Papers

Subpacketization remains a major obstacle to the practical deployment of coded caching (CC) in multi-antenna wireless networks. In this paper, we propose a low-complexity multiple-input multiple-output (MIMO) CC scheme that enables flexible delivery rate adaptation while substantially reducing subpacketization requirements. The proposed design employs a virtual broadcast-channel decomposition to align cache-profile placement with flexible MIMO stream elevation which extends the shared-cache model to multi-antenna receivers. This enables flexible selection of feasible user and stream configurations, thereby providing explicit control over the spatial multiplexing gain under linear decodability constraints. Analytical results show that the proposed framework can asymptotically approach the best-known achievable degrees of freedom (DoF) under linear decodability constraints while requiring orders-of-magnitude lower subpacketization than existing schemes. Numerical evaluations further demonstrate that this flexibility yields notable throughput improvements at practical signal-to-noise ratios.

The integer lattice $\mathbb{Z}^n$ is conjectured to maximize the Gaussian mass $\Theta_L(t)=\sum_{x\in L}e^{-t\|x\|^2}$ over the set of stable lattices in $\mathbb{R}^n$, for every $t>0$. We prove this sharp inequality for every integral unimodular lattice $L$ of rank $n\leq 32$, with equality only at $L\cong\mathbb{Z}^n$, and furthermore obtain the strict inequality for every even unimodular lattice of rank $40$. The proof does not use the classification of unimodular lattices in these ranks; rather, it parametrizes integral unimodular theta series as polynomials in the modular function $u=\Delta_8/\vartheta_3^8\in(0,1/64]$, with the few coefficients that arise controlled by norm-$1$ splitting, ADE root counts, and shadow positivity.

In this work we study the Schwinger-Dyson equations and saddle point equations of matrix integrals that come from type $(0,1)$ random fuzzy geometries coupled to fermions or bosons. Such random fuzzy geometries are bi-tracial Hermitian matrix ensembles with a determinant contribution in the integrand. We derive the Schwinger-Dyson equations using complex analytic techniques from the saddle point equation. For arbitrary potentials with either bosonic or fermionic contributions, their Schwinger-Dyson equations can be solved iteratively. For both the Gaussian models with either one boson or fermion we rigorously derive the formula for the free energy and first moment in terms of elliptic integrals. In the bosonic case this solution is closely related to the Hoppe model and the three-colour model.

The Kleshchev multipartitions arise in the representation theory for the Ariki-Koike algebras. In previous work, Li, Stanton, Xue, Yee, and the author considered a refined enumeration for the $2$-dimensional case, namely, the Kleshchev bipartitions, by invoking the $2$-residue statistic for partitions. In this paper, we make further elaboration by bounding the largest part of the bipartitions and show that the related counting functions are connected with two families of $q$-trinomial coefficients introduced by Andrews and Baxter.

We study the limitations of matrix-free algorithms that access a matrix $A$ only through forward matrix-vector products (matvecs) $x \mapsto Ax$, without access to the transpose $A^\top$ or its action. This setting arises naturally in operator learning, inverse problems, and matrix-free PDE solvers, where adjoint evaluations may be unavailable or prohibitively expensive. We show that the lack of transpose access creates severe and sometimes insurmountable theoretical barriers. For Krylov methods, we prove that the sequence of projected operator norms produced by Arnoldi iteration can follow any prescribed nondecreasing curve, showing that forward matvecs alone provide essentially no reliable information about the spectral norm. For several core problems, including least squares, norm estimation, column subset selection, and local maximum volume, we establish non-identifiability results; distinct matrices can generate identical forward-query transcripts while having fundamentally different solutions. We also prove quantitative lower bounds on the number of forward matvecs required for approximation tasks. In particular, any algorithm that computes a near-optimal rank-$k$ approximation must use at least $n$ queries, and estimating the Frobenius norm to relative accuracy $\eps$ requires $\Omega(\eps^{-2})$ queries when $n$ is sufficiently large, matching the complexity of Hutchinson-type estimators up to constants. Although some problems remain solvable without transpose access, the transpose-free setting is fundamentally more limited in both identifiability and efficiency.

We investigate the ill-posedness mechanism of the hard-sphere Boltzmann equation in $H_{x}^{s}$ Sobolev space. Via a direct construction, we prove a strong-weak type ill-posedness result in the low-regularity regime $s<1$, establishing a sharp threshold in connection to the local $s>1$ well-posedness result [11]. Instead of originating from the large-velocity growth of the collision kernel, this illposedness is generated by the loss term and dispersive effects. Consequently, we prove a dispersion-driven nonlinear instability mechanism for the hard-sphere Boltzmann equation, and provide a capstone of the ill-posedness series [18,20].

The evolution toward 6G wireless networks envisions a seamlessly intelligent, Open-RAN-enabled architecture where unmanned aerial vehicles (UAVs) play a pivotal role in extending coverage, enhancing resilience, and ensuring reliable connectivity for ground users deployment. However, efficiently managing spectrum and resources in such highly dynamic UAV-assisted environments remains a major challenge due to nonlinear system interactions, mobility-induced topology variations, and stringent latency and energy constraints. To address these challenges, we propose a digital twin (DT)-assisted adaptive deep reinforcement learning (DRL) framework that enables intelligent spectrum sharing and resource allocation across distributed ground users. The complex optimization problem is decomposed into UAV trajectory optimization using particle swarm optimization (PSO) and dynamic spectrum-power-association management via multi-agent DRL (MADRL). This hybrid DT-driven approach empowers intelligent, context-aware decision-making and adaptive coordination among UAVs. Extensive simulations demonstrate significant gains in spectral efficiency, data rates, and energy utilization, showcasing a transformative path toward self-evolving, autonomous 6G UAV and ground users (GUs) connectivity.

We consider time-dependent Kohn-Sham equations in dimension $3$ with a fractional dispersion relation $(1-\Delta)^s$, $s\in(0,\frac32)$, and a class of interaction terms including, in particular, external potentials, internal potentials associated to Hartree-type non-linearities, and exchange terms described by energy subcritical pure-power non-linearities. We prove the local existence of weak solutions in $H^s$ using an approximation procedure regularizing the non-linearities. Assuming that the interaction energies can be controlled by the kinetic energy, we show that the solutions can be extended to global solutions using energy estimates. If $s\in[1,\frac32)$, we establish in addition the well-posedness of the time-dependent Kohn-Sham equations using Strichartz estimates.

We present a non-convergence theory for probabilistic direct search, a randomized derivative-free optimization method, where non-convergence means the failure to produce iterates that achieve stationarity asymptotically. The motivation is to understand whether the submartingale-like assumption in the existing convergence theory is essential or merely an artifact of the analysis techniques. For convex objectives, we prove that the probability of non-convergence is positive, provided that the polling directions satisfy a probabilistic ascent condition that is essentially the opposite of the submartingale-like convergence condition. Furthermore, we establish a lower bound for the non-convergence probability. For the typical implementation of this method, where each iteration draws a fixed number of random polling directions independently and uniformly from the unit sphere, our theory implies that the method is not globally convergent if the number of directions is below the threshold specified in the convergence theory, and the submartingale-like assumption is confirmed to be essential for convergence. Our theory is obtained by examining two random series that control the distance from any iterate to the starting point and estimating the probability for these series to stay below certain bounds.

In this note, we show that if the Bergman metric of a pseudoconvex domain in $\mathbb C^n$($n\geq 3$) has constant scalar curvature, then every strongly pseudoconvex boundary point of the domain is spherical.

For an algebraic function field $F$ over a large field $K$, we show: 1) if $F|K$ has a rational place, then there is a finite purely inseparable extension $K'|K$ such that $K'$ is existentially closed in $F.K'$; 2) $F|K$ has a rational place admitting local uniformization if and only if $K$ is existentially closed in $F$.

We develop a multicontinuum Generalized Multiscale Finite Element Method (MC-GMsFEM) for constructing coarse-scale models in heterogeneous media that simultaneously provide accurate numerical approximations and physically consistent macroscopic equations. Classical multiscale methods efficiently approximate fine-scale solutions on coarse grids using localized basis functions, but they do not offer a systematic pathway for deriving macroscopic governing equations. To overcome this limitation, we introduce a unified framework that integrates multiscale finite element constructions with multicontinuum representations. The proposed method builds on the structure of GMsFEM and exploits a representation of multiscale basis functions that separates coarse variables and their gradients. We construct continuum-dependent basis functions using auxiliary fields defined through local problems with integral constraints, ensuring that each basis function is associated with a specific continuum. This leads to a decomposition of the coarse-scale solution into continuum variables and their gradients, establishing a direct connection between multiscale discretizations and multicontinuum homogenization. Compared to existing multicontinuum approaches, the proposed framework provides greater flexibility in handling heterogeneous media with spatially varying numbers of continua and is naturally embedded within a standard finite element setting. This enables both systematic derivation of macroscopic equations and straightforward numerical implementation. We apply the proposed method to the upscaling of two-phase immiscible flow in heterogeneous porous media, where multiple interacting continua, including mobile and trapped phases, arise. With the proposed approaches, we derive new macroscopic models and show that if classical models are used, the errors can be large.

We study the interaction between the group law on an abelian variety and the additive structure induced on its image under a morphism to a projective space. Let $A/F$ be an abelian variety, $f:A \rightarrow \mathbb{P}^n$ be a morphism which is finite onto its image, and $\Gamma \subseteq A(F)$ be a finite-rank subgroup. We show that for any affine chart $\mathbb{A}^n \subseteq \mathbb{P}^n$ and any finite subset $X \subseteq f(\Gamma) \cap \mathbb{A}^n$, the energy satisfies $E(X) \ll \lvert X \rvert^2$ and the sumset satisfies $\lvert X+X \rvert \gg \lvert X \rvert^2$. Thus images of finite-rank subgroups of abelian varieties cannot have strong additive structure in affine space. This removes the simplicity assumption from the author's previous result. The proof combines the uniform Mordell–Lang conjecture of Gao–Ge–Kühne with a refined use of the Ueno locus, Rémond's boundedness theorem for abelian subvarieties of bounded degree, and induction on the dimension of $A$.

We study the behavior of Dehn functions of finitely presentable groups for presentations with finite generating sets and possibly infinite sets of defining relators. For the free abelian group $\mathbb Z^2$ of rank two on generators $a,b$, we prove that the infinite presentation $\langle a,b \mid [a^{2^k},b],\ k=0,1,2,\ldots\rangle$ has Dehn function of order $n\log n$. We also prove that, for every $0<\alpha<2$, the group $\mathbb Z^2$ admits an infinite presentation on the same two generators whose Dehn function satisfies a global upper bound $\delta(n) \le C n^\alpha + C$ and has matching $n_i^\alpha$-order lower-bound peaks along an infinite sequence of lengths $n_i$. We obtain a similar result, for all $0<\alpha<1$ for torsion-free groups $G$ admitting a finite $C'(1/6)$ small cancellation presentation on the given generators $X$. We also show that the same conclusion holds for an arbitrary finitely generated group $G$ and for some finite generating set $X$ of $G$ and for all $0<\alpha<1$. In particular, these produce continuum many distinct growth types of Dehn functions for presentations of $\mathbb Z^2$ on the standard generators $a,b$.

Semantic communication has emerged as a promising paradigm for improving transmission efficiency by conveying task-relevant semantics rather than raw data. Although recent studies have achieved notable gains in communication efficiency and average task performance, reliability remains a fundamental bottleneck in dynamic and uncertain environments. In particular, most existing designs are still optimized mainly for average-case behavior, while lower-tail performance under adverse transmission conditions remains insufficiently understood and inadequately protected. In this article, we present a unified perspective on reliable semantic communication beyond average performance. We first review three reliability-oriented design categories: channel-aware adaptation, robustness-oriented codec design, and hybrid automatic repeat request (HARQ)-based retransmission. We show that these approaches address reliability from complementary perspectives, but each still has inherent limitations. Motivated by these observations, we discuss two solution directions: robust adaptive semantic communication under imperfect CSI, and joint source-channel-check coding with adaptive retransmission for sample-level reliability enhancement. Finally, we outline several future research directions, including the joint design of robustness and retransmission, reliability metrics beyond averages, and compatibility with existing digital wireless networks.

We study the extremality properties of rays generated by tautological Chern classes of matroids in the polyhedral nef cones of the permutohedral variety. Our main result provides necessary and sufficient conditions a matroid has to meet for its tautological Chern classes to generate extremal rays. Our approach yields a new and large family of extremal rays that interpolates between two previously known classes coming from rank functions and Bergman classes of matroids. We further discuss how this theorem extends to the stellahedral variety through augmented tautological Chern classes.

We prove that Thompson's group $F$ satisfies the relative invariant subalgebra rigidity property with respect to its commutator subgroup: every von Neumann subalgebra of $L(F)$ that is invariant under conjugation by $[F,F]$ is of the form $L(N)$ for some normal subgroup $N \trianglelefteq F$. Along the way, we establish a general factoriality criterion for invariant subalgebras whose hypotheses are met whenever the ambient group is i.c.c., simple, and every faithful ergodic measure-preserving action of it on a probability space is essentially free.

The Burrows–Wheeler transform is usually viewed as a clustering transform: it tends to group equal letters into long runs. We study the opposite extremal regime, where the BWT output is completely unclustered, that is, has as many equal-letter runs as positions. Known results imply, on the one hand, that the number of runs in the BWT of a Lyndon word can increase by at most a factor of two, and, on the other hand, that over every alphabet of size at least three completely unclustered BWT images exist in every length. This leads to the extremal problem lying between these two facts. For \(k\ge3\), let \(U_k(n)\) be the minimum cyclic run number of a primitive necklace of length \(n\) whose BWT has \(n\) runs. We prove the universal lower bound \(U_k(n)\ge\lceil n/2\rceil\), reduce the sharpness problem for one-cycle BWT images \(L\) to the Hamming identity \[ \cruns(\BWT^{-1}(L))=\dH(L,\sort(L)), \] and develop a natural multiset-of-necklaces relaxation with an explicit constant-cycle correction. We compute the small values, including the exceptional value \(U_k(6)=4\), prove a parity obstruction for the Parikh vectors of sharp examples, and determine the multiset relaxation exactly. Finally, for every prime \(p\equiv5\pmod8\) for which \(2\) is a primitive root modulo \(p\), we prove sharpness in the adjacent lengths \(p-1\) and \(p\). Under the corresponding Artin-type infinitude hypothesis, this gives infinitely many adjacent sharp pairs.

Gradient-based algorithms are central to modern statistical estimation, yet their statistical analysis is often restricted to fixed-time behavior, such as convergence to a population target or fluctuations at a prescribed iteration. In many applications, however, uncertainty quantification is needed along the entire optimization path, especially when the stopping time is data-dependent or divergent. In this paper, we develop a theory for time-uniform statistical inference on gradient flows arising from empirical risk minimization. We prove a uniform central limit theorem that characterizes the deviation between empirical and population gradient flows as a continuous-time Gaussian process over the entire nonnegative real line. Building on this result, we introduce an algorithm-aware covariance estimator that evolves jointly with the gradient flow and avoids matrix inversion, resampling, or sample splitting. We show that the covariance estimator is uniformly consistent over time and use it to construct confidence intervals for the target parameter with asymptotically valid coverage. Our results connect optimization dynamics with statistical inference and provide practical tools for uncertainty quantification in gradient-based methods.

Motivated by their applications in DNA-based storage systems, codes capable of correcting consecutive deletions have attracted significant attention. An important class of such codes consists of those that can correct multiple consecutive deletion errors, commonly referred to as multiple $b$-burst deletion-correcting codes. In this paper, we investigate the fundamental limits of multiple $b$-burst deletion-correcting codes. Specifically, we first characterize several structural properties of the associated deletion balls. Then, leveraging these properties, we derive several upper bounds and a combinatorial lower bound on the maximum size of such codes. As a consequence, our bounds improve upon the previously known results for general parameter regimes and are shown to be asymptotically optimal for certain cases.

Inspired by the function-space theory of neural networks, we formulate and analyze a variation space for nonlinear operators between Hilbert spaces, defined through vector-valued Borel measures of bounded variation. We characterize its unit ball as the closed convex hull of a vector-valued single-neuron dictionary in Bochner spaces. For the ReLU activation, the bounded linear operators in this space are precisely the Schatten-$1$ operators, with equivalent norms. For operators in this space, we establish encoder–decoder approximation bounds in the Bochner $L^q$-norm, where the error decomposes into input and output encoding errors and a finite-width term of order $N^{-1/2}$. Under sub-Gaussian assumptions on the input and noise, we further derive high-probability generalization bounds for empirical least squares over path-norm-constrained encoder–decoder networks; the finite-sample contribution to the squared prediction error is of order $K^{-1/2}$ up to logarithmic factors. The finite-width and finite-sample constants are independent of the encoding dimensions and bases, with the latter also independent of the network width. When the encoding errors decay algebraically, these bounds yield algebraic approximation and learning rates, in contrast to the complexity barriers for Lipschitz and Fréchet differentiable operator classes.

We study randomized stationary methods for symmetric positive definite linear systems in which component $j$ is selected with probability proportional to $|r_j|^\ell$. This power-weighted family interpolates continuously between uniform randomized Jacobi as $\ell \to 0$ and Gauss–Southwell greedy relaxation as $\ell \to \infty$. For the central case $\ell = 2$, we sharpen the standard one-step convergence analysis using the inverse participation ratio (IPR) $\nu^2(r) = n\|r\|_4^4/\|r\|_2^4$, which equals $1$ when the residual is uniform and grows toward $n$ as it concentrates. The resulting bound amplifies the expected per-step progress by exactly $\nu^2$ over the uniform-sampling baseline. The IPR can be computed online at $O(n)$ cost and doubles as a per-iteration diagnostic. We extend the analysis to asynchronous power-weighted Jacobi via the Avron–Druinsky–Gupta framework, obtaining an epoch-based convergence theorem in which the IPR controls both the progress coefficient and the allowed-delay window. Numerical experiments on shared-memory hardware support the sharpened bound and show the IPR trajectory is essentially concurrency-insensitive. Unexpectedly, consistent-reads execution, the easier case for the ADG analysis, destabilizes power-weighted sampling at high concurrency while inconsistent reads remain stable; the same IPR that amplifies progress amplifies a thread-collision rate that inconsistent reads appear to absorb. We propose a feedback-damping mechanism and verify two predictions about its dependence on problem size.

In this paper, we consider a network transport model in which agents moving along the edges can contribute to dynamics at nodes or bypass them. The model takes the form of a system of first-order partial differential equations coupled with a system of ordinary differential equations, and can describe a range of phenomena, from diseases in metapopulations to migratory systems with delays, to cell differentiation processes, providing a unified platform that includes network transport and delay systems as particular cases. We prove the well-posedness of the model in $ L^p$ spaces, $ 1\leq p<\infty$, study long-term asymptotics, and illustrate the theory using an SIS disease in a metapopulation consisting of several sites where the disease develops, connected by routes along which the population can migrate, as an example.

Collision statistics provide a finite-resolution view of information by measuring how often a fixed number of independent samples fall on the same state. These directly countable quantities form the basis of integer-order Rényi entropies. Here, we use low-order Rényi entropies to approximate Shannon entropy and mutual information, while characterizing what is necessarily lost when only finitely many collision moments are used. We derive interpolation-error bounds showing that approximation error is controlled by the shape of the Rényi entropy path near the Shannon point. We also separate this deterministic error from finite-sample estimation error: for fixed collision order, increasing sample size improves estimation of the finite-resolution target but does not eliminate its deterministic difference from Shannon entropy or mutual information. Finally, we show that finite collision moments do not generally identify Shannon entropy, and that increasing collision order shifts sensitivity toward high-probability events. Numerical experiments illustrate the approximation–estimation tradeoff and compare collision-based approximations with plug-in and Miller–Madow estimators. The framework links collision counts, Rényi entropy, Shannon limits, and mutual information through a finite-resolution view of information, clarifying when low-order coincidence structure is informative and when irreducible information is lost.

The elementwise Hadamard product of two low-rank matrices provides a parameter-efficient model for data with multiplicative structure, but its modeling is challenging due to the presence of additional symmetries under coupled row/column scalings between the two factors. In order to leverage the geometry of the space, we formulate the learning of such matrices as optimization on a Riemannian quotient manifold. We propose a novel block-diagonal Riemannian metric derived from the pullback of the Frobenius inner product. The metric is shown to be invariant under these symmetries. We develop a Riemannian gradient descent algorithm that uses a tuning-free Gauss–Newton step size and scales linearly in the number of observed entries per iteration. The versatile framework of Riemannian quotient optimization enables both first-order and second-order Riemannian methods, the latter through a closed-form connection and the Riemannian Hessian. Experiments on real and synthetic datasets illustrate the efficacy of our proposed Riemannian approach.

We consider an open Jackson network with $n$ exchangeable single-server stations and weak all-to-all interaction through routing: upon service completion at station $i$, a job is routed to station $j$ with probability $p/n$, where $p\in(0,1)$, or leaves the system with probability $q=1-p$. We study a joint asymptotic regime in which the number of stations tends to infinity while the system approaches heavy traffic. Under the critical-load condition and diffusive scaling of time and queue length, we prove propagation of chaos for the queue-length and cumulative-idleness processes. The limiting McKean–Vlasov dynamics are described by the nonlinear reflected Brownian motion \[ \bar X(t)=\bar X_0+\bar W(t)+\hat\gamma t+\bar L(t)-p\,\mathbb{E} \bar L(t), \] where $\bar W$ is a Brownian motion with variance parameter $2$ and $\bar L$ is the reflection term at the origin. The proof proceeds by coupling the rescaled queueing network to a Brownian particle system interacting through boundary local times. A key step is a decoupling construction that replaces the correlated routing noise in the prelimit by asymptotically equivalent independent driving noises.

Let $G$ be a split, simply connected, almost simple algebraic group over a field of characteristic zero, and let $P$ be a maximal parabolic subgroup of $G$. We study the ring of differential operators on $P^{\mathrm{der}}\backslash G$, showing that it shares several key structural properties with classical Weyl algebras. We also develop a corresponding theory of $D$-modules.

Considered herein is a high-order regularization of the nonlinear shallow water equations within the framework of water wave theory. The regularized system is Galilean invariant and its solutions maintain an energy level that closely matches that of the nonlinear shallow water equations. However, in contrast to the classical nonlinear shallow water system, which admits discontinuous shock waves, the regularized formulation gives rise to weakly singular shock waves, which have continuous spatial profiles with unbounded spatial derivatives at isolated points. Using dynamical systems techniques, we establish the existence of such waves. Although weakly singular traveling waves remain continuous over their entire domain, their numerical approximation via finite element or pseudospectral schemes is affected by the emergence of spurious oscillations. To address this issue, we explore several finite volume methods for the accurate numerical approximation of these solutions. Our results demonstrate that the regularized system effectively reproduces the dynamics of the nonlinear shallow water equations in several scenarios. Moreover, our computations indicate that weakly singular shock waves are dynamically stable and can arise from general initial conditions connecting two asymptotic states. In contrast, other weakly singular structures, such as cusped solitons, appear to be structurally unstable, as we were unable to generate them from generic initial data.

Motivated by recent developments in full-duplex radios, cognitive radios, and data-driven signal-processing, we propose a novel class of reactive jamming adversaries wherein the adversary transmits jamming energy on the victim's frequency band while simultaneously monitoring various energy statistics in the network to detect the presence of potential countermeasures, thereby trapping the victim. These adversaries employ generalized energy detectors comprising statistical detectors, based on instantaneous and distributional energy metrics, and data-driven detectors employing machine-learning classifiers to learn patterns in the observed energy sequences. Against such a strong adversary, we propose a family of cooperative mitigation strategies wherein the victim takes assistance from a helper node, with the strategies tailored to operate under a wide range of latency requirements on victim's messages and practical radio hardware constraints at helper node. To provide theoretical guarantees on their efficacy, interesting optimization problems are formulated on the choice of their underlying parameters, followed by extensive mathematical analyses on their error performance and covertness. To assess their practical feasibility, we implement the before-deployment and after-deployment setups on a software-defined-radio-based hardware testbed, and to evaluate their detectability on real energy observations, we collect the corresponding datasets to train and test the data-driven machine-learning classifiers employed by adversary. Both analytical and hardware evaluations show that the proposed strategies cannot be detected with a high-probability under the generalized energy detectors used by adversary.

We investigate several $\sigma$-ideals on the Baire space $\omega^\omega$ $(\mathbb{Z}^\omega)$, introducing and studying the ideals $\mathcal{G}$ and $\mathcal{SMZ}^+$, alongside the classical ideals of meager sets, strong measure zero sets, the eventually different ideal and infinitely equal ideal We establish structural relationships and proper inclusions among these ideals. Also we compute the cardinal invariants of $\mathcal{M}_-$, proving that they are same as invariants of $\sigma$-ideal of meager sets. We further analyze the operation $^*$ on families of sets, establishing dual relationships such as $\mathcal{ED}^* = \mathcal{IE}$, $\mathcal{IE}^*=\mathcal{ED}$, $\mathcal{M}_-^*=\mathcal{SMZ}^+$ and $\mathcal{H}^* = \mathcal{G}$, and derive separations between ideals under additional set-theoretic assumptions. Finally, we prove a tree dichotomy theorem for the ideal $\mathcal{M}_-$ and we study the associated forcing notion.