Papers

This paper develops a Hodge-theoretic feedback framework for flow-valued states, motivated by scalar parabolic thermostat problems with localized sensors and dual sources. On a connected graph, the edge space decomposes into a cut space and a harmonic cycle space. The ideal topological thermostat regulates the cut component toward a prescribed cycle-free transfer target and damps the harmonic component. For realizations through selected physical edge channels, the first Betti number gives the minimal sensing and actuation ranks required for exact compression of the full harmonic sector. Minimal rank does not imply local realization of the nonlocal Hodge projector; the remaining cut-harmonic blocks quantify spillover and forcing. For transmission networks, we formulate ideal and finite-channel nonlinear line-actuated swing closures and linearize the complete closed systems at an angle-cohesive synchronous equilibrium. With the actuator fixed, passive tangent swing motion preserves any pre-existing harmonic line-flow component. The ideal closure replaces this conservation law by exponential decay and appends a stable harmonic block without changing the reduced passive nodal-cut spectrum. Under the channel-rank conditions, finite realizations reproduce the harmonic compression, whereas autonomous harmonic decay and full reduced-state recovery require additional decoupling and Hurwitz conditions. A target-centred Bregman balance yields nonlinear dissipation for the ideal metric-compatible controller and for canonical colocated finite-channel feedback. Explicit theta-network, microgrid, ring, and synthetic IEEE 14-bus examples illustrate transfer regulation, cycle-flow damping, and localization-induced transients. Realistic device dynamics, constraints, and large-scale validation remain open.

The associated, ideal-preserving, and locally associated properties of subrings, first formally defined in 2024, give a way of understanding the multiplicative structure of a subring given information about the larger ring. In this paper, we establish notation and preliminary results on a generalized type of polynomial and power series rings that is often used in the construction of counterexamples in the field of commutative algebra. We then provide sets of necessary and sufficient conditions under which such a polynomial or power series ring may be (locally) associated in a larger such ring. This allows for the production of several informative examples, as well as a better understanding of the circumstances under which the ring of formal power series $R[[x]]$ over an order $R$ in a number field may be half-factorial.

Many dynamical-state reconstruction problems seek to infer a high-dimensional spatial state from measurements at only a few locations. Because the governing dynamics couple evolution across space and time, temporal measurements can contain information about state components beyond the sensor locations. This work develops a theoretical framework comparing the information content of a temporal measurement history with that of a specified instantaneous spatial sensor array. The temporal history and spatial reference array are represented by prior-whitened information operators that account for prior variability, sensor geometry, and measurement noise while preserving the directional distribution of uncertainty reduction. A general space-time information-exchange theorem gives necessary and sufficient conditions for retaining a prescribed fraction of the spatial-reference information in every state direction. It answers three practical questions. First, how many temporal measurements and how much total information are required? Second, is the prescribed fraction attainable, or does a limiting information ceiling rule out every finite history? Third, when attainable, what history length is sufficient? Explicit history-length bounds are derived for contractive, spectrally separated unitary, and simultaneously diagonalizable dissipative linear dynamics, and for stochastic linear estimation under Gaussian process and measurement noise. For nonlinear systems, global delay-map injectivity is combined with a uniform local Fisher-information condition. Numerical experiments test directional exchange lengths, information ceilings, and finite-memory bounds, and assess nonlinear delay-map distinguishability and local Fisher-information predictions. The framework provides an explicit basis for determining when temporal measurements can reduce spatial sensor coverage for full-state reconstruction.

The cosine measure of a set of vectors in $\mathbb{R}^n$ measures how well the set covers all directions in $\mathbb{R}^n$. It identifies the direction furthest, in angle, from the set. It is used in the convergence theory of various optimization algorithms, but also highlights interesting geometric properties of sets. For example, the cosine measure of a set $S$ is greater than zero if and only if given any $\mathcal{C}^1$ function $f$ at a point $\mathbf{x}$ where the gradient is nonzero, $S$ must contain a descent direction of $f$ at $\mathbf{x}$. In this paper, we examine the question of what can be said when the function $f$ is non-differentiable or if it has a gradient equal to the zero vector. To examine these cases, we introduce the novel concept of the cosine measure of a function at a point. This value provides an infimum on the value of the cosine measure that a set of vectors requires to guarantee it contains a descent direction of the function at the point of interest. We present mathematical theory around this concept, including examples showing that the cosine measure of a smooth function can have any value in $[-1,1]$. We further present algorithms to compute the cosine measure of a function, and examples demonstrating the algorithm on smooth and nonsmooth functions. These results also shed light on the the cosine measure of infinite sets and nonconvex cones.

In this paper, we first establish a generalized Cauchy–Kovalevskaya (GCK) extension for axially polyanalytic functions of order $2$. We prove that the extension can be written as a power series involving differential operators acting on two initial functions. We also study the decomposition of the GCK extension in terms of integrals over the sphere 2-sphere $\mathbb{S}$ involving plane-wave type functions. We further establish a connection between the GCK extension for polyanalytic functions of order $2$ and the Fueter theorem. This is one of the most important results in hypercomplex analysis and can be described in two steps. In the first step, starting from holomorphic functions of one complex variable, the application of a suitable operator yields slice hyperholomorphic functions. In the second step, applying the Laplace operator in four real variables (called Fueter map) one obtains axially monogenic functions, i.e. functions in the kernel of the Fueter operator. A suitable factorization of the Fueter map in terms of the Fueter operator and its conjugate gives rise to two intermediate classes of functions between slice hyperholomorphic functions and axially monogenic functions: axially harmonic functions and axially polyanalytic functions of order $2$. Another goal of this paper is to study the invertibility of the factorizations of the Fueter map for harmonic and polyanalytic functions of order 2 on suitable open sets, and to derive integral representation formulas for the inverse of the factorized Fueter map. These integral representations are based on the Cauchy formula for polyanalytic functions of order $2$ and the Poisson integral formula for harmonic functions.

A mathematical model is analyzed for a population of fish exhibiting cannibalism during its larval stage. The population is divided into three classes (vulnerable, cannibals, and non-vulnerable). It is assumed that initially, the entire population is vulnerable, and as they develop, individuals transition into one of the other classes. In addition, the interaction between the non-vulnerable and the cannibals is a predator-prey relationship with a Beddington-DeAngelis functional response. The global stability of the equilibrium point is demonstrated and numerical simulations show how food quality influences the cannibalistic behaviour of the population and its survival. Furthermore, it is shown that the solutions fit the data obtained experimentally for Atractosteus tropicus in its larval state. This model can be adapted for other fish species that exhibit cannibalism.

Hawkes-based microstructural foundations for rough volatility, leverage, and rough Heston-type limits were developed by El Euch et al. (2018, Finance Stoch., 22(2), 241–280) and connected to the affine rough Heston framework of El Euch and Rosenbaum (2019, Math. Finance, 29(1), 3–38). The rough Hawkes–Heston model with common price–volatility jumps of Bondi et al. (2024, Math. Finance, 34(4), 1197–1241) extends this framework by adding state-dependent common jumps to rough affine volatility. We provide a microstructural foundation for its variance and common-jump mechanism by constructing a Poisson-embedded marked Hawkes order-flow model. Ordinary arrivals generate rough continuous volatility and leverage through a nearly unstable heavy-tailed Hawkes mechanism, while rare marked arrivals represent common shock events that produce simultaneous price jumps and volatility excitation. Under the nearly unstable scaling and the reduced-form admissibility conditions, the complete rescaled price/variance/jump system converges along the full sequence to the unique complete canonical rough Hawkes–Heston weak solution. The Hawkes renewal structure yields a Mittag–Leffler Volterra representation, which is then rewritten in Riemann–Liouville fractional form. The limiting coefficients are expressed explicitly in terms of the microscopic parameters. The construction provides a microstructural foundation for the variance and common-jump mechanism of the rough Hawkes–Heston model. Numerical experiments illustrate the convergence of our microstructural foundation to the rough Hawkes-Heston model.

We exhibit an infinite family of disconnected multigraded Hilbert schemes on the biprojective space $\mathbb{P}^2_{\mathbb{k}} \times_\mathbb{k} \mathbb{P}^1_{\mathbb{k}}$. More precisely, for any integer $a \ge 2$ and $p_a(z_1,z_2) = 2az_1+az_2+3a-2a^2 \in \mathbb{Q}[z_1,z_2]$, the multigraded Hilbert scheme ${\rm Hilb}_{p_a}(\mathbb{P}^2_{\mathbb{k}}\times_\mathbb{k} \mathbb{P}^1_{\mathbb{k}})$ is disconnected. As a consequence, there exist infinitely many disconnected Haiman-Sturmfels multigraded Hilbert schemes even for a standard bigrading on a polynomial ring in only five variables.

We introduce a new algorithm for computing Kostka-Foulkes polynomials. It arises from a combinatorial procedure that computes the transition coefficients between successive pre-canonical bases for spherical Hecke algebras, introduced by Libedinsky, Patimo, and the first author, in order to interpolate between the standard and canonical bases of the spherical Hecke algebras. Using this algorithm, we prove that the coefficients of the polynomials in the transition matrix from any pre-canonical basis to the next one has nonnegative coefficients. This establishes the positivity conjecture for pre-canonical bases and, moreover, yields a strictly stronger result that uncovers additional positivity phenomena beyond the original scope of the conjecture. As a by-product, this method yields a combinatorial algorithm for computing Lascoux's atomic decomposition of canonical basis elements. Finally, our results show that, after applying the Satake isomorphism and dualizing with respect to the Hall inner product, pre-canonical bases correspond to a new family of Schur-positive symmetric functions.

We disprove the conjecture that volume is log-submodular under Minkowski addition on the class of zonoids by exhibiting a four-dimensional zonotope $A$ and two segments $B$ and $C$ such that \begin{align*} |A||A+B+C|>|A+B||A+C|, \end{align*} where $|\cdot|$ denotes the volume. Consequently, several related local mixed-volume, local Loomis-Whitney, projection-volume ratio, and volume-to-surface-area conjectures also fail for zonoids. The zonotope in our counterexample is generated by a 2-modular matrix. We also provide a proof of the correctness of the conjecture in the case where $A+B+C$ is a unimodular zonotope as well as a characterization of the equality cases in this setting.

This work investigates recurrent pattern formation in a two–species chemotaxis system with competitive dynamics under the assumption that the carrying capacities of both species are periodic in time. By considering a globally attracting periodic solution to the associated nonautonomous ODE system as a spatially homogeneous base state, we derive sufficient conditions for the existence of a critical chemotactic sensitivity of the first species, denoted by $\chi_{\text{crit}}$. Whenever this threshold is exceeded, the periodic state becomes linearly unstable over a nonempty set of times within each environmental cycle. This gives rise to a recurrent alternation of unstable and stable intervals, during which spatial perturbations are respectively amplified and damped. The analytical derivation is complemented with numerical simulations, in both periodic coexistence and competitive exclusion regimes. Finally, we discuss the temporal location of the instability sets and the possibility of negative instability thresholds.

We consider on Riemannian manifolds the Trudinger equation \begin{equation*}\partial _{t}u=\Delta _{p}u^{\frac{1}{p-1}},\end{equation*} where $p>1$. We prove that a relative $p$–Faber–Krahn inequality is equivalent to the conjunction of volume doubling and a sub-Gaussian upper estimate for non-negative bounded weak subsolutions of Trudinger's equation. We also derive an improved long-time upper estimate under a uniform $p$–Faber–Krahn inequality.

The strong tree property and the super tree property (also called ITP) are generalizations of the tree property that characterize strong compactness and supercompactness up to inaccessibility. That is, an inaccessible cardinal $\kappa$ is strongly compact if and only if the strong tree property holds at $\kappa$, and supercompact if and only if ITP holds at $\kappa$. Generalizing a result of Golshani and Hayut, we show that from large cardinals it is consistent for ITP to hold simultaneously at any countable initial segment of successors of singular cardinals. More formally, given any countable ordinal $\theta$, we construct a forcing extension in which ITP holds at the first $\theta$ successors of singulars. We then extend this result further to obtain the strong tree property on long segments of successors of singular cardinals of multiple cofinalities simultaneously.

A classical geometric result says that every nonzero cycle of the mod-$2$ incidence map from $d$-subsets to $(d-1)$-subsets of $[n]$ has support at least $d+1$, with equality attained by the boundary of a simplex on $d+1$ vertices. We prove an analogous result for the subspace lattice of $\mathbb{F}_q^n$, determining the minimum support size of a nonzero $d$-cycle over a field $K$ of characteristic $p \mid q+1$. Surprisingly, the boundary of a $(d+1)$-space is not always optimal. Shorter cycles occur for $d=1$, and for $d=2$ when $n\ge4$, and otherwise, the boundary of a $(d+1)$-space is shortest. For $d\ge4$, we prove a gap-stability result: every cycle with support less than $(2-10/q)$ times the minimum is a multiple of the boundary of a $d+1$ space. We also construct support-controlled cones, yielding a direct geometric analysis of the dimensions in which the homology groups of the subspace incidence complex vanish and explicit lower bounds on its coboundary expansion. The degree-$1$ expansion estimate is an ingredient in the stability theorem.

We study the long-time behavior of nonnegative solutions to a nonlinear diffusion equation with gradient absorption on infinite weighted graphs. The equation combines a discrete p-Laplacian diffusion term with a nonlinear absorption term of Hamilton-Jacobi type, modeling processes where both diffusion and nonlinear damping occur on discrete structures. Under the assumptions that the graph satisfies a polynomial volume growth condition and that the parameters satisfy certain ordering relations, we establish sharp decay estimates for the total mass of the solution. We prove that the total mass decays to zero as time tends to infinity whenever the absorption exponent lies below a critical threshold. This critical exponent depends explicitly on the volume growth rate of the graph and the diffusion exponent, and coincides with the Fujita-type threshold known for analogous equations in the continuous Euclidean setting. Our proof combines Hardy and Hölder inequalities with a carefully chosen time-dependent scaling parameter adapted to the graph geometry. The results are new even in the linear diffusion case and extend naturally to a broad class of graphs, including integer lattices and Cayley graphs of finitely generated nilpotent groups.

Ischemic heart diseases represent the leading cause of mortality worldwide. Revascularization, the process to restore blood flow in blockages, shows promise. To this end, mathematical models for angiogenesis, the process by which new blood vessels form from existing ones, have been extremely well investigated. In the current work, we consider a classical two species model for angiogenesis, consisting of cell and VEGF populations. However, we assume the cells move according to p-Laplacian diffusion, which could be both “fast" ($1<p<2$) and “slow" ($p>2$), in addition to normal diffusion ($p=2$). We first show that the system is well posed in a weak sense when $p>\frac{3}{2}$, for sufficiently small initial data. Next, we show that the p-Laplacian can lead to several novel dynamics not reported earlier; these include increased cellular proliferation via bi-modal and multi spike solutions, gain of regularity, prevention of finite time blow-up, cell depletion via finite time extinction, and Turing patterns. We discuss applications of these results for cardiac health via a digital twins framework.

This paper investigates the global geometry of the group of Hamiltonian diffeomorphisms $\operatorname{Ham}(M,\omega)$ using random walks. On a large class of symplectic manifolds, we show that the expected Hofer norm of such a random walk grows at least as fast as the square root of the number of steps. Furthermore, we show that if the random walk is restricted to an abelian subgroup, then the growth rate is also bounded from above by the square root of the number of steps. We also provide some numerical evidence suggesting that this upper bound fails away from commutative subgroups. This provides, for subgroups of the group of Hamiltonian diffeomorphisms, a probabilistic version of the flatness observed in commutative finite-dimensional Lie groups. En route, we show that the class of probability measures introduced in the prequel is a class of Borel measures with respect to the $C^\infty$-topology on $\operatorname{Ham}(M,\omega)$, show the measurability of the stable commutator length, and show a central limit theorem for Hofer-Lipschitz quasimorphisms on $\operatorname{Ham}(M,\omega)$.

We prove a Bauer-Poulsen dichotomy theorem for simplices of invariant measures associated with permutation groups. More precisely, let $G$ be a transitive group of permutations of a countable set $\mathcal{S}$, and let $H$ be the stabilizer of a point of $\mathcal{S}$. Let $\overline{G}$ and $\overline{H}$ denote their closures in the topology of pointwise convergence. Assume the Polish group $\overline{H}$ has relative Property (T) in $\overline{G}$. Then the simplex $\mathcal{M}_\mathrm{inv}(2^\mathcal{S})$ of invariant probability measures for the induced action $G\curvearrowright 2^\mathcal{S}$ is Bauer if and only if $\overline{G}$ has Property (T), and is Poulsen otherwise. This addresses some examples and questions considered by Austin. We deduce this result from a more general model-theoretic statement of independent interest. To this end, we initiate the study of existentially closed models in affine logic.

In this note we prove $\mathrm{forb}(n,\mathbf{1}_{r-2}\times C_4)=\Theta(n^{r-1/2})$, thus providing a series of counterexamples of a 2005 conjecture of the first two authors. Here $\mathbf{1}_{r-2}\times C_4$ is the $r$-extension of the 4-cycle, that is the $r$-uniform hypergraph of four edges obtained by joining a set of $r-2$ vertices to each edge of $C_4$. $\mathrm{forb}(n,F)$ is the largest number of edges of a simple hypergraph $G$ on $n$ vertices without trace (induced subhypergraph) $F$.

Stochastic gradient descent for a loss function discontinuous across lower dimensional manifolds is analyzed by studying its differential equation limit.

We give a Lefschetz thimble definition of the Chern–Simons–Kodama wavefunctional in Lorentzian self-dual gravity with positive cosmological constant. By complexifying the connection and choosing the Lefschetz thimble attached to the self-dual de~Sitter saddle, Witten's analytic continuation of Chern–Simons theory replaces the real contour, on which the defining integral is at best conditionally convergent, by an absolutely convergent integration cycle and yields a controlled semiclassical expansion. The relevant integrand is the full holomorphic exponential $e^{I}$, with $I=\kappa Y_{\rm CS}+(\text{source})$, and the steepest descent flow is governed by the Morse function $h=\operatorname{Re} I$. In the homogeneous isotropic de Sitter reduction, the construction reproduces the Hartle–Hawking Airy wavefunction. In the Bianchi~IX truncation, we take the Lefschetz thimble attached to the expanding de~Sitter saddle (the contracting branch defines a conjugate Lefschetz thimble); within this sector, the symmetric configuration is the only critical point for $\Lambda>0$. The corresponding Lefschetz thimble contains one trace direction with the FRW Airy structure and two anisotropic shear directions that are real-Gaussian damped at leading quadratic order. This gives a finite Bianchi~IX wavefunction which, to leading order, is the FRW Airy factor multiplied by a real anisotropy Gaussian. Thus, the Airy behavior is not an artifact of the one-dimensional minisuperspace reduction, but the leading form of the wavefunction on the Lefschetz thimble. We also discuss how the Lefschetz thimble prescription interfaces with Lorentzian reality conditions.

We study $L^p_x L^\infty_t$ maximal estimates for exponential sums associated to $C^2$ graph hypersurfaces, motivated by Schrödinger maximal estimates on $\mathbb{T}^d$. We show that the conjectured maximal estimate for the periodic Schrödinger equation fails when one allows small perturbations of the paraboloid, which can be viewed as a higher-dimensional extension of the phenomenon proved by Fu, Ren, and Wang. Our approach uses new lower bounds for incidence estimates originally proven by Cairo and Zhang, for which we provide an alternative proof based on homogeneous dynamics. Moreover the estimates are essentially sharp at the decoupling endpoint for the paraboloid $p = \frac{2(d+2)}{d}$.

The dual immaculate functions are an example of a Schur-like basis in the algebra of quasisymmetric functions. We classify when the image of a dual immaculate function under one of the involutions $\rho, \psi, \omega$ is equal to a dual immaculate function. As well, the leading term of the transition matrix is identified, and sufficient and necessary conditions for the existence of immaculate tableaux are determined. As a consequence, new maps and canonical tableaux associated with compositions are discovered.

We investigate the analysis and phase ordering dynamics of the active Cahn–Hilliard equation, providing novel results beyond the current state of the art concerning the well-posedness and the characterization of its coarsening dynamics. We consider both regular polynomial and singular logarithmic potentials. In particular, we exploit a new method based on heteroclinic trajectories in the phase plane to characterize static kink profiles and spherical droplet states, recovering the exact values of key quantities related to static phase-separated configurations; moreover, we develop a theory accounting for surface tension modifications driven by activity and local interface curvature, which explains the power-law shift $L(t)\sim t^{\frac{1}{z}}$ from $z=3$ to $z=4$ induced by activity for the characteristic domain length conjectured in the literature. This shows that there is a transitory effect before the attainment of a finite saturation length. We also design an efficient numerical scheme, based on finite elements, to approximate the model, proving its well-posedness and stability both for regular and singular potentials. In dimensions $d=2,3$ with singular potential, the convergence analysis of the finite element approximation proves the local-in-time existence and uniqueness of weak solutions satisfying the physical constraint $\phi \in (-1, 1)$. In dimension $d=1$ with singular potential, we establish global-in-time well-posedness and regularity of weak solutions under a smallness condition on the activity parameter. Finally, we show numerical simulations for different test cases which prove that our numerical algorithm correctly reproduces the expected phase separation dynamics. Moreover, we show the results for coarsening dynamics at late times which present a power law shift from $z=3$ to $z=4$ prior to reaching late-time length saturation, which confirms our theoretical findings.

Wasserstein policy gradient (WPG) updates state-conditional action laws by transport in the action space. We study entropy-regularized discounted linear-quadratic (LQ) control. A Bellman verification argument shows that the unrestricted problem has a linear-Gaussian optimal policy, and the discounted-occupancy-weighted statewise Wasserstein gradient is tangent to this policy class. WPG therefore reduces exactly to a finite-dimensional ODE for the feedback gain and action covariance. We prove that this ODE is globally well posed and converges exponentially from every admissible initialization. For each fixed LQ problem, the exponent has a positive limit as the entropy temperature tends to zero and contains no perturbative factor of the form $\exp(-c/\tau)$, while retaining the usual dependence on the conditioning of the control problem.

We study the fundamental limits of convex-regularized estimation in high-dimensional linear regression with right-rotationally invariant design matrices. We show that the asymptotic risks of convex-penalized least squares estimators are lower bounded by the risk of an approximate message passing algorithm known as Bayes VAMP, and we further characterize when the lower bound is attainable. As a technical ingredient in the proof of our lower bound theorem, we characterize the asymptotic performance of the $\ell_2$-perturbed convex estimator for every fixed perturbation strength $\lambda>0$. This closes a gap in the literature, in which $\lambda$ was required to be sufficiently large or restrictions were imposed on the class of convex estimators. The benefit of our approach is that we can conduct a direct analysis of the spectrum's impact on the lower bound in the high-dimensional limit. In particular, we can show that the lower bound is monotone in an ordering on the limiting spectral distributions of the design matrix. These results isolate how the full singular-value distribution of the design—not merely the aspect ratio or average measurement strength—governs the limitations of convex regularization.

Given integers $k\ge2$ and $t_1, \ldots, t_k\ge2$, we write \emph{$G \rightarrow (K_{t_1}, \ldots, K_{t_k})$} if every $k$-coloring of the edges of a graph $G$ contains a monochromatic copy of $K_{t_i}$ in color $i$ for some $i\in\{1, \ldots, k\}$. A non-complete graph $G$ is \emph{$(K_{t_1}, \ldots, K_{t_k})$-co-critical} if $G \nrightarrow (K_{t_1}, \ldots, K_{t_k})$, but $G+e\rightarrow (K_{t_1}, \ldots, K_{t_k})$ for every edge $e\notin E(G)$. Let $r=R(K_{t_1}, \ldots, K_{t_k})$ denote the Ramsey number. In 1987, Hanson and Toft conjectured that every $(K_{t_1}, \ldots, K_{t_k})$-co-critical graph $G$ on $n\ge r$ vertices satisfies \[|E(G)|\ge (r-2)n- \binom{r- 1}{2}.\] This bound is best possible for every $n\ge r$. More recently, the present author conjectured that every such graph has minimum degree at least $r-2$. Using the $q$-neighbor bootstrap percolation closure method, here we prove that the Hanson-Toft Conjecture holds asymptotically, provided that the minimum-degree conjecture is true; more precisely, If every $(K_{t_1},\ldots,K_{t_k})$-co-critical graph has minimum degree at least $r-2$, then there is a constant $C=C(r,k)$ such that every $(K_{t_1},\ldots,K_{t_k})$-co-critical graph $G$ on $n\ge r$ vertices satisfies $|E(G)|\ge (r-2)n-C$.

We prove a noncommutative topological factor theorem for irreducible lattices in products of real rank-one simple Lie groups. The intermediate C*-subalgebras between the reduced group C*-algebra and the boundary crossed product are exactly the crossed products arising from coordinate subproducts of the Furstenberg boundary. This follows from a more general theorem for product boundary actions, which also yields tree and mixed local-field versions. We finally show that, for lattices in connected semisimple real Lie groups with finite center and no compact factors, the scalar-expectation case of the corresponding classification is equivalent to ordinary ITAP.

The set $A$ is a $B_h$-set if every element of the sumset $hA$ has a unique representation as a sum of $h$ elements of $A$. A $B_2$-set is also called a Sidon set. A $B_{h,\Delta}$-set is a $B_h$-set $A$ whose sumset $hA$ is $\Delta$-separated, that is, $|x-x'| \geq \Delta$ for all $x,x' \in hA$ with $x\neq x'$. Upper and lower bounds are obtained for the cardinality of the largest $B_{2,\Delta}$-sets contained in $\{1,2,\ldots, n\}$, that is, sets $A \subseteq \{1,2,\ldots, n\}$ such that, if $a,b,c,d \in A$ and $\{a,b\} \neq \{c,d\}$, then $|(a+b)-(c+d)| \geq \Delta$.

Building on advances in reconfigurable antenna techniques, movable antennas (MAs) can dynamically reshape antenna arrays and introduce additional spatial degrees of freedom (DoFs), thereby further improving communication performance. Despite these benefits, existing MA design algorithms often entail prohibitively high computational complexity from discrete positioning selection, which prevents practical implementations of MAs. In this paper, we investigate efficient solutions for the mutual information (MI) maximization problem of a multi-user multiple-input multiple-output (MU-MIMO) uplink communication system aided by discrete MAs. To this end, we first formulate the discrete MA positioning problem with the assumption of perfect channel state information (CSI). Then, we prove that the design problem falls into the category of monotone submodular maximization subject to a 2-system constraint. Accordingly, we propose a low-complexity distance-constrained submodular position search algorithm, which is theoretically shown to achieve at least 1/3 of the optimum. Furthermore, we extend our approach to scenarios with imperfect CSI, and show that the proposed submodular optimization-based design remains robust against channel estimation errors. Numerical results demonstrate that the proposed scheme can achieve at least 90% of the optimal solution's MI gain under both perfect and imperfect CSI assumptions. Remarkably, the algorithm achieves orders-of-magnitude complexity reduction (e.g., 34.4x faster than the branch-and-bound approach) while maintaining significant MI gains.