Papers

We prove that, for a compact Hausdorff space $K$, the real Banach space $C(K)$ has the ball fixed point property if and only if $K$ is extremally disconnected. The new implication is obtained by constructing a fixed-point-free nonexpansive self-map of the whole closed unit ball whenever $K$ is a compact $F$-space which is not extremally disconnected. The construction uses a transfinite contractive propagation system with two increasing profiles. A successor–limit delay preserves their tail constraint and has no subsolution in the profile domain. Analysis records boundary deficits; positive operators on ordinal $C_0$-spaces and a common center determined by the two tails yield a nonexpansive synthesis satisfying an order domination inequality. The required topological families are obtained from a gap in a maximal Boolean chain in the zero-dimensional case, and from a transfinite extension of signs otherwise. The argument works in ZFC and resolves the question posed by Avilés, Japón, Lennard, Martínez-Cervantes, and Stawski.

The parasitic mite Varroa destructor poses a severe existential threat to global honey bee Apis mellifera populations. In this paper, we present a dynamical systems model of hive-mite interactions incorporating a eusocial Allee effect to evaluate the efficacy of chemical interventions. We partition treatments into “soft” miticides (targeting phoretic mites) and “harsh” miticides (penetrating the capped brood to target reproductive mites). Our bifurcation analysis reveals a fundamental trade-off: while harsh treatments effectively eradicate the protected mite reservoir by shifting the transcritical bifurcation boundary, they impose sublethal toxicity on the bees. We analytically show that exceeding a critical dosage threshold culminates in a catastrophic saddle-node bifurcation that guarantees colony collapse. To navigate this toxicity limit, we formulate an optimal control problem using Pontryagin's Maximum Principle. Numerical solutions reveal that a dynamic “shock and maintain” cocktail strategy optimally balances reservoir clearance with hive viability. Expanding the model to include treatment-resistant strains demonstrates that single-chemical reliance forces the hive onto a highly toxic, nearly unsustainable chemical treadmill. Finally, our mathematical framework yields a striking ecological prediction: because the eradication boundary scales intimately with the colony's carrying capacity, unchecked mite pressure will likely exert evolutionary forces that select for smaller, naturally resistant hives over the massive colonies favored by commercial agriculture.

We first verify the expected calculations of the topological structure set, in the sense of surgery theory, for the total spaces of $S^{4k-1}$-bundles over $S^{4k}$ for $k\ge3$, in which manifolds homotopy equivalent to the total spaces are organized. Next, we investigate the question of which of the elements in these structure sets can be realized as such bundles. Moreover, we study the forgetful map from the smooth version to the topological version of the structure set; we determine its image. All elements in the image turn out to have the underlying manifold such a bundle.

This paper is concerned with multivariable stochastic Hamiltonian systems governed by an ordinary differential equation for the position and an Ito stochastic differential equation for the momentum. The momentum dynamics involve both conservative and nonconservative forcing including Langevin viscous damping and the Wiener process as a random force. The setting allows for rotational degrees of freedom, a non-quadratic potential energy function and position-dependent mass, diffusion and damping matrices. In addition to modelling nonlinear physical dynamics subject to random forcing, the system is motivated by a stochastic optimisation viewpoint, where the search for global minima of the potential energy (for example, using the gradient descent) is carried out through convergence to a Boltzmann equilibrium measure which favours smaller values of this function. Under certain regularity conditions, we show that the $\chi^2$-divergence or, equivalently, Renyi's second-order relative entropy with respect to the invariant measure has a nonpositive time derivative and thus provides a Lyapunov functional candidate for this convergence. However, this derivative vanishes at some moments of time, which is related to the behaviour of the probability distribution of the momentum conditioned on the position and has a link to a family of auxiliary quantum harmonic oscillator Hamiltonians parameterised by the position space of the underlying classical system. This position-momentum conditioning is used in order to prove that such pre-equilibrium break times in the entropy dissipation are isolated, thus making the $\chi^2$-divergence a strictly decreasing Lyapunov functional for the Fokker-Planck-Kolmogorov equation which governs the joint position-momentum distribution in such systems.

For a totally bounded metric dynamical system $(X, d, T)$ we introduce a new critical value $\mathfrak s(X, d, T)$ which quantifies the asymptotic separation of periodic orbits. More precisely, $\mathfrak s(X,d,T)$ is defined to be the supremum of all $s\ge 0$ for which there exists a sequence of periodic orbits $\{\mathcal O_k\}_{k=1}^\infty$ in $(X,d,T)$ such that \[ \lim_{k\to \infty} \#\mathcal O_k = +\infty \quad \text{and} \quad \liminf_{k\to\infty}\#\mathcal O_k\cdot\eta(\mathcal O_k)^s >0, \] where $\eta(\mathcal O_k)$ denotes the smallest distance between distinct points in $\mathcal O_k$. When the dynamical system is induced by a self-similar iterated function system $\mathcal F$ satisfying the strong separation condition, we prove that this critical value is equal to the Hausdorff dimension $s_{\mathcal F}$ of its self-similar attractor. Furthermore, under a quantitative separation condition on periodic orbits we show that the associated empirical periodic measures converge weakly to the normalized $s_{\mathcal F}$-dimensional Hausdorff measure on the self-similar attractor.

Hypergraphs provide a natural framework for modeling higher-order relationships, but the development of spectral techniques with provable guarantees for general non-uniform hypergraphs remains challenging. Building on Banerjee's normalized adjacency matrix and Spiro's averaging-based diffusion framework, we develop a spectral framework for non-uniform hypergraphs and establish Cheeger's inequality for their conductance. A fundamental result in the spectral theory of hypergraphs asserts that, for every non-covering hypergraph, the second-smallest eigenvalue of its normalized Laplacian is at most one. This spectral characterization yields an improved Cheeger's inequality for non-covering hypergraphs, and we show that the resulting inequality is tight on both sides using cycle and cube hypergraphs. Our framework further yields higher-order Cheeger inequalities and provides theoretical guarantees for Fiedler's spectral partitioning algorithm, all in the setting of hypergraphs. Finally and most notably, we construct a new family of optimal hypergraph expanders that is tight for the Alon–Boppana bound.

We study regular sublattices of lattices that satisfy the Join-Infinite Distributive Law (JID) under Dedekind MacNeille completion. Using order convergence, we identify the lattice of O-closed ideals with Crawleys completion, and compare it with the Dedekind MacNeille completion. When DM(L) satisfies (JID), this comparison yields extension results for lattice homomorphisms. In particular, if Y is a regular sublattice of a complete lattice L, and L and DM(Y) satisfy (JID), then the canonical extension lattice-embeds DM(Y) into L with join-regular image. We also give a sharp finite-dimensional obstruction: there exists an infinitely distributive regular sublattice of R3 whose Dedekind MacNeille completion is not distributive, and therefore cannot be lattice embedded in R3. In contrast, we prove that the inclusion of every join-regular sublattice of R2 with a least and a greatest element extends canonically to a lattice embedding of its MacNeille completion into R2.

Emerging physical AI systems require low-latency, task-oriented video communication over unreliable channels. We propose a semantic-aware multi-level neural video coding method for robust low-latency video transmission over unreliable channels that are abstracted as multi-level packet erasure channels. Built upon the real-time DCVC-RT neural video codec, the proposed framework introduces a semantic- and feature-aware coding strategy that partitions encoded representations into packets carrying different levels of semantic and latent-feature importance and assigns these packets to different streams, each associated with a priority level when transmitted over unreliable communication channels. We also developed an error-resilient entropy model that removes inter-packet dependencies, allowing each packet to be decoded independently under packet losses. The complete system is trained end-to-end over the abstracted multi-level packet erasure channels, enabling learning of channel-aware representations together with importance-aware packet assignment while facilitating the network for differentiated packet prioritization. Experiments show that the proposed framework significantly improves robustness over baseline DCVC-RT under packet erasures, achieving graceful degradation in less important regions while better preserving task-relevant visual content.

A fulfillment center (FC) is a specialized logistics facility where e-commerce inventory is received, stored, and processed. Its internal organization directly impacts space utilization and the performance of put-away and picking operations. Items are stored in bins following strict operational guidelines to ensure productivity in both activities, so bin dimensions must be tailored to the product profile of each facility to maximize space utilization. % We address the bin dimensioning problem arising in e-commerce fulfillment centers: given a set of candidate bin types and a large, heterogeneous inventory, determine the number of bins of each type and assign all items so as to minimize total bin volume, subject to operational constraints governing how products may be combined within a single bin. We develop a dimensional decomposition that exploits these constraints to reduce the three-dimensional packing problem to a one-dimensional block-positioning problem. Building on this decomposition, we formulate the problem as a mixed-integer linear program and propose two solution methods: a Best-Fit-Decreasing (BFD) heuristic and a column generation scheme that produces tight LP lower bounds. Computational experiments on four synthetic datasets, calibrated on proprietary data from an American e-commerce fintech and ranging from 1.5 thousand to 1.5 million SKUs, show that the BFD heuristic scales to instances with up to 1.7 million blocks, and that column generation certifies a BFD–LP gap of 2.26% on the smallest dataset. The experiments further reveal a structural divide in solution quality across datasets, driven by the interaction between operational constraints and each facility's product profile.

We develop a probabilistic approach to the structure of finite skew left braces, motivated both by classical commuting probability in finite group theory and by the role of skew left braces in the study of set-theoretic solutions of the Yang–Baxter equation. We introduce relative commuting probability for skew left braces and establish analogues of several classical structural results, including relative BFC-type theorems. For natural classes of finite skew left braces, we show that a positive lower bound for the commuting probability forces the existence of a large section which is trivial up to bounded subgroups. We also study the probability that two elements generate a trivial sub-skew brace and a Sylow-local version of commuting probability, obtaining further structural consequences. Finally, we introduce a commuting probability for finite non-degenerate set-theoretic solutions of the Yang–Baxter equation. Under natural hypotheses on the associated structure skew brace $G(X,r)$, a positive lower bound for this probability yields a bound on $|G(X,r):\Soc(G(X,r))|$ depending only on the probability and on $|X|$.

The Möbius function of a partition lattice can be used to define interactions of probability measures. We call a lower set a symmetric partition lattice when it is invariant under permutations of the variables. We characterize these lattices through integer partitions and obtain a recursion for their Möbius coefficients at the top element. By constructing probability measures whose partition products are linearly independent, we show that permutation invariance of an interaction is equivalent to that of its underlying lower set, provided the coordinate spaces are sufficiently large. We also characterize the partitions on a lower set that force an interaction to vanish for every probability measure, and construct indecomposable distributions with zero interaction. We define the order of a symmetric partition lattice and relate it to vanishing marginals. We study generalized Lancaster and define the Streitberg and size-limited partition lattices, obtaining explicit Möbius coefficients for them. Finally, we express interaction equations in terms of characteristic functions. For Gaussian distributions, we show that an interaction vanishes exactly when the distribution factorizes according to a nontrivial partition in the underlying lower set. We also give a nonvanishing criterion for radial characteristic functions generated by Bernstein functions.

We establish a quantitative characterization of the Mokobodzki condition for two adapted càdlàg barriers on an arbitrary filtered probability space. For every $p\geq1$, we introduce a mean co-variation functional $Var_p(L,U)$, which for $p=1$ and $L=U$ reduces to Rao's mean variation, and show that it is quantitatively equivalent to the minimal $\underline H^p$-norm among all semimartingales lying between the barriers. The result covers both the case $p>1$ and the endpoint $p=1$, where the natural scale is based on Doob's class $(D)$.

This paper provides a complex-analytic interpretation of certain topologies on the tensor algebra, which are a central tool within strict deformation quantization. We prove that the symmetric tensor algebra $\operatorname{S}(V)$ of a barelled nuclear DF-space $V$ endowed with any of these topologies may be understood as an algebra of bounded Fréchet holomorphic functions on the strong dual space $V'_\beta$. The coarsest of the topologies then corresponds to the topology of uniform convergence on bounded subsets. The finer ones provide an infinite-dimensional generalization of the order of an entire holomorphic function. To facilitate these results, we study the interplay between boundedness and continuity of holomorphic mappings between locally convex spaces. As a byproduct, we establish a simple sufficient criterion for Fréchet holomorphy as well as several completeness results for spaces of bounded functions.

We establish bijections among spherical orbits on loop symmetric spaces, principal bundles on the twistor $\mathbb P^1$, and tempiric Langlands parameters. Using Vogan's theory of minimal $K$-types, we further establish bijections among irreducible equivariant local systems on loop symmetric spaces, irreducible local systems on principal bundles on the twistor $\mathbb P^1$, and irreducible representations of all strong inner forms of the dual symmetric subgroup. The latter bijections may be viewed as a combinatorial shadow of geometric Langlands duality for real reductive groups.

We consider a general class of cascading processes on marked networks, in which vertices are iteratively deleted according to a local vertex property. Under marked local-weak convergence, we establish a necessary and sufficient condition for the proportion of vertices in the final stable set to converge to the survival probability in the limiting network. Under this condition, we also obtain convergence of the proportions of vertices in subsets specified by local properties.

We prove the infinitesimal dynamical spectral rigidity of the circular Bunimovich stadium with sufficiently long flat edges, in the class of domains obtained by replacing its two circular arcs with strictly convex $C^{4}$ arcs and keeping its reflection symmetry across the axis parallel to the flat edges. That is, if a $C^{1}$ one-parameter family in this class passing through the stadium preserves the length spectrum, then its deformation function at the stadium vanishes identically. For flat edges of arbitrary length, the deformation functions of such families still span a finite-dimensional space.

Rotating detonation engines (RDEs) exhibit strongly nonlinear, multiscale wave dynamics that set the observed thermal field. High-fidelity simulations (DNS/LES) resolve these structures but remain computationally prohibitive, while low-order models such as the one-dimensional Koch-Kutz model capture circumferential wave motion yet lack the expressivity for high-frequency content. We use continuous data assimilation (nudging) to synchronize the Koch-Kutz solver with processed high-fidelity temperature data, introducing the prediction-observation mismatch as a relaxation source in the conserved energy equation; where observations are temporally sparse, interpolation supplies a target at every source update. As the nudging strength increases, the reduced model is progressively drawn onto the high-fidelity trajectory, and the forcing recorded along it provides an explicit, state-dependent estimate of the correction the model requires. We then train a Jacobian-regularized closure a priori on this recorded source. With the observation term removed, the corrected model advances autonomously, remains bounded, and recovers the temperature spectrum and the marginal statistics of the conserved variables relative to the baseline.

We prove that a bounded operator $T$ on a complex Hilbert space is self-adjoint whenever $|T|\leq|\re T|$, as conjectured by Fong and Tsui \cite{FT}.

Consider a compact leaf $\mathcal{L}$ of a holomorphic foliation $\mathcal{F}$ such that all the leaves of the restriction of $\mathcal{F}$ to some neighborhood $U$ of $\mathcal{L}$ are closed in $U$. We show that there exists an invariant germ of analytic set $V$ in a neighborhood of $\mathcal{L}$, of dimension higher than $\dim (\mathcal{F})$, consisting of compact leaves, such that the volume of the leaves in $V$ is uniformly bounded by above. We also provide a globalization of this result if the ambient manifold is projective. In order to study closed leaves foliations, and via the holonomy representation, we introduce a new concept, of independent interest, for subgroups $G$ of germs of holomorphic diffeomorphisms, the so called torsion locus. We show that it is non-trivial if $G$ has finite orbits.

We establish level-1 large deviation principles for continuous time suspension flows over countable Markov shifts, a class of non compact dynamical systems. Our results both quantify deviations of Birkhoff averages from their typical values and, importantly, provide a large deviation principle for orbits that at a finite time are close to escaping to infinity. To overcome the challenges posed by non compactness, we introduce the notion of topological pressure at infinity and prove a variational principle linking it to its measure theoretic counterpart. This framework allows the study of a natural class of potentials, known as strongly positive recurrent, which exhibit robust thermodynamic properties. Moreover, we show that compactly supported ergodic measures are pressure dense in this setting. Applications to pressure gaps and examples are given.

This paper proves strong restrictions on the underlying topological space of a closed, orientable hyperbolic 3-orbifold \orbM whose volume satisfies a certain upper bound. For instance, if vol(\orbM) < 0.1491, then the underlying space of \orbM must be either a small Seifert fibered space, or the connected sum of two lens spaces (or of a lens space with S^2 x S^1), or the gluing of one or two highly restricted Seifert fibered spaces along a single incompressible torus. If vol(\orbM) < 0.1571 and the singular locus of \orbM is a link, then the topology of the underlying space is restricted even further. Our methods are primarily topological, and involve studying the underlying topology of orbifold books of I-bundles. Along the way, we provide an exposition of some foundational material about orbifolds that was previously absent from the literature.

We prove that the Hodge filtration on local cohomology is contained in the symbolic Ext filtration, and hence in the usual Ext filtration, answering a question of Mustata and Popa. Our main observation is that the Ext filtration admits a filtered-duality interpretation via Hartshorne's algebraic de Rham complex and its infinitesimal filtration. Comparing this construction with the filtered Du Bois complex yields the containment. The same mechanism proves the higher injectivity conjecture of Popa–Shen–Vo and several variants.

We study combinatorial aspects of hyperplane sections of polytopes, focusing on how much the combinatorics of the sections determines the combinatorics of the original polytope. We show that, in general, combinatorial information about the sections is not enough to determine even the $f$-vector of the polytope. In contrast, for sufficiently generic simple polytopes, the function recording the number of vertices of each central section determines the full combinatorial type. We organize different combinatorial slicing properties into hierarchies, separately for affine and central sections, according to the level of combinatorial structure they determine on the polytope. In analogy with classical metric slicing problems, we formulate combinatorial analogues of the Busemann-Petty problem and Bourgain's slicing problem by replacing volume with face numbers, and show that they fail in every dimension and every face dimension.

Generalized Reed-Solomon codes and twisted generalized Reed-Solomon codes provide important sources of maximum distance separable codes. In this paper, we study two variants obtained by introducing column twists and simultaneous row-column twists into Reed-Solomon-type evaluation codes. For the column-twisted family, we provide necessary and sufficient conditions for the code to be MDS in terms of explicit subset product conditions. Under the stated parameter assumptions, the Schur square has dimension 2k+1, which leads to MDS codes that are not equivalent to Reed-Solomon codes. For the row-column twisted family, we establish necessary and sufficient conditions for the MDS property in terms of elementary symmetric functions. The larger Schur-square dimension provides a further distinction from both Reed-Solomon codes and known twisted families, thereby yielding new non-RS MDS codes. Finally, explicit parity-check matrices and dual descriptions are obtained for both code families. These results provide a foundation for subsequent studies of self-orthogonality, hull dimensions, and applications to quantum-code constructions.

We show that for any positive operator monotone function $f$ the inequality $\int_{[0,\infty)} f(t) \, \mathrm{d} \left\lVert E^{\Delta_{\varphi,\psi}}(t) \xi_\psi \right\rVert^2 + f'_\infty \varphi(1-s(\psi)) \geq \frac{f(1)}{2} \Bigl( \varphi(1) + \psi(1) - \lVert \varphi - \psi \rVert \Bigr)$ holds, where $\varphi, \psi \in \mathcal{M}_{*,+}$ are two normal positive linear functionals on a von Neumann algebra $\mathcal{M}$, and $\Delta_{\varphi,\psi}$ is the associated relative modular operator. Choosing $f(t) = t^s$ with $s \in [0,1]$, the Powers-Størmer-Ogata inequality is recovered.

Recent research has focused on developing GPU-accelerated algorithms for solving linear programs (LPs), with results that are nothing short of extraordinary. But a complete solver pipeline consists of more than the core algorithm, and the elephant in the room is that stages such as presolve have remained on the CPU, where they have become an increasingly large bottleneck. Presolve has stayed on the CPU because of the longstanding view, held by academic and industrial developers alike, that it is inherently sequential and irregular, and therefore difficult to parallelize efficiently. In this paper we challenge this view and show that presolving can benefit substantially from GPU acceleration. We describe simple design principles that avoid severe load imbalance and expose parallelism in reductions that appear inherently sequential. We implement these principles in cuPSLP, a GPU-accelerated version of PSLP, a CPU-based presolver that is itself several times faster than a state-of-the-art commercial presolver. In our experiments, cuPSLP reduces the shifted geometric mean presolve time of PSLP by a factor of 11 on the Mittelmann LP benchmark set and 42 on the GAMS large-scale LP benchmark set, with similar reduction quality.

Non-degenerate periodic orbits in three-dimensional Reeb flows are classified into three types: positive hyperbolic, negative hyperbolic and elliptic. In the present paper, we consider a closed connected contact three-manifold with a non-degenerate contact form. We show that its Reeb flow has a simple positive hyperbolic orbit if it has at least three simple periodic orbits. We mainly study the case in which no elliptic orbit exists. We prove that there is no non-degenerate contact form on a closed connected three-manifold with $b_1=0$ such that all simple periodic orbits are negative hyperbolic. As a corollary, by combining the author's previous result in the presence of an elliptic orbit and the known result for $b_1>0$, we obtain the main result. The proof uses the Weyl law for ECH spectral invariants. We also use compactness for genus zero $J$-holomorphic curves counted by the $U$-map. Under the contrary assumption, the number of simple orbits in each action interval $[L,2L]$ is uniformly bounded with respect to $L$. We use this property to study ECH generators and genus zero $U$-curves.

We show that the satisfiability problem for Strategy Logic introduced by Mogavero, Murano, and Vardi is $\Pi^1_\infty$-complete, and, more strongly, computably isomorphic to true second-order arithmetic. The lower bound is established for the next-time Boolean-goal fragment of Strategy Logic. Consequently, Strategy Logic is not recursively axiomatizable, even with effectively defined $\omega$-rules.

Small high-rate non-CSS stabilizer codes provide compact platforms for encoded quantum computation, but mixed-Pauli checks and limited native transversal logical gates complicate fault-tolerant dynamics. Block-level constructions offer an alternative by mapping an entire logical block to a physical circuit rather than compiling separately protected logical gates. We investigate this approach using the high-rate [[8,3,3]] non-CSS code and logical Trotter circuits as a testbed. We construct flagged syndrome-extraction circuits and establish a circuit-level memory pseudo-threshold near \(1.5\times10^{-3}\). We then apply our symplectic-transvection construction, which maps a logical Trotter circuit to a physical circuit with the same block pattern for any stabilizer code. Although this mapping preserves the intended unitary algebraically, encoded Trotter circuits exhibit asymmetry between logical-\(X\) and logical-\(Z\) failure channels. Single-fault analysis identifies the mechanism: a fault on the shared parity ancilla can propagate through the uncomputation network into an undetectable logical operator, reducing the effective circuit distance in the affected sector. We evaluate flag-conditioned recovery, biased-noise decoding, CliNR resource verification, flag postselection, and asymmetric gate-noise models. These methods suppress propagated faults but do not simultaneously suppress both logical sectors in the realistic configurations studied. A diagnostic protected limit removing the identified malignant first-order locations restores pseudo-threshold behavior in both sectors, approaching memory performance. These results demonstrate the potential of block-level logical constructions for non-CSS codes without rich native transversal gate sets and the joint protection of the parity network, analog rotation, and recovery required to preserve fault-tolerant distance.

Developing general-purpose algorithms with global asymptotic convergence guarantees for Nash equilibria of general-sum games remains a longstanding challenge. This paper extends the fixed-point bundle framework for finite-dimensional variational inequalities (VIs) from simplex domains to product-of-simplex domains, enabling its direct application to Nash equilibrium computation. The resulting fixed-point bundle exhibits a composite fiber bundle structure. The key innovation is to construct an equivalent VI on a simplex and establish the equivalence between the two fixed-point bundle frameworks via a fiber bundle isomorphism. Exploiting this geometric equivalence, we show that the path-following algorithm for VIs on product-of-simplex domains inherits the global linear convergence guarantees of the simplex-domain framework. In numerical experiments, the algorithm attained a precision of $10^{-5}$ on all 5600 randomly generated game instances, with dimension ranging from 2-player 128-action to 128-player 2-action. Code and data are available at https://github.com/shb20tsinghua/FiberBundle_VI.