Papers

Dengue remains a major public health challenge in tropical regions, and recurring outbreaks suggest that current intervention strategies are not yet fully effective. Existing mathematical models typically assume unlimited hospital capacity and continuously applied fogging, neglecting practical constraints that strongly influence disease control. We develop a non-smooth ordinary differential equation model of dengue transmission that incorporates finite hospital capacity and a threshold-triggered fogging strategy activated when reported infections exceed a prescribed fraction of the available capacity. The model exhibits three epidemiologically relevant operating regimes, reflecting changes in hospitalization and vector-control policies as the epidemic progresses. We establish the existence and local stability of the disease-free and endemic equilibria. Numerical continuation confirms the analytical results and reveals boundary-equilibrium bifurcations at the switching thresholds, a Hopf bifurcation after hospital capacity is exceeded leading to sustained oscillatory outbreaks, and a fold bifurcation near the epidemic threshold that generates additional unstable equilibria. We further investigate periodic solutions with respect to the fogging rate and activation threshold, identifying locally optimal intervention regimes that reduce epidemic peaks while avoiding unnecessarily intensive control efforts. The results demonstrate that hospital capacity, reactive fogging, and intervention thresholds fundamentally shape dengue dynamics and provide quantitative insights for designing effective state-dependent control strategies under limited healthcare resources.

We use the tropical trigonal construction to describe the minimal toroidal resolution of the Prym period map $\overline{R}_4\dashrightarrow\overline{A}_3$.

Efficient parametric transient prediction at unseen parameter values and under new operating conditions remains challenging because repeated high-fidelity simulations are computationally prohibitive. Existing data-driven surrogates and parametric reduced-order models perform well within sampled ranges but often lose reliability beyond them. This study proposes a physics-guided spectral parametric reduced-order modeling framework for controlled dynamical systems. Parameter-dependent reduced spectral operators are identified from transient snapshots using Dynamic Mode Decomposition with control, separating intrinsic dynamics from external control effects. After physics-guided parameter transformation, aligned spectral quantities and reduced operator components are propagated across parameter conditions using Secondary Dynamic Mode Decomposition, with linear and radial basis function regressions as baselines. Baseline regularization and nondimensional time mapping improve robustness. Validation uses a mechanical transmission system, a Helium-Xenon closed Brayton cycle, and a Karman vortex street, covering linear transient, nonlinear transient, and nonlinear periodic dynamics. For the mechanical and Brayton systems, system-level multivariable responses at unseen parameter values and under new operating conditions are predicted with relative norm errors below 1%. For the vortex-street system, nondimensional time mapping preserves dominant vortex-shedding structures across Reynolds numbers. Further analyses compare the framework with an LSTM surrogate and assess extrapolation confidence using an auxiliary error-prediction model. Overall, the framework extends parametric reduced-order modeling from in-domain approximation toward extrapolative transient prediction of controlled dynamical systems under unseen conditions.

Motivated by recent results \cite{charamaras_kousek_mountakis_radic2025BBingroups} on infinite sumsets of the form $B+B=\{b_1+b_2:b_1,b_2\in B\}$ in large subsets of abelian groups, and an old problem of Owings \cite[Problem E2494]{Owing_problems} about the partition regularity of $B+B$ in $2$ colours, we show the following theorem. Let $(G,+)$ be a countable abelian group such that the subgroup $\{g+g\colon g\in G\}$ has finite index and the doubling map $D: g\mapsto g+g$ has finite kernel. Let also $\Phi=(\Phi_N)_{N}$ be any Folner sequence in $G$ and $\Phi/2=(D^{-1}(\Phi_N))_{N}$. Then, if $A\subset G$ is such that $d_{\Phi}(A)+d_{\Phi/2}(A)>1$, there is an infinite set $B\subset G$ and some $t\in G$ for which $t+B+B\subset A$. We prove that this result implies the main theorem in \cite{charamaras_kousek_mountakis_radic2025BBingroups}, and construct an example to show the reverse implication does not hold. Moreover, we show that our main theorem is optimal in a strong sense. Namely, for any countable abelian group $(G,+)$ with the aforementioned assumptions – which are necessary – there exists a Folner sequence $\Phi$ and a set $A\subset G$ so that $d_{\Phi}(A)+d_{\Phi/2}(A)=1$, but there is no infinite set $B\subset G$ and $t\in G$ for which $t+B+B\subset A$.

The Kaluzhnin–Krasner embedding establishes that, for a group extension, the middle group can be embedded into the wreath product of its kernel and cokernel. Recently, this construction has been generalised in a categorical framework, recovering both the classical group-theoretic result and its analogue for Lie In this article, we apply this categorical framework to two specific cases: precrossed modules and crossed modules (over groups). For precrossed modules, we introduce a Kaluzhnin–Krasner embedding – previously unformulated – by rigorously following the steps of the categorical procedure. In contrast, for crossed modules, we encounter substantial obstacles: while we present partial positive results, we also explain why constructing a full-fledged Kaluzhnin–Krasner embedding in this context is far more difficult.

Entropy regularization smooths equilibrium policies in time-inconsistent stochastic control. At low temperature, the same Gibbs response can strongly amplify errors in learned rewards and dynamics. We show that the derivative of an exploratory equilibrium is governed by a backward Volterra-parabolic resolvent. Along an aligned positive mode, a lower bound has the same exponential order. A block decomposition identifies the source of the amplification: causal paths contribute powers of 1/tau, whereas a positive feedback cycle can produce exponential growth. At fixed temperature, a local equilibrium branch is twice differentiable with respect to finite-dimensional model parameters, which yields a function-valued delta method. A bounded uniformly elliptic diffusion realizes this path-cycle distinction in every finite dimension. Closing one positive cycle changes the root-n linear-response boundary from a power law to order 1/log n; along the cyclic Perron mode, right-endpoint discretization is relatively consistent exactly when N tau^2 -> infinity. An affine model also gives an exact nonlinear transition at the Lambert-W temperature beta T / W(beta T sqrt(n)). Numerical calculations illustrate these rates.

We introduce and study the so-called {\it generalized $\sqrt{J}$-fine rings}, where every element outside the Jacobson radical is the sum of a unit and an element from the set $\sqrt{J(R)} := \{ x \in R : x^{n} \in J(R) \text{ for some } n \ge 1 \}$. This commonly extends the notions of fine and generalized fine rings defined, respectively, by Călugăreanu-Lam (J. Algebra & Appl., 2016) and Zhou (J. Algebra & Appl., 2022). Specifically, we prove that this class is closed under full matrix rings of any size, as well as we completely characterize when group rings over locally finite groups are generalized $\sqrt{J}$-fine. We also show that every such ring is 2-clean, thus properly placing it between generalized fine rings and 2-clean rings. Several examples are also provided to illustrate the complicated behavior of the introduced concept and its numerous boundaries.

We determine the generalized Turán number of vertex-disjoint paths of length two and characterize all corresponding extremal graphs. Our proof combines the Lovász form of the Kruskal–Katona theorem with a discrete convexity argument.

We revisit the classical Krasnosel'skii-Mann fixed point iteration for contractions and nonexpansive maps in general normed spaces. This iteration is ubiquitous across a wide range of areas, including convex optimization, monotone inclusions, Markov decision processes, under-relaxed methods for nonlinear PDEs, and more. Drawing on a remarkable connection with a Markov chain on $\mathbb{Z}^2$, and using counting arguments from enumerative combinatorics of lattice paths, we derive explicit estimates for the distance between iterates, as well as non-asymptotic error bounds for the fixed point residuals. As the contraction parameter approaches one, these bounds smoothly recover the known estimates for nonexpansive maps. Building upon these estimates, we further derive error bounds for inexact Krasnosel'skii-Mann iterations.

Following approach of Iorgov–Lisovyy–Teschner, we construct solutions of the (modified) Riemann–Hilbert problem using conformal blocks of $(1,k)$ Virasoro models. For $k>1$ case, the solution of this Riemann–Hilbert problem is not unique due to more singular behavior at punctures. On the CFT side the dimension of the space of conformal blocks also increases. We specifically study the $k=2$ case, which corresponds to the central charge $c=-2$ and symplectic fermions. We explicitly construct a corresponding solution of the modified Riemann–Hilbert problem in the case of 3 punctures and prove its uniqueness under suitable initial data conditions. We also obtain new bilinear relations for $c=-2$ tau functions.

Physical artificial intelligence (AI) systems involve distributed sensing agents with embedded AI models that must coordinate to perceive, reason, and act in networked environments. Transmitting raw sensor data incurs significant communication overhead, latency, and redundancy. While semantic communication (SC) mitigates these challenges by transmitting task-relevant information, existing deep learning-based joint source-channel coding approaches exhibit limited adaptability, poor out-of-distribution generalization, and scalability challenges. To address these limitations, this paper proposes a framework for compositional semantic communication (CSC), enabling heterogeneous physical AI sources to transmit semantic representations (SRs) that compose meaningfully at a base station (BS) or edge server for remote inference. First, a category-theoretic measure of compositional semantics is developed to quantify each device's contribution to inference tasks beyond mutual information. Second, Grothendieck topologies and presheaves formalize semantic composition across devices, ensuring consistency and task relevance. Building on these foundations, multi-device coordination is formulated as a Stackelberg game in which devices commit to encoding strategies and the BS optimally composes received SRs. An ADMM-based algorithm computes equilibrium signaling strategies. Equilibrium existence is established under mild conditions and is Pareto optimal when compositional information yields increasing collective benefit. Simulation results demonstrate that the proposed approach achieves up to 17% bandwidth reduction and 53% lower end-to-end latency than cooperative multi-agent, distributed gradient descent, and uniform-selection CSC baselines while maintaining 85% inference accuracy across diverse autonomous driving scenarios.

We investigate the Hales-Jewett numbers and some variants of them, both computationally and via enumerative and probabilistic arguments. In particular, we give the improved lower bound formula on the combinatorial-geometric variant of the Hales-Jewett numbers. {As a consequence of one of the variants investigated, we show that the Milton Bradley game Connect Four played with 2 players on a 5-dimensional hypercube of any side length can end in a draw.

We investigate the topology of the set $\mathsf E$ of entangled bipartite density operators acting on $\mathbb{C}^{n_1}\otimes\mathbb{C}^{n_2}$. We start by showing that $\mathsf E$ is path-connected, and even simply connected except in the two-qubit case. In this exceptional case $\mathsf E$ turns out to be homotopy equivalent to the set of maximally entangled states, which itself is homeomorphic to $\mathbb{RP}^3$. Here we also compute the complete homology of the closure and interior of $\mathsf E$. In all larger dimensions, we show that the homology and homotopy groups of $\mathsf E$ vanish in degrees $1\leq k\leq 2(n_1-1)(n_2-1)-2$, and all homology groups of degree $k\geq (n_1n_2)^2-3$ also vanish. This range is controlled by the space $\mathsf W$ of entanglement witnesses, which we show is highly connected beyond two qubits and homotopy equivalent to $\mathsf E$. By computing the Euler characteristic, using a torus-action fixed point argument together with Alexander duality, we show that $\mathsf E$ nevertheless has non-trivial reduced homology over every field for all $n_1, n_2 \geq 2$.

We construct the periodic directed landscape, which is the conjectured scaling limit for periodic models in the Kardar-Parisi-Zhang universality class. We establish the convergence of periodic exponential last passage percolation to the periodic directed landscape. Moreover, we confirm conjectures by Baik, Liao and Liu on the local structure of the periodic KPZ fixed point, and establish the convergence of periodic ASEPs to periodic KPZ fixed points, coupled according to the same periodic directed landscape. Our main tool to construct the periodic directed landscape (and prove convergence to it) is a technique for gluing full-space directed landscapes (and their prelimits), which is of independent interest.

Fast and accurate fluid simulation of radio-frequency capacitively coupled plasmas (RF CCPs) is of great importance for the iterative design and parameter optimization of modern plasma reactors. This study presents the first successful extension of the time-domain harmonic balance (HB) method to a fully coupled drift-diffusion-Poisson system with complete electron-energy transport for RF plasma simulations. To resolve the severe numerical stiffness arising from highly nonlinear energy-dependent kinetics and dense phase-coupling, a highly efficient spatiotemporal operator-splitting strategy is employed. By sequentially executing a spatial implicit relaxation and a cell-local temporal inversion, this strategy entirely avoids the memory-intensive assembly of global Jacobians while preserving robust numerical stability. The proposed method is rigorously validated against a standard parallel-plate argon CCP benchmark. Evaluated across all discrete temporal collocation points, the HB solution demonstrates that retaining eight harmonics perfectly resolves both the quasi-steady bulk plasma and the highly nonlinear transient sheath dynamics, yielding macroscopic relative errors strictly below 0.3% compared to conventional dual-time stepping (DTS) solutions. Beyond its high physical fidelity, the time-domain HB method completely bypasses the prohibitive physical transients required by conventional time-marching methods. Evaluated on a purely sequential single-core execution, the HB method delivers a greater than 10-fold speedup over fully converged DTS baselines and remains over 5 times faster than the coarsest time-marching configurations. These results establish the time-domain HB framework as a physically rigorous, memory-efficient, and highly accelerated paradigm for practical RF plasma simulations.

We study whether proper homotopy equivalence classifies open contractible manifolds arising as interiors of compact contractible manifolds. In dimensions three and four, it does: any two such interiors that are properly homotopy equivalent are homeomorphic. In every even dimension $N\geq 6$ and every odd dimension $N\geq 9$, however, it does not: a single proper homotopy type contains infinitely many pairwise nonhomeomorphic smooth open contractible $N$-manifolds. The cases of dimensions five and seven remain unresolved.

Let $X$ be a finite set of integers with cardinality $\nu = \kappa \lambda$. A modular balanced partition design is a triplet $(X, \mathcal{A}, \mathcal{B})$ satisfying the following conditions: \begin{itemize} \item $\mathcal{A}$ is a partition of $X$ into $\kappa$ blocks of size $\lambda$, such that every element of $X$ appears in exactly one block. If $\mathcal{A} = \{A_1, A_2, \ldots, A_{\kappa}\}$, then $\sum_{a\in A_i} a \equiv i \lambda \pmod{\nu}$, for each $i=1,2,\ldots,\kappa$ \item $\mathcal{B}$ is a partition of $X$ into $\lambda$ blocks of size $\kappa$, such that every element of $X$ appears in exactly one block. If $\mathcal{B} = \{B_1, B_2, \ldots, B_{\lambda}\}$, then $\sum_{b\in B_j} b \equiv j \kappa \pmod{\nu}$, for each $j=1,2,\ldots,\lambda$ \item $A_i \cap B_j$ has exactly one element, for any $A_i \in \mathcal{A}$, and $B_j \in \mathcal{B}$. \itemize} We prove the necessary conditions for the existence of a modular balanced partition design. Moreover, we investigate and identify a relationship between a modular balanced partition design and a subgroup magic rectangle. Then by using affine automorphisms of an Abelian group, we prove the existence of non-isomorphic modular balanced partition designs. Finally, we provide a method to construct a transversal design via a modular balanced partition design.

We prove the Virtual Surjection Conjecture for discrete groups, both for property $F_{n}$ and for property $FP_{n}$: given a product of groups of type $F_k$ (respectively $FP_k$), a subgroup that virtually surjects onto $k$-tuples must be $F_k$ (respectively $FP_k$) as well. We prove the homological $n$-$(n+1)$-$(n+2)$ Conjecture for discrete groups under the assumption the common quotient is finitely presented, and prove that this assumption cannot be dropped. We deduce the $n$-$(n+1)$-$(n+2)$ Conjecture for property $F_{n}$.

We introduce a family of random connected closed subsets of planar Brownian motion, called Brownian loop-catchers, which interpolate between the continuum loop-erased random-walk (LERW) and the Brownian trace. This provides the canonical continuation of Brownian loop soup clusters to central charges $-2\le c<0$. For each such $c$, the corresponding loop-catcher satisfies the following recovery property: adding all loops from an independent Brownian loop soup of intensity $-c/2$ that intersect it recovers the Brownian trace. Furthermore, no such law exists for $c<-2$. We also show that its outer boundary is locally SLE$_\kappa$ with $\kappa = \frac{1}{3}\left(13 - c - \sqrt{(1-c)(25-c)}\right)\in[2,\frac83)$, and the probability that it intersects an interior ball of radius $\varepsilon$ is asymptotically proportional to $|\log\varepsilon|^{-1+\frac{c}{2}}$ when $-2<c<0$. Therefore, a planar Brownian trace contains an SLE$_\kappa$-type curve for every $\kappa\in[2,\frac83]$. Our construction begins with a random-walk loop-catcher on any finite graph such that recursively inserting loops from an independent random-walk loop soup to it recovers the original random walk. The key ingredient is a new entangled multipath LERW, which recovers a union of independent random-walk paths when decorated with a single common random-walk loop soup. We then prove that the random-walk loop-catcher converges to the Brownian loop-catcher under lattice approximations. To this end, we propose a novel Green function test which converts the recovery property of the Brownian loop-catcher into all mixed moments of Green functions in the remaining domain, based on the entangled multipath LERW. Consequently, the recovery property characterizes the full law of the Brownian loop-catcher, not only its filling. The Green function test also extends to the three-dimensional case.

Let $(0,L)\times E_a\subset \R^{d+1}$ denote the cylinder of length $L$, and base $E_a\subset \R^{d},$ where $E_a$ is an open ellipsoid with semi-axes $a=(a_1,...,a_d)$. Let $w_{(0,L)\times E_a}$ denote its torsion function. Heat equation tools are used to show that (i) if $E_a$ is has small eccentricity and $L$ is sufficiently large, then the maxima of $|\nabla w_{(0,L)\times E_a}|$ are located at the centres $(0,0)$ and $(L,0)$ respectively, (ii) if $E_a$ has large eccentricity and $L$ is sufficiently large, then the maxima are located on the lateral side of the cylinder.

Over-the-air computation (OAC) enables efficient function aggregation in wireless networks by exploiting the superposition property of the multiple-access channel. However, practical deployment of OAC is severely challenged by the reliance on accurate carrier synchronization and coherent reception, which are costly and fragile, especially in short-range and low-complexity systems. In this work, we propose a self-coherent, synthesizer-free over-the-air computation framework based on Kramers–Kronig (KK) reception. By transmitting a biased aggregate waveform and employing direct detection followed by KK phase reconstruction at the receiver, the proposed scheme eliminates the need for explicit carrier recovery while preserving coherent-like signal aggregation. We develop a signal-domain system model for multi-user OAC under KK reception and provide a synchronization-relaxation analysis demonstrating that the proposed architecture fundamentally removes carrier-frequency offset (CFO) sensitivity between transmitters and receiver. By shifting synchronization complexity away from strict carrier-phase tracking and eliminating distributed phase alignment requirements, the framework reduces control overhead and improves scalability in multi-user aggregation. A detailed per-symbol mean-squared error (MSE) characterization isolates the impact of channel mismatch and KK reconstruction noise, showing that the proposed self-coherent architecture approaches the theoretical performance limits of baseband OAC under practical operating conditions. Finally, we demonstrate that the approach is particularly well suited for mmWave and sub-THz systems, where oscillator phase instability otherwise represents a fundamental bottleneck to scalable coherent OAC.

In many quantum applications it is important to know whether or not a Bell nonlocal two-qudit state exhibits its nonlocality under correlation scenarios with some given numbers $S_{1},S_{2}\geq1$ of generalized quantum measurements at two sites. In the present article, we find analytically a new general condition sufficient for a nonseparable Werner state with a dimension $d\leq\min\{S_{1},S_{2}\}$ to satisfy all Bell inequalities under every $S_{1}\times S_{2}$-setting correlation scenario with outcomes of an arbitrary spectral type, discrete or continuous $-$ that is, to be $S_{1}\times S_{2}$-setting Bell local, for short. For a variety of measurement settings $S_{1}, S_{2}$, this new general bound is beyond (i) Werner's and Barrett's locality bounds for a nonseparable Werner state and (ii) the related result by Johnson and Viola [Phys. Rev. A 88, 032323 (2013)] on "sharability" of this state. When specialized to $d = 2$ and $\min\{S_{1},S_{2}\} = 2, 3$, our new general bound improves upon the corresponding results by Terhal et. al. in [Phys. Rev. Lett. 90, 157903 (2003)], moreover, for $d=\min\{S_{1},S_{2}\} = 2$, it reduces exactly to the bound derived for this case via the CHSH inequality. We prove also explicitly in operator terms the result of Terhal et. el. that every nonseparable Werner state with a dimension $d >\min\{S_{1},S_{2}\}$ is $S_{1}\times S_{2}$-setting Bell local. The new results presented in this article are important for both the theory of Bell nonlocality and quantum applications based on this fundamental concept.

A longstanding conjecture of Seymour, called Seymour's second neighborhood conjecture, states that every oriented graph $D$ contains a vertex $x$ with $|N^{++}_D(x)|\geq |N^{+}_D(x)|$. The conjecture was verified in a few special classes of oriented graphs, and it remains open for general oriented graphs. We study a stronger property, asking for a vertex $x$ such that there exists a complete matching from $N^+_D(x)$ to $N^{++}_D(x)$. We prove that this stronger version holds for every oriented graph with minimum out-degree at most $5$, and also for every $5$-anti-transitive oriented graph. This implies that every oriented planar graph satisfies the stronger version.

Let $G$ be an $n$-vertex graph with clique number $\omega(G)$, and let $s^+(G)$ denote the sum of the squared positive adjacency eigenvalues. We prove that $$ \sqrt{s^+(G)}\le\left(1-\frac{1}{\omega(G)}\right)n. $$ This strengthens Wilf's classical spectral Turán theorem and resolves a conjecture of Elphick and Wocjan. Adopting the relaxation of our companion paper on the square-energy conjecture, we reduce the theorem to a Motzkin–Straus inequality for doubly nonnegative matrices, which we prove via a local inverse-probability estimate for the Caro–Wei greedy algorithm on the complement.

Accurately solving partial differential equations (PDEs) on arbitrary geometries and a variety of meshes is an important task in science and engineering applications. In this paper, we propose Adaptive Mamba Neural Operators (AMO), which integrates reproducing kernels for state-space models (SSMs) rather than the kernel integral formulation of SSMs. This is achieved by constructing Takenaka-Malmquist systems for the PDEs. AMO offers new representations that align well with the adaptive Fourier decomposition (AFD) theory and can approximate the solution manifold of PDEs on a wide range of geometries and meshes. In several challenging benchmark PDE problems in the fields of fluid physics, solid physics, and finance on point clouds, structured meshes, regular grids, and irregular domains, AMO consistently outperforms state-of-the-art solvers in terms of relative $L^2$ error. Overall, this work presents a new paradigm for designing explainable neural operator frameworks.

We consider the unitary monodromy problem of the following generalized Lamé equations with apparent parameters \begin{equation*} y^{\prime\prime}(z)=\left(\frac{3}{4}\sum_{k=0}^3\wp(z-\frac{\omega_k}{2};\tau)+\sum_{k=0}^3T_k\zeta(z-\frac{\omega_k}{2};\tau)+B\right)y(z), \end{equation*} where $T_0,\cdots, T_3, B$ are apparent parameters. We first decompose the space of apparent parameters, which turns out to be an algebraic set, into three irreducible components. These three components intersect at $(T_0,\cdots, T_3, B)=(0,\cdots, 0)$, which plays an important role in determining whether the monodromy matrices is unitary or not. Following the approach in KdV theory, we define the spectral polynomial which is a degree 4 polynomial of the apparent parameter. We then obtain that the monodromy is not completely reducible if and only if the apparent parameter is a zero of the spectral polynomial. By introducing a branched double cover of the apparent space, which parametrizes all one-dimensional common eigenspaces, we determine the monodromy data for all apparent parameters. By noticing that the equation under the covering map is exactly the spectral polynomial, we obtain that the generalized Lamé curve is isomorphic to the spectral curve. Finally, with the help of the spectral curve defined by the spectral polynomial, we characterize the conditional stability sets in two directions by making use of the local analytic coordinates of the monodromy data and then prove that the monodromy matrices are unitary if and only if $(T_0,\cdots, T_3, B)=(0,\cdots, 0)$ when the period $\tau\in i\mathbb{R}_{>0}$.

We study the distribution of determinant values on lattices in $\operatorname{M}_n(\mathbb R)$ for $n\ge 2$. Let $\Lambda<\operatorname{M}_n(\mathbb R)$ be a lattice whose elements have algebraic entries. We prove that if $\det (\Lambda)$ is not contained in a scalar multiple of $\mathbb Z$, then for every $a<b$, as $T\to \infty$, $$ \#\{v\in\Lambda:\|v\| <T,\ a<\operatorname{det} v<b,\ \operatorname{det} v\ne0\} \sim \frac{C_n}{\operatorname{covol}(\Lambda)} (b-a)T^{n(n-1)} $$ where $\|\cdot\|$ is the Frobenius norm and $C_n>0$ depends only on $n$. Under an additional hypothesis on the isotropic subspaces associated with $\Lambda$, which is automatic for $n=2,3$ and is satisfied when $n\geq 4$, by every lattice spanned by scalar multiples of the elementary matrices, we also obtain an asymptotic formula for the singular lattice points. Both conclusions extend to the broader class of Diophantine lattices under the corresponding hypotheses. For $n=2$, our theorem recovers the Eskin-Margulis-Mozes theorem on the quantitative Oppenheim problem for quadratic forms of signature $(2,2)$; more generally, our results may be viewed as higher-degree analogues of that theorem.

We give a new proof of the quantitative Oppenheim theorem in signature $(2,2)$, originally proved by Eskin-Margulis-Mozes, by recasting the problem as one about determinant values on lattices in $\operatorname{M}_2(\mathbb R)$.

We consider repulsive Euler-Poisson equations in both one-dimensional space and in the multidimensional case with radial symmetry, assuming a power-law velocity-dependent damping coefficient. We show that under this assumption, in the one-dimensional case the set of initial data corresponding to a globally time-smooth solution expands, whereas in other dimensions (except for dimension 4) the damping does not influence on improving of smooth properties of the solution. Namely, any arbitrary small perturbation of the steady state blow up, despite of its amplitude of oscillations decays.

We prove full replica symmetry breaking for the zero-field Sherrington-Kirkpatrick model at zero temperature: the Parisi minimizer is absolutely continuous, has a smooth density, and has support $[0,1)$. At inverse temperature $\beta>1$, [arxiv.org/abs/2607.11756v3] recently proved that the Parisi measure has support $[0,q_\beta]$. Here, we show $q_\beta$ converges to $1$ as $\beta\to\infty$.