Papers

We prove that the canonical irregular Hodge filtrations in smooth families of Deligne–Mumford Landau–Ginzburg models with fixed pole orders assemble into filtered algebraic vector bundles. More precisely, twisted de Rham cohomology forms a vector bundle carrying a canonical algebraic integrable connection, every rational irregular Hodge level is a subbundle with locally free quotient, and every rational graded piece is a vector bundle. These constructions commute with arbitrary finite-type base change, and the connection satisfies shifted Griffiths transversality. Consequently, all filtered and graded dimensions are locally constant. As an application, for a smooth quasiprojective toric Deligne–Mumford stack and every fixed Newton polytope at infinity satisfying the support condition, these conclusions hold over the full coefficient locus of Laurent polynomials that are nondegenerate at infinity. In particular, their filtered and graded dimensions are independent of the coefficients. When the resulting stacky fans form a Clarke dual pair, the irregular Hodge numbers are computed by Harder–Lee's combinatorial $\Xi$-complex.

We give an approximate sampler for ferromagnetic Ising models with no field on arbitrary graphs that runs in time $\widetilde O(m+n)+\widetilde O_\beta(n^2\log^2 (1 / \varepsilon))$, where $n$ and $m$ are the numbers of vertices and edges, respectively, and $\varepsilon$ is the approximation error. Our approach combines Benczúr–Karger cut sparsification with a new mixing time analysis of the Glauber dynamics for the random-cluster model. The mixing time analysis features a new monotone edge-count Poincaré inequality.

In this article, we prove the sharp form of the fractional isoperimetric inequality in the plane, originally posed by Maz'ya [Problem 1, Integral Equations Operator Theory, 2018]. More precisely, we show that, for any given $s\in(0,1)$ and any bounded domain $\Omega \subset \mathbb{R}^2$ with $C^1$ boundary, $$ P_s(\Omega) \le \frac{\pi^{s-\frac{1}{2}}\Gamma\left(\frac{3-s}{2}\right)}{s(1-s)\Gamma\left(\frac{4-s}{2}\right)}\left[\mathcal{H}^1(\partial\Omega)\right]^{2-s}, $$ where $P_s$ denotes the fractional $s$-perimeter, $\Gamma$ denotes the Gamma function, and $\mathcal{H}^{1}$ denotes the $1$-dimensional Hausdorff measure on $\mathbb{R}^2$. The constant is sharp and equality is attained by the disk. The proof proceeds in three steps: reducing the fractional perimeter to a chord functional, reducing connected domains to convex bodies, and proving the sharp convex chord inequality via a fractional Willmore-type inequality, a variational formula along outer parallel bodies, and asymptotic analysis.

Let $N\in\mathbb N$ and $\gamma\in[-1,0)$. In [Anal. PDE 17 (2024)], Brezis, Seeger, Van~Schaftingen, and Yung asked how, in the exceptional range $\gamma\in[-1,0)$, $\dot{\mathrm{BV}}(\gamma)$ and $\dot W^{1,1}(\gamma)$ on ${\mathbb R}^N$ are related to other function spaces, especially to Hardy–Sobolev spaces, and whether these spaces are normable. In this article, we prove that the homogeneous Hardy–Sobolev space is strictly embedded, respectively, into $\dot{\mathrm{BV}}(\gamma)$ and $\dot W^{1,1}(\gamma)$, and neither $\dot{W}^{1,1}(\gamma)/\mathbb R$ nor $\dot{BV}(\gamma)/\mathbb R$ is normable, which answers the above two questions.

Brezis, Merle, and Riviére [Arch. Rational Mech. Anal. 1994] proved that, if a smooth solution $u:\mathbb{R}^2\to\mathbb{C}$ of the planar entire Ginzburg–Landau equation $$ -\Delta u = u(1-|u|^2)\quad \text{in}\quad \mathbb{R}^2 $$ satisfies the finite potential energy estimate that $$ \int_{\mathbb{R}^2}\left[1-|u(x)|^2\right]^2\,dx < \infty, $$ then it has the asymptotic property that $$ |u(x)|\to 1\quad \text{as}\quad |x|\to\infty. $$ The converse problem whether the asymptotic property implies the finite potential energy estimate was originally posed in their work and later formulated by Brezis as Open Problem 2.5 in [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 2023]. In this article, we give an affirmative answer to this question. Our proof relies on the Kelvin inversion, a Morrey-type energy decay for the translation Jacobi system, the $L^4$-integrability of the phase form, and a linearised amplitude equation; all these tools together imply the $L^2$-integrability of the function $1-|u|$ on an exterior domain, without any a priori integrability assumption.

Graphons are measurable functions used to describe the asymptotic behavior of convergent graph families. Originally motivated by problems in combinatorics and graph theory, graphons have found numerous applications in the modeling and analysis of dynamical processes on networks. In this work, we use graphons to formulate the Ising model on convergent graph sequences, which include many network topologies common in applications. We derive the mean-field limit for the resulting model and obtain exact results for phase transitions in such systems. Specifically, we show that the critical temperatures of the Ising model on graphons are determined by the eigenvalues of the Hilbert-Schmidt operator associated with the graph limit. For many important network topologies, these eigenvalues can be computed explicitly. We illustrate our results with three representative random network models: Erdős-Rényi, small-world, and power-law. In the small-world case, we demonstrate phase transitions to both ferromagnetic and antiferromagnetic phases, as well as coexistence of local minima of the free energy. The latter gives rise to multistability, as confirmed by Monte Carlo simulations. The results of this work demonstrate that the Ising model on graphons combines the analytical tractability of exactly solvable mean-field models with the ability to accommodate a broad range of network topologies. We expect that the use of graphons in spin models will lead to new insights into the statistical physics of interacting systems on complex networks.

For a real polynomial $F$ and $\omega\geq 0$, set $A_\omega(z)=(F(z+i\omega)+F(z-i\omega))/2$ and $\Omega_F=\{\omega\geq0:A_\omega\text{ has only real zeros}\}$. We present an explicit rational even polynomial of degree eight for which $6/25$ and $12/25$ belong to $\Omega_F$, while $3/10$ does not. Exact Sturm certificates give respectively eight, four, and eight distinct real zeros. Consequently $\Omega_F$ is neither an interval nor an up-set. All zeros of $F$ lie in the strip $|\operatorname{Im}z|\leq11/25$, and the classical strip-contraction theorem gives the eventual tail $[11/25,\infty)\subset\Omega_F$. We also include a direct elementary proof of that tail and a standard-library exact verifier.

In this paper we consider nonlinear biharmonic equations with $p$-Laplacian ($p\ge2$) of the form $$ \left\{ \begin{array}{l} \Delta^2 u - \Delta_p u + V (x) u = f (x, u) \text{,} u \in H^2 (\mathbb{R}^N) \text{,} \end{array} \right. $$ where the potential $V(x)$ may be indefinite. Using local linking and Morse theory, nontrivial solutions are obtained. In case the nonlinearity $f(x,\cdot)$ is odd, we obtain a sequence of large energy solutions. In the second part of the paper, for bounded positive potential, we get multiple solutions for the case that $$f(x,u)=\lambda g (x) | u |^{q - 2} u + | u |^{m - 2} u$$ with exponent $m$ critical or subcritical.

The identification and reconstruction of the boundaries separating basins of attraction in multistable, multidimensional dynamical systems presents a fundamental challenge in computational dynamics. These structures govern transition pathways and other important large timescale behavior, yet they remain typically under-sampled since their neighborhood does not get routinely visited during direct simulations. Traditional computational approaches face computational limitations in high-dimensional systems and require a priori knowledge of the dynamical system and its equations. Simplistic sampling methods such as random or uniform sampling of the phase space typically fail to quantitatively approximate separatrices and their structure altogether. We introduce and implement a framework that combines supervised classification with generative modeling to address this challenge. Our approach first trains neural network classifiers on uniformly or randomly sampled initial conditions labeled by their corresponding basins of attraction in the system of interest. Using uncertainty metrics of the trained classifier to quantify decision boundaries, the method then identifies these high uncertainty regions and boundaries of the classifier as preliminary approximate separatrices. Subsequently, score-based generative models are trained specifically on samples from high-uncertainty regions, ultimately generating densities of samples consistent with the empirical density of samples on or close to the manifold that constitutes the separatrix between basins in the sampled region. This approach leverages the complementary strengths of (a) discriminative models for global phase space partitioning and (b) generative models for detailed geometric sampling, resulting in a systematic, iterative, data-driven framework that produces empirically consistent reconstructions of (approximate) separatrix manifolds.

We develop a general framework for $B$-multiplier spaces; these are vector spaces $M = M(V,W)$ obtained from a bilinear operator $B \colon M \times V \longrightarrow W$, where $V$ and $W$ are Banach spaces. We focus on their normability and completeness, mainly in the setting of spaces consisting of functions with values in a Banach space $X$, particularly sequences. Our approach relies on the underlying Banach spaces satisfying the BK property, that is, having continuous evaluations. Classical multiplier spaces arise when $B$ is a pointwise product of scalar functions and we make the point for the case when $V$ or $W$ consists of vector functions. Special attention is given to the sequence spaces $\Sigma \ell_{\infty}(X)$ (bounded partial sums), $\Sigma c(X)$ (summable), and $\ell_u(X)$ (unconditionally summable). Given a BK scalar sequence space $V$, we introduce the multiplier space $M_{\Sigma}(V,X)$ and establish conditions under which it determines a closed subspace of bounded linear operators from $V$ into $X$. The notion of associate space is precised for BK-spaces, linking this construction with classical Köthe duality. We consider what we named strong vectorialization $Y(X)$ and weak vectorialization $Y_w(X)$ of a Banach sequence ideal $Y$. The weak vectorialization is obtained as a multiplier space and employed to describe classical sequence spaces as $\ell_{p,w}(X)$, $1 \leq p \leq \infty$. We introduce the $b$-ideal part of a space and show that the $b$-ideal part of $\Sigma c(X)$ is $\ell_u(X)$ and that of $\Sigma \ell_{\infty}(X)$ is $\ell_{1,w}(X)$. We also study the sequence space $bv(X)$ (bounded variation), proving that $M(\Sigma c(X),\Sigma c(X)) = bv(\mathbb K)$.

This note records a counterexample to Brenti's 1998 conjecture that the nonzero coefficients of $\widetilde{R}$-polynomials form a log-concave sequence.

We establish nondegeneracy and normality conditions for optimal control problems with a single nonregular mixed inequality constraint, where purely finitely additive set functions (charges) appear as multipliers. We first prove for a model problem nonregular at a single instant that degeneration is intrinsic and unavoidable: every admissible process satisfies the necessary conditions degenerately via a pure charge, and normalized multipliers of natural regular approximations admit no weak$^*$ limit in $L^\infty([0,1];\mathbb{R})^*$, yielding only pure charges under every generalized limit, in the sense of Banach limits. Motivated by this analysis, we propose four verifiable conditions that progressively guarantee nondegenerate multipliers, the vanishing of the pure charge, normality, and explicit bounds on charge mass. Furthermore, we show that at a nonregular terminal instant, a balance identity ties the charge mass to the cost multiplier, deciding between complete abnormality and normality. We provide examples confirming the realizability and discussing the limits of each condition.

We characterize all bounded complex symmetric weighted composition operators on the Fock space \(\mathcal F_{\alpha}^{2}\), without prescribing a conjugation a priori. Our approach uses reproducing kernels and two generating functions. The Taylor coefficients of these functions are eigenvectors of the operator and its adjoint, respectively. The conjugations arising from our construction are anti-linear Gaussian integral operators. Some of them are weighted composition conjugations, whereas others are not.

We present an efficient solver framework for the stiff magnetic wave coupling arising in resistive magnetohydrodynamics (MHD) on realistic tokamak geometries. The approach builds on an implicit-implicit (IMIM) time-splitting that separates fast magnetic waves and anisotropic heat transport from slower acoustic dynamics while retaining full coupling (Krzysik et al. 2026). Within this formulation, the magnetic wave subsystem appears as an anisotropic curl-curl operator, enabling the use of scalable auxiliary-space Maxwell (AMS) multigrid solvers. To exploit this structure at the discrete level, we employ curl-conforming finite element spaces for the magnetic field and design the velocity space to preserve the curl-curl structure induced by the Lorentz-force coupling. The resulting compatible discretization preserves the discrete magnetic divergence constraint while producing linear systems directly amenable to efficient AMS-based solvers. We demonstrate solver efficiency as well as the accuracy and stability of the resulting structure-preserving discretization on fully nonlinear three-dimensional tokamak test cases.

We introduce a class of nonautonomous nonlinear oscillator equations of mixed Liénard type that arises from a parametric deformation of the commutative factorization procedure applied to second-order ordinary differential equations with periodic solutions. Their solutions, in particular the isochronous waveforms, are obtained in closed form through a Riccati reduction scheme for the power-law choice of the factorization function, $\phi(x)=kx^{q}$, $k\in \mathbb{R}$, and $q\in\mathbb{N}$, corresponding to the so-called modified Emden oscillators, and do not depend on the arbitrary deformation parameter $A_1$. The variational structure of the equation is characterised by a Lagrangian supplemented with a generalised Rayleigh dissipation function that contains a non-standard cubic term in $\dot{x}$. We also show that multiplying the equation of motion by the Jacobi multiplier $M(x)=x^{-2A_1}$ a position-dependent-mass (PDM) form is obtained with related friction and restoring force.

We develop a filtering and optimal-control framework for partially observable stochastic systems in which each observation identifies a class of a finite measurable partition of the hidden state space. This structure covers regional-information mechanisms associated with threshold, quantized, censored, event-triggered, and intermittent observations, and allows observable classes with atomic, continuous, or mixed components. The formulation is constructed first at the level of measures: for each observable class, we define a class-restricted unnormalized conditional measure, and the posterior distribution is obtained by normalizing with its predictive probability. Based on this recursion, we introduce an information state consisting of the observed class and the conditional measure supported on it, thereby transforming the original problem into a fully observable Markov decision process. We formulate the discounted-cost criterion, derive the Bellman equation, and establish conditions for the existence of stationary optimal policies. To address the infinite-dimensional nature of the information space, we propose class-dependent finite-dimensional approximations capable of preserving both continuous components and atomic masses. We also derive an abstract bound linking the error of the approximate filter to the error of the value function. A reference model illustrates the construction through histogram-based approximations

We study minimax-optimal designs and estimators for estimating the sample average treatment effect in finite population randomized experiments, where both design and estimator are unrestricted. For binary potential outcomes, we show this minimax risk is equivalent to the minimax risk $\rho_n^*$ of an estimation problem with $2$ unknown parameters. We leverage this reduction to establish a second-order risk expansion $\rho_n^* = n^{-1} - Cn^{-4/3} + o_n(n^{-4/3})$ for an explicit constant $C$ related to the Airy function. The minimax risk is attained by Bernoulli randomization with a nonlinear shrinkage estimator. Our results show that standard procedures such as complete randomization with difference in means are only minimax optimal up to first order in $n.$ We derive further results on admissibility of these procedures and discuss the practical implications of our results.

With the help of a $_6\phi_5$ summation formula and Guo and Zudilin's method, we shall establish some Dwork-type $q$-supercongruences in this paper. When $q\to1$, these $q$-supercongruences are able to engender the corresponding supercongruences. One of them may be stated as follows: for any prime $p\geq5$ and any positive integer $s$, \begin{align*} &\sum_{k=0}^{p^s-1}(6k-1)\frac{(-\frac{1}{3})_k^3}{(1)_k^3} \equiv 0\pmod{p^{3s}}. \end{align*}

In this paper, we consider transmissions with superimposed (SI) demodulation-reference-symbol (DMRS) and data in orthogonal frequency-division multiplexing (OFDM) based multiple-input multiple-output (MIMO) systems. First, we derive an analytical framework to characterize the iterative behavior between the mean-square errors (MSEs) of channel estimation (CE) and MIMO detection (MD) within an iterative CE and detection (ICED) process. This framework is subsequently utilized to optimize power allocation and pilot patterns between the DMRS and data symbols for SI-DMRS transmission. Second, we design an artificial intelligence (AI) based receiver built upon Transformer encoders for SI-DMRS transmissions, which incorporates an iterative CE and detection (ICED) structure. Simulation results demonstrate that the proposed AI-ICED receiver, combined with SI-DMRS, effectively increases spectral efficiency (SE) compared to conventional systems using non-overlapped DMRS and data symbols.

A Lie algebra is said to be $1.5$-generated if every nonzero element can be completed to a two-element generating set. We prove that the Lie algebra of polynomial vector fields with constant divergence on the affine space is $1.5$-generated. We also show that the Lie algebra of all polynomial vector fields on the complex affine space is generated by two completely integrable elements.

The diagrams of links (including knots) are characterized in terms of circular rigid disk-chord diagrams of their spun ribbon torus-links in the 4-sphere. As a result, the crossing numbers of links are equal to the chord indexes of their spun ribbon torus-links. By using this result, additivity on the crossing numbers of links under connected sums can be shown.

The Johnson–Lindenstrauss lemma asserts that every set of $n$ points in $d$-dimensional Euclidean space embeds into $O(\varepsilon^{-2}\log n)$-dimensional Euclidean space with distortion at most $1+\varepsilon$. Larsen and Nelson conjectured that the optimal target dimension throughout the full range of the parameters $n,d, \varepsilon$ is \[ \Theta\left(\min\left\{d,n-1,\frac{\log(2+\varepsilon^2n)}{\varepsilon^2}\right\}\right). \] We resolve this conjecture in the affirmative. In fact, we prove the stronger statement that the upper bound is attained by a linear map. The matching lower bound, due to Larsen–Nelson and Alon–Klartag, holds even for nonlinear embeddings.

Let $U$ be a smooth complex affine log Calabi–Yau surface, and let $\widehat{U}$ be its complete finite-type Liouville manifold, equipped with its natural grading structure. We give an algorithm that constructs a finite-type quasi-projective $\mathbb{Z}$-scheme $U^\vee$ such that \[ D^\pi\bigl(\mathcal{W}(\widehat{U},\Bbbk)\bigr) \cong D^b\operatorname{Coh}(U^\vee_{\Bbbk}) \] for every field $\Bbbk$. Our main contribution is to construct an almost toric model encoded by an exact eigenray diagram and, using the symplectic Torelli theorem for symplectic log Calabi–Yau pairs, to prove that this model is grading-preserving strongly exact symplectomorphic to $\widehat{U}$. The result then follows from the homological mirror symmetry theorem of Hacking–Keating.

We study a kinetic–fluid system in which a Vlasov or Vlasov–Fokker–Planck equation is coupled to the compressible Navier–Stokes equations with density-dependent viscosities through a locally averaged Brinkman force. The averaging is chosen in a conservative form so that the coupled system preserves the total momentum and satisfies a natural energy-dissipation balance, while avoiding the pointwise evaluation of a possibly rough fluid velocity. The first main result is a conditional exponential relaxation estimate for sufficiently regular solutions. The estimate is proved under positive and negative density moment bounds and a Muckenhoupt $\calA_2$ condition on a power of the density, which replace uniform pointwise upper and lower bounds. The proof combines a modulated energy and hypocoercivity analysis with a compensating functional for the density fluctuation. A key ingredient is a weighted Calderón–Zygmund estimate for the associated elliptic corrector, which allows us to control the viscous contribution under the $\calA_2$ condition. The second main result concerns global finite-energy Bresch–Desjardins entropy weak solutions in admissible low-dimensional regimes. Using the additional entropy structure associated with the density-dependent viscosities, we verify the density assumptions required by the conditional relaxation theorem and obtain exponential relaxation for the weak solutions. In the diffusionless case, the particle distribution aligns toward a mono-kinetic state, while in the diffusive case the relaxation is towards a Maxwellian equilibrium.

Let $\mathbb{K}$ denote an algebraically closed field of arbitrary characteristic. In this paper we provide bounds for the size of a prime $\ell$ for which there are curves defined over $\mathbb{K}$ admitting automorphism groups of order $\ell^2$. In addition, we also give a classification of the families of curves attaining the upper bounds for $\ell$ in both tame and wild case. Finally, we present the full automorphism groups of the curves attaining the highest bounds.

A feedback loop that returns a message into the process it describes defines a quantum channel on the message. We ask when such a loop preserves entanglement with a reference, that is, when its channel is not entanglement breaking. For any qubit loop, the negativity of the Choi state is at most half the fidelity between the states that the two values of the message leave in the discarded registers, in any basis. This channel analogue of the duality between fringe visibility and which-way information is attained by pure dephasing, and a perfect record of the message in the discarded registers makes the loop entanglement breaking. We then solve a three-qubit loop in which the message steers a future event, a probe reads the event, and a partial swap feeds the probe back. Every partial feedback is a strict contraction, and the loop is entanglement breaking exactly when an explicit trigonometric polynomial in its coupling angles is not positive. A partial record of the message in the future event can restore entanglement that the feedback alone destroys. Stability, a coherent fixed point, information about the future event and entanglement preservation coexist on about three quarters of the parameter space.

This paper will give a definition of belief revision within simplicial semantics. Starting from the work presented in "A Semantics for Belief in Simplicial Complexes" as a baseline, this paper modifies the semantics for belief given there to allow atomic formulae to be assigned to nodes, not facets. Conceptually and philosophically, this version of the semantics is better suited if one wishes to interpret the nodes of a simplicial model of epistemic logic as a "perspective" assigned to a particular agent. If nodes are perspectives, it then follows that worlds, or the facets of a simplicial model, are composed themselves of perspectives. This allows us to say that two worlds are more similar, or "nearer", if they share more perspectives. With this conceptual notion of nearness in hand, two different formal presentations of revision are given. We conclude by exploring some conceptual pitfalls surrounding these definitions under iterated revision, and motivate a few potential solutions that involve giving the agents a "memory" of what has been announced so far.

In the era of modern data science, the rapid proliferation of high-dimensional and functional datasets has fostered increasing interest in the investigation of paradigm shifts and structural breaks. Unlike classical univariate time series, structural changes in functional data need not occur simultaneously across the entire domain; instead, they may emerge locally, producing heterogeneous distortions across the underlying functional structure. The patterns of instability often exhibit sparsity, where it is not known a priori which specific parameters are undergoing a transition. However, in functional contexts, these shifts are often "localised". The difficulty lies in the high dimensionality of the parameter space, where the signal-to-noise ratio may be low for individual components, necessitating the aggregation of information across dimensions to detect a global change. This paper addresses the problem of detecting structural shifts in a functional time series from a Bayesian perspective. We have developed various novel methodologies that capture the inherent structural distortion in a sequence of random functions, both individually and simultaneously. The formulation of the problem is based on the state-space representation of a functional time series. Efficient Blocked Gibbs Sampling algorithms have been proposed to identify these locations accurately. Further, we demonstrate the effectiveness of our methods on several financial and temperature datasets.

We establish a sharp rigidity/flexibility threshold for stationary solutions of the Krieger–Strain equation, an isotropic model of the Landau–Coulomb equation. For every $1< p < \frac65$, we use Nash iteration to construct nontrivial, nonnegative solutions in $L^1(\mathbb{R}^3) \cap L^p(\mathbb{R}^3)$ with arbitrarily strong exponential localization. Conversely, every stationary weak solution in $L^{\frac65}(\mathbb{R}^3)$ is trivial, identifying $L^{\frac65}(\mathbb{R}^3)$ as a new sharp integrability threshold. To our knowledge, this is the first use of Nash iteration for a nonlinear equation from collisional kinetic theory. The construction is based on a high–high–low cancellation within the Krieger–Strain operator and suggests that such mechanisms may occur more broadly in kinetic theory. The construction must accommodate kinetic features unusual for the method including a strongly nonlocal collision operator; an equation fundamentally posed on the whole space—not the periodic box; and a positive scalar unknown. At this low level of regularity, the usual formulations of the collision operator are not a priori well-defined, so a central part of the problem is specifying in what sense the constructed objects solve the equation. We isolate the notion of mollifier confluence, a simple and canonical way to interpret a nonlinearity below naive thresholds related to multiplying distributions. We complement this definition with a systematic treatment of weak solution notions and several explicit formal computations and clarifying examples that may be of independent interest.

This work studies a class of zero-sum stochastic linear quadratic dynamic games (LQDGs) under partial and asymmetric information. Information asymmetry introduces fundamental challenges related to belief representation and theory of mind, where players must impute belief states and estimates of other players to inform their strategies. Existing work highlights the difficulty of applying dynamic programming-like decomposition approach to these problems. An alternative approach based on best response dynamics is proposed, which provides insights into belief representation and theory of mind challenges. Explicit expressions for each player's best response within the class of pure linear dynamic output feedback control strategies are derived, where the internal state dimension of each control is an integer multiple of the system state dimension. As players iteratively update their best responses, they form increasingly higher-order belief states, leading to infinite-dimensional internal states. However, numerical results reveal that the game's value converges after only a few iterations, suggesting that higher-order belief states provide vanishing benefit. This work further conducts numerical experiments to analyze the impact of asymmetric beliefs, belief orders, relative controllability and observability, and direct feed-through on a linear quadratic pursuit-evasion game's value.