Papers

On top of substitution errors, DNA storage channels suffer from both insertions and deletions at the same time. It is therefore important to develop error-correcting codes with efficient encoders and decoders that can combat all three types of noise. This paper introduces the salami-slicing trellis, a decision-feedback trellis that computes bitwise posterior probabilities along each strand and is coupled with polar codes across strands. The decoder alternates between advancing the trellises by one position and polar-decoding the resulting cross-strand slice, feeding the decoded bits back to the trellises for the next position. Simulations suggest that the resulting coding scheme approaches the conjectured capacity of the substitution-insertion-deletion channel.

This paper investigates the role of viscosity in the error upper bounds of a consistent splitting scheme for the Navier-Stokes equations proposed by Huang and Shen [5]. In their original analysis the viscosity is fixed to unity. By following and extending their proof methodology while keeping the viscosity symbolic, we obtain an H1 velocity error bound that contains negative powers of viscosity, indicating that the scheme is not robust as viscosity tends zero. To establish this bound we refine a theorem in [8] on the constant in the Stokes pressure estimate, which is crucial to the error analysis. A targeted numerical experiment based on a perturbation of the Kovasznay flow corroborates this analytical prediction: the scheme of [5] blows up at high Reynolds number, and a comparison with a fully implicit Newton solver and with the time-dependent Stokes counterpart of the same scheme localizes the failure to the explicit treatment of the convection term.

The Bilu-Linial conjecture asserts that every $d$-regular graph admits a signing $\sigma$ such that the spectral radius of the signed adjacency matrix $A_\sigma$ satisfies $\rho(A_\sigma)\le 2\sqrt{d-1}$. Bilu and Linial also proved the weaker bound $O(\sqrt{d\log^3 d})$ for graphs of maximum degree $d$. Marcus, Spielman, and Srivastava confirmed the conjecture in the case of $d$-regular bipartite graphs. In this paper, we prove that every graph of maximum degree $d$ has a signing $\sigma$ such that $$\rho(A_\sigma)\le 2\sqrt{3(d-1)}.$$ This removes the polylogarithmic factor from the estimate of Bilu and Linial and gives an explicit $2\sqrt{3(d-1)}$ two-sided spectral bound. The proof builds on the method of interlacing polynomials introduced by Marcus, Spielman, and Srivastava, together with results on mixed characteristic polynomials established by Marcus, Spielman, and Srivastava and by Bownik.

Machine Learning (ML) algorithms, such as k-Nearest Neighbours (kNN) or random forest, eschew the ideal of true data models in favour of predictive performance. However, minimising the MSE or F-score cannot lead to unbiasedness directly, which is important in many situations such as official statistics. We study the conditions of algorithmic ML, other than the existence and knowledge of true data models, which lead to unbiased prediction or classification for a given finite population, including how the training data may be sampled from the population, how a trained prediction algorithm can be tuned to achieve unbiased prediction or classification for that population, and how the performance of out-of-sample prediction or classification can be assessed unbiasedly. The inference is based on the known probability design of samples and training sets, rather than any assumed distributions or models.

In this paper, we improve the arbitrary Banach space \(n \log n\) bound of Ivanisvili–Volberg \cite{IvanisviliVolberg2022} for the second order projection bound to the order \(\sqrt{n}\) bound. Moreover, we study the lower Riesz estimate with the pointwise square gradient, and prove a fixed chaos characterization: on every fixed homogeneous Walsh chaos $H_k$, the dimension free estimate \[ \|\Delta^{1/2}f\|_{L^p(\Omega_n;X)} \lesssim_{p,k,X} \||\nabla f|_X\|_{L^p(\Omega_n)} \] holds for all $n$ if and only if $X$ has Rademacher type $2$. We also consider an exact tail space norm of the analytic paraproduct $T_\varphi g(z)=\int_0^z g(\zeta)\varphi'(\zeta)\,d\zeta$ on Banach valued \(H^\infty\) spaces. A matching lower bound of Volberg \cite{Volberg2024} \[ \|T_\varphi:H_d^\infty(\mathbb D;Y)\to H^\infty(\mathbb D;Y)\| \asymp_{\alpha,\varphi} d^{-\alpha} \] under a nondegenerate boundary singularity assumption is established.

We obtain an estimate of the rate of convergence on a set of full measure of partial sums of trigonometric Fourier series of functions from Lebesgue classes and construct a counterexample showing the order sharpness of this estimate. We derive a condition for Prinsheim convergence almost everywhere of two-dimensional trigonometric Fourier series of functions from Lebesgue classes in terms of the modulus of continuity.

In next-generation wireless networks, the growing density of devices and limited spectrum resources pose severe jamming challenges to fragile legitimate communication links in the wireless electromagnetic environment. Crucially, when jamming overlaps with pilot and data symbols in both time and frequency domains, it inflicts a severe bottleneck on receiver-side joint estimation and detection. Existing schemes often lack an effective framework to combat such jamming contamination, thereby failing to guarantee reliable transmission. To address this issue, we propose a Brownian bridge diffusion-based joint channel estimation and data detection framework (BBD-JCED) for jamming-resilient receivers. Specifically, the proposed framework comprises two core modules: the first extracts jamming features in the short-time Fourier transform (STFT) domain and suppresses jamming samples, thereby improving the signal-to-jamming-plus-noise ratio (SJNR) of the received signal; the second introduces a Brownian bridge diffusion (BBD) process to model the evolution of the suppressed signal and the encoded bits in the presence of channel estimation errors, thereby enabling enhanced joint channel estimation and data detection. To alleviate the computational burden of the BBD process in the second module, we further derive a fast ordinary differential equation (ODE) solver that enables its low-complexity iterative evolution. Finally, we design a multi-module training algorithm to improve the data recovery capability of the proposed framework. Simulation results demonstrate that the proposed framework achieves superior bit recovery performance compared with baseline schemes while maintaining a lower number of model parameters and competitive computational complexity.

This paper studies conditional entropy realization and weak* approximation by uniquely ergodic measures with compact support. We prove that, after fixing an admissible potential average and an entropy strictly below the entropy supremum over the corresponding average fiber, every invariant measure satisfying these two exact constraints can be approximated weakly* by uniquely ergodic measures with compact support satisfying the same constraints. Each approximating measure is the unique invariant measure on its minimal support, whose topological entropy equals the prescribed metric entropy. This result holds for three broad classes of systems: topologically expanding maps (including topologically Anosov systems), transitive countable Markov shifts, and symbolic systems with non-uniform structure. The proof uses a nested multi-horseshoe construction, with separate arguments addressing non-invertibility, non-compactness and non-uniformity.

In this paper, we establish several finiteness and boundedness theorems for compact positive monotone symplectic manifolds endowed with special Hamiltonian actions, called GKM$_3$, which generalize smooth toric varieties. Specifically, we prove that, for fixed dimension and Euler characteristic, there are only finitely many complex cobordism classes of such spaces. Moreover, modulo lattice transformations, the moment map image can be embedded into a box of explicitly bounded size, and all Chern numbers satisfy quantitative bounds. In particular, this yields a bound on the volume of the underlying symplectic manifold, analogous to the one obtained by Kollár-Miyaoka-Mori for Fano varieties. As a byproduct of our results, we also obtain that there are finitely many combinatorially maximal, labeled, reflexive GKM$_3$ graphs of valency $n$ and with $\chi$ vertices.

We construct infinite-time singularities with vanishing mean curvature for Lagrangian mean curvature flow in Gibbons–Hawking spaces. We consider circle-invariant Lagrangian $2$-spheres whose quotient curves are concave and are $C^2$-close to a collection of consecutive collinear segments. We prove that the corresponding flow exists smoothly for all time and converges to the associated $A_{n-1}$-chain of special Lagrangian spheres. Although the mean curvature converges uniformly to zero, the second fundamental form becomes unbounded. More precisely, $\log\max |A(\,\cdot\,,t)|$ is comparable to $\sqrt{t}$ as $t\to\infty$. The proof is based on a one-parameter family of barrier curves and a detailed analysis of their asymptotics. In this way, we refine the infinite-time convergence picture arising in the work of Lotay and Oliveira by proving curvature blow-up and estimating its rate in this semi-stable case.

Let $p_1,p_2\in(1,\infty)$ and $M=M_1\times M_2$ be the product of two geodesically complete Riemannian manifolds. In this paper, the authors first develop an anisotropic potential-theoretic framework adapted to the Green operator $G^M$ and the mixed-norm Lebesgue space $L^{p_2}(L^{p_1})(M)$, and then demonstrate that the classical equivalence among parabolicity, Green function integrability, and Liouville property persists in this genuinely anisotropic setting. More precisely, the authors establish the following equivalence: $M$ is $L^{p_2}(L^{p_1})$-parabolic if and only if the Green function $G^M(x;\,\cdot\,)$ fails to belong to $L^{p_2'}(L^{p_1'})(M \setminus B(x,\,r))$, which is in turn equivalent to the $L^{p_2'}(L^{p_1'})$-Liouville property, where $p_i'$ denotes the conjugate exponent of $p_i$. Under a weak radial Harnack-type inequality – in particular, under Li–Yau heat kernel estimates, and hence for products of manifolds with nonnegative Ricci curvature – these conditions are further equivalent to the divergence of the nonlinear mixed-potential $\mathcal{G}_{p_1,p_2}(f)$ for every nonzero nonnegative $f\in {\mathcal C}_c^\infty(M)$. A key feature of this anisotropic theory is its sensitivity to the geometry of each factor \(M_i\), rather than merely to that of the total manifold \(M\). In contrast to the isotropic case, where parabolicity and the classical Liouville property holds on \(\mathbb{R}^n\) precisely when \(n \le 2\), the anisotropic setting exhibits a refined threshold: the \(L^{p_2}(L^{p_1})\)-parabolicity and the \(L^{p_2'}(L^{p_1'})\)-Liouville property holds on \(\mathbb{R}^{n_1} \times \mathbb{R}^{n_2}\) if and only if $ D_{\mathrm{eff}} := \frac{n_1}{p_1} + \frac{n_2}{p_2} \le 2. $ This effective dimension $D_{\mathrm{eff}}$ captures the anisotropic interplay between the exponents \(p_1, p_2\) and the geometries of \(M_1, M_2\).

We develop a quantum algorithm for the regularized Wasserstein proximal operator, which is a fundamental tool in optimal transport and mean-field games. The regularization introduces a small diffusive term into the continuity equation of the Benamou-Brenier formulation, which results in a forward-backward PDE system consisting of a Fokker-Planck equation and a viscous Hamilton-Jacobi equation with a quadratic Hamiltonian. Through the Cole-Hopf transformation, both equations are converted to forward heat equations, whose coupling requires a Hadamard division to prepare the initial data for the second heat equation and a Hadamard product to recover the terminal density. We solve these heat equations via the Schrödingerization method and implement the Hadamard division and product operations using simple matrix-vector multiplication representations. The complete quantum algorithm prepares an $\varepsilon$-approximation of the terminal density state with $\mathcal{O}(d N_x T \log^2(1/\varepsilon))$ query complexity, up to constants depending on the potential and initial density, where $d$ is the spatial dimension, $N_x$ is the number of grid points per spatial dimension and $T$ is the evolution time. The complexity depends only linearly on $d N_x$, yielding an exponential speedup over classical methods, whose cost scales as $N_x^d$ per time step. Numerical experiments validate the effectiveness of the proposed algorithm.

We study the mean first capture time of isotropic Lévy flights on Zoll surfaces, namely the expected time for a geodesic Lévy process to reach a shrinking geodesic ball. While the leading-order asymptotics are universal, we prove that the first correction term encodes subtle geometric information. More precisely, it is completely determined by the local singularity type of the conjugate locus, quantified by the degree of the conjugate point. This yields a hierarchy of asymptotic regimes governed by the Lévy exponent.

We give applications of the properties $\mathrm{NSOP}_{r}$ for non-integer values of $r$ to problems on the original hierarchy $\mathrm{NSOP}_{n}$ for integer values of $n$. We first show that the properties $\mathrm{NSOP}_{r}$, previously defined for real values $r \geq 3$, are even well-defined for real values $r \geq 2$, showing that $\mathrm{NSOP}_{2} \subseteq \mathrm{NSOP}_{r}$ for our original definition of $\mathrm{NSOP}_{r}$ even when $2 < r < 3$. As a consequence, newness of all of the well-defined properties $\mathrm{NSOP}_{r}$ for non-integer $r$ would negatively resolve the problem of whether $\mathrm{NSOP}_{2}$ is equal to $\mathrm{NSOP}_{3}$. We then prove an approximate alternative between two possibilities: (1) that in extending Shelah's original $\mathrm{NSOP}_{n}$ hierarchy for integers $n \geq 3$ to the $\mathrm{NSOP}_{r}$ hierarchy for reals $r > 2$, we really did introduce new classification-theoretic properties, and (2) that $\mathrm{NSOP}_{n+1} \cap \mathrm{NTP}_{2} = \mathrm{NSOP}_{n} \cap \mathrm{NTP}_{2}$ for integers $n \geq 3$, which would resolve a central open problem in classification theory. More precisely, we give a rigorous sense in which (1) can fail on particularly general grounds, and then show that if (1) fails for these general reasons, (2) must be true. Finally, we apply cycle-removal techniques from the theory of the properties $\mathrm{NSOP}_{r}$ for real-values of $r$ to make progress on the question of whether $\mathrm{NSOP}_{2}$ is equal to $\mathrm{NSOP}_{3}$. We (a) show that if $\mathcal{H}$ is a hereditary class of structures defined by finitely many forbidden weakly embedded substructures, if every theory whose models have age $\mathcal{H}$ has $\mathrm{SOP}_{2}$, then every theory whose models have age $\mathcal{H}$ has $\mathrm{SOP}_{3}$, and (b) observe that we cannot replace $\mathrm{SOP}_{2}$ with $\mathrm{TP}$ here.

We analyse the space of points of the canonical extension of a coherent doctrine. We first give a full characterisation of doctrine morphisms that are extensible, and relate it to the existing notion of p-model of a coherent category. Through this characterisation, the extensible morphisms are shown to be exactly those which are {\omega}-saturated in the sense of coherent first-order logic. Next, we answer the question: when does a presheaf of models fully describe the canonical extension? We prove a characterisation theorem via two conditions, which are again natural from the perspective of coherent logic, namely, homogeneity and the realisation of all prime types in a strict sense. The characterisation theorem allows us to deduce a reconstruction result for any coherent theory with the property that all prime types can be realised in a countable, saturated model. For instance, {\omega}-stable coherent theories always have this property. We conclude by explaining how our results can be interpreted topos-theoretically, by relating them to the classifying topos and to the topos of types.

The central unsolved problem in the modular representation theory of symmetric groups is to find the decomposition matrices, which describe how irreducible representations in characteristic zero decompose upon reduction modulo a prime characteristic $p$. In this paper we determine a large number of new columns in these decomposition matrices, namely those labeled by partitions whose $p$-divisible hooks have all even arm lengths. In particular in odd characteristic $p$, for every possible block of every possible symmetric group $S_n$, we determine at least one complete column. These columns are multiplicity-free and are described by a recently introduced combinatorial statistic of partitions (depending on $p$), called the odd sequence. As an application, we determine the indecomposable summands of Foulkes modules $H^{(2^m)}$.

The theory of full conformal prediction uses deterministic non-conformity measure, but modern usage of full conformal prediction often relies on machine learning training, making stochasticity inevitable. A simple sufficient condition of almost sure permutation invariance of the non-conformity measure can be too restrictive, so many have suggested the relaxation to permutation in distribution as a condition for full conformal prediction validity. We, however, show that this commonly known condition is actually insufficient. We then provide a correct sufficient condition: Conditional Independence & Permutation Invariance in Distribution, which encompasses several stochastic settings that may be used in machine learning.

In this paper, we study convertible codes in the merge regime and focus on the minimum storage regenerating (MSR) setting, where both the initial codes and the final code admit optimal single-node repair. We propose explicit MSR-to-MSR conversion schemes and analyze their performance in terms of access cost and conversion bandwidth. We first construct convertible MSR codes in the irregular setting, where the $m$ initial codes may have different parameters, achieving optimal access cost. We further consider the practically important same-code setting, where all initial codewords are drawn from the same MSR code. By introducing a row-matching technique, we obtain constructions simultaneously achieving optimal access cost and conversion bandwidth in most parameter regimes.

A graph $G$ is said to be $F$-semi-saturated if the addition of any nonedge $e \not \in E(G)$ would create a new copy of $F$ in $G+e$. The semi-saturation number $ssat(n,F)$ is the minimum number of edges in an $F$-semi-saturated graph of order $n$. In this paper we investigate the semi-saturation number of $K_r$ on $n$ vertices with maximal degree at most $\Delta$, denoted by $ssat^{\Delta}(n,K_r)$. This investigation was suggested by Erd\H os, Rényi and Sós, who in 1966 considered the graph of diameter 2 with degree restrictions, equivalently $ssat^{\Delta}(n,K_3)$. The following are some of our results. For arbitrary $r \geq 4$, we show that the limit $ \lim_{n \rightarrow \infty} ssat^{cn}(n,K_r)/n$ exists for all $0 < c \leq 1$, except for some sparse values of $c$ contained in a countable and rational sequence $c_i \rightarrow 0$. Moreover, we establish the asymptotic behaviour of this limit for $\frac{r}{r+2} < c <1$ and determine the exact value of $ssat^{\Delta}(n,K_r)$ for some specific $\Delta$. As an application, we determine the relation between the saturation number of the join graph $K_r \vee F$ and that of $F$ for a large class of pairs $(r,F)$.

For a strict partition $\lambda$, let $\mathcal Q_\lambda(X;t)=Q_\lambda[X-tX]$ be the shifted $t$-Schur function arising from the modified Greaves–Jing–Zhu operator on the odd power-sum ring. We study transition matrices between the shifted bases with parameters $t$ and $s$. The relative scaling operator is diagonal in the odd power-sum basis, leading to explicit spectral data, determinant and trace formulas, weighted symmetry, a spin-character formula, and a transition Cauchy identity. For the cyclotomic specialization $C_{\lambda\mu}^{[M]}(t)=C_{\lambda\mu}(t^M,t)$, the relative operator becomes plethystic substitution by $1+t+\cdots+t^{M-1}$. We prove Schur $Q$-positivity and reciprocity, derive factorization and root-of-unity rank formulas, and give an exact computation method. For $M=2$, all one-row transitions are computed explicitly, and the nonzero coefficients are unimodal.

Let $T(x)=\prod_{k=0}^{\infty}(1-x^{2^k})$ be the generating function of the Thue–Morse sequence, and write $T(x)^m=\sum_{n\geq 0}t_m(n)x^n$. We prove exact formulas for the $2$-adic valuations of the coefficients $t_5(n)$ and $t_9(n)$: \[ \nu_2\bigl(t_5(4n+j)\bigr) =4\Bigl\lceil\tfrac{\nu_2(n+1)}{2}\Bigr\rceil-\bigl(\nu_2(n+1)\bmod 2\bigr), \quad j\in\{0,1,2,3\}, \] \[ \nu_2\bigl(t_9(8n+j)\bigr) =5\Bigl\lceil\tfrac{\nu_2(n+1)}{2}\Bigr\rceil-2\bigl(\nu_2(n+1)\bmod 2\bigr), \quad j\in\{0,1,\ldots,7\}. \] These formulas confirm Conjecture~5.2 of Gawron–Miska–Ulas~\cite{ga} for $m=5$ and $m=9$, and imply that $t_5(n)\neq 0$ and $t_9(n)\neq 0$ for every $n\geq 0$. A key structural ingredient is a closed-form formula for the determinant of a family of matrices with binomial-coefficient entries.

The dynamical stability of laminates or planar phase boundaries for hyperelastic materials of Hadamard type in two space dimensions is studied. For that purpose, the stability function, known as the Lopatinskii determinant, is computed for states of deformation at both sides of the planar interface that account for the generalized Legendre-Hadamard conditions derived by Grabovsky and Truskinovsky (J. Elast. 123 (2016), 225–243). The sufficient conditions for the dynamical stability of such configurations are described in terms of the physical parameters of the model, such as the shear modulus, and computed under kinetic conditions across the interface of both Maxwell (conservation of energy) or Abeyaratne and Knowles (dissipation of energy) types.

Modern resolution and prudential regimes increasingly wind up a distressed firm not at a single hard threshold but through a graduated, state-dependent process. We study how the design of such a regime shapes the trade-off between shareholder value and financial stability for a firm whose surplus follows a general diffusion. Forced liquidation is modelled in reduced form, arriving at a surplus-dependent hazard rate that rises as the firm's position deteriorates. The framework has three regions: an unregulated region where dividends may be paid, a regulated region where solvency requirements prohibit distributions, and a distress region in which the firm faces the liquidation hazard. To quantify shareholder value we solve the resulting singular stochastic control problem: which is to maximise the expected present value of distributions until liquidation. We establish a verification theorem, prove that a barrier strategy is optimal, and obtain tractable expressions for the value function and the expected survival time, so that alternative designs can be compared at low cost. We show that a distress region placed solely below or solely above the classical ruin threshold does not consistently improve both shareholder value and firm survival, whereas combining the two yields a Pareto improvement. Regulatory design is decisive.

We study exterior Dirichlet problems for \(k\)-Hessian equations with prescribed quadratic asymptotics, allowing the asymptotic matrix to be merely \(k\)-admissible and not necessarily positive definite. The key point is that the correct metric at infinity is not determined by the asymptotic matrix itself, but by the coefficient matrix obtained by linearizing the \(k\)-Hessian operator at this matrix. This gives the exterior barriers and subsolutions needed to solve the Dirichlet problem, both in viscosity and smooth settings, for all sufficiently large asymptotic constants. In the case of smooth, strictly star-shaped domains with strictly \((k-1)\)-convex boundary, we complete the characterization of existence and nonexistence through a linearized capacitary comparison and a tangential-trace contradiction on the inner boundary.

We develop a reduced interface formulation for elliptic interface problems with highly conducting interfaces. The interface condition consists of continuity of the primal variable together with a jump in the normal flux proportional to the surface Laplacian of the interface trace. Instead of using the solution jump as the interface unknown, we employ the common interface trace and derive a trace-based Schur complement formulation. For prescribed interface trace data, independent extension problems are solved in the two subdomains, leading to a reduced interface equation involving the Dirichlet-to-Neumann jump operator and a surface stiffness operator. Finite-dimensional trace approximations produce compact reduced systems posed only on the interface. Numerical experiments for circular, smooth noncircular, and heart-shaped interfaces illustrate the effectiveness of the method and the role of interface-mode enrichment.

We extend the notion of discrete homotopy groups of graphs to arbitrary cubical sets, and show that the discrete homotopy groups of quasisymmetric cubical sets are naturally isomorphic to the homotopy groups of their geometric realizations. Here, quasisymmetric cubical sets are cubical sets equipped with coordinate permutation symmetries that are compatible with faces and degeneracies, but not necessarily with connections. We give a purely combinatorial construction of the left adjoint of the forgetful functor from the category of quasisymmetric cubical sets to the category of cubical sets, and prove that the unit of this adjunction is a natural weak equivalence. As a consequence, we obtain a purely combinatorial description of the homotopy groups of the geometric realizations of arbitrary cubical sets. As an application, we establish the discrete Hurewicz map from the discrete homotopy groups of a cubical set to the homology groups, and prove a Hurewicz theorem with respect to the discrete Hurewicz map whenever the cubical set is quasisymmetric. Applied to the 1-nerve of a directed graph, it also yields a Hurewicz theorem relating the homotopy groups of Li, Wu, Yau, and Zhang to singular cubical homology.

We introduce a general framework for studying natural contravariant adjunctions that refine the adjunction between frames and spaces so that the fixpoints are $T_0$-spaces. Our objects of study are \textit{spatializable $\mathbf{Frm}$-concrete categories}, or SFC-categories. These consist of a faithful functor $\mathcal O:\mathcal C\to \mathbf{Frm}$ equipped with an object $2_{\mathcal C} \in \mathcal C$, satisfying compatibility conditions that ensure that $(2_{\mathcal C},\mathbb{S})$ forms a dualizing object in the sense of Porst and Tholen, where $\mathbb{S}$ denotes the Sierpiński space. Three important instances of pointfree $T_0$ spaces present in the literature fit into this framework: strictly zero-dimensional biframes, MT-algebras, and Raney extensions. We show SFC-categories are assembled in an ordered category – a category enriched in preordered sets – whose morphisms are suitable functors which preserve certain initial liftings. SFC-categories induce natural dual adjunctions, and morphisms between them will respectively induce suitable morphisms between these adjunctions. Motivated by the characterization of sober spaces as maximal objects in the fibers of $\Omega:\mathbf{Top}\to \mathbf{Frm}^{\mathsf{op}}$, and of $T_D$-spaces as the minimal ones, due to Banaschewski and Pultr, we study initial and terminal objects of fibers for an arbitrary SFC-category. We prove that the natural adjunction for fiber-initials has exactly the sober spaces as fixpoints, while for fiber-terminals contains at most $T_D$-spaces, recovering their results of in a much more general setting.

The gravitational field of a distant, isolated system is manifested by the conformally invariant Weyl tensor. Thus the conformal structure far from the system encodes the system's gravitational mass. It also encodes the causal structure, thereby linking it to the mass. For asymptotically anti-de Sitter (AdS) spacetimes, this link led to a novel positive mass theorem of Page, Surya, and the second author \cite{PSW} which did not rely on any traditional energy condition. Here we ask whether that theorem has a rigidity case. Specifically, we consider all null geodesics in an asymptotically AdS spacetime that depart from the Penrose conformal infinity, travel through spacetime, and return to conformal infinity. If all such geodesics from a given point refocus at an antipodal point at infinity, is the spacetime conformal to anti-de Sitter space? It is easy to answer the question if the asymptotically AdS spacetime either (i) obeys the null energy condition in 3 or 4 spacetime dimensions, or (ii) is static (in any dimension), and we give simple proofs in those cases. We then answer the question in the case of globally stationary, asymptotically AdS spacetimes, by applying the theory of magnetic geodesics on the Riemannian manifold-with-boundary obtained by quotienting by the stationary Killing vector field. The question has an analogue for asymptotically flat spacetimes, which we also discuss.

We consider the encoding of graph problems as Quadratic Unconstrained Binary Optimization (QUBO) problems, which are solvable by either quantum or classical annealers. Yet, the class of problems encoded as QUBO problems has not previously included nowhere-zero flows. Nowhere-zero flows are related to Tutte's $5$-flow conjecture and appear in many contexts in graph theory. We provide an encoding of nowhere-zero flows as a QUBO Hamiltonian and prove the correctness of the construction. Our construction yields a Hamiltonian $H_{\mathrm{mod},k}$ whose ground state has zero energy if and only if the graph $G$ has a nowhere-zero $\mathbb Z_k$-flow. By Tutte's equivalence theorem, zero ground energy is equivalent to $\varphi(G)\le k$, and the zero-energy degeneracy is given by the flow polynomial $F(G;k)$. In particular, when the ground-state energy is zero, this is also the ground-state degeneracy. The construction uses one-hot variables to represent the edge flow residues modulo $k$ and auxiliary variables to represent the per-vertex modular quotient. We prove that the correctness of the construction is independent of the choice of orientation, root vertex, and positive penalty weights. We verify the construction on $59$ examples of graphs and values of $k$ that include both yes-instances and no-instances. We exhaustively sweep orientations and root choices on selected robustness instances and test a finite suite of positive penalty weights. The resulting Hamiltonian is implemented using the dimod.BinaryQuadraticModel class, which is compatible with the D-Wave Ocean SDK. Quantum-hardware runs and claims about potential speedup using these devices are left to follow-up work.

Fermionic topological orders can host 't Hooft anomalies with no bosonic counterpart. We identify a new sixteen-fold family of (2+1)D fermionic topological orders, forming a fermionic analogue of Kitaev's sixteen-fold way. This family is distinguished by the mod 16 't Hooft anomaly of a $\mathbb{Z}_2$ one-form symmetry, generated in each theory by a single nontrivial $\mathbb{Z}_2$ anyon. This intrinsically fermionic anomaly permits anyon spins that are forbidden in bosonic phases; the simplest new example is an Abelian fermionic topological order containing a single $\mathbb{Z}_2$ Abelian anyon of spin 1/8. Each theory can be realized as the gapped boundary of a (3+1)D fermionic symmetry-protected topological (SPT) phase protected by the $\mathbb{Z}_2$ one-form symmetry, which acquires a $\mathbb{Z}_{16}$ classification once the spacetime spin structure is twisted by the one-form symmetry. We realize these phases microscopically via lattice models built from Walker-Wang models coupled to local fermions.