Papers

We develop a general framework for understanding when graph alignment is information-theoretically feasible in sparse inhomogeneous random graph models, by studying the set of vertices on which the underlying matching can be recovered. Our main theorem gives a general lower bound on this set by leveraging the balanced load function introduced by Hajek (1990). The corresponding obstruction is captured by a new graph parameter, the self-overlap, which measures the extent to which a graph can imitate itself under a non-trivial relabelling. We then show that this criterion is sharp in a broad class of sparse inhomogeneous models, recovering known Erdős–Rényi phenomena and yielding sharp thresholds for Chung–Lu graphs and stochastic block models.

World Action Models (WAMs) enable semantically- and physically-informed control but are brittle under distribution shift. In this work, we use mechanistic interpretability to study how robustness-relevant perturbations are represented in WAM activation space. Comparing activations across successful and unsuccessful rollouts, we find some WAM architectures exhibit low-dimensional linear separability for robustness-critical features, while others do not. This motivates the use of contrastive activation directions for training-free WAM steering. We also show that local linearity in WAM activation dynamics enables efficient feedback steering via model-based optimal control, yielding World-Action Linear Quadratic Regulator (WA-LQR), a minimally-invasive reduced-order LQR controller. Via mechanistic evaluations, we predict strong steerability in the Cosmos-Policy and DiT4DiT models but weak steerability in LingBot-VA, consistent with steering intervention results. On Cosmos-Policy and DiT4DiT, WA-LQR generalizes contrastive directions to new tasks and improves robustness to camera, gripper, and visual-noise perturbations over unsteered and prompt steering baselines.

We study causal inference under outcome interference for sequential, observational settings. Specifically, we consider settings where the binary outcomes over N units are Markovian across T time steps. At each time step, the outcomes of N units have dependencies captured through an Ising model; each outcome is also impacted through an external field capturing the effects of its treatment as well as latent confounders. Similar to panel data literature, these latent confounders are modeled to have a low-rank factor structure. Our data is a single sample from this high-dimensional distribution. To estimate causal quantities of interest, we provide a computationally efficient method based on Maximum Pseudo-Likelihood Estimation (MPLE) for learning the model parameters. Under mild assumptions, we establish non-asymptotic consistency for parameter estimation and show this translates to faithful estimation of causal quantities of interest after sampling from the learned model. We demonstrate the efficacy of the method through synthetic experiments as well as a real-world case-study investigating causal effects of vaccine rates on COVID-19 death rates within US counties nationwide.

The Schur superalgebra $S(m,n;d)$ contains $S_{m}\times S_{n}$ as a multiplicative subgroup. We consider the problem of determining the restriction coefficients when an irreducible supermodule of $S(m,n;d)$ is restricted to $S_{m}\times S_{n}$. We introduce two notions of superplethysm and provide a superplethystic variant of Littlewood's formula for these restriction coefficients. As an application, we provide several unimodality results for bipartite superpartitions.

Recent foundation models (FMs) for zero-shot reconstruction of dynamical systems (DS) achieve strong out-of-domain generalization but provide little insight into the mechanisms that underlie their forecasts. Such an understanding could help to strip down overladen FM architectures to their bare essence and expose the minimal requirements for in-context learning in the DS domain. Toward this goal, here we iteratively reduce a recent powerful SOTA model for DS reconstruction, DynaMix (Hemmer & Durstewitz, 2025), to a minimal interpretable two-parameter form, which we call DynaBase. DynaBase produces forecasts through a linear blend of the current latent state and the nearest in-context neighbor and its temporal successor. Surprisingly, despite its extreme simplicity, DynaBase produces highly competitive zero-shot DS reconstructions across chaotic and cyclic systems, with a negligible parameter load, many orders of magnitude below that of other FMs. Even more, this extreme simplicity permits direct model optimization on DS reconstruction measures, as well as closed-form one-step analytical solutions on prediction MSE. Theoretical and empirical analysis of DynaBase further leads to a 1-parameter family of maps, with the context-parroting algorithm of (Zhang & Gilpin, 2026) recovered at one end, and chaotic (divergent but bounded) behavior at the other. We further show how different training strategies lead to models either optimal for short-term prediction or for DS reconstruction. Thus, DynaBase not only exposes the minimal mechanisms required for producing zero-shot DS reconstruction, but also reconciles within an accessible mathematical frame divergent observations in the literature.

Here we study the core entropy function $h: PM(d) \rightarrow \mathbb{R}$ defined on the space of degree $d$ primitive majors, a combinatorial model for the space of degree $d$ polynomials. In particular, we introduce techniques to classify the global maximum for each $d$ and the maxima along certain strata. We show that there are $d-1$ global maxima over all of $PM(d)$ and that the maxima on the unicritical stratum is a Cantor set.

We propose a goodness-of-fit test for semiparametric copula regression models. Such models express the regression function in terms of marginal distribution functions and copula densities and therefore provide a flexible way to avoid fully nonparametric estimation in high-dimensional regression problems. Their performance, however, depends crucially on the specification of the parametric copula family. Instead of testing the copula model itself, we assess misspecification directly at the level of the induced regression function. To this end, we introduce a weighted $L^2$-distance between the true regression function and its best approximation within the postulated copula regression model. A kernel-based estimator of this distance is proposed and shown to be consistent and asymptotically normal under both the null hypothesis of correct specification and fixed alternatives. We derive a classical specification test and, using a self-normalized sequential statistic, construct pivotal confidence intervals and tests for relevant deviations from the model. Finite-sample simulations demonstrate accurate level approximation and good power properties of the proposed procedures.

This paper studies the optimal design of Type-I generalized progressive hybrid censoring schemes for life-testing experiments. The design problem involves simultaneously determining the inspection time, the guaranteed number of failures, and the progressive censoring scheme. First we develop a cost-constrained optimization framework for determining the optimal censoring scheme. Structural properties of the A-optimality criterion and the experimental cost with respect to the inspection time and the guaranteed number of failures are established. It reveals that they are conflicting behaviors which enables to develop an efficient search algorithm that substantially reduces the computational burden. Building on these theoretical results, a multi-objective optimization model is proposed to simultaneously minimize A-optimality criterion and the experimental cost. A Variable Neighborhood Search (VNS) algorithm is proposed to efficiently determine the optimal progressive removal vector by exploring the feasible design space while avoiding exhaustive enumeration. The resulting compromise designs simultaneously improve estimation precision and reduce experimental cost. In addition, the Shannon differential entropy of the observed lifetime distribution is derived and employed as a complementary information-theoretic measure for evaluating the selected censoring schemes. Numerical studies show that entropy-optimal designs generally differ from A-optimal designs, indicating that Shannon entropy characterizes uncertainty in the observed data rather than estimation precision. The proposed methodology provides an efficient computational framework for optimal life-test design and offers a foundation for future multi-objective optimization incorporating statistical efficiency, experimental cost, and information-theoretic uncertainty.

The G-Scheme is an explicit, adaptive integration framework for stiff systems of ordinary differential equations that exploits a local time-scale decomposition provided by the eigensystem of the Jacobian (the kernel set). Its computational cost is dominated by the evaluation of a new kernel set at each time step. In a recent paper [R.Malpica Galassi and M.Valorani, Combust. Theory Model. 2025], we introduced a multi-resolution sparse hash-table strategy that replaces online kernel computation with retrieval from a precomputed lookup table, demonstrated in Python. Here we present cpp-gscheme, a production-grade C++ implementation of both the G-Scheme and the multi-resolution hash table, together with the porting methodology that guarantees behavioural equivalence with the Python reference: identical scaling, binning and hashing semantics, a bitwise-inert retrieval hook in the integrator, and a fully native evaluation path for the chemical source terms. On 33 n-heptane autoignition mechanisms spanning 56-459 species plus the detailed 654-species mechanism, the hash-tabulated C++ G-Scheme retrieves 100 percent of the kernel sets, performs zero online eigen decompositions, and outperforms the natively compiled CVODE solver by a factor increasing from 2.7x (56 species) to 9.3x (654 species), with observed cost scaling of approximately N power 1.4 versus CVODE's N power 1.9. A three-metric accuracy assessment (ignition delay, equilibrium state, and accumulated entropy-progress error) shows that the solver contributes negligibly to the overall error budget, which is dominated by skeletal reduction. Consequently, a 5 percent ignition-delay target is achieved by the 80-species skeletal mechanism with a 225x speed-up over CVODE on the detailed mechanism. The software, training-set builders, and benchmark drivers are released as open source.

The integrability of the Kudryashov-Sinelshchikov equation in (1+3) dimension is detected using its point symmetries and the Singularity analysis method. The symmetries mostly points to the existence of infinite-dimensional Lie algebra with the presence of mostly arbitrary functions. The reduction process is initiated by considering these arbitrary functions in the general symmetry vectors. For certain functions, trivial solutions for the general parent equation is obtained whereas for some of the functions, the reductions does not yield an unmitigated result. Finally, a positive result is listed for one of the arbitrary function, for which the singularity analysis method is employed to study the Painleve property. The resultant PDE happens to be a fourth-order equation and its approval of the Painleve property hints at the integrability of the Kudryashov- Sinelshchikov equation in the general case.

We investigate finite-dimensional feedback stabilization for nonlinear nonautonomous stochastic parabolic equations driven by $Q$-Wiener, covering both additive and multiplicative perturbations. The control is given by a finite linear combination of localized indicator-type actuators whose supports are selected as part of the construction and may have arbitrarily small total measure. The feedback law is constructed by means of oblique projections onto suitable finite-dimensional subspaces. Within the variational Gelfand triple framework, we prove well-posedness of the closed-loop system under standard coercivity, growth, and global Lipschitz assumptions. By appropriately choosing the actuator configuration and feedback strength, we establish exponential mean-square stabilization of the stochastic dynamics and, for pure multiplicative noise, almost-sure stabilization. A fully discrete three-layer implementation complements the theoretical results. Numerical experiments illustrate the influence of number of actuators, noise intensity, and nonlinear effects on the closed-loop stabilization behavior.

We present a class of regular axisymmetric black hole geometries fully characterized by the parameters $\{{\cal M},a\}$ and possessing the Kerr event horizon. This family interpolates between regular spacetimes, configurations with integrable singularities, and the Kerr solution as a limiting case. Its main features are: (i) the existence of quasi-extremal configurations without requiring $a \approx {\cal M}$; and (ii) a possible framework toward an analytical description of Kerr black hole formation from an initially regular configuration.

Given $s\in (0,1)$ and a bounded Lipschitz domain $\Omega\subset \mathbb{R}^{n}$, we establish a quantitative Dahlberg theory for the $s$-harmonic measure of $\Omega$, $\omega_s^x$. In the nonlocal setting, the natural reference measure is an integral weight $\sigma_{s}$ in $\Omega^{c}$ that behaves like $(1-s)\text{dist}(\cdot, \partial \Omega)^{-s}$ close to the boundary. Our main result is a scale-invariant reverse-Hölder estimate for the density $d\omega_{s}^{x}/d\sigma_{s}$ on boundary-centered balls. As a consequence, we obtain $L^2(\Omega^c,\sigma_s)$-solvability of the exterior Dirichlet problem, with estimates for a nonlocal non-tangential maximal function and uniqueness in the natural distributional class. A weighted Gehring argument improves the reverse-Hölder exponent beyond $2$ and consequently yields $L^{q}$-solvability for a range of exponents extending strictly below $2$. Our results apply to general symmetric stable operators comparable to the fractional Laplacian. Moreover, the proofs are compatible with the limit $s\to 1^-$ and thus yield the corresponding results for the Laplacian in the nonlocal-to-local limit. The main new step is to convert a fractional Pohozaev identity for the Green function into uniform square estimates on distance level sets of a Lipschitz boundary. As applications, we derive optimal Sobolev regularity estimates for the homogeneous weighted Dirichlet problem and for the inhomogeneous Poisson problem with zero exterior data.

We prove that the recognition problem for algebraic matroids is undecidable. Explicitly, this means that there is no algorithm that takes as input a finite set $S$ and a function $r\colon\mathcal{P}(S) \to \mathbb{Z}_{\ge 0}$ (where $\mathcal{P}(S)$ is the power set) and decides whether there exists a pair of fields $F \subset K$, and a function $f\colon S \to K$, such that for all $A \subseteq S$: $\mathrm{trdeg}_F f(A) = r(A)$. This problem is known to be decidable if the characteristic of the fields involved is constrained to be zero. We prove that it is undecidable if the characteristic is either left unspecified (in which case a realization over any characteristic is accepted) or fixed to be a prime $p$. The proof relies on Hrushovski–Zilber's Group Configuration Theorem and on the work of Evans and Hrushovski on "Projective Planes in Algebraically Closed Fields". We relate two different such projective planes, and eventually construct a reduction from the solvability of Diophantine equations over $\mathbb{F}_p(x)$ ($p$ prime) to algebraicity of matroids. Solvability of Diophantine equations over $\mathbb{F}_p(x)$ was proved to be undecidable by Pheidas for all $p > 2$, and later by Videla for $p=2$. A central part of our proof is a variant of the so-called Field Configuration Theorem.

On $d$-dimensional cylinders $\mathcal{C}= \mathbb{R}^k\times N$, with a closed manifold $N$ as base and large scale dimension $k\in[1,d)$, we prove quadratic estimates in weighted $L^2$ space for Dirac operators perturbed by bounded, measurable and accretive coefficients. This gives in particular homogeneous Kato square root estimates on $\mathcal{C}$ for Riesz transforms associated with second order divergence form elliptic operators, having measurable coefficients with degeneracy governed by a Muckenhoupt $A_2$ weight. By localisation and scaling, it also yields local quadratic estimates for perturbed Dirac operators on general manifolds with locally thin cylindrical geometry, and possibly with zero injectivity radius.

Plug-and-play proximal gradient descent (PnP-PGD) enables flexible image reconstruction by using denoisers as implicit priors. In practice, these denoisers are often deployed outside their training domains. Existing analyses establish convergence under structural assumptions on the deployed denoiser, such as requiring it to be a proximal map or a contraction. However, they do not measure how domain mismatch affects convergence of PnP-PGD. We define this effect as proximal mismatch: the discrepancy between a deployed denoiser $\widehat{\mathsf D}$ and a target-domain reference map $\mathsf D_\star=\operatorname{prox}_{R_\star}$ associated with the underlying regularizer $R_\star$. Under this mismatch, each denoising update becomes an inexact proximal step for the target objective. We further derive a stationarity bound that decays at a rate of $\mathcal{O}(1/K)$, with an additive term proportional to the average squared proximal mismatch. This result motivates adaptation via proximal matching rather than MSE-based adaptation alone. We study this approach with two established denoiser families: learned proximal networks and gradient-step denoisers. Experiments on Gaussian deblurring and super-resolution under substantial domain shift show that proximal matching adaptation improves reconstruction quality significantly over MSE-based adaptation, yielding the largest numerical gains in the few-shot regime.

We study four-wave kinetic equations in space dimension three with power-law dispersion $\omega(p)=|p|^a$ and collision kernels with high-frequency growth measured by $\beta$. In weighted $L^\infty$ spaces, we identify the sharp decay threshold $$ s_c=4\beta+3-\frac a2. $$ For $s>s_c$, we prove local well-posedness by establishing trilinear bounds for the full gain-loss collision operator. For $s<s_c$, we prove ill-posedness by constructing data concentrated near a high-low-low-high resonant configuration. This threshold captures the balance between the high-frequency strength of the kernel and the geometry of the resonant manifold. The proof also shows that gain-loss cancellations are essential in the most delicate regimes.

We study a period-two family of adjacent-sum lattice polytopes whose consecutive-coordinate bounds alternate between $s$ and $s+1$. This provides a simple non-uniform deformation of the classical uniform model while retaining an explicit transfer-matrix structure. The lattice-point counts exhibit a parity split: the odd- and even-dimensional sequences have distinct rational generating functions with a common denominator. The odd-dimensional series satisfies a Möbius recurrence and admits an arctangent closed form, whereas the even-dimensional series obeys a coupled recurrence. Their common dominant pole determines the exponential growth in both parity classes. For the cyclic model obtained by adding a constraint between the first and last coordinates, the count becomes a matrix trace. The two cyclic parity classes again have rational generating functions with the same denominator; the even-dimensional numerator has a Jacobi-derivative form, while the odd-dimensional one is given by an explicit anti-diagonal cofactor expression. We also derive dimension-generating functions for fixed dilations, linear recurrences for lattice-point counts, rational volume-generating functions, and a bivariate identity for the coefficients of the $h^*$-polynomials. When $s=1$, every even-dimensional polytope decomposes into a Cartesian product of unimodular triangles, yielding explicit formulas and the Gorenstein property. For every $s\geq 2$, the Gorenstein property fails in some even dimension.

Given a scalar observable of an ergodic dynamical system with a low-dimensional attractor, two families of methods reconstruct and predict the underlying state: recurrence-based methods (the method of analogues and its descendants), which wait for the trajectory to return to an $\varepsilon$-neighborhood of a previously observed state, and observer-based methods, which fit a converging state estimator on the delay reconstruction. We formalize and empirically verify an exponential separation between the two: the expected cost of recurrence scales as $\varepsilon^{-d}$, where $d$ is the pointwise dimension of the invariant measure (a consequence of the Kac lemma and quantitative Poincare recurrence), whereas a detectable linear observer converges in $\Theta(\log(1/\varepsilon)/(1-\rho(A_{cl})^2))$ steps, where $\rho(A_{cl})$ is the closed-loop spectral radius of the Riccati fixed point. Both laws are verified numerically (return-time exponent $-1.8$ on the Lorenz attractor against the theoretical $-2.05$; observer cost linear in $\log(1/\varepsilon)$ with $R^2=1.000$ and in $(1-\rho^2)^{-1}$ with $R^2=0.985$), yielding a measured cost gap of $\sim 10^{9}$ at $\varepsilon=10^{-6}$ for $d\approx 2$. We complement the theorem with an admission protocol (the Kac-Riccati gate) deciding whether a signal lies inside the theorem's class, via surrogate-data prediction gating; it also explains the folklore of "universal" fractal dimensions as a dataset-size artifact bounded by $2\log_{10}N$. On real data the gate admits the Santa Fe laser benchmark ($\hat D_2=2.0$) and refuses the monthly sunspot series, reproducing the settled resolution of historical low-dimensionality claims. All results reproduce from a single verification script (17/17 checks).

A Hamilton cover of a graph is a collection of Hamilton cycles whose union contains all edges. Since each Hamilton cycle covers two edges at every vertex, every Hamilton cover has size at least $\lceil \Delta(G)/2\rceil$. We prove that this lower bound is tight for binomial random graphs $G(n,p)$ throughout the widest possible range of edge probabilities: if $\omega(n)\to\infty$ and \[ \frac{\log n+\log\log n+\omega(n)}{n} \le p=p(n) \le 1-\frac{\omega(n)}{n^{2}}, \] then $G\sim G(n,p)$ with high probability has a Hamilton cover of size $\left\lceil \frac{\Delta(G)}{2}\right\rceil. $ The main new contribution is the sparse regime near the Hamiltonicity threshold, where we prove a conjecture of Draganić, Glock, Munhá Correia and Sudakov. Our proof develops constructive tools for decomposing such graphs into controlled forest systems and extending them, using reserved pseudorandom structure, into Hamilton cycles. We also prove the corresponding hitting-time result for the random graph process, answering a question of Hefetz, Kühn, Lapinskas and Osthus. Finally, we use our methods to show that $G\sim G(n,p)$ with high probability satisfies the celebrated Linear arboricity conjecture for every $p\leq 1$.

We propose a novel measure of the discrepancy between two probability distributions $f$ and $g$ on a graph - which we call the diffusion distance - that measures the rate of convergence of $f$ to $g$ under a graph-constrained Markov chain with stationary distribution $g$. As a default choice for this Markov chain, we use the Metropolis-Hastings transition matrix targeting $g$ with proposals given by a random walk on the graph. Our primary case of interest is when the second distribution $g$ is uniform, in which case the diffusion distance becomes a measure of spatial clustering in $f$. Used in this way, (Metropolis-Hastings) diffusion distance to uniformity extends Moran's $I$-type measures of spatial autocorrelation by incorporating global graph geometry rather than just local patterns. Indeed, Moran's $I$, the most well-known measure of spatial autocorrelation, can be viewed as a one-step heuristic for diffusion distance, so long as specific spatial weights are used. We establish theoretical bounds and a stability result for our measure, connecting it to graph spectra and optimal transport. We then turn our attention to outlining a statistical test for spatial clustering using diffusion distance. Under permutation null models, we derive high-probability bounds on diffusion distance underpinned by exact spectral formulas for convergence of distributions, enabling an efficient statistical test for spatial clustering on large datasets. We empirically compare diffusion distance to Moran's $I$ both as a numerical measure and as a statistical test. We show that diffusion distance exhibits higher power on synthetic data using a stochastic block model. Empirical analysis of Black population distributions for 100 U.S. cities shows that diffusion distance detects subtle differences in urban segregation patterns that Moran's $I$ does not.

A Banach space $X$ is said to have the ball covering property (BCP) if the unit sphere of $X$ can be covered by countably many open balls $B(x_i, r_i)$ with $r_i\leq \|x_i\|$ for each $i\in\mathbb{N}$. If there are $R, \delta>0$ so that $r_i\leq R$ and $\|x_i\|-r_i>\delta$ for all $i\in\mathbb{N}$, then we say that $X$ has the uniform ball covering property (UBCP). In this paper, we show that if $X$ has an $1$-unconditional basis or $X$ is an $1$-complemented subspace of a Banach space with a shrinking $1$-unconditional basis, then the Calkin algebra $\mathcal{B}(X)/\mathcal{K}(X)$ fails the BCP. It is also shown that if $X$ has a shrinking unconditional basis with unconditional constant less than 2, then $\mathcal{B}(X)$ has the UBCP.

A coned polytope framework (CPF) is the bar-joint framework obtained from the 1-skeleton of a convex polytope by coning over some interior point. It was recently shown that CPFs are rigid, though the exact order of rigidity remained open. In this paper we introduce the Wachspress stress and use it to show that CPFs are prestress stable, in particular, second-order rigid. To this end, we resolve the stress-flex conjecture in the case of the Wachspress stress by identifying its dual formulation as a corollary of a vector-valued Schläfli-type formula introduced by Schlenker and Souam. We give a new and purely discrete-geometric proof of this generalized Schläfli formula.

We give the definitive proof that the Dirac Quantum Cellular Automaton (QCA) used for both quantum simulation and algorithmic foundations of Quantum Field Theory (QFT), and especially of Quantum Electrodynamics (QED), as put forward in References https://doi.org/10.1007/s11128-019-2555-4 and https://doi.org/10.22331/q-2023-11-08-1179, does exhibit Fermion Doubling (FD), albeit thrice as less severe as discrete-time standard Lattice Gauge Theories (LGTs) [arXiv:2505.0790], which are naive regarding the spacetime discretization of differential operators acting on fermionic fields. The proof is done for the (1 + 1)D Dirac-QCA model. We show that the (one-time-step) two-point correlation function, also called Green's function (GF), of the Dirac QCA, is of astonishing simplicity, which is in contrast with the GF of the Dirac equation. We also compare, both qualitatively and quantitatively, this Dirac QCA to the continuous-time-LGT spatial discretization of Dirac fermions regarding how well these two lattice models approximate their naive continuum limit$\unicode{x2014}$which is nothing but the Dirac equation$\unicode{x2014}$even when far away from that limit, a situation which must be considered because of experimental limitations in quantum simulation: the Dirac QCA is better for ultrarelativistic regimes, whereas continuous-time LGT is better for non-relativistic regimes. Then, we compute the GF of the FD-fixed model put forward in the last cited reference, called Flavored Dirac QCA (FDQCA)$\unicode{x2014}$which staggers an extra, artificial flavor $only$, on a diamond spacetime lattice, and does not stagger chirality as staggered fermions in usual LGT. The structure of this FDQCA two-point correlation function is of extreme simplicity, and can be expressed in a very simple manner in terms of the four chiral components of the FD-suffering, original-model GF.

A new kernel method is introduced for isometric actions of groups on real or complex hyperbolic spaces that allows us to classify such representations for $\mathrm{PSL}_2(\R)$ and give sufficient and necessary conditions for two such representations of any group to be conjugated.

Given integers $1\leq t\leq k$, a family of $k$-matchings in a complete $r$-partite $r$-uniform hypergraph is said to be $t$-intersecting if any two of its members share at least $t$ common edges. This concept unifies several well-studied classes of intersecting families, including classical intersecting families, intersecting families of permutations, partial permutations, and generalized permutations, as well as intersecting families of injections. In this paper we employ two approaches to determine the maximum size of $t$-intersecting families of $k$-matchings and to characterize the extremal families that attain this bound. Using a recent result of Keller, Lifshitz, Minzer, and Sheinfeld on $t$-intersecting families of permutations, we obtain Erdős-Ko-Rado-type theorems whose thresholds depend only on $t$. We also develop a $t$-cover-based approach that offers a complementary characterization of the extremal families.

We prove equidistribution and Manin's conjecture for rational points outside the lines on smooth split quintic del Pezzo surfaces over number fields with respect to any anticanonical height. The proof is based on a general theorem that is broadly usable to deduce equidistribution when using the torsor method with several equivalent height functions to treat variants of Manin's problem.

In this paper, we propose a novel tamed stochastic gradient Hamiltonian Monte Carlo (tSGHMC) algorithm for sampling and stochastic optimization problems with superlinearly growing stochastic gradients. Under a certain continuity in average condition and a strong convexity condition, we establish a non-asymptotic error bound in Wasserstein-2 distance for tSGHMC with the rate of convergence equal to $1/4$. Then, we derive an upper estimate for the associated expected excess risk, which provides a theoretical guarantee for the performance of tSGHMC. To illustrate the effectiveness of the proposed algorithm, we apply tSGHMC to practical examples, including a newsvendor problem and a Conditional Value-at-Risk minimization problem, using synthetic and real-world datasets. Numerical results support our theoretical findings. Furthermore, we compare tSGHMC with its first-order counterpart, namely, the tamed unadjusted stochastic Langevin algorithm. Simulation results demonstrate that tSGHMC achieves lower root mean square error and expected excess risk across a range of tasks.

Several vanishing theorems for pseudo-effective line bundles are presented. All related results in the present note can be derived from already known theorems, in particular from the Steenbrink vanishing theorem and Boucksom's generalization of Bogomolov's vanishing theorem. No originality or priority is claimed for these statements; they are listed here merely as conjugate forms of the Kawamata–Viehweg vanishing theorem, for convenience of reference and comparison with existing literature.

We present a general framework for producing families of harmonic maps from the unit ball into the Euclidean sphere starting from a fixed harmonic map. In particular, our analysis provides a theoretical background for a generalized radial projection which was recently introduced by Nakauchi. Our approach is based on work of Toth who established a machinery that generates eigenmaps between spheres from a given eigenmap. Finally, we point out how these families of harmonic maps can be used to construct solutions to associated variational problems such as intrinsic biharmonic and triharmonic maps, $p$-harmonic maps as well as extrinsic polyharmonic maps.