The eigenstate thermalization hypothesis provides a framework for understanding thermalization in isolated quantum many-body systems by characterizing statistical properties of local observables in energy eigenstates. Here we demonstrate that distributions of matrix elements in macroscopic systems may depend not only on the macrostate parameters, such as the densities of local conserved charges, but generally also on the properties of ensembles used in sampling eigenstates. To this end, we depart from the conventional analysis of microcanonical windows and consider statistical ensembles with an adjustable scale parameter prescribing the magnitude of charge fluctuations. We specifically consider an integrable field theory that permits efficient numerical sampling of matrix elements and reliable extrapolation to the thermodynamic limit. Moreover, in this system, we identify a class of states that enables explicit closed-form computation of the suppression rate of matrix elements. Our findings reveal an underlying multiscale structure of matrix elements captured by a non-analytic fluctuation-scale dependence of algebraic exponents governing their statistical properties.

$L_1$-Approximating polynomials, i.e., polynomials that approximate indicator functions in $L_1$-norm under certain distributions, are widely used in computational learning theory. We study the existence of \textit{non-negative} $L_1$-approximating polynomials with respect to Gaussian distributions. This is a stronger requirement than $L_1$-approximation but weaker than sandwiching polynomials (which themselves have many applications). These non-negative approximating polynomials have recently found uses in smoothed learning from positive-only examples. In this short note, we prove that every class of sets with Gaussian surface area (GSA) at most $Γ$ under the standard Gaussian admits degree-$k$ non-negative polynomials that $\eps$-approximate its indicator functions in $L_1$-norm, for $k=\tilde{O}(Γ^2/\varepsilon^2)$. Equivalently, finite GSA implies $L_1$-approximation with the stronger pointwise guarantee that the approximating polynomial has range contained in $[0,\infty)$. Up to a constant-factor, this matches the degree of the best currently known Gaussian $L_1$-approximation degree bound without the non-negativity constraint.

Preventing signal detection in communication and active sensing requires careful control of transmission power. In fact, the square-root laws (SRL) for covert classical and quantum communication and sensing prescribe that the average output power per channel use scales as $1/\sqrt{n}$ for $n$ channel uses. Two strategies for achieving this are diffuse and sparse signaling. The former transmits signals with power decaying as $1/\sqrt{n}$ on all $n$ channel uses, which is convenient for mathematical analysis. The latter transmits constant-power signals rarely, on approximately $\sqrt{n}$ out of $n$ channel uses, while remaining silent on the others. This offers significant practical advantages in compatibility with modern digital transmitters. Here, we study sparse signaling over lossy thermal-noise bosonic channels, which describe quantumly many practical channels (including optical, microwave, and radio-frequency). We characterize the input signal state that minimizes detectability. We find an unintuitive optimal quantum state structure: a mixture of just two consecutive photon-number states. In particular, in the low-brightness regime, the optimal signal state is a mixture of vacuum and a single photon. Since these states are generally suboptimal for both communication and active sensing, we explore the resulting trade-off and identify input-power thresholds for transitions between optimizing for covertness vs. performance in communication and sensing tasks.

We studied the constrained Hamiltonian formulation of a supersymmetric Korteweg-de Vries (KdV) equation, which is observed to be a constrained system similar to its classical version. We found a nontrivial Lagrangian description, where we select $a=2$ for the free parameter $a$ in the supersymmetric extension. The corresponding degenerate Lagrangian requires an exclusive consideration and the utilization of the Dirac-Bergmann algorithm. We explicitly determined the full set of primary and secondary constraints and constructed the total Hamiltonian governing the dynamics of the system. In this analysis, in addition to a nontrivial constraint involving the fermionic fields, the consistency conditions give rise to a nonlocal contribution to the Hamiltonian density. This highlights a distinctive feature of this supersymmetric extension. We showed that the resulting Hamilton equations of motion reproduce the supersymmetric KdV system in the component form. Finally, we derived a compact superspace representation of the Hamiltonian and demonstrated its consistency with the component-level formulation.

A projective hyperkähler manifold of Kummer-type is said to be twisted modular if it is birational to the Albanese fiber of a moduli space of twisted sheaves on an abelian surface. We prove that, with the exception of certain cases of Picard rank 3, any projective Kummer-type manifold admitting a finite-order symplectic birational self-map that acts nontrivially on its second cohomology group is twisted modular. We provide a complete characterization of these exceptions in terms of their Néron-Severi lattices. We then investigate symplectic birational self-maps of modular Kummer-type manifolds, determining exactly which Mukai vectors allow the birational transformation induced by crossing the vertical wall, which acts on cohomology as a reflection, to correspond to a finite-order symplectic birational self-map. Additionally, we prove in an appendix several results concerning moduli spaces of twisted sheaves on abelian surfaces which were not readily available in the literature.

We consider the phase ordering problem for the low-temperature Ising dynamics initialized from a biased and disordered initialization. Work of Fontes, Schonmann, Sidoravicius (2002) showed that at zero-temperature, Ising Glauber dynamics on $\mathbb Z^d$ for $d\ge 2$ initialized from i.i.d. spins on each vertex that are $+1$ with sufficiently large probability, absorbs into the all-plus configuration quickly. We prove that analogous behavior holds throughout the low-temperature regime of the Ising model in two dimensions. Namely, there exists $p_0 <1$ such that Ising Glauber dynamics initialized from i.i.d. spins that are $+1$ with probability $p>p_0$, run at any low temperature $β>β_c$ converges rapidly to the plus phase measure $π^+$. The result is proved using a spacetime multiscale coupling valid in any $d\ge 2$, that boosts a uniform-in-$β$ quasi-polynomial bound on the mixing time of Ising dynamics with plus boundary conditions, into rapid phase ordering from biased initializations with no boundary conditions.

This work concerns the design and analysis of a limiting technique that allows the preservation of invariant domains for high-order numerical approximations of nonlinear hyperbolic systems of conservation laws. The method can be applied to any conservative discretization method in space as well as to a wide range of explicit and implicit time integration schemes. The method limits the high-order solution around a low-order accurate solution that is known to preserve all the invariant domains. It generalizes the flux-corrected transport limiter [J. P. Boris and D. L. Book, J. Comput. Phys., 11, 1973; S. T. Zalesak, J. Comput. Phys., 31, 1979] to systems of conservation laws and relies on the limitation of antidiffusive fluxes, but defines the limiting coefficients so as to express the limited solution as a convex combination of invariant domain preserving quantities similarly to the convex limiting framework [Guermond et al., Comput. Methods Appl. Mech. Engrg., 347, 2019]. We give details on the derivation of this limiting technique and provide some illustration with finite volume or discontinuous Galerkin (DG) space discretizations associated to explicit or implicit Runge-Kutta methods as well as to time DG integrations. The limiter is applied iteratively to refine the limited solution around the high-order one, while preserving the invariant domains, and a heuristic is proposed to accelerate its convergence. Numerical experiments solving one- and two-dimensional problems involving scalar hyperbolic equations and the compressible Euler equations are presented to illustrate the properties of these schemes.

Let $\mathrm{Mac}(W)$ be the MacNeille completion of the Bruhat order of a Coxeter group $W$. We introduce an action of the $0$-Hecke monoid of type $W$ on $\mathrm{Mac}(W)$, which allows us to define a weak order and a descent set statistic on $\mathrm{Mac}(W)$. When $W$ is of type $A$, we recover constructions of Hamaker and Reiner, which were originally formulated in terms of monotone triangles and alternating sign matrices. Using this action, we prove that certain unions of Knutson--Miller subword complexes are vertex-decomposable. By specializing to type $A$, we prove a conjecture of Escobar, Klein, and Weigandt regarding Cohen--Macaulay ASM varieties. Along the way, we also exhibit a counterexample to a conjecture of Hamaker and Reiner regarding the poset topology of intervals in the ASM weak order. Finally, when $W$ is finite and irreducible, we use our $0$-Hecke action to introduce a noninvertible dynamical system on $\mathrm{Mac}(W)$ that we call the MacNeille pop-stack operator, and we prove that the maximum number of iterations of this operator needed to reach the bottom state is $h-1$, where $h$ is the Coxeter number of $W$. This article is meant to serve as a case study in using large language models to automate the workflow of mathematical research. The proof of the conjecture of Escobar--Klein--Weigandt and the disproof of the conjecture of Hamaker--Reiner were obtained autonomously by ChatGPT 5.4 Pro. Other aspects of the paper were obtained mostly by the author, but ChatGPT expedited the process. We provide a detailed account of this interaction, and we speculate on what allowed the model to be successful.

Approximate stationarity conditions provide necessary optimality conditions without requiring additional assumptions by demanding that a perturbed stationarity system possesses solutions as the involved perturbations tend to zero. Together with associated approximate constraint qualifications, which are typically rather mild, they raised much interest in the optimization community during the last decade. In parallel, directional stationarity conditions became quite popular as they sharpen standard stationarity conditions by incorporating data associated with underlying critical directions. The purpose of this paper is twofold. First, we melt the aforementioned concepts of approximate and directional stationarity to formulate and study so-called approximate directional stationarity. For the underlying model problem, an optimization problem with nonsmooth geometric constraints is chosen, which covers diverse practically relevant applications. The role of approximate directional stationarity as a necessary optimality condition is investigated in much detail, complementing results from the literature. Second, we formulate a qualification condition which, based on an approximately directionally stationary point, can be exploited to infer its directional stationarity. The latter condition depends on one particular sequence verifying approximate directional stationarity and merely requires to check a simple condition of Mangasarian--Fromovitz type stated in terms of the directional tools of limiting variational analysis. This contrasts standard approximate constraint qualifications that typically demand a certain stable behavior of all sequences validating approximate stationarity. Throughout, various approaches to verify directional stationarity of local minimizers are established, and illustrative examples are presented to make the theoretical results more accessible.

We derive the exact asymptotic rates of the small eigenvalues of the Laplacian on one-parameter degenerations of compact Kähler manifolds equipped with induced background metrics. This generalizes a recent result of Dai and Yoshikawa to higher dimensions. To achieve this, we combine Li's uniform Skoda inequality with the method of auxiliary Monge-Ampère equations, introduced by Guo--Phong--Song--Sturm--Tong and adapted by Guedj--Tô. As an application, we establish estimates for degenerations of compact Kähler manifolds with reducible singular fibers.

Oscillatory activity in auditory cortex is thought to play a central role in auditory and speech processing by synchronizing neural rhythms to external acoustic features of the speech stream. To support this function, cortical oscillators must flexibly phase-lock to inputs spanning a wide range of timescales, including rhythms substantially slower than their intrinsic frequency. Here we identify a general dynamical mechanism by which intrinsic inhibitory currents operating on multiple timescales enable such flexible phase-locking. Using tools from dynamical systems theory, we show that interactions between slow and superslow inhibitory processes generate prolonged post-input recovery delays through delayed Hopf phenomena, thereby substantially expanding the frequency range over which entrainment can occur. We demonstrate this mechanisms in a biophysically grounded cortical theta oscillator model for speech segmentation. Specifically, we show that both a theta-timescale (4-8 Hz) inhibitory current $I_m$ and a slower delta-timescale (1-4 Hz) inhibitory potassium current $I_{\rm K_{SS}}$ are crucial for entrainment flexibility. Their interaction creates a three-timescale structure that gives rise to pronounced delay phenomena associated with a delayed Hopf bifurcation (DHB). Interestingly, the superslow $I_{\rm K_{SS}}$ and the associated DHB play little role in the unforced oscillatory dynamics, but are recruited to support phase locking under external forcing. Moreover, the intermediate-timescale current $I_m$, rather than being redundant, further expands the phase-locking range by prolonging delayed recovery along the superslow manifold. Together, these results suggest that coordination among intrinsic inhibitory currents operating on multiple timescales may represent a key mechanism supporting flexible phase locking to rhythmic inputs in the brain.

We study a class of bilevel optimization problems in which both the upper- and lower-level problems have minimax structures. This setting captures a broad range of emerging applications. Despite the extensive literature on bilevel optimization and minimax optimization separately, existing methods mainly focus on bilevel optimization with lower-level minimization problems, often under strong convexity assumptions, and are not directly applicable to the minimax lower-level setting considered here. To address this gap, we develop penalty-based first-order methods for bilevel minimax optimization without requiring strong convexity of the lower-level problem. In the deterministic setting, we establish that the proposed method finds an $ε$-KKT point with $\tilde{O}(ε^{-4})$ oracle complexity. We further show that bilevel problems with convex constrained lower-level minimization can be reformulated as special cases of our framework via Lagrangian duality, leading to an $\tilde{O}(ε^{-4})$ complexity bound that improves upon the existing $\tilde{O}(ε^{-7})$ result. Finally, we extend our approach to the stochastic setting, where only stochastic gradient oracles are available, and prove that the proposed stochastic method finds a nearly $ε$-KKT point with $\tilde{O}(ε^{-9})$ oracle complexity.

We prove that the KSGNS construction can be viewed as an endofunctor on a category whose objects are positive $C^*$-correspondences from a fixed $C^*$-algebra and morphisms are given by intertwiners which account for automorphisms of the fixed $C^*$-algebra. Using this perspective, we provide a functorial perspective for strict positive equivariant $C^*$-correspondences of $C^*$-dynamical systems and show every strict positive equivariant $C^*$-correspondence of $C^*$-dynamical systems unitarily uniquely dilates under the KSGNS construction to an equivariant $C^*$-correspondence of the dynamical systems.

Multivariate linear regression is a fundamental statistical task, but classical estimators such as ordinary least squares are highly sensitive to outliers. These may occur as casewise outliers that affect entire observations, or as outlying cells, that are individual contaminated entries in the predictor and/or response matrix. Moreover, modern datasets frequently contain missing values and are high-dimensional. To address these challenges we propose the cellwise multivariate regression (cellMR) estimator, a robust regression method that simultaneously accommodates casewise and cellwise outliers, missing data, and high dimensionality. The approach builds on a cellwise robust covariance estimator and uses ridge regularization for numerical stability. We further introduce cellBoot, a novel bootstrap-based inference procedure tailored to the cellMR framework. Relying on indirect inference, cellBoot provides asymptotically valid confidence intervals that are robust to casewise and cellwise contamination. We derive influence functions of the regression estimator and prove the asymptotic validity of the cellBoot confidence intervals. Simulations and a real genomics application illustrate the strong finite-sample performance of the proposed methods.

Geometric medians on product manifolds are sensitive to the relative scaling of factor metrics because the median objective couples the factors rather than separating them. We study this scale-selection problem and first prove that naive joint minimization over location and scale is degenerate: the scale is driven to the boundary and the problem collapses to a marginal median, effectively discarding one factor. Thus relative scale is not identifiable from the raw median loss alone. We develop three alternatives to mitigate this issue. The first treats scale as indexing a sensitivity path and establishes uniform consistency, a functional central limit theorem, and a derivative-based sensitivity measure. The second constructs a robust scale-calibrated median using marginal radial median scales, yielding unit invariance, consistency, a two-step central limit theorem, and bounded influence. The third introduces a bounded balance equation for direct scale estimation, with monotonicity, uniqueness, joint asymptotic normality, and bounded influence. Simulations illustrate boundary collapse, sensitivity, unit invariance, and balanced estimation in Euclidean and Bures-Wasserstein settings.

We propose semantic smoothing, a smoothing method for language models that uses embeddings to share statistical observations across semantically similar contexts. The starting point is a decomposition of log-perplexity that motivates smoothing as a collection of distribution-estimation problems under Kullback-Leibler (KL) loss. We then show that, under a Lipschitz-logit model for embedding-based language generation, proximity of context embeddings implies proximity of the corresponding next-word distributions in KL divergence. Combining these observations, we formulate semantic smoothing as distribution estimation in KL loss with KL-proximity side information. For $n$ samples on a $d$-symbol alphabet with a side-information distribution at KL distance $Δ$, we give an interpolation estimator with worst-case KL risk $O(\min\{Δ,d/n\})$, and prove a matching-order lower bound for uniform side information. We extend the estimator to multiple and empirically estimated synonymous distributions. Experiments on synthetic Markov data and WikiText-103 bigram models using Word2Vec, GloVe, and GPT-2 embeddings show that semantic smoothing consistently reduces test perplexity when applied to add-constant and Kneser-Ney estimates.

We review the basic theory of bands and band schemes introduced by Baker-Jin-Lorscheid, which is an algebraic framework for tropicalization, analytification, and $\mathbb{F}_1$-geometry. For an affine scheme $X$ over a non-Archimedean valued field $k$, one can associate to every affine embedding $ι$ of $X$ a naturally defined affine band scheme $Y_ι$ whose rational points over the tropical band $\mathbb{T}$ recover the tropicalization $Trop(X,ι)$. We prove that $X$ is the limit of the $Y_ι$ in the category of band schemes, thereby obtaining a scheme-theoretic enhancement of Payne's limit theorem. By taking $\mathbb{T}$-rational points, this recovers Payne's theorem for affine tropicalizations from the perspective of band scheme theory and the same method provides an analogous result in the real tropical setting.

These notes introduce the theory of susceptibilities as developed in [arXiv:2504.18274, arXiv:2601.12703] for interpreting neural networks. The susceptibility of an observable $ϕ$ to a data perturbation is defined as a derivative of a posterior expectation, which by the fluctuation--dissipation theorem equals a posterior covariance. Different choices of $ϕ$ yield different objects: per-sample losses give the influence matrix (the Bayesian influence function of [arXiv:2509.26544]), while component-localized observables give the structural susceptibility matrix that pairs model components with data patterns. The susceptibility matrix is (up to a factor of $nβ$) the Jacobian of the map from data distributions to structural coordinates; its pseudo-inverse provides a linearized solution to the patterning problem of [arXiv:2601.13548]: finding data perturbations that produce a desired structural change. We motivate the theory from its statistical-mechanical foundations, then give a detailed exposition of susceptibilities, their empirical estimators, and their connection to the geometry of the loss landscape.

Let $U\subseteq H^0(\mathcal{O}_{\mathbb{P}^1\times \mathbb{P}^1}(a,b))$ be a four-dimensional vector space and consider the rational map $ϕ_U:\,\mathbb{P}^1\times \mathbb{P}^1 \dashrightarrow \mathbb{P}^3$ defined by its basis of bihomogeneous polynomials. The tensor product surface $X_U\subseteq \mathbb{P}^3$ is the closed image of $ϕ_U$, and a fundamental problem in this setting is to determine its implicit equation. As these surfaces are ubiquitous within the field of geometric modeling and design, knowledge of their implicit equations is particularly advantageous, allowing for more effective and efficient computations. In this article, we expand upon work of Duarte-Schenck and work of the present author to solve this implicitization problem when the bigraded ideal $I_U$ admits a singly graded syzygy.

We define susceptibilities as a measure of the response of an observable quantity of a parameterized statistical model to a perturbation of the data for a general class of observables. We define estimators for these susceptibilities as statistics in a sequence of n data-points and prove that these estimators are consistent and asymptotically unbiased in the large n regime.

Diffusion models perform remarkably well on high-dimensional data such as images, often using only a modest number of reverse-time steps. Despite this practical success, existing convergence theory does not fully explain why such samplers remain efficient in high dimensions. Many prior KL guarantees bound the discretization error in terms of the ambient dimension, while other improved results replace this dependence using intrinsic-dimensional or geometric structure assumptions. In this work, we develop an alternative information-theoretic perspective on diffusion sampler convergence. We prove that, for Gaussian mixture targets, the discretization error is controlled by the Shannon entropy of the latent mixture component rather than by the ambient dimension. Consequently, the leading step complexity scales linearly with latent entropy and depends only logarithmically on the second moment of the data. Our analysis also extends to discrete target distributions, where the relevant complexity is the entropy of the target rather than the dimension of the embedding space. These results suggest that diffusion sampling can remain efficient in high-dimensional spaces when the data distribution admits a compact latent representation, as is widely believed to be the case for natural images.

This paper deals with the kernel density estimator based on the so-called sinc (or Fourier integral) kernel $K(x)=(πx)^{-1}\sin x$. We study in detail both asymptotic and finite sample properties of this estimator. It is shown that, contrary to widespread opinion, the sinc estimator is superior to other estimators in many respects: it is more accurate for quite moderate values of the sample size, has better asymptotics in non-smooth case (the density to be estimated has only first derivative), is more convenient for the bandwidth selection, etc.

Transformers have revolutionized machine learning and deploying attention layers in the model is increasingly standard across a myriad of applications. Further, for large models, it is common to implement Low Rank Adaptation (LoRA), whereby a factorized parameterization of them is trained, to achieve a surprisingly beneficial accuracy-size trade-off. In this work, via a unified framework we rigorously establish trainability of such models under stochastic methods. We prove that for any mild regularization, the empirical regression loss on a attention layer and LoRA on a shallow neural net, both induce Poincaré inequality for the corresponding Gibbs' measure. Then it follows via invoking recent results that a certain SDE, which mimics the SGD, minimizes the corresponding losses. In both the cases, our first-of-its-kind results of trainability on attention and nets, do not rely on any assumptions on the data or the size of the architecture.

We provide a transformation rule for adjoint test modules along Cohen--Macaulay maps between Cohen--Macaulay varieties that have $F$-rational geometric fibers. This is, in part, an effective version of Enescu's theorem on the ascent of $F$-rationality under local maps with $F$-rational geometric fibers.

A stochastic free-boundary problem for the three-dimensional barotropic compressible Navier--Stokes equations is studied. The main feature of the model is that the free boundary is transported by a Stratonovich stochastic flow, so that the noise enters the kinematic boundary condition and hence the evolution of the moving domain. An additional Itô forcing in the momentum equation is also allowed. The problem is transformed by a stochastic Lagrangian map generated by the velocity and the transport vector fields. In these coordinates the density is represented through the Jacobian of the flow, and the remaining system is solved by combining stochastic maximal regularity, deterministic %\rL^p%-%\rL^q$ estimates, and a localized contraction argument. Local pathwise well-posedness is obtained up to an a.s. positive stopping time, with strictly positive density and pathwise uniqueness.

The escalating complexity of modern machine learning necessitates solving challenging non-convex optimization problems, particularly in high-dimensional regimes and scenarios contaminated by gross outliers. Traditional approaches, relying on convex relaxations or specialized local search heuristics, frequently succumb to suboptimal local minima and fail to recover the true underlying discrete structures. In this paper, we propose treating these non-convex challenges as a global search problem and introduce a unified framework based on Quantum-Inspired Evolutionary Optimization (QIEO). By leveraging a probabilistic representation inspired by quantum superposition, QIEO maintains a global view of the search space, enabling it to tunnel through local optima that trap conventional gradient-based and greedy solvers. We comprehensively evaluate QIEO across diverse non-convex applications, including sparse signal recovery (gene expression analysis and compressed sensing) and robust linear regression. Extensive benchmarking against state-of-the-art continuous solvers (ADAM, Differential Evolution), classical metaheuristics (Genetic Algorithms), and specialized non-convex algorithms (Iterative Hard Thresholding) demonstrates that QIEO consistently achieves superior structural fidelity, lower mean squared error, and enhanced robustness without support inflation. Our findings suggest that embracing a quantum-inspired global search provides a resilient, unified paradigm for overcoming the inherent intractability of discrete nonconvex machine learning landscapes.

We classify, up to projective automorphism, all homogeneous pre-foliations of co-degree one and degree four on the complex projective plane $\Ptwo$ whose Legendre transform defines a flat $4$-web. The classification is organized according to the type of the underlying homogeneous foliation $\Hcal$ of degree~$3$, distinguishing the cases $°(\Tcal_{\Hcal})=2$, $3$, and~$4$. The case $°(\Tcal_{\Hcal})=2$ was treated by Bedrouni, while the cases $°(\Tcal_{\Hcal})=3$ and $°(\Tcal_{\Hcal})=4$ are completed here. The proof combines Bedrouni's curvature-holomorphy criteria with explicit normal forms and symbolic computation; the result yields a finite list of explicit one-forms, parametrised by the ramification data of the Gauss map of~$\Hcal$.

Sampling from a high-dimensional probability distribution is a fundamental algorithmic task arising in wide-ranging applications across multiple disciplines, including scientific computing, computational statistics and machine learning. Langevin Monte Carlo (LMC) algorithms are among the most widely used sampling methods in high-dimensional settings. This paper introduces a novel higher-order and Hessian-free LMC sampling algorithm based on an efficient stochastic Runge--Kutta method of strong order $1.5$ for the overdamped Langevin dynamics. In contrast to the existing Runge--Kutta type LMC (Li et al., 2019) involved with three gradient evaluations, the newly proposed algorithm is computationally cheaper and requires only two gradient evaluations for one iteration. Under certain log-smooth conditions, non-asymptotic error bounds of the proposed algorithms are analyzed in $\mathcal{W}_2$-distance. In particular, a uniform-in-time convergence rate of order $O(d ^{\frac32} h^{\frac32})$ is derived in a non-log-concave setting, matching the convergence rate proved in the aforementioned work but under the log-concavity condition. Numerical experiments are finally presented to demonstrate the effectiveness of the new sampling algorithm.

Eugenio Beltrami published his seminal 'Essay on the Interpretation of Non-Euclidean Geometry' in 1868, where he showed that geodesics on a surface of constant negative curvature can be mapped as straight lines on a Euclidean disc. More importantly he showed that figures on the disc would satisfy the identities of hyperbolic geometry characteristic of a surface of negative curvature. However Beltrami did not always give a full explanation of the equations which he used. These notes are an attempt to provide a derivation of some of his principal results, including his formula for hyperbolic distance on the disc, his proof that the sum of the (hyperbolic) angles of a triangle on the disc is less than two right angles and his equations for circles, equidistants and horocycles.

We study connected graphs with a fixed degree sequence, in the sparse setting where the number of edges grows linearly in the number of vertices. Using the relation to the configuration model, we identify the number of such connected graphs up to the exponential order. We do this by viewing a connected graph with a given degree distribution as the realization of the giant component in a larger configuration model, and carefully choosing the degree distribution of the larger graph so that it is likely that its giant component has the required degree distribution. To ensure that the connected graph has exactly the correct degrees, we use a switching argument. Additionally, we obtain results on rare event probabilities and describe the local structure of a uniform connected graph with a fixed degree sequence.