to study dynamical systems, the refined Eberlein decomposition and validity of a Bombieri--Taylor type result in a rather general context. Along the way we also use our approach to (re)prove various characterisations of pure point diffraction., Comment: 39 pages

This article deals with pure point diffraction and its connection to various notions of almost periodicity. We explain why the Fibonacci chain does not fit into the classical class of Bohr almost periodicity and how it fits into the classes of mean, Besicovitch and Weyl almost periodic point sets. We report on recent results which characterize pure point diffraction as mean almost periodicity of the underlying structure, and discuss how the complex amplitudes fit into this picture.

We study the set of intrinsic metrics on a given graph. This is a convex compact set and it carries a natural order. We investigate existence of largest elements with respect to this order. We show that the only locally finite graphs which admit a largest intrinsic metric are certain finite star graphs. In particular all infinite locally finite graphs do not admit a largest intrinsic metric. Moreover, we give a characterization for the existence of intrinsic metrics with finite balls for weakly spherically symmetric graphs.

We develop a new approach to recurrence and the existence of non-constant harmonic functions on infinite weighted graphs. The approach is based on the capacity of subsets of metric boundaries with respect to intrinsic metrics. The main tool is a connection between polar sets in such boundaries and null sets of paths. This connection relies on suitably diverging functions of finite energy.

We characterize all semigroups sandwiched between the semigroup of a Dirichlet form and the semigroup of its active main part. In case the Dirichlet form is regular, we give a more explicit description of the quadratic forms of the sandwiched semigroups in terms of pairs consisting of an open set and a measure on an abstract boundary.

In this paper, we study the properties of the Eberlein convolution of measures and introduce a twisted version of it. For functions we show that the twisted Eberlein convolution can be seen as a translation invariant function-valued inner product. We study its regularity properties and show its existence on suitable sets of functions. For translation bounded measures we show that the (twisted) Eberlein convolution always exists along subsequences of the given sequence, and is a weakly almost periodic and Fourier transformable measure. We prove that if one of the two measures is mean almost periodic, then the (twisted) Eberlein convolution is strongly almost periodic. Moreover, if one of the measures is norm almost periodic, so is the (twisted) Eberlein convolution., Comment: 30 pages

We present a unified theory for the almost periodicity of functions with values in an arbitrary Banach space, measures and distributions via almost periodic elements for the action of a locally compact abelian group on a uniform topological space. We discuss the relation between Bohr and Bochner type almost periodicity, and similar conditions, and how the equivalence among such conditions relates to properties of the group action and the uniformity. We complete the paper by demonstrating how various examples considered earlier all fit in our framework.

We consider continuous $\mathrm{SL}(2,\mathbb{R})$ valued cocycles over general dynamical systems and discuss a variety of uniformity notions. In particular, we provide a description of uniform one-parameter families of continuous $\mathrm{SL}(2,\mathbb{R})$ cocycles as $G_\delta$-sets. These results are then applied to Schr\"odinger operators with dynamically defined potentials. In the case where the base dynamics is given by a subshift satisfying the Boshernitzan condition, we show that for a generic continuous sampling function, the associated Schr\"odinger cocycles are uniform for all energies and, in the aperiodic case, the spectrum is a Cantor set of zero Lebesgue measure., Comment: 24 pages

We study harmonic functions for general Dirichlet forms. First we review consequences of Fukushima's ergodic theorem for the harmonic functions in the domain of the $ L^{p} $ generator. Secondly we prove analogues of Yau's and Karp's Liouville theorems for weakly harmonic functions. Both say that weakly harmonic functions which satisfy certain $ L^{p} $ growth criteria must be constant. As consequence we give an integral criterion for recurrence.

We give a sufficient condition for non-existence of global nonnegative mild solutions of the Cauchy problem for the semilinear heat equation $u' = Lu + f(u)$ in $L^p(X,m)$ for $p \in [1,\infty)$, where $(X,m)$ is a $\sigma$-finite measure space, $L$ is the infinitesimal generator of a sub-Markovian strongly continuous semigroup of bounded linear operators in $L^p(X,m)$, and $f$ is a strictly increasing, convex, continuous function on $[0,\infty)$ with $f(0) = 0$ and $\int_1^\infty 1/f < \infty$. Since we make no further assumptions on the behaviour of the diffusion, our main result can be seen as being about the competition between the diffusion represented by $L$ and the reaction represented by $f$ in a general setting. We apply our result to Laplacians on manifolds, graphs, and, more generally, metric measure spaces with a heat kernel. In the process, we recover and extend some older as well as recent results in a unified framework., Comment: 18 pages; replaced a reference in the introduction, rewrote some paragraphs (outside of proofs) to improve clarity, corrected typos

We consider metrizable ergodic topological dynamical systems over locally compact, $\sigma$-compact abelian groups. We study pure point spectrum via suitable notions of almost periodicity for the points of the dynamical system. More specifically, we characterize pure point spectrum via mean almost periodicity of generic points. We then go on and show how Besicovitch almost periodic points determine both eigenfunctions and the measure in this case. After this, we characterize those systems arising from Weyl almost periodic points and use this to characterize weak and Bohr almost periodic systems. Finally, we consider applications to aperiodic order., Comment: 39 pages

We show that a translation bounded measure has pure point diffraction if and only if it is mean almost periodic. We then go on and show that a translation bounded measure solves what we call the phase problem if and only if it is Besicovitch almost periodic. Finally, we show that a translation bounded measure solves the phase problem independent of the underlying van Hove sequence if and only if it is Weyl almost periodic. These results solve fundamental issues in the theory of pure point diffraction and answer questions of Lagarias., Comment: 87 pages

We compute the deficiency spaces of operators of the form $H_A{\hat{\otimes}} I + I{\hat{\otimes}} H_B$, for symmetric $H_A$ and self-adjoint $H_B$. This enables us to construct self-adjoint extensions (if they exist) by means of von Neumann's theory. The structure of the deficiency spaces for this case was asserted already by Ibort, Marmo and P\'erez-Pardo, but only proven under the restriction of $H_B$ having discrete, non-degenerate spectrum.

We show that a generic quasi-periodic Schr\"odinger operator in $L^2(\mathbb{R})$ has purely singular spectrum. That is, for any minimal translation flow on a finite-dimensional torus, there is a residual set of continuous sampling functions such that for each of these sampling functions, the Schr\"odinger operator with the resulting potential has empty absolutely continuous spectrum., Comment: 11 pages

Modulated crystals and quasicrystals can simultaneously be described as modulated quasicrystals, a class of point sets introduced by de Bruijn in 1987. With appropriate modulation functions, modulated quasicrystals themselves constitute a substantial subclass of strongly almost periodic point measures. We re-analyse these structures using methods from modern mathematical diffraction theory, thereby providing a coherent view over that class. Similarly to de Bruijn's analysis, we find stability with respect to almost periodic modulations., Comment: 33 pages, 1 figure, v2: appendix shortened and restructured, other minor changes

We introduce a class of subshifts governed by finitely many two-sided infinite words. We call these words leading sequences. We show that any locally constant cocycle over such a subshift is uniform. From this we obtain Cantor spectrum of Lebesgue measure zero for associated Jacobi operators if the subshift is aperiodic. Our class covers all simple Toeplitz subshifts as well as all Sturmian subshifts. We apply our results to the spectral theory of Schreier graphs for uncountable families of groups acting on rooted trees.

The cosmic infrared background (CIB) is a powerful probe of large-scale structure across a very large redshift range, and consists of unresolved redshifted infrared emission from dusty galaxies. It can be used to study the astrophysics of galaxies, the star formation history of the universe, and the connection between dark and luminous matter. It can furthermore be used as a tracer of the large-scale structure and thus assist in de-lensing of the cosmic microwave background. The major difficulty in its use lies in obtaining accurate and unbiased large-scale CIB images that are cleaned of the contamination by Galactic dust. We used data on neutral atomic hydrogen from the recently-released HI4PI Survey to create template maps of Galactic dust, allowing us to remove this component from the Planck intensity maps from 353 to 857 GHz for approximately $25\%$ of the sky. This allows us to constrain the CIB power spectrum down to $\ell\gtrsim 70$. We present these CIB maps and the various processing and validation steps that we have performed to ensure their quality, as well as a comparison with previous studies. All our data products are made publicly available at https://doi.org/10.7910/DVN/8A1SR3, thereby enabling the community to investigate a wide range of questions related to the universe's large-scale structure., Comment: 23 pages, 25 figures. Comments welcome!

We address the spatial scale, ionization structure, mass and metal content of gas at the Milky Way disk-halo interface detected as absorption in the foreground of seven closely-spaced, high-latitude halo blue horizontal branch stars (BHBs) with heights z = 3 - 14 kpc. We detect transitions that trace multiple ionization states (e.g. CaII, FeII, SiIV, CIV) with column densities that remain constant with height from the disk, indicating that the gas most likely lies within z < 3.4 kpc. The intermediate ionization state gas traced by CIV and SiIV is strongly correlated over the full range of transverse separations probed by our sightlines, indicating large, coherent structures greater than 1 kpc in size. The low ionization state material traced by CaII and FeII does not exhibit a correlation with either N$_{\rm HI}$ or transverse separation, implying cloudlets or clumpiness on scales less than 10 pc. We find that the observed ratio log(N_SiIV/ N_CIV), with a median value of -0.69+/-0.04, is sensitive to the total carbon content of the ionized gas under the assumption of either photoionization or collisional ionization. The only self-consistent solution for photoionized gas requires that Si be depleted onto dust by 0.35 dex relative to the solar Si/C ratio, similar to the level of Si depletion in DLAs and in the Milky Way ISM. The allowed range of values for the areal mass infall rate of warm, ionized gas at the disk-halo interface is 0.0003 < dM_gas / dtdA [M_sun kpc^-2 yr^-] < 0.006. Our data support a physical scenario in which the Milky Way is fed by complex, multiphase processes at its disk-halo interface that involve kpc-scale ionized envelopes or streams containing pc-scale, cool clumps., Comment: 12 figures, 4 tables, Accepted to ApJ. Revision features areal mass accretion rate calculation

We discuss optimal lower bounds for eigenvalues of Laplacians on weighted graphs. These bounds are formulated in terms of the geometry and, more specifically, the inradius of subsets of the graph. In particular, we study the first non-zero eigenvalue in the finite volume case and the first eigenvalue of the Dirichlet Laplacian on subsets that satisfy natural geometric conditions., Comment: 16 pages

We study mean equicontinuous actions of locally compact $\sigma$-compact amenable groups on compact metric spaces. In this setting, we establish the equivalence of mean equicontinuity and topo-isomorphy to the maximal equicontinuous factor and provide a characterization of mean equicontinuity of an action via properties of its product. This characterization enables us to show the equivalence of mean equicontinuity and the weaker notion of Besicovitch-mean equicontinuity in fairly high generality, including actions of abelian groups as well as minimal actions of general groups. In the minimal case, we further conclude that mean equicontinuity is equivalent to discrete spectrum with continuous eigenfunctions. Applications of our results yield a new class of non-abelian mean equicontinuous examples as well as a characterization of those extensions of mean equicontinuous actions which are still mean equicontinuous., Comment: 33 pages, 1 figure

We consider topological dynamical systems over $\ZZ$ and, more generally, locally compact, $\sigma$-compact abelian groups. We relate spectral theory and diffraction theory. We first use a a recently developed general framework of diffraction theory to associate an autocorrelation and a diffraction measure to any $L^2$-function over such a dynamical system. This diffraction measure is shown to be the spectral measure of the function. If the group has a countable basis of the topology one can also exhibit the underlying autocorrelation by sampling along the orbits. Building on these considerations we then show how the spectral theory of dynamical systems can be reformulated via diffraction theory of function dynamical systems. In particular, we show that the diffraction measures of suitable factors provide a complete spectral invariant., Comment: 22 pages

We study topological Poincar\'e type inequalities on general graphs. We characterize graphs satisfying such inequalities and then turn to the best constants in these inequalities. Invoking suitable metrics we can interpret these constants geometrically as diameters and inradii. Moreover, we can relate them to spectral theory of Laplacians once a probability measure on the graph is chosen. More specifically, we obtain a variational characterization of these constants as infimum over spectral gaps of all Laplacians on the graphs associated to probability measures.

We study pairs of Dirichlet forms related by an intertwining order isomorphisms between the associated $L^2$-spaces. We consider the measurable, the topological and the geometric setting respectively. In the measurable setting, we deal with arbitrary (irreducible) Dirichlet forms and show that any intertwining order isomorphism is necessarily unitary (up to a constant). In the topological setting we deal with quasi-regular forms and show that any intertwining order isomorphism induces a quasi-homeomorphism between the underlying spaces. In the geometric setting we deal with both regular Dirichlet forms as well as resistance forms and essentially show that the geometry defined by these forms is preserved by intertwining order isomorphisms. In particular, we prove in the strongly local regular case that intertwining order isomorphisms induce isometries with respect to the intrinsic metrics between the underlying spaces under fairly mild assumptions. This applies to a wide variety of metric measure spaces including $\mathrm{RCD}(K,N)$-spaces, complete weighted Riemannian manifolds and complete quantum graphs. In the non-local regular case our results cover in particular graphs as well as fractional Laplacians as arising in the treatment of $\alpha$-stable L\'evy processes. For resistance forms we show that intertwining order isomorphisms are isometries with respect to the resistance metrics. Our results can can be understood as saying that diffusion always determines the Hilbert space, and -- under natural compatibility assumptions -- the topology and the geometry respectively. As special instances they cover earlier results for manifolds and graphs.

We study combinatorial properties of the subshift induced by the substitution that describes Lysenok's presentation of Grigorchuk's group of intermediate growth by generators and relators. This subshift has recently appeared in two different contexts: on one hand, it allowed to embed Grigorchuk's group in a topological full group, and on the other hand, it was useful in the spectral theory of Laplacians on the associated Schreier graphs., Comment: 8 pages; part of the material discussed in the paper was originally contained in (earlier versions of) arXiv:1510.00545 and arXiv:1412.6822. In their published versions the latter two articles do not contain this material anymore

We describe the set of all Dirichlet forms associated to a given infinite graph in terms of Dirichlet forms on its Royden boundary. Our approach is purely analytical and uses form methods.

We study domination of quadratic forms in the abstract setting of ordered Hilbert spaces. Our main result gives a characterization in terms of the associated forms. This generalizes and unifies various earlier works. Along the way we present several examples., Comment: The article arXiv:1608.06798v1 has been split, the present article contains the first part. The material presented here is partly based on the third named author's master's thesis (arXiv:1604.05114)

We present a new map of interstellar reddening, covering the 39\% of the sky with low {\rm HI} column densities ($N_{\rm HI} < 4\times10^{20}\,\rm cm^{-2}$ or $E(B-V)\approx 45\rm\, mmag$) at $16\overset{'}{.}1$ resolution, based on all-sky observations of Galactic HI emission by the HI4PI Survey. In this low column density regime, we derive a characteristic value of $N_{\rm HI}/E(B-V) = 8.8\times10^{21}\, \rm\, cm^{2}\, mag^{-1}$ for gas with $|v_{\rm LSR}| < 90\,\rm km\, s^{-1}$ and find no significant reddening associated with gas at higher velocities. We compare our HI-based reddening map with the Schlegel, Finkbeiner, and Davis (1998, SFD) reddening map and find them consistent to within a scatter of $\simeq 5\,\rm mmag$. Further, the differences between our map and the SFD map are in excellent agreement with the low resolution ($4\overset{\circ}{.}5$) corrections to the SFD map derived by Peek and Graves (2010) based on observed reddening toward passive galaxies. We therefore argue that our HI-based map provides the most accurate interstellar reddening estimates in the low column density regime to date. Our reddening map is made publicly available (http://dx.doi.org/10.7910/DVN/AFJNWJ)., Comment: Re-submitted to ApJ. The reddening map is available at http://dx.doi.org/10.7910/DVN/AFJNWJ