Papers

We introduce an Indian-buffet-type model for multi-factorial innovation in which each arriving agent may exhibit both previously observed and new features. The number of new features follows a power-law behavior, while the probability of selecting an old feature combines self-reinforcement, depending on the feature-specific popularity, with a mean-field interaction term depending on the average popularity of all observed features. The model is governed by the usual innovation parameters (mass, discount and concentration), together with two additional parameters: one controlling the strength of reinforcement against a forcing input toward zero, and one regulating the intensity of the mean-field interaction. Although the growth of the total number of distinct observed features has the same behavior as in the three-parameter Indian buffet process, the mean-field interaction mechanism produces new asymptotic regimes. For aggregate quantities, including the predictive mean, the averaged number of features per agent, the mean inclusion probability, and the mean feature popularity, the phase transition is determined by the comparison between the discount parameter and the weight of the forcing input. For feature-specific quantities, a further transition appears according to the comparison between the interaction level and a critical threshold. In particular, high interaction leads to an asymptotic synchronization of feature-specific inclusion probabilities. We establish strong laws and second-order asymptotic results, including central limit theorems in regimes where martingale fluctuations compete with deterministic or random terms. The analysis relies on novel general results for recursive stochastic dynamics, which may be useful beyond the present framework.

In radar sensing, the self-ambiguity function of the probing waveform plays a crucial role in the resolvability and detection of multiple targets. In the recent Zak-OTFS based radar literature, Gaussian pulse shaping filter has been considered, and it has been shown to offer better range/velocity estimation performance compared to the traditional chirp waveform in scenes with multiple targets. While the self-ambiguity function with Gaussian filter has very low side lobes, its main lobe is wide which compromises resolvability and performance. Motivated by this, we seek filters with better ambiguity characteristics. Specifically, we explore two other known filters, namely, sinc and Gaussian-sinc (GS) filters, and demonstrate that these filters offer better performance compared to Gaussian filter under different scenarios and receiver processing. Towards demonstrating this, we derive closed-form expressions for the self-ambiguity functions of Zak-OTFS waveform with sinc and GS filters. The ambiguity functions of sinc and GS filtered waveforms have narrow main lobes, resulting in better resolvability in scenes with densely populated targets for the basic peak-detection based receiver. The ambiguity function of Gaussian filtered waveform has very low sidelobes, resulting in better performance in sparsely populated scenes. When a receiver with inter-target interference mitigation is used, the sinc and GS filters perform better in both dense and sparsely populated scenes compared to Gaussian filter.

High Altitude Platform Stations (HAPS) have emerged as a promising enabler for next-generation wireless networks, offering ubiquitous connectivity to ground users. Operating either in standalone mode or in integration with terrestrial networks, HAPS can significantly enhance both coverage and capacity due to their strategic placement in the stratosphere. However, interference management in HAPS-empowered networks requires special attention due to the unique propagation characteristics of HAPS links. In particular, the strong line-of-sight (LoS) conditions between HAPS and ground users result in limited channel variability, thereby intensifying inter-user interference. In this work, we consider a single HAPS serving multiple ground users through multiple beams over a limited number of orthogonal resource blocks (RBs). To address the resulting interference, we propose a novel angular-aware user clustering and interference-aware RB allocation framework that strategically clusters users, designs beams to serve each cluster, and allocates RBs to users across clusters. To further mitigate intra-RB interference, a rate-splitting multiple access (RSMA) scheme is incorporated. Simulation results demonstrate that the proposed clustering and RSMA-based approach significantly outperforms baseline schemes in terms of achievable per-user spectral efficiency.

We prove the WALA conjecture of G. David and S. Semmes: every AD-regular Radon measure for which all real-valued Lipschitz functions satisfy the weak approximation by affine functions condition is uniformly rectifiable. In order to prove the conjecture, we identify the correct quantitative analogue of the decomposability bundle and establish several structural results for general AD-regular measures.

Let $\mathfrak{g}$ be a symmetrizable Kac-Moody Lie algebra and let $\rho$ denote the sum of the fundamental weights. The irreducible highest weight representations $V(m\rho)$ occupy a distinguished position in representation theory due to their rich symmetry and geometric significance. In this paper, we study the tensor products \[ V(m\rho)\otimes V(n\rho), \quad m,n \in \mathbb{N}, \] and investigate the structure of their irreducible decompositions. Motivated by the classical conjecture of Kostant, which predicts a highly structured behavior in simpler settings, we propose a general framework describing the irreducible components appearing in such tensor products for finite-dimensional semisimple or affine Kac-Moody Lie algebras $\mathfrak{g}$. Our results identify a family of dominant weights governing the decomposition and provide criteria for their occurrence. This work extends the scope of Kostant-type phenomena and reveals new structural patterns in tensor products associated with multiples of the Weyl vector.

The big $(-1)$-Jacobi, Bannai-Ito, and $(-1)$-Meixner-Pollaczek polynomials have appeared in the mathematical physics literature through differential- or difference-reflection eigenvalue problems. We show that these three families are governed by a single alternating mechanism, formulated at the level of orthogonality functionals, transpose operators, and structural identities. Starting from an ordinary orthogonal polynomial sequence in the quadratic variable, the resulting algorithm constructs the direct alternating family and transfers its moments, squared norms, and recurrence coefficients. One regular Geronimus inversion then produces the big $(-1)$-Jacobi and Bannai-Ito families, whereas the $(-1)$-Meixner-Pollaczek family is obtained directly.

Magic relations are a class of integration-by-parts identities where all integrals in the generating sector drop out. Since their presence causes several otherwise successful methods in the Feynman-integral computational pipeline to break down, they are important to detect and understand. In this paper, we take a first step toward a systematic characterization of such identities. Specifically, we observe and argue that the occurrence of magic relations always coincides with the presence of higher-dimensional critical varieties in the generating sector. This provides a practical computational test to check if a family of Feynman integrals can contain magic relations and to find them, which we implement in the ancillary Mathematica file Magic-Test.m. Additionally, we discuss how to count the number of master integrals in the presence of higher-dimensional critical varieties, classify the behavior of magic relations under symmetries, and we discuss their interplay with cuts.

In this article we study the Iwasawa invariants of Bertolini–Darmon theta elements in the anticyclotomic $\mathbb{Z}_p$-extension of an imaginary quadratic field $K$ for weight two modular forms $f\in S_2(\Gamma_0(N))$. We cover both the cases of ordinary and non-ordinary reduction at a prime $p$. Our results extend the known results of Pollack–Weston and Gajek-Leonard–Lei in the cyclotomic setting.

In this paper, we prove a spectral restriction theorem on the three-dimensional Heisenberg nilmanifold. Since this manifold is an $\mathbb S^1$-bundle over the flat torus $\mathbb T^2$, the result provides a sub-elliptic counterpart of Zygmund's restriction theorem on $\mathbb T^2$ \cite{zygmund}. We also establish its sharpness by means of the discrete short-time Fourier transform.

Algebraic representations of time series are symbolic representations whose symbols belong to a finite group. Precisely, the framework of the present paper is the analysis of coupled time series in algebraic representations and, more generally, group-valued time series. The prototype of an algebraic representation is an ordinal representation, whose symbols are permutations, also called ordinal patterns in the context of time series analysis. In fact, permutations, endowed with function composition, build a group called a symmetric group. A simple way to harness the algebraic structure of the alphabet in such cases is the concept of transcript from one group element to another. Since transcripts involve two group elements, they are very suitable for studying couplings between time series in the same algebraic representation. In this paper, we outline several existing entropic and algebraic transcript-based tools for analyzing coupled time series and systems. In addition to entropy, the entropic tools include divergence, statistical complexity and mutual information. The algebraic tools comprise order classes and, most recently, the Cayley and Kendall distances. We use the detection of generalized synchronization in a well-studied coupled system to compare the performances of some of those tools. To this end, we also provide an alternative tool called the similarity distance between times series, which is a mean Kendall distance. We found that the novel similarity distance outperforms the other tools tested.

In this work, we analyze a diffuse-interface model for tumor growth, subject to multiplicative white noises, posed on a bounded domain $\mathcal{O} \subset \mathbb{R}^d$, $d=2,3$. The model couples a stochastic incompressible convective Brinkman-Forchheimer (CBF) equation or Navier-Stokes equation with damping $\eta|v|^{r-1}v $ for the averaged velocity field $v$, to a Cahn-Hilliard (CH) equation for the phase field variable $\phi$ and to a stochastic reaction-diffusion equation governing the nutrient concentration $\sigma$. We establish the existence of local strong solutions , for $ r \geq 1 $ in $d=2$ and $ r \in [1,3] $ in $d=3$. We prove the weak-strong uniqueness holds in both $d = 2, 3$. In addition, for $d = 2$, the uniqueness of weak solutions is obtained for all $\eta,\nu > 0$, and $r \geq 1$, while it holds in $d = 3$ for $r \geq 3$ and $\eta \nu \geq 1$ when $r = 3$ under an assumption on $\sigma$. Moreover, for $d=2$ and $r \in [1,3]$, we obtain that the strong solution exists globally in time.

Convection-dominated problems are known for their slow Kolmogorov $n$-width decays and are challenging for model order reduction (MOR). In this work, we propose a hybrid surrogate modeling approach and a non-intrusive variant that overcome some drawbacks of linear MOR methods. The proposed hybrid surrogate model is a projection-based reduced-order model (ROM), corrected by the error learned from a deep neural network. With the aid of deep learning, the model component of the surrogate model can be kept in a small reduced dimension. The neural network component and the model component are sequentially but separately built during the offline stage. At the online stage, they are easily coupled to output the solution predictions. Due to the intrusive nature of the hybrid-ROM, the numerically discretized operators of the original model must be available. For problems solved using black-box solvers, where the details of the numerical discretization are not accessible, we further propose a non-intrusive variant of the hybrid surrogate. Compared to the existing MOR methods with nonlinear manifolds, the proposed hybrid ROM is more easily built and is also easily assembled for online prediction. In contrast to the surrogate modeling approaches purely based on deep-learning, the proposed non-intrusive variant has a lighter neural network structure with much fewer parameters to be learned. We test the proposed methods on two nonlinear convection parametric problems. The first is the 1D inviscid Burgers' equation with one parameter, and the second is the 2D inviscid Burgers' equation with two parameters. Since both methods are based on error correction, their online predictions exhibit higher accuracy yet with largely reduced prediction time, compared to state-of-the-art methods.

In distributed hypothesis testing, a central server performs hypothesis testing based on information received from distributed sensors/clients. We study a secure variant of this problem in which the central server determines the hypothesis class of an underlying distribution without learning any additional information about the distribution itself. We prove that, in its standard form, this is impossible to achieve, even for simple and highly restricted cases. To bypass this impossibility, we augment the model with a shared secret key available to clients but hidden from the server. We show that a single-bit secret key enables perfectly secure testing for simple classes by reducing the test distributions to a symmetric, canonical instance. Finally, for arbitrary hypothesis classes over finite domains, we establish a reduction to standard hypothesis testing using Private Simultaneous Messages (PSM) protocols, achieving polynomial communication and key lengths.

We propose a new disjunctive regularization for mathematical programs with complementarity constraints (MPCC). Its feasible set coincides with that of the Kanzow-Schwartz regularization. However, their functional descriptions differ considerably. For the disjunctive regularization, the logical operator OR and equivalent max-type constraints are used. Unlike the Kanzow-Schwartz, the disjunctive regularization satisfies the tailored linear independence constraint qualification if the original MPCC does. More than that, the favorable convergence properties - known to hold for the Kanzow-Schwartz regularization - remain valid for the disjunctive regularization as well. In particular, no second order necessary conditions are required to guarantee convergence towards S-stationary points of MPCC. Additionally, we keep track of the topological type of approximating and limiting nondegenerate C-stationary points in terms of their C-indices. Quadratic and biactive parts of the C-indices are shown to generically correspond to each other while regularizing. This is a new phenomenon as compared to the Scholtes or sign-type regularizations studied before. Numerical experiments illustrate that the proposed disjunctive regularization clearly outperforms the Kanzow-Schwartz regularization. Its numerical performance is even better than that of the Scholtes regularization if solving MPCCs with high accuracy.

We consider a smooth compact manifold with boundary, $M$, embedded in a smooth manifold of the same dimension on which an amenable group $\Gamma$ acts by isometries. We do not assume $M$ to be invariant under $\Gamma$. This results in a partial action of $\Gamma$ on $M^\circ$: For $g\in \Gamma$ we let $M^\circ_g = g(M^\circ)\cap M^\circ$ and obtain diffeomorphisms $g:M^\circ_{g^{-1}} \to M^\circ_g$. We assume that any two images of $\partial M$ under $ \Gamma$ either coincide or are disjoint and that only finitely many lie in $M$. The spherical blow-up of these images of $\partial M$ in $M$ yields a manifold $Y$ with boundary consisting of finitely many components. Moreover, $Y$ inherits a partial action of~$\Gamma$. We can then define the $C^*$-algebra $\mathcal A=\overline{\Psi_\Gamma(Y,\partial Y)}$ of operators on $L^2(Y)\oplus L^2(\partial Y)$, generated by the algebra $\Psi(Y,\partial Y)$ of operators of order and type zero in Boutet de Monvel's calculus on $Y$ and partial isometries associated with the partial action. Denote by $\Sigma=\overline{\Psi(Y,\partial Y)}/\mathbb K$ its symbol space. If the partial action of $\Gamma$ on Prim$(\Sigma)$ is topologically free, we find a criterion for the Fredholm property of the operators in $\overline{\Psi_\Gamma(Y,\partial Y)}$. Moreover, we obtain the classification of the elliptic elements in $\overline{\Psi_\Gamma(Y,\partial Y)}$ modulo stable homotopies: For $\mathcal A_0= C(Y\sqcup \partial Y)\rtimes\Gamma$ $${\rm Ell}(\mathcal A_0,\mathcal A)\cong K_0(C_0(T^*Y^\circ)\rtimes\Gamma)\oplus K_0(C(\partial Y)\rtimes \Gamma).$$ If $\Gamma$ is finitely generated and of polynomial growth, then the elements associated with the second summand do not contribute to the index.

We study the temporal dynamics of the first two empirical moments of Brownian traits on phylogenetic trees. For a fixed tree, we characterize the distributions of their empirical mean and empirical variance across all lineages extant at any given time. In particular, we show that the variance of the empirical mean and the expected empirical variance are piecewise linear between diversification events. For lineage-homogeneous random trees, both the variance of the empirical mean and the expected empirical variance can be expressed in terms of the expected age of the most recent common ancestor (MRCA) of a uniformly sampled pair of extant lineages. In this representation, the expected MRCA age enters the two quantities with opposite signs, pointing to a structural opposition between the variance of the empirical mean and the expected empirical variance. For generalized birth-death processes with time-dependent speciation and extinction rates, we derive an explicit formula for the distribution of the MRCA age of a uniformly sampled pair of extant lineages. This yields integral expressions, at any time, for both the variance of the empirical mean and the expected empirical variance. In the constant-rate birth-death case, we further obtain closed-form expressions for the expected empirical variance and describe its asymptotic behavior in the supercritical, critical and subcritical regimes.

The classes of abelian groups that are (uniformly) strongly Hopfian abelian groups, and dually, (uniformly) strongly co-Hopfian abelian groups have been studied by several authors, including Abdelalim (2015) and Abdelalim-Chillali-Essanouni (2015). This paper extends these investivations in the case of (genuinely) mixed groups. For example, it is shown that a reduced group that is (uniformly) strongly co-Hopfian will always be (uniformly) strongly Hopfian. In addition, a result of Chekhlov-Danchev (submitted) characterizing when a torsion-free group is (uniformly) strongly Hopfian is generalized to the case of (global) Warfield groups.

Starting from the exact Projected Central Limit Theorem on hyperspheres, we rederive the Beta distribution for subsystem occupation probabilities and Lubkin's purity formula from elementary hyperspherical moments, quantifying the finite-size “platykurtic” suppression of tails relative to the Gaussian approximation used in standard eigenstate-thermalization and typicality treatments. Our main new result concerns the bipartite quantum mutual information $\langle I(A{:}B)\rangle$ for Haar-random pure states. We show that its full asymptotic expansion in $1/N$ admits a Bernoulli-factorized form in which every order $k \ge 1$ carries the symmetric factor $(d_A^{2k}-1)(d_B^{2k}-1)$ and all higher odd-order corrections vanish identically. Through an exact algebraic reorganization of Page's formula (conjectured in Ref.~\cite{Page1993} and subsequently proven~\cite{Foong1994, SanchezRuiz1995, Sen1996}), we establish that the leading finite-size correction separates into a dominant $\mathfrak{su}(d_A) \otimes \mathfrak{su}(d_B)$ bipartite quantum coherence contribution $(d_A^2 - 1)(d_B^2 - 1)/(2N)$ and a subtracted classical-probability (Cartan $\otimes$ Cartan) contribution $(d_A - 1)(d_B - 1)/(2N)$, and we trace this separation to the difference between diagonal and eigenvalue entropies via Schur's majorisation theorem, with the dimensional counts $(d-1)$ and $(d^2-1)$ acquiring meaning through the Cartan structure of the generalised Bloch decomposition. These results admit a single non-perturbative closed form: the exact typical mutual information factors as $\langle I(A{:}B)\rangle = (d_A^2-1)(d_B^2-1)\,\mathcal{G}(d_A,d_B,d_E)$, with $\mathcal{G}$ given by an explicit Bose–Einstein integral whose asymptotic expansion in $1/N$ reproduces the Bernoulli series.

The average bipartite quantum mutual information $\langle I(A{:}B)\rangle$ of Haar-random pure states can be expressed exactly through Page's formula in terms of digamma functions. We show that this quantity admits a single non-perturbative closed form: $\langle I(A{:}B)\rangle = (d_A^2-1)(d_B^2-1)\,\mathcal{G}(d_A,d_B,d_E)$, where $\mathcal{G}$ is given by an explicit convergent integral over a Bose–Einstein kernel. The overall factor $(d_A^2-1)(d_B^2-1)=\dim[\mathfrak{su}(d_A)]\cdot\dim[\mathfrak{su}(d_B)]$ is exact, not merely asymptotic. The asymptotic expansion of $\mathcal{G}$ in $1/N$ yields a Bernoulli-factorised series whose coefficients involve $\zeta(1{-}2k)$; this series diverges, and our integral is its exact Borel sum. The integral representation also makes $\langle I\rangle < (d_A^2{-}1)(d_B^2{-}1)/(2N)$ manifest via a scale-inversion symmetry of the kernel. Our derivation traces the mutual information's structure to an exact decomposition of Page's entropy into a diagonal (Dirichlet) contribution and a Schur-majorisation eigenvalue correction, whose assembly into the mutual information cleanly separates classical from quantum correlations.

Using a non-commutative analogue of the isomonodromic problem associated with the discrete first Painlevé hierarchy, we construct a non-commutative version of this hierarchy, denoted by $\text{d-PI}_m^{\text{nc}}$. We show that both hierarchies, $\text{d-PI}_m$ and $\text{d-PI}_m^{\text{nc}}$, can be expressed in terms of the polynomials $S_s^k(n)$, which we call the Svinin polynomials. We also derive a reduction of the non-commutative Volterra lattice hierarchy to the $\text{d-PI}_m^{\text{nc}}$ hierarchy and present explicit continuous limits for the first three members of the $\text{d-PI}_m^{\text{nc}}$, thereby recovering non-commutative analogues of the first three members of the differential first Painlevé hierarchy.

We consider the weighted $p$-Laplacian associated with a measure $\mu$ having a continuous positive density with respect to the Lebesgue measure on an open convex subset $X\subset\mathbb{R}^N$. We prove that Talenti's weighted Pólya–Szegő inequality—originally established for Lipschitz functions on $X$—extends to the weighted Sobolev space of functions vanishing on the relative boundary of an open set $\Omega\subset X$, defined as the closure of Lipschitz functions supported away from it. This yields Faber–Krahn-type inequalities for the first $(p,q)$-eigenvalue of the weighted Dirichlet $p$-Laplacian. We present several examples fitting this abstract framework, in which the isoperimetric sets are balls, half-spaces, or complements of balls.

This paper studies Set Shaping Theory (SST) in a database-index setting under a revised interpretation: SST is not treated as a competing hashing method, but as a structural pre processing layer that can be applied before an existing indexing algorithm. The experimental question is therefore whether a method improves when it is used with SST rather than with out it. The study compares linear probing, double hashing, quadratic probing, and Robin Hood hashing against their corresponding SST-augmented variants for shaping orders K = 2,4,8. Beyond mean time, the benchmark reports mean successful probes, 95th and 99th percentile probes, collisions per stored record, and maxi mum cluster length. Experiments cover load factors from 0.75 to 0.95, database sizes from M =5000 to M =500000, query multipliers up to 200 lookups per stored record, and both uniform and hotspot query distributions. The results highlight two fundamental advantages. First, SST reduces the number of RAM accesses required during retrieval. By prevent ing clusters and long probe chains from forming at insertion time, the lookup phase requires fewer memory jumps, lower probe counts, and reduced tail latency. Second, the method introduces a new way of thinking about data storage: the data are not treated as fixed objects that must be placed passively into a table, but as reversible representations that can be struc turally adapted before being written. A small metadata tag records which transformation was selected, allowing the original key to remain recoverable and the lookup process to remain deterministic.This article is connected to the Set Shaping Theory simulator project, available online at https://sst-simulator.github.io/Set-Shaping-Theory-Simulator/ where it is possible to simulate part of the results presented in the article.

Approximately 10% of newborns require assistance to initiate breathing at birth, and around 5% need ventilation support. Fetal heart rate (FHR) monitoring plays a crucial role in assessing fetal well-being during prenatal care, enabling the detection of abnormal patterns and supporting timely obstetric interventions to mitigate fetal risks during labor. Applying artificial intelligence (AI) methods to analyze large datasets of continuous FHR monitoring episodes with diverse outcomes may offer novel insights into predicting the risk of needing breathing assistance or interventions. Recent advances in wearable FHR monitors have enabled continuous fetal monitoring without compromising maternal mobility. However, sensor displacement during maternal movement, as well as changes in fetal or maternal position, often lead to signal dropout, resulting in gaps in recorded FHR data. Such missing data limits the extraction of meaningful insights and complicates automated (AI-based) analysis. Traditional approaches to handling missing data, such as simple interpolation techniques, often fail to preserve the spectral characteristics of the signals. In this paper, we propose a masked transformer-based autoencoder approach to reconstruct missing FHR signals by capturing both local temporal and frequency components of the data. The proposed method demonstrates robustness across varying durations of missing data and can be used for signal inpainting and forecasting. The proposed approach can be applied retrospectively to research datasets to support the development of AI-based risk algorithms. In the future, the proposed method could be integrated into wearable FHR monitoring devices to achieve earlier and more robust risk detection.

We investigate critical polyharmonic equations of the type: $$ Lu = |u|^{2^\sharp-2} u \quad \text{ in } \Omega $$ with Dirichlet boundary conditions, in a smooth bounded domain $\Omega$ of $\mathbb{R}^n$. Here $L$ is an elliptic differential operator of integer order $2 \le 2k < n$ whose leading order term is $(-\Delta)^k$ and $2^\sharp = \frac{2n}{n-2k}$ is the critical Sobolev exponent. Our main result establishes, in large dimensions, uniform a priori bounds in $C^{2k}(\overline{\Omega})$ for bounded-energy solutions of this problem, that only depend on an upper bound on the energy. We prove this under a coercivity assumption of sorts on the lower-order terms of $L$. Our results are sharp, at least when $k=1$. Our approach uses asymptotic analysis techniques and in the course of the proof we obtain in particular a new global pointwise description of bounded-energy blowing-up solutions for this problem, which is of independent interest.

Sepsis remains a diagnostic challenge due to its heterogeneous molecular signatures and complex immune responses. In this study, we develop a logical data analysis framework based on Boolean polynomial rings. This method constructs an ideal $\mathcal{I}$ of selection criteria that isolate empty subsets of previously analyzed patient data. This approach enables the derivation of interpretable classification rules based on biomarker profiles. We demonstrate that logical data analysis identifies distinct logical patterns for positive and negative sepsis classification. For instance, elevated levels of GLP-1 and MyD88 are associated with septic states in our dataset, whereas high TRAIL and low MyD88 concentrations may suggest a non-septic condition. Importantly, a new way to integrate expert knowledge to filter out potential overfitting or dataset-specific artifacts is shown. Our findings highlight the utility of logics in generating transparent, biologically plausible rules for a data-based and expert-based understanding of sepsis. Moreover, we show how data analysis can benefit from algebraic structures.

Given $m \in \mathbb{N}$ and a $p$-random subset $A \subseteq \mathbb{N}$, we asymptotically determine $\log \Pr(|\mathbb{N} \setminus (A + A)| \ge m)$ for $p$ above the threshold for this property. As an application, we show that a uniformly at random chosen subset $A \subseteq \{1, \ldots, n\}$ of size $k$ such that $|A + A| \le \lambda k$, for $\lambda = o(k^{1/4} / \log^{1/2} n)$, with high probability belongs to an arithmetic progression of length $\lambda k / 2 + o(\lambda k)$. Both proofs are based on a bespoke container argument, exploiting the additive robustness of the considered sets.

Fluid antenna systems (FAS) have emerged as a promising technology for next-generation wireless systems. However, practical multiuser multiple-input multiple-output FAS (MIMO-FAS) faces two inherently coupled challenges: acquiring accurate high-dimensional channel state information (CSI) from limited RF chains and solving the combinatorial port selection problem, where the effectiveness of the latter highly depends on the result of the former. In this paper, we propose a unified two-stage diffusion framework that formulates the joint task as a maximum-a-posteriori (MAP) inference problem equivalent to maximizing the posterior expected utility of the port selection, and approximates it through two sequential stages. For Stage I, we employ a continuous flow-based diffusion model as a powerful implicit prior for 2D FAS channels and perform approximate posterior sampling via a parallel guided generation scheme, enabling accurate multiuser channel recovery even under severely low sub-sampling ratios. For Stage II, we train a discrete diffusion model on optimal labels computed by exhaustive search to propose a pool of promising port selection candidates. The final selection is then determined by maximizing the empirical posterior expected utility over this pool, followed by a lightweight local refinement. Extensive simulations demonstrate that the proposed framework simultaneously achieves accurate channel estimation and delivers over 90% of the deployed rate of the computationally expensive exhaustive search over the estimated channel, while substantially improving the minimum achievable rate relative to practical cascaded schemes.

This paper studies frames in Hilbert spaces generated by the orbits of finitely many vectors under a single operator, presenting new results on normal operators and multiplication operators as well as on operators composed of Jordan blocks. These results generalize existing results of Cabrelli, Molter, Paternostro and Philipp by means of techniques which deal with weighted interpolation, weighted composition operators, and Beurling–Lax theory related to shifts of finite multiplicity. Likewise, we discuss Carleson frames and construct counterexamples to a recent conjecture of Aldroubi, Cabrelli, Krishtal and Molter, showing that the relevant necessary condition is of density type rather than the Müntz–Szász condition. Moreover, by using the theory of weighted composition operators, we answer a question of Cheshmavar on frames produced by iterative multiplication. Finally, for orbits of infinitely many vectors the underlying model is a shift of infinite multiplicity which, by a theorem of Rota, is a universal operator; consequently, a complete description of such frames along these lines would encompass the Invariant Subspace Problem.

Recent work in random matrix theory (RMT) has developed the notion of deterministic equivalents: typically linear surrogate models that approximate the spectral behavior of large nonlinear random matrices, such as nonlinear feature maps in neural networks (NNs). Such equivalents make theoretical predictions tractable by reducing a complex model to a simpler one with properties that fall under the umbrella of classical RMT tools. However, this leaves open the question of whether this idealized linear equivalence remains meaningful for classification of high-dimensional nonlinearly separable data. Motivated by this, we consider the conjugate kernel (CK), which is the nonlinear feature map of a one-layer feedforward NN, under a canonical nonlinearly separable dataset for the XOR problem; and we use the study of informative outlier eigenvalues in the CK and whether their corresponding eigenvectors asymptotically align with XOR labels as a proxy for nonlinear learnability. We develop a robust quadratic equivalent of the CK matrix that enables a precise analysis of emergent informative spikes, as one modifies various knobs common in ML practice: sample complexity, signal-to-noise ratio (SNR), nonlinear activation choice, and pretrained features. We identify regimes in which these knobs move the CK beyond the linear equivalent and produce BBP-type transitions to label-aligned outlier eigenspaces. Our analysis helps bring deterministic-equivalence tools from RMT to bear on problems of practical relevance in ML.

Martingales, Markov processes and Laws of Large Numbers have been well studied in the Riesz space (vector lattice) setting. There has, however, been no attention given in the Riesz space setting to Laws of Small Numbers or to the so called Stein-Chen method. Here we adapt the Stein-Chen method to the Riesz space setting and hence give a conditional Laws of Small Numbers for Bernoulli processes in Riesz spaces. This requires extensive use of functional calculus and the associated f-algebra structure.