Papers

We introduce a generalization of generalized Weyl Poisson algebras. This is a Poisson analogue of the twisted generalized Weyl algebras defined by Mazorchuk and Turowska. We prove existence of these algebras in two ways, using Ore extensions and by using skew Laurent Poisson algebras. It is shown that this structure is preserved under tensor products, Poisson twists, and by taking invariant rings. Finally, we prove a simplicity criterion for these Poisson algebras.

It has been a long standing conjecture that the $ \infty $-harmonic functions in the plane have a 1/3-Hölder continuous gradient. It is known that solutions are $ C^1 $ and that the gradient is locally $ \alpha $-Hölder, but $ \alpha $ comes without any positive lower bound. Aronsson's solution $ x^{4/3} - y^{4/3} $ shows that no better general regularity is possible. In the plane there is also a connection between the $ \infty $-Laplace equation and the one-dimensional heat equation, observed already by Aronsson himself. I shall show that this link can be accessed under a certain injectivity condition on the gradient, and that the caloric structure then is enough to prove the 1/3-Hölder continuity. Of course, an injective gradient is by no means a necessary condition, as seen by smooth solutions such as the planes and cones.

For control-affine systems, standard and high-order control barrier function conditions are affine in the control input and are commonly enforced through quadratic-program-based safety filters. Although convex, these optimization problems may be undesirable in embedded, high-rate, or resource-limited implementations. This letter characterizes when the corresponding Euclidean projection can be recovered from the affine inequalities violated by a nominal control input. Given a nominal input, we form the violated set and compute the minimum-norm correction that enforces the violated inequalities with equality. This violated-set correction is closed form, but it need not equal the exact Euclidean projection onto the full feasible set. The main result gives a necessary and sufficient exactness certificate based on primal and dual feasibility, followed by structural sufficient conditions involving interactions among affine-inequality normals. An online certification algorithm is then presented to determine when the closed-form update is exact. When the certificate fails, a finite active-set search can be used to recover the exact projection. Numerical simulations illustrate that the violated-set correction can remain feasible while failing to be the exact projection due to dual infeasibility, and demonstrate computational speedup relative to a standard CBF-QP solver.

We introduce a notion of linkage for sheaves of modules on connected Noetherian schemes, extending classical linkage of modules. Linkage is defined for stable sheaves admitting finite free resolutions via the transpose and syzygy functors. We show that linkedness is a local property and that, on affine schemes, a coherent sheaf is linked if and only if its module of global sections is linked. We further show that linkage is preserved under restriction and, under suitable rank conditions, under gluing over connected schemes. We also obtain criteria for the existence of linked subsheaves when the structure sheaf is not a domain. In the projective setting, we compare invariants of linked sheaves, including graded cohomology modules, Castelnuovo-Mumford regularity, and Hilbert polynomials.

A well-known model in game theory, the Volunteer's Dilemma describes a group of $n$ players who decide whether to volunteer for a collective benefit at a personal cost, or to abstain and risk forfeiting the benefit altogether. A quantum version of this dilemma, developed within the Eisert-Wilkens-Lewenstein framework, allows each player to manipulate one qubit of a shared entangled state, leading to symmetric Nash equilibria with higher expected payoffs than in the classical game. Existing analyses, however, assume maximal entanglement, which is experimentally demanding for hardware-constrained quantum systems. Within the same framework, we introduce a generalized Quantum Volunteer's Dilemma with a tunable entanglement parameter $\gamma$ and study the extent to which equilibrium behavior depends on the level of entanglement. We derive explicit conditions relating $\gamma$, the number of players, and the players' strategies under which symmetric Nash equilibria exist, focusing on two canonical strategy profiles: one for $2\leq n\leq 9$, and one for even $n$. We find that maximal entanglement is not required to sustain symmetric equilibria. Instead, equilibrium behavior persists above a threshold value, which we compute analytically in both cases. We also demonstrate that the threshold value directly depends on system size.

The topology of Kähler manifolds is largely determined by the geometry due to its rigidity. In particular, the cohomology and Hodge theory of compact Kähler manifolds is quite restricted. We prove that within a certain threshold, the odd Betti numbers of any compact almost-hermitian manifold satisfying a degenerate Kähler condition are even, and the even Betti numbers are strictly positive. We call this new type of degenerated Kähler manifold acK. Our approach to proving these results makes use of a compactness theorem for acK manifolds, and a new version of Hodge theory for compact manifolds endowed with a Sobolev regular Kähler strucutre. In addition, we lay out a program to pursue the study of acK geometry that accommodates not only the classical viewpoint, but also constructive and finitary proofs, as well as formalization with proof assistants.

We consider a family of pure Neumann $p$-Laplacian problems, including eigenvalue problems, Lane-Emden type equations, and extremal cases such as sign nonlinearities and the $1$-Laplacian. Using variational methods, we develop a unified framework that establishes existence of solutions and characterizes their asymptotic behavior as the parameters vary. This approach reveals a natural asymptotic connection between pure Neumann $p$-Laplacian equations and a relative isoperimetric problem known as the Neumann-Cheeger problem. We describe the shape of minimizers in domains with different geometries and obtain results on regularity, uniqueness, multiplicity, symmetry, and symmetry breaking phenomena.

In this paper, we establish the equivalence between minimal time and minimal norm control problems for controllable stochastic differential equations (SDEs). The core of this equivalence lies in proving the continuity of the minimal norm function. Unlike the deterministic setting, where the time translation technique is effective, this approach is not applicable due to the adapted nature of the admissible controls in SDEs. To address this challenge, we construct the minimal norm control explicitly, thereby achieving the desired continuity. This constructive approach not only proves theoretical equivalence but also facilitates the design of efficient numerical schemes for minimal norm and time controls.

Compositional priors describe the generic properties of layered functions in deep Bayesian models, where deep neural networks with random weights are a canonical example. In the wide-network limit, the prior is a Gaussian process (GP) with a depth-dependent kernel, and its behaviour as depth grows has been extensively studied through this kernel. Here, we study another case, where each layer itself is a vector valued Gaussian process, and our aim is similarly to understand the limiting behaviour of the prior as depth grows. Previous GP work has established that for the radial basis function (RBF) kernel and a certain range of bandwidths $r$, the prior degenerates in the limit, converging to the set of constant functions, which is not useful as a probabilistic model. In this paper we establish several new results. First, we identify a sharp bandwidth threshold $r_c(d) = \Theta(\sqrt{d})$ above which the limit is degenerate, strengthening the earlier bounds. Second, and more importantly, we show that for $r$ below the threshold $r_c(d)$ the prior converges to a limit distribution $\pi_{\bar{Z}}$. We also prove that these distributions are non-degenerate and non-Gaussian, with non-vanishing dependence between coordinates. In contrast to the previously known degenerate regime, deep Gaussian process priors can therefore admit non-trivial limits. Empirically, we verify the threshold across a range of dimensions $d$, and demonstrate a complex multimodal behaviour of the limit distributions $\pi_{\bar{Z}}$, a regime that becomes increasingly narrow with $d$ and would be hard to identify without knowing the threshold.

For $0\leq k\leq n$, let $u(n,k)$ be the largest divisor of $\binom nk$ whose prime factors are at most $k$. Erdős Problem #684 concerns the special threshold $u(n,k)>n^2$ and asks how early this small-prime part can be forced to become large. We prove the density-one analogue for every fixed power threshold. If $f_c(n)$ is the least $k$ for which $u(n,k)>n^c$, then, for each fixed $c>0$, \[ f_c(n)=\left(\frac{c}{1-\gamma}+o(1)\right)\log n \] for almost all positive integers $n$. In particular, \[ f_2(n)=\left(\frac{2}{1-\gamma}+o(1)\right)\log n =(4.730544237\ldots+o(1))\log n \] for the Erdős #684 threshold. This is a normal-order theorem, not a pointwise resolution of the corresponding worst-case problem. The constant $1-\gamma$ is arithmetic. Kummer's theorem rewrites $\log u(n,k)$ as a sum of carry indicators, and complete-residue averaging gives \[ m(k)=k\sum_{p\leq k}\frac{\log p}{p-1}-\log k!=(1-\gamma)k+o(k). \] The cancellation in this formula moves the typical crossing from the naive scale $c\log n$ to $c(1-\gamma)^{-1}\log n$. We prove the required concentration uniformly for every $k\leq A\log X$ on one dyadic interval, after discarding a zero-density exceptional set caused by large powers of small primes dividing one of the nearby integers $n,n-1,\ldots$. We also prove Gaussian fluctuations in the logarithmic range. If $k=k(X)\to\infty$, $k\leq A\log X$, and $n$ is uniform in $[X,2X)\cap\mathbb Z$, then \[ \frac{\log u(n,k)-m(k)}{\sqrt{V(k)}}\Rightarrow \mathcal N(0,1), \qquad V(k)\sim (2-\log(2\pi))k\log k. \] Higher prime powers are needed for the mean, but after centering their aggregate is $L^2$-negligible on the Gaussian scale; the variance comes only from the prime levels.

In this paper the Clifford groups of general multipartite quantum systems with configuration space $\mathbb{Z}_{n_1}\oplus\cdots\oplus\mathbb{Z}_{n_k}$, $k\geq 1$ are studied. It is known that the Clifford group is a natural semidirect product provided the dimension $N=n_1\cdots n_k$ of the corresponding Hilbert space is an odd number. For even $N$ special results on the Clifford groups are scattered in the mathematical literature, but they do not concern the semidirect product structure. Using the relation of generators of the associated symplectic group $\mathrm{Sp}_{[n_1,\dots,n_k]}$ we prove that for even $N=n_1\cdots n_k$ both Clifford group and the projective Clifford group are natural semidirect products if and only if $N$ is not divisible by four.

We develop a tractable, rigorous approach to risk-aware control for a class of nonlinear systems. While many classical control methods reduce uncertainty to a simple average or a worst-case outcome, risk-aware control aims to equip systems with a refined awareness of uncertainty. Efficient methods for risk-aware control of linear systems are available, but there is a paucity of tools for tractable, risk-aware control of nonlinear systems. To bridge this gap, we develop an analytical, suboptimal controller with respect to a risk-aware performance criterion for systems with nonlinearities characterized by cone-like bounds. Numerical examples demonstrate benefits of the characterization of nonlinearities and risk that we consider.

Filtering-based probabilistic numerical solvers for ordinary differential equations (ODEs) have been established as a flexible and efficient simulation framework with built-in numerical uncertainty quantification. However, problems that are both stiff and high-dimensional remain a challenge, as current methods are either stable and have cubic cost in the ODE dimension, or scale linearly at the expense of stability. In this paper, we close this gap and develop probabilistic ODE solvers that are both stable and scalable. We propose two complementary strategies. First, we develop a matrix-free update step that uses Jacobian-vector products, iterative linear solvers, and stochastic covariance estimation to enable linear scaling, all while retaining stability. Second, we propose iterative re-linearization to further improve stability without sacrificing scalability, turning probabilistic ODE solvers into fully implicit methods. We evaluate the proposed approaches on a range of stiff and high-dimensional problems and demonstrate improved stability and scalability over established probabilistic solvers.

We prove that for any Lie ring $\mathfrak{R}$ and any commutator-preserving bijection $\alpha : \mathfrak{gl}_n(K) \rightarrow \mathfrak{R}$, the map $\alpha$ is additive on $\mathfrak{sl}_n(K)$, where $n \ge 2$ and $K$ is an arbitrary field. Using this result we find criteria for Lie rings $\mathfrak{gl}_n(K)$ and $\mathfrak{sl}_n(K)$ to be unique addition. We also show that any commutator-preserving injection of Lie rings $\beta : \mathfrak{sl}_2(K) \rightarrow \mathfrak{S}$ is additive. This is the first result on additivity of commutator-preserving injections.

We study Birkhoff genericity for a simple uniformly expanding diagonal flow on $\mathrm{SL}_{n+1}(\mathbb R)/\mathrm{SL}_{n+1}(\mathbb Z)$, with initial points restricted to affine subspaces of the expanding horospherical orbit through the identity coset. We prove that almost every point on such an affine subspace is Birkhoff generic, except possibly in two situations: either the defining matrix of the affine subspace has Diophantine exponent at least $n$, or the affine subspace is arbitrarily well approximable by affine subspaces of dimension $(r-1)$ defined over a real number field of degree $m\ge 2$, with $n+1=mr$. As applications, we obtain Dirichlet non-improvability and logarithmic density results for almost every point on these affine subspaces.

We study five classical $t$-norms on the unit interval from the viewpoint of ultrafilter concentration. For a fixed ultrafilter $\mathcal U$ on $[0,1]$, we introduce an equivalence relation identifying two operations whenever they coincide on $A\times A$ for some $A\in\mathcal U$. We show that their asymptotic behavior is governed by two concentration regimes. In the near-$1$ regime, the five operations determine four distinct ultrafilter-equivalence classes. In the low-value regime, the Łukasiewicz, nilpotent minimum, and drastic $t$-norms collapse to the zero operation. We encode these reductions in a discrete quotient category and record simple ultrametric models for the two regimes. We further interpret the classification inside classical ultrapowers: the near-$1$ and near-$0$ regimes become exact algebraic phenomena on infinitesimal monads, and saturation yields a compactness principle for countable systems of asymptotic identities. Finally, we indicate how the same viewpoint interacts with residual fuzzy implications generated by $t$-norms.

This paper establishes for the first time the iterative and rigid theory of weak$^{*}$ slices within a non-metric framework, demonstrating that dual convex sets under the pure weak$^{*}$ topology can achieve localization, diameter control, and fine structural analysis. It fundamentally transforms the traditional understanding of the geometric properties of weak$^{*}$ topology and thereby pioneers a new direction in non-metric weak$^{*}$ slice geometry. By developing a new technique involving intricate manipulations of weak$^{*}$ slices and a carefully designed iterative selection process, we prove that if $M$ is a closed subspace of a G$\mathrm{\hat{a}}$teaux differentiability space $X$, then $M$ is a G$\mathrm{\hat{a}}$teaux differentiability space. As a Corollary, we get that if $X$ is a weak Asplund space and $M$ is a closed subspace of $X$, then $X$ is a G$\mathrm{\hat{a}}$teaux differentiability space. Thus, we definitively solve an open problem raised 46 years ago by D.G. Larman and R.R. Phelps (J. London Math. Soc., 20(1979), 115–127).

Matrix completion has been extensively studied for real-valued data, but existing methods are often limited in handling categorical variables. We propose LCMC, a double-loop optimization framework for categorical matrix completion via latent factorization based on a binary tensor representation. In this setting, each categorical entry is encoded as a one-hot vector along a third tensor mode, thereby preserving its discrete, non-ordinal nature. The outer loop adaptively estimates the latent dimension by iteratively updating it with feedback from the inner loop, while the inner loop reconstructs the categorical matrix through tensor factorization, supported by a corresponding theoretical analysis. To further improve scalability and robustness, we introduce enhancements including a split-merge-refine strategy and an adaptive data reduction technique. Experiments on synthetic and real-world datasets in viral quasispecies reconstruction, demonstrate that LCMC achieves superior accuracy and efficiency compared to existing methods.

We prove an isoperimetric inequality for compact bodies bounded by surfaces embedded into a connected, contact, non-unimodular 3-dimensional Lie group.

In this article, we show that any morphism $f$ from a smooth projective variety $X$ to a simple abelian variety $A$ is smooth, if and only if there exists a holomorphic 1-form $\omega$ on $A$ such that $f^*\omega$ has no zero. As the key ingredient in the proof, we show any $\mathbb{Z}$-homology fibre bundle morphism is without blow-up in codimension 0 in the sense of Sabbah. Furthermore, we investigate the structure of the spaces of holomorphic 1-forms with zeros, and show that they are linear for large classes of varieties. Also, we construct a delicate example of a smooth projective subvariety of an abelian variety for which the holomorphic 1-forms with positive dimensional zero loci do not form a linear subset. Finally, we study algebraic surfaces admitting holomorphic 1-forms that have zeros and do not arise from cohomology jump loci.

We establish a Remez inequality for functions in quasianalytic Denjoy-Carleman classes $\mathcal C_M$ on a large family of fat compact sets $K \subseteq \mathbb R^n$ with tame geometry, including all fat compact sets definable in the o-minimal expansion of $\mathbb R$ by restricted $\mathcal C_M$ functions. The inequality generalizes the classical Remez inequality for polynomials and its quasianalytic versions on convex bodies, replacing the polynomial degree by the Bang degree, an integer associated with the weight $M$ and the size of the function. The constants depend explicitly on the Bang degree and the geometry of $K$. We derive a range of quantitative consequences: estimates for the volume of sublevel sets, comparison of $L^p$-norms, effective inequalities of Lojasiewicz, Harnack, and Markov type with explicit constants, and decay estimates for oscillatory integrals.

Swap algebras generalise Bishop's complemented powerset as Boolean algebras generalise the powerset. Actually, all Boolean algebras are swap algebras. We prove constructively a Stone representation theorem for separated swap algebras of type (II), where the notion of a separated swap algebra generalises the corresponding notion of a separated Boolean algebra. Moreover, we prove a Stone-Cech theorem for swap algebras of type (II), showing that the restriction to separated swap algebras is not a loss of generality from the point of view of the theory of swap characters. A constructive Stone representation theorem and a Stone-Cech theorem for Boolean algebras follow as special cases. We introduce sets with a Boolean inequality, that is sets with an internal falsum. These sets allow a book-keeping of the use of the Ex falso principle in constructive mathematics. If we restrict to swap algebras with a Boolean inequality, then the proof of the Stone representation theorem for swap algebras of type (II) is within minimal logic.

Low-rank approximation based on selected rows and columns is a useful alternative to singular value decompositions when the goal is an interpretable and compact matrix representation. A standard way to choose these rows and columns is the maximum-volume principle: it selects submatrices with large volume, which usually leads to stable interpolation coefficients and accurate CUR-type approximations. In this paper, we study this idea for quaternion matrices. This setting is natural for color images, three-dimensional motion data, and multi-channel signals, but requires care because quaternion multiplication is noncommutative. We define quaternion maximum-volume submatrix selection using quaternion singular values and the Study determinant. We then derive quaternion rank-one update formulas and use them to build two selection procedures: a greedy square-core method for row and column replacement, and a rectangular method that enlarges a selected row set until the interpolation coefficients are controlled. We prove that successful row and column swaps increase the quaternion volume of the selected square core when the exact quaternion inverse is used. We also connect the stopping criterion with quasi-dominance, prove an exact quaternion CUR identity in the full-rank case, and derive an interpolation stability bound. For the rectangular case, we derive an append-row pseudoinverse update and show how it gives a natural right preconditioner for overdetermined quaternion least-squares problems. Finally, we illustrate the methods on three applications: quaternion CUR approximation of RGB images, RectMaxVol-based preconditioning for ill-conditioned quaternion least-squares systems, and row selection in quaternion motion-capture data. The experiments show that the proposed quaternion MaxVol and RectMaxVol methods provide stable and efficient selection routines.

Array superdirectivity is traditionally derived through singular optimization of densely spaced antenna arrays. In this paper, we show that the phenomenon admits a geometric interpretation as a concentration effect induced by spectral collision. As the spacing of an $M$-element linear array tends to zero, the exponential family generated by a linear array undergoes a spectral collision, and the associated finite-dimensional subspaces converge in reproducing kernel to a polynomial jet space. The maximum achievable array gain equals the diagonal evaluation of the reproducing kernel, and is therefore governed by the reciprocal Christoffel function. For the classical flat $L^2([-1,1])$ geometry, the Christoffel–Darboux kernel exhibits boundary concentration, yielding the quadratic $M^2$ superdirective law as a direct consequence of kernel asymptotics. This viewpoint separates intrinsic gain limits from numerical conditioning and identifies superdirectivity as a manifestation of a more general concentration mechanism. The framework further shows that the classical $M^2$ scaling is not universal: alternative spectral geometries produce different concentration laws through their associated Christoffel asymptotics. The results establish a direct connection between superdirectivity, reproducing kernels, orthogonal polynomials, and concentration phenomena arising from singular spectral limits.

This expository paper is a companion to \cite{Rav:gjmcyc}, in which we discuss cyclotomic spectra. Both papers are intended to shed light on the recent resolution of the telescope conjecture by Robert Burklund, Jeremy Hahn, Ishan Levy and Tomer Schlank (hereafter referred to as BHLS) in \cite{BHLS}. Their proof involves both cyclotomic spectra, the subject of \cite{Rav:gjmcyc}, and cyclotomic extensions of spectra, the subject of this paper. Higher cyclotomic extensions of commutative ring spectra are analogous to Galois extensions of $p$-adic number fields (or rings of integers thereof) obtained by adjoining roots of unity.

This paper investigates two problems unified by the study of the uniform boundedness of the dilation operators (UBD) T_r f(z)=f(rz), 0<r<1, acting on weighted Bergman spaces A^p_omega with not necessarily radial weights. We first characterize the random symbol space for A^p_omega under a mild admissible condition (Theorem 1.2). This extends the main result of [7] from radial weights to non-radial weights. We then introduce two new notions, namely non-radial mixed norm spaces M(p,q;omega) and analytic tent spaces A(p,q;omega), and we characterize their corresponding symbol spaces as well (Theorem 1.8, Theorem 1.12). The novelty here is to employ a measure-disintegration framework to prove a general Littlewood-type theorem (M(p,q;omega))* = H(2,q;omega_r), from which the Bergman space result (A^p_omega)* = H(2,p;omega_r) follows as the special case p=q. Among other things, UBD plays a pivotal role in the proofs of the preceding theorems. The second main problem addressed in this paper is to establish a local-to-global criterion for UBD, which remains largely unexplored for non-radial weights. Our principle result in this part (Theorem 1.16) asserts that UBD is guaranteed by two local geometric conditions: bounded hyperbolic oscillation (BHO) and a reverse-Carleson tail condition (RC). This is the technical heart of the paper. In the course of our investigation, three new types of problems arise naturally, each of independent interest: a two-weight top-maximal operator (Theorem 1.17); a truncated maximal operator over hyperbolic balls (Theorem 1.18); and a single-testing Carleson embedding problem (Theorem 1.19).

The 1970 spectral extension of Mantel's theorem, proved by Nosal, states that every graph with $m$ edges and spectral radius $\rho_1>\sqrt{m}$ contains at least one triangle. Its quantitative refinement by Ning and Zhai later established that any graph $G$ with $m$ edges and spectral radius $\rho_1\geq\sqrt{m}$ contains at least $\lfloor\frac{\sqrt{m}-1}{2}\rfloor$ triangles, unless $G$ is a complete bipartite graph. In this paper, we further investigate the minimum number of triangles guaranteed under the strengthened spectral condition $\rho_1\geq\sqrt{m}+c$, where $c$ is a positive constant. We prove that for any constant $c\in (0,\frac{1}{2}]$ and all sufficiently large $m$, if $s=s(m)$ is a real-valued function satisfying $\lim_{m\to\infty} \frac{s}{m}=c$, then every $m$-edge graph $G$ with spectral radius $\rho_1$ satisfying $\rho_1^2\geq m-1+\frac{2s}{\rho_1-1}$ contains at least $s$ triangles. Moreover, we characterize the extremal graph achieving the minimal number of triangles. In particular, when $s=\frac{m-1}2$, our result settles a conjecture proposed by Li, Feng, and Peng.

As capacitance extraction accuracy of rule-based pattern matching becomes difficult to sustain at advanced nodes, a growing trend emerges to develop deep-learning-based 2D capacitance models. However, existing MLP- and CNN-based methods constrain their input to fixed metal-layer combinations in a specific process node, limiting their usability in practice. Recognizing the inherent similarity between capacitance matrix and the prevailing attention mechanism, we propose AttentionCap, a customized Transformer for capacitance matrix learning, with a Gram representation framework, a physics-aligned symmetric-attention output layer, and a novel normalized Laplacian loss. We also introduce a process-node embedding to enable multi-node learning. Trained on synthetic data, AttentionCap attains 0.67%/3.99% self/coupling-capacitance error on unseen real designs under a multi-layer and multi-node setting, surpassing the CNN-Cap baseline with 4.6$\times$/5.7$\times$ lower self/coupling error and 192$\times$ faster inference speed. A pretrained AttentionCap accurately transfers to an unseen node with only 5K samples and 4K finetuning steps. With sufficient accuracy on unseen real designs and strong transferability to new process nodes, AttentionCap offers highly practical value for modern EDA workflows. Code and data are available at https://github.com/THU-numbda/AttentionCap.

In this paper, we introduce and investigate the $\textbf{k}$-neighborhood ideal of a graph, a natural generalization of the closed neighborhood ideal. Let $G$ be a simple graph on the vertex set $[n]$, and let $S=K[x_1,\dots,x_n]$ be the polynomial ring over a field $K$. For a vector $\textbf{k}=(k_1,\ldots,k_n)\in \mathbb{N}^n$ satisfying $1\leq k_i\leq \textrm{deg}_G(i)+1$ for all $i$, the $\textbf{k}$-neighborhood ideal of $G$ is defined as the squarefree monomial ideal $$\textrm{NI}_{\textbf{k}}(G)=\sum_{i=1}^n\, (\textbf{x}_W:\, W\subseteq N_G[i],\, |W|=k_i)$$ of $S$, where $\textbf{x}_W=\prod_{i\in W} x_i$. We study homological invariants and properties of $\textrm{NI}_{\textbf{k}}(G)$ focusing on its Castelnuovo-Mumford regularity, projective dimension and Cohen-Macaulayness. Special attention is devoted to the case where the vector ${\textbf{k}}$ is the degree-vector of the graph, i.e., $k_i=\textrm{deg}_G(i)$ for all vertices $i$, and to the case where $\textrm{NI}_{\textbf{k}}(G)$ coincides with the edge ideal of a graph. In these settings, we provide combinatorial characterizations and bounds for the regularity and projective dimension of $\textrm{NI}_{\textbf{k}}(G)$ for several classes of graphs, and further investigate the Cohen-Macaulay property of these ideals.

This work develops an iterative computational framework to obtain saddle-point solutions for infinite-horizon two-person mean-field linear-quadratic zero-sum stochastic differential games. By generalizing classical iterative framework, we construct a monotonically increasing matrix sequence to decouple the strongly coupled, analytically intractable original problem into a set of tractable subproblems. By sequentially computing the stabilizing solutions of the coupled algebraic Riccati equations for these subproblems, we can further derive the solution to the original problem. Rigorous convergence analysis is established to validate the proposed iterative algorithm. Different from existing algorithms limited to specific simplified setups, this framework proposes the first universal computational paradigm applicable to a broad class of game-driven Riccati equations.