Papers

We compute the cellular $\mathbb{A}^1$-homology of De Concini–Procesi wonderful models of subspace arrangements. For a building set $\mathcal{G}$ over a field $k$, we identify the cellular $\mathbb{A}^1$-chain complex of $\mathbb{P}(\mathcal{G})$ with an $\eta$-twisted nested-set complex carrying Milnor–Witt coefficients and derived orientation data. The key geometric input is a motivic blow-up calculation: for a blow-up along a smooth center of codimension $c$, the relevant connecting class is $(c-1)_\epsilon\eta$, hence it is zero for $c$ odd and $\eta$ for $c$ even. This replaces the parity condition in the computation of Rains by a Milnor–Witt attaching class. As a consequence, the part of cellular $\mathbb{A}^1$-homology surviving after multiplication by $\eta$, and also the homology after inverting $\eta$, are expressed by the interval cohomology of the $2$-divisible subposet of the lattice generated by $\mathcal{G}$. For the braid arrangement, the condition becomes the odd-block condition on partitions, yielding explicit decompositions for the cellular $\mathbb{A}^1$-homology of $\overline{\mathcal M}_{0,N}$ and examples in low rank.

In this note, we show a condition on the derivatives of a multiplier $m\in L^\infty$ that guarantees that $m$ is a multiplier from $L^1(\mathbb{R}^n)$ to the Hardy Space $H^1(\mathbb{R}^n)$.

We develop a unified approach, based on projectively coresolved Gorenstein flat (PGF) objects, for constructing flat model structures on diagram categories. Specifically, we show that PGF objects in such categories are fully determined by their objectwise components, which in turn enables us to establish hereditary abelian model structures whose trivial cofibrant objects are precisely the flat objects. As an application, we reobtain flat model structures on $Q$-shaped derived categories, thereby providing a common framework that subsumes classical constructions for chain complexes. Moreover, we obtain an explicit description of the cofibrant objects in these models.

We study how standard randomization procedures affect the local geometry of quasi-Monte Carlo point sets, as measured by their minimum distance and mesh ratio. Although probabilistic selection within structured lattice families can produce quasi-uniform point sets, randomizing an existing low-discrepancy construction need not preserve quasi-uniformity. We first determine sharp probabilistic orders for Monte Carlo, jittered, and Latin hypercube sampling, whose mesh ratios diverge as positive powers of $N$. The orders are $\Theta_{\mathbb{P}}(N^{1/d}(\log N)^{1/d})$ for Monte Carlo sampling, $\Theta_{\mathbb{P}}(N^{1/(d+1)})$ for jittered sampling, and, for $d\ge 2$, $\Theta_{\mathbb{P}}(N^{1/d}(\log N)^{1/d})$ for Latin hypercube sampling. We also obtain a Weibull limit law for the minimum distance of jittered samples. For full Owen scrambling, every family of fixed-$t$ nets has minimum distance $O_{\mathbb{P}}(N^{-3/(2d)})$, and its mesh ratio is therefore $\Omega_{\mathbb{P}}(N^{1/(2d)})$. Under uniform coincidence and common-prefix conditions, these bounds are sharp up to logarithmic factors. Moreover, a single full Owen scrambling of any $(t,d)$-sequence is almost surely non-quasi-uniform. By contrast, for matrix and linear scrambling of binary digital nets with fixed $t$ in dimension $d\ge 2$, the mesh ratio is $O_{\mathbb{P}}(\log N)$, whereas it is $\Theta_{\mathbb{P}}(\log N)$ in the separate balanced-prefix affine-tail model. The model also yields the exact probabilistic order for one-dimensional binary digital $(0,m,1)$-nets under matrix or linear scrambling. These results demonstrate that the geometric effect of randomization is governed by whether it introduces local independence or shared algebraic randomness.

In this paper, we show that the product, or equivalently the convolutions of two functions in the Hermite-Sobolev spaces $\mathcal{S}_p(\mathbb{R}^d)$ is again in the same space, for $p$ depending on the dimension $d$. As a consequence, for such $p$ we show that the product, or equivalently the convolutions of $\phi \in \mathcal{S}_p(\mathbb{R}^d)$ and $\psi \in \mathcal{S}_{-p}(\mathbb{R}^d)$ is in $\mathcal{S}_{-p}(\mathbb{R}^d)$. As a further consequence, we show that the operators of translation by $x$ on $\mathcal{S}_p(\mathbb{R}^d)$ are bounded uniformly in $x \in \mathbb{R}^d$.

Let $X,X'$ be independent and identically distributed discrete real-valued random variables of finite Shannon entropy, and write $H(X)$ for the Shannon entropy of $X$. We prove that \[ \max\{H(X+X'),\,H(XX')\} \ge \frac87 H(X)-O(\log H(X)). \] This is the entropic analog of the celebrated sum-product phenomenon, and answers a question of Goh, which simply asked for a coefficient strictly larger than 1. An example by the author, Gavalakis, and Kontoyiannis showed the coefficient cannot exceed $\frac43$. Previous work by Gavalakis, Goh, and Kontoyiannis was able to prove a result of a weaker form, which could not translate to a coefficient strictly larger than 1 because of examples where the min-entropy is significantly smaller than the Shannon entropy. By splitting the distribution of $X$ into uniform pieces, which costs $O(\log H(X))$ entropy, we obviate this issue, establishing a coefficient of $\frac{10}{9}$. We augment this to $\frac87$ by adapting the work of Solymosi, which established the combinatorial sum-product phenomenon with coefficient $\frac43$ by bounding the multiplicative energy, to the entropy setting, again via a uniformization technique.

We formulate a baseball plate appearance as a finite two-person, zero-sum, extensive-form game with chance moves and imperfect information. Successive pitch cycles are connected through the evolving count, while the Batter chooses an action without observing the Pitcher's current pitch selection. Terminal outcomes are valued through a run-expectancy utility that incorporates both runs scored during the plate appearance and the continuation value of the half-inning. We implement the sequence-form representation of the game, whose linear-programming formulation grows linearly with the game tree and permits exact minimax equilibrium computations for trees far beyond the practical range of the normal form. Nature's transition probabilities are estimated from MLB Statcast data. We also establish two complementary dynamic-programming interpretations. For general finite two-person zero-sum games with perfect recall, we show that the dual variables of the sequence-form best-response programs decompose into reach weights and conditional continuation values. Under the additional state-Markov assumptions of the baseball model, we prove that the full minimax equilibrium can be computed by backward induction over the twelve non-terminal counts. The resulting computations produce equilibrium strategies, continuation values, and conditional and reach-weighted measures of the strategic cost of non-optimal actions.

Sparse identification of nonlinear dynamics (SINDy) and PDE functional identification (PDE-FIND) recover parsimonious ordinary and partial differential equations (ODEs and PDEs) from data. However, sparse and noisy temporal measurements can make derivative estimates unreliable. To address this problem, we evaluate Koopman-based upsampling techniques implemented with dynamic mode decomposition (DMD), extended DMD (EDMD), and optimized DMD. These methods learn finite-dimensional approximations of Koopman evolution on selected observables and are used to interpolate and denoise snapshots inside the observed time window before derivative estimation and sparse regression. The empirical benchmark comprises two ODE systems, Lorenz-63 and Van der Pol, and three periodic PDE systems, Burgers, Fisher-Kolmogorov-Petrovskii-Piskunov (Fisher-KPP), and linear advection-diffusion, over sparse and noisy sampling regimes. Polynomial EDMD gives the strongest ODE results, especially in coefficient accuracy. The PDE results are system-dependent: low-rank DMD-assisted reconstructions improve Burgers and advection-diffusion discovery, while the raw baseline (without upsampling) remains competitive for the Fisher-KPP data. A comparison against linear and smoothing-spline interpolation techniques shows that the selected Koopman-based preprocessors provide overall performance gains over these non-dynamical alternatives. We also demonstrate that DMD-assisted upsampling can stabilize Pareto-based non-oracle support-size selection. Overall, Koopman-based upsampling is best viewed as a dynamics-aware preprocessing step that can reduce derivative-estimation error when its observable representation and low-rank structure are appropriate for the data.

This paper investigates a generalization of the notion of the Weyl group of a real reductive Lie group to a semisimple symmetric space $G/H$. First, we show that the group structure of the Weyl group of $G/H$ is independent of the choice of split Cartan subalgebras of $G/H$ and the choice of Cartan involutions of $G$ stabilizing $H$. After that, we can understand all elements of the Weyl group of $G/H$ by comparing the Weyl group of some reductive Lie group associated to $G/H$. With the aim of extending these results to reductive real spherical homogeneous spaces, this work serves as a first step in this direction.

This paper studies a low-rank Kalman-Bucy filtering framework for linear time-varying systems through the tracking analysis of Oja's principal component flow. Under structured assumptions on the eigenspaces and their time variation, we show that the Oja flow can remain in a neighborhood of the time-varying dominant subspace by tuning a parameter of the flow, rather than tracking it exactly. These restrictive assumptions identify a tractable class of linear time-varying systems and provide a theoretical basis for low-rank filtering, as illustrated by a numerical experiment.

We investigate the wave interactions and stability of Riemann solutions for a nonautonomous chromatography-type system of Langmuir isotherm with time-dependent damping and flux. The system models two-component chromatographic separation with a time-dependent saturation capacity $n(t)$, leading to a nonautonomous hyperbolic system of balance laws. We consider a perturbed Riemann problem with piecewise constant initial data having two jump discontinuities at $x = \pm\epsilon$, and construct the global weak solution by analyzing all possible wave interactions, both classical (shock waves, rarefaction waves, contact discontinuities) and nonclassical (delta shock waves). We prove that as $\epsilon \to 0$, the solution of the perturbed Riemann problem converges to the solution of the corresponding Riemann problem in the space of Radon measures, establishing the stability of Riemann solutions under small perturbations of the initial data. To the best of our knowledge, this is the first instance of wave interaction and stability analysis for a nonautonomous chromatography-type system with time-dependent coefficients. Numerical experiments using a Lax-Friedrichs type scheme illustrate the wave interaction structure, the profiles at selected times, and the asymptotic convergence as $\epsilon \to 0$.

In this paper, we study generalized Wronskian-Hermite (WH) polynomials associated with arithmetic-progression index sets. We verify the conjecture that all nonzero roots are simple for three subclasses of these polynomials, which, in a natural sense, cover more than half of the relevant parameter range. We also provide an interpretation of the root multiplicity at $z=0$ in terms of Young diagrams. In addition, we show that certain members of generalized WH polynomials provide representations of rational solutions of the Noumi-Yamada systems, a family of higher-order Painlevé equations. Finally, we apply our results to the large-parameter asymptotic analysis of rogue wave patterns for the multi-component Hirota equation.

This paper is concerned with the non-Markovian stochastic optimal control problems in which the value function is a random field characterized by a stochastic Hamilton-Jacobi-Bellman (SHJB) equation. When the stochastic integration coefficients are not controlled, the SHJB equation takes a semilinear form, which is subject to computational challenges compared to the Markovian case due to the measurable randomness. We introduce a policy-iteration algorithm based on successive linearization that reduces the nonlinear SHJB equation to a sequence of linear ones. Furthermore, we prove that the resulting approximation sequence converges monotonically to the value function in the mean-square sense with an exponential rate.

A parameterized matrix family $x\mapsto H(x)$ on a configuration domain is determined by its physical content only up to a constant orthogonal change of basis. This gauge ambiguity is intrinsic to data-driven Hamiltonian models, such as tight-binding parameterizations, reduced-order electronic structure methods, or excited-state models. It raises a basic inverse problem: what observations of $H(x)$ suffice to identify the family up to this gauge? The pointwise spectrum is incomplete already for linear families on $\mathbb{R}$. Here, we prove that, under natural non-degeneracy and connectivity assumptions, augmenting the spectrum with loop products of the coupling matrices in the instantaneous eigenframe yields a complete invariant and that inversion is stable. We support the theory with numerical experiments.

An asymmetric random variable $X$ in the reals is said to be variance symmetrization resistant if every independent random variable $Y$ in the reals that produces a symmetric sum $X+Y$ has a greater variance than that of $X$. Asymmetric Bernoulli random variables were shown to be variance symmetrization resistant by Kagan, Mallows, Shepp, Vanderbei, and Vardi (1999); Pal (2008) gave a proof using stochastic calculus. We introduce the notion of entropic symmetrization resistance on locally compact groups– this means that the entropy of any independent symmetrizer $Y$ must exceed that of $X$. We show that asymmetric Bernoulli random variables exhibit entropic symmetrization resistance, and show a multidimensional generalization. We also explore basic aspects of the entropic symmetrization resistance problem in compact groups. In particular, we show that any distribution on a finite group that is entropic symmetrization resistant must lie on the boundary of the probability simplex, and describe precisely the class of all entropic symmetrization resistant distributions on $\mathbb{Z}_3$ and $\mathbb{Z}_4$.

Aldous' spectral gap conjecture states that the random walk and the interchange process on any connected graph have the same spectral gap, or, equivalently, the second largest eigenvalue of any connected Cayley graph on the symmetric group $S_n$ with respect to a set of transpositions is achieved by the standard representation of $S_n$. This celebrated conjecture, proved in its general form in 2010, has inspired much interest in searching for other Cayley graphs on $S_n$ possessing this property, now known as the Aldous property. In this paper, we first prove that for $n \ge 5$ at most one of a normal Cayley graph on $S_n$ and its complement can possess the Aldous property except when these two graphs are $2K_{n!/2}$ and $K_{n!/2,n!/2}$ respectively. We then determine, for sufficiently large $n$, all normal Cayley graphs $\mathrm{Cay}(S_n, S)$ that have the Aldous property, except for the case when $S$ contains a permutation with support size in $\{2, 3, \dots, n-2\}$ and a permutation with support size in $\{n-1, n\}$, but not all permutations with support size $n$ are contained in $S$. In particular, we show that a non-complete normal Cayley graph $\mathrm{Cay}(S_n, S)$ does not have the Aldous property if all permutations in $S$ have support size $n-1$ or $n$, or all permutations with support size $n$ are contained in $S$, thereby solving an open problem posed by Li, Xia and Zhou in 2023. Along the way we determine all normal Cayley graphs on $S_n$ that are line graphs, and classify all normal Cayley graphs on $S_n$ with the strictly second largest eigenvalue at most $1$.

In recent years, nonlinear dynamics derived from kinetic theory have gained attention in the context of sampling configurations of spin systems such as the Ising model. We focus on nonlinear dynamics for the hard-core model, a canonical spin system with hard constraints that specifies a distribution over independent sets in a graph, weighted by their sizes. We explore two distinct types of nonlinear dynamics: the mean-field dynamics, which preserves the density (or average size) of independent sets, and the single-site dynamics, which preserves the marginal vector (i.e., the occupancy probabilities of the vertices). These dynamics are natural stochastic processes for sampling from the hard-core model with a specified density or marginal vector, respectively, both of which are canonical instances of maximum entropy distributions that have been studied in various contexts. In contrast to linear Markov chains, there is a significant lack of a fundamental theoretical framework for nonlinear dynamics. We develop foundational theoretical tools for analyzing nonlinear dynamics within the context of the hard-core model. We establish almost linear convergence of both the mean-field and single-site dynamics at sufficiently low density through novel coupling arguments. We also establish exponential decay of relative entropy for the mean-field dynamics all the way up to the critical density. Additionally, we design new algorithms for sampling from the hard-core distribution with either a specified density or a specified marginal vector. These algorithms are based on a related linear Markov chain, called the particle-system dynamics and inspired by the so-called Kac's program, that approximates the associated nonlinear dynamics. As we demonstrate in the paper, they are comparable in time complexity, but simpler to implement, than traditional approaches based on learning parameter values.

We develop new characterizations of both global and local error bounds for general functions, using directional derivatives and tangent cones without imposing convexity or linear structure. We first establish several equivalent conditions for the global error bound of a nonnegative lower semicontinuous function. These equivalences hold for general, possibly nonconvex and nonsmooth functions. We further link the error bound with perturbation stability, Hausdorff stability of sublevel sets, and an inverse-sublevel-set estimate. Turning to directional derivatives, we introduce the minimal unit-sphere directional derivative \(\varphi(x)\) on the tangent cone and clarify its exact relation with the global slope. For Lipschitz continuous functions we prove that \(\sup_{x\notin S_0} \varphi(x) < 0\) is sufficient for an error bound, and for convex functions on convex sets this condition is also necessary, In finite dimensions we obtain sharp local results: if \(\varphi(\bar{x}) > 0\) at a solution \(\bar{x}\), then a local error bound holds and the optimal constant is exactly \(1/\varphi(\bar{x})\); if \(\varphi(\bar{x}) = 0\) and a suitable direction exists outside the tangent cone of the solution set, the local error bound fails. A general estimate relating the directional derivative to the distance from the tangent cone of the solution set is also derived.

We review recent developments in the study of spreading phenomena in reaction–diffusion equations arising from ecological invasion models. Motivated by the conjecture of Shigesada and Kawasaki on staged invasions, we discuss how competition systems can lead to effective scalar models with shifting habitats. We present the Hamilton–Jacobi approach for determining spreading speeds and revisit the result of {Li–Bewick–Shang–Fagan} (2014) from this perspective. We then describe the emergence of nonlocally pulled fronts when the shifting habitat connects regions of distinct positive growth rates. Recent results including the works of Lam–Yu (2022) and Lam–Nadin–Yu (2025) are surveyed. We also discuss spreading phenomena in competition systems, predator-prey systems, and the existence of various classes of entire solutions. Finally, we discuss the logarithmic correction for invasion waves in moving environments and prove a new result.

Given a permutation $\sigma$, its corresponding inversion graph is obtained by adding an edge between $i<j$ if and only if $\sigma(i)>\sigma(j)$. The first results on random inversion graphs come from Acan and Pittel, who studied the connected threshold for a uniform permutation with fixed inversion number, and Bhattacharya and Mukherjee, who mostly focused on the degrees of the graph when the permutation is chosen uniformly at random. In this work, we call minimal inversion the minimal degree of the inversion graph and extend a theorem from Bhattacharya and Mukherjee to the case where the permutation is not only uniform, but obtained as the ordering of points sampled according to some distribution on the plane. Under regularity assumptions on the distribution, and for the appropriate $\alpha>0$, we show that the probability that the minimal inversion rescaled by $n^{\alpha/(\alpha+1)}$ is larger than $t$ behaves like $\exp(-ct^{\alpha+1})$ for some constant $c>0$ depending on the distribution. We further show that every $\alpha>0$ admits at least one corresponding distribution, thus proving that the minimal inversion can asymptotically scale as $n^\beta$ for any $\beta\in[0,1]$ (the cases $\beta=0$ and $\beta=1$ being obtained via the identity and anti-identity permutations, among others).

Let $2\le k\le n$, let $\Omega\subset\mathbb{R}^n$ be open and convex, and let $u$ be a convex viscosity solution of $\sigma_k(D^2u)=1$ in $\Omega$. We prove that the set on which $u$ fails to be locally $C^2$ has vanishing $(n-1)$-dimensional Hausdorff measure. In the intermediate range $3\le k<n$, this gives a codimension-one refinement of the known almost-everywhere partial regularity, and the exponent is sharp. More generally, for a convex viscosity subsolution of $\sigma_k(D^2u)\ge\lambda>0$, we obtain Hausdorff bounds for strata defined by the affine dimension of all supporting contact sets. The proof combines a support-dependent Chou–Wang barrier argument, an estimate for the product of the smallest $k$ semiaxes of a John ellipsoid, and Mooney's convex section-covering theorem. As a direct analytical consequence, the full distributional Hessian is absolutely continuous and $u\in W^{2,1}_{\mathrm{loc}}(\Omega)$, yielding a $k$-Hessian counterpart of the $W^{2,1}$ regularity known for singular Monge–Ampère solutions. In a logically separate structural part, we characterize the distinguished number of flat directions, $n-k+1$, by an asymptotic infimum mean-value formula over affine sections, and explain how this mean-value heuristic leads to the supporting-contact geometry used in the proof.

We realize the GML estimator by an expectation-maximization algorithm initialized near the true positions and verify the theory by simulation.

Testing independence or conditional independence is fundamental to statistical inference, yet existing methods for non-Euclidean random objects often face a difficult trade-off between geometric flexibility and theoretical tractability. We introduce the Distance Profile Embedding (DPE), a novel representation that maps random objects from general metric spaces into a Hilbert space of square-integrable functions. We prove that this mapping is injective and preserves full distributional information without requiring isometric Hilbert embeddings or one-to-one correspondence conditions. Leveraging the DPE, we develop a unified framework for marginal and conditional independence testing of random objects that enjoys a rigorous asymptotic theory for both size and power. Notably, our framework is the first in the literature to accommodate object-valued conditioning variables when testing conditional independence, overcoming the Euclidean or Hilbertian constraints of existing methodologies. We facilitate the calculation of analytic $p$-values using closed-form asymptotic null distributions, which avoids the computational burden of permutation tests common in existing metric-based methods. The numerical properties of our methods are demonstrated through both simulations and two real-world applications involving gut microbiome compositions and global human mortality distributions, respectively.

This paper proposes a novel Pseudo-time Proper Orthogonal Decomposition (POD) framework to enable model reduction for stationary problems lacking temporal snapshot data. By recasting static operator equations into a pseudo-dynamic evolution form, we artificially generate temporal data while preserving the system's intrinsic spectral properties. Mathematically, we rigorously prove the exponential convergence of the pseudo-time trajectory to the exact stationary solution. Furthermore, the approximation properties of the generated POD basis functions are rigorously established. Finally, the universality and accuracy of the framework are validated both theoretically and numerically across two representative settings: elliptic inverse source problems and Fredholm integral equations of the first kind.

The upper-mid band, particularly the 7-8 GHz range within frequency range 3 (FR3), has emerged as a leading spectrum candidate for wide-area sixth-generation (6G) cellular networks. Its shorter wavelength enables hundreds of antenna elements to be integrated within the physical aperture of an existing 5G base-station panel. In principle, the resulting aperture gain can compensate for the increased path loss and enable extreme MIMO (E-MIMO) with 256 or more antenna ports while reusing current cell sites. In practice, however, simply scaling the 5G New Radio (NR) architecture from tens to hundreds of ports encounters fundamental system-level limitations. This paper identifies where 5G-style MIMO scaling breaks and develops a research roadmap for practical upper-mid-band E-MIMO. We first review the evolution of FR3 spectrum, its propagation and channel characteristics, and the emerging 6G system requirements. We then organize the principal challenges into four coupled areas: maintaining effective coverage across all physical channels and protocol states; implementing wideband, energy-efficient RF devices and radio units; developing new low-power array and beamforming architectures; and acquiring sufficiently refined channel state information with manageable sounding and feedback overhead. Representative system studies illustrate the coverage asymmetry between user-specific data transmission and common or channel-acquisition signals, as well as the spectral- and energy-efficiency tradeoffs among fully digital, hybrid, tri-hybrid, dynamic-metasurface, and fluid-antenna architectures. Finally, we discuss how distributed apertures, integrated sensing, AI-assisted channel acquisition, and environment-aware operation can transform fixed-aperture scaling into a deployable 6G E-MIMO architecture.

In this paper, we study the Klein-Gordon-Schrödinger system in the nonrelativistic regime $\epsilon \to 0$, where $\epsilon$ is proportional to the inverse of the speed of light. We show that the Klein-Gordon-Schrödinger system converges to a system of decoupled linear Schrödinger equations over a long time interval of order $\epsilon^{-1}$ with error estimates of the form $(1+t)\epsilon^2$; in particular, the error estimate for the Schrödinger component is uniform in time of the form $\epsilon^{2}$ The specific forms of the error estimates coincide with the numerical results shown by Bao et a.l., and the $O(\epsilon^{2})$ convergence rates coincide with the order of initial error, and thus are optimal.

In this work, we study the nonlinear stability of linearly expanding Goldreich-Weber (GW) solutions for the gravitational Navier-Stokes-Poisson system with pressure law $p(\rho)=\rho^{\frac{4}{3}}$ and degenerate viscosity. It is well known that linearly expanding GW solutions are special solutions to the Euler-Poisson system with $\gamma=\frac{4}{3}$. With bulk viscosity equal to 0, linearly expanding GW solutions are also solutions to the Navier-Stokes-Poisson equations. Choosing shear viscosity proportional to $\rho^{\alpha}$ with $0<\alpha\le\frac{2}{3}$ and zero bulk viscosity, we prove the nonlinear stability of linearly expanding GW solutions under radial perturbations.

I present a very simple example in which a full-support prior over distribution functions has an inconsistent posterior, which I believe has instructive value. The example is $Y_i \stackrel{\text{iid}}{\sim} P_0$ under the Bayesian hierarchical model $[Y_i \mid P] \stackrel{\text{iid}}{\sim} P$ and $P \sim \int \mbox{DP}(\alpha, H) \, \pi_\alpha(\alpha) \ d\alpha$ when $\pi_\alpha(\cdot)$ has exponential (or heavier) tails, where $\mbox{DP}(\alpha, H)$ denotes a Dirichlet process with concentration parameter $\alpha$ and mean $H(\cdot)$; on the other hand, consistency is obtained under a light-tailed prior. This unifies and generalizes a consistency result of Freedman and Diaconis (1983) with an inconsistency result described by Ferguson et al. (1992).

We prove that every arithmetic lattice in PSL$(2,\mathbb{C})$ and every arithmetic lattice of the simplest type in PO$(n,1)$, $n\ge 2$, is the normalizer of arbitrarily many of its sublattices. Combined with previous work, this result implies that every lattice in PSL$(2,\mathbb{C})$ has this property. In this way, we prove that the set of profinitely flexible lattices in PSL$(2,\mathbb{C})$ is either empty or countably infinite. Another result is that every finite group is realized as the full isometry group of an arithmetic hyperbolic $n$-manifold. The proof of this theorem is based on study of normalizers of lattices and subgroup growth theory.

Inspired by a classical algorithm for online prediction, we propose a novel denoising scheme which is universal for families of probability distributions both in terms of the source that generates the noiseless signal and in terms of the channel that generates the noisy signal. The new denoising scheme does not rely on assumptions of exact channel knowledge or finite signal alphabets. Our analysis provides an upper bound for the performance gap compared with the Bayes envelope. For the special case in which the signal is an i.i.d. sequence and the family is countable, we characterize the consistency condition under which our scheme approaches the Bayes envelope as the length of the sequence tends to infinity. We also show that in general, approaching the Bayes envelope is not possible for a universal denoising scheme when this consistency condition is not satisfied. Furthermore, we show that, as in the online prediction case, the so-called plug-in approach (which relies on a maximum likelihood estimation of the underlying distribution parameter) does not approach the Bayes envelope in general. We also include numerical evaluations of our scheme for denoising of a Poisson signal.