Motivated by approaches to the word problem for one-relation monoids arising from work of Adian and Oganesian (1987), Guba (1997), and Ivanov, Margolis and Meakin (2001), we study the submonoid and rational subset membership problems in one-relation monoids and in positive one-relator groups. We give the first known examples of positive one-relator groups with undecidable submonoid membership problem, and apply this to give the first known examples of one-relation monoids with undecidable submonoid membership problem. We construct several infinite families of one-relation monoids with undecidable submonoid membership problem, including examples that are defined by relations of the form $w=1$ but which are not groups, and examples defined by relations of the form $u=v$ where both of $u$ and $v$ are non-empty. As a consequence we obtain a classification of the right-angled Artin groups that can arise as subgroups of one-relation monoids. We also give examples of monoids with a single defining relation of the form $aUb = a$, and examples of the form $aUb=aVa$, with undecidable rational subset membership problem. We give a one-relator group defined by a freely reduced word of the form $uv^{-1}$ with $u, v$ positive words, in which the prefix membership problem is undecidable. Finally, we prove the existence of a special two-relator inverse monoid with undecidable word problem, and in which both the relators are positive words. As a corollary, we also find a positive two-relator group with undecidable prefix membership problem. In proving these results, we introduce new methods for proving undecidability of the rational subset membership problem in monoids and groups, including by finding suitable embeddings of certain trace monoids.

We define and study graphs associated to hexagon decompositions of surfaces by curves and arcs. One of the variants is shown to be quasi-isometric to the pants graph, whereas the other variant is quasi-isometric to (a Cayley graph of) the mapping class group.

We present a direct lifetime measurement of the $5s5p~^1P_1$ state of strontium using time-correlated single-photon counting of laser induced fluorescence in a hot atomic beam. To achieve fast switch-off times and a high signal-to-noise ratio, we excite the strontium atoms with a femtosecond pulsed laser at $\approx$461 nm and collect the fluorescence onto a hybrid single-photon detector. Analysis of the measured exponential decay gives a lifetime of the $^1P_1$ state of $τ= (5.216 \pm 0.006_{stat} \pm 0.013_{sys})$ ns, where all the systematic effects have been thoroughly considered.

The concept of ideology is central to political discourse and dynamics, and is often cast as falling primarily on a one-dimensional scale from "left-wing/liberal" to "right-wing/conservative", but the validity of this simple quantitative treatment is uncertain. Here we investigate and compare various high-resolution measures of ideology, both internal (self-identification and policy-stance agreements) and external (estimating the ideological position of political opinion statements). We find strong consistency between internal measures, although policy-stance agreement ideology yields a systematically centralizing and liberalizing portrait relative to more abstract "liberal/conservative" measures. More remarkably, we find that external assessments of ideology, while noisy, are largely consistent across observers, even for highly dissonant ideas and regardless of speaker identity markers. This supports the use of these responses as meaningful, comparable quantities, which general members of the public reliably project from the abstract space of political thought onto a shared one-dimensional domain.

Let $\mathcal E$ be a complex elliptic curve and $S$ be a non-empty finite subset of $\mathcal E$. We show that the functions $\tildeΓ$ introduced in arXiv:1712.07089 out of string theory motivations give rise to a basis of the minimal algebra $A_{\mathcal E\smallsetminus S}$ of holomorphic multivalued functions on $\mathcal E\smallsetminus S$ which is stable under integration, introduced in arXiv:2212.03119; this basis is alternative to the basis of $A_{\mathcal E\smallsetminus S}$ constructed in loc. cit. using elliptic analogues of the hyperlogarithm functions.

We study analytic properties of graph product of finite groups with a hyperbolic defining graph. This is done by studying dynamics on the Bowditch compactification of the extension graph, or the crossing graph, of graph product. In particular, we provide a new class of convergence groups and identify the if and only if condition for this convergence action to be geometrically finite. We also provide a new class of properly proximal groups, relatively bi-exact groups, and groups with strongly solid group von Neumann algebras.

We consider the community detection problem in sparse random hypergraphs under the non-uniform hypergraph stochastic block model (HSBM), a general model of random networks with community structure and higher-order interactions. When the random hypergraph has bounded expected degrees, we provide a spectral algorithm that outputs a partition with at least a $γ$ fraction of the vertices classified correctly, where $γ\in (0.5,1)$ depends on the signal-to-noise ratio (SNR) of the model. When the SNR grows slowly as the number of vertices goes to infinity, our algorithm achieves weak consistency, which improves the previous results in Ghoshdastidar and Dukkipati (2017) for non-uniform HSBMs. Our spectral algorithm consists of three major steps: (1) Hyperedge selection: select hyperedges of certain sizes to provide the maximal signal-to-noise ratio for the induced sub-hypergraph; (2) Spectral partition: construct a regularized adjacency matrix and obtain an approximate partition based on singular vectors; (3) Correction and merging: incorporate the hyperedge information from adjacency tensors to upgrade the error rate guarantee. The theoretical analysis of our algorithm relies on the concentration and regularization of the adjacency matrix for sparse non-uniform random hypergraphs, which can be of independent interest.

We prove that triangle Artin groups of the type $A_{2,3,2n}$ are residually finite for all $n\geq4$. This requires splitting these triangle Artin groups as graphs of groups and then proving that each of these graphs of groups has finite stature with respect to its vertex groups.

We consider an inhomogeneous $XX$ spin chain which interpolates between the Krawtchouk one with perfect state transfer and the homogeneous $XX$ chain. This model can be used to perform qubit state transfer with sufficiently high fidelity. The advantage of this model with respect to the Krawtchouk chain is that while maintaining high transfer fidelity, the coupling strengths are capped and do not become excessively large as the number of sites grows. The construction is fully analytic and is based on spectral surgery transformations of the homogeneous chain.

We investigate quantum cellular automata (QCA) on one-dimensional spin systems defined over a subalgebra of the full local operator algebra - the symmetric subalgebra under a finite Abelian group symmetry $G$. For systems where each site carries a regular representation of $G$, we establish a complete classification of such subalgebra QCAs based on two topological invariants: (1) a surjective homomorphism from the group of subalgebra QCAs to the group of anyon permutation symmetries in a $(2+1)d$ $G$ gauge theory; and (2) a generalization of the Gross-Nesme-Vogts-Werner (GNVW) index that characterizes the flow of the symmetric subalgebra. Specifically, two subalgebra QCAs correspond to the same anyon permutation and share the same index if and only if they differ by a finite-depth unitary circuit composed of $G$-symmetric local gates. We also identify a set of operations that generate all subalgebra QCAs through finite compositions. As an example, we examine the Kramers-Wannier duality on a $\mathbb{Z}_2$ symmetric subalgebra, demonstrating that it maps to the $e$-$m$ permutation in the two-dimensional toric code and has an irrational index of $\sqrt{2}$. Therefore, it cannot be extended to a QCA over the full local operator algebra and mixes nontrivially with lattice translations.

Bounded minimizers of double phase problems at nearly linear growth have locally Hölder continuous gradient within the sharp maximal nonuniformity range $q<1+α$.

We introduce the concept of "irrational paths" for a given subshift and use it to characterize all minimal left ideals in the associated unital subshift algebra. Consequently, we characterize the socle as the sum of the ideals generated by irrational paths. Proceeding, we construct a graph such that the Leavitt path algebra of this graph is graded isomorphic to the socle. This realization allows us to show that the graded structure of the socle serves as an invariant for the conjugacy of Ott-Tomforde-Willis subshifts and for the isometric conjugacy of subshifts constructed with the product topology. Additionally, we establish that the socle of the unital subshift algebra is contained in the socle of the corresponding unital subshift C*-algebra.

In the present paper, we study compact complex manifolds admitting a Hermitian metric which is SKT and CYT and whose Bismut torsion is parallel. We first obtain a characterization of the universal cover of such manifolds as a product of a Kaehler Ricci-flat manifold with a Bismut flat one. Then, using a mapping torus construction, we provide non-Bismut flat examples. The existence of generalized Kaehler structures is also investigated.

We study the topological structure of the space $\mathcal{X}$ of isomorphism classes of metric measure spaces equipped with the box or concentration topologies. We consider the scale-change action of the multiplicative group $\mathbb{R}_+$ of positive real numbers on $\mathcal{X}$, which has a one-point metric measure space, say $*$, as only one fixed-point. We prove that the $\mathbb{R}_+$-action on $\mathcal{X}_* := \mathcal{X} \setminus \{*\}$ admits the structure of nontrivial and locally trivial principal $\mathbb{R}_+$-bundle over the quotient space. Our bundle $\mathbb{R}_+ \to \mathcal{X}_* \to \mathcal{X}_*/\mathbb{R}_+$ is a curious example of a nontrivial principal fiber bundle with contractible fiber. A similar statement is obtained for the pyramidal compactification of $\mathcal{X}$, where we completely determine the structure of the fixed-point set of the $\mathbb{R}_+$-action on the compactification.

We develop a new method suitable for establishing lower bounds on the ball measure of noncompactness of operators acting between considerably general quasinormed function spaces. This new method removes some of the restrictions oft-presented in the previous work. Most notably, the target function space need not be disjointly superadditive nor equipped with a norm. Instead, a property that is far more often at our disposal is exploited, namely the absolute continuity of the target quasinorm. We use this new method to prove that limiting Sobolev embeddings into spaces of Brezis--Wainger type are so-called maximally noncompact, i.e., their ball measure of noncompactness is the worst possible.

Attosecond transient absorption spectroscopy (ATAS) is becoming an indispensable and powerful tool in the emerging field of attochemistry, while the interpretation of measurements often requires full considerations of the coupling among various freedoms of motion. Here we develop the ionization-coupling model to incorporate the transient absorption and explore the coupled electronic-vibrational dynamics of nitrogen ions induced by strong-field ionization (SFI), which has been investigated in the recent transient x-ray K-edge absorption experiment [PRL 129, 123002 (2022)]. It is found the coherent vibrational wave packet on the involved electronic state is created with broad distribution of vibrational levels which leads to the spectral overlap on the K-edge absorption. By identifying the contributions of each electronic state, the study provides a different interpretation revealing the significant role of the excited state $A^2Π_u$ arising from the electronic-vibrational coupling in strong laser fields. We uncover new features of absorption from forbidden transitions during the laser pulse and confirm the vibronic coherence induced modulations of absorbance after SFI. A new scheme is proposed to avoid the spectral overlap and determine the population among the states which is crucial to resolve the debate on nitrogen air lasing. The work lays down the framework to research the ionic coherence in ATAS and offers valuable insights into the intricate interplay between electronic and vibrational dynamics.

Let $p\geq 3$ be a prime. The hyper-algebraic elements in the $p$-adic Mal'cev-Neumann field $\mathbb{L}_p$ form an algebraically closed subfield $\mathbb{L}_p^{\operatorname{ha}}$. In this article, we clarify the relations among the fields $\mathbb{L}_p^{\operatorname{ha}}$, $\overline{\mathbb{Q}}_p$ and $\mathbb{C}_p$. We introduce two arithmetic invariants (hyper-tame index and hyper-inertia index) of hyper-algebraic elements and study the relation between these invariants and classical arithmetic invariants of $p$-adic algebraic numbers. Finally, we give a criterion for hyper-algebraic elements to be tamely ramified over $\mathbb{Q}_p$.

In this study, we present a new closed form for the generalized integral $$\int_0^1 \frac{\mathrm{Li}_2(z) \ln(1+az)}{z}\, \mathrm{d}z,$$ where $a \in \mathbb{C} \setminus(-\infty, -1)$ and $\mathrm{Li}_2(z)$ is the dilogarithm function. This generalization is achieved by leveraging our established findings in conjunction with Vălean's results. Furthermore, we provide explicit closed forms for associated integrals, prove a transformation formula for double infinite series, expressing them as the sum of the square of an infinite series and another infinite series. We utilize this relationship to derive a novel closed form for the generalized series $$\sum_{k=1}^\infty \frac{ ζ\left(m, \frac{rk-s}{r}\right) }{(rk-s)^m},$$ for $\Re(m) > 1$, $r, s \in \mathbb{C}$, where $r \neq 0$, $rk \neq s$, for any positive integer $k$, and $ζ(s, z)$ denotes the Hurwitz zeta function. Utilizing Hermite's integral representation for $ζ(s, z)$, we derive a family of integrals from this series.

This paper discusses generalizations of logarithmic and hyperbolic integrals. A new proof for an integral presented by Vardi and several other integrals in relation to known mathematical constants are discovered. We introduce the signed generalized Stirling polynomials of the first kind from the generalized Stirling polynomials of the first kind, and we give new expressions for the signed generalized Stirling polynomials of the first kind in terms of the Stirling cycle numbers and complete Bell polynomials. We establish the role of the signed generalized Stirling polynomials of the first kind and complete Bell polynomials in generalizing Malmsten's integral for all natural powers of the hyperbolic secant function, and we derive a reduction formula for the integral sequence. We give expressions for new integral sequences, which possess similar properties with Malmsten's integral, in terms of the signed generalized Stirling polynomials of the first kind, and we discover identities and a functional equation for the signed generalized Stirling polynomials of the first kind.

We are considering the problem of optimal portfolio delegation between an investor and a portfolio manager under a random default time. We focus on a novel variation of the Principal-Agent problem adapted to this framework. We address the challenge of an uncertain investment horizon caused by an exogenous random default time, after which neither the agent nor the principal can access the market. This uncertainty introduces significant complexities in analyzing the problem, requiring distinct mathematical approaches for two cases: when the random default time falls within the initial time frame [0,T] and when it extends beyond this period. We develop a theoretical framework to model the stochastic dynamics of the investment process, incorporating the random default time. We then analyze the portfolio manager's investment decisions and compensation mechanisms for both scenarios. In the first case, where the default time could be unbounded, we apply traditional results from Backward Stochastic Differential Equations (BSDEs) and control theory to address the agent problem. In the second case, where the default time is within the interval [0,T], the problem becomes more intricate due to the degeneracy of the BSDE's driver. For both scenarios, we demonstrate that the contracting problem can be resolved by examining the existence of solutions to integro-partial Hamilton-Jacobi-Bellman (HJB) equations in both situations. We develop a deep-learning algorithm to solve the problem in high-dimension with no access to the optimizer of the Hamiltonian function.

In this contribution, kernel approximations are applied as ansatz functions within the Deep Ritz method. This allows to approximate weak solutions of elliptic partial differential equations with weak enforcement of boundary conditions using Nitsche's method. A priori error estimates are proven in different norms leveraging both standard results for weak solutions of elliptic equations and well-established convergence results for kernel methods. This availability of a priori error estimates renders the method useful for practical purposes. The procedure is described in detail, meanwhile providing practical hints and implementation details. By means of numerical examples, the performance of the proposed approach is evaluated numerically and the results agree with the theoretical findings.

We prove global existence and decay for small-data solutions to a class of quasilinear wave equations on a wide variety of asymptotically flat spacetime backgrounds, allowing in particular for the presence of horizons, ergoregions and trapped null geodesics, and including as a special case the Schwarzschild and very slowly rotating $\vert a \vert \ll M$ Kerr family of black holes in general relativity. There are two distinguishing aspects of our approach. The first aspect is its dyadically localised nature: The nontrivial part of the analysis is reduced entirely to time-translation invariant $r^p$-weighted estimates, in the spirit of [DR09], to be applied on dyadic time-slabs which for large $r$ are outgoing. Global existence and decay then both immediately follow by elementary iteration on consecutive such time-slabs, without further global bootstrap. The second, and more fundamental, aspect is our direct use of a "blackbox" linear inhomogeneous energy estimate on exactly stationary metrics, together with a novel but elementary physical space top order identity that need not capture the structure of trapping and is robust to perturbation. In the specific example of Kerr black holes, the required linear inhomogeneous estimate can then be quoted directly from the literature [DRSR16], while the additional top order physical space identity can be shown easily in many cases (we include in the Appendix a proof for the Kerr case $\vert a \vert \ll M$, which can in fact be understood in this context simply as a perturbation of Schwarzschild). In particular, the approach circumvents the need either for producing a purely physical space identity capturing trapping or for a careful analysis of the commutation properties of frequency projections with the wave operator of time-dependent metrics.

We define duality triples and duality pairs in compactly generated triangulated categories and investigate their properties. This enables us to give an elementary way to determine whether a class is closed under pure subobjects, pure quotients and pure extensions, as well as providing a way to show the existence of approximations. One key ingredient is a new characterisation of phantom maps. We then introduce an axiomatic form of Auslander-Gruson-Jensen duality, from which we define dual definable categories, and show that these coincide with symmetric coproduct closed duality pairs. This framework is ubiquitous, encompassing both algebraic triangulated categories and stable homotopy theories. Accordingly, we provide many applications in both settings, with a particular emphasis on silting theory and stratified tensor-triangulated categories.

We consider difference-in-differences identification and estimation strategies when the parallel trends assumption holds conditional on covariates, which can be time-varying, time-invariant, or both. We uncover several weaknesses of two-way fixed effects (TWFE) regressions in this context. The most important, which we call \textit{hidden linearity bias}, arises because transformations that eliminate unit fixed effects also transform the covariates, either implicitly changing the identification strategy or relying on correct model specification. We provide diagnostics for assessing a TWFE regression's susceptibility to hidden linearity bias and propose alternative estimation strategies that circumvent these issues.

The entropic doubling $σ_{\operatorname{ent}}[X]$ of a random variable $X$ taking values in an abelian group $G$ is a variant of the notion of the doubling constant $σ[A]$ of a finite subset $A$ of $G$, but it enjoys somewhat better properties; for instance, it contracts upon applying a homomorphism. In this paper we develop further the theory of entropic doubling and give various applications, including: (1) A new proof of a result of Pálvölgyi and Zhelezov on the ``skew dimension'' of subsets of $\mathbf{Z}^D$ with small doubling; (2) A new proof, and an improvement, of a result of the second author on the dimension of subsets of $\mathbf{Z}^D$ with small doubling; (3) A proof that the Polynomial Freiman--Ruzsa conjecture over $\mathbf{F}_2$ implies the (weak) Polynomial Freiman--Ruzsa conjecture over $\mathbf{Z}$.

As the amount and complexity of available data increases, the need for robust statistical learning becomes more pressing. To enhance resilience against model misspecification, the generalized posterior inference method adjusts the likelihood term by exponentiating it with a learning rate, thereby fine-tuning the dispersion of the posterior distribution. This study proposes a computationally efficient strategy for selecting an appropriate learning rate. The proposed approach builds upon the generalized posterior calibration (GPC) algorithm, which is designed to select a learning rate that ensures nominal frequentist coverage. This algorithm, which evaluates the coverage probability using bootstrap samples, has high computational costs because of the repeated posterior simulations needed for bootstrap samples. To address this limitation, the study proposes an algorithm that combines elements of the GPC algorithm with the sequential Monte Carlo (SMC) sampler. By leveraging the similarity between the learning rate in generalized posterior inference and the inverse temperature in SMC sampling, the proposed algorithm efficiently calibrates the posterior distribution with a reduced computational cost. For demonstration, the proposed algorithm was applied to several statistical learning models and shown to be significantly faster than the original GPC.

Consider the community detection problem in random hypergraphs under the non-uniform hypergraph stochastic block model (HSBM), where each hyperedge appears independently with some given probability depending only on the labels of its vertices. We establish, for the first time in the literature, a sharp threshold for exact recovery under this non-uniform case, subject to minor constraints; in particular, we consider the model with multiple communities. One crucial point here is that by aggregating information from all the uniform layers, we may obtain exact recovery even in cases when this may appear impossible if each layer were considered alone. Besides that, we prove a wide-ranging, information-theoretic lower bound on the number of misclassified vertices \emph{for any algorithm}, depending on a \emph{generalized Chernoff-Hellinger} divergence involving model parameters. We provide two efficient algorithms which successfully achieve exact recovery when above the threshold, and attain the lowest possible mismatch ratio when the exact recovery is impossible, proved to be optimal. The theoretical analysis of our algorithms relies on the concentration and regularization of the adjacency matrix for non-uniform random hypergraphs, which could be of independent interest. We also address some open problems regarding parameter knowledge and estimation.

We consider a general Kinetic Fokker-Planck (KFP) equation in a domain with Maxwell reflection condition on the boundary, not necessarily with conservation of mass. We establish the wellposedness in many spaces including Radon measures spaces, and in particular the existence and uniqueness of fundamental solutions. We also establish a Krein-Rutman theorem with constructive rate of convergence in an abstract setting that we use for proving that the solutions to the KFP equation converge toward the conveniently normalized first eigenfunction. Both results use the ultracontractivity of the associated semigroup in a fundamental way.

We present a Chebyshev collocation method for linear ODE and DDE problems. We first give a posteriori estimates for the accuracy of the approximate solution of a scalar ODE initial value problem. Examples of the success of the estimate are given. For linear, periodic DDEs with integer delays we define and discuss the monodromy operator U as our main goal is reliable estimation of the stability of such DDEs. We prove a theorem which gives a posteriori estimates for eigenvalues of U, our main result. This result is based on a generalization to operators on Hilbert spaces of the Bauer-Fike theorem for (matrix) eigenvalue perturbation problems. We generalize these results to systems of DDEs. A delayed, damped Mathieu equation example is given. The computation of good bounds on ODE fundamental solutions is an important technical issue; an a posteriori method for such bounds is given. Certain technical issues are also addressed, namely the evaluation of polynomials and the estimation of L^\infty norms of analytic functions. Generalization to the non-integer delays case is also considered.

We construct a $d$-dimensional Eddy Damped Quasi-Normal Markovian (EDQNM) Closure Model to study dynamo action in arbitrary dimensions. In particular, we find lower $d_L$ and upper $d_U$ critical dimensions for sustained dynamo action in this incompressible problem. Our model is adaptable for future studies incorporating helicity, compressible effects and a wide range of magnetic Reynolds and Prandtl numbers.