Papers

The signless Laplacian matrix of a graph $G$ is $Q(G)=D(G)+A(G)$, where $D(G)$ and $A(G)$ are the diagonal degree matrix and the adjacency matrix of $G$, respectively. The signless Laplacian spectral radius of $G$ is the largest eigenvalue of $Q(G)$. For a positive integer $t$, a graph is called $tK_3$-free if it contains no $t$ vertex-disjoint triangles. In this paper, for every fixed $t\geq 2$ and all $n\geq 28t-17$, we determine the unique graph achieving the maximum signless Laplacian spectral radius among all $tK_3$-free graphs of order $n$.

The Gauss-Seidel projection method (GSPM) constitutes an efficient and numerically stable numerical framework for micromagnetic simulations of ferromagnetic media. This scheme attains first-order temporal accuracy and second-order spatial accuracy. Fast Fourier transform (FFT) techniques can be incorporated to accelerate both the solution of the arising linear algebraic systems and the evaluation of stray magnetic fields. The conventional GSPM relies on a single-sided Gauss-Seidel iteration, which leverages the latest updated state variables associated with the heat-diffusion subproblem. In this work, we develop a symmetric Gauss-Seidel projection method (SGSPM) that retains first-order temporal accuracy and second-order spatial consistency. The proposed symmetric variant exhibits superior stability properties relative to the standard GSPM. Specifically, SGSPM adopts a two-pass symmetric Gauss-Seidel iteration, where updated information from the heat-diffusion stage is fully exploited to rigorously guarantee discrete energy stability. We validate the performance of the devised scheme through numerical investigations of magnetization dynamic evolution and magnetic domain-wall propagation. Numerical evidence demonstrates that the improved symmetric scheme delivers enhanced stability for capturing magnetization motion dynamics.

We present a systematic procedure for constructing brane brick models corresponding to abelian orbifolds of toric Calabi-Yau 4-folds, extending the orbifold construction beyond the well-studied case of abelian orbifolds of C^4. Given a parent brane brick model corresponding to a toric Calabi-Yau 4-fold M, we show that the action of an abelian orbifold group Gamma on the generators of M induces an action on the chiral and Fermi fields as well as the J- and E-terms of the associated 2d (0,2) supersymmetric gauge theory. The requirement that the orbifolded brane brick model remains consistent with the closed paths associated with the J- and E-terms, together with the chiral cycles formed by their products, precisely reproduces the Calabi-Yau condition on the orbifold action. This procedure yields an explicit formula for the J- and E-terms of the orbifolded brane brick model in terms of those of the parent theory. We apply our construction to the brane brick models corresponding to Q^{1,1,1} and D_3, and present the resulting families of 2d (0,2) quiver gauge theories. We also present explicit expressions for generating functions that count distinct abelian orbifolds of Q^{1,1,1} and D_3.

Let $\mathcal Y(z;t)$ be the modified Greaves–Jing–Zhu operator on the odd power-sum ring. We first point out that this operator can be obtained from the classical neutral operator by a simple diagonal change of variables. We then study products in which the two deformation parameters are not necessarily the same. For two parameters $t$ and $s$, we compute the scalar factor that appears in the mixed product. This factor has an explicit exponential form and, in a completed setting, can also be written as a quotient of infinite $t$-Pochhammer products. We also give a recurrence for its coefficients, a product formula for several mixed operators, and formulas for the coefficients obtained after applying the operators to $\mathbf 1$. A particularly simple case occurs when $s=t^M$. In this case the scalar factor becomes the finite quotient $(u;t)_M/(-u;t)_M$. Its coefficients are signed principal specializations of one-row Schur $Q$-functions. As a result, after removing the signs, these coefficients are nonnegative palindromic polynomials. We also give a Gaussian-binomial formula and a finite-order recurrence.

We study a family of generalizations of the notion of Euler characteristic of discrete groups (or of orbifolds, depending on one's perspective) indexed on the natural numbers. For $n=0$, this is the classical orbifold Euler characteristic as studied by Wall and Serre, whereas for $n \geq 1$ and finite groups, this is the chromatic cardinality as studied by Ben-Moshe–Carmeli–Schlank–Yanovski. For general $n$, we show that our generalized Euler characteristic admits a natural interpretation in terms of the Morava $E$-theories. Our work involves showing that the generalized cohomology of infinite groups $G$ with finite universal space for proper actions $\underline{E}G$ has a good theory of duality, as expressed by a new duality functor on the category of proper $G$-equivariant spectra. In particular, for such groups we prove the vanishing of Klein's generalized Farrell–Tate cohomology with $T(n)$-local coefficients. We compute our generalized orbifold Euler characteristics in a large number of examples. This includes many mapping class groups, where the classical calculation is a result of Harer–Zagier, and many arithmetic groups, whose classical orbifold Euler characteristics were computed by Harder.

We study the local controllability near zero of the Burgers equation with a scalar control and a fixed space-dependent source profile, in the case where the linearized system fails to be controllable and a second-order analysis is therefore required. We prove that quadratic obstructions to finite-time controllability can be quantified by Sobolev norms of the control with fractional negative exponents ranging over a full interval. To our knowledge, this is the first example, for a natural physical PDE, of a continuous scale of fractional quadratic obstructions, and the first such continuous scale for finite-time local controllability. Our explicit constructions shed light on the origin of fractional obstructions for partial differential equations, by relating the obstruction exponent to the regularity, and in some cases to the physical-space singularity, of the source profile. We identify the natural structural conditions on the source profile leading to obstructions quantified by the $H^{-1}$ and $H^{-5/4}$ norms of the control, thereby providing a general framework in which the previously studied case of a constant source profile fits naturally. In this constant-profile case, we improve existing results by identifying the arithmetic condition on the Fourier mode which ensures that a small-time obstruction actually persists in finite time. Finally, we derive sharp nonlinear remainder estimates adapted to the precise regularity of the source profile. These estimates make most of our obstruction results optimal with respect to the smallness assumption imposed on the control.

Strategic behaviour in queueing systems has been studied extensively in the behavioural queueing literature, but almost exclusively for systems that admit closed-form expressions for the cost or utility experienced by a strategic user. Evolutionary game theory offers a mature framework for analysing populations whose individual payoffs depend on the composition of the population itself, and would in principle apply to a much wider class of queueing systems; its application has, however, been constrained by the same closed-form requirement. We introduce Discrete Event Population Updates (DEPU), a general algorithmic framework that couples a single long run of a discrete event simulation (DES) directly to an evolutionary population update rule, removing that constraint. We present two implementations: Discrete Event Replicator Dynamics (DERD), which follows an Euler discretisation of the replicator dynamics equation, and Discrete Event Moran Replacement (DEMR), which maintains a finite population updated via Moran-style copying events. Both are applied to a multi-server jockeying model for which no closed-form fitness expressions are available. On the jockeying model considered, DEPU reaches comparable precision tens of times faster than the standard practice of nesting short simulations inside an outer evolutionary loop, and because each operating point then costs only a single simulation run it also makes systematic parameter sweeps tractable. This brings the toolkit of evolutionary dynamics within reach of any system a modeller can build in a discrete event simulator.

We study $\mathfrak{sl}(2)$-modules that are free of finite rank over $U(\mathfrak h)$, where $\mathfrak h$ is a fixed Cartan subalgebra of $\mathfrak{sl}(2)$. These modules form a natural class of non-weight modules. The coherent families obtained from this class via the weighting functor are identified. We also study a distinguished class of indecomposable $U(\mathfrak h)$-free modules defined in terms of Jordan blocks and give a recursive description of their socle filtrations. Finally, we apply the general results to exponential modules arising from the first Weyl algebra and obtain simplicity criteria for these modules.

We prove the stability of the Langlands-Shahidi local $\gamma$-factor for the exterior cube representation of $\mathrm{GL}_6$. More precisely, if $\pi_1$ and $\pi_2$ are irreducible admissible generic representations of $\mathrm{GL}_6(F)$ with the same central character, then \[ \gamma(s,\pi_1\otimes\chi,\wedge^3,\psi)= \gamma(s,\pi_2\otimes\chi,\wedge^3,\psi) \] for every sufficiently ramified character $\chi$ of $F^\times$, where $\chi$ is regarded as a character of $\mathrm{GL}_6(F)$ through the determinant. The proof uses the realization of the exterior cube representation by the maximal parabolic subgroup of the simply connected group of type $E_6$. We give an explicit description of the relevant geometric quotient $U_M\backslash N'$, compute its invariant measure, and relate Shahidi's partial Bessel functions to partial Bessel integrals on the Levi subgroup. The desired stability then follows from an asymptotic expansion of these partial Bessel integrals and the vanishing of highly ramified Mellin transforms.

Let $q$ range over odd prime powers and let $G_q=\mathrm{GL}_2(\mathbb{F}_q)$. Fix a prime number $\ell$. Motivated by work of Peluse and Soundararajan on Miller's conjecture for character tables of symmetric groups, we study the proportion of entries in the character table of $G_q$ which are not divisible by $\ell$, in the sense of divisibility in the ring of algebraic integers. We prove that $N_\ell(q)=\frac{q^4}{2}+O_\epsilon(q^{3+\epsilon})$ for every $\epsilon>0$, where $N_\ell(q)$ denotes the number of entries which are not divisible by $\ell$. We also show that the number of zero entries is $\frac{q^4}{2}+O_\epsilon(q^{3+\epsilon})$. Consequently, the proportion of all entries not divisible by $\ell$ tends to $1/2$, while the proportion of nonzero entries not divisible by $\ell$ tends to $1$. This differs significantly from the symmetric-group case, where almost every character-table entry is divisible by any fixed prime. We also prove an angular equidistribution result for the nonzero character values as $q\to\infty$. We show that the arguments become equidistributed in $[0,2\pi]$. This proves an analogue of Miller's question on the distribution of signs among the nonzero entries in character tables of symmetric groups.

We introduce the notion of sectional indecomposability and study it for finite groups: a group $H$ is sectionally indecomposable if, whenever $H$ is a section of a direct product $A \times B$, then $H$ is already a section of $A$ or of $B$. We show that the study of sectionally indecomposable finite groups reduces to the monolithic case. Our main result is a complete characterisation of sectional indecomposability for monolithic primitive groups: such a group $G$ with $N = \mathrm{soc}(G)$ is sectionally indecomposable if and only if either $N$ is non-abelian, or $N$ is a $p$-group and $O_{p'}(G/N) \neq 1$. The proof relies on the introduction of the notion of an $H$-Frattini module and on the theory of the universal $p$-Frattini cover, together with a result of Griess–Schmid. As a corollary, every monolithic primitive solvable group is sectionally indecomposable. We also discuss the non-primitive case, which appears significantly harder, and highlight open questions concerning monolithic $p$-groups.

Mixed-precision variants of the Jacobi algorithm for symmetric positive definite eigenproblems and the one-sided Jacobi algorithm for singular value decompositions have recently been shown to compute eigenvalues and singular values to high relative accuracy. However, these analyses do not address the accuracy of the computed eigenvectors and singular vectors. In this paper, we prove error bounds for the computed eigenvectors and singular vectors, where the error is measured by the sine of the angle between the vector and its computed counterpart. The obtained bounds preserve the relative gap structure of the bounds for Jacobi algorithms proved by Demmel and Veselić, but involve the scaled condition number of the preconditioned matrix rather than that of the original matrix (the former of which is typically much smaller). Numerical experiments support our theoretical bounds and demonstrate that the mixed-precision preconditioned Jacobi algorithms are especially effective for ill-conditioned matrices with small absolute gaps and moderate relative gaps between eigenvalues or singular values.

This paper studies the approximation of invariant distributions for a broad class of law-dependent dynamics, including McKean-Vlasov stochastic differential equations and Boltzmann-type equations. We consider discrete-time approximation schemes with decreasing time steps and analyse the convergence of their associated ergodic (or occupation) measure towards the invariant distribution of the underlying continuous-time process. Under a general coupling assumption, we prove convergence in the expected $p$-Wasserstein distance ($p \ge 1$) and derive explicit convergence rates. Our approach combines estimates on ergodic averages, regularization techniques for discrete measures, and a generalized discrete Gronwall lemma to control the error between the self-interacting scheme and the target invariant measure. We show that our framework applies to a wide range of models: McKean-Vlasov SDEs, a Boltzmann type equation, and a neuronal model.

Search and recommender systems have produced highly relevant search results. A natural next step in the development of such systems in e-commerce is to rerank these results to increase the platform's revenue from paid promotion products. However, maximizing revenue alone may degrade the user experience by reducing relevance or increasing fraud risk. To avoid this, we state the reranking problem as an integer linear program ($ILP$) that maximizes revenue subject to per-query constraints on other metrics, e.g., relevance. Since solving $ILP$ exactly for every query is slow for deployment to the online service, we propose a lightweight permutation-based reranking approximation algorithm PermR. At each step, the algorithm selects a pair of neighboring items and swaps them to either improve the objective or repair a violated constraint. We evaluate PermR across multiple categories of a large classified platform in offline and online settings. PermR achieves about 63% of the ILP revenue improvement, within production latency limits, preserving all constraints. In a 14-day online A/B test over 56 million search queries, PermR increased revenue by $2$%.

The classification of mixed-state phases requires criteria beyond two-point correlation functions, such as the decay of the mutual information (MI) and the conditional mutual information (CMI), with the latter encapsulated in the notion of Markov length. Here we show how such static properties of the fixed point of a Lindbladian follow from natural dynamical features of its generators: rapid mixing and frustration-freeness. We focus on systems with long-range interactions, and prove (i) that local Lindbladians satisfying (global) rapid mixing and frustration-freeness have fixed-points whose CMI decays with the shielding distance, and (ii) that (local) rapid mixing together with primitivity and regularity implies global decay of MI. For long-range interactions decaying with a power law with rate $\alpha$, both quantities decay polynomially rather than exponentially, in contrast to the finite- and short-range regimes where exponential decay (a finite Markov length) is expected within a phase. We further show that Gibbs states of long-range, non-commuting Hamiltonians satisfy a local Markov property at any temperatures, extending the recent results (Chen–Rouzé, 2025) for short-range systems to the long-range regime relevant to a variety of experimental platforms. As a numerical example, we study the long-range Ising model both with and without a transverse field. We find regimes in which the polynomial decay of the CMI holds, in accordance with the bounds proven.

The theoretical foundations of the EM algorithm are often thought of in the context of Gaussian mixture models, However, the practical use cases of the EM algorithm span beyond Gaussian models. This paper establishes the first step towards understanding the behavior of the EM algorithm under mixtures of non-Gaussian densities. We show that a mixture of two Exponential distributions can be approximated by the EM algorithm at the sub-Exponential rate of convergence in at most $\log(n)$ iterations. The results here show that extending away from Gaussian mixture models does not affect the statistical performance of the EM algorithm. Furthermore, we present generalizations of typical assumptions in the Gaussian setting like minimum mean-separation and signal-to-noise ratio to the sub-Exponential setting. A simulation study is used to highlight the empirical performance of EM for mixtures of exponentials with promising results for the extension of existing theory to a larger class of mixture models.

The celebrated (homological) nerve theorem makes use of spectral sequences to determine the homology of a simplicial complex. However, this theorem cannot effectively compute the homology in every circumstance. In this paper, we develop an effective version of the nerve theorem, yielding a new and powerful tool for homology computation. The essence of our theorem can be formulated in the following manner. Suppose, $X$ is a simplicial complex with covering subcomplexes $A_1, \dots ,A_k$, that is, $X= \cup_{i=1}^k A_i$ and $\mathcal{N}(X)$ is the nerve of $X$ with respect to its covering. Let $\mathcal{W}_\alpha$ be a given gradient vector field on $A_{\alpha}(=\cap_{i \in \alpha} A_i)$ for each $\alpha \in \mathcal{N}(X)$. Then, we use the mere information of the gradient trajectories in $A_{\alpha}$ for each $\alpha \in \mathcal{N}(X)$ to explicitly compute the homology groups of $X$. Furthermore, we point out here, that these gradient vector fields do not need to be coherent, that is, they do not need to coincide on the intersections, which gives us ample flexibility to apply our theorem. Moreover, we can further simplify the computation of the homology groups using a gradient vector field on the nerve of $X$. Our approach is purely combinatorial, in the sense that it does not involve any notions of geometric realisation, continuity or homotopy, which makes it more amenable to computation and coding.

Erdős conjectured that every triangle-free graph on $N$ vertices can be made bipartite by deleting at most $N^2/25$ edges; the bound would be sharp, attained by the balanced blow-up $C_5[N/5]$. Writing $\beta(G)$ for the minimum number of edges whose deletion makes $G$ bipartite and $a(N) = \max\{\beta(G):G$ triangle-free on $N$ vertices$\}$, the conjecture is $a(N)\le N^2/25$, and for $N=5n$ it reads $a(5n)\le n^2$. Balogh, Clemen and Lidícký proved it for large $N$ in the two density tails (edge density at most $0.2486$ or at least $0.3197$) and proved the global bound $a(N)\le N^2/23.5$; the medium-density band remains open. We prove \[ a(5n) = n^2 \qquad \text{for every } 1 \le n \le 40, \quad \text{i.e. } N \in \{5,10,\dots,200\}. \] The proof is computer-assisted and combines three ingredients. (i) A per-root-MaxCut envelope: for the $107$ triangle-free $7$-root types, the mean over types of the best per-type cut is an upper bound $d_{\rm mono}(W)\le U_7(W)$ that is tight at the $C_5$-blow-up. (ii) An order-$10$ flag-algebra certificate – the per-root-MaxCut rows at $7$ and $8$ roots together with rooted-Horn cuts and a manifestly-PSD moment block – bounds the envelope on the medium band, $U_7(W)\le \tfrac{2}{25}+\delta$ with an explicit rational $\delta\approx 4.8558\times10^{-5}$, for every triangle-free graphon $W$ of edge density in $[0.2486,0.3197]$. (iii) The blow-up identity $\beta(G[t])=t^2\beta(G)$ plus integrality of $\beta$ turns this into $\beta(G)\le n^2+\tfrac{25}{2}n^2\delta$ for any $5n$-vertex band-density $G$, and $\tfrac{25}{2}n^2\delta<1$ for $n\le 40$; the two density tails are handled by the Balogh-Clemen-Lidícký bounds, transferred to finite $N$ by the same blow-up. The envelope bound $d_{\rm mono} \le U_7$ is a genuine graphon upper bound (each per-root rule is one global $2$-colouring), the certificate is verified in exact rational arithmetic, the moment positivity is Razborov's flag-algebra theorem exhibited as an exact Gram factorization, and the bound is cross-checked against brute-force max-cut on all triangle-free graphs of order at most $12$. The same envelope at orders $9$ and $10$ provably does not reach the constant needed for larger $n$; we explain why, and locate the all-$n$ conjecture at a single self-tight obstruction.

This work introduces a novel high-order numerical framework for solving kinetic equations, designed to remain uniformly valid across all regimes of the mean free path, spanning from the rarefied kinetic scale to the incompressible hydrodynamic limit. The method is built upon a micro-macro decomposition, which reformulates the underlying kinetic equation into a coupled system consisting of a macroscopic part, representing the fluid-dynamic evolution, and a microscopic part, describing the non-equilibrium deviations. The proposed framework ensures high-order temporal accuracy through the use of Implicit-Explicit Runge-Kutta methods, which provide stability and efficiency in stiff regimes, while spatial resolution is enhanced by combining finite-difference WENO reconstructions with high-order central difference approximations. A key feature of the proposed methodology is its Asymptotic-Preserving (AP) property. We demonstrate that, in the appropriate asymptotic limit as the mean free path tends to zero, the scheme consistently reduces to a high-order finite-difference formulation of the incompressible Navier-Stokes equations. To support the theoretical findings, a set of numerical experiments are performed on one- and two-dimensional benchmark problems, which confirm the accuracy, stability, and versatility of the method across different flow regimes.

Third medium contact provides a smooth continuum alternative to classical contact algorithms by replacing explicit contact constraints with a highly compliant fictitious medium. In this work, an auxiliary-field stabilization is introduced in which a deformation-gradient-like field is treated as an independent unknown in the third medium and coupled to the physical deformation gradient by a penalty term. A gradient contribution acting on the auxiliary field provides the regularization mechanism without requiring a direct evaluation of higher displacement derivatives. Linear and quadratic interpolation spaces are investigated, including continuous and element-wise discontinuous auxiliary-field approximations. The numerical results show that continuous low-order auxiliary fields provide an effective gradient-type stabilization of the third medium, even when the displacement field is approximated by first-order finite elements. For element-wise discontinuous auxiliary fields, the additional unknowns remain local to each element and can be eliminated locally by static condensation, so that the global system does not necessarily contain additional auxiliary degrees of freedom. Benchmark problems involving large deformation, progressive self-contact and severe third-medium compression are used to assess the formulation.

Epidemic dynamics introduce time-varying heterogeneity into insured populations, as individuals' risk profiles depend on their evolving health status, thereby challenging classical insurance models based on homogeneity. Motivated by this phenomenon, we develop in this paper actuarial risk models in which the claim process is directly linked to an underlying general stochastic population dynamics, resulting in interacting subpopulations with different claim frequencies and severities. We propose both collective and individual modeling frameworks that track the changing composition of the insured population and its impact on aggregate claims. For these models, we derive scaling limits and provide bounds and approximations for ruin probabilities, offering tractable tools for risk assessment. Applications in an SIS context are detailed for different classes of risk processes, thereby contributing to a better understanding of how contagion-driven changes in population structure affect insurer solvency and supporting more realistic modeling of insurance portfolios in the presence of epidemic risk.

Guessing Random Additive Noise Decoding (GRAND) performs decoding by sequentially guessing channel error patterns (EPs). Ordered Reliability Bits GRAND (ORBGRAND) is a notable instance suitable for efficient implementation, as it schedules EPs solely according to the ranking of soft channel outputs. In this paper, we generalize this principle to a broader class of GRAND algorithms whose testing order depends only on reliability ranking, referred to as ORB-type GRAND. We develop a unified analytical framework based on a key quantity termed the average guessing posterior (AGP), which captures the effectiveness of each EP and reduces decoding into an ordering problem over the EP space. For random code ensembles, we derive exact expressions for the block error rate (BLER), stopping-time distribution, and average number of tests under a fixed test budget. The analysis separates target-miss and target-preemption errors and shows that ordering EPs by non-increasing AGP is optimal over the EP set under consideration. For fixed linear block codes, we derive the BLER expression that isolates the code-dependent target-preemption term and characterize this term through higher-order weight relationships of codeword tuples, with a computable first-order upper bound as a useful special case. Guided by these insights, we formulate ReShuffled-ORBGRAND (RS-ORBGRAND) as an offline AGP-based reshuffling scheme. Numerical results for the Bose–Chaudhuri–Hocquenghem (BCH)$(127,113)$ code show that RS-ORBGRAND consistently improves existing ORB-type GRAND algorithms and lies within $0.1$~dB of a maximum-likelihood decoding lower-bound benchmark at a BLER of $10^{-6}$.

We develop a framework for constructing and rigorously validating controls for semilinear wave equations with localized interior control. We analyze the continuous energy-space problem and certify exact control for finite-dimensional Galerkin systems. Under explicit smallness and neighborhood conditions, our a posteriori theorem proves the existence of an exact Galerkin control near a computed candidate. It combines a validated terminal-state error, a bounded right inverse of the derivative of the control-to-terminal-state map, and a Lipschitz bound for that derivative. Under additional uniform right-inverse and neighborhood conditions, we also show that Newton correction converges with quadratic decay of terminal-state errors. We use residual estimates and interval arithmetic to enclose the nonlinear trajectory and its linearized controllability Gramian. For the specified eight-mode tanh problem, we rigorously bound a source-response operator to prove contraction of the exact source-frozen iteration. We compare deterministic controls based on the Hilbert Uniqueness Method with neural approximations of temporal modal coefficients. Two Newton corrections reduce the computed terminal-state errors of all six tested neural candidates below $10^{-8}$. For the three certified candidates, we obtain smaller bounds on the distance to an exact Galerkin steering control by refining the validation mesh while keeping the computed controls fixed.

We study small–data solutions of a nonlinear scalar field equation on spatially flat $d$–dimensional FLRW spacetimes ($d\ge4$). In conformal time $\tau$ the field satisfies a damped semilinear wave/Klein–Gordon equation with time–dependent coefficients determined by the scale factor $a(\tau)$ and the conformal Hubble rate $H(\tau)=\dot a/a$. We focus on accelerated conformal expansion of the form $H(\tau)=H_0(1+\tau)^{-\alpha}$ with $H_0>0$ and $0\le\alpha<1$, for which $a(\tau)$ grows stretched–exponentially, and we assume a power potential $V(\varphi)=-\frac{\varepsilon}{m+1}|\varphi|^{m+1}$. For global solutions arising from sufficiently small, spatially localized initial data, we introduce the conformal rescaling $\phi=a^{(d-2)/2}\varphi$, which removes the first–order Hubble damping and exposes the interaction as a time–dependent coupling. In the rescaled equation the nonlinearity is weighted by $g(\tau)=a(\tau)^{\sigma}$ with $\sigma=\frac{d+2-(d-2)m}{2}$, so the conformal power $m_{\mathrm{conf}}=\frac{d+2}{d-2}$ is the sharp threshold for redshift suppression: $g$ decays for $m>m_{\mathrm{conf}}$, is constant for $m=m_{\mathrm{conf}}$ (classical conformal invariance), and grows for $1<m<m_{\mathrm{conf}}$. For accelerated conformal expansion $H(\tau)=H_0(1+\tau)^{-\alpha}$ with $0\le\alpha<1$ and for superconformal interactions $m>m_{\mathrm{conf}}$, we prove that $g\in L^1([0,\infty))$ and deduce small–data global existence together with scattering/asymptotic linearization for $\phi$. As a complementary result in the diffusion–dominated regime $1<m<1+\frac{2}{d-1}$, we adapt a weighted energy method for variable damping to deduce explicit $L^2$ and $L^1$ decay rates. These bounds provide a quantitative PDE formulation of redshift–induced suppression of nonlinear scalar self–interactions at late conformal times.

The natural extension of the triangle map (a type of multi-dimensional continued fraction algorithm) is completely described in all possible dimensions. The motivation and inspiration for this natural extension stems from the triangle map's recent link to the classical study of integer partitions. Inspired by Young conjugation for integer partitions, we show that the natural extension has an internal symmetry and allows the natural extension to be subdivided into four natural subdomains. This appears to be new even for the classical case of the natural extension for continued fractions, namely for both the classical Gauss map and the classical Farey map.

In this paper, we study a broad class of structured monotone inclusion problems in real Hilbert spaces. We propose a novel primal-dual splitting algorithm for solving such inclusions, which accommodates multiple monotone operators and cocoercive terms, as well as a composite monotone operator involving the linear map. The algorithm combines forward evaluations for the cocoercive components with backward resolvent steps for the monotone operators and employs a dual update for the linear composition term. It generalizes and unifies several existing methods, while requiring only a single resolvent or operator evaluation per iteration. We prove weak convergence of the iterates under standard assumptions on monotonicity and cocoercivity. Furthermore, we establish strong convergence under a mild regularity condition, such as uniform monotonicity. Numerical experiments on image deblurring and denoising problems demonstrate the efficiency and flexibility of the proposed algorithm.

A Gibbs point process models particles interacting in the continuum through a potential. Among the most classical examples is the hard-sphere model, where given an activity parameter $\lambda$, a radius $r$, and a bounded set $\Lambda \subset \mathbb{R}^d$ one samples a Poisson process of intensity $\lambda$ in $\Lambda$ conditioned on the points forming the centers of an $r$-sphere packing. We prove uniqueness of infinite-volume Gibbs measure, analyticity of the pressure, and various notions of spatial and temporal mixing for activities up to what we define as the spectral threshold $\lambda_{spec}$ of the potential. For each fixed dimension $d \geq 2$, this improves the uniqueness and analyticity bounds for the hard-sphere model. As $d \to \infty$, our improvement over the classical bounds grows exponentially. We also prove an optimal mixing time bound for heat bath dynamics for the hard-sphere model up to an expected density of $\Theta(d / 2^d)$, the first result that asymptotically matches the maximum density for rapid mixing predicted by Parisi and Zamponi. We also exhibit repulsive, radial pair potentials for which $\lambda_{spec} = + \infty$, showing that the corresponding Gibbs point processes have no phase transition at any activity $\lambda > 0$. Further, in dimensions $8$ and $24$ we exhibit such a potential with no phase transition for which the work of Cohn-Kumar-Miller-Radchenko-Viazovska proves that the unique ground state at any fixed density is given by the $E_8$ and Leech lattices, respectively. Our work builds upon a 2013 work of Kondratiev-Kuna-Ohlerich that implicitly defined $\lambda_{spec}$ and proved a spectral gap for a Glauber-like continuum birth-death dynamics. Our main work shows that such a spectral gap implies several strong notions of absence of phase transition and analyzes the behavior of $\lambda_{spec}$ for interesting potentials.

We study wave-type equations on dynamical spacetimes that settle down to a subextremal Kerr black hole spacetime. We prove strong estimates for solutions of (tensorial) linear wave-type equations when the time-translation-invariant model satisfies a spectral assumption of mode stability type. We allow for this model to admit zero energy bound states; besides the scalar wave operator (which has no bound states), examples include the wave operator on 1-forms and the linearization of the Einstein field equations in generalized harmonic gauge. We demonstrate the utility of our estimates by proving the global existence of solutions to some quasilinear wave equations, including in the presence of zero energy bound states. The results proved here are, moreover, crucial ingredients in the author's proof of the nonlinear stability of subextremal Kerr black holes. Our key novel linear estimate controls linear waves in weighted $L^2$-based spacetime Sobolev spaces that encode b-regularity, by which we mean regularity with respect to spacetime scaling, spatial scaling (in a hyperboloidal foliation of spacetime), and angular derivatives; this estimate is moreover tame in the b-regularity order, as needed for its applicability in a Nash-Moser iteration scheme. Its proof combines four main ingredients: microlocal propagation estimates in the edge-b-setting near null infinity (as introduced by the author with Vasy) and in the author's 3b-setting in the forward cone; estimates for the stationary model operator; energy estimates on edge-b-spaces on finite time intervals; and commutations with b-vector fields. For the nonlinear applications, we moreover develop a dictionary between decay rates in different spacetime regimes on the one hand and weighted low-energy resolvent estimates on the other hand. This paper builds on Part I only a broad conceptual level, and is largely self-contained.

Reid–Smith parametrised ($P$)-closed groups acting on trees using graph-based combinatorial structures known as local action diagrams. Properties of the acting (topological) group, such as being locally compact, compactly generated, discrete or simple, are reflected in its local action diagram. In this article, we describe the translations of ($P$)-closed groups and their axes in terms of local action diagrams. As applications, we determine the scale function of ($P$)-closed groups and characterise unimodular as well as uniscalar ($P$)-closed groups. The latter provides one possible answer to a question of Thomas Weigel.

Let $G$ be a closed type-preserving subgroup of the automorphism group of a thick locally finite building $X$ of finite rank, and assume that $G$ acts Weyl-transitively. We prove that every unitary representation of $G$ is mixing, unless its restriction to a parabolic subgroup of minimal non-spherical type is amenable in the sense of Bekka. It follows that every unitary representation of $G$ that is weakly contained in the regular representation, is mixing. In case $X$ is of minimal non-spherical type and its thickness satisfies some modest lower bound, we deduce that $G$ has the Howe–Moore property provided its only compact quotient is trivial. We also obtain results on rigidity of invariant random subgroups for Kac–Moody lattices of compact hyperbolic type, yielding examples of infinite finitely presented Kazhdan groups with exactly two ergodic invariant random subgroups.