Papers

We determine both the excluded vertex-minors and the excluded pivot-minors for the class of graphs of rank-width at most two. Up to local equivalence and graph isomorphism, there are exactly 25 excluded vertex-minors: 1 graph on 8 vertices, 18 on 9 vertices, and 6 on 10 vertices. Up to pivot equivalence and graph isomorphism, there are exactly 609 excluded pivot-minors: 2 on 8 vertices, 447 on 9 vertices, 146 on 10 vertices, 10 on 11 vertices, and 4 on 12 vertices. No excluded vertex-minor occurs on 11–16 vertices, and no excluded pivot-minor occurs on 13–16 vertices; the author's 16-vertex bound makes both lists complete. The proof is computer-assisted. Instead of enumerating all graphs, we reverse the one-vertex reduction theorem for prime graphs. For each $n$, we retain exactly the prime $n$-vertex graphs of rank-width at most two, modulo local equivalence and isomorphism, and extend those graphs by one vertex. Local-equivalence classes are identified by an exact canonical key obtained from the associated isotropic system, the binary row space of $[I\mid A(G)]$. A restricted version of the same key classifies pivot equivalence exactly. The vertex-minor and pivot-minor computations examine, respectively, more than $9.0\times 10^{10}$ and $4.9\times 10^{11}$ prime extensions in their 16-vertex final layers.

Let $$ f_R(x)=\frac{x}{1+\norm{x}/R},\qquad R>0, $$ be the radial cone projection of the plane onto the open disk $\DR=\{x:\norm{x}<R\}$. Previous work established the Self-Directrix and Confocal-Codirectrix Theorems, according to which $f_R$ maps focal conic arcs to focal conic arcs while preserving the distinguished focus and directrix. This note combines those results with three classical facts recorded by W.~H.~Besant. First, reciprocal polarity with respect to $\partial\DR$ represents every focal conic under consideration as the reciprocal polar of a unique circle, and in this circle representation the nonlinear action of $f_R$ becomes the elementary radius translation $q\mapsto q+R$. Second, the normals at the intersections of a fixed ray with the self-directrix family envelope an explicit parabola. Third, the tangent at $f_R(z)$ is constructed directly from the original point $z$ and the original line, without differentiating $f_R$ or solving for the conic.

In this paper, we introduce Siu's curvature operator \(A^E_{p,q}\) for vector-bundle-valued differential forms on Kähler manifolds. When $p=n$, this operator reduces to the classical Akizuki–Nakano curvature operator. We first characterize the semipositivity of \(A^E_{p,q}\) in terms of an optimal \(L^2\)-estimate condition for the \(\bar\partial\)-operator, and then prove an Ohsawa–Takegoshi-type extension theorem for \(E\)-valued \((p,q)\)-forms under the curvature condition \(A^E_{p,q+1}\geq0\), using a new twisted basic estimate adapted to this setting. As an application, we prove the local freeness of the higher direct image sheaf \(R^q s_*(\Omega^p_{X/ B_m}\otimes E)\) under the curvature conditions $A^E_{p,q+1}\geq0$ and $A^E_{p,q}\geq0$, where $s: X \to B_m:=\{t\in\mathbb C^m:\ |t|<1\}$ is a proper holomorphic submersion from a Kähler manifold $X$, and $E$ is a Hermitian holomorphic vector bundle.

We study the Cauchy problem for the generalized porous medium equation on infinite weighted graphs. For a general nonlinearity, we establish Dirichlet comparison and weak maximum principles on finite subgraphs and, through an exhaustion argument, construct minimal and maximal pointwise solutions for arbitrary $\ell^\infty$ initial data, controlled by explicit, possibly time-dependent, barriers. For the porous nonlinearity $\phi(s)=s|s|^{m-1}$, assuming a $\nu$-Sobolev inequality with $\nu>2$, we derive quantitative energy estimates for $\ell^1$-mild solutions. These yield finite-time extinction in the fast diffusion range $0<m<2/\nu$ and $\ell^1$-$\ell^q$ smoothing in the range $m>2/\nu$. Interestingly, we recover the Euclidean critical exponent for several model graphs. Finally, we prove an exact generalized mass balance for nonnegative $\ell^1$-mild solutions on graphs that are stochastically complete at infinity, allowing for an arbitrary killing term. The same balance is established for suitable classical and bounded pointwise solutions. In the absence of killing, these reduce to conservation of mass.

We derive the $\mathcal{P}(\Phi)_2$ measure on the two-dimensional unit torus as the rigorous limit of many-body quantum Gibbs states. In particular, the quantum model corresponding to the $\Phi^{2p}_2$ measure is formulated in the grand-canonical ensemble with a general symmetric $p$-body interaction potential which is not required to have a factorized form. Unlike in the $\Phi^4_2$ case investigated by Fröhlich–Knowles–Schlein–Sohinger in \cite{FKSS25}, in the treatment of higher-order interactions, Wick renormalization generates a full hierarchy of lower-order interactions which we organize systematically using a graphical formalism. One key ingredient of our analysis is a logarithmic stability estimate for both the nonlocal Hartree functional and the many-body Hamiltonian that is uniform in the interaction range, allowing us to use the positivity of the leading $p$-body interaction to control all lower-order terms generated by the Wick counterterms.

For a simple graph $G$ of order $n$, let $S_2(G)=\lambda_1(G)+\lambda_2(G)$ denote its spectral sum. We determine, for every $n\geq5$, the exact maximum of $S_2(G)$ and all equality cases. The unique maximizer, up to isomorphism, is the complement of the disjoint union of a suitably balanced complete bipartite graph and isolated vertices, with the sizes of its three parts determined by $n$ modulo $7$. Denoting this graph by $K_n^\star$, we further show that $ S_2(K_n^\star)\leq\frac{8n}{7}-2,$ with equality exactly when $7\mid n$. This proves a conjecture of Kumar, Liu, Monterde, Pragada and Tait, which strengthens the Aouchiche–Hansen 2010 conjecture by extending it from connected graphs to all graphs and by asserting uniqueness of the extremal graph. The result also subsumes the 2008 conjecture of Ebrahimi B., Mohar, Nikiforov, and Ahmady. The proof combines Ky Fan's variational principle with a spectral inequality for weighted Ferrers quotients to reduce the problem to an explicit family whose complements have incidence rank one. Exact integer optimization and a separate equality analysis then yield the maximum and uniqueness.

We propose a mixed residual method (MIM) for numerically solving the biharmonic equation with nonhomogeneous clamped boundary conditions. By establishing the well-posedness of the biharmonic equation in spectral Barron spaces, we derive an error bound for MIM that relates shallow neural network approximations to the exact solution and overcomes the curse of dimensionality. This error bound consists of two components: the first corresponds to the approximation error of the neural network, while the second represents the generalization error arising from randomly sampled training data. Several numerical experiments are presented to demonstrate the effectiveness of the proposed method.

Inference for epidemic transmission on dynamic networks is fundamentally limited by latent infection times, incomplete contact histories, imperfect observation, and external sources of infection. Although coherent likelihood formulations are available for partially observed epidemic processes, considerably less is known about the theoretical limits of statistical inference under such observation mechanisms. This paper develops a unified framework for studying identifiability and Fisher information in epidemic transmission models observed on dynamic contact networks. We establish conditions for structural and local identifiability, derive observed and complete-data information matrices, and quantify information loss arising from unobserved transmission events and missing network information through a missing-information decomposition. We further investigate how observation frequency, network coverage, and measurement accuracy influence parameter estimability and statistical efficiency, providing a principled basis for evaluating surveillance strategies. Simulation studies demonstrate that the proposed framework accurately characterises the relationship between observation design, statistical information, and parameter estimation, with theoretical predictions closely matching finite-sample performance. The proposed framework clarifies the relationship between observation design, identifiability, and inferential precision, and provides a theoretical foundation for statistical inference in partially observed epidemic transmission models.

Given a numerical semigroup $S$ and a positive integer $p$, the quotient $\frac{S}{p}=\{n\in \mathbb{N} \mid pn\in S\}$ also forms a numerical semigroup. When $S=\langle a,b\rangle$ with $\gcd(a,b)=1$, a well-known open problem is to find a closed-form formula for the Frobenius number $g\!\left(\frac{\langle a,b\rangle}{p}\right)$, which remains open even in the special case $b=a+1$. Inspired by Curtis's theorem on the non-existence of polynomial formulas for the Frobenius number $g(\langle s_1,s_2,s_3\rangle)$, we provide a negative answer to this open problem in a certain sense. Concretely, we obtain the following three main results. (i): The Frobenius number $g\!\left(\frac{\langle a,b\rangle}{p}\right)$ cannot be represented, uniformly in $a,b,p$, by any finite collection of polynomial (or rational) formulas. (ii): For each fixed $p$, the function $a\mapsto g\!\left(\frac{\langle a,a+1\rangle}{p}\right)$ is a quadratic quasi-polynomial with period dividing $p$. (iii): There is no nonzero polynomial $F\in \mathbb{C}[X_1,X_2,X_3]$ satisfying $F\left(a,p,g\!\left(\frac{\langle a,a+1\rangle}{p}\right)\right)=0$ for all primes $a,p$ with $2<p<a$; the same conclusion already holds if only $p$ is required to be prime and $a$ ranges over all integers greater than $p$. While (iii) is stronger than (i), the proofs of the two results reveal different insights. Dirichlet's theorem on primes in arithmetic progressions plays a crucial role in our arguments.

The necessary and sufficient conditions for the Moran-type measures generated by two-element and three-element digit sets to be spectral have been given. The main result indicates that the spectrality of such a Moran-type measure is completely determined by the number of 2-factors and 3-factors in the zero set of the Fourier transform of each term in the convolution. The proof shows that, without the existence of Hadamard triples, the problem becomes much more complicated and difficult than the case with the existence of Hadamard triples.

Learned PDE solvers often entangle governing operators with the geometry, boundary conditions, and discretization used for training. This limits reuse when the same physics is posed on new domains, and it also makes physical-law discovery geometry-dependent. We introduce Geometry-aware LegONet (gLegONet), a boundary-manifold extension of Lego-like operator learning. Physical mechanisms are pretrained once as modular variational blocks on an ambient spectral domain. For a target geometry, sampled boundary constraints define an affine admissible manifold. Its mass-orthonormal tangent coordinates are used to evolve the dynamics and evaluate candidate law-discovery features directly. Changing the domain therefore changes only an algebraic coordinate interface, not the learned operator blocks. This converts arbitrary-domain PDE learning from geometry-specific retraining or soft penalty enforcement into boundary-guaranteed assembly of reusable mechanisms. In forward simulations and sparse identification tests on unseen domains, the method maintains boundary residuals near the algebraic tolerance and yields predictive governing laws from short-time observations.

Several open problems in quantum information theory can be formulated as equipartition problems for positive operators, asking for a decomposition into bounded-rank positive parts under uniform constraints. The existence problems for SIC-POVMs and MUBs are of this type, as is the Ruskai-Audenaert conjecture. We first show that certain problems of this form can be attacked using equivariant cohomology, and then present new results on the Ruskai-Audenaert conjecture. In its weak form, this conjecture asserts that every quantum channel admits a convex decomposition into a minimal number of generalized extreme points; in its strong form, one with equal weights. We prove the strong conjecture in all dimensions for a set of channels of nonzero measure, including all cq- and qc-channels, as well as for all channels with qubit inputs. We also prove the weak conjecture for all qutrit channels, along with further results on convex decompositions of quantum channels.

We determine the sharp asymptotic behavior of the optimal constants in the discrete Hardy-Rellich inequalities on the lattice $\mathbb{Z}^d$. For every fixed integer $m\ge 1$, let ${\mathcal C}_{m,d}$ be the best constant in the $m$-th order Hardy-Rellich inequality. We prove that $$\lim_{d\to\infty}\frac{\mathcal C_{m,d}}{d^m}=2^m.$$ Our approach combines a Fourier reduction to weighted inequalities with the flat torus and general weighted Hardy-Rellich identities of first and second order. A key novelty is the use of probabilistic concentration estimates, specifically Hoeffding's inequality and entropy methods, to handle estimates involving the anisotropic weight $\omega^\gamma$ $(\gamma\geq 1)$ where $$\omega(x)=\sum_{j=1}^{d} \Big(\sin\frac{x_j}{2}\Big)^2$$ in a dimension-uniform manner. These tools yield asymptotically sharp weighted estimates on the torus, from which the lattice inequalities follow by iteration.

Quantum devices are open systems whose dynamics interleave coherent evolution with dissipation, and benchmarking, error mitigation, and error correction all rest on a faithful model of both. Existing characterization protocols either assume prior knowledge of the interaction and noise structure, or demand ancillas, entangled probes, or mid-circuit control, or capture only the Pauli-diagonal part of the noise. Here, we present a protocol that reconstructs an arbitrary sparse Markovian generator, including every Hamiltonian together with the jump operator coefficients, using only product Pauli state preparation, single uninterrupted forward evolutions, and product Pauli measurements. Given a sparsity budget $M_0$ and a strength bound $\Gamma$ of the Lindbladian, every coefficient is learned to precision $\epsilon$ from $\widetilde{O}(\Gamma^2M_0^2/\epsilon^4)$ experiments and $\widetilde{O}(\Gamma M_0^2/\epsilon^2)$ total evolution time, with both supports identified from data without locality assumptions. The protocol runs at a logarithmic number of positive evolution times on a hardware clock lattice and is provably robust to calibrated state-preparation and measurement errors.

We study the high-temperature mean-field limit of grand-canonical bosonic Gibbs states on the torus with renormalized nonlocal three-body interactions. In dimensions two and three, we construct the limiting nonlinear classical Gibbs measure and prove convergence of the relative free energy and of the reduced density matrices of every fixed order; in three dimensions, a smallness condition on the interaction is imposed. The proof combines the density-channel representation of the interaction with a coherent-state variational method based on the upper-symbol representation of the free Gibbs state. The same framework also contains, as a special case, the homogeneous positive-type model studied by Lewin, Nam, and Rougerie (2021).

For any Tychonoff space $X$ let $C_p(X)$ (resp., $C^*_p(X)$) be the set of all continuous (resp., and bounded) functions on $X$ with the pointwise convergence topology. Given Tychonoff spaces $X$ and $Y$, Uspenskij \cite{us} proved that if $C_p(X)$ is uniformly homeomorphic to $C_p(Y)$, then $X$ is pseudocompact if and only if $Y$ is pseudocompact. The second author and Vuma \cite{valvu} have shown that linear homeomorphisms between $C_p^*(X)$ and $C_p^*(Y)$ also preserve pseudocompactness. Recently Baars-van Mill-Tkachuk \cite{bmt} gave another proof of that result and raised the question if the same remains true provided $C_p^*(X)$ and $C_p^*(Y)$ are uniformly homeomorphic. In the present paper we introduce the notion of bounded uniformly continuous maps and show that every bounded uniform homeomorphism between $C_p^*(X)$ and $C_p^*(Y)$ preserve pseudocompactness. Moreover, we show that if there is a uniform homeomorphism $\varphi:C_p^*(X)\to C_p^*(Y)$, where $X$ is pseudocompact and $Y$ is normal, then $Y$ is $\sigma$-pseudocompact, i.e. $Y$ is a countable union of closed pseudocompact subsets. It is also shown that a continuous linear map between $C_p$-spaces is norm-bounded if and only if it is bounded in our sense.

This paper addresses the output regulation problem for 1-D diffusion-reaction system, where both the disturbance and reference signals are generated by an unstable exosystem. We propose a constructive approach to the design of a finite-dimensional tracking error-based regulator for this class of possibly unstable systems. Based on the regulator equations, a combined plant is derived, converting the output regulation problem into a partial stabilization problem for the combined system. Using the modal decomposition method, the observability of the truncated modes is characterized under appropriate transmission zeros and observability conditions. For the controller design, unlike in the standard stabilization case, where the observer gain is dependent on the unstable modes only, the observer gain here has full order due to the coupling introduced by the exosystem. We prove that the observer gain can be designed such that its norm remains uniformly bounded with respect to the dimension of the observer. LMI-based conditions are provided for determining the observer dimension, and it is shown that the LMI is feasible for a sufficiently large dimension. Finally, the output regulation problem in the presence of unknown time-varying measurement delays is analyzed. Numerical examples are provided to validate the theoretical results.

Developing efficient function-approximation methods for policy evaluation is a fundamental challenge in risk-aware reinforcement learning. Existing approaches either focus on restrictive classes of risk measures or rely on access to a simulator, limiting their applicability in fully online settings. In this work, we propose computationally efficient online learning algorithms for policy evaluation in Markov decision processes (MDPs) with dynamic utility-based shortfall risk (UBSR) measures under linear function approximation. Specifically, we introduce the UBSR-TD algorithm, establish conditions under which it converges almost surely, and develop several variants designed to accelerate convergence. Our formulation shows that existing policy evaluation algorithms for risk-neutral MDPs can be readily adapted to dynamic UBSR settings by incorporating a loss function into the temporal-difference error. Numerical experiments support our theoretical findings, and an application to a perishable inventory management problem with shelf-life uncertainty demonstrates the practical effectiveness of the proposed methods.

In this paper, we use the adherence semigroup to describe finite-sums configurations in topological dynamical systems. We establish a correspondence between exact finite sumsets and powers of adherence elements. As applications, we give dynamical formulations of density finite-sums results, characterize total minimality through the density of adherence-power orbits, and give a positive answer to the ultrafilter question in [Question~8.9, 6]. We also characterize the sets that belong to the common sum of two commuting nonprincipal ultrafilters.

Accurate and real-time direction of arrival (DOA) estimation is crucial for beamforming in unmanned aerial vehicle (UAV) communication systems. However, the existing high-precision DOA estimation algorithms encounter high computational complexity when implemented on a UAV with on-board signal processing constraints. To tackle this issue, an electromagnetic neural network (EMNN) is developed for DOA estimation, which is capable of generating the angular spectrum of the incident signal based solely on amplitude observation. Specifically, the proposed EMNN consists of two components: a stacked intelligent metasurfaces (SIM) is mounted on the UAV, and each meta-atom is an artificial neuron that can process signals in the electromagnetic domain with low energy consumption and ultra-fast computing speed. Furthermore, a fully connected layer is cascaded to process the received amplitude signal, enhancing the non-linear extraction and representational ability of EMNN. Moreover, to reduce the computational complexity and observation snapshots required for high-resolution DOA estimation, we develop a hierarchical DOA estimation framework, which involves two stages for conducting coarse and fine DOA estimation, respectively. For each stage, EMNN is trained on randomly generated training samples and their corresponding spectra to achieve the desired estimation goal. Finally, the simulation results validate that the proposed EMNN achieves approximately 13 dB gain in classification error reduction over the conventional beamforming (CBF) method in dual-signal scenarios, albeit its lower cost and radio frequency (RF)-related power consumption.

Let $\Pi$ be a $k\times n$ sparse random matrix. For a fixed $r$-dimensional subspace $V\subset{\mathbb R}^n$, let $U_V:{\mathbb R}^r\to{\mathbb R}^n$ denote an isometry from ${\mathbb R}^r$ onto $V$. The product $\Pi U_V$ is a central model in randomized dimension reduction and has been studied primarily through trace and Gaussian comparison inequalities. In this work, we develop an approach to the spectral norm of the matrix product $\Pi U_V$, based on entropy estimates for level sets of vectors $x\in V$. Combining the method with existing estimates, we show the following. Assume that \[ k\ge C\,r(\log\log r)^2,\qquad p\ge (\log k)/k. \] Let $\Pi$ be a $k\times n$ matrix with i.i.d. entries equidistributed with the product $b\,\xi$, where $b$ is a Bernoulli($p$) random variable and $\xi$ is mean-zero, independent of $b$, and satisfies $|\xi|\le1$ almost surely. Then with high probability \[ \|\Pi U_V\|\le C\sqrt{kp}. \] Matching results hold for other random models with negatively associated entries.

The plunge region records the spectral transition of a time-frequency localization operator. Let $A_0,B_0\subset\mathbb{R}$ be bounded measurable sets of positive measure with finite boundaries, and let $S_{cA_0,B_0}=P_{cA_0}Q_{B_0}P_{cA_0}$. We prove that, for every $c>0$ and $0<\varepsilon<1/2$, $\Lambda_\varepsilon\le C(A_0,B_0)\widetilde L[1+\ln_+(ca/\widetilde L)]$, where $\widetilde L=\ln(1/[\varepsilon(1-\varepsilon)])$, and $\ln_+x=\max(0,\ln x)$. If $M$ and $K$ count the interval components and $a$ and $b$ are their maximal lengths, one may take $C(A_0,B_0)=63MK(1+b)$. The estimate is uniform in $c$ and $\varepsilon$ and becomes $O(\widetilde L)$ when $\widetilde L\ge ca$. On the range $c\ge2$ and $\alpha^{-c}<\varepsilon<1/2$ covered by the parallelepiped theorem of Kulikov and Dam Larsen (2026), this recovers its $d=1$ order; their complementary very-small-threshold result is sharper. We give an independent direct proof with explicit geometry dependence and a single formulation for all $c$ and $\varepsilon$. It acts on the off-diagonal factor of the localization operator: an exact one-dimensional oscillation factorization reduces each far-field piece to two modulated copies of the scale-free Hankel kernel $1/[2\pi(s+r)]$, Bernstein-ellipse approximation gives uniform singular-value decay, and a Taylor-rank estimate controls the boundary layer. Choosing this width at the spectral depth absorbs finer scales and leaves only $\ln_+(ca/\widetilde L)$ dyadic scales, yielding the self-improving logarithm. It also identifies tangential phase dependence as an obstruction to this factorization in nonproduct higher-dimensional geometry.

In hypothesis testing, Monte Carlo tests are usually justified either by exact null simulation or by joint exchangeability of the observed data and its simulated copies. This leaves a gap for common computational procedures, such as parallel MCMC sampling initialized at the observed data, where each copy may be marginally null and even pairwise exchangeable with the observation, but the full collection is not jointly exchangeable. In such cases the usual empirical p-value can be invalid when the chain has not mixed, while exactly exchangeable constructions such as the Besag–Clifford hub-and-spoke sampler may suffer from high conditional Monte Carlo variability. We give finite-sample guarantees for this intermediate regime. If, under the null, the observed data $X$ and a copy $X'\sim P(\cdot\mid X)$ are conditionally i.i.d.\ given a latent variable, then for any prespecified statistic and any finite number $m$ of conditionally independent Monte Carlo copies, the resulting empirical p-value obeys $\mathbb P\{p_m\le \alpha\}\le 2\alpha$. This guarantee requires no mixing conditions and holds for any number of copies $m$, and it explains finite-sample oscillatory behavior in inference via MCMC sampling. In addition, we further show that the guarantee provides insights into inference problems arising in other settings, including inference on Bayesian models (recovering a classical result showing validity up to a factor of $2$ for posterior predictive p-values), and inference via balanced permutation tests.

Optimization over probability measures has become an increasingly important paradigm in modern machine learning, scientific computing, and uncertainty quantification. Motivated by Nesterov's accelerated gradient method in Euclidean space, we develop Heavy-ball and Nesterov acceleration methods over the probability measure space $\mathcal{P}_2$ and establish non-asymptotic convergence guarantees that match their Euclidean counterparts. In particular, we derive convergence rates with respect to both the number of iterations and the number of particles used to represent the underlying probability distributions. Extending accelerated optimization from Euclidean space to probability measures is challenging. The natural notion of momentum requires concepts such as tangent bundles of the set of probability space and they are hard to operate numerically. To overcome these difficulties, we introduce two complementary lifting procedures. The first lifts probability measures to phase space through a Hamiltonian formulation, introducing momentum variables into the dynamics. The second lifts probability measures to a common Hilbert space, restoring the linear structure required for convergence analysis while simultaneously yielding executable particle dynamics. Together, these two complementary lifting procedures provide a systematic methodology for designing, analyzing, and implementing momentum-based accelerated optimization methods over probability measure spaces.

Let $t(N)$ be the largest $t$ for which there exist distinct sets $A_1,\dots,A_t \subseteq \{1,\dots,N\}$ such that $A_i \cap A_j$ is a nonempty arithmetic progression for all $i \neq j$ (Erdos Problem #272). Simonovits and Sos proved $t(N)=O(N^2)$ and conjectured $\binom{N}{2}+1$ is best possible; Szabo disproved this by a construction giving $t(N) \geq \binom{N}{2}+1+\lfloor(N-1)/4\rfloor$, proved the asymptotics $t(N)=N^2/2+O(N^{5/3}(\log N)^3)$, and asked whether $t(N)=\binom{N}{2}+O(N)$ and whether some element lies in all sets of any extremal family (the kernel question). We determine $t(N)$ exactly for all $3 \leq N \leq 12$ by exhaustive computation: in this entire range Szabo's lower bound is exact, and we conjecture that $t(N)=\binom{N}{2}+1+\lfloor(N-1)/4\rfloor$ for every $N$. Towards the matching upper bound we prove, for every $N$, that Szabo's bound is the exact maximum over all families with a common element (starred families). The proof combines a self-contained “defect-one” counting inequality for staircase regions with a new structural theorem: every non-progression member of such a family contains a bad pair that no other member can share. Consequently the sharpened conjecture reduces to a single remaining statement, namely Szabo's kernel conjecture that some element lies in all sets of an extremal family, and we prove first structural constraints on putative non-starred extremal families.

We develop a theory of approximate quantum error correction (QEC) based on the error-set model, complemented by general methods for code construction. Exact QEC has a powerful error-set structure: by the Knill-Laflamme conditions, a code correcting a given error set automatically protects against every channel whose Kraus operators lie in their linear span. This linearity gives rise to code distance, the equivalence between erasures and general errors, and a theory of asymptotically good codes. A longstanding view has been that these features do not extend to AQEC, leaving the theory essentially channel-by-channel. We show instead that, although full Knill–Laflamme linearity fails, a restricted form survives and suffices to extend all three structural features to the approximate setting. Specifically, a common error-set criterion governs families of channels whose Kraus operators are linear combinations of a given error set and whose coefficient matrices satisfy a spectral constraint. Using the Bény-Oreshkov worst-case and Petz average-case frameworks, we derive uniform fidelity guarantees for these families in terms of two new code parameters–the environment-leakage distance, controlling worst-case performance, and the Knill-Laflamme Hellinger distance, characterizing the average-case performance of Petz recovery. To demonstrate the scope of this model, we develop partition-based constructions across diverse quantum systems and geometries, placing exact and approximate correction on equal footing. These constructions lead to a metric–error alignment hierarchy for Hilbert spaces, metrics, and error families, which in turn characterizes the resulting recovery guarantees. They yield the first known asymptotically good code families for fermionic systems, one-dimensional Rydberg-blockaded systems, and deletion errors, and extend to other physical platforms.

We give a new proof of Maurer's homotopy theorem for matroids using polyhedral methods, in contrast to Maurer's original combinatorial proof. The same polyhedral approach yields a homotopy theorem for delta-matroids, from which the corresponding results for matroids, even delta-matroids, and antisymmetric matroids follow. We further prove an analogous theorem for integral polymatroids.

The symmetric exclusion process (SEP) is a classical model of interacting particles on a graph, in which multiple particles execute random walks subject to the constraint that no two particles may occupy the same vertex. The version of the process in which the number of particles is fixed, is by now very well understood, through the study of reversible Markov chains. In this paper, we study the non-equilibrium version of the SEP, in which some vertices of the graph interact with external heat baths, held at different temperatures, that mediate transport through the graph. The resulting steady-state distribution is still the stationary distribution of a natural Markov chain, but the chain is no longer reversible. This means that even the steady-state distribution is very hard to describe, and indeed is the subject of a rich literature, which has focused almost exclusively on graphs which are finite subsets of the lattice $Z^d$. Very little is known about other important quantities, notably the mixing time. We derive various quantitative results about the non-equilibrium SEP for arbitrary graphs with arbitrary heat baths. Our main result is a bound on the mixing time in terms of the worst-case hitting time of a single particle to any of the heat baths; this bound is tight up to two logarithmic factors in the size of the graph. We then use this result to derive a simple algorithm, running in time roughly $n^{O(k)}$, that computes the steady-state joint occupation distribution for any set of $k$ vertices; since the steady-state distribution is very elusive, this provides a potentially useful tool to study its $k$-wise correlations for small values of $k$. Finally, we prove that the spectrum of the Markov chain associated with the non-reversible dynamics for the SEP is independent of the heat bath temperatures; thus in particular the spectrum is always real.

In Problem 6 of his 1988 paper on differential posets, Stanley asked for the least possible cardinality of a fixed rank of an $r$-differential poset and suggested that the minimum should be attained by $Y^r$, the $r$-fold Cartesian power of Young's lattice. We disprove the resulting universal coefficientwise lower bound. For every $r\geq 3$, we construct an infinite $r$-differential poset $P^{(r)}$ satisfying $\lvert P^{(r)}_4\rvert=\lvert (Y^r)_4\rvert-\lfloor r/3\rfloor$. For $r=3$, the construction replaces thirteen rank-four lower-cover blocks of $Y^3$ by twelve blocks with the same point and pair incidence multiplicities, producing the initial rank sequence $1,3,9,22,50$ instead of $1,3,9,22,51$. A reflection extension then yields an infinite differential poset. The construction does not address the cases $r=1$ and $r=2$.

We study a parabolic-elliptic forager-exploiter model describing the cascade interaction between two species through a shared environmental resource with a constant renewal rate. We first establish the global existence and uniform boundedness of classical solutions in arbitrary dimensions {\color{black}for large initial population data and large taxis sensitivity coefficients. We then investigate the long-time behavior of solutions.} When the resource renewal rate is small, all solutions will converge exponentially to the unique constant steady state. Furthermore, for large resource renewal rates, the parabolic-elliptic system does not undergo Hopf bifurcation, which is in sharp contrast to the corresponding fully parabolic case, where temporal oscillations arise via Hopf bifurcation.