Papers

We construct singular weak solutions to scalar, linear, uniformly parabolic equations with complex coefficients in one space dimension. Our examples show that the space-time integrability provided by the energy estimates is sharp: for every $p>6$, there exists such an equation admitting an energy solution that fails to be $p$-integrable on a compact subset of the space-time domain. In particular, the local boundedness of weak solutions from the De Giorgi–Nash–Moser theory fails for complex coefficients already in one space dimension.

Independently at every vertex $x$ of a large core graph $\mathbb G$, we draw a random graph and connect $x$ to a subset of its vertices. This model of a randomly sprinkled graph captures qualitative features of commonly observed graphs; it can also be regarded as a model of quantum disorder, where disorder arises from local perturbations to the graph geometry. We show that it is naturally connected both to the Anderson model and to site percolation on $\mathbb G$ by deriving an Anderson-percolation representation for its spectrum. We characterize the support of the spectrum and derive quantitative bounds on the integrated density of states. We also analyse in detail the regime of sparse sprinkling, in particular investigating the behaviour of the density of states in the vicinity of atoms.

In 1986, Füredi and Kahn showed that the dimension $\dim(P)$ of any finite poset $P$ satisfies $\dim(P) = O(d \log^2 d)$, where $d$ is the maximum degree of the comparability graph of $P$. Scott and Wood more recently improved this bound to one of the form $d \log^{1+o(1)} d$. We show that $\dim(P) = O(d \log d)$, thus confirming that the corresponding lower bound of Erdős, Kierstead, and Trotter is tight up to the implicit constant.

We present two results concerning the decidability of (strong) quasipositivity of links in the 3-sphere. First, using a result of Dynnikov–Prasolov, we provide an algorithm to decide whether or not a link is strongly quasipositive. Second, we reduce the decision problem of the quasipositivity of links to that of braids.

We consider an application of the Atiyah-Bott fixed point theorem in quaternionic geometry. We review the relation of the general Salamon complex of a quaternionic manifold and the Dolbeault complex with coefficients in a certain holomorphic vector bundle of its twistor space. Furthermore, we investigate which maps are suitable for the application of the fixed point theorem. As a result, we obtain a fixed point formula for both complexes and apply it to Wolf spaces.

We investigate the zero-Hopf bifurcation and the local $C^{1}$-non-integrability of the original Rössler system. First, we characterize the unique zero-Hopf equilibrium of the system and study its perturbation by means of the first-order averaging theory. We prove that, under suitable assumptions on the bifurcation parameters, three distinct families of periodic orbits bifurcate from the zero-Hopf equilibrium, substantially extending the previously known result in which only one bifurcating periodic orbit was obtained. In the second part of the paper, these bifurcating periodic solutions are used to investigate the integrability properties of the Rössler system. By combining the first-order averaging theory with the classical Poincaré criterion for first integrals, we prove that the bifurcating periodic orbits constitute a generic obstruction to the existence of local first integrals of class $C^{1}$. This demonstrates that averaging theory provides not only an effective tool for detecting periodic solutions but also a powerful method for the integrability analysis.

We present a generating-function approach to the topological entropy of finite strongly connected digraphs, working throughout with digraphs in which loops are allowed but multiple edges are excluded. Using dominant singularities of path-generating functions, we recover by new analytic–combinatorial arguments the known first and second positive-entropy minimizers from the previously studied loopless spectral setting, and extend the corresponding extremal statements to the present framework. For the class $\mathcal{SC}_{m+1}(m)$ of strongly connected digraphs with $m$ vertices and $m+1$ edges, we introduce a unified $(t,k_1,k_2)$-butterfly parametrization. The entropy of $\mathcal{B}^{\,t}_{k_1,k_2}$ depends only on $(k_1,k_2)$ and is determined by the unique root $R\in(0,1)$ of $1-z^{k_1}-z^{k_2}=0$ via $h=-\ln R$. This parametrization yields a detailed entropy hierarchy within $\mathcal{SC}_{m+1}(m)$. We introduce the Pyramidal Entropy Diagram, determine the entropy order completely for $m\leq 7$, and identify its first structural bifurcation at $m=8$. We further establish maximal stable initial and terminal segments consisting of ten entropy minima and two entropy maxima, respectively, and derive an explicit formula for the minimum order required to realize a positive entropy not exceeding a prescribed threshold.

We prove a spectral gap inequality for Gaussian measures modified by singular attractive pair interactions. The spectral gap constant is explicit and uniform in the number of particles and the regularization parameter. As an application, we use a corresponding spectral gap inequality for a cutoff interaction weight to obtain two-sided bounds on the transition density of the attractive logarithmic gas.

Let $p>3$ be a prime number. We prove that every principally polarized superspecial abelian surface over $\mathbb{F}_{p^{2}}$ with $p^{2}$-Frobenius $[-p]$ admits a separable polarized isogeny to its Frobenius conjugate with multiplier at most $(p^{3}/2)^{1/5}$. This generalizes a recent result of Aubry, Oyono, and Vincent (2026, arXiV:2607.14624) who proved that every supersingular elliptic curve defined over $\bar{\mathbb{F}}_{p}$ admits an isogeny to its Frobenius conjugate with degree at most $(p/2)^{1/3}$.

It is an established idea in concurrency theory that every Petri net admits an unfolding semantics. This is a denotational object that represents its domain of possible executions. Unfoldings play an important role in practical analysis and verification. This paper is concerned with the following well-known problem: while the unfolding resembles a universal construction in the category of Petri nets, it generally fails to satisfy the expected universal property. This is because the unfolding construction overlooks the net's internal symmetries. There are two solutions: make these symmetries explicit to obtain a weak universal property (one that holds only ”up to symmetry”); or break the symmetries by assigning individual identities to components of the net. We review these two solutions and establish, in each case, a universal unfolding of Petri nets to event structures. This paper demonstrates a 2-categorical approach to Petri net unfoldings. We show that each unfolding semantics determines a 2-categorical relative adjunction involving Petri nets and event structures. Viewed in this way, the above two constructions can be related formally via an appropriate morphism of adjunctions. We exhibit a 2-density property of event structures which implies that unfolding functors are essentially unique.

The derivative-free loss method (DFLM) is a mesh-free neural network approach for solving partial differential equations by simulating stochastic walkers rather than computing derivatives directly. Although DFLM has previously been applied to the Navier–Stokes equations, extending it to turbulent flows reveals two limitations of its simplest implementation. The walkers do not account for the swirling, directionally stretching motion of the flow, and randomly sampling the walkers used to evaluate the target introduces noise into the learned solution. We address both limitations with an analytic approximation of the walkers' local motion that captures directional stretching and eliminates this sampling noise, while remaining computationally efficient. We study the resulting method as a non-intrusive multiscale solver capable of learning macroscopic flow behavior without resolving every scale on a fine grid or explicitly coupling coarse- and fine-scale models. For two turbulent two-dimensional flows, the proposed method reproduces the macroscopic energy spectrum of fully resolved reference simulations more accurately than the standard method, particularly where directional stretching is strongest, confirming DFLM's efficacy as a multiscale solver for turbulent fluid systems.

We establish a class formula with variable for arbitrary effective Anderson motives. This pursues the pioneering work of Taelman and extends the scope of the work of Anglès-Ngo Dac-Tavares Ribeiro, removing the admissibility assumption and allowing for more general Anderson modules. In the same setting, we further conjecture a ”nondeterminantal” version of the class formula which holds before taking determinants and prove a weak version of it. Finally, we prove an overconvergent version of the class formula and derive from it strong convergence properties for the associated L-series.

We construct an irreducible character of $\operatorname{GL}_{n}(q)$ whose square contains every irreducible character that is trivial on the centre of $\operatorname{GL}_{n}(q)$. This character is related to, but in general not equal to, the Steinberg character. A key ingredient, which we also prove, is that every nonlinear irreducible character of $\operatorname{GL}_{n}(q)$ that is trivial on the centre contains the trivial character when restricted to the diagonal subgroup $T$.

We consider the Cauchy problem $$u_t-\Delta u=|u|^p,\qquad (t,x)\in(0,T)\times\R^n, \quad u(0,x)=\eps u_0(x),\qquad x\in\R^n,$$ with small sign-changing initial data having zero total mass. We assume $$u_0\in L^1(\R^n)\cap L^\infty(\R^n),\quad |x|u_0\in L^1(\R^n),\quad \int_{\R^n}u_0(x)\,dx=0,$$ and, for the sharp subcritical upper bounds, that $\int_{\R^n}x\,u_0(x)\,dx\neq0.$ Let $T_\eps$ denote the maximal lifespan. We identify a new transition exponent $p_m=1+\frac1{n+1}$ inside the Fujita range $1<p\le p_F:=1+2/n$, and establish the sharp lifespan estimates $$T_\eps\asymp \begin{cases} \eps^{-\frac{2(p-1)}{2-(n+1)(p-1)}}, &1<p<p_m,\\[2mm] \eps^{-\frac2{n+1}} \bigl(\log\frac1\eps\bigr)^{-\frac2{n+2}}, &p=p_m,\\[2mm] \eps^{-p\left(\frac{1}{p-1}-\frac{n}{2}\right)^{-1}}, &p_m<p<p_F,\\[2mm] \exp\!\left(\eps^{-p(p-1)}\right), & p=p_F. \end{cases}$$ These estimates reveal a lifespan phenomenon that is different from the classical Lee–Ni law for initial data with positive mass. In the zero-mass setting, the leading linear contribution is dipole-like, while the nonlinear source subsequently generates a positive mass of size \(O(\eps^p)\). Their competition produces the additional threshold \(p_m\), the logarithmic correction at $p=p_m$, and, at $p=p_F$, a substantially longer critical lifespan with exponent $p_F(p_F-1)$ instead of the classical Lee–Ni exponent $p_F-1$. The upper estimates are obtained by backward-Gaussian and critical scale-ODE test-function arguments, whereas the lower estimates follow from a unified $L^1$–$L^\infty$ bootstrap preserving the zero-mass cancellation of the linear flow and controlling the mass generated by the nonlinear source. Thus, although $p_F$ remains the Fujita critical exponent, zero initial mass creates a new quantitative lifespan regime below and at the critical exponent.

We establish the sharp CFL condition for the piecewise constant sparse grid discontinuous Galerkin (DG) method with forward Euler time stepping, applied to transport equations with constant coefficients on periodic domains in arbitrary dimensions. For the transport velocity $\boldsymbol c=(c_1,\ldots,c_d)$ and a uniform mesh of size $h$, we prove that the scheme is $L^2$ stable if and only if $\Delta t \leq {h}/{\max_{1\leq \ell\leq d}|c_\ell|}$, whereas the corresponding full grid upwind scheme is well-known to require $\Delta t\leq h/\sum_{\ell=1}^d |c_\ell|$. The sparse grid discretization therefore enlarges the admissible time step by a factor of ${(\sum_{\ell=1}^d |c_\ell|)}/{(\max_{1\leq \ell\leq d}|c_\ell|)}$, which lies between $1$ and $d$ and reaches $d$ for isotropic transport. The proof of sufficiency relies on projection leakage identities for the multilevel Haar decomposition, which allow the mixed directional terms in the energy estimate to be absorbed by the energy discarded by the sparse grid projection. The proof of sharpness follows from alternating modes in one dimension at the finest level. As a by-product, we obtain explicit formulas for the $L^2$ operator norm and the spectral radius of the amplification operator. For spaces over general downward closed index sets, we derive an explicit sufficient CFL condition and a geometric criterion for its sharpness. Numerical experiments in two and four dimensions confirm the theoretical results.

By consuming multiple copies of an unknown qubit state, one can modify its purity while preserving the direction of its Bloch vector. We determine the maximum linear rate at which qubit states of different purities can be interconverted, allowing a nonzero error, quantified, for instance, by the trace distance, provided that it vanishes in the limit of infinitely many copies. Interestingly, the optimal conversion rate is determined by the two eigenvalues of the complex right-logarithmic-derivative (RLD) Fisher information matrix associated with $\mathrm{SU}(2)$ rotations of the qubit state. When the output qubits have higher purity, corresponding to concentration, the optimal rate is given by the ratio of the maximum eigenvalues of the input and output RLD matrices. In contrast, when the output qubits have lower purity, corresponding to dilution, the optimal rate is given by the ratio of their minimum eigenvalues. Remarkably, both concentration and dilution can be implemented using SWAP tests as the only nontrivial two-qubit measurement primitive, together with ancillary qubits initially prepared in maximally mixed states, without requiring any additional two-qubit gates. Our work thus provides a novel operational interpretation of the full complex RLD Fisher information matrix. Crucially, its antisymmetric, purely imaginary part encodes geometric information beyond the statistical distance between density operators and plays an essential role in determining the optimal state-conversion rates.

The generalised quantum Stein's lemma is a key result in quantum hypothesis testing, and connects this fundamental primitive of quantum information processing with quantum resource manipulation, a task that is central for technological applications. Prior works have proved this statement in the idealised setting of independent and identically distributed (i.i.d.) sequences of quantum states, and recent extensions consider also sources that are 'close' to i.i.d., according to the strict notion put forth by Mazzola, Sutter, and Renner. For several applications, however, one would need to consider yet more general sources. We establish a version of the generalised quantum Stein's lemma that is conceptually much simpler and general, as it applies to any source that is asymptotically close to an i.i.d. state with respect to the normalised quantum Wasserstein distance of order 1. As an immediate consequence, we solve the Stein exponent of a scenario where the null hypothesis is arbitrarily varying, expressing it in terms of i.i.d. Stein exponents corresponding to arbitrary states in the convex hull of the null hypothesis base set.

In 1989, Roth and Lempel constructed a well-known family of non-Reed-Solomon maximum distance separable (MDS) codes. For decades, this family of codes has attracted extensive research attention due to its algebraic structure, low-complexity decoding, and broad applications in cryptography and data storage. In this paper, we present a class of twisted Roth-Lempel codes. We investigate their minimum distance, MDS and NMDS properties. Specifically, we determine the necessary and sufficient conditions for the TRL codes to have minimum distance n-k or n-k+1. Furthermore, we determine the necessary and sufficient conditions for the TRL code to be an MDS or NMDS code. Moreover, we show that the dimension of the Schur square of the TRL code is at least 2k+1, and thus the TRL code is a non-RS code inequivalent to the corresponding RL code.

We give a variational comparison proof of the sharp logarithmic Sobolev inequality with constant $2$ for the word-length Poisson semigroup on free group von Neumann algebras. Our approach extends the Lagrange method for cyclic groups in [Yao26] to a framework in finite tracial von Neumann algebras. The inequality was recently established by Xie and Zhang in [XZ26] using a completely different approach.

Policy learning aims to determine who should be treated based on individual characteristics. In high-stakes settings such as medicine and public policy where safety is a central concern, improving the average outcomes alone may not be sufficient: decision makers may also seek to protect individuals from harm, in line with the Hippocratic principle of “do no harm.” In this paper, we propose conformal policy learning (CPL), a policy learning procedure with a new distribution-free safety guarantee that controls the probability of assigning treatment to an individual who would be harmed relative to control. CPL views each treatment decision as testing a hypothesis of counterfactual harm and assigns treatment by thresholding conformal p-values. These p-values use observable proxies and selective calibration to address the challenge that the potential outcomes under comparison are never simultaneously observed. For randomized experiments, under standard exchangeability conditions, CPL provides finite-sample safety guarantee at a user-specified level, without imposing any outcome modeling assumptions. Moreover, when the outcome model is consistently estimated, CPL achieves asymptotically optimal welfare subject to the safety constraint. In observational studies, CPL with learn-then-balance weights achieves doubly robust safety guarantees. We evaluate CPL through extensive simulations and apply it to an empirical study of AI-powered interventions designed to reduce conspiracy beliefs.

Recently, dynamic state feedback controllers that are based on dynamic extensions have been presented for heterodirectional hyperbolic systems. In this paper, a similar concept for the control of coupled diffusion-reaction systems is suggested. The introduction of a specific controller dynamics leads to homogenized diffusion coefficients for the extended system. Then, a backstepping-based static state feedback for the dynamically extended system is designed, which, overall, results in a dynamic state feedback. Such a design allows stabilizing a more general class of parabolic systems as well as assigning arbitrary closed-loop dynamics. This can be used, e.g., to achieve a decoupled input-output behavior, which is, in general, not possible with a static state feedback. A simulation example illustrates the results.

Two features intrinsic to survey sampling complicate semiparametric efficiency analysis: design-induced dependence among sampling indicators and the randomness of finite-population targets under the superpopulation law. For general semiparametric full-data models, we show that the observed-data experiment under a broad class of dependent designs is locally asymptotically normal with the tangent-space structure of a reference Poisson experiment. Under the joint superpopulation-design law, first-order efficiency depends on the design only through the limiting inclusion-probability function. Standard missing-at-random projection in the reference experiment characterizes the observed-data efficient influence function. Finite-population targets are treated through first-order asymptotic expansions, extending the analysis beyond exact random sums to nonlinear census characteristics. Superpopulation and finite-population centerings yield equivalent notions of local regularity and efficiency, with their bounds linked by a Pythagorean decomposition that gives a generalized finite-population correction. We then give general and design-specific conditions under which cross-fitted estimators with estimated nuisance functions attain both efficiency bounds. For the finite-population mean, the bound equals the large-sample limit of the Godambe-Joshi anticipated-variance lower bound. For scalar targets, we characterize optimal limiting inclusion probabilities. Simulations and California Academic Performance Index data illustrate the theory.

In flow matching (FM), a velocity model $v_{\theta}$ is trained using a predefined path $g_t$ that connects data and noise samples (e.g., $g_t(x_0, x_1) = (1 - t) x_0 + t x_1$). In this work, we study the choice of this path from an optimization perspective by analyzing the variance of stochastic gradients. We consider the class $G(p_t,v^\star_t)$ of paths that induce the same marginal distributions $p_t$ and marginal velocity field $v^\star_t$, and therefore the same FM objective. Our main finding is that the choice of path $g_t$ can fundamentally change the convergence rate of SGD, even when the FM objective remains exactly the same. (i) For a linear velocity model and one-dimensional Gaussian data, we derive a tight bound on the SGD iteration complexity up to logarithmic factors and find an analytically optimal path that minimizes this bound among linear paths inducing the same FM problem. (ii) We then extend the variance analysis to general FM problems and formulate path selection at a fixed $\theta$ as the variance-minimization problem PathOpt$_\theta$, constrained to $g_t\in G(p_t,v^\star_t)$. We show that this constraint is essential: reducing variance without it can lead to slower convergence. (iii) Since the constraint $g_t \in G(p_t,v^\star_t)$ cannot generally be verified directly, we derive an equivalent formulation with constraints that can be estimated from samples, allowing paths to be found numerically. Our theoretical results are supported by experiments with Gaussian data, Gaussian mixture models, and real datasets.

Answering a question of Chepoi and Hagen, we give two constructions of bounded degree CAT(0) square complexes that cannot be isometrically embedded into any finite product of trees. The first is based on Burling graphs and has degree at most five, which is optimal. These square complexes topologically embed into $\R^3$ since we further prove that every CAT(0) square complex whose vertices have degree at most five can be topologically embedded into the product of a line and a star. It turns out that in this setting, Burling graphs are the only obstruction: we prove that CAT(0) square complexes that topologically embed into $\R^3$ and whose crossing graph forbids some induced Burling graph can be isometrically embedded into a finite product of trees. Our second CAT(0) square complex has degree at most six and its crossing graph forbids an induced Burling graph. Of independent graph-theoretic interest, along the way we construct intersections of chordal and interval graphs with arbitrarily large girth and chromatic number. We also discuss connections to nice labellings of event structures.

We construct two complemented subspaces of $L_1[0,1]$. The first has the Schur property but fails the Radon–Nikodým property. The second contains a copy of $\ell_2$ but no copy of $L_1[0,1]$. Neither space is isomorphic to a Banach lattice. This gives a negative answer to the complemented-subspace question of Lindenstrauss and Rosenthal. A Lean 4 formalisation of the main results accompanies the paper.

In this work, we develop a fibrational approach to freely adjoining adjoints in an $(\infty,2)$-category. We construct the universal bifibration obtained from a cocartesian fibration of $(\infty,2)$-categories by adjoining cartesian lifts over a chosen class of $1$-morphisms in the base. We then use this construction to provide an explicit model for freely adjoining adjoints, together with a zig-zag formula for the resulting mapping $(\infty,1)$-categories. As applications, we give a model-independent proof of the universal property of the walking adjunction, providing an alternative proof of a theorem of Riehl–Verity, and establish the universal characterization of the simplex $2$-category conjectured by Dyckerhoff–Kapranov–Schechtman–Soibelman.

Conner, Gesmundo, Landsberg and Ventura (2019) determined the largest stabilizer dimension of concise $n\times n \times n$ tensors that are binding, and they determined the corresponding maximizing tensors to be the null algebra tensors. They left as an open problem to extend this to all concise $n \times n \times n$ tensors (i.e. dropping binding). We solve this problem: We prove that the largest stabilizer dimension of concise $n\times n\times n$ tensors is $n^2 + 1$ and the maximizers are the null algebra tensors (as in the binding case) and the skew symmetric tensor $e_1 \wedge e_2 \wedge e_3$. As part of our approach we obtain upper bounds on the stabilizer dimension of matrix tuples under left-right action (generalized Kronecker quiver representations), which we think are of independent interest.

We compute the $\operatorname{RO}^{\sigma}(C_p)$-graded Mackey functors $\underline{\pi}_* L_{\operatorname{KU}_{C_p}/p}\mathbb{S}_{C_p}$.

Digital halftoning aims to represent continuous-tone images by binary patterns while preserving their visually relevant low-frequency content. Among the many available approaches, error-diffusion methods implement noise shaping through causal feedback filters and can be interpreted as two-dimensional versions of the signal quantization paradigm Sigma–Delta modulation. On closed domains, however, the terminal state of the underlying recurrence relation need not match the initial one, producing boundary artifacts. We study this problem for bandlimited functions on the two-dimensional torus. By arranging all pixels along a single closed rank-one lattice, we replace the multiple mismatches associated with separately processed rows and columns with a single terminal contribution, while retaining exact reconstruction. For a uniform lattice with \(N=M^2+1\) points, we obtain first- and second-order error bounds of order \(N^{-1/2}\) and \(N^{-1}\). A suitable constant update eliminates the terminal mismatch and reduces the spatial localization of the error without changing these asymptotic orders. For fixed-direction rank-one lattices, the corrected first- and second-order reconstructions instead achieve rates \(N^{-1}\) and \(N^{-2}\). Numerical experiments illustrate a reduction in boundary artifacts compared with classical schemes applied on the Cartesian grid.

We use an analogue of the Springer resolution to describe the $G$-module structure on the ring of regular functions on the universal cover $\widetilde{\mathcal{O}}$ of any nilpotent orbit for $G = SL_n$. Building on previous work on the extended Springer resolution, we construct a variety $\widetilde{\mathcal{M}}$ that is finite over the cotangent bundle of a partial flag variety $G/P$, and proper and birational over the affinization $\mathcal{M}$ of $\widetilde{\mathcal{O}}$. We use techniques in birational geometry to show that $\widetilde{\mathcal{M}}$ has rational singularities, which provides the cohomology vanishing needed to describe the ring of functions on $\widetilde{\mathcal{O}}$ as an induced representation from a Levi subgroup of $G$. Our results also yield a description of the structure of $R(\widetilde{\mathcal{O}})$ as a graded $G$-module. We describe the minimal embedding of $\mathcal{M}$, study the lifting of characters of the component group of $\widetilde{\mathcal{O}}$ to parabolics and Levi subgroups, and make a more general vanishing conjecture.