Papers

The paper proposes an approach for verifying integral persistent excitation, which is important in problems of parameter identification and adaptive control in nonlinear dynamical systems. The approach works for conservative polynomial ODEs a priori without knowledge of the parameters. Rigorous proofs of the corresponding theorems are provided. An example of a nonlinear dynamical system with higher-order interactions and the application of the proposed method to it are analyzed. The proof of the main result is formalized in the Lean formal verification language.

Steinberg described the group of components of the centralizer of a semisimple element of a connected semisimple algebraic group $G$ as a subgroup of the fundamental group of $G$. We show that this description can be generalized to explain the fact that centralizers of unipotent elements can fail to be reduced when the universal cover of $G$ is not étale. As applications, we compute generic multiplicities in the special fibers of moduli spaces of L-parameters and universal deformation rings, and we show there is no Springer isomorphism for $\mathrm{PGL}_p$ in characteristic $p$.

We study the exact partition function of 4d $\mathcal N=1$ supersymmetric gauge theories on a torus times a cylinder $\mathrm{Cyl}=I\times S^1$, where $I$ is a finite interval carrying two boundary components. Each endpoint supports an independent Dirichlet or Robin-like boundary polarization, so that the partition function is a boundary-to-boundary elliptic kernel. We construct the rigid supersymmetric geometry, determine the BPS locus, and compute the chiral-multiplet 1-loop determinants for the four possible boundary polarizations via equivariant localization. The resulting elementary building blocks are theta functions dressed by cubic phases. We then prove rank-changing Seiberg-type dualities as identities of Jeffrey–Kirwan residues of these elliptic kernels. We also discuss factorization into holomorphic-block cap wavefunctions represented by elliptic Gamma functions, dimensional reductions to three and two dimensions, complete-intersection gauged linear sigma models, and elliptic kernels for 4d $\mathcal N=4$ super Yang–Mills and the Klebanov–Witten theory, useful for holographic applications.

The Boolean lattice $BL_n$, $n\geq 3$, is the graph whose vertex set is the collection of all subsets of $[n]=\{1,2,\ldots,n\}$, where two subsets $U$ and $W$ are adjacent if and only if their symmetric difference has precisely one element. In the graph $BL_n$, the layer $L_k$ is the family of all $k$-element subsets of $[n]$. The subgraph $BL_n(k-1,k)$ is the induced subgraph of $BL_n$ on layers $L_{k-1}$ and $L_{k}$. This graph is bipartite and, when $n=2k-1$, is $k$-regular and isomorphic to the bipartite double cover $2{\cdot}O_k$ of the odd graph $O_k$. In this paper, we determine the full adjacency spectrum – eigenvalues together with their multiplicities – of the line graph $L(BL_n(k-1,k))$ for all admissible values of $n$ and $k$. As a consequence, we show that $L(BL_n(k-1,k))$ is an integral graph whenever $n = 2k-1$, and we recover as a special case the spectrum of the line graph $L(n)$ of $BL_n(1,2)$ established by Mirafzal~\cite{pap-sm-1}.

In their study of the ring of integer-valued polynomials in non-commutative algebra, Peruginelli and Werner characterized the algebras for which this ring is a Prüfer domain. Here, we apply their results to the case of group algebras.

In this paper, a fractal–fractional HIV model with the Mittag–Leffler kernel is proposed using the Atangana–Baleanu–Caputo operator to capture the memory and hereditary properties of the disease dynamics. The existence and uniqueness of the solutions are investigated using suitable analytical techniques, and the Hyers–Ulam stability analysis is carried out to verify the stability behavior of the proposed system. For the numerical simulations, the Newton polynomial approximation method together with the Atangana–Toufik numerical scheme is employed to obtain approximate solutions for different parameter settings. Furthermore, several visualization techniques, including sensitivity heatmap representation and tornado diagram analysis, are utilized to study the influence of model parameters on the HIV dynamics. The obtained numerical results demonstrate that the proposed fractal–fractional framework provides an effective and reliable approach for analyzing the transient and long-term behavior of HIV transmission dynamics.

This paper studies energy efficient tracking of power-limited mobile users with the assistance of a Reconfigurable Intelligent Surface (RIS). Since localization pilot transmissions dominate the energy budget of power-constrained devices, we introduce a low-overhead feedback link from the Base Station (BS) to the user to enable dynamic uplink power control. To navigate the discrete and decentralized nature of this active sensing problem, we propose a novel Dual-Agent (DA) deep learning framework that jointly optimizes the discrete RIS phase profiles and the UE's transmit power in real time. Specifically, our approach employs a hybrid training methodology integrating the neuroevolution paradigm with supervised learning, effectively overcoming the non-differentiability of discrete phase responses from the RIS unit elements and the strict information bottleneck of single-bit feedback messages for pilot power control. The proposed DA active sensing framework can be applied with both single- and multi-antenna BSs, the latter with only minor modifications in the structure of one NN: an additional output branch with appropriate structure is included for the latter case to select a valid digital combiner from a finite set. Extensive numerical simulations demonstrate that the proposed scheme achieves highly accurate and robust tracking across diverse target motion models, outperforming extended Kalman and particle filters, as well as, machine learning-based trackers. Furthermore, in static localization, it is shown to significantly outperform traditional fingerprinting schemes, deep reinforcement learning baselines, and standard backpropagation-based estimators.

In this work we generalize two recently proved intersection theorems for DG-rings. The Derived Improved New Intersection Theorem concerns the length of semi-free DG-modules over DG-rings and it was recently proved by the second author. We show that it holds under weaker hypotheses. Foxby's Intersection Theorem was generalized to DG-rings by Yang and we improve the inequality that they provided. As an application we prove a DG version of the classic result that finite length modules of finite projective dimension only exist over Cohen-Macaulay rings, generalizing another result of Yang.

We prove the sharp diagonal spectral correlation conjecture of Friedgut, Kahn, Kalai and Keller, proposed in their Fourier-analytic approach to Chvátal's conjecture. For every pair of increasing Boolean functions $f,g:\{0,1\}^n\to\{0,1\}$, $$\mathrm{Cov}(f,g)\ge4\sum_{\varnothing\ne S\subseteq[n]}|S|\hat{f}(S)^2\hat{g}(S)^2.$$ Thus covariance controls the degree-weighted collision of the two nonconstant Fourier spectra, giving a sharp Fourier strengthening of the Harris–Kleitman inequality. The theorem also implies the unweighted diagonal conjecture of Friedgut–Kahn–Kalai–Keller for an increasing family and a maximal intersecting family. The factor $4$ is optimal, and we determine all equality cases. Apart from pairs whose relevant coordinate sets are disjoint, equality occurs only for a common dictatorship and, up to relabelling coordinates and interchanging $f$ and $g$, for the two-coordinate AND-OR pair $(f,g)=(x_i x_j,\,x_i\vee x_j).$ The main novelty is a correlated four-restriction induction and a sharp endpoint convolution inequality. The usual two-restriction induction behind Harris–Kleitman sees only the parallel restricted pairs and loses the mixed Fourier information needed to control the degree-weighted diagonal spectral energy. We instead couple the four codimension-one restricted pairs with correlation $1/2$; this precise correlation extracts the missing degree-weighted energy as a nonnegative square.

Consider the following online filling game. An array of length $n$ is initially empty. At each time step one observes an independent sample from $\mathrm{Unif}[0,1]$ and must either discard it or place it irrevocably into an empty position of the array, while preserving the constraint that the occupied entries are non-decreasing from left to right. Among all possible strategies, what is the optimal expected time required to fill the array? Let $v_n$ denote this optimal expected completion time. Our main result determines $v_n$ up to lower-order terms: \[ v_n=\left(\frac12+o(1)\right)n\log n. \] More precisely, no strategy, even if randomized and adaptive, can have expected completion time below $\left(\frac12-o(1)\right)n\log n$, while we provide an explicit deterministic strategy whose expected completion time is at most $\left(\frac12+o(1)\right)n\log n$. For comparison, the natural coupon-collector strategy, which partitions $[0,1]$ into $n$ equal intervals and reserves one array position for each interval, has expected completion time $(1+o(1))n\log n$. We also consider a with-replacement version of the game, in which previously placed entries may be overwritten. For this variant, we give a deterministic strategy with expected completion time $O(n\sqrt{\log n})$, thereby establishing a separation between the two models.

We study dynamic random graphs in which the set of nodes is fixed, but edges evolve over time according to an underlying stochastic mechanism. Using a maximum-entropy approach, we define a probability distribution on graph trajectories that is consistent with observed constraints, capturing the inherent uncertainty in partially observed networks. We introduce a moment-based estimator for the parameters of this distribution and establish its statistical properties, such as consistency and asymptotic normality, with explicit formulas for the covariance structure. Numerical experiments demonstrate the estimator's accuracy and robustness across various dynamic network scenarios. Our framework bridges probabilistic modeling and statistical inference in time-varying networks, providing practical tools for understanding and predicting complex edge dynamics.

For a remote estimation system, we study the optimization of age of incorrect information (AoII), which is a recently proposed semantic-aware information freshness metric. In particular, we assume an information source that observes a discrete-time finite-state Markov chain (DTMC), and occasionally transmits status update packets to a remote monitor which is tasked with remote estimation of the source. For the forward channel from the source to the monitor, we assume the channel delay to be modeled by a general discrete-time phase-type (DPH) distribution, whereas the reverse channel from the monitor to the source is assumed to be perfect, ensuring that the source has perfect information on the AoII and the remote estimate at the monitor, at all times. Push-based transmissions are initiated when AoII exceeds a threshold depending on the current estimation value, i.e., multi-threshold policy. In this very general setting, our goal is to minimize a weighted sum of the time average of a polynomial function of AoII, depending on the remote estimate, and energy consumption from transmissions. We formulate the problem as a semi-Markov decision process (SMDP) with the same state-space of the original DTMC to obtain the optimal multi-threshold policy, whereas the parameters of the SMDP are obtained by using a novel stochastic tool called dual-regime absorbing Markov chain (DR-AMC), and its corresponding absorption time distribution named as dual-regime DPH (DR-DPH). The proposed method is validated with numerical examples using comparisons against other policies obtained by exhaustive search, and also various benchmark policies.

Stochastic Gradient Descent ($\textsf{SGD}$) is one of the most classical optimization algorithms with favorable theoretical guarantees, yet the practical implementation of $\textsf{SGD}$ differs subtly from its well-known form and is often referred to as Shuffling Stochastic Gradient Descent ($\textsf{Shuffling SGD}$). A particularly popular strategy in $\textsf{Shuffling SGD}$ is Random Reshuffling ($\textsf{RR}$), which has achieved great empirical success across numerous experiments. Despite its strong performance, $\textsf{RR}$ has long been considered a heuristic due to a lack of theoretical support. Over the last decade, people have finally established provable convergence rates for $\textsf{RR}$, thus justifying its observed superiority. However, for smooth convex optimization, two clouds over the convergence theory of $\textsf{RR}$ remain to this day. More precisely, according to the current theory, $\textsf{Shuffling SGD}$ under $\textsf{RR}$ converges only when the stepsize is smaller than a threshold proportional to $1/n$, where $n$ is the number of summands in the objective (or the number of data points). Consequently, the optimally tuned theoretical rate of $\textsf{Shuffling SGD}$ under $\textsf{RR}$ is strictly worse than that of $\textsf{SGD}$ when the number of epochs is smaller than another threshold proportional to $n$. These two restrictions heavily limit the applicability of existing theories and leave a critical mismatch with practice. In this work, for the first time, we prove that $\textsf{RR}$ dominates $\textsf{SGD}$ in smooth convex optimization under any reasonable stepsize after any finite number of epochs, thereby addressing a longstanding open question.

We study hard-erasure belief-propagation (BP), understood as the sum-product algorithm on erasure messages, for a punctured MacKay-Neal/Hsu-Anastasopoulos (MN/HA) representation of Calderbank-Shor-Steane (CSS) codes. For every integer degree triple $2\leq j_Z<j_X<k\leq60$, we prove that the constituent potential thresholds are $j_Z/k$ and $1-j_X/k$. The proof combines classical HA-MN duality with a completed exact rational certificate for all 1711 MN degree pairs in this range. The certificate establishes strict positivity on every physical nontrivial fixed-point branch, at every erasure probability in $[0,1]$, and has an independent integer-arithmetic verifier. A reduction of the Z-side recursion and a fixed-channel vector-potential argument then prove seeded density-evolution convergence below the smaller constituent threshold for sufficiently large coupling width and an ideal seed interval at least that wide. Under $j_Z+j_X=k$, this threshold equals the quantum-erasure hashing-bound parameter determined by the CSS design rate. In particular, the degree triple $(4,8,12)$ has potential threshold $1/3$ without a positivity hypothesis. The numerical example at erasure probability $0.3325$ reproduces the resulting decoding waves. The theorem uses an ideal auxiliary-message seed; its finite-code realization and logical block-error convergence are separate questions.

We investigate the field of a point-like electric charge freely falling in a gravitational wave. In the presence of a gravitational wave, the initially static Coulomb field of the charge becomes time-dependent and generates corresponding radiation. The gravitational wave is treated as a weak perturbation of the Minkowski metric. The electromagnetic four-potential of the charge is sought as a solution to Maxwell's equations in the gravitational wave metric, to first order in perturbation theory. The potentials of the point charge are found in quadratures throughout the space. To regularize the potentials, an approach motivated by the Shapiro effect for the time delay of radiation in a gravitational field is used. The potentials of the charge in the far zone are calculated explicitly for a monochromatic, arbitrarily polarized gravitational wave. The angular distribution of the electromagnetic radiation induced by the gravitational wave is obtained.

High-order accurate simulations of special relativistic hydrodynamics (RHD) are prone to numerical breakdown if intrinsic physical constraints (positive rest-mass density/pressure and subluminal velocity) are violated near strong discontinuities. In this work, we develop a robust and efficient physical-constraint-preserving (PCP) flux-limiting framework for high-order schemes, using finite-difference WENO as a representative example. By leveraging the geometric quasilinearization (GQL) representation, which equivalently reformulates the nonlinear RHD constraints into a family of linear inequalities, we integrate a Zalesak-type Flux-Corrected Transport (FCT) update into a scalar-style limiter that acts directly on conservative variables. A critical innovation is the explicit, non-iterative determination of limiting parameters via a rational stereographic parameterization of the GQL normal vector. This technique transforms the required worst-case minimization over auxiliary variables into a generalized Rayleigh-quotient formulation, allowing the optimal parameters to be obtained by solving small symmetric eigenvalue problems ($2\times2$ in 1D; $(d+1)\times(d+1)$ in $d$ dimensions). Relaxed variants are further introduced to reduce computational costs in multidimensions while retaining the PCP guarantee. Extensive numerical benchmarks ranging from 1D to 3D, including ultra-relativistic Riemann problems and astrophysical jets, demonstrate that the proposed method robustly enforces physical admissibility, sharply resolves discontinuities, and maintains design-order accuracy for smooth solutions.

Semantic communication (SemCom) has emerged as a promising paradigm in which the transmission of task-relevant information is prioritized over raw data, enabling efficient and robust communication under resource and channel constraints. In this paper, the privacy implications of relay-assisted SemCom systems are studied, where the intermediate relay node operates directly on learned latent representations. It is shown that the relay, even without access to source data, can reliably infer semantic meaning and reconstruct signals with performance comparable to that of the legitimate receiver, revealing a fundamental privacy vulnerability of semantic representations. To address this issue, an iterative adversarial training framework is proposed in which a strong, adaptively trained eavesdropper at the relay is explicitly accounted for. The proposed approach alternates between optimizing the relay's eavesdropping function and the legitimate system, resulting in representations that preserve semantic decoding performance at the intended receiver while degrading semantic inference at the relay. The semantic accuracy gap between the legitimate receiver and the eavesdropper is significantly enlarged across channel conditions. Importantly, this protection is achieved in a stealthy manner, with high reconstruction fidelity maintained while semantic leakage is selectively suppressed.

Insurance payments may depend on latent micro states although only macro states and realized payments are observed. We study a sojourn-payment model for such aggregated multi-state systems under left-truncation and right-censoring. Starting from a micro-to-macro projection, we establish strong consistency and weak convergence for inverse-probability-weighted estimators of state-specific cumulative payment processes.

An intuitive scheme to assign a temperature to an arbitrary state of a quantum system is to investigate the heat flow resulting from the coupling to a thermometer. We introduce a simple model of a universal thermometer with the following property. When it is prepared in a Gibbs equilibrium state at inverse temperature $\beta\in\mathbb R$ and brought into thermal contact with a system in any state, the heat flow between the system and thermometer vanishes for a unique value of $\beta$. We call this value the contact temperature $\beta_{\rm op}\in\mathbb R$ of the system state. The thermometer is universal in that it yields a unique contact temperature for arbitrary states of finite dimensional quantum systems.

The generalized Burgers-Huxley (GBH) equation is a prototype model that describes the interplay among reaction, convection, and diffusion. In this article, we explore the controllability of this model by means of an interior control supported in an arbitrary non-empty open subset of the domain. We establish that the GBH equation is not globally approximately controllable in a given time. However, it is possible to steer the system from any steady state to an arbitrarily small neighborhood of another steady state in some suitable time by means of a localized interior control, provided that both steady states lie in the same connected component of the set of steady states.

Let $k_r(G)$ denote the number of $r$-cliques in a graph $G$ and let $F_r(\cdot)$ be the Lovász–Simonovits $r$-clique density function. For any integers $2\le s<t$, we determine the asymptotically sharp lower bound on $k_t(G)$ in an $n$-vertex graph $G$ with a prescribed number $k_s(G)$, by showing that \[ \frac{k_t(G)}{n^t}\ge F_t\!\left(F_s^{-1}\!\left(\frac{k_s(G)}{n^s}\right)\right), \] where $F_s^{-1}$ denotes the generalized inverse. This strengthens Bollobás's piecewise-linear interpolation bound and, in the case $s=2$, recovers Reiher's clique density theorem via a new inductive proof.

Motivated by applications of algebraic geometry to reconstruction problems in computer vision, we initiate a study of the equations of degeneracy loci associated with linearly dependent points on Segre varieties. When these points are constrained to lie on a common hyperplane, we prove that the vanishing ideals of these loci are prime, Cohen-Macaulay, and generated by the natural maximal minors, and that these minors form a universal Gröbner basis.

We introduce the Beckmann boundary of a Boolean function \[ \mathsf{B}(f)=\inf_{\operatorname{div} V=Lf}\mathbb E\|V(x)\|_2. \] Here \[ L=\sum_iD_i,\qquad D_i f(x)=\frac{f(x)-f(x^{\oplus i})}{2}, \] and $\operatorname{div} V(x)=\sum_i (V_{i}(x)-V_{i}(x^{\oplus i}))$. This nonlocal quantity is no larger than the usual two-sided, one-sided, colored, optimized colored, or optimized fractional colored boundaries. Nevertheless, every nonconstant Boolean $f$ satisfies \[ \mathsf{B}(f)\gtrsim \operatorname{Var}(f) \sqrt{\log\!\left(1+\frac{1}{\sum_i\operatorname{Inf}_i(f)^2}\right)}. \] We also prove strong one-sided fractional spectral estimates. If $A\subset\{-1,1\}^n$ and \[ h_{A}(x)=\#\{i:x\in A,\ x^{\oplus i}\notin A\}, \] then, for $0<\alpha<1$, \[ \sum_{S\ne\varnothing}|S|^\alpha\widehat{\mathbf 1_{A}}(S)^2 \lesssim_\alpha \mathbb E\omega_\alpha(h_{A}), \] where $\omega_\alpha(m)=\sqrt m$ for $\alpha<1/2$, $\omega_{1/2}(m)=\sqrt m\log(e+m)$, and $\omega_\alpha(m)=m^\alpha$ for $\alpha>1/2$. These profiles are sharp, up to $\alpha$-dependent constants, for majority. We also show that the comparison is genuinely nonreversible: an explicit quotient-cube family makes the optimized fractional, and hence optimized colored, boundary exceed $\mathsf{B}$ by a factor $\gtrsim\sqrt{\log n}$. We further obtain a driftless Bernstein-multiplier inequality.

We study the scalar proximity operator of the nonconvex power penalty with exponent between zero and one. Lagrange–Bürmann inversion gives an exact series for the larger stationary root, and the series converges uniformly throughout the active domain of the proximity operator. For rational exponents, an explicit residue-class decomposition reorganizes the coefficients into finite sums of generalized hypergeometric functions, recovering the familiar special cases and producing an additional formula for \(p=\tfrac13\). A Mellin–Barnes formula yields an equivalent residue-corrected representation and analytic truncation bounds. We distinguish the proximal activation threshold from the lower saddle-node threshold and prove that the active proximal branch is uniformly well-conditioned; the practical difficulty near activation is slow series convergence when the exponent approaches one, rather than singular root sensitivity. We also formulate an inexact proximal-gradient step using a computable optimality residual and a descent test. With summable descent errors and vanishing residuals, the objective values converge, the steps vanish, and every cluster point is critical. Tests over a broad parameter grid and in sparse-recovery proximal-gradient runs illustrate both evaluator accuracy and the implementability of the acceptance certificates.

We give algebraic models for the tame homotopy type of the configuration spaces of certain algebraic varieties of Tate type. Such tame models carry information on the l-adic homotopy type. Our method uses the theory of weights in étale cohomology, and also produces models for more general arrangement complements, both in the tame sense and over the rationals.

In this paper, we establish the relation between the Ekedahl-Kottwitz-Oort-Rapoport stratification and the Bruhat-Tits stratification on basic unramified $\mathrm{GU}(1,n-1)$ Rapoport-Zink spaces with arbitrary parahoric level. More precisely, we prove that every basic EKOR stratum is a disjoint union of copies of an explicitly defined fine Deligne-Lusztig variety. Using this comparison and the closure relations for fine Deligne-Lusztig varieties, we give an explicit combinatorial description of the closure of every basic EKOR stratum. As further applications, we determine which KR strata are entirely contained in the basic locus, and we prove the smoothness of the irreducible components of the closure of certain EKOR strata.

We develop maximal inequalities for empirical processes indexed by graph-dependent observations. Our bounds separate the complexity of the indexing class from two features specific to graph dependence: the geometry of the underlying graph and the cost of coupling graph-separated blocks to independent copies. The coupling construction combines a novel graph-adapted dependence coefficient with a coloring of a block partition. As an application, we derive Glivenko–Cantelli results and characterize the associated effective sample size. A central implication is that graph-dependent empirical processes need not exhibit a generic root-$n$ rate: convergence is jointly determined by function-class complexity, graph geometry, and the decay of dependence with graph distance. We specialize the results to graphs with polynomial and exponential growth and to directed dyadic graphs. Finally, we apply the results to network autoregressive models, nonlinear local-propagation models, and treatment-interference settings.

This paper establishes the first rigorous superconvergence theory for semidiscrete and fully discrete central discontinuous Galerkin (CDG) methods for linear hyperbolic equations on overlapping meshes. While the optimal $L^2$ convergence of $\mathbb{Q}^k$ CDG schemes was established on uniform Cartesian meshes by Liu, Shu, and Zhang [ SIAM J. Numer. Anal.}, 56 (2018), pp. 520–541], their observed $\mathcal{O}(h^{k+2})$ pointwise superconvergence has remained unproven, due to the loss of standard single-mesh Galerkin orthogonality inherent in the CDG overlapping structure. To overcome this fundamental barrier, we introduce a projection-correction framework that identifies a hidden superconvergent mechanism: an asymptotic weak residual cancellation in one dimension, and a high-order cancellation-by-aggregation (HOCA) mechanism in multiple dimensions. This HOCA approach overcomes the analytical challenge posed by coupled primal-dual directional residuals, recovering critical error cancellation properties absent from the standard variational formulation. Consequently, we provide the rigorous proof of the conjectured $\mathcal{O}(h^{k+2})$ pointwise superconvergence in the discrete $\ell^{\infty}$ norm across all superconvergent points. Furthermore, we reveal that under a systematically corrected initialization, this framework yields a previously undiscovered, stronger cell-average superconvergence estimate of order $\mathcal{O}(h^{\min\{2k+1,k+3\}})$. The theory is extended to fully discrete explicit Runge–Kutta CDG schemes, where stagewise corrected errors are constructed to preserve spatial superconvergence up to temporal truncation errors, yielding a stable reconstruction-based postprocessing estimate. Numerical experiments in one and two spatial dimensions confirm the sharpness of the theoretical rates.

We introduce a class of separable II$_1$ factors $M$ admitting no non-trivial crossed product decompositions: $M\not\cong B\rtimes_\sigma G$, for any trace preserving action $G\curvearrowright^\sigma (B,\tau)$ of an infinite countable group $G$ on a tracial von Neumann algebra $(B,\tau)$. These provide the first examples of II$_1$ factors that do not arise as crossed products of noncommutative dynamical systems. Our approach relies on a novel construction of separable II$_1$ factors $M$ whose embeddings into their tensor product square $M\overline{\otimes}M$ all arise from the canonical embeddings $x\mapsto x\otimes 1$ and $x\mapsto 1\otimes x$.

We study the solvability of some linear nonhomogeneous elliptic problems and establish that under certain technical assumptions the convergence in $L^2$ of their right-hand sides yields the existence and the convergence in $H^4$ of the solutions. The equations contain fourth order differential operators with or without the Fredholm property, in particular the second and the fourth derivative operators, on the whole real line or on a finite interval with periodic boundary conditions. We establish that the transport term involved in these problems provides the regularization of the solutions.